Subgroups — #35204138

Order 20288.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₁₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#23778851	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778803	C₂	1	1	yes	maximal, characteristic
4	#19099584	—	2	1	no	—
5	#19099577	—	2	1	no	—
6	#19099575	—	2	1	no	—
7	#19099561	—	2	1	no	—
8	D₂₅₃₆	C₄	1	1	yes	characteristic
9	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₁₇	C₄	1	1	yes	characteristic
11	#14579268	—	4	1	no	—
12	D₁₂₆₈	—	2	1	no	—
13	D₁₂₆₈	—	4	1	no	—
14	C₄×D₃₁₇	—	4	1	no	—
15	C₄×D₃₁₇	D₄	1	1	yes	characteristic
16	#14579261	D₄	1	1	yes	characteristic
17	Q₈×C₃₁₇	—	2	1	no	—
18	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₁₇	—	4	1	no	—
20	C₂×D₃₁₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₃₁₇	—	4	1	no	—
22	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₃₁₇	D₈	1	1	yes	characteristic
24	C₂×C₃₁₇	#102	1	1	yes	characteristic
25	D₃₁₇	—	2	1	no	—
26	D₃₁₇	—	8	1	no	—
27	C₃₁₇	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	317	1	no	maximal, Sylow
29	#135	—	317	1	no	—
30	C₂×C₁₆	—	317	1	no	—
31	#108	—	317	1	no	—
32	#41	—	634	1	no	—
33	Q₁₆	#11929639	1	1	yes	characteristic
34	SD₁₆	—	634	1	no	—
35	D₈	—	317	1	no	—
36	M₁₆	—	634	1	no	—
37	C₂×C₈	—	317	1	no	—
38	C₁₆	—	634	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	634	1	no	—
41	D₄	—	1268	1	no	—
42	C₂×C₄	—	317	1	no	—
43	C₂×C₄	—	1268	1	no	—
44	C₈	—	317	1	no	—
45	C₈	—	1268	1	no	—
46	C₈	C₂×#11929639	1	1	yes	characteristic
47	C₂²	—	317	1	no	—
48	C₂²	—	1268	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	317	1	no	—
51	C₄	C₂²⋊#11929639	1	1	yes	characteristic
52	C₂	—	634	1	no	—
53	C₂	—	2536	1	no	—
54	C₂	#23778842	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778842
commutator	C₈×C₃₁₇	C₂×C₄
Frattini	C₈	C₂×#11929639
Fitting	Q₁₆×C₃₁₇	C₄
socle	C₂×C₃₁₇	#102
prime radical	#23778803	C₂
cosocle	C₈×D₃₁₇	C₂²
hypercenter	Q₁₆	#11929639
hypocenter	C₃₁₇	#359
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. D₃₁₇⋊C₁₆	C₂	semidirect
2. #23778851	C₂	semidirect
27. C₃₁₇	#359	semidirect

Maximal subgroups

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

subgroup	index
D₃₁₇⋊C₁₆	2
#23778851	2
#23778803	2
#359	317