Group #24908562 — subgroups

Subgroups — #24908562

Order 10624.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×#13050580	C₂	1	2	yes	maximal
2	C₂×#14581792	C₂	1	1	yes	maximal, characteristic
3	C₂×#14581715	C₂	1	2	yes	maximal
4	#19127456	C₂	1	1	yes	maximal, characteristic
5	#391×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	2	yes	—
7	C₂×#13050580	—	2	4	no	—
8	C₂×#13050580	—	2	4	no	—
9	C₂×#13050580	—	2	4	no	—
10	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
11	#14581792	—	2	4	no	—
12	C₂×C₄⋊Dic₈₃	—	2	2	no	—
13	#14581715	—	2	4	no	—
14	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	#95×C₈₃	—	2	2	no	—
18	C₂×Dic₁₆₆	—	2	2	no	—
19	C₂×Dic₁₆₆	—	2	2	no	—
20	C₂×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₁₆₆	—	2	2	no	—
22	C₂×Dic₁₆₆	—	2	2	no	—
23	C₂×Dic₁₆₆	D₄	1	4	yes	—
24	C₂²×Dic₈₃	—	2	2	no	—
25	#13050580	—	2	16	no	—
26	C₂×C₈₃⋊C₈	—	2	2	no	—
27	C₂×C₈₃⋊C₈	—	2	2	no	—
28	C₂×C₈₃⋊C₈	D₄	1	4	yes	—
29	C₄⋊Dic₈₃	—	2	8	no	—
30	C₂×Q₈×C₈₃	—	2	1	no	—
31	C₂×Q₈×C₈₃	—	2	2	no	—
32	C₂×Q₈×C₈₃	—	2	2	no	—
33	C₂×Q₈×C₈₃	D₄	1	2	yes	—
34	C₂²×C₄×C₈₃	—	2	1	no	—
35	C₂²×C₄×C₈₃	—	2	1	no	—
36	C₂²×C₄×C₈₃	D₄	1	2	yes	—
37	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₈₃	—	2	2	no	—
39	C₄²×C₈₃	—	2	2	no	—
40	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₈₃	—	2	4	no	—
42	Q₈×C₈₃	—	4	2	no	—
43	C₂×C₄×C₈₃	—	2	2	no	—
44	C₂×C₄×C₈₃	—	2	2	no	—
45	C₂×C₄×C₈₃	—	2	2	no	—
46	C₂×C₄×C₈₃	—	4	1	no	—
47	C₂×C₄×C₈₃	—	4	1	no	—
48	C₂×C₄×C₈₃	—	4	1	no	—
49	C₂×C₄×C₈₃	—	4	2	no	—
50	C₂×C₄×C₈₃	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
55	C₂×Dic₈₃	—	2	4	no	—
56	C₂×Dic₈₃	—	4	2	no	—
57	C₂×Dic₈₃	—	4	2	no	—
58	Dic₁₆₆	—	2	4	no	—
59	Dic₁₆₆	—	2	4	no	—
60	Dic₁₆₆	—	4	2	no	—
61	Dic₁₆₆	—	4	2	no	—
62	C₈₃⋊C₈	—	2	4	no	—
63	C₂²×C₈₃	#120	1	1	yes	characteristic
64	C₂²×C₈₃	#121	1	1	yes	characteristic
65	C₂²×C₈₃	#127	1	1	yes	characteristic
66	C₂²×C₈₃	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₈₃	#137	1	2	yes	—
68	C₄×C₈₃	—	2	4	no	—
69	C₄×C₈₃	—	4	2	no	—
70	C₄×C₈₃	—	8	1	no	—
71	C₄×C₈₃	#120	1	2	yes	—
72	C₄×C₈₃	#121	1	2	yes	—
73	Dic₈₃	—	4	4	no	—
74	C₂×C₈₃	#392	1	1	yes	characteristic
75	C₂×C₈₃	#451	1	2	yes	—
76	C₂×C₈₃	#462	1	2	yes	—
77	C₂×C₈₃	#494	1	1	yes	characteristic
78	C₂×C₈₃	#497	1	1	yes	characteristic
79	#2016	—	83	1	no	maximal, Sylow
80	C₈₃	#2016	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₁₆	—	83	2	no	—
82	C₂×C₄⋊Q₈	—	83	1	no	—
83	C₂×#103	—	83	2	no	—
84	#391	D₈₃	1	1	yes	characteristic
85	#336	—	83	1	no	—
86	C₂²×Q₈	—	83	2	no	—
87	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
88	C₂×Q₁₆	—	166	4	no	—
89	C₂×Q₁₆	—	166	4	no	—
90	C₂×Q₁₆	—	166	4	no	—
91	C₂²×C₈	—	83	2	no	—
92	C₄⋊Q₈	—	166	4	no	—
93	C₂×C₄⋊C₄	—	166	2	no	—
94	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
96	#103	—	166	4	no	—
97	#95	—	2	2	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	83	4	no	—
102	C₂×Q₈	—	166	2	no	—
103	C₂×Q₈	—	166	2	no	—
104	C₂×Q₈	—	166	2	no	—
105	C₂×Q₈	—	166	2	no	—
106	C₂×Q₈	—	166	2	no	—
107	C₂×Q₈	D₃₃₂	1	2	yes	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	166	2	no	—
111	C₂²×C₄	C₂²⋊D₈₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
113	Q₁₆	—	166	16	no	—
114	C₂×C₈	—	83	4	no	—
115	C₂×C₈	—	166	2	no	—
116	C₂×C₈	—	166	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	166	8	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	332	2	no	—
124	Q₈	—	332	2	no	—
125	Q₈	—	166	4	no	—
126	Q₈	—	166	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	332	2	no	—
135	C₂×C₄	—	332	2	no	—
136	C₂×C₄	—	166	4	no	—
137	C₂×C₄	#13050580	1	4	yes	complemented
138	C₂×C₄	D₄×D₈₃	1	2	yes	—
139	C₂×C₄	D₄×D₈₃	1	2	yes	—
140	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
141	C₂×C₄	#13050590	1	1	yes	characteristic
142	C₈	—	166	4	no	—
143	C₂²	#14581760	1	1	yes	characteristic
144	C₂²	#14581766	1	1	yes	characteristic
145	C₂²	#14581788	1	1	yes	characteristic
146	C₂²	C₂×#13050580	1	2	yes	—
147	C₂²	#14581822	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	332	4	no	—
152	C₄	#14581762	1	2	yes	—
153	C₄	#14581765	1	2	yes	—
154	C₂	#19127610	1	1	yes	characteristic
155	C₂	#19127856	1	2	yes	—
156	C₂	#19127911	1	2	yes	—
157	C₂	#19128080	1	1	yes	characteristic
158	C₂	#19128096	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13050580

C₂

yes

maximal

C₂×#14581792

C₂

yes

maximal, characteristic

C₂×#14581715

C₂

yes

maximal

#19127456

C₂

yes

maximal, characteristic

#391×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

—

C₂×#13050580

—

C₂×#13050580

—

C₂×#13050580

—

C₂²×C₈₃⋊C₈

C₂²

yes

—

#14581792

—

C₂×C₄⋊Dic₈₃

—

#14581715

—

C₂²×Q₈×C₈₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

#95×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

D₄

yes

—

C₂²×Dic₈₃

—

#13050580

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

Q₁₆

yes

complemented

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

#41

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

C₂×Q₁₆

yes

—

C₂²×C₈₃

#137

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#120

yes

—

C₄×C₈₃

#121

yes

—

Dic₈₃

—

C₂×C₈₃

#392

yes

characteristic

C₂×C₈₃

#451

yes

—

C₂×C₈₃

#462

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₈₃

yes

characteristic

#336

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂²×C₈

—

C₄⋊Q₈

—

166

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂×D₈₃

yes

characteristic

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

166

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

166

—

103

C₂×Q₈

—

166

—

104

C₂×Q₈

—

166

—

105

C₂×Q₈

—

166

—

106

C₂×Q₈

—

166

—

107

C₂×Q₈

D₃₃₂

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

166

—

111

C₂²×C₄

C₂²⋊D₈₃

yes

—

112

C₂²×C₄

C₂²×D₈₃

yes

characteristic

113

Q₁₆

—

166

—

114

C₂×C₈

—

115

C₂×C₈

—

166

—

116

C₂×C₈

—

166

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

166

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₈₃

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

332

—

124

Q₈

—

332

—

125

Q₈

—

166

—

126

Q₈

—

166

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

332

—

135

C₂×C₄

—

332

—

136

C₂×C₄

—

166

—

137

C₂×C₄

#13050580

yes

complemented

138

C₂×C₄

D₄×D₈₃

yes

—

139

C₂×C₄

D₄×D₈₃

yes

—

140

C₂×C₄

C₂×D₃₃₂

yes

characteristic

141

C₂×C₄

#13050590

yes

characteristic

142

C₈

—

166

—

143

C₂²

#14581760

yes

characteristic

144

C₂²

#14581766

yes

characteristic

145

C₂²

#14581788

yes

characteristic

146

C₂²

C₂×#13050580

yes

—

147

C₂²

#14581822

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

332

—

152

C₄

#14581762

yes

—

153

C₄

#14581765

yes

—

154

C₂

#19127610

yes

characteristic

155

C₂

#19127856

yes

—

156

C₂

#19127911

yes

—

157

C₂

#19128080

yes

characteristic

158

C₂

#19128096

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₃
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#391×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#391	D₈₃
hypocenter	C₈₃	#2016
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₃

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#391×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
50. C₂×C₄×C₈₃	Q₁₆	semidirect
80. C₈₃	#2016	semidirect
137. C₂×C₄	#13050580	semidirect

H ≅ G/N

type

50. C₂×C₄×C₈₃

Q₁₆

semidirect

80. C₈₃

#2016

semidirect

137. C₂×C₄

#13050580

semidirect

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

subgroup	index
C₂²×#13050580	2
C₂×#14581792	2
C₂×#14581715	2
#19127456	2
#391×C₈₃	2
#2016	83

subgroup

index

#391×C₈₃

Subgroups — #24908562

Conjugacy classes of subgroups

Complements

Maximal subgroups