Subgroups — #35203665

Order 20288.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778803	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778726	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778725	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778723	C₂	1	1	yes	maximal, characteristic, complemented
6	#23778719	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#19099575	—	2	1	no	—
9	Q₈×D₃₁₇	—	2	1	no	—
10	Dic₁₂₆₈	C₂²	1	1	yes	characteristic
11	#19099564	—	2	1	no	—
12	#19099561	—	2	1	no	—
13	D₂₅₃₆	C₂²	1	1	yes	characteristic
14	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
15	#19099549	C₂²	1	1	yes	characteristic
16	Q₁₆×C₃₁₇	C₂²	1	1	yes	characteristic
17	Q₁₆×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₁₆×C₃₁₇	C₂²	1	1	yes	characteristic
19	Dic₆₃₄	—	2	1	no	—
20	Dic₆₃₄	—	4	1	no	—
21	D₁₂₆₈	—	2	1	no	—
22	D₁₂₆₈	—	4	1	no	—
23	C₄×D₃₁₇	—	4	1	no	—
24	C₄×D₃₁₇	—	4	1	no	—
25	C₄×D₃₁₇	D₄	1	1	yes	characteristic
26	#14579261	D₄	1	1	yes	characteristic
27	Q₈×C₃₁₇	—	2	1	no	—
28	Q₈×C₃₁₇	—	2	1	no	—
29	C₈×C₃₁₇	C₂³	1	1	yes	characteristic
30	C₂×D₃₁₇	—	4	1	no	—
31	C₂×D₃₁₇	D₈	1	1	yes	characteristic
32	C₄×C₃₁₇	—	4	1	no	—
33	C₄×C₃₁₇	—	4	1	no	—
34	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
35	Dic₃₁₇	—	4	1	no	—
36	Dic₃₁₇	D₈	1	1	yes	characteristic
37	C₂×C₃₁₇	C₂×D₈	1	1	yes	characteristic
38	D₃₁₇	—	2	1	no	—
39	D₃₁₇	—	8	1	no	—
40	C₃₁₇	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	317	1	no	maximal, Sylow
42	#135	—	317	1	no	—
43	C₂×Q₁₆	—	317	1	no	—
44	Q₃₂	—	317	1	no	—
45	Q₃₂	D₃₁₇	1	1	yes	characteristic, complemented
46	SD₃₂	—	317	1	no	—
47	SD₃₂	—	317	1	no	—
48	M₃₂	—	317	1	no	—
49	#41	—	634	1	no	—
50	C₂×Q₈	—	634	1	no	—
51	Q₁₆	—	317	1	no	—
52	Q₁₆	—	634	1	no	—
53	Q₁₆	C₂×D₃₁₇	1	1	yes	characteristic
54	Q₁₆	C₂×D₃₁₇	1	1	yes	characteristic
55	SD₁₆	—	634	1	no	—
56	D₈	—	317	1	no	—
57	C₂×C₈	—	317	1	no	—
58	C₁₆	—	317	1	no	—
59	C₁₆	C₂×D₃₁₇	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	2	1	no	—
62	Q₈	—	634	1	no	—
63	Q₈	—	1268	1	no	—
64	D₄	—	634	1	no	—
65	D₄	—	1268	1	no	—
66	C₂×C₄	—	317	1	no	—
67	C₂×C₄	—	1268	1	no	—
68	C₂×C₄	—	1268	1	no	—
69	C₈	—	317	1	no	—
70	C₈	C₂²×D₃₁₇	1	1	yes	characteristic
71	C₂²	—	317	1	no	—
72	C₂²	—	1268	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	317	1	no	—
76	C₄	—	1268	1	no	—
77	C₄	D₄×D₃₁₇	1	1	yes	characteristic
78	C₂	—	634	1	no	—
79	C₂	—	2536	1	no	—
80	C₂	D₈×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₁₇
commutator, cosocle	C₈×C₃₁₇	C₂³
Frattini	C₈	C₂²×D₃₁₇
Fitting	Q₃₂×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×D₈
prime radical	#23778803	C₂
hypercenter	Q₃₂	D₃₁₇
hypocenter	C₃₁₇	#510
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₃₁₇	C₂	semidirect
3. #23778726	C₂	semidirect
4. #23778725	C₂	semidirect
5. #23778723	C₂	semidirect
6. #23778719	C₂	semidirect
7. Q₃₂×C₃₁₇	C₂	semidirect
40. C₃₁₇	#510	semidirect
45. Q₃₂	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778803	2
Q₁₆×D₃₁₇	2
#23778726	2
#23778725	2
#23778723	2
#23778719	2
Q₃₂×C₃₁₇	2
#510	317