Subgroups — #35203160

Order 20288.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₃₁₇	C₂	1	3	yes	maximal, complemented
2	C₂×#19099551	C₂	1	3	yes	maximal
3	#95×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₁₇	C₂²	1	1	yes	characteristic
5	C₂²×Dic₃₁₇	C₂²	1	3	yes	—
6	C₄⋊Dic₃₁₇	—	2	6	no	—
7	#19099551	—	2	6	no	—
8	C₂²×C₄×C₃₁₇	C₂²	1	3	yes	—
9	C₂²×D₃₁₇	—	2	3	no	—
10	C₂²×D₃₁₇	—	2	3	no	—
11	C₂²×D₃₁₇	Q₈	1	2	yes	—
12	C₂×Dic₃₁₇	—	2	3	no	—
13	C₂×Dic₃₁₇	—	2	6	no	—
14	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₃₁₇	—	2	6	no	—
16	C₂×C₄×C₃₁₇	D₄	1	6	yes	—
17	C₂²×C₃₁₇	C₂×D₄	1	3	yes	—
18	C₂²×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
19	C₂²×C₃₁₇	#41	1	3	yes	—
20	C₂×D₃₁₇	—	2	6	no	—
21	C₂×D₃₁₇	—	4	3	no	—
22	C₂×D₃₁₇	—	4	1	no	—
23	C₄×C₃₁₇	—	2	6	no	—
24	Dic₃₁₇	—	4	3	no	—
25	C₂×C₃₁₇	C₂²⋊Q₈	1	3	yes	—
26	C₂×C₃₁₇	#123	1	3	yes	—
27	C₂×C₃₁₇	#127	1	1	yes	characteristic
28	D₃₁₇	—	4	2	no	—
29	C₃₁₇	#399	1	1	yes	characteristic, complemented, Sylow
30	#399	—	317	1	no	maximal, Sylow
31	C₂×C₄⋊C₄	—	317	3	no	—
32	C₂×C₂²⋊C₄	—	317	3	no	—
33	#95	D₃₁₇	1	1	yes	characteristic, complemented
34	C₂⁴	—	317	1	no	—
35	C₂²×C₄	—	317	3	no	—
36	C₂²×C₄	C₂×D₃₁₇	1	3	yes	—
37	C₄⋊C₄	—	634	6	no	—
38	C₂²⋊C₄	—	634	6	no	—
39	C₂³	—	317	2	no	—
40	C₂³	—	634	3	no	—
41	C₂³	—	634	3	no	—
42	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
43	C₂×C₄	—	2	6	no	—
44	C₂×C₄	—	634	3	no	—
45	C₂×C₄	—	634	6	no	—
46	C₂×C₄	D₁₂₆₈	1	6	yes	—
47	C₂²	—	634	6	no	—
48	C₂²	—	1268	3	no	—
49	C₂²	—	1268	1	no	—
50	C₂²	C₂×D₁₂₆₈	1	3	yes	—
51	C₂²	Q₈×D₃₁₇	1	1	yes	characteristic
52	C₂²	#19099573	1	3	yes	—
53	C₄	—	2	6	no	—
54	C₄	—	1268	3	no	—
55	C₂	—	1268	2	no	—
56	C₂	#23778760	1	3	yes	—
57	C₂	#23778764	1	3	yes	—
58	C₂	#23778777	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₁₇
commutator, socle, cosocle	C₂³×C₃₁₇	C₂³
Fitting	#95×C₃₁₇	C₂
prime radical	C₂³×D₃₁₇	C₂²
hypercenter	#95	D₃₁₇
hypocenter	C₃₁₇	#399
LCS	G ▷ C₂³×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₃₁₇	C₂	semidirect
3. #95×C₃₁₇	C₂	semidirect
29. C₃₁₇	#399	semidirect
33. #95	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₃₁₇	2
C₂×#19099551	2
#95×C₃₁₇	2
#399	317