Subgroups — #35203133

Order 20288.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₆₈	C₂	1	3	yes	maximal, complemented
2	C₂×#19099551	C₂	1	3	yes	maximal, complemented
3	#95×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₁₇	C₂²	1	3	yes	—
5	C₂×D₁₂₆₈	—	2	12	no	—
6	C₂×D₁₂₆₈	—	2	6	no	—
7	C₂²×Dic₃₁₇	C₂²	1	1	yes	characteristic
8	#19099551	—	2	6	no	—
9	C₂²×C₄×C₃₁₇	C₂²	1	3	yes	—
10	C₂²×D₃₁₇	—	2	3	no	—
11	C₂²×D₃₁₇	—	2	6	no	—
12	C₂²×D₃₁₇	—	2	6	no	—
13	C₂²×D₃₁₇	—	2	3	no	—
14	C₂²×D₃₁₇	D₄	1	6	yes	complemented
15	D₁₂₆₈	—	2	24	no	—
16	C₂×Dic₃₁₇	—	2	3	no	—
17	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₃₁₇	—	2	6	no	—
19	C₂×C₄×C₃₁₇	D₄	1	6	yes	complemented
20	C₂²×C₃₁₇	C₂×D₄	1	3	yes	—
21	C₂²×C₃₁₇	C₂×D₄	1	3	yes	—
22	C₂²×C₃₁₇	#41	1	1	yes	characteristic
23	C₂×D₃₁₇	—	2	12	no	—
24	C₂×D₃₁₇	—	2	6	no	—
25	C₂×D₃₁₇	—	4	3	no	—
26	C₂×D₃₁₇	—	4	6	no	—
27	C₂×D₃₁₇	—	4	3	no	—
28	C₄×C₃₁₇	—	2	6	no	—
29	Dic₃₁₇	—	4	1	no	—
30	C₂×C₃₁₇	#120	1	3	yes	—
31	C₂×C₃₁₇	#121	1	3	yes	—
32	C₂×C₃₁₇	#127	1	1	yes	characteristic
33	D₃₁₇	—	4	6	no	—
34	C₃₁₇	#392	1	1	yes	characteristic, complemented, Sylow
35	#392	—	317	1	no	maximal, Sylow
36	C₂²×D₄	—	317	3	no	—
37	C₂×C₂²⋊C₄	—	317	3	no	—
38	#95	D₃₁₇	1	1	yes	characteristic, complemented
39	C₂⁴	—	317	3	no	—
40	C₂×D₄	—	634	12	no	—
41	C₂×D₄	—	634	6	no	—
42	C₂²×C₄	—	317	1	no	—
43	C₂²×C₄	C₂×D₃₁₇	1	3	yes	—
44	C₂²⋊C₄	—	634	6	no	—
45	C₂³	—	317	6	no	—
46	C₂³	—	634	6	no	—
47	C₂³	—	634	3	no	—
48	C₂³	—	634	6	no	—
49	C₂³	—	634	3	no	—
50	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
51	D₄	—	634	24	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	634	3	no	—
54	C₂×C₄	D₁₂₆₈	1	6	yes	complemented
55	C₂²	—	634	12	no	—
56	C₂²	—	634	6	no	—
57	C₂²	—	1268	3	no	—
58	C₂²	—	1268	6	no	—
59	C₂²	—	1268	3	no	—
60	C₂²	D₄×D₃₁₇	1	3	yes	—
61	C₂²	C₂×D₁₂₆₈	1	3	yes	—
62	C₂²	#19099575	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	1268	1	no	—
65	C₂	—	1268	6	no	—
66	C₂	#23778750	1	3	yes	—
67	C₂	#23778755	1	3	yes	—
68	C₂	#23778777	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₁₇
commutator, socle, cosocle	C₂³×C₃₁₇	C₂³
Fitting	#95×C₃₁₇	C₂
hypercenter	#95	D₃₁₇
hypocenter	C₃₁₇	#392
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. C₂²×D₁₂₆₈	C₂	semidirect
2. C₂×#19099551	C₂	semidirect
3. #95×C₃₁₇	C₂	semidirect
14. C₂²×D₃₁₇	D₄	semidirect
19. C₂×C₄×C₃₁₇	D₄	semidirect
34. C₃₁₇	#392	semidirect
38. #95	D₃₁₇	semidirect
54. C₂×C₄	D₁₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₆₈	2
C₂×#19099551	2
#95×C₃₁₇	2
#392	317