Subgroups — #35203398

Order 20288.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778828	C₂	1	1	yes	maximal, characteristic, complemented
2	#23778808	C₂	1	2	yes	maximal, complemented
3	#23778769	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778706	C₂	1	2	yes	maximal, complemented
5	#101×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099575	—	2	2	no	—
7	#19099574	—	2	2	no	—
8	#19099574	C₂²	1	2	yes	—
9	Q₈×D₃₁₇	—	2	2	no	—
10	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
11	#19099563	—	2	2	no	—
12	D₂₅₃₆	—	2	2	no	—
13	#19099551	—	2	2	no	—
14	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₁₇	C₂²	1	1	yes	characteristic
16	M₁₆×C₃₁₇	C₂²	1	2	yes	—
17	C₂²×D₃₁₇	—	2	1	no	—
18	Dic₆₃₄	—	2	2	no	—
19	Dic₆₃₄	D₄	1	2	yes	—
20	D₁₂₆₈	—	2	2	no	—
21	D₁₂₆₈	—	2	2	no	—
22	D₁₂₆₈	D₄	1	2	yes	—
23	C₂²⋊D₃₁₇	—	2	2	no	—
24	C₂×Dic₃₁₇	—	2	1	no	—
25	C₄×D₃₁₇	—	2	4	no	—
26	C₄×D₃₁₇	—	2	2	no	—
27	Q₈×C₃₁₇	—	2	2	no	—
28	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
29	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
30	C₈×C₃₁₇	—	2	2	no	—
31	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₁₇	—	2	2	no	—
33	C₂×D₃₁₇	—	4	2	no	—
34	C₂×D₃₁₇	—	4	1	no	—
35	C₄×C₃₁₇	—	2	2	no	—
36	C₄×C₃₁₇	C₂×D₄	1	2	yes	—
37	Dic₃₁₇	—	2	2	no	—
38	Dic₃₁₇	—	4	1	no	—
39	C₂×C₃₁₇	—	2	1	no	—
40	C₂×C₃₁₇	#120	1	1	yes	characteristic
41	D₃₁₇	—	4	2	no	—
42	D₃₁₇	—	8	1	no	—
43	C₃₁₇	#454	1	1	yes	characteristic, complemented, Sylow
44	#454	—	317	1	no	maximal, Sylow
45	2₋¹⁺⁴	—	317	1	no	—
46	AGL(1,ℤ/8)	—	317	2	no	—
47	#124	—	317	1	no	—
48	#104	—	317	2	no	—
49	#101	D₃₁₇	1	1	yes	characteristic, complemented
50	#41	—	317	2	no	—
51	#41	—	634	2	no	—
52	#41	—	634	2	no	—
53	C₂×Q₈	—	634	2	no	—
54	C₂×Q₈	C₂×D₃₁₇	1	1	yes	characteristic
55	C₂×D₄	—	317	1	no	—
56	SD₁₆	—	634	2	no	—
57	D₈	—	634	2	no	—
58	M₁₆	C₂×D₃₁₇	1	2	yes	—
59	C₂²⋊C₄	—	634	2	no	—
60	C₄²	—	317	1	no	—
61	C₂³	—	634	1	no	—
62	Q₈	—	2	2	no	—
63	Q₈	—	317	2	no	—
64	Q₈	—	634	2	no	—
65	D₄	—	317	2	no	—
66	D₄	—	634	2	no	—
67	D₄	—	634	2	no	—
68	D₄	—	634	2	no	—
69	C₂×C₄	—	634	4	no	—
70	C₂×C₄	—	634	2	no	—
71	C₂×C₄	—	634	1	no	—
72	C₂×C₄	D₁₂₆₈	1	2	yes	—
73	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₂²	—	634	2	no	—
76	C₂²	—	1268	2	no	—
77	C₂²	—	1268	1	no	—
78	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	634	2	no	—
81	C₄	—	1268	1	no	—
82	C₄	D₄×D₃₁₇	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	2536	1	no	—
85	C₂	—	1268	2	no	—
86	C₂	#23778750	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778750
commutator, cosocle	C₂×C₄×C₃₁₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	#101×C₃₁₇	C₂
socle	C₂×C₃₁₇	#120
hypercenter	#101	D₃₁₇
hypocenter	C₃₁₇	#454
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

Complements

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

N	H ≅ G/N	type
1. #23778828	C₂	semidirect
2. #23778808	C₂	semidirect
3. #23778769	C₂	semidirect
4. #23778706	C₂	semidirect
5. #101×C₃₁₇	C₂	semidirect
43. C₃₁₇	#454	semidirect
49. #101	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778828	2
#23778808	2
#23778769	2
#23778706	2
#101×C₃₁₇	2
#454	317