Galois group search results

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Label	Name	Order	Parity	Solvable	Nil. class	Conj. classes	Subfields	Low Degree Siblings
30T1	$C_{30}$	$30$	$-1$	✓	$1$	$30$	$C_2$, $C_3$, $C_5$, $C_6$, $C_{10}$, $C_{15}$
30T2	$C_5\times S_3$	$30$	$-1$	✓	$-1$	$15$	$C_2$, $S_3$, $C_5$, $S_3$, $C_{10}$, $S_3 \times C_5$	15T4
30T3	$D_{15}$	$30$	$-1$	✓	$-1$	$9$	$C_2$, $S_3$, $D_{5}$, $S_3$, $D_5$, $D_{15}$	15T2
30T4	$C_3\times D_5$	$30$	$-1$	✓	$-1$	$12$	$C_2$, $C_3$, $D_{5}$, $C_6$, $D_5$, $D_5\times C_3$	15T3
30T5	$C_3\times D_{10}$	$60$	$-1$	✓	$-1$	$24$	$C_2$, $C_3$, $D_{5}$, $C_6$, $D_{10}$, $D_5\times C_3$	30T5
30T6	$C_{15}:C_4$	$60$	$-1$	✓	$-1$	$9$	$C_2$, $S_3$, $F_5$, $S_3$, $F_5$, $C_{15} : C_4$	15T6
30T7	$C_3\times F_5$	$60$	$-1$	✓	$-1$	$15$	$C_2$, $C_3$, $F_5$, $C_6$, $F_5$, $F_5\times C_3$	15T8
30T8	$S_3\times D_5$	$60$	$-1$	✓	$-1$	$12$	$C_2$, $S_3$, $D_{5}$, $S_3$, $D_{10}$, $D_5\times S_3$	15T7, 30T10, 30T13
30T9	$A_5$	$60$	$1$		$-1$	$5$	$A_5$, $\PSL(2,5)$, $A_{5}$, $A_5$	5T4, 6T12, 10T7, 12T33, 15T5, 20T15
30T10	$S_3\times D_5$	$60$	$-1$	✓	$-1$	$12$	$C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_5$, $D_5\times S_3$	15T7, 30T8, 30T13
30T11	$C_5\times A_4$	$60$	$1$	✓	$-1$	$20$	$C_3$, $C_5$, $A_4$, $C_{15}$	20T14
30T12	$S_3\times C_{10}$	$60$	$-1$	✓	$-1$	$30$	$C_2$, $S_3$, $C_5$, $D_{6}$, $C_{10}$, $S_3 \times C_5$	30T12
30T13	$S_3\times D_5$	$60$	$-1$	✓	$-1$	$12$	$C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_{10}$, $D_5\times S_3$	15T7, 30T8, 30T10
30T14	$D_{30}$	$60$	$-1$	✓	$-1$	$18$	$C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_{10}$, $D_{15}$	30T14
30T15	$S_3\times C_{15}$	$90$	$-1$	✓	$-1$	$45$	$C_2$, $C_5$, $S_3\times C_3$, $C_{10}$	45T3
30T16	$C_3\times D_{15}$	$90$	$-1$	✓	$-1$	$27$	$C_2$, $D_{5}$, $S_3\times C_3$, $D_5$	45T5
30T17	$C_{30}:C_4$	$120$	$-1$	✓	$-1$	$18$	$C_2$, $S_3$, $F_5$, $D_{6}$, $F_{5}\times C_2$, $C_{15} : C_4$	30T17
30T18	$C_{10}\times A_4$	$120$	$-1$	✓	$-1$	$40$	$C_3$, $C_5$, $A_4\times C_2$, $C_{15}$	40T59
30T19	$C_5:S_4$	$120$	$1$	✓	$-1$	$13$	$S_3$, $D_{5}$, $S_4$, $D_{15}$	20T33, 30T31, 40T63
30T20	$D_5\times A_4$	$120$	$-1$	✓	$-1$	$16$	$C_3$, $D_{5}$, $A_4\times C_2$, $D_5\times C_3$	20T37, 30T28, 40T65
30T21	$S_3\times D_{10}$	$120$	$-1$	✓	$-1$	$24$	$C_2$, $S_3$, $D_{5}$, $D_{6}$, $D_{10}$, $D_5\times S_3$	30T21 x 3
30T22	$S_5$	$120$	$-1$		$-1$	$7$	$S_5$, $\PGL(2,5)$, $S_5$	5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T25, 30T27, 40T62
30T23	$S_3\times F_5$	$120$	$-1$	✓	$-1$	$15$	$C_2$, $S_3$, $F_5$, $D_{6}$, $F_{5}\times C_2$, $F_5 \times S_3$	15T11, 30T24, 30T32
30T24	$S_3\times F_5$	$120$	$-1$	✓	$-1$	$15$	$C_2$, $S_3$, $F_5$, $D_{6}$, $F_5$, $F_5 \times S_3$	15T11, 30T23, 30T32
30T25	$S_5$	$120$	$-1$		$-1$	$7$	$C_2$, $S_5$, $S_5$, $S_5$	5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T27, 40T62
30T26	$C_6\times F_5$	$120$	$-1$	✓	$-1$	$30$	$C_2$, $C_3$, $F_5$, $C_6$, $F_{5}\times C_2$, $F_5\times C_3$	30T26
30T27	$S_5$	$120$	$1$		$-1$	$7$	$S_5$, $S_5$, $S_5$	5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 40T62
30T28	$D_5\times A_4$	$120$	$1$	✓	$-1$	$16$	$C_3$, $D_{5}$, $A_4$, $D_5\times C_3$	20T37, 30T20, 40T65
30T29	$C_2\times A_5$	$120$	$-1$		$-1$	$10$	$A_5$, $A_5$	10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T30, 40T61
30T30	$C_2\times A_5$	$120$	$-1$		$-1$	$10$	$C_2$, $A_5$, $A_5\times C_2$, $A_5$	10T11, 12T75, 12T76, 20T31, 20T36, 24T203, 30T29, 40T61
30T31	$C_5:S_4$	$120$	$-1$	✓	$-1$	$13$	$S_3$, $D_{5}$, $S_4$, $D_{15}$	20T33, 30T19, 40T63
30T32	$S_3\times F_5$	$120$	$-1$	✓	$-1$	$15$	$C_2$, $S_3$, $F_5$, $S_3$, $F_{5}\times C_2$, $F_5 \times S_3$	15T11, 30T23, 30T24
30T33	$C_5\times S_4$	$120$	$1$	✓	$-1$	$25$	$S_3$, $C_5$, $S_4$, $S_3 \times C_5$	20T34, 30T34, 40T64
30T34	$C_5\times S_4$	$120$	$-1$	✓	$-1$	$25$	$S_3$, $C_5$, $S_4$, $S_3 \times C_5$	20T34, 30T33, 40T64
30T35	$C_5^2:C_6$	$150$	$-1$	✓	$-1$	$10$	$C_2$, $C_3$, $C_6$, $(C_5^2 : C_3):C_2$	15T12 x 2, 25T15, 30T35
30T36	$C_5\times D_{15}$	$150$	$-1$	✓	$-1$	$45$	$C_2$, $S_3$, $S_3$, $D_5\times C_5$	30T36
30T37	$C_5^2:S_3$	$150$	$-1$	✓	$-1$	$13$	$C_2$, $S_3$, $S_3$, $(C_5^2 : C_3):C_2$	15T13, 15T14, 25T16, 30T38
30T38	$C_5^2:S_3$	$150$	$-1$	✓	$-1$	$13$	$C_2$, $S_3$, $S_3$, $(C_5^2 : C_3):C_2$	15T13, 15T14, 25T16, 30T37
30T39	$D_5\times C_{15}$	$150$	$-1$	✓	$-1$	$60$	$C_2$, $C_3$, $C_6$, $D_5\times C_5$	30T39
30T40	$C_5^2:C_6$	$150$	$-1$	✓	$-1$	$22$	$C_2$, $C_3$, $C_6$, $C_5^2 : C_3$	30T40
30T41	$C_5\times S_3^2$	$180$	$-1$	✓	$-1$	$45$	$C_2$, $C_5$, $S_3^2$, $C_{10}$	45T15
30T42	$S_3\times D_{15}$	$180$	$-1$	✓	$-1$	$27$	$C_2$, $D_{5}$, $S_3^2$, $D_{10}$	45T13
30T43	$C_{15}:D_6$	$180$	$-1$	✓	$-1$	$21$	$C_2$, $D_{5}$, $S_3^2$, $D_5$	45T21
30T44	$C_{15}:D_6$	$180$	$-1$	✓	$-1$	$36$	$C_2$, $D_{5}$, $S_3\times C_3$, $D_{10}$	45T14
30T45	$\GL(2,4)$	$180$	$1$		$-1$	$15$	$C_3$, $A_{5}$	15T15 x 2, 15T16, 18T90, 36T176, 45T16
30T46	$C_3^2:F_5$	$180$	$1$	✓	$-1$	$15$	$C_2$, $F_5$, $C_3^2:C_4$, $F_5$	30T46, 45T27
30T47	$C_{15}:C_{12}$	$180$	$-1$	✓	$-1$	$27$	$C_2$, $F_5$, $S_3\times C_3$, $F_5$	45T18
30T48	$(C_3\times C_{15}):C_4$	$180$	$1$	✓	$-1$	$18$	$C_2$, $D_{5}$, $C_3^2:C_4$, $D_5$	30T48, 45T26
30T49	$C_3^2:C_{20}$	$180$	$1$	✓	$-1$	$30$	$C_2$, $C_5$, $C_3^2:C_4$, $C_{10}$	30T49, 45T25
30T50	$F_{16}$	$240$	$1$	✓	$-1$	$16$	$C_3$, $C_5$, $C_{15}$	16T447, 20T67

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Results are complete for degrees $\leq 23$.