Subgroups — #35204189

Order 20288.

130 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	#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	M₁₆×D₃₁₇	C₂	1	1	yes	maximal, characteristic
6	D₃₁₇⋊C₈	C₄	1	2	yes	complemented
7	D₃₁₇⋊C₈	C₂²	1	1	yes	characteristic
8	D₃₁₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14579268	C₄	1	2	yes	complemented
10	C₄⋊#11929639	C₄	1	2	yes	—
11	C₂²⋊#11929639	C₄	1	2	yes	—
12	C₄×#11929639	C₂²	1	2	yes	—
13	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
14	#19099558	C₄	1	1	yes	characteristic, complemented
15	#19099557	C₄	1	2	yes	complemented
16	C₈×D₃₁₇	C₂²	1	2	yes	—
17	M₁₆×C₃₁₇	C₄	1	1	yes	characteristic, complemented
18	C₂×#11929639	C₂×C₄	1	4	yes	—
19	#14579268	C₂×C₄	1	2	yes	—
20	#14579268	C₂×C₄	1	2	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	2	yes	—
28	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₃₁₇	C₄²	1	1	yes	characteristic
31	C₂×D₃₁₇	—	2	2	no	—
32	C₂×D₃₁₇	C₄²	1	1	yes	characteristic
33	C₂×D₃₁₇	C₂²×C₄	1	1	yes	characteristic
34	#11929639	—	2	4	no	—
35	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
36	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
37	Dic₃₁₇	C₄²	1	1	yes	characteristic
38	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃₁₇	—	2	1	no	—
40	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
41	D₃₁₇	—	2	1	no	—
42	D₃₁₇	#131	1	2	yes	—
43	C₃₁₇	#405	1	1	yes	characteristic, complemented, Sylow
44	#405	—	317	1	no	maximal, Sylow
45	C₂×M₁₆	—	317	1	no	—
46	C₂²×C₈	—	317	1	no	—
47	#117	—	317	1	no	—
48	#97	—	317	2	no	—
49	C₄×C₈	—	317	2	no	—
50	C₂²×C₄	—	317	1	no	—
51	M₁₆	—	317	1	no	—
52	M₁₆	—	317	2	no	—
53	M₁₆	#11929639	1	1	yes	characteristic, complemented
54	C₂×C₈	—	317	1	no	—
55	C₂×C₈	—	317	1	no	—
56	C₂×C₈	—	317	2	no	—
57	C₂×C₈	—	317	2	no	—
58	C₂×C₈	—	317	2	no	—
59	C₄⋊C₄	—	317	2	no	—
60	C₂²⋊C₄	—	317	2	no	—
61	C₄²	—	317	2	no	—
62	C₂³	—	317	1	no	—
63	C₂×C₄	—	317	1	no	—
64	C₂×C₄	—	317	1	no	—
65	C₂×C₄	—	317	1	no	—
66	C₂×C₄	—	317	1	no	—
67	C₂×C₄	—	317	1	no	—
68	C₂×C₄	—	317	4	no	—
69	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
70	C₈	—	317	2	no	—
71	C₈	—	317	2	no	—
72	C₈	—	317	2	no	—
73	C₈	C₂×#11929639	1	2	yes	—
74	C₂²	—	317	1	no	—
75	C₂²	—	317	1	no	—
76	C₂²	—	634	2	no	—
77	C₂²	C₄×#11929639	1	1	yes	characteristic
78	C₄	—	317	1	no	—
79	C₄	—	317	1	no	—
80	C₄	—	634	4	no	—
81	C₄	C₄×#11929639	1	1	yes	characteristic
82	C₄	C₂²×#11929639	1	1	yes	characteristic
83	C₂	—	2	1	no	—
84	C₂	—	317	2	no	—
85	C₂	—	634	1	no	—
86	C₂	C₂×C₄×#11929639	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#11929639
commutator, socle	C₂×C₃₁₇	C₂×C₄²
Fitting	M₁₆×C₃₁₇	C₄
prime radical	C₂²×D₃₁₇	C₂×C₄
cosocle	C₄×D₃₁₇	C₂³
hypercenter	M₁₆	#11929639
hypocenter	C₃₁₇	#405
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ M₁₆

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. D₃₁₇⋊C₈	C₄	semidirect
9. C₂×#14579268	C₄	semidirect
14. #19099558	C₄	semidirect
15. #19099557	C₄	semidirect
17. M₁₆×C₃₁₇	C₄	semidirect
43. C₃₁₇	#405	semidirect
53. M₁₆	#11929639	semidirect

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
#23778859	2
#23778834	2
C₈×#11929639	2
M₁₆×D₃₁₇	2
#405	317