Subgroups — #35204282

Order 20288.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778855	C₂	1	2	yes	maximal, complemented
2	#23778854	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃₁₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×#19099577	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778788	C₂	1	1	yes	maximal, characteristic
6	#19099577	C₂²	1	1	yes	characteristic
7	#19099577	C₂²	1	1	yes	characteristic
8	#19099577	C₂²	1	2	yes	—
9	#19099574	C₄	1	1	yes	characteristic
10	#19099557	C₄	1	2	yes	—
11	C₂×#14579261	C₂²	1	1	yes	characteristic
12	C₈×D₃₁₇	C₂²	1	2	yes	—
13	M₁₆×C₃₁₇	C₄	1	1	yes	characteristic
14	Dic₆₃₄	C₈	1	1	yes	characteristic
15	D₁₂₆₈	C₈	1	1	yes	characteristic
16	C₂²⋊D₃₁₇	C₈	1	2	yes	—
17	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
18	#14579261	C₂×C₄	1	1	yes	characteristic
19	#14579261	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
21	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₃₁₇	C₂×C₈	1	1	yes	characteristic
23	C₂×D₃₁₇	C₂×C₈	1	2	yes	—
24	C₄×C₃₁₇	C₂×C₈	1	1	yes	characteristic
25	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
26	Dic₃₁₇	C₂×C₈	1	2	yes	—
27	C₂×C₃₁₇	—	2	1	no	—
28	C₂×C₃₁₇	C₂²×C₈	1	1	yes	characteristic
29	D₃₁₇	—	2	2	no	—
30	C₃₁₇	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	317	1	no	maximal, Sylow
32	#131	—	317	1	no	—
33	M₃₂	—	317	1	no	—
34	M₃₂	—	317	2	no	—
35	C₂×C₁₆	—	317	1	no	—
36	C₂×C₁₆	—	317	2	no	—
37	#41	—	317	1	no	—
38	M₁₆	—	317	2	no	—
39	M₁₆	#11929639	1	1	yes	characteristic
40	C₂×C₈	—	317	1	no	—
41	C₂×C₈	—	317	2	no	—
42	C₁₆	—	317	1	no	—
43	C₁₆	—	317	1	no	—
44	C₁₆	—	317	2	no	—
45	Q₈	—	317	1	no	—
46	D₄	—	317	2	no	—
47	D₄	—	317	1	no	—
48	C₂×C₄	—	317	2	no	—
49	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
50	C₈	—	317	1	no	—
51	C₈	—	317	1	no	—
52	C₈	C₂×#11929639	1	2	yes	—
53	C₂²	—	317	2	no	—
54	C₂²	D₃₁₇⋊C₈	1	1	yes	characteristic
55	C₄	—	317	2	no	—
56	C₄	D₃₁₇⋊C₈	1	1	yes	characteristic
57	C₄	C₂²×#11929639	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	634	2	no	—
60	C₂	C₂×D₃₁₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11929639
commutator, socle	C₂×C₃₁₇	C₂²×C₈
Fitting	M₁₆×C₃₁₇	C₄
prime radical	#19099574	C₄
cosocle	#14579261	C₂³
hypercenter	M₁₆	#11929639
hypocenter	C₃₁₇	#504
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #23778855	C₂	semidirect
2. #23778854	C₂	semidirect
3. D₃₁₇⋊C₁₆	C₂	semidirect
4. C₂×#19099577	C₂	semidirect
30. C₃₁₇	#504	semidirect

Maximal subgroups

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

subgroup	index
#23778855	2
#23778854	2
D₃₁₇⋊C₁₆	2
C₂×#19099577	2
#23778788	2
#504	317