Subgroups — C₂×#23778834 #35204180

Order 20288.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₁₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11929639	C₂	1	1	yes	maximal, characteristic
3	#23778834	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929639	C₄	1	2	yes	—
6	D₃₁₇⋊C₈	C₄	1	2	yes	complemented
7	D₃₁₇⋊C₈	C₂²	1	2	yes	—
8	C₂×#14579268	C₄	1	2	yes	complemented
9	C₄×#11929639	C₄	1	2	yes	—
10	C₄×#11929639	C₂²	1	2	yes	—
11	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
12	C₂×#14579261	C₄	1	1	yes	characteristic, complemented
13	C₈×D₃₁₇	C₄	1	2	yes	complemented
14	C₈×D₃₁₇	C₂²	1	2	yes	—
15	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11929639	—	2	4	no	—
17	C₂×#11929639	C₂×C₄	1	4	yes	—
18	#14579268	C₂×C₄	1	4	yes	complemented
19	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
25	#14579261	C₂×C₄	1	2	yes	complemented
26	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
27	C₈×C₃₁₇	C₂×C₄	1	2	yes	complemented
28	C₂²×C₃₁₇	C₄²	1	1	yes	characteristic
29	C₂×D₃₁₇	C₄²	1	1	yes	characteristic
30	C₂×D₃₁₇	M₁₆	1	4	yes	—
31	C₂×D₃₁₇	C₂²×C₄	1	1	yes	characteristic
32	#11929639	—	2	4	no	—
33	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
34	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
35	Dic₃₁₇	C₄²	1	1	yes	characteristic
36	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₃₁₇	#97	1	2	yes	complemented
38	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
39	D₃₁₇	#97	1	2	yes	complemented
40	D₃₁₇	C₂×M₁₆	1	2	yes	—
41	C₃₁₇	C₂×#97	1	1	yes	characteristic, complemented, Sylow
42	C₂×#97	—	317	1	no	maximal, Sylow
43	C₂²×C₈	—	317	1	no	—
44	C₂²×C₈	—	317	1	no	—
45	C₂×C₄²	—	317	1	no	—
46	#97	—	317	4	no	—
47	C₂²×C₄	—	317	1	no	—
48	C₂²×C₄	—	317	2	no	—
49	C₂×C₈	—	317	2	no	—
50	C₂×C₈	—	317	2	no	—
51	C₂×C₈	—	317	2	no	—
52	C₂×C₈	—	317	1	no	—
53	C₂×C₈	—	317	2	no	—
54	C₂×C₈	—	317	2	no	—
55	C₂×C₈	#11929639	1	1	yes	characteristic, complemented
56	C₄²	—	317	2	no	—
57	C₄²	—	317	2	no	—
58	C₂³	—	317	1	no	—
59	C₂×C₄	—	317	1	no	—
60	C₂×C₄	—	317	1	no	—
61	C₂×C₄	—	317	1	no	—
62	C₂×C₄	—	317	1	no	—
63	C₂×C₄	—	317	1	no	—
64	C₂×C₄	—	317	4	no	—
65	C₂×C₄	—	634	4	no	—
66	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
67	C₈	—	317	4	no	—
68	C₈	—	317	2	no	—
69	C₈	C₂×#11929639	1	2	yes	complemented
70	C₂²	—	317	4	no	—
71	C₂²	—	317	1	no	—
72	C₂²	—	317	1	no	—
73	C₂²	C₄×#11929639	1	1	yes	characteristic
74	C₄	—	317	1	no	—
75	C₄	—	317	1	no	—
76	C₄	—	634	4	no	—
77	C₄	C₄×#11929639	1	1	yes	characteristic
78	C₄	C₂²×#11929639	1	1	yes	characteristic
79	C₂	—	317	2	no	—
80	C₂	—	317	2	no	—
81	C₂	#23778834	1	2	yes	direct product factor
82	C₂	C₂×C₄×#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11929639
commutator	C₂×C₃₁₇	C₂×C₄²
Frattini	C₄	C₂²×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₄²
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₄×D₃₁₇	C₂³
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	C₂×#97
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. #23778834	C₂	direct (N × H)
6. D₃₁₇⋊C₈	C₄	semidirect
8. C₂×#14579268	C₄	semidirect
12. C₂×#14579261	C₄	semidirect
13. C₈×D₃₁₇	C₄	semidirect
15. C₂×C₈×C₃₁₇	C₄	semidirect
18. #14579268	C₂×C₄	semidirect
25. #14579261	C₂×C₄	semidirect
27. C₈×C₃₁₇	C₂×C₄	semidirect
37. C₂×C₃₁₇	#97	semidirect
39. D₃₁₇	#97	semidirect
41. C₃₁₇	C₂×#97	semidirect
55. C₂×C₈	#11929639	semidirect
69. C₈	C₂×#11929639	semidirect
81. C₂	#23778834	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×D₃₁₇⋊C₈	2
C₂×C₄×#11929639	2
#23778834	2
C₂×C₈×D₃₁₇	2
C₂×#97	317

Subgroups — C2×#23778834 #35204180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778834 #35204180