Subgroups — #35203302

Order 20288.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099557	C₂	1	2	yes	maximal, complemented
2	C₄×D₁₂₆₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#14579261	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778692	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₆₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₁₇	C₂²	1	2	yes	—
8	#19099557	—	2	4	no	—
9	C₂×#14579261	C₂²	1	2	yes	—
10	C₄⋊Dic₃₁₇	C₄	1	1	yes	characteristic
11	#19099551	C₄	1	2	yes	—
12	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
14	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
15	C₂²×D₃₁₇	C₂×C₄	1	2	yes	—
16	D₁₂₆₈	—	2	2	no	—
17	C₂×Dic₃₁₇	C₂×C₄	1	2	yes	—
18	C₄×D₃₁₇	—	2	2	no	—
19	C₄×D₃₁₇	—	2	2	no	—
20	#14579261	—	2	2	no	—
21	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
24	C₈×C₃₁₇	—	2	1	no	—
25	C₈×C₃₁₇	D₄	1	2	yes	complemented
26	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₁₇	—	2	2	no	—
28	C₂×D₃₁₇	—	2	2	no	—
29	C₂×D₃₁₇	—	2	2	no	—
30	C₄×C₃₁₇	—	2	1	no	—
31	C₄×C₃₁₇	M₁₆	1	2	yes	complemented
32	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
33	C₄×C₃₁₇	#41	1	1	yes	characteristic
34	Dic₃₁₇	—	2	2	no	—
35	C₂×C₃₁₇	C₄×D₄	1	1	yes	characteristic
36	C₂×C₃₁₇	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₃₁₇	#131	1	1	yes	characteristic
38	D₃₁₇	—	4	2	no	—
39	C₃₁₇	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	317	1	no	maximal, Sylow
41	C₂×M₁₆	—	317	2	no	—
42	C₄×D₄	—	317	1	no	—
43	C₄⋊C₈	—	317	1	no	—
44	C₂²⋊C₈	—	317	2	no	—
45	C₄×C₈	D₃₁₇	1	1	yes	characteristic, complemented
46	C₂×D₄	—	317	1	no	—
47	C₂²×C₄	—	317	2	no	—
48	M₁₆	—	634	4	no	—
49	C₂×C₈	—	317	2	no	—
50	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
51	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
52	C₄⋊C₄	—	317	1	no	—
53	C₂²⋊C₄	—	317	2	no	—
54	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
55	C₂³	—	317	2	no	—
56	D₄	—	634	2	no	—
57	C₂×C₄	—	317	2	no	—
58	C₂×C₄	—	634	2	no	—
59	C₂×C₄	—	634	2	no	—
60	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
61	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	634	2	no	—
65	C₈	D₁₂₆₈	1	2	yes	complemented
66	C₂²	—	634	2	no	—
67	C₂²	—	634	2	no	—
68	C₂²	—	634	2	no	—
69	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	634	2	no	—
72	C₄	#19099557	1	2	yes	complemented
73	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
74	C₄	#19099574	1	1	yes	characteristic
75	C₂	—	1268	2	no	—
76	C₂	C₄×D₁₂₆₈	1	1	yes	characteristic
77	C₂	C₂×#19099557	1	1	yes	characteristic
78	C₂	#23778786	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₁₇
commutator, socle	C₂²×C₃₁₇	C₂²×C₄
Fitting	C₄×C₈×C₃₁₇	C₂
prime radical	C₂×D₁₂₆₈	C₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₄×C₈	D₃₁₇
hypocenter	C₃₁₇	#436
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19099557	C₂	semidirect
3. C₄⋊#14579261	C₂	semidirect
4. #23778692	C₂	semidirect
5. C₄×C₈×C₃₁₇	C₂	semidirect
25. C₈×C₃₁₇	D₄	semidirect
31. C₄×C₃₁₇	M₁₆	semidirect
39. C₃₁₇	#436	semidirect
45. C₄×C₈	D₃₁₇	semidirect
65. C₈	D₁₂₆₈	semidirect
72. C₄	#19099557	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099557	2
C₄×D₁₂₆₈	2
C₄⋊#14579261	2
#23778692	2
C₄×C₈×C₃₁₇	2
#436	317