Group #25110294 — subgroups

Subgroups — #25110294

Order 11008.

586 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 243 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	#19143530	C₂	1	2	yes	maximal
2	#19135577	C₂	1	2	yes	maximal
3	#19135196	C₂	1	8	yes	maximal
4	#19134736	C₂	1	1	yes	maximal, characteristic
5	#19134348	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#3162	C₂	1	1	yes	maximal, characteristic
7	#14583030	—	2	2	no	—
8	#14583030	—	2	2	no	—
9	Q₈×C₄₃⋊C₈	—	2	2	no	—
10	Q₈×C₄₃⋊C₈	—	2	2	no	—
11	#14582926	C₂²	1	4	yes	—
12	#14582923	C₂²	1	2	yes	—
13	#14582855	C₂²	1	4	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	#14582745	C₂²	1	1	yes	characteristic
16	#14582664	C₂²	1	4	yes	—
17	#14582658	C₂²	1	8	yes	—
18	#14582653	C₂²	1	4	yes	—
19	C₄₃×#533	C₂²	1	2	yes	—
20	C₄₃×#501	C₄	1	4	yes	complemented
21	C₄₃×#501	C₄	1	4	yes	complemented
22	C₄₃×#427	C₂²	1	4	yes	—
23	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×#10621877	C₂³	1	4	yes	—
25	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
26	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
27	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
28	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
31	#13063236	—	2	4	no	—
32	C₄×C₄₃⋊C₈	—	2	4	no	—
33	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
34	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
35	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
36	C₄₃⋊#95	—	2	2	no	—
37	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
40	#125×C₄₃	C₂×C₄	1	4	yes	—
41	C₂²⋊Q₈×C₄₃	—	2	2	no	—
42	C₂²⋊Q₈×C₄₃	—	2	2	no	—
43	C₄×Q₈×C₄₃	—	2	2	no	—
44	C₄×Q₈×C₄₃	—	2	2	no	—
45	#117×C₄₃	C₂³	1	4	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	#107×C₄₃	C₂×C₄	1	8	yes	—
48	#106×C₄₃	C₂×C₄	1	4	yes	—
49	#106×C₄₃	C₂×C₄	1	4	yes	—
50	#97×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×Q₈×C₄₃	—	2	2	no	—
52	C₂×Q₈×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	#41	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₃	—	2	2	no	—
56	C₂×C₈×C₄₃	—	2	4	no	—
57	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
58	C₄⋊C₄×C₄₃	—	2	4	no	—
59	C₄⋊C₄×C₄₃	—	2	4	no	—
60	C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
63	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
64	C₂²⋊C₄×C₄₃	—	2	4	no	—
65	C₄²×C₄₃	—	2	4	no	—
66	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₄×Dic₄₃	—	2	4	no	—
70	#10621877	—	2	8	no	—
71	C₂×C₄₃⋊C₈	—	2	4	no	—
72	C₂×C₄₃⋊C₈	—	2	4	no	—
73	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
74	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
75	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
76	Q₈×C₄₃	—	4	4	no	—
77	Q₈×C₄₃	—	4	4	no	—
78	C₂×C₄×C₄₃	—	2	4	no	—
79	C₂×C₄×C₄₃	—	2	4	no	—
80	C₂×C₄×C₄₃	—	4	2	no	—
81	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
86	C₂×Dic₄₃	—	4	4	no	—
87	C₂×Dic₄₃	—	4	2	no	—
88	C₈×C₄₃	—	2	4	no	—
89	C₄₃⋊C₈	—	2	2	no	—
90	C₄₃⋊C₈	—	2	2	no	—
91	C₄₃⋊C₈	—	4	4	no	—
92	#22257	—	43	1	no	maximal, Sylow
93	C₂²×C₄₃	—	2	2	no	—
94	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
95	C₂²×C₄₃	#541	1	2	yes	—
96	C₄×C₄₃	—	2	2	no	—
97	C₄×C₄₃	—	2	2	no	—
98	C₄×C₄₃	—	4	4	no	—
99	C₄×C₄₃	—	4	4	no	—
100	C₄×C₄₃	#519	1	2	yes	—
101	C₄×C₄₃	#536	1	2	yes	—
102	Dic₄₃	—	8	2	no	—
103	#3162	D₄₃	1	1	yes	characteristic
104	#2960	—	43	2	no	—
105	#1995	—	43	2	no	—
106	#1922	—	43	8	no	—
107	#1838	—	43	1	no	—
108	#1746	—	43	1	no	—
109	C₂×C₄₃	—	2	2	no	—
110	C₂×C₄₃	#2327	1	1	yes	characteristic
111	C₂×C₄₃	#2955	1	2	yes	—
112	#533	C₂×D₄₃	1	2	yes	—
113	#501	Dic₄₃	1	4	yes	complemented
114	#501	Dic₄₃	1	4	yes	complemented
115	#446	—	86	2	no	—
116	#446	—	86	2	no	—
117	C₈×Q₈	—	86	2	no	—
118	C₈×Q₈	—	86	2	no	—
119	#427	C₂×D₄₃	1	4	yes	—
120	#424	—	43	4	no	—
121	#423	—	43	2	no	—
122	#405	—	43	4	no	—
123	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
124	C₂×C₄×C₈	—	43	1	no	—
125	#376	—	43	1	no	—
126	#341	—	43	4	no	—
127	#339	—	43	8	no	—
128	#336	—	43	4	no	—
129	C₄₃	#22257	1	1	yes	characteristic, complemented, Sylow
130	C₂×M₁₆	—	43	4	no	—
131	C₂²×C₈	—	43	2	no	—
132	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
133	C₄⋊Q₈	C₂×Dic₄₃	1	2	yes	—
134	C₄⋊Q₈	C₂×Dic₄₃	1	2	yes	—
135	#125	C₂×Dic₄₃	1	4	yes	—
136	C₂²⋊Q₈	—	2	2	no	—
137	C₂²⋊Q₈	—	2	2	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	#117	C₂²×D₄₃	1	4	yes	—
141	C₂×C₄²	—	43	2	no	—
142	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
143	#107	C₂×Dic₄₃	1	8	yes	—
144	#106	C₂×Dic₄₃	1	4	yes	—
145	#106	C₂×Dic₄₃	1	4	yes	—
146	C₄⋊C₈	—	86	4	no	—
147	C₄⋊C₈	—	86	4	no	—
148	C₄⋊C₈	—	86	2	no	—
149	C₄⋊C₈	—	86	2	no	—
150	#97	—	86	4	no	—
151	#97	C₂×Dic₄₃	1	4	yes	—
152	C₄×C₈	—	43	1	no	—
153	C₄×C₈	—	43	2	no	—
154	C₄×C₈	—	43	1	no	—
155	C₄×C₈	—	86	4	no	—
156	#95	—	86	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂²×C₄	—	86	2	no	—
160	C₂²×C₄	#10621900	1	2	yes	—
161	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
162	M₁₆	—	86	8	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	43	2	no	—
165	C₂×C₈	—	43	2	no	—
166	C₂×C₈	—	86	4	no	—
167	C₂×C₈	—	86	4	no	—
168	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
174	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	86	4	no	—
178	C₄²	D₄×D₄₃	1	1	yes	characteristic
179	C₄²	D₄×D₄₃	1	1	yes	characteristic
180	C₄²	C₂²×Dic₄₃	1	2	yes	—
181	C₂³	C₂×#10621900	1	1	yes	characteristic
182	Q₈	—	4	4	no	—
183	Q₈	—	4	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	172	4	no	—
188	C₂×C₄	—	172	2	no	—
189	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
190	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
191	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
192	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
193	C₂×C₄	#13063373	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	86	2	no	—
196	C₈	—	86	2	no	—
197	C₈	—	172	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
200	C₂²	#14583510	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	4	no	—
204	C₄	—	4	4	no	—
205	C₄	—	344	2	no	—
206	C₄	#14583377	1	2	yes	—
207	C₄	#14583474	1	2	yes	—
208	C₂	—	2	2	no	—
209	C₂	#19137315	1	1	yes	characteristic
210	C₂	#19143498	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19143530

C₂

yes

maximal

#19135577

C₂

yes

maximal

#19135196

C₂

yes

maximal

#19134736

C₂

yes

maximal, characteristic

#19134348

C₂

yes

maximal, characteristic

C₄₃×#3162

C₂

yes

maximal, characteristic

#14583030

—

#14583030

—

Q₈×C₄₃⋊C₈

—

Q₈×C₄₃⋊C₈

—

#14582926

C₂²

yes

—

#14582923

C₂²

yes

—

#14582855

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

#14582745

C₂²

yes

characteristic

#14582664

C₂²

yes

—

#14582658

C₂²

yes

—

#14582653

C₂²

yes

—

C₄₃×#533

C₂²

yes

—

C₄₃×#501

C₄

yes

complemented

C₄₃×#501

C₄

yes

complemented

C₄₃×#427

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂×#10621877

C₂³

yes

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

C₂×C₄×Dic₄₃

C₂³

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

D₄

yes

characteristic

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

D₄

yes

characteristic

C₄₃⋊#95

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₄⋊Q₈×C₄₃

C₂×C₄

yes

—

C₄⋊Q₈×C₄₃

C₂×C₄

yes

—

#125×C₄₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#107×C₄₃

C₂×C₄

yes

—

#106×C₄₃

C₂×C₄

yes

—

#106×C₄₃

C₂×C₄

yes

—

#97×C₄₃

C₂×C₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

#41

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂²×C₄

yes

—

C₄⋊C₄×C₄₃

C₂²×C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂³×C₄₃

C₂×#41

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

2₋¹⁺⁴

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#22257

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#541

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

#519

yes

—

101

C₄×C₄₃

#536

yes

—

102

Dic₄₃

—

103

#3162

D₄₃

yes

characteristic

104

#2960

—

105

#1995

—

106

#1922

—

107

#1838

—

108

#1746

—

109

C₂×C₄₃

—

110

C₂×C₄₃

#2327

yes

characteristic

111

C₂×C₄₃

#2955

yes

—

112

#533

C₂×D₄₃

yes

—

113

#501

Dic₄₃

yes

complemented

114

#501

Dic₄₃

yes

complemented

115

#446

—

116

#446

—

117

C₈×Q₈

—

118

C₈×Q₈

—

119

#427

C₂×D₄₃

yes

—

120

#424

—

121

#423

—

122

#405

—

123

C₂×#97

C₂×D₄₃

yes

characteristic

124

C₂×C₄×C₈

—

125

#376

—

126

#341

—

127

#339

—

128

#336

—

129

C₄₃

#22257

yes

characteristic, complemented, Sylow

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₂²×C₈

C₂²×D₄₃

yes

—

133

C₄⋊Q₈

C₂×Dic₄₃

yes

—

134

C₄⋊Q₈

C₂×Dic₄₃

yes

—

135

#125

C₂×Dic₄₃

yes

—

136

C₂²⋊Q₈

—

137

C₂²⋊Q₈

—

138

C₄×Q₈

—

139

C₄×Q₈

—

140

#117

C₂²×D₄₃

yes

—

141

C₂×C₄²

—

142

C₂×C₄²

C₂²×D₄₃

yes

characteristic

143

#107

C₂×Dic₄₃

yes

—

144

#106

C₂×Dic₄₃

yes

—

145

#106

C₂×Dic₄₃

yes

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

#97

—

151

#97

C₂×Dic₄₃

yes

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

#95

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂²×C₄

—

160

C₂²×C₄

#10621900

yes

—

161

C₂²×C₄

C₂³×D₄₃

yes

characteristic

162

M₁₆

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

C₂²×Dic₄₃

yes

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

—

172

C₄⋊C₄

—

173

C₄⋊C₄

C₂²×Dic₄₃

yes

—

174

C₄⋊C₄

C₂²×Dic₄₃

yes

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

—

178

C₄²

D₄×D₄₃

yes

characteristic

179

C₄²

D₄×D₄₃

yes

characteristic

180

C₄²

C₂²×Dic₄₃

yes

—

181

C₂³

C₂×#10621900

yes

characteristic

182

Q₈

—

183

Q₈

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

172

—

188

C₂×C₄

—

172

—

189

C₂×C₄

D₄×Dic₄₃

yes

—

190

C₂×C₄

D₄×Dic₄₃

yes

—

191

C₂×C₄

C₂³×Dic₄₃

yes

characteristic

192

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

193

C₂×C₄

#13063373

yes

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

172

—

198

C₂²

—

199

C₂²

C₂×D₄×Dic₄₃

yes

characteristic

200

C₂²

#14583510

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

344

—

206

C₄

#14583377

yes

—

207

C₄

#14583474

yes

—

208

C₂

—

209

C₂

#19137315

yes

characteristic

210

C₂

#19143498

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#3162	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#3162	D₄₃
hypocenter	C₄₃	#22257
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#3162

C₂

socle

C₂²×C₄₃

C₂×C₄×D₄

prime radical

C₂³×C₄₃

C₂×#41

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
20. C₄₃×#501	C₄	semidirect
21. C₄₃×#501	C₄	semidirect
113. #501	Dic₄₃	semidirect
114. #501	Dic₄₃	semidirect
129. C₄₃	#22257	semidirect

H ≅ G/N

type

20. C₄₃×#501

C₄

semidirect

21. C₄₃×#501

C₄

semidirect

113. #501

Dic₄₃

semidirect

114. #501

Dic₄₃

semidirect

129. C₄₃

#22257

semidirect

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

subgroup	index
#19143530	2
#19135577	2
#19135196	2
#19134736	2
#19134348	2
C₄₃×#3162	2
#22257	43

subgroup

index

C₄₃×#3162

Subgroups — #25110294

Conjugacy classes of subgroups

Complements

Maximal subgroups