Group #24983229 — subgroups

Subgroups — #24983229

Order 11008.

188 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 47 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	#19135534	C₂	1	1	yes	maximal, characteristic
2	#19132535	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#1322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₈₆	C₄	1	1	yes	characteristic
5	C₂×#13063236	C₄	1	1	yes	characteristic
6	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
7	#14582653	—	2	1	no	—
8	#14582653	—	2	1	no	—
9	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₄₃×C₂²⋊C₁₆	—	2	1	no	—
11	C₂²×Dic₈₆	C₈	1	1	yes	characteristic
12	C₂²×C₄₃⋊C₈	—	2	1	no	—
13	C₄×Dic₈₆	—	4	1	no	—
14	C₄×Dic₈₆	—	4	1	no	—
15	C₂×C₄⋊Dic₄₃	C₈	1	1	yes	characteristic
16	C₂×Dic₄₃⋊C₄	—	2	1	no	—
17	C₂×C₄×Dic₄₃	—	2	1	no	—
18	C₂×C₄₃⋊C₁₆	—	4	1	no	—
19	#13063236	—	4	1	no	—
20	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₃	—	4	1	no	—
24	C₄⋊C₈×C₄₃	—	4	1	no	—
25	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
28	C₂²×Dic₄₃	—	2	1	no	—
29	C₂²×Dic₄₃	—	2	1	no	—
30	C₂×Dic₈₆	—	4	1	no	—
31	C₂×Dic₈₆	—	4	2	no	—
32	C₂×C₈×C₄₃	—	2	1	no	—
33	C₂×C₈×C₄₃	—	2	1	no	—
34	C₂×C₈×C₄₃	—	4	1	no	—
35	C₂×C₈×C₄₃	—	4	1	no	—
36	C₄²×C₄₃	—	2	1	no	—
37	C₄²×C₄₃	—	2	1	no	—
38	C₄⋊Dic₄₃	—	4	1	no	—
39	Dic₄₃⋊C₄	—	4	2	no	—
40	C₄×Dic₄₃	—	4	1	no	—
41	C₄×Dic₄₃	—	4	1	no	—
42	C₂×C₄₃⋊C₈	—	2	2	no	—
43	C₂×C₄₃⋊C₈	—	4	1	no	—
44	C₂×C₄₃⋊C₈	—	4	1	no	—
45	C₁₆×C₄₃	—	8	1	no	—
46	C₄₃⋊C₁₆	—	8	1	no	—
47	C₂³×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₄₃	—	2	1	no	—
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	1	no	—
54	C₂×C₄×C₄₃	—	4	1	no	—
55	C₂×C₄×C₄₃	#99	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#100	1	1	yes	characteristic
57	C₂×Dic₄₃	—	2	2	no	—
58	C₂×Dic₄₃	—	4	1	no	—
59	C₂×Dic₄₃	—	4	1	no	—
60	C₂×Dic₄₃	—	4	1	no	—
61	C₂×Dic₄₃	—	4	1	no	—
62	C₂×Dic₄₃	—	4	1	no	—
63	Dic₈₆	—	4	2	no	—
64	Dic₈₆	—	4	2	no	—
65	C₈×C₄₃	—	4	1	no	—
66	C₈×C₄₃	—	4	1	no	—
67	C₄₃⋊C₈	—	4	2	no	—
68	#7061	—	43	1	no	maximal, Sylow
69	C₂²×C₄₃	—	2	1	no	—
70	C₂²×C₄₃	—	2	1	no	—
71	C₂²×C₄₃	#323	1	1	yes	characteristic
72	C₂²×C₄₃	#350	1	1	yes	characteristic
73	C₂²×C₄₃	#350	1	1	yes	characteristic
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	4	1	no	—
76	C₄×C₄₃	—	4	1	no	—
77	C₄×C₄₃	#354	1	1	yes	characteristic
78	C₄×C₄₃	#356	1	1	yes	characteristic
79	Dic₄₃	—	4	2	no	—
80	Dic₄₃	—	8	1	no	—
81	#1985	—	43	1	no	—
82	#1322	—	43	1	no	—
83	#1322	D₄₃	1	1	yes	characteristic
84	C₂×C₄₃	—	2	1	no	—
85	C₂×C₄₃	—	2	1	no	—
86	C₂×C₄₃	#1315	1	1	yes	characteristic
87	C₂×C₄₃	#1319	1	1	yes	characteristic
88	C₂×C₄₃	#1320	1	1	yes	characteristic
89	C₂×C₄×Q₈	—	43	1	no	—
90	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
91	C₂×#97	—	43	1	no	—
92	C₂²⋊C₁₆	—	2	1	no	—
93	C₂²⋊C₁₆	—	86	1	no	—
94	#336	—	86	1	no	—
95	#336	—	86	1	no	—
96	C₄₃	#7061	1	1	yes	characteristic, complemented, Sylow
97	C₂²×Q₈	—	43	1	no	—
98	C₂²×C₈	—	86	1	no	—
99	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
101	C₄×Q₈	—	172	1	no	—
102	C₄×Q₈	—	172	1	no	—
103	C₂×C₄⋊C₄	—	43	1	no	—
104	C₂×C₄⋊C₄	—	86	1	no	—
105	C₂×C₄²	—	86	1	no	—
106	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
107	C₂×C₁₆	—	4	1	no	—
108	C₂×C₁₆	—	172	1	no	—
109	C₄⋊C₈	—	4	1	no	—
110	#97	—	172	1	no	—
111	C₂×Q₈	—	172	1	no	—
112	C₂×Q₈	—	172	2	no	—
113	C₂²×C₄	—	86	1	no	—
114	C₂²×C₄	—	86	1	no	—
115	C₂²×C₄	C₈×D₄₃	1	1	yes	characteristic
116	C₂²×C₄	#10621872	1	1	yes	characteristic
117	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	4	1	no	—
122	C₂×C₈	—	86	2	no	—
123	C₂×C₈	—	172	1	no	—
124	C₂×C₈	—	172	1	no	—
125	C₄⋊C₄	—	172	1	no	—
126	C₄⋊C₄	—	172	2	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	172	1	no	—
130	C₄²	—	172	1	no	—
131	C₁₆	—	8	1	no	—
132	C₁₆	—	344	1	no	—
133	C₂³	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
134	Q₈	—	172	2	no	—
135	Q₈	—	172	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	86	2	no	—
144	C₂×C₄	—	172	1	no	—
145	C₂×C₄	—	172	1	no	—
146	C₂×C₄	—	172	1	no	—
147	C₂×C₄	—	172	1	no	—
148	C₂×C₄	—	172	1	no	—
149	C₂×C₄	#13063239	1	1	yes	characteristic
150	C₂×C₄	#13063241	1	1	yes	characteristic
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	172	2	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#14582622	1	1	yes	characteristic
157	C₂²	#14582682	1	1	yes	characteristic
158	C₂²	#14582684	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	344	1	no	—
163	C₄	—	172	2	no	—
164	C₄	#14582693	1	1	yes	characteristic
165	C₄	#14582697	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	#19132520	1	1	yes	characteristic
169	C₂	#19132528	1	1	yes	characteristic
170	C₂	#19132530	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19135534

C₂

yes

maximal, characteristic

#19132535

C₂

yes

maximal, characteristic

C₄₃×#1322

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₄

yes

characteristic

C₂×#13063236

C₄

yes

characteristic

C₂²⋊(C₄₃⋊C₁₆)

—

#14582653

—

#14582653

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

characteristic

C₄₃×C₂²⋊C₁₆

—

C₂²×Dic₈₆

C₈

yes

characteristic

C₂²×C₄₃⋊C₈

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₈

yes

characteristic

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

C₂×C₄₃⋊C₁₆

—

#13063236

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₃

M₁₆

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#7061

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

#1985

—

#1322

—

#1322

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

C₂×D₄₃

yes

characteristic

C₂×#97

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#336

—

#336

—

C₄₃

#7061

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

D₁₇₂

yes

characteristic

100

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

101

C₄×Q₈

—

172

—

102

C₄×Q₈

—

172

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

—

106

C₂×C₄²

C₄×D₄₃

yes

characteristic

107

C₂×C₁₆

—

108

C₂×C₁₆

—

172

—

109

C₄⋊C₈

—

110

#97

—

172

—

111

C₂×Q₈

—

172

—

112

C₂×Q₈

—

172

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₈×D₄₃

yes

characteristic

116

C₂²×C₄

#10621872

yes

characteristic

117

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

172

—

124

C₂×C₈

—

172

—

125

C₄⋊C₄

—

172

—

126

C₄⋊C₄

—

172

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

172

—

130

C₄²

—

172

—

131

C₁₆

—

132

C₁₆

—

344

—

133

C₂³

(C₂×D₄₃)⋊C₈

yes

characteristic

134

Q₈

—

172

—

135

Q₈

—

172

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

172

—

145

C₂×C₄

—

172

—

146

C₂×C₄

—

172

—

147

C₂×C₄

—

172

—

148

C₂×C₄

—

172

—

149

C₂×C₄

#13063239

yes

characteristic

150

C₂×C₄

#13063241

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

172

—

154

C₂²

—

155

C₂²

—

156

C₂²

#14582622

yes

characteristic

157

C₂²

#14582682

yes

characteristic

158

C₂²

#14582684

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

344

—

163

C₄

—

172

—

164

C₄

#14582693

yes

characteristic

165

C₄

#14582697

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

#19132520

yes

characteristic

169

C₂

#19132528

yes

characteristic

170

C₂

#19132530

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063239
commutator	C₂²×C₄×C₄₃	C₂×C₈
Frattini	C₂×C₄⋊C₈	C₂×D₄₃
Fitting	C₄₃×#1322	C₂
socle	C₂²×C₄₃	#323
prime radical	C₂³×C₄₃	C₂²⋊C₈
cosocle	C₂×C₄⋊C₈×C₄₃	C₂²
hypercenter	#1322	D₄₃
hypocenter	C₄₃	#7061
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

subgroup

G/H

center

C₂×C₄

#13063239

commutator

C₂²×C₄×C₄₃

C₂×C₈

Frattini

C₂×C₄⋊C₈

C₂×D₄₃

Fitting

C₄₃×#1322

C₂

socle

C₂²×C₄₃

#323

prime radical

C₂³×C₄₃

C₂²⋊C₈

cosocle

C₂×C₄⋊C₈×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ #1322

N	H ≅ G/N	type
96. C₄₃	#7061	semidirect

H ≅ G/N

type

96. C₄₃

#7061

semidirect

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

subgroup	index
#19135534	2
#19132535	2
C₄₃×#1322	2
#7061	43

subgroup

index

#19135534

#19132535

C₄₃×#1322

#7061

Subgroups — #24983229

Conjugacy classes of subgroups

Complements

Maximal subgroups