Subgroups — #35204094

Order 20288.

110 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	C₄²×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#19099584	C₄	1	2	yes	complemented
5	#19099583	—	2	1	no	—
6	D₃₁₇⋊C₈	—	2	1	no	—
7	C₄⋊#11929639	C₄	1	2	yes	complemented
8	C₂²⋊#11929639	—	2	1	no	—
9	C₄×#11929639	—	2	1	no	—
10	C₂×C₄×D₃₁₇	—	2	1	no	—
11	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₁₇	—	2	1	no	—
13	C₄×Dic₃₁₇	C₄	1	1	yes	characteristic, complemented
14	C₄²×C₃₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#11929639	—	2	2	no	—
16	#14579268	—	2	2	no	—
17	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
18	C₂×Dic₃₁₇	—	2	1	no	—
19	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
20	C₄×D₃₁₇	—	2	4	no	—
21	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₁₇	D₄	1	1	yes	characteristic
24	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₃₁₇	—	2	1	no	—
26	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₃₁₇	—	2	2	no	—
29	C₂×D₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
31	#11929639	—	4	2	no	—
32	C₄×C₃₁₇	—	2	2	no	—
33	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
34	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
35	Dic₃₁₇	—	2	2	no	—
36	Dic₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₁₇	—	2	1	no	—
39	C₂×C₃₁₇	#95	1	1	yes	characteristic
40	D₃₁₇	—	2	1	no	—
41	D₃₁₇	#104	1	2	yes	—
42	C₃₁₇	#339	1	1	yes	characteristic, complemented, Sylow
43	#339	—	317	1	no	maximal, Sylow
44	C₂×M₁₆	—	317	1	no	—
45	#117	—	317	1	no	—
46	C₂×C₄²	—	317	1	no	—
47	C₂²×C₄	—	317	1	no	—
48	C₂²×C₄	—	634	1	no	—
49	M₁₆	—	317	2	no	—
50	M₁₆	—	634	1	no	—
51	C₂×C₈	—	634	1	no	—
52	C₄⋊C₄	—	317	2	no	—
53	C₂²⋊C₄	—	634	1	no	—
54	C₄²	—	317	1	no	—
55	C₄²	—	634	1	no	—
56	C₄²	—	634	1	no	—
57	C₄²	#11929639	1	1	yes	characteristic, complemented
58	C₂³	—	317	1	no	—
59	C₂×C₄	—	2	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	1	no	—
65	C₂×C₄	—	634	4	no	—
66	C₂×C₄	—	634	1	no	—
67	C₂×C₄	—	634	2	no	—
68	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
69	C₈	—	634	2	no	—
70	C₂²	—	317	1	no	—
71	C₂²	—	317	1	no	—
72	C₂²	—	634	2	no	—
73	C₂²	C₂²⋊#11929639	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	317	1	no	—
76	C₄	—	317	1	no	—
77	C₄	—	634	2	no	—
78	C₄	—	1268	2	no	—
79	C₄	C₄×#11929639	1	1	yes	characteristic
80	C₄	C₄⋊#11929639	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	317	2	no	—
83	C₂	—	634	1	no	—
84	C₂	(C₂×#11929639)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#11929639
commutator	C₄×C₃₁₇	C₄²
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₄²×C₃₁₇	C₄
socle	C₂×C₃₁₇	#95
prime radical	C₂²×D₃₁₇	D₄
cosocle	C₂×C₄×D₃₁₇	C₂²
hypercenter	C₄²	#11929639
hypocenter	C₃₁₇	#339
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. #19099584	C₄	semidirect
7. C₄⋊#11929639	C₄	semidirect
13. C₄×Dic₃₁₇	C₄	semidirect
14. C₄²×C₃₁₇	C₄	semidirect
42. C₃₁₇	#339	semidirect
57. C₄²	#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
C₄²×D₃₁₇	2
#339	317