Subgroups — #35203350

Order 20288.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₃₄	C₂	1	1	yes	maximal, characteristic
2	#23778709	C₂	1	2	yes	maximal
3	#23778690	C₂	1	2	yes	maximal
4	C₄×#14579261	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₃₄	C₄	1	1	yes	characteristic
7	C₂×#14579261	C₂²	1	1	yes	characteristic
8	C₂×#14579261	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₃₁₇	C₄	1	1	yes	characteristic
10	#19099553	C₄	1	2	yes	—
11	C₄×Dic₃₁₇	C₂²	1	2	yes	—
12	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
13	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
14	Dic₆₃₄	—	2	2	no	—
15	C₂×Dic₃₁₇	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₁₇	C₂×C₄	1	2	yes	—
17	#14579261	—	2	1	no	—
18	#14579261	Q₈	1	2	yes	complemented
19	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
22	C₈×C₃₁₇	—	2	2	no	—
23	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₁₇	—	2	1	no	—
25	C₄×C₃₁₇	M₁₆	1	2	yes	—
26	C₄×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₃₁₇	#41	1	1	yes	characteristic
28	Dic₃₁₇	—	2	2	no	—
29	Dic₃₁₇	—	2	2	no	—
30	C₂×C₃₁₇	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₃₁₇	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₃₁₇	#131	1	1	yes	characteristic
33	C₃₁₇	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	317	1	no	maximal, Sylow
35	C₄×Q₈	—	317	1	no	—
36	C₄⋊C₈	—	317	2	no	—
37	C₄⋊C₈	D₃₁₇	1	1	yes	characteristic
38	#97	—	317	2	no	—
39	C₄×C₈	—	317	1	no	—
40	C₂×Q₈	—	317	1	no	—
41	C₂×C₈	—	317	1	no	—
42	C₂×C₈	—	317	1	no	—
43	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
44	C₄⋊C₄	—	317	2	no	—
45	C₄⋊C₄	—	317	1	no	—
46	C₄²	—	317	2	no	—
47	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
48	Q₈	—	634	2	no	—
49	C₂×C₄	—	317	2	no	—
50	C₂×C₄	—	317	2	no	—
51	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
52	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	317	2	no	—
56	C₈	—	634	1	no	—
57	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	634	2	no	—
60	C₄	—	634	2	no	—
61	C₄	#19099557	1	2	yes	—
62	C₄	Q₈×D₃₁₇	1	1	yes	characteristic
63	C₄	#19099573	1	1	yes	characteristic
64	C₂	#23778745	1	1	yes	characteristic
65	C₂	C₂×#19099557	1	1	yes	characteristic
66	C₂	#23778788	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₁₇
commutator, socle, prime radical	C₂²×C₃₁₇	C₂²×C₄
Fitting	C₄⋊C₈×C₃₁₇	C₂
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₄⋊C₈	D₃₁₇
hypocenter	C₃₁₇	#446
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
18. #14579261	Q₈	semidirect
33. C₃₁₇	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₃₄	2
#23778709	2
#23778690	2
C₄×#14579261	2
C₄⋊C₈×C₃₁₇	2
#446	317