Subgroups — #35204204

Order 20288.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778859	C₂	1	2	yes	maximal, complemented
2	#23778839	C₂	1	4	yes	maximal, complemented
3	C₂×#19099575	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11929639	—	2	2	no	—
5	C₂²⋊#11929639	C₂²	1	4	yes	—
6	C₄×#11929639	—	2	2	no	—
7	#19099575	—	2	2	no	—
8	#19099575	—	2	2	no	—
9	C₂×D₁₂₆₈	C₄	1	1	yes	characteristic, complemented
10	C₂×D₁₂₆₈	C₂²	1	2	yes	—
11	C₂×C₄×D₃₁₇	C₄	1	2	yes	complemented
12	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
14	C₂×#11929639	—	2	4	no	—
15	C₂²×D₃₁₇	C₂×C₄	1	2	yes	—
16	C₂²×D₃₁₇	C₂³	1	1	yes	characteristic
17	D₁₂₆₈	—	2	4	no	—
18	D₁₂₆₈	—	2	2	no	—
19	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₄×D₃₁₇	—	2	2	no	—
21	C₄×D₃₁₇	—	2	2	no	—
22	C₄×D₃₁₇	D₄	1	2	yes	—
23	C₄×D₃₁₇	D₄	1	2	yes	—
24	Q₈×C₃₁₇	—	2	2	no	—
25	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₃₁₇	—	2	2	no	—
29	C₂×D₃₁₇	—	4	1	no	—
30	C₂×D₃₁₇	—	4	2	no	—
31	C₂×D₃₁₇	C₂×D₄	1	2	yes	—
32	#11929639	—	4	4	no	—
33	C₄×C₃₁₇	—	2	2	no	—
34	C₄×C₃₁₇	C₂²⋊C₄	1	2	yes	—
35	Dic₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₃₁₇	—	2	1	no	—
38	C₂×C₃₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	D₃₁₇	—	2	2	no	—
40	D₃₁₇	—	4	2	no	—
41	C₃₁₇	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	317	1	no	maximal, Sylow
43	C₂×#41	—	317	1	no	—
44	#117	—	317	2	no	—
45	#99	—	317	4	no	—
46	#41	—	634	2	no	—
47	#41	—	634	2	no	—
48	C₂×Q₈	#11929639	1	1	yes	characteristic, complemented
49	C₂×D₄	—	317	2	no	—
50	C₂×D₄	—	317	1	no	—
51	C₂²×C₄	—	317	1	no	—
52	C₂²×C₄	—	317	2	no	—
53	C₄⋊C₄	—	634	2	no	—
54	C₂²⋊C₄	—	317	4	no	—
55	C₄²	—	634	2	no	—
56	C₂³	—	317	1	no	—
57	C₂³	—	317	2	no	—
58	Q₈	—	2	2	no	—
59	D₄	—	634	4	no	—
60	D₄	—	634	2	no	—
61	C₂×C₄	—	317	1	no	—
62	C₂×C₄	—	317	2	no	—
63	C₂×C₄	—	317	2	no	—
64	C₂×C₄	—	634	2	no	—
65	C₂×C₄	—	634	2	no	—
66	C₂×C₄	—	634	4	no	—
67	C₂×C₄	C₂×#11929639	1	2	yes	—
68	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
69	C₂²	—	317	2	no	—
70	C₂²	—	634	2	no	—
71	C₂²	—	1268	1	no	—
72	C₂²	—	1268	2	no	—
73	C₂²	C₂²×#11929639	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	317	1	no	—
76	C₄	—	317	1	no	—
77	C₄	—	1268	4	no	—
78	C₄	C₂²⋊#11929639	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	634	2	no	—
81	C₂	—	1268	2	no	—
82	C₂	C₂×C₂²⋊#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11929639
commutator	C₂²×C₃₁₇	C₂²×C₄
Frattini	C₂²	C₂²×#11929639
Fitting	C₂×Q₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	C₂×C₂²⋊C₄
prime radical	C₂×#19099575	C₂
cosocle	C₂²×D₃₁₇	C₂³
hypercenter	C₂×Q₈	#11929639
hypocenter	C₃₁₇	#410
LCS	G ▷ C₂²×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #23778859	C₂	semidirect
2. #23778839	C₂	semidirect
9. C₂×D₁₂₆₈	C₄	semidirect
11. C₂×C₄×D₃₁₇	C₄	semidirect
13. C₂×Q₈×C₃₁₇	C₄	semidirect
41. C₃₁₇	#410	semidirect
48. C₂×Q₈	#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
#23778839	2
C₂×#19099575	2
#410	317