Subgroups — #24909340

Order 10624.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃₂₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₆₆₄	C₂	1	2	yes	maximal, complemented
3	#19127529	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127503	C₂	1	2	yes	maximal, complemented
5	#346×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₆₆₄	C₂²	1	2	yes	—
7	C₄×D₃₃₂	—	2	2	no	—
8	D₁₃₂₈	C₄	1	4	yes	complemented
9	#14581726	C₂²	1	2	yes	—
10	#14581714	—	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×D₃₃₂	—	2	2	no	—
15	C₂×C₄×D₈₃	—	4	2	no	—
16	D₆₆₄	—	2	2	no	—
17	D₆₆₄	C₂×C₄	1	4	yes	complemented
18	C₄⋊Dic₈₃	—	2	2	no	—
19	(C₂×D₈₃)⋊C₄	—	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	complemented
25	C₂²×D₈₃	—	4	2	no	—
26	C₂×C₄×C₈₃	D₈	1	2	yes	—
27	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₈₃	—	4	2	no	—
29	D₃₃₂	—	2	4	no	—
30	D₃₃₂	—	4	2	no	—
31	C₄×D₈₃	—	8	4	no	—
32	C₈×C₈₃	—	2	1	no	—
33	C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₈₃	#41	1	1	yes	characteristic
35	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
36	C₂×D₈₃	—	4	4	no	—
37	C₂×D₈₃	—	8	2	no	—
38	C₂×D₈₃	—	8	2	no	—
39	C₄×C₈₃	—	2	2	no	—
40	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
41	C₄×C₈₃	#135	1	1	yes	characteristic
42	Dic₈₃	—	8	2	no	—
43	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
44	C₂×C₈₃	#509	1	2	yes	—
45	D₈₃	—	8	4	no	—
46	#2177	—	83	1	no	maximal, Sylow
47	C₈₃	#2177	1	1	yes	characteristic, complemented, Sylow
48	C₂×D₁₆	—	83	1	no	—
49	C₄×D₈	—	83	2	no	—
50	#367	—	83	1	no	—
51	#357	—	83	2	no	—
52	#346	D₈₃	1	1	yes	characteristic, complemented
53	C₂×D₈	—	83	2	no	—
54	C₄×D₄	—	166	2	no	—
55	D₁₆	—	83	4	no	—
56	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
57	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
58	#107	—	83	2	no	—
59	#102	—	166	2	no	—
60	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
61	C₂×D₄	—	166	2	no	—
62	C₂²×C₄	—	332	2	no	—
63	D₈	—	83	4	no	—
64	D₈	—	166	2	no	—
65	C₂×C₈	D₃₃₂	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
67	C₄⋊C₄	—	166	2	no	—
68	C₂²⋊C₄	—	332	2	no	—
69	C₄²	D₃₃₂	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₄×D₈₃	1	2	yes	complemented
72	C₂³	—	332	2	no	—
73	D₄	—	332	2	no	—
74	D₄	—	166	4	no	—
75	C₂×C₄	—	332	2	no	—
76	C₂×C₄	—	664	4	no	—
77	C₂×C₄	D₆₆₄	1	2	yes	—
78	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
81	C₈	#13050590	1	1	yes	characteristic
82	C₂²	—	332	4	no	—
83	C₂²	—	664	2	no	—
84	C₂²	—	664	2	no	—
85	C₂²	C₂×D₆₆₄	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	664	2	no	—
88	C₄	C₄×D₃₃₂	1	1	yes	characteristic
89	C₄	#14581814	1	1	yes	characteristic
90	C₂	—	664	4	no	—
91	C₂	C₄×D₆₆₄	1	1	yes	characteristic
92	C₂	#19128139	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	C₂²×C₈₃	C₂×D₈
prime radical	C₂×D₁₃₂₈	C₂
cosocle	C₂×C₈×C₈₃	C₂³
hypercenter	#346	D₈₃
hypocenter	C₈₃	#2177
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
2. C₄×D₆₆₄	C₂	semidirect
3. #19127529	C₂	semidirect
4. #19127503	C₂	semidirect
5. #346×C₈₃	C₂	semidirect
8. D₁₃₂₈	C₄	semidirect
17. D₆₆₄	C₂×C₄	semidirect
24. C₁₆×C₈₃	C₂×C₄	semidirect
47. C₈₃	#2177	semidirect
52. #346	D₈₃	semidirect
71. C₁₆	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃₂₈	2
C₄×D₆₆₄	2
#19127529	2
#19127503	2
#346×C₈₃	2
#2177	83