Group #26111230 — subgroups

Subgroups — #26111230

Order 12032.

376 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20253910	C₂	1	1	yes	maximal, characteristic
2	#20252494	C₂	1	1	yes	maximal, characteristic
3	#20241862	C₂	1	2	yes	maximal
4	#20241288	C₂	1	2	yes	maximal
5	C₄₇×#1544	C₂	1	1	yes	maximal, characteristic
6	#15689680	C₂²	1	1	yes	characteristic
7	#15689533	C₄	1	4	yes	—
8	#15689529	—	2	2	no	—
9	#15689254	—	2	4	no	—
10	#15689184	—	2	2	no	—
11	#15689180	C₂²	1	2	yes	—
12	#15689159	—	2	2	no	—
13	C₂×#13093935	C₂²	1	1	yes	characteristic
14	#15688855	—	2	4	no	—
15	C₄₇×#533	C₂²	1	1	yes	characteristic
16	C₄₇×#431	C₂²	1	2	yes	—
17	C₄₇×#332	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	1	no	—
19	#13094024	C₂×C₄	1	2	yes	—
20	#13094016	—	2	1	no	—
21	#13094003	—	2	2	no	—
22	C₄×Dic₉₄	—	2	2	no	—
23	#13093983	—	2	1	no	—
24	#13093957	—	2	4	no	—
25	#13093955	—	2	4	no	—
26	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
27	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
28	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
29	#13093935	—	2	2	no	—
30	#13093935	—	2	2	no	—
31	#13093935	—	2	1	no	—
32	#13093935	C₂×C₄	1	2	yes	—
33	C₄×C₄₇⋊C₈	—	2	2	no	—
34	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
35	#125×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²⋊Q₈×C₄₇	—	2	2	no	—
37	C₄×Q₈×C₄₇	—	2	2	no	—
38	#117×C₄₇	D₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₇	—	2	2	no	—
41	C₂²⋊C₈×C₄₇	—	2	2	no	—
42	C₄×C₈×C₄₇	—	2	2	no	—
43	C₂×Q₈×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	#41	1	2	yes	—
46	C₂×Dic₉₄	—	2	2	no	—
47	C₂×C₈×C₄₇	—	4	2	no	—
48	C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₄⋊C₄×C₄₇	—	2	2	no	—
50	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₂²⋊C₄×C₄₇	—	2	2	no	—
52	C₄²×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²⋊Dic₄₇	—	4	1	no	—
57	C₄⋊Dic₄₇	—	2	2	no	—
58	C₄⋊Dic₄₇	—	2	2	no	—
59	Dic₄₇⋊C₄	—	4	2	no	—
60	C₄×Dic₄₇	—	4	1	no	—
61	C₂×C₄₇⋊C₈	—	2	4	no	—
62	C₂×C₄₇⋊C₈	—	2	2	no	—
63	C₂×C₄₇⋊C₈	—	2	4	no	—
64	C₂³×C₄₇	#123	1	1	yes	characteristic
65	Q₈×C₄₇	—	8	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	4	no	—
68	C₂×C₄×C₄₇	—	4	2	no	—
69	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#121	1	2	yes	—
73	C₂×Dic₄₇	—	4	2	no	—
74	Dic₉₄	—	4	4	no	—
75	C₈×C₄₇	—	8	2	no	—
76	C₄₇⋊C₈	—	2	4	no	—
77	C₄₇⋊C₈	—	4	4	no	—
78	#9006	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	—	2	2	no	—
80	C₂²×C₄₇	—	2	1	no	—
81	C₂²×C₄₇	#386	1	1	yes	characteristic
82	C₄×C₄₇	—	2	2	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	2	1	no	—
85	C₄×C₄₇	—	4	4	no	—
86	C₄×C₄₇	#456	1	2	yes	—
87	Dic₄₇	—	8	2	no	—
88	#3162	—	47	1	no	—
89	#2987	—	47	1	no	—
90	#1544	D₄₇	1	1	yes	characteristic
91	#1497	—	47	2	no	—
92	#1302	—	47	2	no	—
93	C₂×C₄₇	—	4	1	no	—
94	C₂×C₄₇	#1881	1	2	yes	—
95	C₂×C₄₇	#1886	1	1	yes	characteristic
96	#533	—	47	1	no	—
97	#533	C₂×D₄₇	1	1	yes	characteristic
98	#501	—	47	4	no	—
99	#501	—	94	2	no	—
100	#446	—	94	4	no	—
101	#433	—	94	2	no	—
102	#432	—	47	2	no	—
103	#431	C₂×D₄₇	1	2	yes	—
104	#427	—	94	2	no	—
105	C₂×#97	—	47	1	no	—
106	#332	—	2	2	no	—
107	Q₈⋊C₈	—	94	4	no	—
108	C₄₇	#9006	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₈	—	94	1	no	—
110	C₄⋊Q₈	—	47	2	no	—
111	C₄⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
112	#125	—	94	1	no	—
113	#125	C₄×D₄₇	1	2	yes	—
114	C₂²⋊Q₈	—	2	2	no	—
115	C₂²⋊Q₈	—	94	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	C₄×Q₈	—	94	2	no	—
118	#117	—	94	1	no	—
119	#117	D₁₈₈	1	2	yes	—
120	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
121	#107	—	94	4	no	—
122	#106	—	94	4	no	—
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	—	47	4	no	—
125	C₄⋊C₈	—	94	4	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	C₂²⋊C₈	—	94	2	no	—
128	#97	—	47	2	no	—
129	#97	—	94	2	no	—
130	#97	—	94	2	no	—
131	#97	—	94	1	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	94	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	94	2	no	—
136	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
137	C₂²×C₄	#10625352	1	2	yes	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	94	4	no	—
140	C₂×C₈	—	94	2	no	—
141	C₂×C₈	—	94	4	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	94	2	no	—
145	C₄⋊C₄	—	94	2	no	—
146	C₄⋊C₄	—	188	2	no	—
147	C₄⋊C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	188	1	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	188	1	no	—
152	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
153	C₄²	D₄×D₄₇	1	2	yes	—
154	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
155	C₂³	#13094008	1	1	yes	characteristic
156	Q₈	—	8	2	no	—
157	Q₈	—	188	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	188	2	no	—
162	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
163	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
164	C₂×C₄	#13093994	1	1	yes	characteristic
165	C₂×C₄	#13093996	1	2	yes	—
166	C₈	—	8	2	no	—
167	C₈	—	94	4	no	—
168	C₈	—	188	4	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#15689000	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	1	no	—
175	C₄	—	4	4	no	—
176	C₄	—	376	2	no	—
177	C₄	#15689308	1	2	yes	—
178	C₂	—	4	1	no	—
179	C₂	#20243748	1	2	yes	—
180	C₂	#20243779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20253910

C₂

yes

maximal, characteristic

#20252494

C₂

yes

maximal, characteristic

#20241862

C₂

yes

maximal

#20241288

C₂

yes

maximal

C₄₇×#1544

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

characteristic

#15689533

C₄

yes

—

—

—

—

C₂²

yes

—

#15689159

—

C₂×#13093935

C₂²

yes

characteristic

#15688855

—

C₄₇×#533

C₂²

yes

characteristic

C₄₇×#431

C₂²

yes

—

C₄₇×#332

—

C₂²×C₄₇⋊C₈

—

#13094024

C₂×C₄

yes

—

#13094016

—

#13094003

—

C₄×Dic₉₄

—

#13093983

—

#13093957

—

#13093955

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

—

—

—

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#125×C₄₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9006

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#386

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#456

yes

—

Dic₄₇

—

#3162

—

#2987

—

#1544

D₄₇

yes

characteristic

#1497

—

#1302

—

C₂×C₄₇

—

C₂×C₄₇

#1881

yes

—

C₂×C₄₇

yes

—

C₂×D₄₇

yes

characteristic

#501

—

#501

—

100

#446

—

101

#433

—

102

#432

—

103

#431

C₂×D₄₇

yes

—

104

#427

—

105

C₂×#97

—

106

#332

—

107

Q₈⋊C₈

—

108

C₄₇

#9006

yes

characteristic, complemented, Sylow

109

C₂²×C₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

C₂²⋊D₄₇

yes

—

112

#125

—

113

#125

C₄×D₄₇

yes

—

114

C₂²⋊Q₈

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

C₄×Q₈

—

118

#117

—

119

#117

D₁₈₈

yes

—

120

C₂×C₄²

C₂²×D₄₇

yes

characteristic

121

#107

—

122

#106

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

128

#97

—

129

#97

—

130

#97

—

131

#97

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

137

C₂²×C₄

#10625352

yes

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

188

—

147

C₄⋊C₄

(C₂×D₄₇)⋊C₄

yes

—

148

C₂²⋊C₄

—

149

C₂²⋊C₄

—

188

—

150

C₄²

—

151

C₄²

—

188

—

152

C₄²

C₂×C₄×D₄₇

yes

characteristic

153

C₄²

D₄×D₄₇

yes

—

154

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

155

C₂³

#13094008

yes

characteristic

156

Q₈

—

157

Q₈

—

188

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

188

—

162

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

163

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

164

C₂×C₄

#13093994

yes

characteristic

165

C₂×C₄

#13093996

yes

—

166

C₈

—

167

C₈

—

168

C₈

—

188

—

169

C₂²

—

170

C₂²

—

171

C₂²

#15689000

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

376

—

177

C₄

#15689308

yes

—

178

C₂

—

179

C₂

#20243748

yes

—

180

C₂

#20243779

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093994
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1544	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂³×C₄₇	#123
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1544	D₄₇
hypocenter	C₄₇	#9006
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

subgroup

G/H

center

C₂×C₄

#13093994

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1544

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

N	H ≅ G/N	type
108. C₄₇	#9006	semidirect

H ≅ G/N

type

108. C₄₇

#9006

semidirect

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

subgroup	index
#20253910	2
#20252494	2
#20241862	2
#20241288	2
C₄₇×#1544	2
#9006	47

subgroup

index

C₄₇×#1544

Subgroups — #26111230

Conjugacy classes of subgroups

Complements

Maximal subgroups