Group #24921957 — subgroups

Subgroups — #24921957

Order 10624.

890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃⋊#564	C₂	1	3	yes	maximal, complemented
2	#19128428	C₂	1	12	yes	maximal, complemented
3	#19128310	C₂	1	12	yes	maximal
4	#19128257	C₂	1	3	yes	maximal
5	#564×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#14581791	C₂²	1	6	yes	—
7	#14581791	C₂²	1	12	yes	—
8	#14581787	C₂²	1	12	yes	—
9	#14581786	C₂²	1	4	yes	—
10	#14581784	C₂²	1	3	yes	—
11	#14581783	C₂²	1	12	yes	—
12	#14581782	C₂²	1	12	yes	—
13	#14581782	C₂²	1	12	yes	—
14	#14581782	C₂²	1	12	yes	—
15	#14581770	C₂²	1	12	yes	—
16	C₄×Dic₁₆₆	C₂²	1	12	yes	—
17	#14581757	C₂²	1	12	yes	—
18	#14581757	C₂²	1	12	yes	—
19	#14581750	C₂²	1	6	yes	—
20	C₄²×D₈₃	C₂²	1	1	yes	characteristic
21	#125×C₈₃	C₂²	1	12	yes	—
22	#125×C₈₃	C₂²	1	3	yes	—
23	C₂×Dic₁₆₆	C₂³	1	12	yes	—
24	C₂×C₄×D₈₃	C₂³	1	3	yes	—
25	C₄⋊Dic₈₃	C₂³	1	6	yes	—
26	C₄⋊Dic₈₃	C₂³	1	12	yes	—
27	Dic₈₃⋊C₄	C₂³	1	12	yes	—
28	Dic₈₃⋊C₄	C₂³	1	12	yes	—
29	Dic₈₃⋊C₄	C₂³	1	12	yes	—
30	Dic₈₃⋊C₄	C₂³	1	12	yes	—
31	(C₂×D₈₃)⋊C₄	C₂³	1	12	yes	—
32	C₄×Dic₈₃	C₂³	1	12	yes	—
33	C₄×Dic₈₃	C₂³	1	12	yes	—
34	C₄×Dic₈₃	C₂³	1	3	yes	—
35	C₄⋊C₄×C₈₃	C₂³	1	12	yes	—
36	C₄⋊C₄×C₈₃	C₂³	1	6	yes	—
37	C₄⋊C₄×C₈₃	C₂³	1	12	yes	—
38	C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	C₄²×C₈₃	C₂³	1	4	yes	—
40	C₂²×D₈₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂⁴	1	12	yes	—
42	C₂×C₄×C₈₃	C₂⁴	1	3	yes	—
43	C₂×Dic₈₃	C₂⁴	1	12	yes	—
44	C₂×Dic₈₃	C₂⁴	1	3	yes	—
45	C₄×D₈₃	—	2	6	no	—
46	Dic₁₆₆	—	2	24	no	—
47	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
48	C₂×D₈₃	—	2	3	no	—
49	C₄×C₈₃	—	2	12	no	—
50	C₄×C₈₃	—	2	3	no	—
51	Dic₈₃	—	2	12	no	—
52	Dic₈₃	2₋¹⁺⁴	1	6	yes	—
53	C₂×C₈₃	C₂×2₋¹⁺⁴	1	3	yes	—
54	D₈₃	—	4	1	no	—
55	#3556	—	83	1	no	maximal, Sylow
56	C₈₃	#3556	1	1	yes	characteristic, complemented, Sylow
57	#564	—	83	3	no	—
58	#564	D₈₃	1	1	yes	characteristic, complemented
59	#557	—	83	12	no	—
60	#541	—	83	12	no	—
61	#533	—	83	3	no	—
62	C₄⋊Q₈	—	83	6	no	—
63	C₄⋊Q₈	—	83	12	no	—
64	#126	—	83	12	no	—
65	#126	—	83	4	no	—
66	#125	—	83	12	no	—
67	#125	—	83	12	no	—
68	#125	—	83	12	no	—
69	#125	—	83	12	no	—
70	#125	—	83	3	no	—
71	#125	C₂×D₈₃	1	3	yes	—
72	#125	C₂×D₈₃	1	12	yes	—
73	C₂²⋊Q₈	—	83	12	no	—
74	C₄×Q₈	—	83	12	no	—
75	C₄×Q₈	—	83	12	no	—
76	C₄×Q₈	—	83	12	no	—
77	#117	—	83	6	no	—
78	C₂×C₄²	—	83	1	no	—
79	C₂×Q₈	—	83	12	no	—
80	C₂²×C₄	—	83	3	no	—
81	C₄⋊C₄	—	83	6	no	—
82	C₄⋊C₄	—	83	12	no	—
83	C₄⋊C₄	—	83	12	no	—
84	C₄⋊C₄	—	83	12	no	—
85	C₄⋊C₄	—	83	12	no	—
86	C₄⋊C₄	—	83	12	no	—
87	C₄⋊C₄	C₂²×D₈₃	1	6	yes	—
88	C₄⋊C₄	C₂²×D₈₃	1	12	yes	—
89	C₄⋊C₄	C₂²×D₈₃	1	12	yes	—
90	C₂²⋊C₄	—	83	12	no	—
91	C₄²	—	83	3	no	—
92	C₄²	—	83	12	no	—
93	C₄²	—	83	12	no	—
94	C₄²	C₂²×D₈₃	1	1	yes	characteristic
95	C₄²	C₂²×D₈₃	1	4	yes	—
96	C₂³	—	83	1	no	—
97	Q₈	—	166	24	no	—
98	C₂×C₄	—	83	3	no	—
99	C₂×C₄	—	83	12	no	—
100	C₂×C₄	—	166	6	no	—
101	C₂×C₄	C₂³×D₈₃	1	12	yes	—
102	C₂×C₄	C₂³×D₈₃	1	3	yes	—
103	C₂²	—	166	3	no	—
104	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
105	C₄	—	2	3	no	—
106	C₄	—	2	12	no	—
107	C₄	—	83	6	no	—
108	C₄	—	166	12	no	—
109	C₂	—	332	1	no	—
110	C₂	2₋¹⁺⁴×D₈₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₃⋊#564

C₂

yes

maximal, complemented

#19128428

C₂

yes

maximal, complemented

#19128310

C₂

yes

maximal

#19128257

C₂

yes

maximal

#564×C₈₃

C₂

yes

maximal, characteristic, complemented

#14581791

C₂²

yes

—

#14581791

C₂²

yes

—

#14581787

C₂²

yes

—

#14581786

C₂²

yes

—

#14581784

C₂²

yes

—

#14581783

C₂²

yes

—

#14581782

C₂²

yes

—

#14581782

C₂²

yes

—

#14581782

C₂²

yes

—

#14581770

C₂²

yes

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581757

C₂²

yes

—

#14581757

C₂²

yes

—

#14581750

C₂²

yes

—

C₄²×D₈₃

C₂²

yes

characteristic

#125×C₈₃

C₂²

yes

—

#125×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

C₂³

yes

—

C₂×C₄×D₈₃

C₂³

yes

—

C₄⋊Dic₈₃

C₂³

yes

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

C₂³

yes

—

Dic₈₃⋊C₄

C₂³

yes

—

Dic₈₃⋊C₄

C₂³

yes

—

Dic₈₃⋊C₄

C₂³

yes

—

(C₂×D₈₃)⋊C₄

C₂³

yes

—

C₄×Dic₈₃

C₂³

yes

—

C₄×Dic₈₃

C₂³

yes

—

C₄×Dic₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₄²×C₈₃

C₂³

yes

characteristic

C₄²×C₈₃

C₂³

yes

—

C₂²×D₈₃

C₂⁴

yes

characteristic

C₂×C₄×C₈₃

C₂⁴

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

—

C₂×Dic₈₃

C₂⁴

yes

—

C₂×Dic₈₃

C₂⁴

yes

—

C₄×D₈₃

—

Dic₁₆₆

—

C₂²×C₈₃

C₂⁵

yes

characteristic

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

Dic₈₃

—

Dic₈₃

2₋¹⁺⁴

yes

—

C₂×C₈₃

C₂×2₋¹⁺⁴

yes

—

D₈₃

—

#3556

—

maximal, Sylow

C₈₃

#3556

yes

characteristic, complemented, Sylow

#564

—

#564

D₈₃

yes

characteristic, complemented

#557

—

#541

—

#533

—

C₄⋊Q₈

—

C₄⋊Q₈

—

—

—

—

—

—

—

—

C₂×D₈₃

yes

—

#125

C₂×D₈₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₈₃

yes

—

C₄⋊C₄

C₂²×D₈₃

yes

—

C₄⋊C₄

C₂²×D₈₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₈₃

yes

characteristic

C₄²

C₂²×D₈₃

yes

—

C₂³

—

Q₈

—

166

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

166

—

101

C₂×C₄

C₂³×D₈₃

yes

—

102

C₂×C₄

C₂³×D₈₃

yes

—

103

C₂²

—

166

—

104

C₂²

C₂⁴×D₈₃

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

166

—

109

C₂

—

332

—

110

C₂

2₋¹⁺⁴×D₈₃

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₃
commutator, socle, cosocle	C₂²×C₈₃	C₂⁵
Fitting	#564×C₈₃	C₂
prime radical	C₂²×D₈₃	C₂⁴
hypercenter	#564	D₈₃
hypocenter	C₈₃	#3556
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #564

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₈₃

commutator, socle, cosocle

C₂²×C₈₃

C₂⁵

Fitting

#564×C₈₃

C₂

prime radical

C₂²×D₈₃

C₂⁴

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂² ◁ #564

N	H ≅ G/N	type
1. C₈₃⋊#564	C₂	semidirect
2. #19128428	C₂	semidirect
5. #564×C₈₃	C₂	semidirect
56. C₈₃	#3556	semidirect
58. #564	D₈₃	semidirect

H ≅ G/N

type

1. C₈₃⋊#564

C₂

semidirect

2. #19128428

C₂

semidirect

5. #564×C₈₃

C₂

semidirect

56. C₈₃

#3556

semidirect

58. #564

D₈₃

semidirect

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

subgroup	index
C₈₃⋊#564	2
#19128428	2
#19128310	2
#19128257	2
#564×C₈₃	2
#3556	83

subgroup

index

C₈₃⋊#564

#564×C₈₃

Subgroups — #24921957

Conjugacy classes of subgroups

Complements

Maximal subgroups