Subgroups — #35204303

Order 20288.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778868	C₂	1	8	yes	maximal, complemented
2	D₃₁₇⋊M₁₆	C₂	1	6	yes	maximal, complemented
3	C₂×#19099575	C₂	1	1	yes	maximal, characteristic
4	#19099584	C₂²	1	12	yes	complemented
5	#19099583	C₂²	1	4	yes	—
6	D₃₁₇⋊C₈	C₂²	1	12	yes	complemented
7	#19099575	C₄	1	4	yes	—
8	#19099575	C₂²	1	4	yes	—
9	C₂×D₁₂₆₈	C₄	1	3	yes	—
10	C₂×C₄×D₃₁₇	C₂²	1	3	yes	—
11	C₂×Q₈×C₃₁₇	C₄	1	1	yes	characteristic
12	#14579268	C₂³	1	8	yes	—
13	C₂²×D₃₁₇	C₂×C₄	1	3	yes	—
14	D₁₂₆₈	C₂×C₄	1	12	yes	—
15	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
16	C₄×D₃₁₇	C₂×C₄	1	6	yes	—
17	C₄×D₃₁₇	C₂³	1	6	yes	—
18	Q₈×C₃₁₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₃₁₇	C₂×C₄	1	3	yes	—
20	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₃₁₇	—	4	3	no	—
22	C₂×D₃₁₇	C₂²×C₄	1	6	yes	—
23	C₄×C₃₁₇	C₂²×C₄	1	6	yes	—
24	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
25	Dic₃₁₇	C₂⁴	1	1	yes	characteristic
26	C₂×C₃₁₇	—	2	1	no	—
27	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
28	D₃₁₇	—	2	6	no	—
29	C₃₁₇	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	317	1	no	maximal, Sylow
31	C₂×#41	—	317	1	no	—
32	#131	—	317	8	no	—
33	C₂×M₁₆	—	317	6	no	—
34	#41	—	317	4	no	—
35	#41	—	317	4	no	—
36	C₂×Q₈	#11929639	1	1	yes	characteristic
37	C₂×D₄	—	317	3	no	—
38	C₂²×C₄	—	317	3	no	—
39	M₁₆	—	317	4	no	—
40	M₁₆	—	317	12	no	—
41	C₂×C₈	—	317	12	no	—
42	C₂³	—	317	3	no	—
43	Q₈	C₂×#11929639	1	4	yes	—
44	D₄	—	317	12	no	—
45	C₂×C₄	—	317	1	no	—
46	C₂×C₄	—	317	6	no	—
47	C₂×C₄	—	317	6	no	—
48	C₂×C₄	C₂×#11929639	1	3	yes	—
49	C₈	—	317	8	no	—
50	C₂²	—	317	6	no	—
51	C₂²	—	1268	3	no	—
52	C₂²	C₂²×#11929639	1	1	yes	characteristic
53	C₄	—	317	1	no	—
54	C₄	—	317	1	no	—
55	C₄	C₂²×#11929639	1	6	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	634	6	no	—
58	C₂	C₂³×#11929639	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11929639
commutator, socle	C₂×C₃₁₇	C₂³×C₄
Fitting	C₂×Q₈×C₃₁₇	C₄
prime radical	C₂×#19099575	C₂
cosocle	Dic₃₁₇	C₂⁴
hypercenter	C₂×Q₈	#11929639
hypocenter	C₃₁₇	#568
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #23778868	C₂	semidirect
2. D₃₁₇⋊M₁₆	C₂	semidirect
4. #19099584	C₂²	semidirect
6. D₃₁₇⋊C₈	C₂²	semidirect
29. C₃₁₇	#568	semidirect

Maximal subgroups

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

subgroup	index
#23778868	2
D₃₁₇⋊M₁₆	2
C₂×#19099575	2
#568	317