Group #27085497 — subgroups

Subgroups — C₂²×#15689491 #27085497

Order 12032.

2142 conjugacy classes of subgroups; only the 747 normal subgroups are shown.

normal subgroups only

This group has 2142 conjugacy classes of subgroups — too many to draw. Only its 747 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	C₂²×#13094016	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689491	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13093948	C₂	1	4	yes	maximal
4	C₂²×#13093935	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×#13094016	—	2	6	no	—
8	C₂×#13094016	C₂²	1	12	yes	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
11	#15689491	C₂²	1	64	yes	direct product factor
12	C₂×#13093948	C₂²	1	48	yes	—
13	C₂×#13093935	C₂²	1	12	yes	—
14	C₂³×Q₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₄₇	—	2	6	no	—
16	C₂×C₄⋊Q₈×C₄₇	C₂²	1	12	yes	—
17	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×Dic₄₇	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	8	no	—
21	C₂²×C₄₇⋊C₈	C₂³	1	12	yes	—
22	#13094016	—	2	24	no	—
23	#13094016	C₂³	1	16	yes	—
24	C₂×C₄⋊Dic₄₇	—	2	6	no	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
26	C₂×Dic₄₇⋊C₄	—	2	24	no	—
27	#13093948	C₂³	1	64	yes	—
28	#13093935	C₂³	1	16	yes	—
29	C₂²×Q₈×C₄₇	—	2	6	no	—
30	C₂²×Q₈×C₄₇	—	2	16	no	—
31	C₂²×Q₈×C₄₇	C₂³	1	12	yes	—
32	C₂³×C₄×C₄₇	—	2	2	no	—
33	C₂³×C₄×C₄₇	D₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₄₇	—	2	24	no	—
36	C₄⋊Q₈×C₄₇	C₂³	1	16	yes	—
37	C₂×C₄⋊C₄×C₄₇	—	2	24	no	—
38	C₂×C₄²×C₄₇	D₄	1	6	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	6	yes	—
40	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Q₈×C₄₇	—	2	12	no	—
42	C₂×Q₈×C₄₇	—	2	48	no	—
43	C₂×Q₈×C₄₇	—	4	24	no	—
44	C₂×Q₈×C₄₇	C₂⁴	1	8	yes	—
45	C₂²×C₄×C₄₇	—	2	8	no	—
46	C₂²×C₄×C₄₇	—	2	12	no	—
47	C₂²×C₄×C₄₇	—	4	8	no	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
49	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
50	C₂²×C₄×C₄₇	#41	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
52	C₂²×Dic₄₇	—	2	12	no	—
53	C₂²×Dic₄₇	—	4	8	no	—
54	C₄⋊C₄×C₄₇	—	2	32	no	—
55	C₄²×C₄₇	C₂×D₄	1	12	yes	—
56	C₄²×C₄₇	C₂⁴	1	4	yes	—
57	C₄⋊Dic₄₇	—	2	12	no	—
58	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
59	Dic₄₇⋊C₄	—	2	32	no	—
60	C₂×C₄₇⋊C₈	—	2	24	no	—
61	C₂×C₄₇⋊C₈	C₂⁴	1	8	yes	—
62	C₂³×C₄₇	#124	1	4	yes	—
63	C₂³×C₄₇	#137	1	8	yes	—
64	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
65	Q₈×C₄₇	—	2	16	no	—
66	Q₈×C₄₇	—	4	24	no	—
67	C₂×C₄×C₄₇	—	2	24	no	—
68	C₂×C₄×C₄₇	—	2	8	no	—
69	C₂×C₄×C₄₇	—	4	24	no	—
70	C₂×C₄×C₄₇	#124	1	12	yes	—
71	C₂×C₄×C₄₇	C₂²×D₄	1	8	yes	—
72	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
73	C₂×C₄×C₄₇	C₂×#41	1	12	yes	—
74	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
75	C₂×Dic₄₇	—	2	8	no	—
76	C₂×Dic₄₇	—	4	24	no	—
77	C₄₇⋊C₈	—	2	8	no	—
78	C₂²×#491	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	#491	1	16	yes	complemented
80	C₂²×C₄₇	C₂×#124	1	6	yes	—
81	C₂²×C₄₇	C₂×#137	1	12	yes	—
82	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
83	C₄×C₄₇	—	2	8	no	—
84	C₄×C₄₇	—	4	8	no	—
85	C₄×C₄₇	C₂×#124	1	6	yes	—
86	C₄×C₄₇	C₂²×#41	1	2	yes	—
87	Dic₄₇	—	4	8	no	—
88	C₂²×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
89	C₂²×#125	—	47	1	no	—
90	C₂×#491	—	47	24	no	—
91	C₂²×#103	—	47	4	no	—
92	C₂²×#97	—	47	1	no	—
93	C₂×C₄₇	C₂×#491	1	12	yes	complemented
94	C₂×C₄₇	C₂²×#124	1	1	yes	characteristic
95	C₂×C₄₇	C₂²×#137	1	2	yes	—
96	C₂³×Q₈	C₂×D₄₇	1	2	yes	—
97	C₂³×C₈	—	47	2	no	—
98	C₂×C₄⋊Q₈	—	2	6	no	—
99	C₂×C₄⋊Q₈	C₂×D₄₇	1	12	yes	—
100	C₂×#125	—	47	12	no	—
101	C₂×#125	—	94	6	no	—
102	C₂²×C₄⋊C₄	—	2	2	no	—
103	C₂²×C₄⋊C₄	—	47	2	no	—
104	C₂²×C₄⋊C₄	—	94	2	no	—
105	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
106	#491	—	47	64	no	—
107	C₂×#103	—	47	48	no	—
108	C₂×#97	—	47	12	no	—
109	C₄₇	C₂²×#491	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	—	2	6	no	—
111	C₂²×Q₈	—	2	16	no	—
112	C₂²×Q₈	C₂²×D₄₇	1	12	yes	—
113	C₂³×C₄	—	2	2	no	—
114	C₂³×C₄	—	94	2	no	—
115	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
116	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
117	C₂²×C₈	—	47	12	no	—
118	C₂²×C₈	—	94	8	no	—
119	C₄⋊Q₈	—	2	24	no	—
120	C₄⋊Q₈	C₂²×D₄₇	1	16	yes	—
121	#125	—	47	16	no	—
122	#125	—	94	24	no	—
123	C₂×C₄⋊C₄	—	2	24	no	—
124	C₂×C₄⋊C₄	—	47	12	no	—
125	C₂×C₄⋊C₄	—	94	6	no	—
126	C₂×C₄⋊C₄	—	94	24	no	—
127	C₂×C₄²	C₂²⋊D₄₇	1	6	yes	—
128	C₂×C₄²	C₂²×D₄₇	1	6	yes	—
129	#103	—	47	64	no	—
130	#97	—	47	16	no	—
131	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₂×Q₈	—	2	12	no	—
133	C₂×Q₈	—	2	48	no	—
134	C₂×Q₈	—	4	24	no	—
135	C₂×Q₈	C₂³×D₄₇	1	8	yes	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	2	12	no	—
138	C₂²×C₄	—	4	8	no	—
139	C₂²×C₄	—	94	12	no	—
140	C₂²×C₄	—	188	8	no	—
141	C₂²×C₄	#10625352	1	8	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
144	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
145	C₂×C₈	—	47	8	no	—
146	C₂×C₈	—	94	24	no	—
147	C₄⋊C₄	—	2	32	no	—
148	C₄⋊C₄	—	47	8	no	—
149	C₄⋊C₄	—	94	32	no	—
150	C₄⋊C₄	—	94	12	no	—
151	C₄²	C₂×C₂²⋊D₄₇	1	12	yes	—
152	C₄²	C₂³×D₄₇	1	4	yes	—
153	C₂³	#13094011	1	4	yes	—
154	C₂³	#13094055	1	8	yes	—
155	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
156	Q₈	—	2	16	no	—
157	Q₈	—	4	24	no	—
158	C₂×C₄	—	2	24	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	24	no	—
161	C₂×C₄	—	94	8	no	—
162	C₂×C₄	—	188	24	no	—
163	C₂×C₄	#13094011	1	12	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₄₇	1	8	yes	—
165	C₂×C₄	C₂²×C₂²⋊D₄₇	1	3	yes	—
166	C₂×C₄	C₂×#10625352	1	12	yes	—
167	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
168	C₈	—	94	8	no	—
169	C₂²	#15689491	1	16	yes	direct product factor
170	C₂²	C₂×#13094011	1	6	yes	—
171	C₂²	C₂×#13094055	1	12	yes	—
172	C₂²	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
173	C₄	—	2	8	no	—
174	C₄	—	4	8	no	—
175	C₄	—	188	8	no	—
176	C₄	C₂×#13094011	1	6	yes	—
177	C₄	C₂²×#10625352	1	2	yes	—
178	C₂	C₂×#15689491	1	12	yes	direct product factor
179	C₂	C₂²×#13094011	1	1	yes	characteristic
180	C₂	C₂²×#13094055	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13094016

C₂

yes

maximal, characteristic

C₂×#15689491

C₂

yes

maximal, direct product factor

C₂²×#13093948

C₂

yes

maximal

C₂²×#13093935

C₂

yes

maximal, characteristic

C₂²×C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂³×C₄₇⋊C₈

C₂²

yes

—

C₂×#13094016

—

C₂×#13094016

C₂²

yes

—

C₂²×C₄⋊Dic₄₇

C₂²

yes

—

C₂²×Dic₄₇⋊C₄

—

#15689491

C₂²

yes

direct product factor

C₂×#13093948

C₂²

yes

—

C₂×#13093935

C₂²

yes

—

C₂³×Q₈×C₄₇

C₂²

yes

—

C₂×C₄⋊Q₈×C₄₇

—

C₂×C₄⋊Q₈×C₄₇

C₂²

yes

—

C₂²×C₄⋊C₄×C₄₇

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×Dic₄₇

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

C₂³

yes

—

#13094016

—

#13094016

C₂³

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

C₂×Dic₄₇⋊C₄

—

#13093948

C₂³

yes

—

#13093935

C₂³

yes

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

C₂³

yes

—

C₂³×C₄×C₄₇

—

C₂³×C₄×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂³

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

—

C₂⁴×C₄₇

C₂×D₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂⁴

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂⁴

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂⁴

yes

—

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂⁴

yes

—

C₂³×C₄₇

#124

yes

—

C₂³×C₄₇

#137

yes

—

C₂³×C₄₇

C₂²×D₄

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×C₄×C₄₇

C₂⁵

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×#491

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#137

yes

—

C₂²×C₄₇

C₂³×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#124

yes

—

C₄×C₄₇

C₂²×#41

yes

—

Dic₄₇

—

C₂²×C₄⋊Q₈

D₄₇

yes

characteristic

C₂²×#125

—

C₂×#491

—

C₂²×#103

—

C₂²×#97

—

C₂×C₄₇

C₂×#491

yes

complemented

C₂×C₄₇

C₂²×#124

yes

characteristic

C₂×C₄₇

C₂²×#137

yes

—

C₂³×Q₈

C₂×D₄₇

yes

—

C₂³×C₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

C₂×D₄₇

yes

—

100

C₂×#125

—

101

C₂×#125

—

102

C₂²×C₄⋊C₄

—

103

C₂²×C₄⋊C₄

—

104

C₂²×C₄⋊C₄

—

105

C₂²×C₄²

C₂×D₄₇

yes

characteristic

106

#491

—

107

C₂×#103

—

108

C₂×#97

—

109

C₄₇

C₂²×#491

yes

characteristic, complemented, Sylow

110

C₂²×Q₈

—

111

C₂²×Q₈

—

112

C₂²×Q₈

C₂²×D₄₇

yes

—

113

C₂³×C₄

—

114

C₂³×C₄

—

115

C₂³×C₄

C₂²⋊D₄₇

yes

—

116

C₂³×C₄

C₂²×D₄₇

yes

characteristic

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₄⋊Q₈

—

120

C₄⋊Q₈

C₂²×D₄₇

yes

—

121

#125

—

122

#125

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄⋊C₄

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

C₂²⋊D₄₇

yes

—

128

C₂×C₄²

C₂²×D₄₇

yes

—

129

#103

—

130

#97

—

131

C₂⁴

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

C₂³×D₄₇

yes

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

188

—

141

C₂²×C₄

#10625352

yes

—

142

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

144

C₂²×C₄

C₂³×D₄₇

yes

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄²

C₂×C₂²⋊D₄₇

yes

—

152

C₄²

C₂³×D₄₇

yes

—

153

C₂³

#13094011

yes

—

154

C₂³

#13094055

yes

—

155

C₂³

C₂²×C₂²⋊D₄₇

yes

—

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

188

—

163

C₂×C₄

#13094011

yes

—

164

C₂×C₄

C₂²×C₂²⋊D₄₇

yes

—

165

C₂×C₄

C₂²×C₂²⋊D₄₇

yes

—

166

C₂×C₄

C₂×#10625352

yes

—

167

C₂×C₄

C₂⁴×D₄₇

yes

characteristic

168

C₈

—

169

C₂²

#15689491

yes

direct product factor

170

C₂²

C₂×#13094011

yes

—

171

C₂²

C₂×#13094055

yes

—

172

C₂²

C₂³×C₂²⋊D₄₇

yes

characteristic

173

C₄

—

174

C₄

—

175

C₄

—

188

—

176

C₄

C₂×#13094011

yes

—

177

C₄

C₂²×#10625352

yes

—

178

C₂

C₂×#15689491

yes

direct product factor

179

C₂

C₂²×#13094011

yes

characteristic

180

C₂

C₂²×#13094055

yes

—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting	C₂²×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	C₂²×#491
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

subgroup

G/H

center

C₂⁴

C₂×C₂²⋊D₄₇

commutator, cosocle

C₂×C₄×C₄₇

C₂⁵

Frattini

C₂×C₄

C₂⁴×D₄₇

Fitting

C₂²×C₄⋊Q₈×C₄₇

C₂

socle, prime radical

C₂⁴×C₄₇

C₂×D₄

hypercenter

C₂²×C₄⋊Q₈

D₄₇

hypocenter

C₄₇

C₂²×#491

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×#15689491	C₂	direct (N × H)
11. #15689491	C₂²	direct (N × H)
79. C₂²×C₄₇	#491	semidirect
93. C₂×C₄₇	C₂×#491	semidirect
109. C₄₇	C₂²×#491	semidirect
169. C₂²	#15689491	direct (N × H)
178. C₂	C₂×#15689491	direct (N × H)

H ≅ G/N

type

2. C₂×#15689491

C₂

direct (N × H)

11. #15689491

C₂²

direct (N × H)

79. C₂²×C₄₇

#491

semidirect

93. C₂×C₄₇

C₂×#491

semidirect

109. C₄₇

C₂²×#491

semidirect

169. C₂²

#15689491

direct (N × H)

178. C₂

C₂×#15689491

direct (N × H)

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

subgroup	index
C₂²×#13094016	2
C₂×#15689491	2
C₂²×#13093948	2
C₂²×#13093935	2
C₂²×C₄⋊Q₈×C₄₇	2
C₂²×#491	47

subgroup

index

C₂²×C₄⋊Q₈×C₄₇

C₂²×#491

Subgroups — C₂²×#15689491 #27085497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×#15689491 #27085497

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689491 #27085497