Group #26368867 — subgroups

Subgroups — #26368867

Order 12032.

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

normal subgroups only

This group has 806 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×#15689522	C₂	1	4	yes	maximal
2	#20251751	C₂	1	1	yes	maximal, characteristic
3	#20243557	C₂	1	4	yes	maximal
4	#20243554	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×#13094016	C₂²	1	2	yes	—
8	#15689522	—	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	#15689021	—	2	4	no	—
15	#15688993	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×C₄₇⋊C₈	C₂³	1	4	yes	—
23	#13094016	—	2	8	no	—
24	C₂×C₄⋊Dic₄₇	—	2	4	no	—
25	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
26	C₂×Dic₄₇⋊C₄	—	2	4	no	—
27	#13093957	—	2	16	no	—
28	#13093955	—	2	16	no	—
29	#13093935	Q₈	1	8	yes	—
30	#13093935	Q₈	1	8	yes	—
31	C₄×C₄₇⋊C₈	D₄	1	8	yes	—
32	C₄₇⋊#95	—	2	4	no	—
33	#125×C₄₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₄₇	D₄	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	#95×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂²×Dic₄₇	—	2	4	no	—
48	C₄⋊C₄×C₄₇	—	2	8	no	—
49	C₄⋊C₄×C₄₇	—	4	8	no	—
50	C₄⋊C₄×C₄₇	—	4	8	no	—
51	C₄²×C₄₇	—	4	4	no	—
52	C₄²×C₄₇	C₂×D₄	1	4	yes	—
53	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
54	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
55	C₄⋊Dic₄₇	—	2	8	no	—
56	C₄⋊Dic₄₇	—	4	8	no	—
57	Dic₄₇⋊C₄	—	4	8	no	—
58	C₂×C₄₇⋊C₈	C₂×D₄	1	8	yes	—
59	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
60	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
61	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	2	4	no	—
65	C₂×C₄×C₄₇	—	4	4	no	—
66	C₂×C₄×C₄₇	—	4	8	no	—
67	C₂×C₄×C₄₇	#127	1	4	yes	—
68	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
69	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
70	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
71	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
72	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
73	C₂×Dic₄₇	—	4	4	no	—
74	C₂×Dic₄₇	—	4	8	no	—
75	C₄₇⋊C₈	C₄⋊Q₈	1	16	yes	—
76	#33315	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	C₂×#127	1	1	yes	characteristic
78	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₄₇	#575	1	4	yes	—
81	C₄×C₄₇	—	2	4	no	—
82	C₄×C₄₇	—	2	4	no	—
83	C₄×C₄₇	—	2	4	no	—
84	C₄×C₄₇	—	8	4	no	—
85	C₄×C₄₇	C₂×C₄⋊Q₈	1	4	yes	—
86	Dic₄₇	—	8	4	no	—
87	C₂×#499	—	47	4	no	—
88	#2859	—	47	1	no	—
89	#2859	D₄₇	1	1	yes	characteristic
90	#1847	—	47	4	no	—
91	#1847	—	47	4	no	—
92	C₄×#97	—	47	1	no	—
93	C₂×C₄₇	#2868	1	1	yes	characteristic
94	C₂×C₄₇	#3150	1	2	yes	—
95	C₂×C₄₇	#3166	1	2	yes	—
96	C₂×C₄₇	#3166	1	2	yes	—
97	C₂×#125	—	47	2	no	—
98	C₂×#125	C₂×D₄₇	1	2	yes	—
99	#499	—	94	16	no	—
100	C₂×#107	—	47	8	no	—
101	C₂×#106	—	47	8	no	—
102	C₂×#97	—	47	2	no	—
103	C₂×#97	—	47	2	no	—
104	C₂×C₄×C₈	—	47	2	no	—
105	#389	—	2	4	no	—
106	#389	—	94	4	no	—
107	#384	—	47	2	no	—
108	#384	C₂×D₄₇	1	2	yes	—
109	#382	—	47	2	no	—
110	#382	C₂×D₄₇	1	2	yes	—
111	C₄³	C₂×D₄₇	1	1	yes	characteristic
112	C₄₇	#33315	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	47	4	no	—
114	#125	—	2	8	no	—
115	#125	—	94	8	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₄⋊C₄	—	47	4	no	—
119	C₂×C₄⋊C₄	—	94	4	no	—
120	C₂×C₄⋊C₄	—	94	4	no	—
121	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
122	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
123	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	#107	—	94	16	no	—
127	#106	—	94	16	no	—
128	#97	—	47	8	no	—
129	#97	—	47	8	no	—
130	C₄×C₈	—	47	8	no	—
131	#95	—	2	4	no	—
132	#95	—	94	4	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	94	4	no	—
135	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
138	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
139	C₂×C₈	—	47	8	no	—
140	C₂×C₈	—	47	8	no	—
141	C₂×C₈	—	47	8	no	—
142	C₄⋊C₄	—	2	8	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	8	no	—
145	C₄⋊C₄	—	94	8	no	—
146	C₄⋊C₄	—	188	8	no	—
147	C₄⋊C₄	—	188	8	no	—
148	C₄²	—	4	4	no	—
149	C₄²	D₄×D₄₇	1	4	yes	—
150	C₄²	Q₈×D₄₇	1	4	yes	—
151	C₄²	Q₈×D₄₇	1	4	yes	—
152	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	—	188	4	no	—
159	C₂×C₄	—	188	8	no	—
160	C₂×C₄	#13094020	1	4	yes	—
161	C₂×C₄	#13094022	1	4	yes	—
162	C₂×C₄	#13094022	1	4	yes	—
163	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
164	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
165	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
166	C₈	—	47	16	no	—
167	C₂²	C₂×#13094020	1	1	yes	characteristic
168	C₂²	C₂×#13094022	1	1	yes	characteristic
169	C₂²	C₂×#13094022	1	1	yes	characteristic
170	C₂²	#15689921	1	4	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	4	no	—
173	C₄	—	2	4	no	—
174	C₄	—	8	4	no	—
175	C₄	—	376	4	no	—
176	C₄	C₄⋊Q₈×D₄₇	1	4	yes	—
177	C₂	#20251784	1	1	yes	characteristic
178	C₂	#20253833	1	2	yes	—
179	C₂	#20253937	1	2	yes	—
180	C₂	#20253941	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689522

C₂

yes

maximal

#20251751

C₂

yes

maximal, characteristic

#20243557

C₂

yes

maximal

#20243554

C₂

yes

maximal

C₄×#13093935

C₂

yes

maximal, characteristic

C₄₇×#2859

C₂

yes

maximal, characteristic

C₂×#13094016

C₂²

yes

—

#15689522

—

C₂×#13093957

C₂²

yes

—

C₂×#13093955

C₂²

yes

—

C₂×#13093935

C₂²

yes

—

C₂×#13093935

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15689021

—

#15688993

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×C₄₇⋊C₈

C₂³

yes

—

#13094016

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

C₂×Dic₄₇⋊C₄

—

#13093957

—

#13093955

—

#13093935

Q₈

yes

—

#13093935

Q₈

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

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

—

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

—

Dic₄₇⋊C₄

—

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

—

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×#499

—

#2859

—

#2859

D₄₇

yes

characteristic

#1847

—

#1847

—

C₄×#97

—

C₂×C₄₇

#2868

yes

characteristic

C₂×C₄₇

#3150

yes

—

C₂×C₄₇

#3166

yes

—

C₂×C₄₇

#3166

yes

—

C₂×#125

—

C₂×#125

C₂×D₄₇

yes

—

#499

—

100

C₂×#107

—

101

C₂×#106

—

102

C₂×#97

—

103

C₂×#97

—

104

C₂×C₄×C₈

—

105

#389

—

106

#389

—

107

#384

—

108

#384

C₂×D₄₇

yes

—

109

#382

—

110

#382

C₂×D₄₇

yes

—

111

C₄³

C₂×D₄₇

yes

characteristic

112

C₄₇

#33315

yes

characteristic, complemented, Sylow

113

C₂²×C₈

—

114

#125

—

115

#125

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂²×D₄₇

yes

—

122

C₂×C₄²

C₂²⋊D₄₇

yes

—

123

C₂×C₄²

C₂²×D₄₇

yes

characteristic

124

C₂×C₄²

C₂²×D₄₇

yes

characteristic

125

C₂×C₄²

C₂²×D₄₇

yes

characteristic

126

#107

—

127

#106

—

128

#97

—

129

#97

—

130

C₄×C₈

—

131

#95

—

132

#95

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

136

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

138

C₂²×C₄

C₂³×D₄₇

yes

characteristic

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

—

188

—

148

C₄²

—

149

C₄²

D₄×D₄₇

yes

—

150

C₄²

Q₈×D₄₇

yes

—

151

C₄²

Q₈×D₄₇

yes

—

152

C₂³

C₂²×C₂²⋊D₄₇

yes

characteristic

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

188

—

159

C₂×C₄

—

188

—

160

C₂×C₄

#13094020

yes

—

161

C₂×C₄

#13094022

yes

—

162

C₂×C₄

#13094022

yes

—

163

C₂×C₄

C₂×D₄×D₄₇

yes

—

164

C₂×C₄

C₂×Q₈×D₄₇

yes

—

165

C₂×C₄

C₂×Q₈×D₄₇

yes

—

166

C₈

—

167

C₂²

C₂×#13094020

yes

characteristic

168

C₂²

C₂×#13094022

yes

characteristic

169

C₂²

C₂×#13094022

yes

characteristic

170

C₂²

#15689921

yes

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

376

—

176

C₄

C₄⋊Q₈×D₄₇

yes

—

177

C₂

#20251784

yes

characteristic

178

C₂

#20253833

yes

—

179

C₂

#20253937

yes

—

180

C₂

#20253941

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	#33315
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
112. C₄₇	#33315	semidirect

H ≅ G/N

type

112. C₄₇

#33315

semidirect

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

subgroup	index
C₂×#15689522	2
#20251751	2
#20243557	2
#20243554	2
C₄×#13093935	2
C₄₇×#2859	2
#33315	47

subgroup

index

C₄₇×#2859

Subgroups — #26368867

Conjugacy classes of subgroups

Complements

Maximal subgroups