Subgroups — #24909335

Order 10624.

148 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
2	C₄×Dic₃₃₂	C₂	1	2	yes	maximal
3	#19127529	C₂	1	1	yes	maximal, characteristic
4	#19127506	C₂	1	2	yes	maximal
5	#346×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₃₂	C₂²	1	2	yes	—
7	C₄×Dic₁₆₆	—	2	2	no	—
8	Dic₆₆₄	C₄	1	4	yes	complemented
9	#14581726	C₂²	1	2	yes	—
10	#14581717	—	2	2	no	—
11	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₆₆	—	2	2	no	—
15	Dic₃₃₂	—	2	2	no	—
16	Dic₃₃₂	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₈₃	—	2	2	no	—
18	Dic₈₃⋊C₄	—	4	2	no	—
19	C₄×Dic₈₃	—	4	2	no	—
20	C₂×C₈×C₈₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
22	C₄²×C₈₃	D₄	1	1	yes	characteristic
23	C₁₆×C₈₃	—	2	1	no	—
24	C₁₆×C₈₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈₃	D₈	1	2	yes	—
26	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₈₃	—	4	2	no	—
28	C₂×Dic₈₃	—	4	2	no	—
29	Dic₁₆₆	—	2	4	no	—
30	Dic₁₆₆	—	4	2	no	—
31	C₈×C₈₃	—	2	1	no	—
32	C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₈₃	#41	1	1	yes	characteristic
34	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
35	C₄×C₈₃	—	2	2	no	—
36	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
37	C₄×C₈₃	#135	1	1	yes	characteristic
38	Dic₈₃	—	4	4	no	—
39	Dic₈₃	—	8	2	no	—
40	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
41	C₂×C₈₃	#510	1	2	yes	—
42	#2176	—	83	1	no	maximal, Sylow
43	C₈₃	#2176	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	83	1	no	—
45	C₄×Q₁₆	—	83	2	no	—
46	#367	—	83	1	no	—
47	#358	—	83	2	no	—
48	#346	D₈₃	1	1	yes	characteristic
49	C₂×Q₁₆	—	83	2	no	—
50	C₄×Q₈	—	166	2	no	—
51	Q₃₂	—	83	4	no	—
52	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
54	#107	—	83	2	no	—
55	#103	—	166	2	no	—
56	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
57	C₂×Q₈	—	166	2	no	—
58	Q₁₆	—	83	4	no	—
59	Q₁₆	—	166	2	no	—
60	C₂×C₈	D₃₃₂	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
62	C₄⋊C₄	—	332	2	no	—
63	C₄⋊C₄	—	166	2	no	—
64	C₄²	—	332	2	no	—
65	C₄²	D₃₃₂	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₈₃	1	2	yes	—
68	Q₈	—	332	2	no	—
69	Q₈	—	166	4	no	—
70	C₂×C₄	—	332	2	no	—
71	C₂×C₄	—	332	2	no	—
72	C₂×C₄	D₆₆₄	1	2	yes	—
73	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
76	C₈	#13050590	1	1	yes	characteristic
77	C₂²	C₂×D₆₆₄	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	332	4	no	—
80	C₄	—	664	2	no	—
81	C₄	C₄×D₃₃₂	1	1	yes	characteristic
82	C₄	#14581814	1	1	yes	characteristic
83	C₂	C₄×D₆₆₄	1	1	yes	characteristic
84	C₂	#19128144	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₆₆₄
commutator	C₈×C₈₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#346×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₈
cosocle	C₂×C₈×C₈₃	C₂³
hypercenter	#346	D₈₃
hypocenter	C₈₃	#2176
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₆₆₄	C₄	semidirect
43. C₈₃	#2176	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
C₄×Dic₃₃₂	2
#19127529	2
#19127506	2
#346×C₈₃	2
#2176	83