Group #10695316 — subgroups

Subgroups — #10695316

Order 768.

430 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 163 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	#73307	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#69392	C₂	1	1	yes	maximal, characteristic
4	#67702	C₂	1	4	yes	maximal
5	#9899	—	3	1	no	maximal, Sylow
6	C₂×#870	—	2	1	no	—
7	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	—	2	1	no	—
9	C₃×C₄³	C₂²	1	1	yes	characteristic
10	#4936	—	2	1	no	—
11	#4935	C₂²	1	2	yes	—
12	#4928	—	2	2	no	—
13	#4928	C₄	1	4	yes	—
14	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
15	C₃×#336	—	2	1	no	—
16	#4524	—	2	2	no	—
17	#2859	—	3	1	no	—
18	#1831	—	3	1	no	—
19	C₄×C₄⋊C₈	D₃	1	1	yes	characteristic
20	#1296	—	3	4	no	—
21	C₂²×C₃×C₈	—	2	2	no	—
22	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄²	D₄	1	2	yes	—
24	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
25	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₃	—	2	2	no	—
28	C₂×C₄⋊Dic₃	—	2	2	no	—
29	C₂×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
30	C₂×Dic₃⋊C₄	—	2	4	no	—
31	C₂²×C₃⋊C₈	—	2	2	no	—
32	#870	—	4	2	no	—
33	#870	—	4	2	no	—
34	C₃×C₄⋊C₈	—	2	2	no	—
35	C₃×C₄⋊C₈	Q₈	1	4	yes	—
36	C₃×C₄×C₈	—	4	2	no	—
37	C₃⋊#95	—	2	4	no	—
38	C₄⋊(C₃⋊C₈)	—	2	4	no	—
39	C₂×#125	—	6	1	no	—
40	C₂×C₄⋊C₈	—	3	2	no	—
41	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
42	C₂×C₄×C₈	—	2	1	no	—
43	#389	—	3	4	no	—
44	#389	—	6	2	no	—
45	#384	—	3	2	no	—
46	#382	—	6	1	no	—
47	C₄³	C₂×D₃	1	1	yes	characteristic
48	#336	—	2	1	no	—
49	#336	—	6	2	no	—
50	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×Dic₃	—	2	4	no	—
56	C₂×C₃×C₈	—	2	4	no	—
57	C₂×C₃×C₈	—	4	2	no	—
58	C₂×C₃×C₈	—	4	2	no	—
59	C₃×C₄²	—	2	4	no	—
60	C₃×C₄²	—	2	2	no	—
61	C₃×C₄²	—	4	2	no	—
62	C₃×C₄²	D₈	1	2	yes	—
63	C₃×C₄²	SD₁₆	1	2	yes	—
64	C₃×C₄²	C₂×Q₈	1	2	yes	—
65	C₃×C₄²	#41	1	2	yes	—
66	C₄⋊Dic₃	—	4	2	no	—
67	C₄⋊Dic₃	—	4	2	no	—
68	C₄⋊Dic₃	—	4	4	no	—
69	C₄⋊Dic₃	—	4	4	no	—
70	Dic₃⋊C₄	—	4	8	no	—
71	C₂×C₃⋊C₈	—	4	2	no	—
72	C₂×C₃⋊C₈	—	4	4	no	—
73	C₂²×C₈	—	2	2	no	—
74	C₂²×C₈	—	6	2	no	—
75	#125	—	12	2	no	—
76	#125	—	12	2	no	—
77	C₂×C₄⋊C₄	—	3	4	no	—
78	C₂×C₄⋊C₄	—	6	4	no	—
79	C₂×C₄⋊C₄	—	6	2	no	—
80	C₂×C₄⋊C₄	—	6	2	no	—
81	C₂×C₄²	C₄×D₃	1	2	yes	—
82	C₂×C₄²	D₁₂	1	2	yes	—
83	C₂×C₄²	#67	1	1	yes	characteristic
84	C₂×C₄²	#67	1	1	yes	characteristic
85	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	6	4	no	—
88	C₄⋊C₈	Dic₆	1	4	yes	—
89	C₄×C₈	—	4	2	no	—
90	#95	—	6	4	no	—
91	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	2	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	#102	1	4	yes	—
99	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
100	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₃×C₄	#124	1	1	yes	characteristic
102	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₃×C₄	#135	1	4	yes	—
106	C₂×C₃×C₄	#137	1	2	yes	—
107	C₂×Dic₃	—	4	4	no	—
108	C₂×Dic₃	—	4	4	no	—
109	C₂×Dic₃	—	4	4	no	—
110	C₃×C₈	—	4	4	no	—
111	C₃⋊C₈	—	8	2	no	—
112	C₂²×C₄	—	6	4	no	—
113	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
114	C₂²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
115	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
117	C₂²×C₄	C₂×#67	1	1	yes	characteristic
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	12	2	no	—
122	C₂×C₈	—	12	4	no	—
123	C₄⋊C₄	—	12	4	no	—
124	C₄⋊C₄	—	12	8	no	—
125	C₄⋊C₄	—	12	4	no	—
126	C₄⋊C₄	—	12	2	no	—
127	C₄⋊C₄	—	12	2	no	—
128	C₄²	—	2	4	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	4	2	no	—
131	C₄²	#213	1	2	yes	—
132	C₄²	#215	1	2	yes	—
133	C₄²	C₂×Dic₆	1	2	yes	—
134	C₄²	#235	1	2	yes	—
135	C₂²×C₃	#391	1	1	yes	characteristic
136	C₂²×C₃	#413	1	2	yes	—
137	C₂²×C₃	C₂×#102	1	1	yes	characteristic
138	C₂²×C₃	#416	1	2	yes	—
139	C₂²×C₃	#419	1	1	yes	characteristic
140	C₃×C₄	—	2	4	no	—
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	4	2	no	—
143	C₃×C₄	#474	1	2	yes	—
144	C₃×C₄	#475	1	2	yes	—
145	C₃×C₄	#488	1	2	yes	—
146	C₃×C₄	#499	1	2	yes	—
147	Dic₃	—	8	4	no	—
148	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	12	4	no	—
156	C₂×C₄	—	12	4	no	—
157	C₂×C₄	—	12	4	no	—
158	C₂×C₄	#809	1	4	yes	—
159	C₂×C₄	C₄×Dic₆	1	2	yes	—
160	C₂×C₄	#869	1	1	yes	characteristic
161	C₂×C₄	#875	1	1	yes	characteristic
162	C₂×C₄	#904	1	4	yes	—
163	C₂×C₄	#924	1	2	yes	—
164	C₂×C₄	C₂×#213	1	1	yes	characteristic
165	C₂×C₄	C₂×#215	1	1	yes	characteristic
166	C₂×C₄	#951	1	2	yes	—
167	C₈	—	4	4	no	—
168	C₈	—	24	2	no	—
169	C₂×C₃	#1540	1	2	yes	—
170	C₂×C₃	#1554	1	2	yes	—
171	C₂×C₃	#1988	1	1	yes	characteristic
172	C₂×C₃	#1991	1	1	yes	characteristic
173	C₂×C₃	#1998	1	1	yes	characteristic
174	C₂²	#4929	1	1	yes	characteristic
175	C₂²	C₂×#809	1	1	yes	characteristic
176	C₂²	#5004	1	1	yes	characteristic
177	C₂²	#5113	1	2	yes	—
178	C₂²	#5132	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	2	2	no	—
181	C₄	—	4	2	no	—
182	C₄	—	24	4	no	—
183	C₄	#4683	1	2	yes	—
184	C₄	#4697	1	2	yes	—
185	C₄	#5007	1	2	yes	—
186	C₄	#5020	1	2	yes	—
187	C₃	#9899	1	1	yes	characteristic, complemented, Sylow
188	C₂	#69490	1	1	yes	characteristic
189	C₂	#69778	1	1	yes	characteristic
190	C₂	#69880	1	1	yes	characteristic
191	C₂	#70431	1	2	yes	—
192	C₂	#70781	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#73307

C₂

yes

maximal, characteristic

C₃×C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

#69392

C₂

yes

maximal, characteristic

#67702

C₂

yes

—

—

C₂×C₃×C₄⋊C₈

C₂²

yes

—

C₂×C₃×C₄×C₈

—

C₃×C₄³

C₂²

yes

characteristic

#4936

—

#4935

C₂²

yes

—

#4928

—

#4928

C₄

yes

—

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

—

—

—

—

C₄×C₄⋊C₈

D₃

yes

characteristic

#1296

—

C₂²×C₃×C₈

—

C₂×C₃×C₄²

C₂×C₄

yes

—

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

characteristic

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂×C₄

yes

—

C₂×Dic₃⋊C₄

—

C₂²×C₃⋊C₈

—

#870

—

#870

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

Q₈

yes

—

C₃×C₄×C₈

—

C₃⋊#95

—

C₄⋊(C₃⋊C₈)

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₃

yes

—

C₂×C₄×C₈

—

—

—

—

—

C₄³

C₂×D₃

yes

characteristic

#336

—

#336

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

D₈

yes

—

C₃×C₄²

SD₁₆

yes

—

C₃×C₄²

C₂×Q₈

yes

—

C₃×C₄²

#41

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₄×D₃

yes

—

C₂×C₄²

D₁₂

yes

—

C₂×C₄²

yes

C₂×C₄²

yes

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₆

yes

—

C₄×C₈

—

#95

—

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#102

yes

—

C₂×C₃×C₄

C₄×Q₈

yes

—

100

C₂×C₃×C₄

C₂²⋊Q₈

yes

—

101

C₂×C₃×C₄

#124

yes

characteristic

102

C₂×C₃×C₄

C₄⋊Q₈

yes

characteristic

103

C₂×C₃×C₄

C₂×D₈

yes

characteristic

104

C₂×C₃×C₄

C₂×SD₁₆

yes

characteristic

105

C₂×C₃×C₄

#135

yes

—

106

C₂×C₃×C₄

#137

yes

—

107

C₂×Dic₃

—

108

C₂×Dic₃

—

109

C₂×Dic₃

—

110

C₃×C₈

—

111

C₃⋊C₈

—

112

C₂²×C₄

—

113

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

114

C₂²×C₄

(C₂×D₃)⋊C₄

yes

—

115

C₂²×C₄

C₂×C₄×D₃

yes

characteristic

116

C₂²×C₄

C₂×D₁₂

yes

characteristic

117

C₂²×C₄

C₂×#67

yes

characteristic

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

#213

yes

—

132

C₄²

#215

yes

—

133

C₄²

C₂×Dic₆

yes

—

134

C₄²

#235

yes

—

135

C₂²×C₃

#391

yes

characteristic

136

C₂²×C₃

#413

yes

—

137

C₂²×C₃

C₂×#102

yes

characteristic

138

C₂²×C₃

#416

yes

—

139

C₂²×C₃

#419

yes

characteristic

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

#474

yes

—

144

C₃×C₄

#475

yes

—

145

C₃×C₄

#488

yes

—

146

C₃×C₄

#499

yes

—

147

Dic₃

—

148

C₂³

C₂×(C₂×D₃)⋊C₄

yes

characteristic

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

#809

yes

—

159

C₂×C₄

C₄×Dic₆

yes

—

160

C₂×C₄

#869

yes

characteristic

161

C₂×C₄

#875

yes

characteristic

162

C₂×C₄

#904

yes

—

163

C₂×C₄

#924

yes

—

164

C₂×C₄

C₂×#213

yes

characteristic

165

C₂×C₄

C₂×#215

yes

characteristic

166

C₂×C₄

#951

yes

—

167

C₈

—

168

C₈

—

169

C₂×C₃

#1540

yes

—

170

C₂×C₃

#1554

yes

—

171

C₂×C₃

#1988

yes

characteristic

172

C₂×C₃

#1991

yes

characteristic

173

C₂×C₃

#1998

yes

characteristic

174

C₂²

#4929

yes

characteristic

175

C₂²

C₂×#809

yes

characteristic

176

C₂²

#5004

yes

characteristic

177

C₂²

#5113

yes

—

178

C₂²

#5132

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

#4683

yes

—

184

C₄

#4697

yes

—

185

C₄

#5007

yes

—

186

C₄

#5020

yes

—

187

C₃

#9899

yes

characteristic, complemented, Sylow

188

C₂

#69490

yes

characteristic

189

C₂

#69778

yes

characteristic

190

C₂

#69880

yes

characteristic

191

C₂

#70431

yes

—

192

C₂

#70781

yes

—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₃)⋊C₄
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊C₈	D₃
hypocenter	C₃	#9899
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×(C₂×D₃)⋊C₄

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×C₄×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

hypercenter

C₄×C₄⋊C₈

D₃

hypocenter

C₃

#9899

LCS

G ▷ C₂²×C₃×C₄ ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

N	H ≅ G/N	type
187. C₃	#9899	semidirect

H ≅ G/N

type

187. C₃

#9899

semidirect

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

subgroup	index
#73307	2
C₃×C₄×C₄⋊C₈	2
#69392	2
#67702	2
#9899	3

subgroup

index

C₃×C₄×C₄⋊C₈

Subgroups — #10695316

Conjugacy classes of subgroups

Complements

Maximal subgroups