Subgroups — #35204076

Order 20288.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778837	C₂	1	1	yes	maximal, characteristic
2	(C₂×#11929639)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#23778760	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929639	—	2	1	no	—
5	C₂×#14579268	—	2	1	no	—
6	C₂×Dic₆₃₄	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃₁₇	—	2	1	no	—
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	D₄	1	1	yes	characteristic
16	Dic₆₃₄	—	4	1	no	—
17	C₂×Dic₃₁₇	—	2	1	no	—
18	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
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	1	no	—
24	C₂×D₃₁₇	—	2	1	no	—
25	C₂×D₃₁₇	SD₁₆	1	1	yes	characteristic
26	C₂×D₃₁₇	Q₁₆	1	1	yes	characteristic
27	#11929639	—	4	2	no	—
28	C₄×C₃₁₇	—	2	1	no	—
29	C₄×C₃₁₇	—	4	1	no	—
30	Dic₃₁₇	—	2	1	no	—
31	Dic₃₁₇	—	4	1	no	—
32	C₂×C₃₁₇	#99	1	1	yes	characteristic
33	C₂×C₃₁₇	#103	1	1	yes	characteristic
34	C₂×C₃₁₇	#104	1	1	yes	characteristic
35	D₃₁₇	—	2	1	no	—
36	D₃₁₇	—	2	1	no	—
37	C₃₁₇	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	317	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	317	1	no	—
40	C₂²⋊C₈	—	317	1	no	—
41	#95	—	317	1	no	—
42	C₂×Q₈	—	317	1	no	—
43	C₂²×C₄	—	317	1	no	—
44	C₂²×C₄	—	634	1	no	—
45	C₂×C₈	—	634	1	no	—
46	C₄⋊C₄	—	634	1	no	—
47	C₄⋊C₄	#11929639	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	634	1	no	—
49	C₂³	—	317	1	no	—
50	Q₈	—	1268	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	317	1	no	—
53	C₂×C₄	—	634	2	no	—
54	C₂×C₄	—	634	1	no	—
55	C₂×C₄	—	1268	1	no	—
56	C₂×C₄	—	1268	1	no	—
57	C₂×C₄	—	1268	1	no	—
58	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
59	C₈	—	1268	1	no	—
60	C₂²	—	317	1	no	—
61	C₂²	—	317	1	no	—
62	C₂²	—	634	1	no	—
63	C₂²	—	634	1	no	—
64	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	634	1	no	—
68	C₄	—	1268	2	no	—
69	C₄	—	1268	1	no	—
70	C₂	—	634	1	no	—
71	C₂	—	634	1	no	—
72	C₂	#23778839	1	1	yes	characteristic
73	C₂	#23778843	1	1	yes	characteristic
74	C₂	#23778844	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	C₂²×D₃₁₇	D₄
cosocle	C₂×C₄×D₃₁₇	C₂²
hypercenter	C₄⋊C₄	#11929639
hypocenter	C₃₁₇	#328
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
6. C₂×Dic₆₃₄	C₄	semidirect
10. C₄⋊C₄×C₃₁₇	C₄	semidirect
37. C₃₁₇	#328	semidirect
47. 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
#23778760	2
#328	317