Group #26116308 — subgroups

Subgroups — #26116308

Order 12032.

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

normal subgroups only

This group has 486 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×#13094016	C₂	1	1	yes	maximal, characteristic
2	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
3	#20241227	C₂	1	4	yes	maximal
4	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	C₄	1	1	yes	characteristic
6	C₂×#13093935	—	2	2	no	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15689021	C₄	1	2	yes	—
9	#15689007	—	2	2	no	—
10	#15688989	C₄	1	1	yes	characteristic
11	C₄×Dic₄₇⋊C₄	—	2	2	no	—
12	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
14	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
15	C₄³×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	#13094016	—	2	4	no	—
18	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×Dic₄₇⋊C₄	—	2	2	no	—
21	C₂×Dic₄₇⋊C₄	—	2	2	no	—
22	C₂×C₄×Dic₄₇	—	2	2	no	—
23	#13093935	—	2	8	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
26	C₄₇⋊#95	—	2	2	no	—
27	C₄₇⋊#95	—	2	2	no	—
28	C₄₇⋊#95	—	2	2	no	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₇	—	2	4	no	—
36	C₄⋊C₈×C₄₇	—	4	4	no	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₄₇	—	2	2	no	—
43	C₂²×Dic₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₂×C₈×C₄₇	—	4	4	no	—
46	C₄²×C₄₇	—	2	4	no	—
47	C₄²×C₄₇	—	2	2	no	—
48	C₄²×C₄₇	—	2	2	no	—
49	C₄²×C₄₇	—	2	2	no	—
50	C₄²×C₄₇	—	2	2	no	—
51	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
52	C₄²×C₄₇	#41	1	2	yes	—
53	C₄⋊Dic₄₇	—	2	2	no	—
54	C₄⋊Dic₄₇	—	2	4	no	—
55	C₄⋊Dic₄₇	M₁₆	1	4	yes	—
56	Dic₄₇⋊C₄	—	2	8	no	—
57	Dic₄₇⋊C₄	—	4	4	no	—
58	C₄×Dic₄₇	—	4	2	no	—
59	C₄×Dic₄₇	—	4	2	no	—
60	C₂×C₄₇⋊C₈	—	2	8	no	—
61	C₂×C₄₇⋊C₈	—	2	4	no	—
62	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	2	2	no	—
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	#104	1	4	yes	—
70	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₄₇	#131	1	2	yes	—
76	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₄₇	#137	1	2	yes	—
78	C₂×Dic₄₇	—	2	4	no	—
79	C₂×Dic₄₇	—	4	2	no	—
80	C₂×Dic₄₇	—	4	2	no	—
81	C₂×Dic₄₇	—	4	2	no	—
82	C₂×Dic₄₇	—	4	2	no	—
83	C₂×Dic₄₇	—	4	2	no	—
84	C₈×C₄₇	—	8	2	no	—
85	C₄₇⋊C₈	—	2	8	no	—
86	#9876	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	#391	1	1	yes	characteristic
88	C₂²×C₄₇	#407	1	1	yes	characteristic
89	C₂²×C₄₇	#408	1	1	yes	characteristic
90	C₂²×C₄₇	#417	1	2	yes	—
91	C₂²×C₄₇	C₂×#104	1	2	yes	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	2	4	no	—
94	C₄×C₄₇	—	4	2	no	—
95	C₄×C₄₇	#446	1	4	yes	—
96	C₄×C₄₇	#490	1	1	yes	characteristic
97	C₄×C₄₇	#491	1	1	yes	characteristic
98	C₄×C₄₇	#501	1	2	yes	—
99	Dic₄₇	—	4	4	no	—
100	Dic₄₇	—	8	2	no	—
101	C₄×#125	—	47	1	no	—
102	#1842	D₄₇	1	1	yes	characteristic
103	C₄×#97	—	47	1	no	—
104	#1279	—	47	4	no	—
105	C₂×C₄₇	#1483	1	2	yes	—
106	C₂×C₄₇	#1489	1	2	yes	—
107	C₂×C₄₇	#1985	1	1	yes	characteristic
108	C₂×C₄₇	#1992	1	1	yes	characteristic
109	C₂×C₄₇	#1994	1	1	yes	characteristic
110	C₂×#125	—	47	1	no	—
111	C₂×C₄⋊C₈	—	2	2	no	—
112	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
113	C₂×#97	—	94	2	no	—
114	C₂×C₄×C₈	—	47	2	no	—
115	#389	—	47	2	no	—
116	#387	—	94	2	no	—
117	#382	—	47	1	no	—
118	C₄×C₄⋊C₄	—	47	2	no	—
119	C₄×C₄⋊C₄	—	94	2	no	—
120	C₄³	C₂×D₄₇	1	1	yes	characteristic
121	C₄₇	#9876	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	94	2	no	—
124	#125	—	94	4	no	—
125	C₂×C₄⋊C₄	—	47	2	no	—
126	C₂×C₄⋊C₄	—	47	2	no	—
127	C₂×C₄⋊C₄	—	94	2	no	—
128	C₂×C₄⋊C₄	—	94	2	no	—
129	C₂×C₄²	—	94	2	no	—
130	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
131	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
132	C₂×C₄²	D₁₈₈	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
134	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
135	C₄⋊C₈	—	2	4	no	—
136	C₄⋊C₈	—	4	4	no	—
137	#97	—	94	8	no	—
138	C₄×C₈	—	47	4	no	—
139	C₄×C₈	—	94	2	no	—
140	#95	—	94	2	no	—
141	#95	—	94	2	no	—
142	#95	—	94	2	no	—
143	C₂²×C₄	—	94	2	no	—
144	C₂²×C₄	—	94	2	no	—
145	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
146	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	94	8	no	—
153	C₂×C₈	—	94	4	no	—
154	C₄⋊C₄	—	47	4	no	—
155	C₄⋊C₄	—	94	8	no	—
156	C₄⋊C₄	—	94	2	no	—
157	C₄⋊C₄	—	94	4	no	—
158	C₄⋊C₄	—	188	4	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	188	2	no	—
165	C₄²	—	188	2	no	—
166	C₄²	#10625352	1	2	yes	—
167	C₄²	Q₈×D₄₇	1	2	yes	—
168	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
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	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	94	4	no	—
176	C₂×C₄	—	188	2	no	—
177	C₂×C₄	—	188	2	no	—
178	C₂×C₄	—	188	2	no	—
179	C₂×C₄	—	188	2	no	—
180	C₂×C₄	—	188	2	no	—
181	C₂×C₄	#13093950	1	4	yes	—
182	C₂×C₄	#13093989	1	2	yes	—
183	C₂×C₄	#13094004	1	2	yes	—
184	C₂×C₄	#13094011	1	1	yes	characteristic
185	C₂×C₄	#13094022	1	1	yes	characteristic
186	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
187	C₂×C₄	#13094030	1	2	yes	—
188	C₂×C₄	#13094050	1	2	yes	—
189	C₂×C₄	#13094055	1	2	yes	—
190	C₈	—	8	2	no	—
191	C₈	—	94	8	no	—
192	C₂²	#15689028	1	1	yes	characteristic
193	C₂²	#15689086	1	1	yes	characteristic
194	C₂²	#15689089	1	1	yes	characteristic
195	C₂²	#15689122	1	2	yes	—
196	C₂²	C₂×#13093950	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	376	2	no	—
201	C₄	—	188	4	no	—
202	C₄	#15689252	1	4	yes	—
203	C₄	#15689487	1	1	yes	characteristic
204	C₄	#15689491	1	1	yes	characteristic
205	C₄	#15689529	1	2	yes	—
206	C₂	#20241799	1	2	yes	—
207	C₂	#20241825	1	2	yes	—
208	C₂	#20244327	1	1	yes	characteristic
209	C₂	#20244356	1	1	yes	characteristic
210	C₂	#20244364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13094016

C₂

yes

maximal, characteristic

C₄×#13093935

C₂

yes

maximal, characteristic

#20241227

C₂

yes

maximal

C₄₇×#1842

C₂

yes

maximal, characteristic

C₂×#13094016

C₄

yes

characteristic

C₂×#13093935

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15689021

C₄

yes

—

#15689007

—

#15688989

C₄

yes

characteristic

C₄×Dic₄₇⋊C₄

—

C₄×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

#13094016

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₄₇⋊#95

—

C₄₇⋊#95

—

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

—

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×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

M₁₆

yes

—

Dic₄₇⋊C₄

—

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×Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

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

#446

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

100

Dic₄₇

—

101

C₄×#125

—

102

#1842

D₄₇

yes

characteristic

103

C₄×#97

—

104

#1279

—

105

C₂×C₄₇

#1483

yes

—

106

C₂×C₄₇

#1489

yes

—

107

C₂×C₄₇

#1985

yes

characteristic

108

C₂×C₄₇

#1992

yes

characteristic

109

C₂×C₄₇

#1994

yes

characteristic

110

C₂×#125

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

113

C₂×#97

—

114

C₂×C₄×C₈

—

115

#389

—

116

#387

—

117

#382

—

118

C₄×C₄⋊C₄

—

119

C₄×C₄⋊C₄

—

120

C₄³

C₂×D₄₇

yes

characteristic

121

C₄₇

#9876

yes

characteristic, complemented, Sylow

122

C₂²×C₈

—

123

C₂²×C₈

—

124

#125

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄²

—

130

C₂×C₄²

C₄×D₄₇

yes

characteristic

131

C₂×C₄²

C₄×D₄₇

yes

characteristic

132

C₂×C₄²

D₁₈₈

yes

—

133

C₂×C₄²

C₂²⋊D₄₇

yes

—

134

C₂×C₄²

C₂²×D₄₇

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

#97

—

138

C₄×C₈

—

139

C₄×C₈

—

140

#95

—

141

#95

—

142

#95

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

146

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

147

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

148

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

149

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

188

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

188

—

165

C₄²

—

188

—

166

C₄²

#10625352

yes

—

167

C₄²

Q₈×D₄₇

yes

—

168

C₂³

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

188

—

177

C₂×C₄

—

188

—

178

C₂×C₄

—

188

—

179

C₂×C₄

—

188

—

180

C₂×C₄

—

188

—

181

C₂×C₄

#13093950

yes

—

182

C₂×C₄

#13093989

yes

—

183

C₂×C₄

#13094004

yes

—

184

C₂×C₄

#13094011

yes

characteristic

185

C₂×C₄

#13094022

yes

characteristic

186

C₂×C₄

M₁₆×D₄₇

yes

—

187

C₂×C₄

#13094030

yes

—

188

C₂×C₄

#13094050

yes

—

189

C₂×C₄

#13094055

yes

—

190

C₈

—

191

C₈

—

192

C₂²

#15689028

yes

characteristic

193

C₂²

#15689086

yes

characteristic

194

C₂²

#15689089

yes

characteristic

195

C₂²

#15689122

yes

—

196

C₂²

C₂×#13093950

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

376

—

201

C₄

—

188

—

202

C₄

#15689252

yes

—

203

C₄

#15689487

yes

characteristic

204

C₄

#15689491

yes

characteristic

205

C₄

#15689529

yes

—

206

C₂

#20241799

yes

—

207

C₂

#20241825

yes

—

208

C₂

#20244327

yes

characteristic

209

C₂

#20244356

yes

characteristic

210

C₂

#20244364

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#9876
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1842

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 ◁ #1842

N	H ≅ G/N	type
121. C₄₇	#9876	semidirect

H ≅ G/N

type

121. C₄₇

#9876

semidirect

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

subgroup	index
C₄×#13094016	2
C₄×#13093935	2
#20241227	2
C₄₇×#1842	2
#9876	47

subgroup

index

C₄₇×#1842

Subgroups — #26116308

Conjugacy classes of subgroups

Complements

Maximal subgroups