Subgroups — #35204127

Order 20288.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778839	C₂	1	2	yes	maximal, complemented
2	#23778776	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11929639	—	2	2	no	—
4	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊D₃₁₇	—	2	2	no	—
6	C₄×Dic₃₁₇	C₄	1	1	yes	characteristic, complemented
7	C₂×D₄×C₃₁₇	C₄	1	1	yes	characteristic, complemented
8	C₂×#11929639	—	4	2	no	—
9	C₂²×D₃₁₇	D₄	1	2	yes	—
10	D₁₂₆₈	—	4	1	no	—
11	C₂²⋊D₃₁₇	—	4	4	no	—
12	C₂×Dic₃₁₇	—	2	1	no	—
13	C₂³×C₃₁₇	—	2	1	no	—
14	D₄×C₃₁₇	—	4	1	no	—
15	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃₁₇	—	4	2	no	—
17	C₂²×C₃₁₇	—	4	1	no	—
18	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×D₃₁₇	—	2	2	no	—
20	C₂×D₃₁₇	—	4	2	no	—
21	#11929639	—	8	2	no	—
22	C₄×C₃₁₇	—	2	1	no	—
23	Dic₃₁₇	—	2	2	no	—
24	C₂×C₃₁₇	—	2	1	no	—
25	C₂×C₃₁₇	—	8	1	no	—
26	C₂×C₃₁₇	#99	1	1	yes	characteristic
27	D₃₁₇	—	4	2	no	—
28	C₃₁₇	#353	1	1	yes	characteristic, complemented, Sylow
29	#353	—	317	1	no	maximal, Sylow
30	#127	—	317	1	no	—
31	#99	—	317	2	no	—
32	C₂×D₄	—	317	1	no	—
33	C₂×D₄	—	634	2	no	—
34	C₂×D₄	#11929639	1	1	yes	characteristic, complemented
35	C₂²⋊C₄	—	634	2	no	—
36	C₄²	—	317	1	no	—
37	C₂³	—	2	1	no	—
38	C₂³	—	317	2	no	—
39	D₄	—	4	1	no	—
40	D₄	—	1268	1	no	—
41	D₄	—	1268	4	no	—
42	C₂×C₄	—	634	1	no	—
43	C₂×C₄	—	1268	2	no	—
44	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
45	C₂²	—	4	2	no	—
46	C₂²	—	4	1	no	—
47	C₂²	—	634	2	no	—
48	C₂²	—	1268	2	no	—
49	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	634	2	no	—
52	C₄	—	2536	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	8	1	no	—
55	C₂	—	1268	2	no	—
56	C₂	#23778838	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778838
commutator	C₂×C₄×C₃₁₇	C₂×C₄
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₂×D₄×C₃₁₇	C₄
socle	C₂×C₃₁₇	#99
prime radical	#23778776	C₂
cosocle	C₂×D₁₂₆₈	C₂²
hypercenter	C₂×D₄	#11929639
hypocenter	C₃₁₇	#353
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂²×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #23778839	C₂	semidirect
6. C₄×Dic₃₁₇	C₄	semidirect
7. C₂×D₄×C₃₁₇	C₄	semidirect
28. C₃₁₇	#353	semidirect
34. C₂×D₄	#11929639	semidirect

Maximal subgroups

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

subgroup	index
#23778839	2
#23778776	2
#353	317