Subgroups — #24909772

Order 10624.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128133	C₂	1	2	yes	maximal, complemented
2	Dic₁₃₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#19127540	C₂	1	2	yes	maximal, complemented
4	D₂₆₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581814	—	2	2	no	—
7	Dic₆₆₄	C₂²	1	2	yes	—
8	#14581737	—	2	2	no	—
9	D₁₃₂₈	C₂²	1	2	yes	—
10	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
11	C₃₂×C₈₃	C₂²	1	1	yes	characteristic
12	C₃₂×C₈₃	C₂²	1	1	yes	characteristic
13	#13050590	—	4	2	no	—
14	Dic₃₃₂	—	2	2	no	—
15	#13050579	—	4	2	no	—
16	D₆₆₄	—	2	2	no	—
17	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
18	C₁₆×C₈₃	D₄	1	1	yes	characteristic
19	C₁₆×C₈₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
21	C₂²⋊D₈₃	—	8	2	no	—
22	D₃₃₂	—	4	2	no	—
23	C₄×D₈₃	—	8	2	no	—
24	Dic₁₆₆	—	4	2	no	—
25	C₈×C₈₃	D₈	1	1	yes	characteristic
26	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₈₃	D₁₆	1	1	yes	characteristic
28	C₂×D₈₃	—	8	2	no	—
29	C₄×C₈₃	D₁₆	1	1	yes	characteristic
30	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
31	Dic₈₃	—	8	2	no	—
32	C₂×C₈₃	—	2	1	no	—
33	C₂×C₈₃	C₂×D₁₆	1	1	yes	characteristic
34	D₈₃	—	16	2	no	—
35	#2262	—	83	1	no	maximal, Sylow
36	C₈₃	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	83	2	no	—
38	Q₆₄	—	83	1	no	—
39	SD₆₄	—	83	2	no	—
40	D₃₂	—	83	1	no	—
41	C₂×C₃₂	D₈₃	1	1	yes	characteristic, complemented
42	#135	—	166	2	no	—
43	Q₃₂	—	83	2	no	—
44	SD₃₂	—	166	2	no	—
45	D₁₆	—	83	2	no	—
46	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
47	C₃₂	C₂×D₈₃	1	1	yes	characteristic
48	C₃₂	C₂×D₈₃	1	1	yes	characteristic
49	#41	—	332	2	no	—
50	Q₁₆	—	166	2	no	—
51	SD₁₆	—	332	2	no	—
52	D₈	—	166	2	no	—
53	C₂×C₈	D₃₃₂	1	1	yes	characteristic
54	C₁₆	D₃₃₂	1	1	yes	characteristic
55	C₁₆	C₂²×D₈₃	1	1	yes	characteristic
56	Q₈	—	332	2	no	—
57	D₄	—	332	2	no	—
58	D₄	—	664	2	no	—
59	C₂×C₄	—	664	2	no	—
60	C₂×C₄	D₆₆₄	1	1	yes	characteristic
61	C₈	D₆₆₄	1	1	yes	characteristic
62	C₈	C₂×D₃₃₂	1	1	yes	characteristic
63	C₂²	—	664	2	no	—
64	C₂²	D₁₃₂₈	1	1	yes	characteristic
65	C₄	—	664	2	no	—
66	C₄	D₁₃₂₈	1	1	yes	characteristic
67	C₄	C₂×D₆₆₄	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1328	2	no	—
70	C₂	C₂×D₁₃₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₃₂₈
commutator, cosocle	C₁₆×C₈₃	C₂³
Frattini	C₁₆	C₂²×D₈₃
Fitting	C₂×C₃₂×C₈₃	C₂
socle	C₂×C₈₃	C₂×D₁₆
hypercenter	C₂×C₃₂	D₈₃
hypocenter	C₈₃	#2262
LCS	G ▷ C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #19128133	C₂	semidirect
2. Dic₁₃₂₈	C₂	semidirect
3. #19127540	C₂	semidirect
4. D₂₆₅₆	C₂	semidirect
5. C₂×C₃₂×C₈₃	C₂	semidirect
36. C₈₃	#2262	semidirect
41. C₂×C₃₂	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128133	2
Dic₁₃₂₈	2
#19127540	2
D₂₆₅₆	2
C₂×C₃₂×C₈₃	2
#2262	83