Group #11800171 — subgroups

Subgroups — #11800171

Order 1152.

396 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 107 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	#10592882	C₂	1	1	yes	maximal, characteristic
2	#10592577	C₂	1	1	yes	maximal, characteristic
3	C₃²×#358	C₂	1	1	yes	maximal, characteristic
4	#10589430	C₂	1	1	yes	maximal, characteristic
5	#10589369	C₂	1	1	yes	maximal, characteristic
6	#10589368	C₂	1	1	yes	maximal, characteristic
7	#10589358	C₂	1	1	yes	maximal, characteristic
8	#69218	—	3	4	no	maximal
9	C₂×C₃²×Q₁₆	C₂²	1	1	yes	characteristic
10	#64153	—	2	1	no	—
11	C₂×#3756	—	2	1	no	—
12	C₂×#3746	C₂²	1	1	yes	characteristic
13	#64099	—	2	1	no	—
14	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
15	C₃²×#107	C₂²	1	1	yes	characteristic
16	#63645	C₂²	1	1	yes	characteristic
17	C₈×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
19	#63635	—	2	1	no	—
20	#5183	—	3	4	no	—
21	#4878	—	3	4	no	—
22	C₃×#358	D₃	1	4	yes	—
23	#4564	—	3	4	no	—
24	#4503	—	3	4	no	—
25	#4502	—	3	4	no	—
26	#4492	—	3	4	no	—
27	C₂×C₃²×Q₈	—	2	1	no	—
28	C₂×#634	—	2	1	no	—
29	C₃²×Q₁₆	—	2	1	no	—
30	C₃²×Q₁₆	—	2	1	no	—
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₃²×C₄⋊C₄	—	2	1	no	—
33	#3756	—	4	2	no	—
34	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
35	C₄⋊(C₃⋊Dic₃)	—	4	1	no	—
36	#3750	—	4	1	no	—
37	#3750	—	4	1	no	—
38	#3750	—	4	1	no	—
39	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
40	C₂×C₃⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
41	#3746	—	2	1	no	—
42	#3746	—	2	1	no	—
43	C₃²×C₁₆	—	2	1	no	—
44	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
45	#2245	—	9	1	no	Sylow
46	C₂×C₃×Q₁₆	C₂×D₃	1	4	yes	—
47	#944	—	6	4	no	—
48	C₂×#216	—	6	4	no	—
49	C₂×Dic₁₂	—	3	4	no	—
50	#890	—	6	4	no	—
51	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
52	C₃×#107	C₂×D₃	1	4	yes	—
53	#818	—	3	4	no	—
54	C₈×Dic₃	—	3	4	no	—
55	C₂×C₃⋊C₁₆	—	3	4	no	—
56	#808	—	6	4	no	—
57	C₃²×Q₈	—	4	1	no	—
58	C₃²×Q₈	—	4	1	no	—
59	C₂×C₃²×C₄	—	4	1	no	—
60	C₂×C₃²×C₄	—	4	1	no	—
61	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃⋊Dic₃	—	4	1	no	—
63	C₂×C₃⋊Dic₃	—	4	1	no	—
64	C₂×C₃⋊Dic₃	D₈	1	1	yes	characteristic
65	C₂×C₃⋊Dic₃	D₈	1	1	yes	characteristic
66	#634	—	4	1	no	—
67	#634	—	4	1	no	—
68	C₃²×C₈	#41	1	1	yes	characteristic
69	C₃²×C₈	#41	1	1	yes	characteristic
70	C₃⋊(C₃⋊C₈)	SD₁₆	1	2	yes	—
71	#499	—	9	1	no	—
72	#494	—	9	1	no	—
73	#358	—	9	1	no	—
74	#358	—	9	1	no	—
75	#358	—	9	1	no	—
76	#358	C₃⋊D₃	1	1	yes	characteristic
77	#346	—	9	1	no	—
78	C₂×C₃×Q₈	—	2	4	no	—
79	C₂×Dic₆	—	6	4	no	—
80	C₃×Q₁₆	—	2	4	no	—
81	C₃×Q₁₆	—	2	4	no	—
82	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
83	C₃×C₄⋊C₄	—	2	4	no	—
84	#216	—	12	8	no	—
85	C₄⋊Dic₃	—	6	4	no	—
86	C₄⋊Dic₃	—	12	4	no	—
87	Dic₃⋊C₄	—	12	4	no	—
88	Dic₃⋊C₄	—	12	4	no	—
89	Dic₃⋊C₄	—	12	4	no	—
90	C₄×Dic₃	—	3	4	no	—
91	C₂×C₃⋊C₈	—	3	4	no	—
92	Dic₁₂	—	6	4	no	—
93	Dic₁₂	—	6	4	no	—
94	C₃×C₁₆	—	2	4	no	—
95	C₃⋊C₁₆	—	6	4	no	—
96	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
97	C₃²×C₄	—	8	1	no	—
98	C₃²×C₄	—	8	1	no	—
99	C₃²×C₄	#124	1	1	yes	characteristic
100	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
101	C₃⋊Dic₃	—	2	1	no	—
102	C₃⋊Dic₃	—	2	1	no	—
103	C₃⋊Dic₃	—	8	1	no	—
104	C₃⋊Dic₃	—	8	1	no	—
105	C₂×Q₁₆	—	9	1	no	—
106	C₂×Q₁₆	—	18	1	no	—
107	C₂×Q₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
108	C₄⋊Q₈	—	18	1	no	—
109	#125	—	18	1	no	—
110	C₂×C₁₆	—	9	1	no	—
111	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
112	#107	—	9	1	no	—
113	#107	C₂×C₃⋊D₃	1	1	yes	characteristic
114	#106	—	18	1	no	—
115	C₄×C₈	—	9	1	no	—
116	C₃×Q₈	—	4	4	no	—
117	C₃×Q₈	—	4	4	no	—
118	C₂×C₃×C₄	—	4	4	no	—
119	C₂×C₃×C₄	—	4	4	no	—
120	C₂×C₃×C₄	D₃×D₄	1	4	yes	—
121	C₂×Dic₃	—	3	4	no	—
122	C₂×Dic₃	—	3	4	no	—
123	C₂×Dic₃	—	12	4	no	—
124	C₂×Dic₃	—	12	4	no	—
125	Dic₆	—	12	4	no	—
126	Dic₆	—	12	4	no	—
127	C₃×C₈	#235	1	4	yes	—
128	C₃×C₈	#237	1	4	yes	—
129	C₃⋊C₈	—	3	8	no	—
130	C₂×C₃²	#486	1	1	yes	characteristic
131	C₂×C₃²	#510	1	1	yes	characteristic
132	C₂×C₃²	#510	1	1	yes	characteristic
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	18	1	no	—
135	Q₁₆	—	2	1	no	—
136	Q₁₆	—	2	1	no	—
137	Q₁₆	—	18	1	no	—
138	Q₁₆	—	18	1	no	—
139	Q₁₆	—	36	2	no	—
140	C₂×C₈	—	9	1	no	—
141	C₂×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	18	1	no	—
144	C₄⋊C₄	—	36	1	no	—
145	C₄⋊C₄	—	36	1	no	—
146	C₄⋊C₄	—	36	1	no	—
147	C₄⋊C₄	—	36	1	no	—
148	C₄²	—	9	1	no	—
149	C₁₆	—	2	1	no	—
150	C₁₆	—	18	1	no	—
151	C₂²×C₃	D₃×D₈	1	4	yes	—
152	C₃×C₄	—	8	4	no	—
153	C₃×C₄	—	8	4	no	—
154	C₃×C₄	#885	1	4	yes	—
155	C₃×C₄	D₃×SD₁₆	1	4	yes	—
156	Dic₃	—	6	4	no	—
157	Dic₃	—	6	4	no	—
158	Dic₃	—	24	4	no	—
159	Dic₃	—	24	4	no	—
160	C₃²	#2245	1	1	yes	characteristic, complemented, Sylow
161	Q₈	—	4	1	no	—
162	Q₈	—	4	1	no	—
163	Q₈	—	36	1	no	—
164	Q₈	—	36	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	9	1	no	—
168	C₂×C₄	—	9	1	no	—
169	C₂×C₄	—	36	1	no	—
170	C₂×C₄	—	36	1	no	—
171	C₂×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
172	C₈	—	9	2	no	—
173	C₈	#3828	1	1	yes	characteristic
174	C₈	#3830	1	1	yes	characteristic
175	C₂×C₃	#4760	1	4	yes	—
176	C₂×C₃	#4915	1	4	yes	—
177	C₂×C₃	#4918	1	4	yes	—
178	C₂²	D₈×C₃⋊D₃	1	1	yes	characteristic
179	C₄	—	8	1	no	—
180	C₄	—	8	1	no	—
181	C₄	—	18	1	no	—
182	C₄	—	18	1	no	—
183	C₄	—	72	1	no	—
184	C₄	—	72	1	no	—
185	C₄	#64094	1	1	yes	characteristic
186	C₄	SD₁₆×C₃⋊D₃	1	1	yes	characteristic
187	C₃	#69218	1	4	yes	complemented
188	C₂	#10592459	1	1	yes	characteristic
189	C₂	#10592614	1	1	yes	characteristic
190	C₂	#10592617	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10592882

C₂

yes

maximal, characteristic

#10592577

C₂

yes

maximal, characteristic

C₃²×#358

C₂

yes

maximal, characteristic

#10589430

C₂

yes

maximal, characteristic

#10589369

C₂

yes

maximal, characteristic

#10589368

C₂

yes

maximal, characteristic

#10589358

C₂

yes

maximal, characteristic

#69218

—

maximal

C₂×C₃²×Q₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

#64099

—

C₂×C₃²×C₁₆

C₂²

yes

characteristic

C₃²×#107

C₂²

yes

characteristic

#63645

C₂²

yes

characteristic

C₈×C₃⋊Dic₃

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₂²

yes

—

—

—

D₃

yes

—

—

—

—

—

C₂×C₃²×Q₈

—

C₂×#634

—

C₃²×Q₁₆

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

C₂³

yes

characteristic

C₃²×C₄⋊C₄

—

#3756

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

#3750

—

#3750

—

C₄×C₃⋊Dic₃

D₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

D₄

yes

characteristic

#3746

—

#3746

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#2245

—

Sylow

C₂×C₃×Q₁₆

C₂×D₃

yes

—

#944

—

C₂×#216

—

C₂×Dic₁₂

—

#890

—

C₂×C₃×C₁₆

C₂×D₃

yes

—

C₃×#107

C₂×D₃

yes

—

#818

—

C₈×Dic₃

—

C₂×C₃⋊C₁₆

—

#808

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

D₈

yes

characteristic

C₂×C₃⋊Dic₃

D₈

yes

characteristic

#634

—

#634

—

C₃²×C₈

yes

C₃²×C₈

yes

C₃⋊(C₃⋊C₈)

SD₁₆

yes

—

—

—

—

—

—

C₃⋊D₃

yes

characteristic

#346

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₂²×D₃

yes

—

C₃×C₄⋊C₄

—

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

Dic₁₂

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

C₂×D₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#124

yes

characteristic

100

C₃²×C₄

C₂×SD₁₆

yes

characteristic

101

C₃⋊Dic₃

—

102

C₃⋊Dic₃

—

103

C₃⋊Dic₃

—

104

C₃⋊Dic₃

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

C₂×C₃⋊D₃

yes

characteristic

108

C₄⋊Q₈

—

109

#125

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂×C₃⋊D₃

yes

characteristic

112

#107

—

113

#107

C₂×C₃⋊D₃

yes

characteristic

114

#106

—

115

C₄×C₈

—

116

C₃×Q₈

—

117

C₃×Q₈

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

—

120

C₂×C₃×C₄

D₃×D₄

yes

—

121

C₂×Dic₃

—

122

C₂×Dic₃

—

123

C₂×Dic₃

—

124

C₂×Dic₃

—

125

Dic₆

—

126

Dic₆

—

127

C₃×C₈

#235

yes

—

128

C₃×C₈

#237

yes

—

129

C₃⋊C₈

—

130

C₂×C₃²

#486

yes

characteristic

131

C₂×C₃²

#510

yes

characteristic

132

C₂×C₃²

#510

yes

characteristic

133

C₂×Q₈

—

134

C₂×Q₈

—

135

Q₁₆

—

136

Q₁₆

—

137

Q₁₆

—

138

Q₁₆

—

139

Q₁₆

—

140

C₂×C₈

—

141

C₂×C₈

C₂²×C₃⋊D₃

yes

characteristic

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

—

149

C₁₆

—

150

C₁₆

—

151

C₂²×C₃

D₃×D₈

yes

—

152

C₃×C₄

—

153

C₃×C₄

—

154

C₃×C₄

#885

yes

—

155

C₃×C₄

D₃×SD₁₆

yes

—

156

Dic₃

—

157

Dic₃

—

158

Dic₃

—

159

Dic₃

—

160

C₃²

#2245

yes

characteristic, complemented, Sylow

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

D₄×C₃⋊D₃

yes

characteristic

172

C₈

—

173

C₈

#3828

yes

characteristic

174

C₈

#3830

yes

characteristic

175

C₂×C₃

#4760

yes

—

176

C₂×C₃

#4915

yes

—

177

C₂×C₃

#4918

yes

—

178

C₂²

D₈×C₃⋊D₃

yes

characteristic

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

#64094

yes

characteristic

186

C₄

SD₁₆×C₃⋊D₃

yes

characteristic

187

C₃

#69218

yes

complemented

188

C₂

#10592459

yes

characteristic

189

C₂

#10592614

yes

characteristic

190

C₂

#10592617

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×C₃⋊D₃
commutator, cosocle	C₂×C₃²×C₈	C₂³
Frattini	C₂×C₈	C₂²×C₃⋊D₃
Fitting	C₃²×#358	C₂
socle, prime radical	C₂²×C₃²	C₂×D₈
hypercenter	#358	C₃⋊D₃
hypocenter	C₃²	#2245
LCS	G ▷ C₂×C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×C₃⋊D₃

commutator, cosocle

C₂×C₃²×C₈

C₂³

Frattini

C₂×C₈

C₂²×C₃⋊D₃

Fitting

C₃²×#358

C₂

socle, prime radical

C₂²×C₃²

C₂×D₈

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
160. C₃²	#2245	semidirect
187. C₃	#69218	semidirect

H ≅ G/N

type

160. C₃²

#2245

semidirect

187. C₃

#69218

semidirect

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

subgroup	index
#10592882	2
#10592577	2
C₃²×#358	2
#10589430	2
#10589369	2
#10589368	2
#10589358	2
#69218	3

subgroup

index

C₃²×#358

Subgroups — #11800171

Conjugacy classes of subgroups

Complements

Maximal subgroups