Subgroups — #35203666

Order 20288.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778802	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₅₃₆	C₂	1	2	yes	maximal, complemented
4	#23778725	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19099574	—	2	1	no	—
7	C₂×Dic₆₃₄	—	2	1	no	—
8	Dic₁₂₆₈	—	2	1	no	—
9	Dic₁₂₆₈	C₂²	1	2	yes	—
10	Dic₁₂₆₈	C₂²	1	1	yes	characteristic
11	#19099563	—	2	1	no	—
12	D₂₅₃₆	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
14	C₁₆×C₃₁₇	C₂²	1	2	yes	—
15	Dic₆₃₄	—	2	1	no	—
16	Dic₆₃₄	—	2	2	no	—
17	Dic₆₃₄	—	4	1	no	—
18	D₁₂₆₈	—	2	1	no	—
19	C₂²⋊D₃₁₇	—	4	1	no	—
20	C₂×Dic₃₁₇	—	4	1	no	—
21	C₄×D₃₁₇	—	4	1	no	—
22	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
23	C₈×C₃₁₇	D₄	1	1	yes	characteristic
24	C₈×C₃₁₇	C₂³	1	1	yes	characteristic
25	C₂²×C₃₁₇	D₈	1	1	yes	characteristic
26	C₂×D₃₁₇	—	4	1	no	—
27	C₄×C₃₁₇	D₈	1	1	yes	characteristic
28	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
29	Dic₃₁₇	—	4	1	no	—
30	Dic₃₁₇	—	4	2	no	—
31	C₂×C₃₁₇	—	2	1	no	—
32	C₂×C₃₁₇	C₂×D₈	1	1	yes	characteristic
33	D₃₁₇	—	8	1	no	—
34	C₃₁₇	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	317	1	no	maximal, Sylow
36	#135	—	317	1	no	—
37	C₂×Q₁₆	—	317	1	no	—
38	Q₃₂	—	317	2	no	—
39	SD₃₂	—	317	2	no	—
40	M₃₂	D₃₁₇	1	1	yes	characteristic, complemented
41	#41	—	634	1	no	—
42	C₂×Q₈	—	634	1	no	—
43	Q₁₆	—	317	2	no	—
44	Q₁₆	—	317	1	no	—
45	Q₁₆	—	634	1	no	—
46	SD₁₆	—	634	1	no	—
47	D₈	—	317	1	no	—
48	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
49	C₁₆	C₂×D₃₁₇	1	2	yes	—
50	Q₈	—	634	1	no	—
51	Q₈	—	634	2	no	—
52	Q₈	—	1268	1	no	—
53	D₄	—	634	1	no	—
54	D₄	—	1268	1	no	—
55	C₂×C₄	—	1268	1	no	—
56	C₂×C₄	—	1268	1	no	—
57	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
58	C₈	D₁₂₆₈	1	1	yes	characteristic
59	C₈	C₂²×D₃₁₇	1	1	yes	characteristic
60	C₂²	—	1268	1	no	—
61	C₂²	D₂₅₃₆	1	1	yes	characteristic
62	C₄	—	1268	1	no	—
63	C₄	—	1268	2	no	—
64	C₄	D₂₅₃₆	1	1	yes	characteristic
65	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	2536	1	no	—
68	C₂	C₂×D₂₅₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₅₃₆
commutator, cosocle	C₈×C₃₁₇	C₂³
Frattini	C₈	C₂²×D₃₁₇
Fitting	M₃₂×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×D₈
prime radical	#23778802	C₂
hypercenter	M₃₂	D₃₁₇
hypocenter	C₃₁₇	#510
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₂₆₈	C₂	semidirect
3. Dic₂₅₃₆	C₂	semidirect
4. #23778725	C₂	semidirect
5. M₃₂×C₃₁₇	C₂	semidirect
34. C₃₁₇	#510	semidirect
40. M₃₂	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778802	2
C₂×Dic₁₂₆₈	2
Dic₂₅₃₆	2
#23778725	2
M₃₂×C₃₁₇	2
#510	317