Subgroups — #35204251

Order 20288.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₁₇⋊C₈	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14579268	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778837	C₂	1	2	yes	maximal, complemented
4	C₄×#14579268	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778742	C₂	1	1	yes	maximal, characteristic
6	D₃₁₇⋊C₈	—	2	4	no	—
7	C₂×#14579268	C₂²	1	1	yes	characteristic
8	C₂×#14579268	C₂²	1	2	yes	—
9	C₂×#14579268	C₂²	1	1	yes	characteristic
10	C₂×D₁₂₆₈	C₄	1	1	yes	characteristic
11	C₂×C₄×D₃₁₇	C₂²	1	2	yes	—
12	#19099551	C₄	1	2	yes	—
13	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₃₁₇	C₄	1	1	yes	characteristic
15	#14579268	—	2	2	no	—
16	#14579268	—	2	1	no	—
17	#14579268	D₄	1	2	yes	complemented
18	C₂²×D₃₁₇	C₂×C₄	1	2	yes	—
19	D₁₂₆₈	C₈	1	4	yes	complemented
20	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
22	C₄×D₃₁₇	—	2	2	no	—
23	C₄×D₃₁₇	—	2	2	no	—
24	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₁₇	—	2	2	no	—
28	C₂×D₃₁₇	—	2	2	no	—
29	C₂×D₃₁₇	C₂×C₈	1	4	yes	complemented
30	C₄×C₃₁₇	—	2	2	no	—
31	C₄×C₃₁₇	C₂×C₈	1	2	yes	complemented
32	Dic₃₁₇	—	2	1	no	—
33	Dic₃₁₇	C₂×D₄	1	1	yes	characteristic
34	Dic₃₁₇	#41	1	1	yes	characteristic
35	C₂×C₃₁₇	C₄×D₄	1	1	yes	characteristic
36	C₂×C₃₁₇	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₃₁₇	#131	1	1	yes	characteristic
38	D₃₁₇	—	2	4	no	—
39	C₃₁₇	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₈×D₄	—	317	1	no	maximal, Sylow
41	C₂²×C₈	—	317	2	no	—
42	C₄×D₄	—	317	1	no	—
43	C₄⋊C₈	—	317	1	no	—
44	C₂²⋊C₈	—	317	2	no	—
45	C₄×C₈	—	317	1	no	—
46	C₂×D₄	—	317	1	no	—
47	C₂²×C₄	—	317	2	no	—
48	C₂×C₈	—	317	1	no	—
49	C₂×C₈	—	317	1	no	—
50	C₂×C₈	—	317	2	no	—
51	C₂×C₈	—	634	4	no	—
52	C₄⋊C₄	#11929639	1	1	yes	characteristic
53	C₂²⋊C₄	—	317	2	no	—
54	C₄²	—	317	1	no	—
55	C₂³	—	317	2	no	—
56	D₄	—	317	4	no	—
57	C₂×C₄	—	317	1	no	—
58	C₂×C₄	—	317	1	no	—
59	C₂×C₄	—	634	2	no	—
60	C₂×C₄	—	634	2	no	—
61	C₂×C₄	C₂×#11929639	1	2	yes	—
62	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
63	C₈	—	317	2	no	—
64	C₈	—	634	2	no	—
65	C₈	—	634	1	no	—
66	C₂²	—	317	4	no	—
67	C₂²	—	634	2	no	—
68	C₂²	—	634	2	no	—
69	C₂²	C₂²×#11929639	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	317	1	no	—
72	C₄	—	317	1	no	—
73	C₄	—	634	1	no	—
74	C₄	D₃₁₇⋊C₈	1	2	yes	complemented
75	C₂	—	634	4	no	—
76	C₂	D₄×#11929639	1	1	yes	characteristic
77	C₂	C₂×D₃₁₇⋊C₈	1	1	yes	characteristic
78	C₂	#23778868	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11929639
commutator	C₂×C₃₁₇	C₂²×C₈
Fitting	C₄⋊C₄×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²×C₄
prime radical	C₂×D₁₂₆₈	C₄
cosocle	C₂×Dic₃₁₇	C₂³
hypercenter	C₄⋊C₄	#11929639
hypocenter	C₃₁₇	C₈×D₄
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₃₁₇⋊C₈	C₂	semidirect
2. C₄⋊#14579268	C₂	semidirect
3. #23778837	C₂	semidirect
4. C₄×#14579268	C₂	semidirect
17. #14579268	D₄	semidirect
19. D₁₂₆₈	C₈	semidirect
29. C₂×D₃₁₇	C₂×C₈	semidirect
31. C₄×C₃₁₇	C₂×C₈	semidirect
39. C₃₁₇	C₈×D₄	semidirect
74. C₄	D₃₁₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₁₇⋊C₈	2
C₄⋊#14579268	2
#23778837	2
C₄×#14579268	2
#23778742	2
C₈×D₄	317