Subgroups — #35204105

Order 20288.

90 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	C₂×#19099557	C₂	1	1	yes	maximal, characteristic
4	#19099584	—	2	1	no	—
5	#19099583	C₄	1	2	yes	complemented
6	D₃₁₇⋊C₈	—	2	1	no	—
7	C₄⋊#11929639	—	2	1	no	—
8	C₂²⋊#11929639	C₄	1	2	yes	—
9	C₄×#11929639	—	2	1	no	—
10	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
11	#19099557	—	2	1	no	—
12	#19099557	—	2	1	no	—
13	C₂×#14579261	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#11929639	—	2	2	no	—
16	#14579268	—	2	2	no	—
17	C₂²×D₃₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₁₇	D₄	1	1	yes	characteristic
20	C₄×D₃₁₇	D₄	1	1	yes	characteristic
21	C₄×D₃₁₇	D₄	1	1	yes	characteristic
22	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
23	#14579261	—	2	1	no	—
24	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₁₇	—	2	1	no	—
26	C₂²×C₃₁₇	C₄²	1	1	yes	characteristic
27	C₂×D₃₁₇	—	4	1	no	—
28	C₂×D₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
30	#11929639	—	4	2	no	—
31	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
33	Dic₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₃₁₇	—	2	1	no	—
36	C₂×C₃₁₇	#95	1	1	yes	characteristic
37	D₃₁₇	—	2	1	no	—
38	D₃₁₇	—	2	1	no	—
39	C₃₁₇	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	317	1	no	maximal, Sylow
41	C₂×M₁₆	—	317	1	no	—
42	C₂×M₁₆	—	317	1	no	—
43	#117	—	317	1	no	—
44	C₂²×C₄	—	317	1	no	—
45	M₁₆	—	317	2	no	—
46	M₁₆	—	634	1	no	—
47	M₁₆	—	634	1	no	—
48	M₁₆	—	634	1	no	—
49	C₂×C₈	—	317	1	no	—
50	C₂×C₈	—	634	1	no	—
51	C₂×C₈	#11929639	1	1	yes	characteristic, complemented
52	C₄⋊C₄	—	634	1	no	—
53	C₂²⋊C₄	—	317	2	no	—
54	C₄²	—	634	1	no	—
55	C₂³	—	317	1	no	—
56	C₂×C₄	—	317	1	no	—
57	C₂×C₄	—	317	1	no	—
58	C₂×C₄	—	317	1	no	—
59	C₂×C₄	—	317	1	no	—
60	C₂×C₄	—	317	1	no	—
61	C₂×C₄	—	634	2	no	—
62	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	634	2	no	—
65	C₈	—	634	1	no	—
66	C₂²	—	317	1	no	—
67	C₂²	—	317	1	no	—
68	C₂²	—	1268	1	no	—
69	C₂²	C₄×#11929639	1	1	yes	characteristic
70	C₄	—	317	1	no	—
71	C₄	—	317	1	no	—
72	C₄	—	1268	2	no	—
73	C₄	C₂²⋊#11929639	1	1	yes	characteristic
74	C₄	C₄⋊#11929639	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	634	1	no	—
77	C₂	—	634	1	no	—
78	C₂	(C₂×#11929639)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#11929639
commutator	C₂²×C₃₁₇	C₄²
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	#95
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₂×C₄×D₃₁₇	C₂²
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	#344
LCS	G ▷ C₂²×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. #19099583	C₄	semidirect
13. C₂×#14579261	C₄	semidirect
14. C₂×C₈×C₃₁₇	C₄	semidirect
39. C₃₁₇	#344	semidirect
51. 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
C₂×#19099557	2
#344	317