Group #24989364 — subgroups

Subgroups — #24989364

Order 11008.

456 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 151 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	#19134448	C₂	1	2	yes	maximal
2	C₂×#14582641	C₂	1	2	yes	maximal
3	#19132476	C₂	1	2	yes	maximal
4	C₄×#125×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
6	C₂×#13063236	C₂²	1	4	yes	—
7	#14582654	—	2	2	no	—
8	#14582641	—	2	4	no	—
9	C₂×#125×C₄₃	C₂²	1	1	yes	characteristic
10	C₄₃×#389	C₄	1	2	yes	—
11	C₄₃×#387	—	2	2	no	—
12	C₄₃×#382	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₄₃	—	2	2	no	—
14	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
15	C₄³×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₈	—	2	4	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
18	#13063236	—	2	4	no	—
19	#13063236	—	2	4	no	—
20	#125×C₄₃	—	2	2	no	—
21	#125×C₄₃	D₄	1	2	yes	—
22	#125×C₄₃	Q₈	1	2	yes	—
23	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₃	—	2	2	no	—
28	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₃	D₄	1	2	yes	—
31	C₂×C₄²×C₄₃	D₄	1	2	yes	—
32	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
33	#95×C₄₃	—	2	2	no	—
34	#95×C₄₃	—	2	2	no	—
35	#95×C₄₃	—	2	2	no	—
36	C₂²×C₄×C₄₃	—	2	2	no	—
37	C₂²×C₄×C₄₃	—	2	2	no	—
38	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₄₃	—	2	2	no	—
44	C₄⋊C₄×C₄₃	—	2	4	no	—
45	C₄⋊C₄×C₄₃	—	2	4	no	—
46	C₄⋊C₄×C₄₃	—	4	2	no	—
47	C₄⋊C₄×C₄₃	—	4	4	no	—
48	C₄⋊C₄×C₄₃	C₄⋊C₄	1	4	yes	—
49	C₄²×C₄₃	—	2	2	no	—
50	C₄²×C₄₃	—	2	4	no	—
51	C₄²×C₄₃	—	4	1	no	—
52	C₄²×C₄₃	—	4	2	no	—
53	C₄²×C₄₃	—	4	4	no	—
54	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₄₃	#41	1	2	yes	—
57	C₂×C₄₃⋊C₈	—	2	8	no	—
58	C₂×C₄₃⋊C₈	—	4	4	no	—
59	C₂×C₄₃⋊C₈	—	4	4	no	—
60	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₃	—	2	1	no	—
62	C₂×C₄×C₄₃	—	2	1	no	—
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	4	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	4	2	no	—
74	C₂×C₄×C₄₃	—	4	2	no	—
75	C₂×C₄×C₄₃	—	4	2	no	—
76	C₂×C₄×C₄₃	#101	1	2	yes	—
77	C₂×C₄×C₄₃	#104	1	4	yes	—
78	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#121	1	2	yes	—
81	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
82	C₄₃⋊C₈	—	4	8	no	—
83	#8580	—	43	1	no	maximal, Sylow
84	C₂²×C₄₃	#380	1	1	yes	characteristic
85	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
86	C₂²×C₄₃	#413	1	1	yes	characteristic
87	C₂²×C₄₃	C₂×#104	1	2	yes	—
88	C₂²×C₄₃	#421	1	2	yes	—
89	C₄×C₄₃	—	2	1	no	—
90	C₄×C₄₃	—	2	1	no	—
91	C₄×C₄₃	—	2	2	no	—
92	C₄×C₄₃	—	2	2	no	—
93	C₄×C₄₃	—	4	1	no	—
94	C₄×C₄₃	—	4	4	no	—
95	C₄×C₄₃	—	4	2	no	—
96	C₄×C₄₃	—	8	2	no	—
97	C₄×C₄₃	#330	1	4	yes	complemented
98	C₄×#125	D₄₃	1	1	yes	characteristic
99	#1767	—	43	2	no	—
100	C₂×#330	—	43	2	no	—
101	#1295	—	43	2	no	—
102	C₂×C₄₃	C₂×#330	1	1	yes	characteristic
103	C₂×C₄₃	#1529	1	2	yes	—
104	C₂×C₄₃	#1537	1	1	yes	characteristic
105	C₂×C₄₃	#1801	1	1	yes	characteristic
106	C₂×C₄₃	#1806	1	2	yes	—
107	C₂×#125	C₂×D₄₃	1	1	yes	characteristic
108	C₂×C₄⋊C₈	—	86	2	no	—
109	C₂×#97	—	43	4	no	—
110	#389	Dic₄₃	1	2	yes	—
111	#387	—	2	2	no	—
112	#382	C₂×D₄₃	1	1	yes	characteristic
113	C₄×C₄⋊C₄	—	2	2	no	—
114	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
115	C₄³	C₂×D₄₃	1	1	yes	characteristic
116	#336	—	86	2	no	—
117	#330	—	86	4	no	—
118	C₄₃	#8580	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	86	4	no	—
120	#125	—	2	2	no	—
121	#125	Dic₈₆	1	2	yes	—
122	#125	D₁₇₂	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
127	C₂×C₄²	—	2	2	no	—
128	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
129	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
131	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
132	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
133	C₄⋊C₈	—	86	8	no	—
134	#97	—	86	4	no	—
135	#97	—	86	4	no	—
136	#95	—	2	2	no	—
137	#95	—	2	2	no	—
138	#95	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
143	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
146	C₂×C₈	—	86	8	no	—
147	C₂×C₈	—	172	4	no	—
148	C₂×C₈	—	172	4	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	C₄⋊Dic₄₃	1	4	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	4	2	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
161	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
162	C₄²	#10621898	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#13063244	1	2	yes	—
180	C₂×C₄	#13063252	1	4	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
182	C₂×C₄	#13063284	1	1	yes	characteristic
183	C₂×C₄	#13063299	1	2	yes	—
184	C₂×C₄	#13063304	1	2	yes	—
185	C₈	—	172	8	no	—
186	C₂²	#14582759	1	1	yes	characteristic
187	C₂²	C₂×#13063244	1	1	yes	characteristic
188	C₂²	#14582883	1	1	yes	characteristic
189	C₂²	C₂×#13063252	1	2	yes	—
190	C₂²	#14582916	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	1	no	—
197	C₄	—	4	4	no	—
198	C₄	—	8	2	no	—
199	C₄	#14582641	1	4	yes	complemented
200	C₂	C₂×#14582641	1	1	yes	characteristic
201	C₂	#19133202	1	2	yes	—
202	C₂	#19133233	1	1	yes	characteristic
203	C₂	#19134591	1	1	yes	characteristic
204	C₂	#19134615	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19134448

C₂

yes

maximal

C₂×#14582641

C₂

yes

maximal

#19132476

C₂

yes

maximal

C₄×#125×C₄₃

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

—

C₂×#13063236

C₂²

yes

—

#14582654

—

#14582641

—

C₂×#125×C₄₃

C₂²

yes

characteristic

C₄₃×#389

C₄

yes

—

C₄₃×#387

—

C₄₃×#382

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₄₃

—

C₄×C₄⋊C₄×C₄₃

C₄

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

—

#125×C₄₃

—

#125×C₄₃

D₄

yes

—

#125×C₄₃

Q₈

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

—

#95×C₄₃

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×C₂²⋊C₄

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

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#101

yes

—

C₂×C₄×C₄₃

#104

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#421

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#330

yes

complemented

C₄×#125

D₄₃

yes

characteristic

#1767

—

100

C₂×#330

—

101

#1295

—

102

C₂×C₄₃

C₂×#330

yes

characteristic

103

C₂×C₄₃

#1529

yes

—

104

C₂×C₄₃

#1537

yes

characteristic

105

C₂×C₄₃

#1801

yes

characteristic

106

C₂×C₄₃

#1806

yes

—

107

C₂×#125

C₂×D₄₃

yes

characteristic

108

C₂×C₄⋊C₈

—

109

C₂×#97

—

110

#389

Dic₄₃

yes

—

111

#387

—

112

#382

C₂×D₄₃

yes

characteristic

113

C₄×C₄⋊C₄

—

114

C₄×C₄⋊C₄

Dic₄₃

yes

—

115

C₄³

C₂×D₄₃

yes

characteristic

116

#336

—

117

#330

—

118

C₄₃

#8580

yes

characteristic, complemented, Sylow

119

C₂²×C₈

—

120

#125

—

121

#125

Dic₈₆

yes

—

122

#125

D₁₇₂

yes

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

126

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

127

C₂×C₄²

—

128

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

129

C₂×C₄²

C₂×Dic₄₃

yes

characteristic

130

C₂×C₄²

C₂²⋊D₄₃

yes

—

131

C₂×C₄²

C₂²⋊D₄₃

yes

—

132

C₂×C₄²

C₂²×D₄₃

yes

characteristic

133

C₄⋊C₈

—

134

#97

—

135

#97

—

136

#95

—

137

#95

—

138

#95

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₄₃

yes

—

143

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

144

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

146

C₂×C₈

—

147

C₂×C₈

—

172

—

148

C₂×C₈

—

172

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₄⋊Dic₄₃

yes

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×Dic₈₆

yes

characteristic

161

C₄²

C₂×D₁₇₂

yes

characteristic

162

C₄²

#10621898

yes

—

163

C₂³

C₂×C₂²⋊Dic₄₃

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#13063244

yes

—

180

C₂×C₄

#13063252

yes

—

181

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

characteristic

182

C₂×C₄

#13063284

yes

characteristic

183

C₂×C₄

#13063299

yes

—

184

C₂×C₄

#13063304

yes

—

185

C₈

—

172

—

186

C₂²

#14582759

yes

characteristic

187

C₂²

C₂×#13063244

yes

characteristic

188

C₂²

#14582883

yes

characteristic

189

C₂²

C₂×#13063252

yes

—

190

C₂²

#14582916

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

#14582641

yes

complemented

200

C₂

C₂×#14582641

yes

characteristic

201

C₂

#19133202

yes

—

202

C₂

#19133233

yes

characteristic

203

C₂

#19134591

yes

characteristic

204

C₂

#19134615

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄×#125×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄×#125	D₄₃
hypocenter	C₄₃	#8580
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₃

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄×#125×C₄₃

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 ◁ C₄×#125

N	H ≅ G/N	type
97. C₄×C₄₃	#330	semidirect
118. C₄₃	#8580	semidirect
199. C₄	#14582641	semidirect

H ≅ G/N

type

97. C₄×C₄₃

#330

semidirect

118. C₄₃

#8580

semidirect

199. C₄

#14582641

semidirect

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

subgroup	index
#19134448	2
C₂×#14582641	2
#19132476	2
C₄×#125×C₄₃	2
#8580	43

subgroup

index

C₄×#125×C₄₃

Subgroups — #24989364

Conjugacy classes of subgroups

Complements

Maximal subgroups