Subgroups — #35204075

Order 20288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778837	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×#11929639)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778755	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929639	—	2	1	no	—
5	C₂×#14579268	—	2	1	no	—
6	C₂×D₁₂₆₈	—	2	1	no	—
7	C₂×D₁₂₆₈	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
9	#19099551	—	2	1	no	—
10	C₄⋊C₄×C₃₁₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#11929639	—	2	2	no	—
12	C₂×#11929639	—	4	1	no	—
13	C₂×#11929639	—	4	1	no	—
14	#14579268	—	4	1	no	—
15	C₂²×D₃₁₇	—	2	1	no	—
16	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
17	D₁₂₆₈	—	4	1	no	—
18	D₁₂₆₈	—	4	1	no	—
19	D₁₂₆₈	—	4	1	no	—
20	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
21	C₄×D₃₁₇	—	4	1	no	—
22	C₂×C₄×C₃₁₇	—	2	1	no	—
23	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×D₃₁₇	—	2	1	no	—
26	C₂×D₃₁₇	—	2	1	no	—
27	C₂×D₃₁₇	—	4	1	no	—
28	C₂×D₃₁₇	—	4	1	no	—
29	C₂×D₃₁₇	—	4	1	no	—
30	C₂×D₃₁₇	D₈	1	1	yes	characteristic
31	C₂×D₃₁₇	SD₁₆	1	1	yes	characteristic
32	#11929639	—	4	2	no	—
33	C₄×C₃₁₇	—	2	1	no	—
34	C₄×C₃₁₇	—	4	1	no	—
35	Dic₃₁₇	—	2	1	no	—
36	C₂×C₃₁₇	#99	1	1	yes	characteristic
37	C₂×C₃₁₇	#102	1	1	yes	characteristic
38	C₂×C₃₁₇	#104	1	1	yes	characteristic
39	D₃₁₇	—	2	1	no	—
40	D₃₁₇	—	2	1	no	—
41	D₃₁₇	—	8	1	no	—
42	C₃₁₇	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	317	1	no	maximal, Sylow
44	#121	—	317	1	no	—
45	C₂²⋊C₈	—	317	1	no	—
46	#95	—	317	1	no	—
47	C₂×D₄	—	317	1	no	—
48	C₂×D₄	—	634	1	no	—
49	C₂²×C₄	—	317	1	no	—
50	C₂²×C₄	—	634	1	no	—
51	C₂×C₈	—	634	1	no	—
52	C₄⋊C₄	#11929639	1	1	yes	characteristic, complemented
53	C₂²⋊C₄	—	634	1	no	—
54	C₂³	—	317	1	no	—
55	C₂³	—	634	1	no	—
56	D₄	—	1268	1	no	—
57	D₄	—	1268	1	no	—
58	D₄	—	1268	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	317	1	no	—
61	C₂×C₄	—	634	2	no	—
62	C₂×C₄	—	1268	1	no	—
63	C₂×C₄	—	1268	1	no	—
64	C₂×C₄	—	1268	1	no	—
65	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
66	C₈	—	1268	1	no	—
67	C₂²	—	317	1	no	—
68	C₂²	—	317	1	no	—
69	C₂²	—	634	1	no	—
70	C₂²	—	634	1	no	—
71	C₂²	—	1268	1	no	—
72	C₂²	—	1268	1	no	—
73	C₂²	—	1268	1	no	—
74	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	634	1	no	—
78	C₄	—	1268	2	no	—
79	C₂	—	634	1	no	—
80	C₂	—	634	1	no	—
81	C₂	—	2536	1	no	—
82	C₂	#23778839	1	1	yes	characteristic
83	C₂	#23778842	1	1	yes	characteristic
84	C₂	#23778845	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11929639
commutator	C₂×C₄×C₃₁₇	C₂×C₄
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₄⋊C₄×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²⋊C₄
prime radical	#23778755	C₂
cosocle	C₂×C₄×D₃₁₇	C₂²
hypercenter	C₄⋊C₄	#11929639
hypocenter	C₃₁₇	#327
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂²×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. #23778837	C₂	semidirect
2. (C₂×#11929639)⋊C₄	C₂	semidirect
7. C₂×D₁₂₆₈	C₄	semidirect
10. C₄⋊C₄×C₃₁₇	C₄	semidirect
42. C₃₁₇	#327	semidirect
52. C₄⋊C₄	#11929639	semidirect

Maximal subgroups

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

subgroup	index
#23778837	2
(C₂×#11929639)⋊C₄	2
#23778755	2
#327	317