Subgroups — #35204186

Order 20288.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₁₇⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#23778859	C₂	1	1	yes	maximal, characteristic
3	#23778834	C₂	1	2	yes	maximal, complemented
4	C₈×#11929639	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	#19099584	C₄	1	2	yes	complemented
7	#19099583	C₄	1	2	yes	complemented
8	D₃₁₇⋊C₈	C₂²	1	2	yes	—
9	C₄⋊#11929639	C₄	1	2	yes	—
10	C₂²⋊#11929639	C₄	1	2	yes	—
11	C₄×#11929639	C₂²	1	2	yes	—
12	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
13	C₂×#14579261	C₄	1	1	yes	characteristic, complemented
14	C₈×D₃₁₇	C₄	1	1	yes	characteristic, complemented
15	C₈×D₃₁₇	C₄	1	1	yes	characteristic, complemented
16	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
17	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
19	C₂×#11929639	C₂×C₄	1	4	yes	—
20	#14579268	C₂×C₄	1	4	yes	—
21	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
27	#14579261	C₂×C₄	1	1	yes	characteristic
28	#14579261	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
30	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
31	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃₁₇	C₄²	1	1	yes	characteristic
33	C₂×D₃₁₇	—	2	2	no	—
34	C₂×D₃₁₇	C₄²	1	1	yes	characteristic
35	C₂×D₃₁₇	C₂²×C₄	1	1	yes	characteristic
36	#11929639	—	2	4	no	—
37	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
38	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
39	Dic₃₁₇	C₄²	1	1	yes	characteristic
40	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃₁₇	—	2	1	no	—
42	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
43	D₃₁₇	—	2	1	no	—
44	D₃₁₇	#131	1	2	yes	—
45	C₃₁₇	#405	1	1	yes	characteristic, complemented, Sylow
46	#405	—	317	1	no	maximal, Sylow
47	C₂×M₁₆	—	317	1	no	—
48	C₂²×C₈	—	317	1	no	—
49	#117	—	317	1	no	—
50	#97	—	317	2	no	—
51	C₄×C₈	—	317	2	no	—
52	C₂²×C₄	—	317	1	no	—
53	M₁₆	—	317	2	no	—
54	M₁₆	—	317	2	no	—
55	C₂×C₈	—	317	2	no	—
56	C₂×C₈	—	317	1	no	—
57	C₂×C₈	—	317	1	no	—
58	C₂×C₈	—	317	1	no	—
59	C₂×C₈	—	317	1	no	—
60	C₂×C₈	—	317	1	no	—
61	C₂×C₈	#11929639	1	1	yes	characteristic, complemented
62	C₄⋊C₄	—	317	2	no	—
63	C₂²⋊C₄	—	317	2	no	—
64	C₄²	—	317	2	no	—
65	C₂³	—	317	1	no	—
66	C₂×C₄	—	317	1	no	—
67	C₂×C₄	—	317	1	no	—
68	C₂×C₄	—	317	1	no	—
69	C₂×C₄	—	317	1	no	—
70	C₂×C₄	—	317	1	no	—
71	C₂×C₄	—	317	4	no	—
72	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
73	C₈	—	317	4	no	—
74	C₈	—	317	1	no	—
75	C₈	—	317	1	no	—
76	C₈	C₂×#11929639	1	1	yes	characteristic
77	C₈	C₂×#11929639	1	1	yes	characteristic
78	C₂²	—	317	1	no	—
79	C₂²	—	317	1	no	—
80	C₂²	—	634	2	no	—
81	C₂²	C₄×#11929639	1	1	yes	characteristic
82	C₄	—	317	1	no	—
83	C₄	—	317	1	no	—
84	C₄	—	634	4	no	—
85	C₄	C₄×#11929639	1	1	yes	characteristic
86	C₄	C₂²×#11929639	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	317	2	no	—
89	C₂	—	634	1	no	—
90	C₂	C₂×C₄×#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#11929639
commutator, socle	C₂×C₃₁₇	C₂×C₄²
Frattini	C₄	C₂²×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₄×D₃₁₇	C₂³
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	#405
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. #23778834	C₂	semidirect
4. C₈×#11929639	C₂	semidirect
6. #19099584	C₄	semidirect
7. #19099583	C₄	semidirect
13. C₂×#14579261	C₄	semidirect
14. C₈×D₃₁₇	C₄	semidirect
15. C₈×D₃₁₇	C₄	semidirect
18. C₂×C₈×C₃₁₇	C₄	semidirect
45. C₃₁₇	#405	semidirect
61. C₂×C₈	#11929639	semidirect

Maximal subgroups

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

subgroup	index
D₃₁₇⋊M₁₆	2
#23778859	2
#23778834	2
C₈×#11929639	2
C₂×C₈×D₃₁₇	2
#405	317