Group #26103499 — subgroups

Subgroups — #26103499

Order 12032.

322 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20241706	C₂	1	2	yes	maximal
2	C₄₇×#1955	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
4	#15688849	—	2	2	no	—
5	#15688847	C₄	1	4	yes	—
6	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
9	C₂×#97×C₄₇	C₄	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₈	—	4	2	no	—
11	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
13	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	D₄	1	2	yes	—
15	C₂²×C₈×C₄₇	—	2	1	no	—
16	C₂²×C₈×C₄₇	—	2	1	no	—
17	C₄×D₄×C₄₇	—	4	1	no	—
18	C₄×D₄×C₄₇	—	4	1	no	—
19	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	—	4	1	no	—
23	C₂²⋊C₈×C₄₇	—	4	4	no	—
24	#97×C₄₇	—	4	1	no	—
25	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
26	C₂×D₄×C₄₇	—	2	4	no	—
27	C₂×D₄×C₄₇	—	4	1	no	—
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	—	2	2	no	—
30	C₂²×C₄×C₄₇	—	4	2	no	—
31	C₂²×C₄×C₄₇	—	4	2	no	—
32	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	4	1	no	—
38	C₂×C₈×C₄₇	—	4	1	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	4	1	no	—
41	C₄⋊C₄×C₄₇	—	4	1	no	—
42	C₂²⋊C₄×C₄₇	—	4	2	no	—
43	C₄²×C₄₇	—	2	1	no	—
44	C₄²×C₄₇	—	2	1	no	—
45	C₂×C₄₇⋊C₈	—	4	4	no	—
46	C₂×C₄₇⋊C₈	—	8	2	no	—
47	C₂×C₄₇⋊C₈	—	8	2	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	—	4	2	no	—
51	C₂³×C₄₇	—	4	2	no	—
52	C₂³×C₄₇	#95	1	1	yes	characteristic
53	C₂³×C₄₇	#108	1	4	yes	—
54	D₄×C₄₇	—	4	4	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	4	2	no	—
62	C₂×C₄×C₄₇	—	4	2	no	—
63	C₂×C₄×C₄₇	—	4	2	no	—
64	C₂×C₄×C₄₇	—	4	1	no	—
65	C₂×C₄×C₄₇	—	8	2	no	—
66	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
68	C₈×C₄₇	—	4	2	no	—
69	C₈×C₄₇	—	8	1	no	—
70	C₄₇⋊C₈	—	8	4	no	—
71	#7287	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	2	1	no	—
73	C₂²×C₄₇	—	2	1	no	—
74	C₂²×C₄₇	—	2	4	no	—
75	C₂²×C₄₇	—	4	4	no	—
76	C₂²×C₄₇	—	4	2	no	—
77	C₂²×C₄₇	—	4	2	no	—
78	C₂²×C₄₇	—	8	2	no	—
79	C₂²×C₄₇	#341	1	2	yes	—
80	C₂²×C₄₇	#342	1	1	yes	characteristic
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	4	1	no	—
83	C₄×C₄₇	—	4	1	no	—
84	C₄×C₄₇	—	4	2	no	—
85	C₄×C₄₇	#353	1	1	yes	characteristic
86	C₄×C₄₇	#354	1	1	yes	characteristic
87	#1955	D₄₇	1	1	yes	characteristic
88	#1459	—	47	2	no	—
89	C₂×C₄₇	—	2	2	no	—
90	C₂×C₄₇	—	4	4	no	—
91	C₂×C₄₇	#1298	1	1	yes	characteristic
92	C₂×C₄₇	#1392	1	1	yes	characteristic
93	C₂×C₄₇	#1393	1	1	yes	characteristic
94	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
95	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
96	C₂×C₂²⋊C₈	—	2	2	no	—
97	C₂×C₂²⋊C₈	—	94	2	no	—
98	C₂×#97	Dic₄₇	1	1	yes	characteristic
99	#324	—	94	2	no	—
100	#323	—	47	4	no	—
101	C₄₇	#7287	1	1	yes	characteristic, complemented, Sylow
102	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
103	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	188	2	no	—
107	C₄×D₄	—	4	1	no	—
108	C₄×D₄	—	4	1	no	—
109	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
111	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
112	C₄⋊C₈	—	4	1	no	—
113	C₂²⋊C₈	—	4	4	no	—
114	C₂²⋊C₈	—	94	4	no	—
115	C₂²⋊C₈	—	188	2	no	—
116	#97	—	4	1	no	—
117	C₂⁴	Dic₄₇⋊C₄	1	2	yes	—
118	C₂×D₄	—	2	4	no	—
119	C₂×D₄	—	4	1	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
126	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	376	2	no	—
134	C₂×C₈	—	376	2	no	—
135	C₂×C₈	—	188	4	no	—
136	C₄⋊C₄	—	4	1	no	—
137	C₂²⋊C₄	—	4	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	—	4	2	no	—
143	C₂³	—	4	2	no	—
144	C₂³	C₄₇⋊#95	1	1	yes	characteristic
145	C₂³	#13093959	1	4	yes	—
146	D₄	—	4	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	#13093939	1	1	yes	characteristic
159	C₂×C₄	#13093939	1	1	yes	characteristic
160	C₈	—	4	2	no	—
161	C₈	—	8	1	no	—
162	C₈	—	376	4	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	4	4	no	—
167	C₂²	—	4	2	no	—
168	C₂²	—	4	2	no	—
169	C₂²	—	8	2	no	—
170	C₂²	#15688887	1	2	yes	—
171	C₂²	#15688890	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	4	1	no	—
174	C₄	—	4	2	no	—
175	C₄	—	4	1	no	—
176	C₄	#15688916	1	1	yes	characteristic
177	C₄	#15688918	1	1	yes	characteristic
178	C₂	—	2	2	no	—
179	C₂	—	4	4	no	—
180	C₂	#20241275	1	1	yes	characteristic
181	C₂	#20241511	1	1	yes	characteristic
182	C₂	#20241513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241706

C₂

yes

maximal

C₄₇×#1955

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₇⋊C₈)

—

#15688849

—

#15688847

C₄

yes

—

C₂×C₄×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₄₇

C₄

yes

characteristic

C₂×C₂²⋊C₈×C₄₇

—

C₂×#97×C₄₇

C₄

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²×D₄×C₄₇

Q₈

yes

characteristic

C₂³×C₄×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

—

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₂⁴×C₄₇

C₄⋊C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#95

yes

characteristic

C₂³×C₄₇

#108

yes

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7287

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#341

yes

—

C₂²×C₄₇

#342

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#1459

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×D₄

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₈

Dic₄₇

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

Dic₄₇

yes

characteristic

#324

—

100

#323

—

101

C₄₇

#7287

yes

characteristic, complemented, Sylow

102

C₂²×D₄

Dic₉₄

yes

characteristic

103

C₂³×C₄

C₂²⋊D₄₇

yes

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

188

—

107

C₄×D₄

—

108

C₄×D₄

—

109

C₂×C₄⋊C₄

D₁₈₈

yes

characteristic

110

C₂×C₂²⋊C₄

C₄×D₄₇

yes

—

111

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

C₂²⋊C₈

—

115

C₂²⋊C₈

—

188

—

116

#97

—

117

C₂⁴

Dic₄₇⋊C₄

yes

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

125

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

126

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

127

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

376

—

134

C₂×C₈

—

376

—

135

C₂×C₈

—

188

—

136

C₄⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

C₄₇⋊#95

yes

characteristic

145

C₂³

#13093959

yes

—

146

D₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#13093939

yes

characteristic

159

C₂×C₄

#13093939

yes

characteristic

160

C₈

—

161

C₈

—

162

C₈

—

376

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

#15688887

yes

—

171

C₂²

#15688890

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

#15688916

yes

characteristic

177

C₄

#15688918

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

#20241275

yes

characteristic

181

C₂

#20241511

yes

characteristic

182

C₂

#20241513

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093939
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₂×C₄×D₄	C₂×D₄₇
Fitting	C₄₇×#1955	C₂
socle	C₂²×C₄₇	#342
prime radical	C₂²×D₄×C₄₇	Q₈
cosocle	C₂×C₄×D₄×C₄₇	C₂²
hypercenter	#1955	D₄₇
hypocenter	C₄₇	#7287
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ #1955

subgroup

G/H

center

C₂×C₄

#13093939

commutator

C₂²×C₄×C₄₇

C₄²

Frattini

C₂×C₄×D₄

C₂×D₄₇

Fitting

C₄₇×#1955

C₂

socle

C₂²×C₄₇

#342

prime radical

C₂²×D₄×C₄₇

Q₈

cosocle

C₂×C₄×D₄×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ #1955

N	H ≅ G/N	type
101. C₄₇	#7287	semidirect

H ≅ G/N

type

101. C₄₇

#7287

semidirect

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

subgroup	index
#20241706	2
C₄₇×#1955	2
#7287	47

subgroup

index

#20241706

C₄₇×#1955

#7287

Subgroups — #26103499

Conjugacy classes of subgroups

Complements

Maximal subgroups