Group #26110084 — subgroups

Subgroups — #26110084

Order 12032.

322 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 143 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	#20243525	C₂	1	2	yes	maximal
2	#20242146	C₂	1	2	yes	maximal
3	#20242144	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1566	C₂	1	1	yes	maximal, characteristic
6	#15689147	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	#15688887	—	2	4	no	—
10	#15688875	—	2	4	no	—
11	#15688874	—	2	2	no	—
12	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
15	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
16	C₄₇×#423	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#334	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
21	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
29	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
30	C₂×M₁₆×C₄₇	—	2	2	no	—
31	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	—	2	2	no	—
34	C₄⋊C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	D₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	M₁₆×C₄₇	—	4	4	no	—
42	C₂×C₈×C₄₇	—	2	4	no	—
43	C₂×C₈×C₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	#10625329	—	4	4	no	—
51	C₂×C₄₇⋊C₈	—	2	4	no	—
52	C₂×C₄₇⋊C₈	—	2	4	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
57	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
58	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	#135	1	4	yes	—
68	C₈×C₄₇	—	2	2	no	—
69	C₈×C₄₇	—	8	2	no	—
70	C₈×C₄₇	#108	1	4	yes	—
71	C₄₇⋊C₈	—	2	2	no	—
72	C₄₇⋊C₈	—	2	2	no	—
73	C₄₇⋊C₈	—	2	2	no	—
74	C₄₇⋊C₈	—	8	2	no	—
75	C₄₇⋊C₈	#108	1	4	yes	—
76	#8824	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	1	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	#384	1	1	yes	characteristic
80	C₂²×C₄₇	#499	1	2	yes	—
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	#427	1	2	yes	—
84	C₄×C₄₇	C₂×#108	1	2	yes	—
85	C₄×C₄₇	C₂×#108	1	2	yes	—
86	C₄×C₄₇	#495	1	2	yes	—
87	#1837	—	47	2	no	—
88	#1567	—	47	2	no	—
89	#1566	—	47	1	no	—
90	#1566	D₄₇	1	1	yes	characteristic
91	C₂×C₈²	—	47	1	no	—
92	C₂×C₄₇	—	2	2	no	—
93	C₂×C₄₇	#1846	1	1	yes	characteristic
94	C₂×C₄₇	#1848	1	1	yes	characteristic
95	C₂×C₄₇	#1848	1	1	yes	characteristic
96	#423	—	47	2	no	—
97	#423	C₂×D₄₇	1	2	yes	—
98	C₂×C₄×C₈	—	47	1	no	—
99	C₂×C₄×C₈	—	47	1	no	—
100	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
101	#341	—	94	4	no	—
102	#335	—	94	4	no	—
103	#334	—	2	2	no	—
104	#334	—	94	2	no	—
105	C₈²	—	47	1	no	—
106	C₈²	—	47	1	no	—
107	C₈²	—	47	1	no	—
108	C₈²	—	47	1	no	—
109	C₄₇	#8824	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	2	2	no	—
111	C₂×M₁₆	—	94	2	no	—
112	C₂²×C₈	—	47	2	no	—
113	C₂²×C₈	—	47	2	no	—
114	C₂²×C₈	Dic₉₄	1	2	yes	—
115	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	94	2	no	—
119	C₄⋊C₈	—	94	2	no	—
120	C₄×C₈	—	47	1	no	—
121	C₄×C₈	—	47	1	no	—
122	C₄×C₈	—	47	1	no	—
123	C₄×C₈	—	47	1	no	—
124	C₄×C₈	—	47	2	no	—
125	C₄×C₈	—	47	2	no	—
126	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
127	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
129	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
130	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
131	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
132	M₁₆	—	4	4	no	—
133	M₁₆	—	188	4	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	47	2	no	—
137	C₂×C₈	—	47	2	no	—
138	C₂×C₈	—	47	2	no	—
139	C₂×C₈	—	47	2	no	—
140	C₂×C₈	—	94	4	no	—
141	C₂×C₈	—	94	4	no	—
142	C₂×C₈	—	188	2	no	—
143	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
144	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
145	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
146	C₄²	D₄×D₄₇	1	1	yes	characteristic
147	C₄²	Q₈×D₄₇	1	1	yes	characteristic
148	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
149	C₂³	#13094023	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
154	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#13094020	1	1	yes	characteristic
156	C₂×C₄	#13094022	1	1	yes	characteristic
157	C₂×C₄	#13094023	1	1	yes	characteristic
158	C₂×C₄	#13094045	1	4	yes	—
159	C₈	—	2	2	no	—
160	C₈	—	8	2	no	—
161	C₈	—	47	4	no	—
162	C₈	—	94	2	no	—
163	C₈	—	94	2	no	—
164	C₈	—	94	2	no	—
165	C₈	—	376	2	no	—
166	C₈	#13093959	1	4	yes	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	#15688992	1	1	yes	characteristic
170	C₂²	#15689523	1	2	yes	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	#15689158	1	2	yes	—
174	C₄	#108×D₄₇	1	2	yes	—
175	C₄	C₂×#13093959	1	2	yes	—
176	C₄	#15689504	1	2	yes	—
177	C₂	—	2	2	no	—
178	C₂	#20243549	1	1	yes	characteristic
179	C₂	#20243558	1	1	yes	characteristic
180	C₂	#20243559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243525

C₂

yes

maximal

#20242146

C₂

yes

maximal

#20242144

C₂

yes

maximal, characteristic

C₂×C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#1566

C₂

yes

maximal, characteristic

#15689147

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

—

—

C₈×C₄₇⋊C₈

C₄

yes

characteristic

C₈×C₄₇⋊C₈

C₄

yes

characteristic

C₈×C₄₇⋊C₈

C₄

yes

characteristic

C₈×C₄₇⋊C₈

C₄

yes

characteristic

C₄₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#334

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂²×C₄₇⋊C₈

Q₈

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#108

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#108

yes

—

#8824

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#499

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#427

yes

—

C₄×C₄₇

C₂×#108

yes

—

C₄×C₄₇

C₂×#108

yes

—

C₄×C₄₇

#495

yes

—

—

—

—

D₄₇

yes

characteristic

C₂×C₈²

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

100

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

101

#341

—

102

#335

—

103

#334

—

104

#334

—

105

C₈²

—

106

C₈²

—

107

C₈²

—

108

C₈²

—

109

C₄₇

#8824

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

Dic₉₄

yes

—

115

C₂×C₄²

C₂²×D₄₇

yes

characteristic

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₄×D₄₇

yes

characteristic

127

C₄×C₈

C₄×D₄₇

yes

characteristic

128

C₄×C₈

C₂²⋊D₄₇

yes

—

129

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

130

C₂²×C₄

D₄×D₄₇

yes

characteristic

131

C₂²×C₄

Q₈×D₄₇

yes

characteristic

132

M₁₆

—

133

M₁₆

—

188

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

188

—

143

C₂×C₈

Dic₄₇⋊C₄

yes

—

144

C₂×C₈

Dic₄₇⋊C₄

yes

—

145

C₄²

C₂×C₄×D₄₇

yes

characteristic

146

C₄²

D₄×D₄₇

yes

characteristic

147

C₄²

Q₈×D₄₇

yes

characteristic

148

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

149

C₂³

#13094023

yes

characteristic

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

153

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

154

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

155

C₂×C₄

#13094020

yes

characteristic

156

C₂×C₄

#13094022

yes

characteristic

157

C₂×C₄

#13094023

yes

characteristic

158

C₂×C₄

#13094045

yes

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

376

—

166

C₈

#13093959

yes

—

167

C₂²

—

168

C₂²

—

169

C₂²

#15688992

yes

characteristic

170

C₂²

#15689523

yes

—

171

C₄

—

172

C₄

—

173

C₄

#15689158

yes

—

174

C₄

#108×D₄₇

yes

—

175

C₄

C₂×#13093959

yes

—

176

C₄

#15689504

yes

—

177

C₂

—

178

C₂

#20243549

yes

characteristic

179

C₂

#20243558

yes

characteristic

180

C₂

#20243559

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1566	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1566	D₄₇
hypocenter	C₄₇	#8824
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1566

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
109. C₄₇	#8824	semidirect

H ≅ G/N

type

109. C₄₇

#8824

semidirect

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

subgroup	index
#20243525	2
#20242146	2
#20242144	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1566	2
#8824	47

subgroup

index

#20243525

#20242146

#20242144

C₂×C₈×C₄₇⋊C₈

C₄₇×#1566

#8824

Subgroups — #26110084

Conjugacy classes of subgroups

Complements

Maximal subgroups