Group #24995679 — subgroups

Subgroups — #24995679

Order 11008.

538 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 175 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	#19134712	C₂	1	1	yes	maximal, characteristic
2	#19134448	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582648	C₂	1	4	yes	maximal
4	C₄₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13063236	—	2	1	no	—
8	#14582654	—	2	1	no	—
9	#14582654	—	2	2	no	—
10	#14582648	—	2	8	no	—
11	C₂×C₄⋊Q₈×C₄₃	C₄	1	4	yes	—
12	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
13	C₄₃×#384	—	2	1	no	—
14	C₄₃×#384	—	2	2	no	—
15	C₄₃×#384	—	2	1	no	—
16	C₄³×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₈	—	2	2	no	—
18	C₂²×C₄₃⋊C₈	—	2	2	no	—
19	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
21	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
22	#13063236	—	4	2	no	—
23	C₂²×Q₈×C₄₃	C₂×C₄	1	4	yes	—
24	C₄⋊Q₈×C₄₃	—	2	4	no	—
25	C₄⋊Q₈×C₄₃	—	2	4	no	—
26	C₄⋊Q₈×C₄₃	—	4	8	no	—
27	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
31	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	D₄	1	2	yes	—
34	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	C₂×Q₈×C₄₃	—	2	8	no	—
37	C₂×Q₈×C₄₃	—	4	8	no	—
38	C₂²×C₄×C₄₃	—	2	4	no	—
39	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₄⋊C₄×C₄₃	—	4	8	no	—
45	C₄⋊C₄×C₄₃	—	4	8	no	—
46	C₄⋊C₄×C₄₃	—	4	4	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄⋊C₄×C₄₃	—	4	2	no	—
49	C₄²×C₄₃	—	2	4	no	—
50	C₄²×C₄₃	—	2	2	no	—
51	C₄²×C₄₃	—	2	4	no	—
52	C₄²×C₄₃	SD₁₆	1	2	yes	—
53	C₄²×C₄₃	Q₁₆	1	2	yes	—
54	C₄²×C₄₃	C₂×D₄	1	2	yes	—
55	C₄²×C₄₃	#41	1	2	yes	—
56	C₂×C₄₃⋊C₈	—	2	4	no	—
57	C₂×C₄₃⋊C₈	—	4	2	no	—
58	C₂×C₄₃⋊C₈	—	4	2	no	—
59	C₂×C₄₃⋊C₈	—	4	2	no	—
60	C₂×C₄₃⋊C₈	—	4	4	no	—
61	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	Q₈×C₄₃	—	8	8	no	—
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	4	4	no	—
69	C₂×C₄×C₄₃	—	4	8	no	—
70	C₂×C₄×C₄₃	#100	1	4	yes	—
71	C₂×C₄×C₄₃	#103	1	4	yes	—
72	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
73	C₂×C₄×C₄₃	#121	1	2	yes	—
74	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₃	#137	1	2	yes	—
79	C₂×C₄×C₄₃	#137	1	4	yes	—
80	C₄₃⋊C₈	—	4	4	no	—
81	C₄₃⋊C₈	—	8	2	no	—
82	#9702	—	43	1	no	maximal, Sylow
83	C₂²×C₄₃	#390	1	1	yes	characteristic
84	C₂²×C₄₃	C₂×#100	1	2	yes	—
85	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
86	C₂²×C₄₃	#419	1	2	yes	—
87	C₂²×C₄₃	#419	1	1	yes	characteristic
88	C₄×C₄₃	—	2	4	no	—
89	C₄×C₄₃	—	2	4	no	—
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	—	8	4	no	—
92	C₄×C₄₃	#461	1	2	yes	—
93	C₄×C₄₃	#462	1	2	yes	—
94	C₄×C₄₃	#491	1	2	yes	—
95	C₄×C₄₃	#497	1	2	yes	—
96	#2868	D₄₃	1	1	yes	characteristic
97	#1831	—	43	1	no	—
98	#1767	—	43	1	no	—
99	C₂×#333	—	43	4	no	—
100	C₂×C₄₃	#1548	1	2	yes	—
101	C₂×C₄₃	#1558	1	2	yes	—
102	C₂×C₄₃	#1960	1	1	yes	characteristic
103	C₂×C₄₃	#1965	1	1	yes	characteristic
104	C₂×C₄₃	#1981	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	Dic₄₃	1	4	yes	—
106	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
107	C₂×C₄⋊C₈	—	43	2	no	—
108	C₂×C₄⋊C₈	—	43	2	no	—
109	C₂×#97	—	86	1	no	—
110	#384	—	2	1	no	—
111	#384	—	2	2	no	—
112	#384	—	2	1	no	—
113	C₄³	C₂×D₄₃	1	1	yes	characteristic
114	#336	—	86	2	no	—
115	#336	—	86	1	no	—
116	#333	—	86	8	no	—
117	C₄₃	#9702	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₈	C₂×Dic₄₃	1	4	yes	—
119	C₂²×C₈	—	86	2	no	—
120	C₂²×C₈	—	86	2	no	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	4	8	no	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
129	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
133	C₄⋊C₈	—	43	4	no	—
134	C₄⋊C₈	—	86	4	no	—
135	C₄⋊C₈	—	86	2	no	—
136	#97	—	172	2	no	—
137	C₂×Q₈	—	2	8	no	—
138	C₂×Q₈	—	4	8	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
141	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
145	C₂×C₈	—	86	4	no	—
146	C₂×C₈	—	172	2	no	—
147	C₂×C₈	—	172	4	no	—
148	C₂×C₈	—	172	2	no	—
149	C₂×C₈	—	172	2	no	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	4	no	—
158	C₄²	#10621884	1	2	yes	—
159	C₄²	#10621885	1	2	yes	—
160	C₄²	D₄×D₄₃	1	2	yes	—
161	C₄²	#10621900	1	2	yes	—
162	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
163	Q₈	—	8	8	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	8	no	—
171	C₂×C₄	#13063242	1	4	yes	—
172	C₂×C₄	#13063249	1	4	yes	—
173	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
174	C₂×C₄	#13063296	1	2	yes	—
175	C₂×C₄	#13063312	1	1	yes	characteristic
176	C₂×C₄	#13063321	1	1	yes	characteristic
177	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
178	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
179	C₂×C₄	#13063356	1	2	yes	—
180	C₂×C₄	#13063356	1	4	yes	—
181	C₈	—	344	2	no	—
182	C₈	—	172	4	no	—
183	C₂²	#14582802	1	1	yes	characteristic
184	C₂²	C₂×#13063242	1	2	yes	—
185	C₂²	C₂×#13063249	1	1	yes	characteristic
186	C₂²	#14582908	1	1	yes	characteristic
187	C₂²	#14582908	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	8	4	no	—
192	C₄	#14583101	1	2	yes	—
193	C₄	#14583106	1	2	yes	—
194	C₄	#14583267	1	2	yes	—
195	C₄	#14583291	1	2	yes	—
196	C₂	#19133276	1	2	yes	—
197	C₂	#19133321	1	2	yes	—
198	C₂	#19135416	1	1	yes	characteristic
199	C₂	#19135436	1	1	yes	characteristic
200	C₂	#19135519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134712

C₂

yes

maximal, characteristic

#19134448

C₂

yes

maximal, characteristic

C₂×#14582648

C₂

yes

maximal

C₄₃×#2868

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₄₃

C₄

yes

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

—

C₄₃×#384

—

C₄₃×#384

—

C₄₃×#384

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

#13063236

—

C₂²×Q₈×C₄₃

C₂×C₄

yes

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#100

yes

—

C₂×C₄×C₄₃

#103

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

#137

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

C₂×#103

yes

characteristic

C₂²×C₄₃

#419

yes

—

C₂²×C₄₃

#419

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#461

yes

—

C₄×C₄₃

#462

yes

—

C₄×C₄₃

#491

yes

—

C₄×C₄₃

#497

yes

—

#2868

D₄₃

yes

—

—

—

100

C₂×C₄₃

#1548

yes

—

101

C₂×C₄₃

#1558

yes

—

102

C₂×C₄₃

#1960

yes

characteristic

103

C₂×C₄₃

#1965

yes

characteristic

104

C₂×C₄₃

#1981

yes

characteristic

105

C₂×C₄⋊Q₈

Dic₄₃

yes

—

106

C₂×C₄⋊Q₈

C₂×D₄₃

yes

—

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

—

109

C₂×#97

—

110

#384

—

111

#384

—

112

#384

—

113

C₄³

C₂×D₄₃

yes

characteristic

114

#336

—

115

#336

—

116

#333

—

117

C₄₃

#9702

yes

characteristic, complemented, Sylow

118

C₂²×Q₈

C₂×Dic₄₃

yes

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₄⋊Q₈

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄²

C₂×Dic₄₃

yes

—

129

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

130

C₂×C₄²

C₂²⋊D₄₃

yes

—

131

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

132

C₂×C₄²

C₂²×D₄₃

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

#97

—

172

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²⋊Dic₄₃

yes

—

141

C₂²×C₄

C₂²⋊Dic₄₃

yes

—

142

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

144

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

172

—

147

C₂×C₈

—

172

—

148

C₂×C₈

—

172

—

149

C₂×C₈

—

172

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

#10621884

yes

—

159

C₄²

#10621885

yes

—

160

C₄²

D₄×D₄₃

yes

—

161

C₄²

#10621900

yes

—

162

C₂³

C₂×C₂²⋊Dic₄₃

yes

characteristic

163

Q₈

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

#13063242

yes

—

172

C₂×C₄

#13063249

yes

—

173

C₂×C₄

D₄×Dic₄₃

yes

—

174

C₂×C₄

#13063296

yes

—

175

C₂×C₄

#13063312

yes

characteristic

176

C₂×C₄

#13063321

yes

characteristic

177

C₂×C₄

C₂×#10621884

yes

characteristic

178

C₂×C₄

C₂×#10621885

yes

characteristic

179

C₂×C₄

#13063356

yes

—

180

C₂×C₄

#13063356

yes

—

181

C₈

—

344

—

182

C₈

—

172

—

183

C₂²

#14582802

yes

characteristic

184

C₂²

C₂×#13063242

yes

—

185

C₂²

C₂×#13063249

yes

characteristic

186

C₂²

#14582908

yes

characteristic

187

C₂²

#14582908

yes

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

#14583101

yes

—

193

C₄

#14583106

yes

—

194

C₄

#14583267

yes

—

195

C₄

#14583291

yes

—

196

C₂

#19133276

yes

—

197

C₂

#19133321

yes

—

198

C₂

#19135416

yes

characteristic

199

C₂

#19135436

yes

characteristic

200

C₂

#19135519

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2868	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#2868	D₄₃
hypocenter	C₄₃	#9702
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₃

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#2868

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

N	H ≅ G/N	type
117. C₄₃	#9702	semidirect

H ≅ G/N

type

117. C₄₃

#9702

semidirect

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

subgroup	index
#19134712	2
#19134448	2
C₂×#14582648	2
C₄₃×#2868	2
#9702	43

subgroup

index

C₄₃×#2868

Subgroups — #24995679

Conjugacy classes of subgroups

Complements

Maximal subgroups