Subgroups — #35203177

Order 20288.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099557	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
3	#23778737	C₂	1	1	yes	maximal, characteristic
4	#23778691	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778690	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
9	#19099557	C₄	1	4	yes	complemented
10	C₂×#14579261	C₂²	1	1	yes	characteristic
11	C₂×#14579261	C₂²	1	1	yes	characteristic
12	C₈×D₃₁₇	C₄	1	4	yes	complemented
13	#19099553	C₄	1	2	yes	—
14	#19099551	C₄	1	2	yes	—
15	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₃₁₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
23	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
24	#14579261	C₂×C₄	1	2	yes	—
25	#14579261	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
28	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
29	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₃₁₇	—	2	2	no	—
32	C₂×D₃₁₇	C₄²	1	2	yes	—
33	C₄×C₃₁₇	—	2	2	no	—
34	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
36	Dic₃₁₇	—	2	2	no	—
37	Dic₃₁₇	C₄²	1	2	yes	—
38	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₃₁₇	#131	1	2	yes	—
40	D₃₁₇	—	2	2	no	—
41	C₃₁₇	#405	1	1	yes	characteristic, complemented, Sylow
42	#405	—	317	1	no	maximal, Sylow
43	C₂×M₁₆	—	317	1	no	—
44	C₂²×C₈	—	317	1	no	—
45	#117	—	317	1	no	—
46	#97	—	317	1	no	—
47	#97	—	317	1	no	—
48	C₄×C₈	—	317	1	no	—
49	C₄×C₈	D₃₁₇	1	1	yes	characteristic, complemented
50	C₂²×C₄	—	317	1	no	—
51	M₁₆	—	317	4	no	—
52	C₂×C₈	—	317	1	no	—
53	C₂×C₈	—	317	1	no	—
54	C₂×C₈	—	317	4	no	—
55	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
56	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
57	C₄⋊C₄	—	317	2	no	—
58	C₂²⋊C₄	—	317	2	no	—
59	C₄²	—	317	1	no	—
60	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
61	C₂³	—	317	1	no	—
62	C₂×C₄	—	317	1	no	—
63	C₂×C₄	—	317	2	no	—
64	C₂×C₄	—	317	2	no	—
65	C₂×C₄	—	317	2	no	—
66	C₂×C₄	C₄×D₃₁₇	1	2	yes	—
67	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
68	C₈	—	317	2	no	—
69	C₈	—	317	2	no	—
70	C₈	C₄×D₃₁₇	1	2	yes	—
71	C₈	C₄×D₃₁₇	1	2	yes	—
72	C₂²	—	317	2	no	—
73	C₂²	—	634	2	no	—
74	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	317	2	no	—
77	C₄	—	634	2	no	—
78	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
79	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
80	C₂	—	634	2	no	—
81	C₂	C₄²×D₃₁₇	1	1	yes	characteristic
82	C₂	#23778786	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₁₇
commutator	C₂×C₃₁₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₄×C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂²×C₄
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₄×C₈	D₃₁₇
hypocenter	C₃₁₇	#405
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #23778691	C₂	semidirect
5. #23778690	C₂	semidirect
6. C₈×Dic₃₁₇	C₂	semidirect
7. C₄×C₈×C₃₁₇	C₂	semidirect
9. #19099557	C₄	semidirect
12. C₈×D₃₁₇	C₄	semidirect
41. C₃₁₇	#405	semidirect
49. C₄×C₈	D₃₁₇	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
C₂×C₈×D₃₁₇	2
#23778737	2
#23778691	2
#23778690	2
C₈×Dic₃₁₇	2
C₄×C₈×C₃₁₇	2
#405	317