Group #26115372 — subgroups

Subgroups — #26115372

Order 12032.

538 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 171 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	#20251786	C₂	1	1	yes	maximal, characteristic
2	#20243505	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688870	C₂	1	4	yes	maximal
4	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	—	2	1	no	—
6	C₂×#13094024	—	2	1	no	—
7	C₂×#13094024	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15688993	—	2	2	no	—
10	#15688993	C₄	1	4	yes	—
11	#15688878	—	2	2	no	—
12	#15688870	—	2	8	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×#97×C₄₇	—	2	1	no	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	1	no	—
17	C₂²×Dic₉₄	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	#13094024	—	2	4	no	—
20	#13094024	—	2	4	no	—
21	#13094024	—	4	4	no	—
22	#13094024	—	4	2	no	—
23	#13094024	—	4	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄⋊Dic₄₇	—	2	4	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
28	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₇	—	2	2	no	—
36	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
37	#97×C₄₇	—	4	2	no	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₄₇	—	2	4	no	—
44	C₂×Dic₉₄	—	4	4	no	—
45	C₂×Dic₉₄	—	4	2	no	—
46	C₂×Dic₉₄	—	4	2	no	—
47	C₂×Dic₉₄	—	4	4	no	—
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	—	4	2	no	—
50	C₂×C₈×C₄₇	—	4	2	no	—
51	C₄²×C₄₇	—	2	4	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	—	2	4	no	—
54	C₄²×C₄₇	D₈	1	2	yes	—
55	C₄²×C₄₇	SD₁₆	1	2	yes	—
56	C₄²×C₄₇	C₂×D₄	1	2	yes	—
57	C₄²×C₄₇	#41	1	2	yes	—
58	C₄⋊Dic₄₇	—	2	8	no	—
59	C₄⋊Dic₄₇	—	4	8	no	—
60	C₄⋊Dic₄₇	—	4	8	no	—
61	C₄⋊Dic₄₇	—	4	2	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	4	4	no	—
65	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	#101	1	4	yes	—
72	C₂×C₄×C₄₇	#102	1	4	yes	—
73	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₄₇	#121	1	2	yes	—
75	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₄₇	#137	1	4	yes	—
81	C₂×Dic₄₇	—	4	4	no	—
82	C₂×Dic₄₇	—	4	4	no	—
83	C₂×Dic₄₇	—	4	4	no	—
84	Dic₉₄	—	8	4	no	—
85	Dic₉₄	—	8	4	no	—
86	C₈×C₄₇	—	4	4	no	—
87	C₄₇⋊C₈	—	8	2	no	—
88	#9705	—	47	1	no	maximal, Sylow
89	C₂²×C₄₇	#390	1	1	yes	characteristic
90	C₂²×C₄₇	C₂×#101	1	2	yes	—
91	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
92	C₂²×C₄₇	#418	1	1	yes	characteristic
93	C₂²×C₄₇	#419	1	2	yes	—
94	C₄×C₄₇	—	2	4	no	—
95	C₄×C₄₇	—	2	4	no	—
96	C₄×C₄₇	—	2	2	no	—
97	C₄×C₄₇	#459	1	2	yes	—
98	C₄×C₄₇	#460	1	2	yes	—
99	C₄×C₄₇	#491	1	2	yes	—
100	C₄×C₄₇	#497	1	2	yes	—
101	Dic₄₇	—	8	4	no	—
102	#2868	—	47	1	no	—
103	#1831	—	47	1	no	—
104	#1767	D₄₇	1	1	yes	characteristic
105	C₂×#332	—	47	4	no	—
106	C₂×C₄₇	#1546	1	2	yes	—
107	C₂×C₄₇	#1556	1	2	yes	—
108	C₂×C₄₇	#1960	1	1	yes	characteristic
109	C₂×C₄₇	#1966	1	1	yes	characteristic
110	C₂×C₄₇	#1982	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	—	47	2	no	—
112	C₂×C₄⋊Q₈	—	94	1	no	—
113	C₂×C₄⋊Q₈	—	94	1	no	—
114	C₂×C₄⋊C₈	—	47	2	no	—
115	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
116	C₂×#97	—	2	1	no	—
117	#384	—	47	4	no	—
118	#384	—	94	2	no	—
119	C₄³	C₂×D₄₇	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#336	—	94	2	no	—
122	#332	—	94	8	no	—
123	C₄₇	#9705	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	94	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	94	2	no	—
127	C₄⋊Q₈	—	94	4	no	—
128	C₄⋊Q₈	—	94	4	no	—
129	C₄⋊Q₈	—	188	4	no	—
130	C₄⋊Q₈	—	188	2	no	—
131	C₄⋊Q₈	—	188	2	no	—
132	C₂×C₄⋊C₄	—	47	4	no	—
133	C₂×C₄⋊C₄	—	94	4	no	—
134	C₂×C₄⋊C₄	—	94	4	no	—
135	C₂×C₄⋊C₄	—	94	2	no	—
136	C₂×C₄²	C₄×D₄₇	1	2	yes	—
137	C₂×C₄²	D₁₈₈	1	2	yes	—
138	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	94	4	no	—
143	C₄⋊C₈	D₁₈₈	1	4	yes	—
144	#97	—	4	2	no	—
145	C₂×Q₈	—	188	4	no	—
146	C₂×Q₈	—	188	2	no	—
147	C₂×Q₈	—	188	2	no	—
148	C₂×Q₈	—	188	4	no	—
149	C₂²×C₄	—	94	4	no	—
150	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
151	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	188	2	no	—
159	C₂×C₈	—	188	4	no	—
160	C₄⋊C₄	—	94	8	no	—
161	C₄⋊C₄	—	188	8	no	—
162	C₄⋊C₄	—	188	8	no	—
163	C₄⋊C₄	—	188	2	no	—
164	C₄⋊C₄	—	188	2	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	4	no	—
168	C₄²	#10625334	1	2	yes	—
169	C₄²	#10625336	1	2	yes	—
170	C₄²	C₂×D₁₈₈	1	2	yes	—
171	C₄²	#10625350	1	2	yes	—
172	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
173	Q₈	—	376	4	no	—
174	Q₈	—	376	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	188	4	no	—
181	C₂×C₄	—	188	4	no	—
182	C₂×C₄	—	188	4	no	—
183	C₂×C₄	#13093942	1	4	yes	—
184	C₂×C₄	#13093944	1	4	yes	—
185	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
186	C₂×C₄	#13093998	1	2	yes	—
187	C₂×C₄	#13094012	1	1	yes	characteristic
188	C₂×C₄	#13094021	1	1	yes	characteristic
189	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
190	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
191	C₂×C₄	#13094049	1	2	yes	—
192	C₂×C₄	#13094057	1	4	yes	—
193	C₈	—	4	4	no	—
194	C₈	—	376	2	no	—
195	C₂²	#15689025	1	1	yes	characteristic
196	C₂²	C₂×#13093942	1	2	yes	—
197	C₂²	C₂×#13093944	1	1	yes	characteristic
198	C₂²	#15689127	1	1	yes	characteristic
199	C₂²	#15689133	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	2	4	no	—
203	C₄	—	376	4	no	—
204	C₄	#15689318	1	2	yes	—
205	C₄	#15689323	1	2	yes	—
206	C₄	#15689492	1	2	yes	—
207	C₄	#15689517	1	2	yes	—
208	C₂	#20242061	1	2	yes	—
209	C₂	#20242106	1	2	yes	—
210	C₂	#20244208	1	1	yes	characteristic
211	C₂	#20244232	1	1	yes	characteristic
212	C₂	#20244315	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20251786

C₂

yes

maximal, characteristic

#20243505

C₂

yes

maximal, characteristic

C₂×#15688870

C₂

yes

maximal

C₄₇×#1767

C₂

yes

maximal, characteristic

C₂×#13094024

—

C₂×#13094024

—

C₂×#13094024

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688993

—

#15688993

C₄

yes

—

#15688878

—

#15688870

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₂×#97×C₄₇

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×Dic₉₄

—

C₂²×C₄₇⋊C₈

—

—

—

—

—

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

D₄

yes

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₈

yes

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#101

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#459

yes

—

C₄×C₄₇

#460

yes

—

C₄×C₄₇

#491

yes

—

100

C₄×C₄₇

#497

yes

—

101

Dic₄₇

—

102

#2868

—

103

#1831

—

104

#1767

D₄₇

yes

characteristic

105

C₂×#332

—

106

C₂×C₄₇

#1546

yes

—

107

C₂×C₄₇

#1556

yes

—

108

C₂×C₄₇

#1960

yes

characteristic

109

C₂×C₄₇

#1966

yes

characteristic

110

C₂×C₄₇

#1982

yes

characteristic

111

C₂×C₄⋊Q₈

—

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

—

114

C₂×C₄⋊C₈

—

115

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

116

C₂×#97

—

117

#384

—

118

#384

—

119

C₄³

C₂×D₄₇

yes

characteristic

120

#336

—

121

#336

—

122

#332

—

123

C₄₇

#9705

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

188

—

130

C₄⋊Q₈

—

188

—

131

C₄⋊Q₈

—

188

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄²

C₄×D₄₇

yes

—

137

C₂×C₄²

D₁₈₈

yes

—

138

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

139

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

140

C₂×C₄²

C₂²×D₄₇

yes

characteristic

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

D₁₈₈

yes

—

144

#97

—

145

C₂×Q₈

—

188

—

146

C₂×Q₈

—

188

—

147

C₂×Q₈

—

188

—

148

C₂×Q₈

—

188

—

149

C₂²×C₄

—

150

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

151

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

152

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

153

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

188

—

159

C₂×C₈

—

188

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

—

188

—

163

C₄⋊C₄

—

188

—

164

C₄⋊C₄

—

188

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

#10625334

yes

—

169

C₄²

#10625336

yes

—

170

C₄²

C₂×D₁₈₈

yes

—

171

C₄²

#10625350

yes

—

172

C₂³

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

173

Q₈

—

376

—

174

Q₈

—

376

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

188

—

181

C₂×C₄

—

188

—

182

C₂×C₄

—

188

—

183

C₂×C₄

#13093942

yes

—

184

C₂×C₄

#13093944

yes

—

185

C₂×C₄

C₄×D₁₈₈

yes

—

186

C₂×C₄

#13093998

yes

—

187

C₂×C₄

#13094012

yes

characteristic

188

C₂×C₄

#13094021

yes

characteristic

189

C₂×C₄

C₂×#10625334

yes

characteristic

190

C₂×C₄

C₂×#10625336

yes

characteristic

191

C₂×C₄

#13094049

yes

—

192

C₂×C₄

#13094057

yes

—

193

C₈

—

194

C₈

—

376

—

195

C₂²

#15689025

yes

characteristic

196

C₂²

C₂×#13093942

yes

—

197

C₂²

C₂×#13093944

yes

characteristic

198

C₂²

#15689127

yes

characteristic

199

C₂²

#15689133

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

376

—

204

C₄

#15689318

yes

—

205

C₄

#15689323

yes

—

206

C₄

#15689492

yes

—

207

C₄

#15689517

yes

—

208

C₂

#20242061

yes

—

209

C₂

#20242106

yes

—

210

C₂

#20244208

yes

characteristic

211

C₂

#20244232

yes

characteristic

212

C₂

#20244315

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1767	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1767	D₄₇
hypocenter	C₄₇	#9705
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

subgroup

G/H

center

C₂³

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1767

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

N	H ≅ G/N	type
123. C₄₇	#9705	semidirect

H ≅ G/N

type

123. C₄₇

#9705

semidirect

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

subgroup	index
#20251786	2
#20243505	2
C₂×#15688870	2
C₄₇×#1767	2
#9705	47

subgroup

index

C₄₇×#1767

Subgroups — #26115372

Conjugacy classes of subgroups

Complements

Maximal subgroups