Subgroups — #35204141

Order 20288.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778855	C₂	1	1	yes	maximal, characteristic, complemented
2	#23778850	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#19099584	—	2	1	no	—
5	#19099577	—	2	1	no	—
6	D₄×D₃₁₇	—	2	1	no	—
7	D₂₅₃₆	C₄	1	1	yes	characteristic
8	#19099559	—	2	1	no	—
9	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
10	D₈×C₃₁₇	C₄	1	1	yes	characteristic
11	#14579268	—	4	1	no	—
12	C₂²×D₃₁₇	—	4	1	no	—
13	D₁₂₆₈	—	2	1	no	—
14	C₂²⋊D₃₁₇	—	4	1	no	—
15	C₄×D₃₁₇	D₄	1	1	yes	characteristic
16	#14579261	D₄	1	1	yes	characteristic
17	D₄×C₃₁₇	—	2	1	no	—
18	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃₁₇	—	4	1	no	—
20	C₂×D₃₁₇	—	4	1	no	—
21	C₂×D₃₁₇	—	8	2	no	—
22	C₂×D₃₁₇	SD₁₆	1	1	yes	characteristic
23	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₃₁₇	D₈	1	1	yes	characteristic
25	C₂×C₃₁₇	—	8	1	no	—
26	C₂×C₃₁₇	#102	1	1	yes	characteristic
27	D₃₁₇	—	2	1	no	—
28	D₃₁₇	—	8	1	no	—
29	C₃₁₇	#361	1	1	yes	characteristic, complemented, Sylow
30	#361	—	317	1	no	maximal, Sylow
31	C₂×D₈	—	317	1	no	—
32	M₃₂	—	317	1	no	—
33	#108	—	317	1	no	—
34	C₂×D₄	—	634	1	no	—
35	D₈	—	317	1	no	—
36	D₈	—	634	1	no	—
37	D₈	#11929639	1	1	yes	characteristic
38	M₁₆	—	634	1	no	—
39	C₂×C₈	—	317	1	no	—
40	C₁₆	—	634	1	no	—
41	C₂³	—	1268	1	no	—
42	D₄	—	2	1	no	—
43	D₄	—	634	1	no	—
44	D₄	—	1268	1	no	—
45	C₂×C₄	—	317	1	no	—
46	C₈	—	317	1	no	—
47	C₈	—	1268	1	no	—
48	C₈	C₂×#11929639	1	1	yes	characteristic
49	C₂²	—	4	1	no	—
50	C₂²	—	317	1	no	—
51	C₂²	—	2536	2	no	—
52	C₂²	—	1268	1	no	—
53	C₄	—	317	1	no	—
54	C₄	C₂²⋊#11929639	1	1	yes	characteristic
55	C₂	—	8	1	no	—
56	C₂	—	634	1	no	—
57	C₂	—	2536	1	no	—
58	C₂	#23778842	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778842
commutator	C₈×C₃₁₇	C₂×C₄
Frattini	C₈	C₂×#11929639
Fitting	D₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	#102
prime radical	D₈×D₃₁₇	C₂
cosocle	C₈×D₃₁₇	C₂²
hypercenter	D₈	#11929639
hypocenter	C₃₁₇	#361
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #23778855	C₂	semidirect
2. #23778850	C₂	semidirect
29. C₃₁₇	#361	semidirect

Maximal subgroups

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

subgroup	index
#23778855	2
#23778850	2
D₈×D₃₁₇	2
#361	317