Group #24906582 — subgroups

Subgroups — #24906582

Order 10624.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19128437	C₂	1	1	yes	maximal, characteristic
2	#19127993	C₂	1	2	yes	maximal, complemented
3	#19127445	C₂	1	1	yes	maximal, characteristic
4	#19127433	C₂	1	2	yes	maximal, complemented
5	#492×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#14581779	—	2	2	no	—
7	#14581773	—	2	2	no	—
8	C₄×Dic₁₆₆	C₂²	1	2	yes	—
9	D₄×Dic₈₃	—	2	2	no	—
10	#14581725	—	2	2	no	—
11	C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
12	#14581715	—	2	2	no	—
13	C₂×SD₁₆×C₈₃	—	2	2	no	—
14	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
15	#127×C₈₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₆₆	D₄	1	2	yes	—
18	C₂²×Dic₈₃	—	2	2	no	—
19	C₂×C₈₃⋊C₈	—	2	2	no	—
20	C₄⋊Dic₈₃	D₄	1	2	yes	—
21	Dic₈₃⋊C₄	—	2	2	no	—
22	C₂²⋊Dic₈₃	—	2	2	no	—
23	C₂²⋊Dic₈₃	—	2	4	no	—
24	C₄×Dic₈₃	—	2	2	no	—
25	C₂×Q₈×C₈₃	—	2	1	no	—
26	C₂×D₄×C₈₃	—	2	2	no	—
27	C₂×D₄×C₈₃	D₄	1	2	yes	—
28	SD₁₆×C₈₃	—	4	4	no	—
29	C₂×C₈×C₈₃	—	2	1	no	—
30	C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₂³×C₈₃	—	2	2	no	—
33	Q₈×C₈₃	—	4	2	no	—
34	D₄×C₈₃	—	4	2	no	—
35	D₄×C₈₃	—	4	2	no	—
36	D₄×C₈₃	—	4	2	no	—
37	C₂×C₄×C₈₃	—	4	1	no	—
38	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₈₃	—	2	2	no	—
41	C₂×Dic₈₃	—	2	2	no	—
42	C₂×Dic₈₃	—	8	2	no	—
43	Dic₁₆₆	—	2	2	no	—
44	Dic₁₆₆	—	2	2	no	—
45	C₈×C₈₃	—	2	2	no	—
46	C₈₃⋊C₈	—	4	2	no	—
47	C₂²×C₈₃	—	4	4	no	—
48	C₂²×C₈₃	—	4	2	no	—
49	C₂²×C₈₃	#120	1	1	yes	characteristic
50	C₄×C₈₃	—	2	1	no	—
51	C₄×C₈₃	—	8	1	no	—
52	C₄×C₈₃	#135	1	2	yes	—
53	C₄×C₈₃	#137	1	2	yes	—
54	Dic₈₃	—	4	2	no	—
55	Dic₈₃	—	4	2	no	—
56	C₂×C₈₃	—	8	2	no	—
57	C₂×C₈₃	#452	1	2	yes	—
58	C₂×C₈₃	#453	1	1	yes	characteristic
59	#1630	—	83	1	no	maximal, Sylow
60	C₈₃	#1630	1	1	yes	characteristic, complemented, Sylow
61	#559	—	83	1	no	—
62	#492	D₈₃	1	1	yes	characteristic
63	#477	—	83	2	no	—
64	#331	—	83	1	no	—
65	Q₈⋊C₈	—	83	2	no	—
66	C₂×SD₁₆	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
68	#127	C₂×D₈₃	1	1	yes	characteristic
69	#124	—	166	2	no	—
70	#123	—	166	2	no	—
71	C₄×Q₈	—	83	2	no	—
72	C₄×D₄	—	166	2	no	—
73	#107	—	166	2	no	—
74	C₄⋊C₈	—	83	2	no	—
75	#103	—	166	2	no	—
76	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	83	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	C₂²⋊D₈₃	1	2	yes	—
81	C₂²×C₄	—	166	2	no	—
82	SD₁₆	—	4	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	166	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	83	2	no	—
87	C₄⋊C₄	—	166	2	no	—
88	C₂²⋊C₄	—	166	4	no	—
89	C₂²⋊C₄	—	166	2	no	—
90	C₄²	—	166	2	no	—
91	C₄²	C₂²×D₈₃	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	166	2	no	—
95	Q₈	—	166	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	664	2	no	—
101	C₂×C₄	—	166	2	no	—
102	C₂×C₄	—	166	2	no	—
103	C₂×C₄	D₄×D₈₃	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	332	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#14581760	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	332	2	no	—
113	C₄	—	332	2	no	—
114	C₄	#14581813	1	2	yes	—
115	C₄	#14581821	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#19127860	1	2	yes	—
118	C₂	#19127868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19128437

C₂

yes

maximal, characteristic

#19127993

C₂

yes

maximal, complemented

#19127445

C₂

yes

maximal, characteristic

#19127433

C₂

yes

maximal, complemented

#492×C₈₃

C₂

yes

maximal, characteristic

#14581779

—

#14581773

—

C₄×Dic₁₆₆

C₂²

yes

—

D₄×Dic₈₃

—

#14581725

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581715

—

C₂×SD₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

C₂²

yes

characteristic

#127×C₈₃

C₂²

yes

characteristic

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

D₄

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

D₄

yes

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

D₄

yes

—

SD₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂³×C₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#120

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#135

yes

—

C₄×C₈₃

#137

yes

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#452

yes

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#559

—

#492

D₈₃

yes

characteristic

#477

—

#331

—

Q₈⋊C₈

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₈₃

yes

characteristic

#127

C₂×D₈₃

yes

characteristic

#124

—

166

—

#123

—

166

—

C₄×Q₈

—

C₄×D₄

—

166

—

#107

—

166

—

C₄⋊C₈

—

#103

—

166

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₈₃

yes

—

C₂²×C₄

—

166

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

C₂²⋊C₄

—

166

—

C₄²

—

166

—

C₄²

C₂²×D₈₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

166

—

Q₈

—

166

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

664

—

101

C₂×C₄

—

166

—

102

C₂×C₄

—

166

—

103

C₂×C₄

D₄×D₈₃

yes

—

104

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

105

C₈

—

106

C₈

—

332

—

107

C₂²

—

108

C₂²

—

109

C₂²

#14581760

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

332

—

113

C₄

—

332

—

114

C₄

#14581813

yes

—

115

C₄

#14581821

yes

—

116

C₂

—

117

C₂

#19127860

yes

—

118

C₂

#19127868

yes

characteristic

	subgroup	G/H
center	C₂²	#14581760
commutator, cosocle	C₄²×C₈₃	C₂³
Frattini	C₄²	C₂²×D₈₃
Fitting	#492×C₈₃	C₂
socle	C₂²×C₈₃	#120
prime radical	#127×C₈₃	C₂²
hypercenter	#492	D₈₃
hypocenter	C₈₃	#1630
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14581760

commutator, cosocle

C₄²×C₈₃

C₂³

Frattini

C₄²

C₂²×D₈₃

Fitting

#492×C₈₃

C₂

socle

C₂²×C₈₃

#120

prime radical

#127×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #19127993	C₂	semidirect
4. #19127433	C₂	semidirect
60. C₈₃	#1630	semidirect

H ≅ G/N

type

2. #19127993

C₂

semidirect

4. #19127433

C₂

semidirect

60. C₈₃

#1630

semidirect

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

subgroup	index
#19128437	2
#19127993	2
#19127445	2
#19127433	2
#492×C₈₃	2
#1630	83

subgroup

index

#492×C₈₃

Subgroups — #24906582

Conjugacy classes of subgroups

Complements

Maximal subgroups