Group #26115093 — subgroups

Subgroups — #26115093

Order 12032.

558 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 155 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	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688853	C₂	1	4	yes	maximal
4	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
7	C₂×#13093935	—	2	2	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15689029	C₄	1	2	yes	—
10	#15689021	—	2	2	no	—
11	#15688993	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₄₇	—	2	2	no	—
13	#15688853	—	2	8	no	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	—	2	2	no	—
19	#13094024	—	2	4	no	—
20	C₄×Dic₉₄	—	2	4	no	—
21	C₄×Dic₉₄	—	2	4	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₄₇⋊C₄	—	2	2	no	—
26	C₂×C₄×Dic₄₇	—	2	2	no	—
27	#13093935	—	2	8	no	—
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
30	C₄₇⋊#95	—	2	4	no	—
31	C₂²×C₈×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₇	D₄	1	2	yes	—
35	C₂×C₄²×C₄₇	D₄	1	2	yes	—
36	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₄₇	—	2	4	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	2	no	—
44	C₂²×Dic₄₇	—	2	2	no	—
45	C₂×Dic₉₄	—	2	4	no	—
46	C₂×Dic₉₄	—	2	4	no	—
47	C₂×Dic₉₄	—	2	2	no	—
48	C₂×Dic₉₄	M₁₆	1	4	yes	—
49	C₂×C₈×C₄₇	—	4	2	no	—
50	C₂×C₈×C₄₇	—	4	4	no	—
51	C₄²×C₄₇	—	2	4	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	—	2	2	no	—
54	C₄²×C₄₇	—	2	4	no	—
55	C₄²×C₄₇	C₂×D₄	1	2	yes	—
56	C₄²×C₄₇	#41	1	2	yes	—
57	C₄⋊Dic₄₇	—	2	8	no	—
58	C₄⋊Dic₄₇	—	2	4	no	—
59	C₄⋊Dic₄₇	—	4	4	no	—
60	Dic₄₇⋊C₄	—	4	4	no	—
61	C₄×Dic₄₇	—	4	2	no	—
62	C₄×Dic₄₇	—	4	2	no	—
63	C₂×C₄₇⋊C₈	—	2	8	no	—
64	C₂×C₄₇⋊C₈	—	2	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	—	2	2	no	—
72	C₂×C₄×C₄₇	#104	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×M₁₆	1	2	yes	—
78	C₂×C₄×C₄₇	#131	1	2	yes	—
79	C₂×C₄×C₄₇	#137	1	4	yes	—
80	C₂×Dic₄₇	—	2	4	no	—
81	C₂×Dic₄₇	—	4	2	no	—
82	C₂×Dic₄₇	—	4	2	no	—
83	C₂×Dic₄₇	—	4	2	no	—
84	C₂×Dic₄₇	—	4	4	no	—
85	Dic₉₄	—	2	8	no	—
86	Dic₉₄	—	4	4	no	—
87	C₈×C₄₇	—	8	2	no	—
88	C₄₇⋊C₈	—	2	8	no	—
89	#9647	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	#390	1	1	yes	characteristic
91	C₂²×C₄₇	#407	1	1	yes	characteristic
92	C₂²×C₄₇	#408	1	1	yes	characteristic
93	C₂²×C₄₇	#419	1	2	yes	—
94	C₂²×C₄₇	C₂×#104	1	2	yes	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	2	4	no	—
97	C₄×C₄₇	—	4	2	no	—
98	C₄×C₄₇	#435	1	4	yes	—
99	C₄×C₄₇	#491	1	2	yes	—
100	C₄×C₄₇	#497	1	2	yes	—
101	Dic₄₇	—	4	4	no	—
102	Dic₄₇	—	8	2	no	—
103	C₄×C₄⋊Q₈	—	47	1	no	—
104	#1831	D₄₇	1	1	yes	characteristic
105	C₄×#97	—	47	1	no	—
106	C₂×Q₈⋊C₈	—	47	4	no	—
107	C₂×C₄₇	#1485	1	2	yes	—
108	C₂×C₄₇	#1491	1	2	yes	—
109	C₂×C₄₇	#1955	1	1	yes	characteristic
110	C₂×C₄₇	#1960	1	1	yes	characteristic
111	C₂×C₄₇	#1962	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	47	1	no	—
113	C₂×C₄×Q₈	—	47	2	no	—
114	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
115	C₂×#97	—	94	2	no	—
116	C₂×C₄×C₈	—	47	2	no	—
117	#391	—	47	2	no	—
118	#389	—	94	2	no	—
119	#384	—	47	1	no	—
120	C₄×C₄⋊C₄	—	94	2	no	—
121	C₄³	C₂×D₄₇	1	1	yes	characteristic
122	#336	—	2	2	no	—
123	Q₈⋊C₈	—	94	8	no	—
124	C₄₇	#9647	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₈	—	47	2	no	—
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	94	2	no	—
128	C₄⋊Q₈	—	94	4	no	—
129	C₄×Q₈	—	94	4	no	—
130	C₄×Q₈	—	94	4	no	—
131	C₂×C₄⋊C₄	—	47	2	no	—
132	C₂×C₄⋊C₄	—	94	2	no	—
133	C₂×C₄⋊C₄	—	94	2	no	—
134	C₂×C₄⋊C₄	—	94	2	no	—
135	C₂×C₄²	—	94	2	no	—
136	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
137	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄²	D₁₈₈	1	2	yes	—
139	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
140	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
141	C₄⋊C₈	—	2	4	no	—
142	#97	—	94	8	no	—
143	C₄×C₈	—	47	4	no	—
144	C₄×C₈	—	94	2	no	—
145	#95	—	94	4	no	—
146	C₂×Q₈	—	47	4	no	—
147	C₂×Q₈	—	94	2	no	—
148	C₂×Q₈	—	94	4	no	—
149	C₂×Q₈	—	94	4	no	—
150	C₂²×C₄	—	94	2	no	—
151	C₂²×C₄	—	94	2	no	—
152	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
153	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	4	no	—
159	C₂×C₈	—	94	8	no	—
160	C₂×C₈	—	94	4	no	—
161	C₄⋊C₄	—	94	8	no	—
162	C₄⋊C₄	—	94	4	no	—
163	C₄⋊C₄	—	188	4	no	—
164	C₄⋊C₄	—	188	4	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	4	no	—
169	C₄²	—	188	2	no	—
170	C₄²	—	188	2	no	—
171	C₄²	D₄×D₄₇	1	2	yes	—
172	C₄²	#10625354	1	2	yes	—
173	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
174	Q₈	—	94	8	no	—
175	Q₈	—	188	4	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	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	94	4	no	—
183	C₂×C₄	—	188	2	no	—
184	C₂×C₄	—	188	4	no	—
185	C₂×C₄	—	188	2	no	—
186	C₂×C₄	—	188	2	no	—
187	C₂×C₄	#13093950	1	4	yes	—
188	C₂×C₄	#13093986	1	2	yes	—
189	C₂×C₄	#13093999	1	2	yes	—
190	C₂×C₄	#13094013	1	1	yes	characteristic
191	C₂×C₄	#13094020	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
193	C₂×C₄	#13094030	1	2	yes	—
194	C₂×C₄	#13094054	1	4	yes	—
195	C₈	—	8	2	no	—
196	C₈	—	94	8	no	—
197	C₂²	#15689024	1	1	yes	characteristic
198	C₂²	#15689086	1	1	yes	characteristic
199	C₂²	#15689089	1	1	yes	characteristic
200	C₂²	#15689131	1	2	yes	—
201	C₂²	C₂×#13093950	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	376	2	no	—
206	C₄	—	188	4	no	—
207	C₄	#15689193	1	4	yes	—
208	C₄	#15689493	1	2	yes	—
209	C₄	#15689518	1	2	yes	—
210	C₂	#20241808	1	2	yes	—
211	C₂	#20241834	1	2	yes	—
212	C₂	#20244185	1	1	yes	characteristic
213	C₂	#20244207	1	1	yes	characteristic
214	C₂	#20244215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13094024

C₂

yes

maximal, characteristic

C₄×#13093935

C₂

yes

maximal, characteristic

C₂×#15688853

C₂

yes

maximal

C₄₇×#1831

C₂

yes

maximal, characteristic

C₂×#13094024

C₄

yes

characteristic

C₂×C₄×Dic₉₄

C₂²

yes

—

C₂×#13093935

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15689029

C₄

yes

—

#15689021

—

#15688993

C₄

yes

characteristic

C₄×C₄⋊Dic₄₇

—

#15688853

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×Dic₉₄

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

—

#13094024

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×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

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

M₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

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

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

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

C₂×#104

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#435

yes

—

C₄×C₄₇

#491

yes

—

100

C₄×C₄₇

#497

yes

—

101

Dic₄₇

—

102

Dic₄₇

—

103

C₄×C₄⋊Q₈

—

104

#1831

D₄₇

yes

characteristic

105

C₄×#97

—

106

C₂×Q₈⋊C₈

—

107

C₂×C₄₇

#1485

yes

—

108

C₂×C₄₇

#1491

yes

—

109

C₂×C₄₇

#1955

yes

characteristic

110

C₂×C₄₇

#1960

yes

characteristic

111

C₂×C₄₇

#1962

yes

characteristic

112

C₂×C₄⋊Q₈

—

113

C₂×C₄×Q₈

—

114

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

115

C₂×#97

—

116

C₂×C₄×C₈

—

117

#391

—

118

#389

—

119

#384

—

120

C₄×C₄⋊C₄

—

121

C₄³

C₂×D₄₇

yes

characteristic

122

#336

—

123

Q₈⋊C₈

—

124

C₄₇

#9647

yes

characteristic, complemented, Sylow

125

C₂²×Q₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₄⋊Q₈

—

129

C₄×Q₈

—

130

C₄×Q₈

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

C₄×D₄₇

yes

characteristic

137

C₂×C₄²

C₄×D₄₇

yes

characteristic

138

C₂×C₄²

D₁₈₈

yes

—

139

C₂×C₄²

C₂²⋊D₄₇

yes

—

140

C₂×C₄²

C₂²×D₄₇

yes

characteristic

141

C₄⋊C₈

—

142

#97

—

143

C₄×C₈

—

144

C₄×C₈

—

145

#95

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

153

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

154

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

155

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

156

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

188

—

164

C₄⋊C₄

—

188

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

—

188

—

170

C₄²

—

188

—

171

C₄²

D₄×D₄₇

yes

—

172

C₄²

#10625354

yes

—

173

C₂³

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

174

Q₈

—

175

Q₈

—

188

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

188

—

184

C₂×C₄

—

188

—

185

C₂×C₄

—

188

—

186

C₂×C₄

—

188

—

187

C₂×C₄

#13093950

yes

—

188

C₂×C₄

#13093986

yes

—

189

C₂×C₄

#13093999

yes

—

190

C₂×C₄

#13094013

yes

characteristic

191

C₂×C₄

#13094020

yes

characteristic

192

C₂×C₄

M₁₆×D₄₇

yes

—

193

C₂×C₄

#13094030

yes

—

194

C₂×C₄

#13094054

yes

—

195

C₈

—

196

C₈

—

197

C₂²

#15689024

yes

characteristic

198

C₂²

#15689086

yes

characteristic

199

C₂²

#15689089

yes

characteristic

200

C₂²

#15689131

yes

—

201

C₂²

C₂×#13093950

yes

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

376

—

206

C₄

—

188

—

207

C₄

#15689193

yes

—

208

C₄

#15689493

yes

—

209

C₄

#15689518

yes

—

210

C₂

#20241808

yes

—

211

C₂

#20241834

yes

—

212

C₂

#20244185

yes

characteristic

213

C₂

#20244207

yes

characteristic

214

C₂

#20244215

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#9647
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1831

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
124. C₄₇	#9647	semidirect

H ≅ G/N

type

124. C₄₇

#9647

semidirect

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

subgroup	index
C₄×#13094024	2
C₄×#13093935	2
C₂×#15688853	2
C₄₇×#1831	2
#9647	47

subgroup

index

C₄₇×#1831

Subgroups — #26115093

Conjugacy classes of subgroups

Complements

Maximal subgroups