Subgroups — #24907216

Order 10624.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#19127547	C₂	1	1	yes	maximal, characteristic
3	#19127455	C₂	1	4	yes	maximal
4	C₂×#97×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₈₃	C₂²	1	2	yes	—
7	C₄×C₈₃⋊C₈	C₄	1	4	yes	complemented
8	C₈₃⋊#95	C₄	1	4	yes	—
9	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
11	#97×C₈₃	C₄	1	4	yes	complemented
12	C₂²×Dic₈₃	C₂×C₄	1	4	yes	—
13	C₂×C₈₃⋊C₈	—	2	4	no	—
14	C₂×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×Dic₈₃	—	2	4	no	—
16	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₈₃	—	2	4	no	—
19	C₂×C₈×C₈₃	C₂×C₄	1	4	yes	—
20	C₄²×C₈₃	C₂×C₄	1	2	yes	—
21	C₄²×C₈₃	C₂×C₄	1	2	yes	—
22	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈₃	—	2	4	no	—
24	C₂×C₄×C₈₃	C₄²	1	4	yes	—
25	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈₃	#41	1	4	yes	—
28	C₂×Dic₈₃	—	2	4	no	—
29	C₂×Dic₈₃	—	2	8	no	—
30	C₈×C₈₃	—	2	4	no	—
31	C₈₃⋊C₈	—	2	4	no	—
32	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₈₃	#117	1	2	yes	—
34	C₂²×C₈₃	#131	1	4	yes	—
35	C₄×C₈₃	—	2	4	no	—
36	C₄×C₈₃	#117	1	2	yes	—
37	C₄×C₈₃	#117	1	2	yes	—
38	Dic₈₃	—	4	4	no	—
39	C₂×C₈₃	#376	1	1	yes	characteristic
40	C₂×C₈₃	#405	1	2	yes	—
41	C₂×C₈₃	#433	1	4	yes	—
42	#1746	—	83	1	no	maximal, Sylow
43	C₈₃	#1746	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	D₈₃	1	1	yes	characteristic
45	C₂×C₄×C₈	—	83	1	no	—
46	#376	—	83	1	no	—
47	#336	—	83	4	no	—
48	C₂²×C₈	—	83	2	no	—
49	C₂²×C₈	C₂×D₈₃	1	2	yes	—
50	C₂×C₄²	—	83	2	no	—
51	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
52	#97	Dic₈₃	1	4	yes	complemented
53	C₄×C₈	—	83	4	no	—
54	#95	—	83	4	no	—
55	C₂²×C₄	—	83	4	no	—
56	C₂²×C₄	C₄×D₈₃	1	2	yes	—
57	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	83	4	no	—
60	C₂×C₈	—	166	4	no	—
61	C₂×C₈	C₂×Dic₈₃	1	4	yes	—
62	C₄²	—	166	4	no	—
63	C₄²	C₄×D₈₃	1	2	yes	—
64	C₄²	C₂×Dic₈₃	1	2	yes	—
65	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	166	8	no	—
68	C₂×C₄	—	166	4	no	—
69	C₂×C₄	C₄×Dic₈₃	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
72	C₂×C₄	#13050590	1	4	yes	—
73	C₈	—	2	4	no	—
74	C₈	—	166	4	no	—
75	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
76	C₂²	#14581749	1	2	yes	—
77	C₂²	#14581800	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	332	4	no	—
80	C₄	#14581749	1	2	yes	—
81	C₄	#14581751	1	2	yes	—
82	C₂	#19127547	1	1	yes	characteristic
83	C₂	#19127659	1	2	yes	—
84	C₂	#19127763	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂×C₄²
Fitting	C₂×#97×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×#97	D₈₃
hypocenter	C₈₃	#1746
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
7. C₄×C₈₃⋊C₈	C₄	semidirect
11. #97×C₈₃	C₄	semidirect
43. C₈₃	#1746	semidirect
52. #97	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈₃⋊C₈	2
#19127547	2
#19127455	2
C₂×#97×C₈₃	2
#1746	83