Subgroups — #24909780

Order 10624.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#19127541	C₂	1	2	yes	maximal, complemented
3	#19127539	C₂	1	2	yes	maximal, complemented
4	#19127535	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	—	2	1	no	—
7	Dic₆₆₄	—	2	1	no	—
8	Dic₆₆₄	C₂²	1	2	yes	—
9	C₈₃⋊C₃₂	C₂²	1	2	yes	—
10	#135×C₈₃	—	2	1	no	—
11	Q₃₂×C₈₃	C₂²	1	1	yes	characteristic
12	SD₃₂×C₈₃	—	2	1	no	—
13	D₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₆₆	—	4	1	no	—
16	Dic₃₃₂	—	2	2	no	—
17	Dic₃₃₂	—	4	1	no	—
18	#41×C₈₃	—	4	1	no	—
19	Q₁₆×C₈₃	—	2	1	no	—
20	SD₁₆×C₈₃	—	4	1	no	—
21	D₈×C₈₃	—	2	1	no	—
22	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
23	C₁₆×C₈₃	D₄	1	1	yes	characteristic
24	C₁₆×C₈₃	C₂³	1	1	yes	characteristic
25	Q₈×C₈₃	—	4	1	no	—
26	D₄×C₈₃	—	4	1	no	—
27	D₄×C₈₃	—	8	1	no	—
28	C₂×C₄×C₈₃	—	8	1	no	—
29	C₂×C₄×C₈₃	D₈	1	1	yes	characteristic
30	C₂×Dic₈₃	—	8	1	no	—
31	Dic₁₆₆	—	4	2	no	—
32	Dic₁₆₆	—	8	1	no	—
33	C₈×C₈₃	D₈	1	1	yes	characteristic
34	C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₈₃	—	8	1	no	—
36	C₂²×C₈₃	D₁₆	1	1	yes	characteristic
37	C₄×C₈₃	—	8	1	no	—
38	C₄×C₈₃	D₁₆	1	1	yes	characteristic
39	C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
40	Dic₈₃	—	8	2	no	—
41	C₂×C₈₃	—	2	1	no	—
42	C₂×C₈₃	—	16	1	no	—
43	C₂×C₈₃	C₂×D₁₆	1	1	yes	characteristic
44	#2264	—	83	1	no	maximal, Sylow
45	C₈₃	#2264	1	1	yes	characteristic, complemented, Sylow
46	#508	D₈₃	1	1	yes	characteristic
47	C₂×Q₃₂	—	83	1	no	—
48	Q₆₄	—	83	2	no	—
49	SD₆₄	—	83	2	no	—
50	M₆₄	—	83	1	no	—
51	#135	—	2	1	no	—
52	C₂×Q₁₆	—	166	1	no	—
53	Q₃₂	—	83	2	no	—
54	Q₃₂	—	166	1	no	—
55	Q₃₂	C₂×D₈₃	1	1	yes	characteristic
56	SD₃₂	—	2	1	no	—
57	D₁₆	C₂×D₈₃	1	1	yes	characteristic
58	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
59	C₃₂	—	83	2	no	—
60	#41	—	4	1	no	—
61	C₂×Q₈	—	332	1	no	—
62	Q₁₆	—	2	1	no	—
63	Q₁₆	—	332	1	no	—
64	Q₁₆	—	166	2	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	C₂×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
68	C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
69	C₁₆	C₂²×D₈₃	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	Q₈	—	332	2	no	—
72	Q₈	—	664	1	no	—
73	D₄	—	4	1	no	—
74	D₄	—	8	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	664	1	no	—
77	C₂×C₄	#13050575	1	1	yes	characteristic
78	C₈	#13050575	1	1	yes	characteristic
79	C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
80	C₂²	—	8	1	no	—
81	C₂²	#14581733	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	664	2	no	—
84	C₄	#14581733	1	1	yes	characteristic
85	C₄	C₂×#13050575	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	16	1	no	—
88	C₂	C₂×#14581733	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14581733
commutator, cosocle	C₁₆×C₈₃	C₂³
Frattini	C₁₆	C₂²×D₈₃
Fitting, prime radical	#508×C₈₃	C₂
socle	C₂×C₈₃	C₂×D₁₆
hypercenter	#508	D₈₃
hypocenter	C₈₃	#2264
LCS	G ▷ C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₆₆₄	C₂	semidirect
2. #19127541	C₂	semidirect
3. #19127539	C₂	semidirect
4. #19127535	C₂	semidirect
45. C₈₃	#2264	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₆₄	2
#19127541	2
#19127539	2
#19127535	2
#508×C₈₃	2
#2264	83