Subgroups — #24909773

Order 10624.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128134	C₂	1	2	yes	maximal, complemented
2	#19127538	C₂	1	2	yes	maximal, complemented
3	D₂₆₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581815	—	2	2	no	—
7	#14581735	—	2	2	no	—
8	D₁₃₂₈	C₂²	1	2	yes	—
9	C₁₆×D₈₃	C₂²	1	1	yes	characteristic
10	C₈₃⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₈₃	C₂²	1	2	yes	—
12	C₃₂×C₈₃	C₂²	1	1	yes	characteristic
13	#13050591	—	4	2	no	—
14	#13050577	—	4	2	no	—
15	D₆₆₄	—	2	2	no	—
16	C₈×D₈₃	D₄	1	1	yes	characteristic
17	C₈₃⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₈₃	—	2	2	no	—
19	C₁₆×C₈₃	C₂³	1	1	yes	characteristic
20	Q₈×C₈₃	—	4	2	no	—
21	D₃₃₂	—	4	2	no	—
22	D₃₃₂	—	8	2	no	—
23	C₄×D₈₃	—	8	2	no	—
24	C₄×D₈₃	D₈	1	1	yes	characteristic
25	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
26	C₈₃⋊C₈	D₈	1	1	yes	characteristic
27	C₂×D₈₃	—	8	2	no	—
28	C₂×D₈₃	D₁₆	1	1	yes	characteristic
29	C₄×C₈₃	—	8	2	no	—
30	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
31	Dic₈₃	D₁₆	1	1	yes	characteristic
32	C₂×C₈₃	C₂×D₁₆	1	1	yes	characteristic
33	D₈₃	—	2	1	no	—
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	D₈₃	1	1	yes	characteristic, complemented
39	SD₆₄	—	83	2	no	—
40	D₃₂	—	83	1	no	—
41	C₂×C₃₂	—	83	1	no	—
42	#135	—	166	2	no	—
43	Q₃₂	C₂×D₈₃	1	2	yes	—
44	SD₃₂	—	166	2	no	—
45	D₁₆	—	83	2	no	—
46	C₂×C₁₆	—	83	1	no	—
47	C₃₂	—	83	1	no	—
48	C₃₂	C₂×D₈₃	1	1	yes	characteristic
49	#41	—	332	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	332	2	no	—
52	D₈	—	166	2	no	—
53	C₂×C₈	—	83	1	no	—
54	C₁₆	—	83	1	no	—
55	C₁₆	C₂²×D₈₃	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	332	2	no	—
58	D₄	—	664	2	no	—
59	C₂×C₄	—	83	1	no	—
60	C₂×C₄	—	664	2	no	—
61	C₈	—	83	1	no	—
62	C₈	D₄×D₈₃	1	1	yes	characteristic
63	C₂²	—	83	1	no	—
64	C₂²	—	664	2	no	—
65	C₄	—	8	2	no	—
66	C₄	—	83	1	no	—
67	C₄	D₈×D₈₃	1	1	yes	characteristic
68	C₂	—	1328	2	no	—
69	C₂	—	166	1	no	—
70	C₂	D₁₆×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₈₃
commutator, cosocle	C₁₆×C₈₃	C₂³
Frattini	C₁₆	C₂²×D₈₃
Fitting	Q₆₄×C₈₃	C₂
socle	C₂×C₈₃	C₂×D₁₆
hypercenter	Q₆₄	D₈₃
hypocenter	C₈₃	#2262
LCS	G ▷ C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #19128134	C₂	semidirect
2. #19127538	C₂	semidirect
3. D₂₆₅₆	C₂	semidirect
4. C₃₂×D₈₃	C₂	semidirect
5. Q₆₄×C₈₃	C₂	semidirect
36. C₈₃	#2262	semidirect
38. Q₆₄	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128134	2
#19127538	2
D₂₆₅₆	2
C₃₂×D₈₃	2
Q₆₄×C₈₃	2
#2262	83