Group #26244411 — subgroups

Subgroups — #26244411

Order 12032.

1162 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 291 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	#20245870	C₂	1	3	yes	maximal
2	#20244740	C₂	1	8	yes	maximal
3	#20243121	C₂	1	3	yes	maximal
4	C₄₇×#3458	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093983	C₂²	1	3	yes	—
6	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
7	#15689019	—	2	6	no	—
8	#15688995	C₂²	1	12	yes	—
9	C₄×Dic₄₇⋊C₄	—	2	6	no	—
10	#15688890	C₂²	1	12	yes	—
11	C₄₇×#544	—	2	6	no	—
12	C₄₇×#543	C₂²	1	4	yes	—
13	C₄₇×#533	—	2	6	no	—
14	C₂×#117×C₄₇	C₂²	1	3	yes	—
15	C₂³×Dic₄₇	C₂³	1	2	yes	—
16	#13093983	—	2	6	no	—
17	#13093983	Q₈	1	12	yes	complemented
18	C₂×C₄⋊Dic₄₇	—	2	3	no	—
19	C₂×Dic₄₇⋊C₄	—	2	12	no	—
20	C₂×C₂²⋊Dic₄₇	C₂³	1	6	yes	—
21	C₂×C₄×Dic₄₇	—	2	3	no	—
22	C₂×C₄×Dic₄₇	—	2	3	no	—
23	C₂×C₄×Dic₄₇	—	2	3	no	—
24	C₄₇⋊#95	—	2	12	no	—
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₄⋊Q₈×C₄₇	—	2	12	no	—
27	#125×C₄₇	—	2	12	no	—
28	C₂²⋊Q₈×C₄₇	—	2	24	no	—
29	C₄×Q₈×C₄₇	—	2	8	no	—
30	#117×C₄₇	—	2	6	no	—
31	#117×C₄₇	—	2	6	no	—
32	C₂×C₄⋊C₄×C₄₇	D₄	1	6	yes	—
33	C₂×C₂²⋊C₄×C₄₇	C₂³	1	6	yes	—
34	C₂×C₄²×C₄₇	D₄	1	6	yes	—
35	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
36	C₂×Q₈×C₄₇	—	2	8	no	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	12	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
40	C₂²×C₄×C₄₇	C₂×Q₈	1	6	yes	—
41	C₂²×C₄×C₄₇	#41	1	2	yes	—
42	C₂²×Dic₄₇	—	2	6	no	—
43	C₂²×Dic₄₇	—	2	6	no	—
44	C₂²×Dic₄₇	—	4	4	no	—
45	C₄⋊C₄×C₄₇	—	2	24	no	—
46	C₄⋊C₄×C₄₇	—	2	12	no	—
47	C₂²⋊C₄×C₄₇	—	2	12	no	—
48	C₄²×C₄₇	—	2	6	no	—
49	C₄²×C₄₇	—	2	6	no	—
50	C₂²⋊Dic₄₇	—	2	6	no	—
51	C₂²⋊Dic₄₇	C₂×Q₈	1	12	yes	—
52	C₄⋊Dic₄₇	—	2	12	no	—
53	Dic₄₇⋊C₄	—	4	24	no	—
54	C₄×Dic₄₇	—	2	12	no	—
55	C₄×Dic₄₇	—	4	3	no	—
56	C₄×Dic₄₇	—	4	1	no	—
57	C₂³×C₄₇	—	4	2	no	—
58	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
59	C₂³×C₄₇	C₂²×Q₈	1	3	yes	—
60	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
61	Q₈×C₄₇	—	8	8	no	—
62	C₂×C₄×C₄₇	—	2	12	no	—
63	C₂×C₄×C₄₇	—	4	12	no	—
64	C₂×C₄×C₄₇	#120	1	3	yes	—
65	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂²⋊Q₈	1	6	yes	—
67	C₂×C₄×C₄₇	C₂²⋊Q₈	1	6	yes	—
68	C₂×C₄×C₄₇	C₄⋊Q₈	1	6	yes	—
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	6	yes	—
70	C₂×Dic₄₇	—	2	12	no	—
71	C₂×Dic₄₇	—	2	12	no	—
72	C₂×Dic₄₇	—	4	12	no	—
73	C₂×Dic₄₇	—	4	6	no	—
74	C₂×Dic₄₇	—	4	2	no	—
75	#23564	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	—	2	6	no	—
77	C₂²×C₄₇	—	4	3	no	—
78	C₂²×C₄₇	—	4	1	no	—
79	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
80	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	3	yes	—
81	C₂²×C₄₇	C₂×C₄⋊Q₈	1	3	yes	—
82	C₄×C₄₇	—	4	12	no	—
83	C₄×C₄₇	#395	1	6	yes	—
84	C₄×C₄₇	#395	1	2	yes	—
85	Dic₄₇	—	4	12	no	—
86	Dic₄₇	—	4	4	no	—
87	#3458	D₄₇	1	1	yes	characteristic
88	#2296	—	47	3	no	—
89	#2062	—	47	8	no	—
90	#1740	—	47	3	no	—
91	C₂×C₄₇	—	2	3	no	—
92	C₂×C₄₇	—	2	3	no	—
93	C₂×C₄₇	C₂×#395	1	1	yes	characteristic
94	C₂×C₄₇	#3025	1	2	yes	—
95	#544	—	2	6	no	—
96	#543	C₂×D₄₇	1	4	yes	—
97	#533	—	2	6	no	—
98	C₂×#117	—	47	3	no	—
99	C₂×#117	C₂×D₄₇	1	3	yes	—
100	C₂²×C₄²	—	47	1	no	—
101	#389	—	94	6	no	—
102	#385	—	47	12	no	—
103	C₄×C₄⋊C₄	—	94	6	no	—
104	#342	—	47	12	no	—
105	C₄₇	#23564	1	1	yes	characteristic, complemented, Sylow
106	C₂³×C₄	—	47	2	no	—
107	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
108	C₄⋊Q₈	—	2	12	no	—
109	#125	—	2	12	no	—
110	C₂²⋊Q₈	—	2	24	no	—
111	C₄×Q₈	—	2	8	no	—
112	#117	—	2	6	no	—
113	#117	—	2	6	no	—
114	#117	—	47	12	no	—
115	#117	—	94	6	no	—
116	C₂×C₄⋊C₄	—	94	3	no	—
117	C₂×C₄⋊C₄	—	94	12	no	—
118	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	6	yes	—
119	C₂×C₂²⋊C₄	—	47	6	no	—
120	C₂×C₂²⋊C₄	C₂²×D₄₇	1	6	yes	—
121	C₂×C₄²	—	94	3	no	—
122	C₂×C₄²	—	94	3	no	—
123	C₂×C₄²	—	94	3	no	—
124	C₂×C₄²	C₂²⋊D₄₇	1	6	yes	—
125	#95	—	94	12	no	—
126	C₂⁴	C₂³×D₄₇	1	1	yes	characteristic
127	C₂×Q₈	—	2	8	no	—
128	C₂²×C₄	—	94	6	no	—
129	C₂²×C₄	—	94	6	no	—
130	C₂²×C₄	—	188	4	no	—
131	C₂²×C₄	Q₈×D₄₇	1	6	yes	—
132	C₂²×C₄	#10625354	1	2	yes	—
133	C₂²×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
136	C₄⋊C₄	—	2	24	no	—
137	C₄⋊C₄	—	2	12	no	—
138	C₄⋊C₄	—	94	12	no	—
139	C₄⋊C₄	—	188	24	no	—
140	C₂²⋊C₄	—	2	12	no	—
141	C₂²⋊C₄	—	47	12	no	—
142	C₂²⋊C₄	—	94	6	no	—
143	C₄²	—	2	6	no	—
144	C₄²	—	2	6	no	—
145	C₄²	—	94	12	no	—
146	C₄²	—	188	3	no	—
147	C₄²	—	188	1	no	—
148	C₂³	—	4	2	no	—
149	C₂³	C₂²×C₂²⋊D₄₇	1	3	yes	—
150	C₂³	C₂×Q₈×D₄₇	1	3	yes	—
151	C₂³	C₂×#10625354	1	1	yes	characteristic
152	Q₈	—	8	8	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	4	12	no	—
155	C₂×C₄	—	94	12	no	—
156	C₂×C₄	—	94	12	no	—
157	C₂×C₄	—	188	12	no	—
158	C₂×C₄	—	188	6	no	—
159	C₂×C₄	—	188	2	no	—
160	C₂×C₄	#13093993	1	3	yes	—
161	C₂×C₄	#13093993	1	1	yes	characteristic
162	C₂×C₄	#13094000	1	6	yes	—
163	C₂×C₄	#13094000	1	6	yes	—
164	C₂×C₄	#13094022	1	6	yes	—
165	C₂×C₄	#13094022	1	6	yes	—
166	C₂²	—	2	6	no	—
167	C₂²	—	4	3	no	—
168	C₂²	—	4	1	no	—
169	C₂²	C₂×#13093993	1	1	yes	characteristic
170	C₂²	C₂×#13094000	1	3	yes	—
171	C₂²	C₂×#13094022	1	3	yes	—
172	C₄	—	4	12	no	—
173	C₄	—	188	12	no	—
174	C₄	—	188	4	no	—
175	C₄	#15689042	1	6	yes	—
176	C₄	#15689042	1	2	yes	—
177	C₂	—	2	3	no	—
178	C₂	—	2	3	no	—
179	C₂	C₂×#15689042	1	1	yes	characteristic
180	C₂	#20252780	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245870

C₂

yes

maximal

#20244740

C₂

yes

maximal

#20243121

C₂

yes

maximal

C₄₇×#3458

C₂

yes

maximal, characteristic

C₂×#13093983

C₂²

yes

—

C₂²×C₄×Dic₄₇

C₂²

yes

characteristic

#15689019

—

#15688995

C₂²

yes

—

C₄×Dic₄₇⋊C₄

—

#15688890

C₂²

yes

—

C₄₇×#544

—

C₄₇×#543

C₂²

yes

—

C₄₇×#533

—

C₂×#117×C₄₇

C₂²

yes

—

C₂³×Dic₄₇

C₂³

yes

—

#13093983

—

#13093983

Q₈

yes

complemented

C₂×C₄⋊Dic₄₇

—

C₂×Dic₄₇⋊C₄

—

C₂×C₂²⋊Dic₄₇

C₂³

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

—

C₄₇⋊#95

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂⁴

yes

characteristic

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×D₄

yes

—

C₂³×C₄₇

C₂²×Q₈

yes

—

C₂³×C₄₇

C₂×#41

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#23564

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#120

yes

characteristic

C₂²×C₄₇

C₂×C₂²⋊Q₈

yes

—

C₂²×C₄₇

C₂×C₄⋊Q₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

#395

yes

—

C₄×C₄₇

#395

yes

—

Dic₄₇

—

Dic₄₇

—

#3458

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#395

yes

characteristic

C₂×C₄₇

#3025

yes

—

#544

—

#543

C₂×D₄₇

yes

—

#533

—

C₂×#117

—

C₂×#117

C₂×D₄₇

yes

—

100

C₂²×C₄²

—

101

#389

—

102

#385

—

103

C₄×C₄⋊C₄

—

104

#342

—

105

C₄₇

#23564

yes

characteristic, complemented, Sylow

106

C₂³×C₄

—

107

C₂³×C₄

C₂²×D₄₇

yes

characteristic

108

C₄⋊Q₈

—

109

#125

—

110

C₂²⋊Q₈

—

111

C₄×Q₈

—

112

#117

—

113

#117

—

114

#117

—

115

#117

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₂²⋊C₄

C₂²×D₄₇

yes

—

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

—

124

C₂×C₄²

C₂²⋊D₄₇

yes

—

125

#95

—

126

C₂⁴

C₂³×D₄₇

yes

characteristic

127

C₂×Q₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

188

—

131

C₂²×C₄

Q₈×D₄₇

yes

—

132

C₂²×C₄

#10625354

yes

—

133

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

134

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

135

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

188

—

140

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

—

188

—

147

C₄²

—

188

—

148

C₂³

—

149

C₂³

C₂²×C₂²⋊D₄₇

yes

—

150

C₂³

C₂×Q₈×D₄₇

yes

—

151

C₂³

C₂×#10625354

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

188

—

158

C₂×C₄

—

188

—

159

C₂×C₄

—

188

—

160

C₂×C₄

#13093993

yes

—

161

C₂×C₄

#13093993

yes

characteristic

162

C₂×C₄

#13094000

yes

—

163

C₂×C₄

#13094000

yes

—

164

C₂×C₄

#13094022

yes

—

165

C₂×C₄

#13094022

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂×#13093993

yes

characteristic

170

C₂²

C₂×#13094000

yes

—

171

C₂²

C₂×#13094022

yes

—

172

C₄

—

173

C₄

—

188

—

174

C₄

—

188

—

175

C₄

#15689042

yes

—

176

C₄

#15689042

yes

—

177

C₂

—

178

C₂

—

179

C₂

C₂×#15689042

yes

characteristic

180

C₂

#20252780

yes

—

	subgroup	G/H
center	C₂×C₄	#13093993
commutator, prime radical, cosocle	C₂⁴×C₄₇	C₂⁴
Frattini	C₂⁴	C₂³×D₄₇
Fitting	C₄₇×#3458	C₂
socle	C₂²×C₄₇	C₂×#120
hypercenter	#3458	D₄₇
hypocenter	C₄₇	#23564
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

subgroup

G/H

center

C₂×C₄

#13093993

commutator, prime radical, cosocle

C₂⁴×C₄₇

C₂⁴

Frattini

C₂⁴

C₂³×D₄₇

Fitting

C₄₇×#3458

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

N	H ≅ G/N	type
17. #13093983	Q₈	semidirect
105. C₄₇	#23564	semidirect

H ≅ G/N

type

17. #13093983

Q₈

semidirect

105. C₄₇

#23564

semidirect

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

subgroup	index
#20245870	2
#20244740	2
#20243121	2
C₄₇×#3458	2
#23564	47

subgroup

index

C₄₇×#3458

Subgroups — #26244411

Conjugacy classes of subgroups

Complements

Maximal subgroups