Group #13158852 — subgroups

Subgroups — #13158852

Order 1728.

764 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 164 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	#11723911	C₂	1	1	yes	maximal, characteristic
2	C₃×#64078	C₂	1	1	yes	maximal, characteristic
3	C₃²×#803	C₂	1	1	yes	maximal, characteristic
4	#11722040	C₂	1	4	yes	maximal
5	#10592402	—	3	1	no	maximal
6	#10590289	—	3	4	no	maximal
7	C₂×C₃×#634	C₂²	1	2	yes	—
8	C₃³×C₄²	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
10	#88857	—	2	2	no	—
11	C₃×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
12	C₂×C₃²×C₃⋊C₈	C₂²	1	2	yes	—
13	#64079	—	3	1	no	—
14	#64079	—	3	4	no	—
15	#64079	—	6	4	no	—
16	#64078	D₃	1	1	yes	characteristic
17	C₃×#869	—	3	4	no	—
18	C₃²×#97	—	3	1	no	—
19	#63643	—	3	4	no	—
20	C₃×#803	D₃	1	4	yes	—
21	#63566	—	3	16	no	—
22	C₂×C₃³×C₄	D₄	1	2	yes	—
23	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
25	C₂×C₃×C₃⋊Dic₃	—	2	2	no	—
26	C₃×#634	—	2	4	no	—
27	C₃²×C₃⋊C₈	—	2	2	no	—
28	#5082	—	9	1	no	—
29	#4703	—	9	4	no	—
30	C₂×#634	C₂×D₃	1	2	yes	—
31	C₂×C₃×Dic₆	—	3	8	no	—
32	C₂×C₃²×C₈	—	3	2	no	—
33	C₃²×C₄²	—	2	4	no	—
34	C₃²×C₄²	C₂×D₃	1	4	yes	—
35	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
36	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
37	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	3	8	no	—
39	C₄⋊(C₃⋊Dic₃)	—	6	8	no	—
40	#3750	—	6	16	no	—
41	#3750	—	6	8	no	—
42	#3750	—	6	2	no	—
43	C₃×C₄⋊Dic₃	—	6	8	no	—
44	C₂×C₃×C₃⋊C₈	C₂×D₃	1	8	yes	—
45	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
46	C₃³×C₄	—	2	2	no	—
47	C₃³×C₄	#41	1	2	yes	—
48	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
49	C₃×C₃⋊Dic₃	—	4	2	no	—
50	C₃×C₄⋊Q₈	—	9	1	no	—
51	#870	—	9	4	no	—
52	#870	—	9	1	no	—
53	#870	—	18	4	no	—
54	#869	—	3	4	no	—
55	C₃×#97	—	3	4	no	—
56	#835	—	9	4	no	—
57	#816	—	9	16	no	—
58	#803	C₃⋊D₃	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	8	no	—
60	C₂×C₃²×C₄	—	2	4	no	—
61	C₂×C₃²×C₄	D₁₂	1	8	yes	—
62	C₂×C₃²×C₄	#67	1	2	yes	—
63	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
64	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₂×C₃⋊Dic₃	—	2	2	no	—
66	C₂×C₃⋊Dic₃	—	6	16	no	—
67	C₂×C₃⋊Dic₃	—	6	8	no	—
68	C₂×C₃⋊Dic₃	—	6	2	no	—
69	#634	—	2	4	no	—
70	C₂×C₃×Dic₃	—	6	8	no	—
71	C₃×Dic₆	—	6	16	no	—
72	C₃²×C₈	—	6	2	no	—
73	C₃×C₃⋊C₈	—	2	8	no	—
74	#491	—	27	1	no	Sylow
75	C₂×C₃³	#124	1	1	yes	characteristic
76	C₂×C₃³	#137	1	2	yes	—
77	C₂×C₃×Q₈	—	9	2	no	—
78	C₂×Dic₆	—	3	8	no	—
79	C₂×C₃×C₈	—	3	8	no	—
80	C₃×C₄⋊C₄	—	18	2	no	—
81	C₃×C₄²	—	2	4	no	—
82	C₃×C₄²	D₃²	1	4	yes	—
83	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
84	C₄⋊Dic₃	—	6	8	no	—
85	C₄⋊Dic₃	—	9	8	no	—
86	C₄⋊Dic₃	—	9	2	no	—
87	C₄⋊Dic₃	—	18	8	no	—
88	Dic₃⋊C₄	—	18	16	no	—
89	Dic₃⋊C₄	—	18	8	no	—
90	Dic₃⋊C₄	—	18	2	no	—
91	C₂×C₃⋊C₈	C₂×C₃⋊D₃	1	2	yes	—
92	C₂²×C₃²	—	2	4	no	—
93	C₂²×C₃²	C₂×D₁₂	1	4	yes	—
94	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
95	C₃²×C₄	—	2	2	no	—
96	C₃²×C₄	—	2	8	no	—
97	C₃²×C₄	—	2	8	no	—
98	C₃²×C₄	—	4	8	no	—
99	C₃²×C₄	#235	1	8	yes	—
100	C₃²×C₄	#237	1	2	yes	—
101	C₃⋊Dic₃	—	4	2	no	—
102	C₃⋊Dic₃	—	12	16	no	—
103	C₃⋊Dic₃	—	12	8	no	—
104	C₃⋊Dic₃	—	12	2	no	—
105	C₃×Dic₃	—	12	8	no	—
106	C₄⋊Q₈	—	9	1	no	—
107	#125	—	27	1	no	—
108	#103	—	27	4	no	—
109	#97	—	3	1	no	—
110	C₃³	#491	1	1	yes	characteristic, complemented, Sylow
111	C₃×Q₈	—	18	4	no	—
112	C₂×C₃×C₄	—	2	8	no	—
113	C₂×C₃×C₄	—	2	4	no	—
114	C₂×C₃×C₄	—	18	2	no	—
115	C₂×C₃×C₄	#626	1	8	yes	—
116	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
117	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
118	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
119	C₂×Dic₃	—	6	8	no	—
120	C₂×Dic₃	—	18	16	no	—
121	C₂×Dic₃	—	18	8	no	—
122	C₂×Dic₃	—	18	2	no	—
123	Dic₆	—	6	16	no	—
124	C₃×C₈	—	6	8	no	—
125	C₃⋊C₈	—	2	2	no	—
126	C₂×C₃²	—	2	8	no	—
127	C₂×C₃²	—	2	4	no	—
128	C₂×C₃²	#875	1	4	yes	—
129	C₂×C₃²	#909	1	8	yes	—
130	C₂×C₃²	#936	1	1	yes	characteristic
131	C₂×C₃²	#942	1	2	yes	—
132	C₂×Q₈	—	9	2	no	—
133	C₂×C₈	—	3	2	no	—
134	C₄⋊C₄	—	18	2	no	—
135	C₄⋊C₄	—	27	2	no	—
136	C₄⋊C₄	—	54	2	no	—
137	C₄²	D₃×C₃⋊D₃	1	1	yes	characteristic
138	C₂²×C₃	—	2	4	no	—
139	C₂²×C₃	C₂×#626	1	4	yes	—
140	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
141	C₃×C₄	—	2	8	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	8	no	—
144	C₃×C₄	—	4	8	no	—
145	C₃×C₄	—	36	2	no	—
146	C₃×C₄	#3795	1	8	yes	—
147	C₃×C₄	#3828	1	2	yes	—
148	Dic₃	—	12	8	no	—
149	Dic₃	—	36	16	no	—
150	Dic₃	—	36	8	no	—
151	Dic₃	—	36	2	no	—
152	C₃²	—	2	4	no	—
153	C₃²	#4703	1	4	yes	complemented
154	C₃²	#5082	1	1	yes	characteristic, complemented
155	Q₈	—	18	4	no	—
156	C₂×C₄	—	18	2	no	—
157	C₂×C₄	—	54	2	no	—
158	C₂×C₄	#6292	1	2	yes	—
159	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
160	C₈	—	6	2	no	—
161	C₂×C₃	—	2	8	no	—
162	C₂×C₃	—	2	4	no	—
163	C₂×C₃	#63833	1	8	yes	—
164	C₂×C₃	#63860	1	4	yes	—
165	C₂×C₃	#64084	1	1	yes	characteristic
166	C₂×C₃	#64118	1	2	yes	—
167	C₂²	C₂×#6292	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	36	2	no	—
170	C₄	—	108	2	no	—
171	C₄	#89021	1	2	yes	—
172	C₃	—	2	4	no	—
173	C₃	#10590289	1	4	yes	complemented
174	C₃	#10592402	1	1	yes	characteristic, complemented
175	C₂	#11723387	1	2	yes	—
176	C₂	#11723439	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11723911

C₂

yes

maximal, characteristic

C₃×#64078

C₂

yes

maximal, characteristic

C₃²×#803

C₂

yes

maximal, characteristic

#11722040

C₂

yes

—

—

C₂×C₃×#634

C₂²

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

—

C₃×C₄⋊(C₃⋊Dic₃)

—

C₂×C₃²×C₃⋊C₈

C₂²

yes

—

—

—

—

D₃

yes

characteristic

C₃×#869

—

C₃²×#97

—

#63643

—

C₃×#803

D₃

yes

—

#63566

—

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃×C₃⋊Dic₃

—

C₃×#634

—

C₃²×C₃⋊C₈

—

#5082

—

#4703

—

C₂×#634

C₂×D₃

yes

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

—

C₃²×C₄²

—

C₃²×C₄²

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

#3750

—

#3750

—

C₃×C₄⋊Dic₃

—

C₂×C₃×C₃⋊C₈

C₂×D₃

yes

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

#41

yes

—

C₃⋊(C₃⋊Dic₃)

—

C₃×C₃⋊Dic₃

—

C₃×C₄⋊Q₈

—

—

—

—

—

C₃×#97

—

#835

—

#816

—

#803

C₃⋊D₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃²×C₈

—

C₃×C₃⋊C₈

—

—

C₂×C₃³

yes

C₂×C₃³

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

D₃²

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

C₂×C₃⋊D₃

yes

—

C₂²×C₃²

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#235

yes

—

100

C₃²×C₄

#237

yes

—

101

C₃⋊Dic₃

—

102

C₃⋊Dic₃

—

103

C₃⋊Dic₃

—

104

C₃⋊Dic₃

—

105

C₃×Dic₃

—

106

C₄⋊Q₈

—

107

#125

—

108

#103

—

109

#97

—

110

C₃³

#491

yes

characteristic, complemented, Sylow

111

C₃×Q₈

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

#626

yes

—

116

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

117

C₂×C₃×C₄

C₂×D₃²

yes

—

118

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

119

C₂×Dic₃

—

120

C₂×Dic₃

—

121

C₂×Dic₃

—

122

C₂×Dic₃

—

123

Dic₆

—

124

C₃×C₈

—

125

C₃⋊C₈

—

126

C₂×C₃²

—

127

C₂×C₃²

—

128

C₂×C₃²

#875

yes

—

129

C₂×C₃²

#909

yes

—

130

C₂×C₃²

#936

yes

characteristic

131

C₂×C₃²

#942

yes

—

132

C₂×Q₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

D₃×C₃⋊D₃

yes

characteristic

138

C₂²×C₃

—

139

C₂²×C₃

C₂×#626

yes

—

140

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

characteristic

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

#3795

yes

—

147

C₃×C₄

#3828

yes

—

148

Dic₃

—

149

Dic₃

—

150

Dic₃

—

151

Dic₃

—

152

C₃²

—

153

C₃²

#4703

yes

complemented

154

C₃²

#5082

yes

characteristic, complemented

155

Q₈

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#6292

yes

—

159

C₂×C₄

C₂×D₃×C₃⋊D₃

yes

characteristic

160

C₈

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

#63833

yes

—

164

C₂×C₃

#63860

yes

—

165

C₂×C₃

#64084

yes

characteristic

166

C₂×C₃

#64118

yes

—

167

C₂²

C₂×#6292

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

108

—

171

C₄

#89021

yes

—

172

C₃

—

173

C₃

#10590289

yes

complemented

174

C₃

#10592402

yes

characteristic, complemented

175

C₂

#11723387

yes

—

176

C₂

#11723439

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6292
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂×D₃×C₃⋊D₃
Fitting	C₃³×C₄²	C₂²
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₄²	D₃×C₃⋊D₃
hypocenter	C₃³	#491
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6292

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂×D₃×C₃⋊D₃

Fitting

C₃³×C₄²

C₂²

socle, prime radical

C₂²×C₃³

C₂×D₄

hypercenter

C₄²

D₃×C₃⋊D₃

hypocenter

C₃³

#491

LCS

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
110. C₃³	#491	semidirect
153. C₃²	#4703	semidirect
154. C₃²	#5082	semidirect
173. C₃	#10590289	semidirect
174. C₃	#10592402	semidirect

H ≅ G/N

type

110. C₃³

#491

semidirect

153. C₃²

#4703

semidirect

154. C₃²

#5082

semidirect

173. C₃

#10590289

semidirect

174. C₃

#10592402

semidirect

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

subgroup	index
#11723911	2
C₃×#64078	2
C₃²×#803	2
#11722040	2
#10592402	3
#10590289	3

subgroup

index

C₃²×#803

Subgroups — #13158852

Conjugacy classes of subgroups

Complements

Maximal subgroups