Subgroups — #35204071

Order 20288.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14579268	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14579268	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778742	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579268	—	2	1	no	—
5	C₂×#14579268	—	2	1	no	—
6	C₂×D₁₂₆₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₁₇	—	2	1	no	—
8	#19099551	—	2	1	no	—
9	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₁₇	C₄	1	1	yes	characteristic
11	#14579268	—	2	2	no	—
12	#14579268	—	4	1	no	—
13	C₂²×D₃₁₇	—	2	1	no	—
14	D₁₂₆₈	—	2	1	no	—
15	D₁₂₆₈	C₈	1	2	yes	complemented
16	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
17	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
18	C₄×D₃₁₇	—	4	1	no	—
19	C₄×D₃₁₇	—	4	1	no	—
20	C₂×C₄×C₃₁₇	—	2	1	no	—
21	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₃₁₇	—	2	2	no	—
24	C₂×D₃₁₇	—	4	1	no	—
25	C₂×D₃₁₇	—	4	1	no	—
26	C₄×C₃₁₇	—	4	1	no	—
27	C₄×C₃₁₇	C₂×C₈	1	1	yes	characteristic
28	C₄×C₃₁₇	M₁₆	1	1	yes	characteristic
29	Dic₃₁₇	—	2	1	no	—
30	Dic₃₁₇	D₈	1	1	yes	characteristic
31	Dic₃₁₇	SD₁₆	1	1	yes	characteristic
32	C₂×C₃₁₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₃₁₇	#102	1	1	yes	characteristic
34	C₂×C₃₁₇	#104	1	1	yes	characteristic
35	D₃₁₇	—	4	2	no	—
36	C₃₁₇	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
37	D₄⋊C₈	—	317	1	no	maximal, Sylow
38	C₄×D₄	—	317	1	no	—
39	C₄⋊C₈	—	317	1	no	—
40	C₄×C₈	—	317	1	no	—
41	C₂×D₄	—	317	1	no	—
42	C₂²×C₄	—	634	1	no	—
43	C₂×C₈	—	634	1	no	—
44	C₂×C₈	—	634	1	no	—
45	C₄⋊C₄	#11929639	1	1	yes	characteristic
46	C₂²⋊C₄	—	634	1	no	—
47	C₄²	—	317	1	no	—
48	C₂³	—	634	1	no	—
49	D₄	—	317	2	no	—
50	D₄	—	634	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	317	1	no	—
53	C₂×C₄	—	317	1	no	—
54	C₂×C₄	—	1268	1	no	—
55	C₂×C₄	—	1268	1	no	—
56	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
57	C₈	—	634	2	no	—
58	C₈	—	1268	1	no	—
59	C₂²	—	634	2	no	—
60	C₂²	—	1268	1	no	—
61	C₂²	—	1268	1	no	—
62	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
63	C₄	—	4	1	no	—
64	C₄	—	317	1	no	—
65	C₄	—	317	1	no	—
66	C₄	—	634	1	no	—
67	C₄	D₃₁₇⋊C₈	1	1	yes	characteristic
68	C₄	#19099584	1	1	yes	characteristic
69	C₂	—	1268	2	no	—
70	C₂	#23778837	1	1	yes	characteristic
71	C₂	#23778842	1	1	yes	characteristic
72	C₂	#23778845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11929639
commutator	C₄×C₃₁₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₄⋊C₄×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²⋊C₄
prime radical	C₂×D₁₂₆₈	C₄
cosocle	C₄×Dic₃₁₇	C₂²
hypercenter	C₄⋊C₄	#11929639
hypocenter	C₃₁₇	D₄⋊C₈
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#14579268	C₂	semidirect
2. C₄×#14579268	C₂	semidirect
15. D₁₂₆₈	C₈	semidirect
36. C₃₁₇	D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14579268	2
C₄×#14579268	2
#23778742	2
D₄⋊C₈	317