Subgroups — #35204269

Order 20288.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778868	C₂	1	2	yes	maximal, complemented
2	#23778850	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778845	C₂	1	2	yes	maximal, complemented
4	#23778834	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778803	C₂	1	1	yes	maximal, characteristic
6	#19099584	—	2	2	no	—
7	#19099584	C₂²	1	2	yes	—
8	D₃₁₇⋊C₈	—	2	2	no	—
9	D₃₁₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×#11929639	C₂²	1	1	yes	characteristic
11	#19099575	C₂²	1	2	yes	—
12	#19099561	C₄	1	2	yes	complemented
13	D₂₅₃₆	C₄	1	1	yes	characteristic, complemented
14	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
15	Q₁₆×C₃₁₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#11929639	—	2	1	no	—
17	#14579268	—	2	2	no	—
18	#14579268	D₄	1	2	yes	—
19	D₁₂₆₈	—	2	2	no	—
20	D₁₂₆₈	C₂×C₄	1	2	yes	—
21	C₄×D₃₁₇	—	2	2	no	—
22	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
23	#14579261	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₃₁₇	C₂×C₄	1	2	yes	—
25	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×D₃₁₇	—	2	2	no	—
27	C₂×D₃₁₇	#41	1	1	yes	characteristic
28	#11929639	—	4	1	no	—
29	C₄×C₃₁₇	—	2	2	no	—
30	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
31	Dic₃₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃₁₇	C₄×D₄	1	1	yes	characteristic
33	D₃₁₇	—	2	1	no	—
34	D₃₁₇	—	4	2	no	—
35	C₃₁₇	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	317	1	no	maximal, Sylow
37	#135	—	317	1	no	—
38	#131	—	317	2	no	—
39	#108	—	317	1	no	—
40	#104	—	317	2	no	—
41	#97	—	317	1	no	—
42	#41	—	317	2	no	—
43	Q₁₆	#11929639	1	1	yes	characteristic, complemented
44	SD₁₆	—	317	2	no	—
45	D₈	—	317	1	no	—
46	M₁₆	—	317	2	no	—
47	M₁₆	—	634	2	no	—
48	C₂×C₈	—	317	1	no	—
49	C₂×C₈	—	317	1	no	—
50	C₂×C₈	—	634	2	no	—
51	C₄²	—	317	1	no	—
52	Q₈	C₂×#11929639	1	2	yes	—
53	D₄	—	317	2	no	—
54	D₄	—	634	2	no	—
55	C₂×C₄	—	317	1	no	—
56	C₂×C₄	—	634	2	no	—
57	C₂×C₄	—	634	1	no	—
58	C₈	—	317	2	no	—
59	C₈	—	317	1	no	—
60	C₈	—	634	2	no	—
61	C₈	C₂×#11929639	1	1	yes	characteristic
62	C₂²	—	317	1	no	—
63	C₂²	—	634	2	no	—
64	C₄	—	2	2	no	—
65	C₄	—	317	1	no	—
66	C₄	—	1268	1	no	—
67	C₄	C₂²×#11929639	1	1	yes	characteristic
68	C₂	—	634	1	no	—
69	C₂	—	1268	2	no	—
70	C₂	D₄×#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11929639
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×#11929639
Fitting	Q₁₆×C₃₁₇	C₄
socle	C₂×C₃₁₇	C₄×D₄
prime radical	#23778803	C₂
cosocle	C₄×D₃₁₇	C₂³
hypercenter	Q₁₆	#11929639
hypocenter	C₃₁₇	#444
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #23778868	C₂	semidirect
2. #23778850	C₂	semidirect
3. #23778845	C₂	semidirect
4. #23778834	C₂	semidirect
12. #19099561	C₄	semidirect
13. D₂₅₃₆	C₄	semidirect
15. Q₁₆×C₃₁₇	C₄	semidirect
35. C₃₁₇	#444	semidirect
43. Q₁₆	#11929639	semidirect

Maximal subgroups

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

subgroup	index
#23778868	2
#23778850	2
#23778845	2
#23778834	2
#23778803	2
#444	317