Subgroups — #35204233

Order 20288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778859	C₂	1	2	yes	maximal
2	#23778849	C₂	1	2	yes	maximal, complemented
3	#23778848	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11929639	C₂²	1	4	yes	—
6	C₂²⋊#11929639	—	2	2	no	—
7	C₄×#11929639	—	2	2	no	—
8	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
9	C₂×#14579261	C₄	1	1	yes	characteristic, complemented
10	C₈×D₃₁₇	C₄	1	1	yes	characteristic, complemented
11	C₈×D₃₁₇	C₄	1	1	yes	characteristic, complemented
12	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
13	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#11929639	—	2	4	no	—
16	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
17	C₂×Dic₃₁₇	Q₈	1	1	yes	characteristic
18	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₁₇	D₄	1	1	yes	characteristic
20	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
21	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
22	#14579261	C₂×C₄	1	1	yes	characteristic
23	#14579261	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₃₁₇	—	2	2	no	—
29	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×D₃₁₇	C₂×D₄	1	1	yes	characteristic
31	#11929639	—	4	4	no	—
32	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
34	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
35	Dic₃₁₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₃₁₇	—	2	1	no	—
37	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	D₃₁₇	—	2	1	no	—
39	D₃₁₇	#135	1	2	yes	—
40	C₃₁₇	#427	1	1	yes	characteristic, complemented, Sylow
41	#427	—	317	1	no	maximal, Sylow
42	C₂²×C₈	—	317	1	no	—
43	#117	—	317	2	no	—
44	#107	—	317	2	no	—
45	#106	—	317	2	no	—
46	C₂²×C₄	—	317	1	no	—
47	C₂×C₈	—	317	1	no	—
48	C₂×C₈	—	317	1	no	—
49	C₂×C₈	—	317	1	no	—
50	C₂×C₈	—	317	1	no	—
51	C₂×C₈	—	317	1	no	—
52	C₂×C₈	#11929639	1	1	yes	characteristic, complemented
53	C₄⋊C₄	—	317	4	no	—
54	C₂²⋊C₄	—	634	2	no	—
55	C₄²	—	634	2	no	—
56	C₂³	—	317	1	no	—
57	C₂×C₄	—	317	1	no	—
58	C₂×C₄	—	317	1	no	—
59	C₂×C₄	—	317	1	no	—
60	C₂×C₄	—	317	1	no	—
61	C₂×C₄	—	317	1	no	—
62	C₂×C₄	—	634	4	no	—
63	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
64	C₈	—	317	1	no	—
65	C₈	—	317	1	no	—
66	C₈	C₂×#11929639	1	1	yes	characteristic
67	C₈	C₂×#11929639	1	1	yes	characteristic
68	C₂²	—	317	1	no	—
69	C₂²	—	317	1	no	—
70	C₂²	—	634	2	no	—
71	C₂²	C₄⋊#11929639	1	1	yes	characteristic
72	C₄	—	317	1	no	—
73	C₄	—	317	1	no	—
74	C₄	—	1268	4	no	—
75	C₄	C₄⋊#11929639	1	1	yes	characteristic
76	C₄	C₂²×#11929639	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	317	2	no	—
79	C₂	—	634	1	no	—
80	C₂	C₂×C₄⋊#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#11929639
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₁₇	D₄
cosocle	C₄×D₃₁₇	C₂³
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	#427
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #23778849	C₂	semidirect
3. #23778848	C₂	semidirect
9. C₂×#14579261	C₄	semidirect
10. C₈×D₃₁₇	C₄	semidirect
11. C₈×D₃₁₇	C₄	semidirect
14. C₂×C₈×C₃₁₇	C₄	semidirect
40. C₃₁₇	#427	semidirect
52. C₂×C₈	#11929639	semidirect

Maximal subgroups

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

subgroup	index
#23778859	2
#23778849	2
#23778848	2
C₂×C₈×D₃₁₇	2
#427	317