Subgroups — #35204136

Order 20288.

58 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
2	#23778849	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11929639	—	2	1	no	—
5	#19099577	—	2	1	no	—
6	Q₈×D₃₁₇	—	2	1	no	—
7	Dic₁₂₆₈	C₄	1	1	yes	characteristic, complemented
8	#19099564	—	2	1	no	—
9	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₁₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#11929639	—	4	1	no	—
12	Dic₆₃₄	—	2	1	no	—
13	Dic₆₃₄	—	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	D₈	1	1	yes	characteristic
20	#11929639	—	8	1	no	—
21	C₄×C₃₁₇	—	4	1	no	—
22	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₃₁₇	—	4	1	no	—
24	Dic₃₁₇	SD₁₆	1	1	yes	characteristic
25	C₂×C₃₁₇	#102	1	1	yes	characteristic
26	D₃₁₇	SD₃₂	1	1	yes	characteristic
27	D₃₁₇	Q₃₂	1	1	yes	characteristic
28	C₃₁₇	#358	1	1	yes	characteristic, complemented, Sylow
29	#358	—	317	1	no	maximal, Sylow
30	C₂×Q₁₆	—	317	1	no	—
31	C₂×C₁₆	—	317	1	no	—
32	#107	—	317	1	no	—
33	C₂×Q₈	—	634	1	no	—
34	Q₁₆	—	317	1	no	—
35	Q₁₆	—	634	1	no	—
36	Q₁₆	#11929639	1	1	yes	characteristic, complemented
37	C₂×C₈	—	317	1	no	—
38	C₄⋊C₄	—	634	1	no	—
39	C₁₆	—	634	1	no	—
40	Q₈	—	2	1	no	—
41	Q₈	—	634	1	no	—
42	Q₈	—	1268	1	no	—
43	C₂×C₄	—	317	1	no	—
44	C₂×C₄	—	1268	1	no	—
45	C₂×C₄	—	1268	1	no	—
46	C₈	—	317	1	no	—
47	C₈	C₂×#11929639	1	1	yes	characteristic
48	C₂²	—	317	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	317	1	no	—
51	C₄	—	2536	1	no	—
52	C₄	—	1268	1	no	—
53	C₄	C₂²⋊#11929639	1	1	yes	characteristic
54	C₂	—	317	1	no	—
55	C₂	—	317	1	no	—
56	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	C₂×D₃₁₇	D₈
cosocle	C₈×D₃₁₇	C₂²
hypercenter	Q₁₆	#11929639
hypocenter	C₃₁₇	#358
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
7. Dic₁₂₆₈	C₄	semidirect
10. Q₁₆×C₃₁₇	C₄	semidirect
28. C₃₁₇	#358	semidirect
36. Q₁₆	#11929639	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
#23778849	2
Q₁₆×D₃₁₇	2
#358	317