Group #24909331 — subgroups

Subgroups — #24909331

Order 10624.

148 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#14581738	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#19127795	C₂	1	1	yes	maximal, characteristic
4	#19127528	C₂	1	1	yes	maximal, characteristic
5	#19127506	C₂	1	1	yes	maximal, characteristic
6	#19127476	C₂	1	1	yes	maximal, characteristic
7	#358×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
9	Q₈×Dic₈₃	—	2	1	no	—
10	#14581757	—	2	1	no	—
11	#14581738	C₄	1	4	yes	complemented
12	C₂×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14581726	C₂²	1	1	yes	characteristic
14	#14581716	—	2	1	no	—
15	#14581715	—	2	1	no	—
16	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
19	#107×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₆₆	—	2	1	no	—
21	Dic₃₃₂	—	2	1	no	—
22	Dic₃₃₂	C₂×C₄	1	2	yes	—
23	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₈₃	—	2	1	no	—
25	C₄⋊Dic₈₃	—	4	1	no	—
26	Dic₈₃⋊C₄	—	4	1	no	—
27	C₄×Dic₈₃	—	4	1	no	—
28	C₄×Dic₈₃	—	4	1	no	—
29	C₄×Dic₈₃	D₄	1	1	yes	characteristic
30	C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₈₃	—	2	1	no	—
32	Q₁₆×C₈₃	—	2	1	no	—
33	Q₁₆×C₈₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₈₃	—	2	1	no	—
36	C₁₆×C₈₃	—	2	1	no	—
37	Q₈×C₈₃	—	2	2	no	—
38	Q₈×C₈₃	—	4	1	no	—
39	C₂×C₄×C₈₃	—	4	1	no	—
40	C₂×C₄×C₈₃	—	4	1	no	—
41	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₈₃	—	4	1	no	—
43	C₂×Dic₈₃	—	4	1	no	—
44	C₂×Dic₈₃	D₈	1	1	yes	characteristic
45	C₂×Dic₈₃	D₈	1	1	yes	characteristic
46	Dic₁₆₆	—	2	2	no	—
47	Dic₁₆₆	—	4	1	no	—
48	C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₈₃	#41	1	1	yes	characteristic
50	C₈₃⋊C₈	—	2	1	no	—
51	C₂²×C₈₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₈₃	—	4	2	no	—
53	C₄×C₈₃	—	8	1	no	—
54	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
55	C₄×C₈₃	#135	1	1	yes	characteristic
56	Dic₈₃	—	2	1	no	—
57	Dic₈₃	—	2	1	no	—
58	Dic₈₃	—	4	2	no	—
59	Dic₈₃	—	8	1	no	—
60	C₂×C₈₃	C₄×D₈	1	1	yes	characteristic
61	C₂×C₈₃	#510	1	1	yes	characteristic
62	C₂×C₈₃	#510	1	1	yes	characteristic
63	#2176	—	83	1	no	maximal, Sylow
64	C₈₃	#2176	1	1	yes	characteristic, complemented, Sylow
65	C₂×Q₃₂	—	83	1	no	—
66	C₄×Q₁₆	—	83	1	no	—
67	C₄×Q₁₆	—	83	1	no	—
68	#367	—	83	1	no	—
69	#358	—	83	1	no	—
70	#358	D₈₃	1	1	yes	characteristic
71	#346	—	83	1	no	—
72	C₂×Q₁₆	—	83	1	no	—
73	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
74	C₄×Q₈	—	166	1	no	—
75	C₄×Q₈	—	166	1	no	—
76	Q₃₂	—	83	4	no	—
77	C₂×C₁₆	—	83	1	no	—
78	C₂×C₁₆	C₂×D₈₃	1	1	yes	characteristic
79	#107	—	83	1	no	—
80	#107	C₂×D₈₃	1	1	yes	characteristic
81	#103	—	166	1	no	—
82	#103	—	166	1	no	—
83	C₄×C₈	—	83	1	no	—
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	166	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	83	2	no	—
88	Q₁₆	—	166	1	no	—
89	Q₁₆	C₄×D₈₃	1	2	yes	—
90	C₂×C₈	—	83	1	no	—
91	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	332	1	no	—
94	C₄⋊C₄	—	332	1	no	—
95	C₄⋊C₄	—	166	1	no	—
96	C₄²	—	332	1	no	—
97	C₄²	—	332	1	no	—
98	C₄²	—	83	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	83	2	no	—
101	Q₈	—	2	2	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	332	1	no	—
104	Q₈	—	166	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	332	1	no	—
108	C₂×C₄	—	332	1	no	—
109	C₂×C₄	—	83	1	no	—
110	C₂×C₄	—	83	1	no	—
111	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
112	C₈	—	166	1	no	—
113	C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
114	C₈	#13050589	1	1	yes	characteristic
115	C₂²	D₈×D₈₃	1	1	yes	characteristic
116	C₄	—	4	2	no	—
117	C₄	—	8	1	no	—
118	C₄	—	332	2	no	—
119	C₄	—	664	1	no	—
120	C₄	—	166	1	no	—
121	C₄	—	166	1	no	—
122	C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
123	C₄	#14581813	1	1	yes	characteristic
124	C₂	#19127781	1	1	yes	characteristic
125	C₂	#19128140	1	1	yes	characteristic
126	C₂	#19128143	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581738

C₂

yes

maximal, characteristic

Q₁₆×Dic₈₃

C₂

yes

maximal, characteristic

#19127795

C₂

yes

maximal, characteristic

#19127528

C₂

yes

maximal, characteristic

#19127506

C₂

yes

maximal, characteristic

#19127476

C₂

yes

maximal, characteristic

#358×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₃₃₂

C₂²

yes

characteristic

Q₈×Dic₈₃

—

#14581757

—

#14581738

C₄

yes

complemented

C₂×C₈₃⋊C₁₆

C₂²

yes

characteristic

#14581726

C₂²

yes

characteristic

#14581716

—

#14581715

—

C₈×Dic₈₃

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₂²

yes

characteristic

C₂×C₁₆×C₈₃

C₂²

yes

characteristic

#107×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₄

yes

characteristic

C₈₃⋊C₁₆

C₂×C₄

yes

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₁₆×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

D₈

yes

characteristic

C₂×Dic₈₃

D₈

yes

characteristic

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

C₂²×C₄

yes

characteristic

C₈×C₈₃

#41

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×D₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

#135

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₄×D₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#367

—

#358

—

#358

D₈₃

yes

characteristic

#346

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₈₃

yes

characteristic

C₄×Q₈

—

166

—

C₄×Q₈

—

166

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₃

yes

characteristic

#107

—

#107

C₂×D₈₃

yes

characteristic

#103

—

166

—

#103

—

166

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

166

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

166

—

Q₁₆

C₄×D₈₃

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

166

—

C₄²

—

332

—

C₄²

—

332

—

C₄²

—

C₁₆

—

100

C₁₆

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

332

—

104

Q₈

—

166

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

332

—

108

C₂×C₄

—

332

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

D₄×D₈₃

yes

characteristic

112

C₈

—

166

—

113

C₈

C₂×C₄×D₈₃

yes

characteristic

114

C₈

#13050589

yes

characteristic

115

C₂²

D₈×D₈₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

332

—

119

C₄

—

664

—

120

C₄

—

166

—

121

C₄

—

166

—

122

C₄

(C₂²⋊D₈₃)⋊C₄

yes

characteristic

123

C₄

#14581813

yes

characteristic

124

C₂

#19127781

yes

characteristic

125

C₂

#19128140

yes

characteristic

126

C₂

#19128143

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₃
commutator	C₈×C₈₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₈₃
Fitting	#358×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₈
cosocle	C₂×C₈×C₈₃	C₂³
hypercenter	#358	D₈₃
hypocenter	C₈₃	#2176
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₈₃

commutator

C₈×C₈₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₈₃

Fitting

#358×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₈

cosocle

C₂×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
11. #14581738	C₄	semidirect
64. C₈₃	#2176	semidirect

H ≅ G/N

type

11. #14581738

C₄

semidirect

64. C₈₃

#2176

semidirect

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

subgroup	index
C₂×#14581738	2
Q₁₆×Dic₈₃	2
#19127795	2
#19127528	2
#19127506	2
#19127476	2
#358×C₈₃	2
#2176	83

subgroup

index

Q₁₆×Dic₈₃

#358×C₈₃

Subgroups — #24909331

Conjugacy classes of subgroups

Complements

Maximal subgroups