Group #26368882 — subgroups

Subgroups — #26368882

Order 12032.

822 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 267 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	C₂×#15689533	C₂	1	4	yes	maximal
2	#20251766	C₂	1	1	yes	maximal, characteristic
3	#20243554	C₂	1	4	yes	maximal
4	#20243550	C₂	1	4	yes	maximal
5	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#2859	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094024	C₂²	1	2	yes	—
8	#15689533	—	2	16	no	—
9	C₂×#13093957	C₂²	1	8	yes	—
10	C₂×#13093955	C₂²	1	8	yes	—
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×#13093935	C₂²	1	2	yes	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
14	#15689029	—	2	4	no	—
15	#15688989	C₂²	1	2	yes	—
16	#15688989	C₂²	1	2	yes	—
17	C₂×#125×C₄₇	C₂²	1	2	yes	—
18	C₄₇×#389	—	2	4	no	—
19	C₄₇×#384	C₂²	1	2	yes	—
20	C₄₇×#382	C₂²	1	2	yes	—
21	C₄³×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₉₄	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
24	#13094024	—	2	8	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
27	#13093957	—	2	16	no	—
28	#13093955	—	2	16	no	—
29	#13093935	D₄	1	8	yes	—
30	#13093935	Q₈	1	8	yes	complemented
31	C₄×C₄₇⋊C₈	Q₈	1	8	yes	complemented
32	C₄₇⋊#95	—	2	4	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	#125×C₄₇	—	2	8	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
37	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₄₇	D₄	1	4	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#95×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂²×Dic₄₇	—	2	4	no	—
49	C₂×Dic₉₄	—	4	8	no	—
50	C₂×Dic₉₄	—	4	4	no	—
51	C₄⋊C₄×C₄₇	—	2	4	no	—
52	C₄⋊C₄×C₄₇	—	2	4	no	—
53	C₄⋊C₄×C₄₇	—	4	8	no	—
54	C₄⋊C₄×C₄₇	—	4	8	no	—
55	C₄²×C₄₇	—	4	4	no	—
56	C₄²×C₄₇	C₂×D₄	1	4	yes	—
57	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
58	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
59	C₄⋊Dic₄₇	—	2	8	no	—
60	C₄⋊Dic₄₇	—	4	4	no	—
61	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
62	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
63	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
64	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	—	2	4	no	—
66	C₂×C₄×C₄₇	—	2	4	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	4	4	no	—
69	C₂×C₄×C₄₇	—	4	8	no	—
70	C₂×C₄×C₄₇	#127	1	4	yes	—
71	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
72	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
73	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
74	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
75	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
76	C₂×Dic₄₇	—	4	4	no	—
77	C₂×Dic₄₇	—	4	8	no	—
78	Dic₉₄	—	8	8	no	—
79	C₄₇⋊C₈	C₄⋊Q₈	1	16	yes	—
80	#33317	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
82	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₄₇	#575	1	4	yes	—
85	C₄×C₄₇	—	2	4	no	—
86	C₄×C₄₇	—	2	4	no	—
87	C₄×C₄₇	—	2	4	no	—
88	C₄×C₄₇	—	8	4	no	—
89	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
90	Dic₄₇	—	8	4	no	—
91	C₂×#501	—	47	4	no	—
92	#2862	—	47	1	no	—
93	#2859	D₄₇	1	1	yes	characteristic
94	#1847	—	47	4	no	—
95	#1846	—	47	4	no	—
96	C₄×#97	—	47	1	no	—
97	C₂×C₄₇	#2868	1	1	yes	characteristic
98	C₂×C₄₇	#3153	1	2	yes	—
99	C₂×C₄₇	#3162	1	2	yes	—
100	C₂×C₄₇	#3166	1	2	yes	—
101	C₂×C₄⋊Q₈	—	47	2	no	—
102	C₂×#125	C₂×D₄₇	1	2	yes	—
103	#501	—	94	16	no	—
104	C₂×#107	—	47	8	no	—
105	C₂×#106	—	47	8	no	—
106	C₂×#97	—	47	2	no	—
107	C₂×#97	—	47	2	no	—
108	C₂×C₄×C₈	—	47	2	no	—
109	#391	—	94	4	no	—
110	#389	—	2	4	no	—
111	#384	C₂×D₄₇	1	2	yes	—
112	#382	—	47	2	no	—
113	#382	—	47	2	no	—
114	#382	C₂×D₄₇	1	2	yes	—
115	C₄³	C₂×D₄₇	1	1	yes	characteristic
116	C₄₇	#33317	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	—	94	2	no	—
118	C₂²×C₈	—	47	4	no	—
119	C₄⋊Q₈	—	94	8	no	—
120	#125	—	2	8	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	—	2	4	no	—
123	C₂×C₄⋊C₄	—	47	4	no	—
124	C₂×C₄⋊C₄	—	94	2	no	—
125	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
126	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
127	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
129	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
130	#107	—	94	16	no	—
131	#106	—	94	16	no	—
132	#97	—	47	8	no	—
133	#97	—	47	8	no	—
134	C₄×C₈	—	47	8	no	—
135	#95	—	2	4	no	—
136	#95	—	94	4	no	—
137	#95	—	94	4	no	—
138	C₂×Q₈	—	188	8	no	—
139	C₂×Q₈	—	188	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	94	4	no	—
142	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
144	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
145	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
146	C₂×C₈	—	47	8	no	—
147	C₂×C₈	—	47	8	no	—
148	C₂×C₈	—	47	8	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	94	8	no	—
154	C₄⋊C₄	—	188	4	no	—
155	C₄²	—	4	4	no	—
156	C₄²	D₄×D₄₇	1	4	yes	—
157	C₄²	Q₈×D₄₇	1	4	yes	—
158	C₄²	Q₈×D₄₇	1	4	yes	—
159	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
160	Q₈	—	376	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	8	no	—
166	C₂×C₄	—	188	4	no	—
167	C₂×C₄	—	188	8	no	—
168	C₂×C₄	#13094020	1	4	yes	—
169	C₂×C₄	#13094022	1	4	yes	—
170	C₂×C₄	#13094022	1	4	yes	—
171	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
172	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
173	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
174	C₈	—	47	16	no	—
175	C₂²	C₂×#13094020	1	1	yes	characteristic
176	C₂²	C₂×#13094022	1	1	yes	characteristic
177	C₂²	C₂×#13094022	1	1	yes	characteristic
178	C₂²	#15689921	1	4	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	376	4	no	—
184	C₄	C₄⋊Q₈×D₄₇	1	4	yes	—
185	C₂	#20251784	1	1	yes	characteristic
186	C₂	#20253856	1	2	yes	—
187	C₂	#20253910	1	2	yes	—
188	C₂	#20253937	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689533

C₂

yes

maximal

#20251766

C₂

yes

maximal, characteristic

#20243554

C₂

yes

maximal

#20243550

C₂

yes

maximal

C₄×#13093935

C₂

yes

maximal, characteristic

C₄₇×#2859

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

#15689533

—

C₂×#13093957

C₂²

yes

—

C₂×#13093955

C₂²

yes

—

C₂×#13093935

C₂²

yes

—

C₂×#13093935

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15689029

—

#15688989

C₂²

yes

—

#15688989

C₂²

yes

—

C₂×#125×C₄₇

C₂²

yes

—

C₄₇×#389

—

C₄₇×#384

C₂²

yes

—

C₄₇×#382

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

#13094024

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

#13093957

—

#13093955

—

#13093935

D₄

yes

—

#13093935

Q₈

yes

complemented

C₄×C₄₇⋊C₈

Q₈

yes

complemented

C₄₇⋊#95

—

C₄₇⋊#95

—

#125×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

C₂×D₄

yes

—

C₂×C₄₇⋊C₈

C₂×Q₈

yes

—

C₂×C₄₇⋊C₈

C₂×Q₈

yes

—

C₂³×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

—

C₂×C₄×C₄₇

C₂²×Q₈

yes

—

C₂×C₄×C₄₇

C₂²×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₄₇⋊C₈

C₄⋊Q₈

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₄₇

#575

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊Q₈

yes

—

Dic₄₇

—

C₂×#501

—

#2862

—

#2859

D₄₇

yes

characteristic

#1847

—

#1846

—

C₄×#97

—

C₂×C₄₇

#2868

yes

characteristic

C₂×C₄₇

#3153

yes

—

C₂×C₄₇

#3162

yes

—

100

C₂×C₄₇

#3166

yes

—

101

C₂×C₄⋊Q₈

—

102

C₂×#125

C₂×D₄₇

yes

—

103

#501

—

104

C₂×#107

—

105

C₂×#106

—

106

C₂×#97

—

107

C₂×#97

—

108

C₂×C₄×C₈

—

109

#391

—

110

#389

—

111

#384

C₂×D₄₇

yes

—

112

#382

—

113

#382

—

114

#382

C₂×D₄₇

yes

—

115

C₄³

C₂×D₄₇

yes

characteristic

116

C₄₇

#33317

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

—

118

C₂²×C₈

—

119

C₄⋊Q₈

—

120

#125

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

C₂²×D₄₇

yes

—

126

C₂×C₄²

C₂²⋊D₄₇

yes

—

127

C₂×C₄²

C₂²×D₄₇

yes

characteristic

128

C₂×C₄²

C₂²×D₄₇

yes

characteristic

129

C₂×C₄²

C₂²×D₄₇

yes

characteristic

130

#107

—

131

#106

—

132

#97

—

133

#97

—

134

C₄×C₈

—

135

#95

—

136

#95

—

137

#95

—

138

C₂×Q₈

—

188

—

139

C₂×Q₈

—

188

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

144

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

145

C₂²×C₄

C₂³×D₄₇

yes

characteristic

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

188

—

155

C₄²

—

156

C₄²

D₄×D₄₇

yes

—

157

C₄²

Q₈×D₄₇

yes

—

158

C₄²

Q₈×D₄₇

yes

—

159

C₂³

C₂²×C₂²⋊D₄₇

yes

characteristic

160

Q₈

—

376

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

188

—

167

C₂×C₄

—

188

—

168

C₂×C₄

#13094020

yes

—

169

C₂×C₄

#13094022

yes

—

170

C₂×C₄

#13094022

yes

—

171

C₂×C₄

C₂×D₄×D₄₇

yes

—

172

C₂×C₄

C₂×Q₈×D₄₇

yes

—

173

C₂×C₄

C₂×Q₈×D₄₇

yes

—

174

C₈

—

175

C₂²

C₂×#13094020

yes

characteristic

176

C₂²

C₂×#13094022

yes

characteristic

177

C₂²

C₂×#13094022

yes

characteristic

178

C₂²

#15689921

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

376

—

184

C₄

C₄⋊Q₈×D₄₇

yes

—

185

C₂

#20251784

yes

characteristic

186

C₂

#20253856

yes

—

187

C₂

#20253910

yes

—

188

C₂

#20253937

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#2859	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#2859	D₄₇
hypocenter	C₄₇	#33317
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂²×C₂²⋊D₄₇

commutator, cosocle

C₂²×C₄×C₄₇

C₂⁴

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#2859

C₂

socle, prime radical

C₂³×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
30. #13093935	Q₈	semidirect
31. C₄×C₄₇⋊C₈	Q₈	semidirect
116. C₄₇	#33317	semidirect

H ≅ G/N

type

30. #13093935

Q₈

semidirect

31. C₄×C₄₇⋊C₈

Q₈

semidirect

116. C₄₇

#33317

semidirect

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

subgroup	index
C₂×#15689533	2
#20251766	2
#20243554	2
#20243550	2
C₄×#13093935	2
C₄₇×#2859	2
#33317	47

subgroup

index

C₄₇×#2859

Subgroups — #26368882

Conjugacy classes of subgroups

Complements

Maximal subgroups