Subgroups — #34088015

Order 19648.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₁₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	Dic₃₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	Dic₃₀₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#23754170	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₃₀₇	C₂	1	1	yes	maximal, characteristic
7	#97×C₃₀₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₁₄	C₄	1	1	yes	characteristic
9	C₂×C₃₀₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₀₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₀₇	C₄	1	1	yes	characteristic
12	Dic₃₀₇⋊C₄	C₄	1	1	yes	characteristic
13	Dic₃₀₇⋊C₄	C₄	1	1	yes	characteristic
14	C₄×Dic₃₀₇	C₂²	1	1	yes	characteristic
15	C₄×Dic₃₀₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₃₀₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₀₇	C₂²	1	1	yes	characteristic
18	C₄²×C₃₀₇	C₂²	1	1	yes	characteristic
19	Dic₆₁₄	—	2	2	no	—
20	C₂×Dic₃₀₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₃₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₃₀₇	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₃₀₇	C₂×C₄	1	1	yes	characteristic
24	C₃₀₇⋊C₈	—	2	1	no	—
25	C₃₀₇⋊C₈	—	2	1	no	—
26	C₂×C₄×C₃₀₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₃₀₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₀₇	C₂³	1	1	yes	characteristic
29	C₈×C₃₀₇	—	2	1	no	—
30	C₈×C₃₀₇	Q₈	1	2	yes	complemented
31	C₂²×C₃₀₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₃₀₇	—	2	1	no	—
33	C₄×C₃₀₇	—	2	1	no	—
34	C₄×C₃₀₇	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₃₀₇	#41	1	1	yes	characteristic
36	Dic₃₀₇	—	2	1	no	—
37	Dic₃₀₇	—	2	1	no	—
38	Dic₃₀₇	—	2	1	no	—
39	Dic₃₀₇	M₁₆	1	2	yes	—
40	C₂×C₃₀₇	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₃₀₇	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₃₀₇	#131	1	1	yes	characteristic
43	C₃₀₇	#446	1	1	yes	characteristic, complemented, Sylow
44	#446	—	307	1	no	maximal, Sylow
45	C₄×Q₈	—	307	1	no	—
46	C₄⋊C₈	—	307	1	no	—
47	C₄⋊C₈	—	307	1	no	—
48	C₄⋊C₈	—	307	1	no	—
49	#97	—	307	1	no	—
50	#97	D₃₀₇	1	1	yes	characteristic
51	C₄×C₈	—	307	1	no	—
52	C₂×Q₈	—	307	1	no	—
53	C₂×C₈	—	307	1	no	—
54	C₂×C₈	—	307	1	no	—
55	C₂×C₈	C₂×D₃₀₇	1	1	yes	characteristic
56	C₂×C₈	C₂×D₃₀₇	1	1	yes	characteristic
57	C₄⋊C₄	—	307	1	no	—
58	C₄⋊C₄	—	307	1	no	—
59	C₄⋊C₄	—	307	1	no	—
60	C₄²	—	307	1	no	—
61	C₄²	—	307	1	no	—
62	C₄²	C₂×D₃₀₇	1	1	yes	characteristic
63	Q₈	—	614	2	no	—
64	C₂×C₄	—	307	1	no	—
65	C₂×C₄	—	307	1	no	—
66	C₂×C₄	—	307	1	no	—
67	C₂×C₄	—	307	1	no	—
68	C₂×C₄	C₄×D₃₀₇	1	1	yes	characteristic
69	C₂×C₄	C₄×D₃₀₇	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₃₀₇	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	614	1	no	—
73	C₈	—	614	1	no	—
74	C₈	Dic₆₁₄	1	2	yes	complemented
75	C₂²	C₂×C₄×D₃₀₇	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	307	2	no	—
79	C₄	—	614	1	no	—
80	C₄	—	614	1	no	—
81	C₄	—	614	1	no	—
82	C₄	C₂×Dic₆₁₄	1	1	yes	characteristic
83	C₄	#19096826	1	1	yes	characteristic
84	C₂	C₄×Dic₆₁₄	1	1	yes	characteristic
85	C₂	M₁₆×D₃₀₇	1	1	yes	characteristic
86	C₂	#23754268	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₀₇
commutator, socle, prime radical	C₂²×C₃₀₇	C₂²×C₄
Fitting	#97×C₃₀₇	C₂
cosocle	C₂×C₄×C₃₀₇	C₂³
hypercenter	#97	D₃₀₇
hypocenter	C₃₀₇	#446
LCS	G ▷ C₂²×C₃₀₇ ▷ C₃₀₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
30. C₈×C₃₀₇	Q₈	semidirect
43. C₃₀₇	#446	semidirect
74. C₈	Dic₆₁₄	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×Dic₆₁₄	2
C₄⋊(C₃₀₇⋊C₈)	2
Dic₃₀₇⋊C₈	2
#23754170	2
C₈×Dic₃₀₇	2
#97×C₃₀₇	2
#446	307