Subgroups — #35204159

Order 20288.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11929639	C₂	1	2	yes	maximal
2	(C₂×#11929639)⋊C₄	C₂	1	4	yes	maximal
3	C₄²×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929639	C₂²	1	4	yes	—
5	C₄×#11929639	C₄	1	8	yes	complemented
6	C₂×C₄×D₃₁₇	C₂²	1	2	yes	—
7	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
8	C₄×Dic₃₁₇	C₄	1	2	yes	complemented
9	C₄×Dic₃₁₇	C₄	1	1	yes	characteristic, complemented
10	C₄²×C₃₁₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#11929639	—	2	8	no	—
12	C₂×#11929639	C₂×C₄	1	8	yes	—
13	C₂²×D₃₁₇	C₂³	1	1	yes	characteristic
14	C₂×Dic₃₁₇	C₂×C₄	1	2	yes	—
15	C₂×Dic₃₁₇	C₂×C₄	1	1	yes	characteristic
16	C₄×D₃₁₇	—	2	4	no	—
17	C₄×D₃₁₇	C₂×C₄	1	4	yes	—
18	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×D₃₁₇	C₂²×C₄	1	2	yes	—
22	C₂×D₃₁₇	#41	1	4	yes	—
23	#11929639	—	2	8	no	—
24	C₄×C₃₁₇	—	2	2	no	—
25	C₄×C₃₁₇	C₄²	1	2	yes	—
26	Dic₃₁₇	—	2	2	no	—
27	Dic₃₁₇	C₄²	1	2	yes	—
28	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₃₁₇	#117	1	2	yes	—
30	D₃₁₇	#117	1	4	yes	—
31	C₃₁₇	#376	1	1	yes	characteristic, complemented, Sylow
32	#376	—	317	1	no	maximal, Sylow
33	C₂×C₄²	—	317	2	no	—
34	C₂×C₄²	—	317	1	no	—
35	#95	—	317	4	no	—
36	C₂²×C₄	—	317	1	no	—
37	C₂²×C₄	—	317	4	no	—
38	C₂²×C₄	—	317	2	no	—
39	C₄²	—	317	8	no	—
40	C₄²	—	317	2	no	—
41	C₄²	—	317	1	no	—
42	C₄²	#11929639	1	1	yes	characteristic, complemented
43	C₂³	—	317	1	no	—
44	C₂×C₄	—	317	1	no	—
45	C₂×C₄	—	317	4	no	—
46	C₂×C₄	—	317	8	no	—
47	C₂×C₄	—	317	2	no	—
48	C₂×C₄	—	634	4	no	—
49	C₂×C₄	—	634	8	no	—
50	C₂×C₄	C₂×#11929639	1	2	yes	—
51	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
52	C₂²	—	317	2	no	—
53	C₂²	—	317	4	no	—
54	C₂²	C₂²×#11929639	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	317	2	no	—
57	C₄	—	634	8	no	—
58	C₄	—	634	2	no	—
59	C₄	C₄×#11929639	1	2	yes	—
60	C₂	—	317	4	no	—
61	C₂	C₂×C₄×#11929639	1	1	yes	characteristic
62	C₂	#23778859	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11929639
commutator	C₂×C₃₁₇	C₂×C₄²
Frattini	C₂²	C₂²×#11929639
Fitting	C₄²×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²×C₄
prime radical, cosocle	C₂²×D₃₁₇	C₂³
hypercenter	C₄²	#11929639
hypocenter	C₃₁₇	#376
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
5. C₄×#11929639	C₄	semidirect
8. C₄×Dic₃₁₇	C₄	semidirect
9. C₄×Dic₃₁₇	C₄	semidirect
10. C₄²×C₃₁₇	C₄	semidirect
31. C₃₁₇	#376	semidirect
42. C₄²	#11929639	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11929639	2
(C₂×#11929639)⋊C₄	2
C₄²×D₃₁₇	2
#376	317