Subgroups — #35203088

Order 20288.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
3	#23778687	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₇	C₂²	1	2	yes	—
6	C₂²×Dic₃₁₇	C₂²	1	2	yes	—
7	C₄⋊Dic₃₁₇	—	2	2	no	—
8	C₄×Dic₃₁₇	C₄	1	4	yes	complemented
9	C₂²×C₄×C₃₁₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₃₁₇	—	2	2	no	—
12	C₂×Dic₃₁₇	—	2	2	no	—
13	C₂×Dic₃₁₇	—	2	2	no	—
14	C₂×Dic₃₁₇	—	2	2	no	—
15	C₂×Dic₃₁₇	—	2	4	no	—
16	C₂×Dic₃₁₇	C₂×C₄	1	4	yes	—
17	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₃₁₇	—	2	2	no	—
19	C₂×C₄×C₃₁₇	—	2	4	no	—
20	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
22	C₂×C₄×C₃₁₇	Q₈	1	2	yes	—
23	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₃₁₇	#41	1	2	yes	—
27	C₂²×C₃₁₇	#41	1	2	yes	—
28	C₄×C₃₁₇	—	4	2	no	—
29	C₄×C₃₁₇	C₄⋊C₄	1	4	yes	—
30	Dic₃₁₇	—	2	4	no	—
31	Dic₃₁₇	—	4	2	no	—
32	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃₁₇	#117	1	2	yes	—
34	C₂×C₃₁₇	#124	1	1	yes	characteristic
35	C₂×C₃₁₇	#124	1	1	yes	characteristic
36	C₂×C₃₁₇	#125	1	2	yes	—
37	C₃₁₇	#382	1	1	yes	characteristic, complemented, Sylow
38	#382	—	317	1	no	maximal, Sylow
39	C₂×C₄⋊C₄	—	317	1	no	—
40	C₂×C₄⋊C₄	D₃₁₇	1	1	yes	characteristic
41	C₂×C₄²	—	317	1	no	—
42	#95	—	317	4	no	—
43	C₂²×C₄	—	317	2	no	—
44	C₂²×C₄	—	317	2	no	—
45	C₂²×C₄	C₂×D₃₁₇	1	2	yes	—
46	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	—	634	2	no	—
49	C₄²	—	317	4	no	—
50	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	—	317	4	no	—
54	C₂×C₄	—	634	2	no	—
55	C₂×C₄	—	634	2	no	—
56	C₂×C₄	—	634	2	no	—
57	C₂×C₄	—	634	4	no	—
58	C₂×C₄	C₄×D₃₁₇	1	2	yes	—
59	C₂×C₄	C₂²⋊D₃₁₇	1	2	yes	—
60	C₂×C₄	Dic₆₃₄	1	2	yes	—
61	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
62	C₂²	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
63	C₂²	C₂×Dic₆₃₄	1	1	yes	characteristic
64	C₂²	#19099573	1	2	yes	—
65	C₂²	#19099575	1	2	yes	—
66	C₄	—	4	2	no	—
67	C₄	—	634	4	no	—
68	C₄	—	1268	2	no	—
69	C₄	#19099553	1	4	yes	—
70	C₂	C₂×#19099553	1	1	yes	characteristic
71	C₂	#23778738	1	2	yes	—
72	C₂	#23778767	1	1	yes	characteristic
73	C₂	#23778769	1	1	yes	characteristic
74	C₂	#23778771	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₁₇
commutator	C₂²×C₃₁₇	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₃₁₇	C₂
socle, prime radical, cosocle	C₂³×C₃₁₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₃₁₇
hypocenter	C₃₁₇	#382
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. C₄×Dic₃₁₇	C₄	semidirect
37. C₃₁₇	#382	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×C₄×Dic₃₁₇	2
#23778687	2
C₂×C₄⋊C₄×C₃₁₇	2
#382	317