Group #13158864 — subgroups

Subgroups — #13158864

Order 1728.

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

normal subgroups only

This group has 812 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	#11723910	C₂	1	1	yes	maximal, characteristic
2	C₃²×#870	C₂	1	1	yes	maximal, characteristic
3	C₃×#63630	C₂	1	1	yes	maximal, characteristic
4	#11722041	C₂	1	4	yes	maximal
5	#10592768	—	3	1	no	maximal
6	#10590295	—	3	4	no	maximal
7	C₂×#6308	C₂²	1	2	yes	—
8	C₃³×C₄²	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	2	no	—
10	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
11	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₃²×Dic₃⋊C₄	—	2	2	no	—
13	C₃²×#125	—	3	1	no	—
14	#64078	—	3	1	no	—
15	#64078	—	3	4	no	—
16	#64078	—	6	4	no	—
17	C₃×#870	D₃	1	4	yes	—
18	#63637	—	3	4	no	—
19	#63630	D₃	1	1	yes	characteristic
20	C₃×#803	—	3	4	no	—
21	#63567	—	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	#6308	—	2	4	no	—
26	C₂×C₃²×Dic₃	—	2	2	no	—
27	C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
28	#5069	—	9	4	no	—
29	#4703	—	9	1	no	—
30	C₂×#634	—	3	2	no	—
31	C₂×#634	—	3	8	no	—
32	C₂×#634	—	6	8	no	—
33	C₃²×C₄⋊C₄	—	3	2	no	—
34	C₃²×C₄⋊C₄	—	6	2	no	—
35	C₃²×C₄²	—	2	4	no	—
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₃²×C₄²	C₂×D₃	1	4	yes	—
38	C₄⋊(C₃⋊Dic₃)	—	6	16	no	—
39	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
40	C₄⋊(C₃⋊Dic₃)	—	6	8	no	—
41	C₂×C₃⋊(C₃⋊C₈)	C₂×D₃	1	2	yes	—
42	C₃×C₄⋊Dic₃	C₂×D₃	1	8	yes	—
43	C₃×Dic₃⋊C₄	—	2	8	no	—
44	C₂×C₃×C₃⋊C₈	—	3	8	no	—
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	C₃²×Dic₃	—	4	2	no	—
50	C₃×#125	—	3	4	no	—
51	#870	C₃⋊D₃	1	1	yes	characteristic
52	#869	—	9	4	no	—
53	#869	—	9	1	no	—
54	#869	—	18	4	no	—
55	C₃×#97	—	9	1	no	—
56	#816	—	9	4	no	—
57	#810	—	9	16	no	—
58	#803	—	3	4	no	—
59	C₂×C₃²×C₄	—	2	8	no	—
60	C₂×C₃²×C₄	—	2	4	no	—
61	C₂×C₃²×C₄	—	6	2	no	—
62	C₂×C₃²×C₄	D₁₂	1	2	yes	—
63	C₂×C₃²×C₄	#67	1	8	yes	—
64	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
65	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
66	C₂×C₃⋊Dic₃	—	6	16	no	—
67	C₂×C₃⋊Dic₃	—	6	2	no	—
68	C₂×C₃⋊Dic₃	—	6	8	no	—
69	#634	—	6	4	no	—
70	#634	—	6	16	no	—
71	#634	—	12	16	no	—
72	C₂×C₃×Dic₃	—	2	8	no	—
73	C₃⋊(C₃⋊C₈)	—	2	2	no	—
74	C₃×C₃⋊C₈	—	6	8	no	—
75	#491	—	27	1	no	Sylow
76	C₂×C₃³	#124	1	1	yes	characteristic
77	C₂×C₃³	#137	1	2	yes	—
78	C₂×Dic₆	—	9	8	no	—
79	C₂×Dic₆	—	9	2	no	—
80	C₂×Dic₆	—	18	8	no	—
81	C₂×C₃×C₈	—	9	2	no	—
82	C₃×C₄⋊C₄	—	3	8	no	—
83	C₃×C₄⋊C₄	—	6	8	no	—
84	C₃×C₄²	—	2	4	no	—
85	C₃×C₄²	D₃²	1	4	yes	—
86	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
87	C₄⋊Dic₃	—	18	16	no	—
88	C₄⋊Dic₃	—	18	8	no	—
89	C₄⋊Dic₃	—	18	2	no	—
90	C₄⋊Dic₃	C₂×C₃⋊D₃	1	2	yes	—
91	Dic₃⋊C₄	—	2	2	no	—
92	C₂×C₃⋊C₈	—	3	8	no	—
93	C₂²×C₃²	—	2	4	no	—
94	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
95	C₂²×C₃²	C₂×#67	1	4	yes	—
96	C₃²×C₄	—	2	16	no	—
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	—	2	8	no	—
99	C₃²×C₄	—	2	8	no	—
100	C₃²×C₄	—	12	2	no	—
101	C₃²×C₄	#235	1	2	yes	—
102	C₃²×C₄	#239	1	8	yes	—
103	C₃⋊Dic₃	—	12	16	no	—
104	C₃⋊Dic₃	—	12	2	no	—
105	C₃⋊Dic₃	—	12	8	no	—
106	C₃×Dic₃	—	4	8	no	—
107	C₄⋊Q₈	—	27	1	no	—
108	#125	—	3	1	no	—
109	#103	—	27	4	no	—
110	#97	—	9	1	no	—
111	C₃³	#491	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₃×C₄	—	2	8	no	—
113	C₂×C₃×C₄	—	2	4	no	—
114	C₂×C₃×C₄	—	6	8	no	—
115	C₂×C₃×C₄	#626	1	8	yes	—
116	C₂×C₃×C₄	#638	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	—	2	2	no	—
120	C₂×Dic₃	—	18	16	no	—
121	C₂×Dic₃	—	18	8	no	—
122	C₂×Dic₃	—	18	2	no	—
123	Dic₆	—	18	16	no	—
124	Dic₆	—	18	4	no	—
125	Dic₆	—	36	16	no	—
126	C₃×C₈	—	18	2	no	—
127	C₃⋊C₈	—	6	8	no	—
128	C₂×C₃²	—	2	8	no	—
129	C₂×C₃²	—	2	4	no	—
130	C₂×C₃²	#875	1	1	yes	characteristic
131	C₂×C₃²	#909	1	2	yes	—
132	C₂×C₃²	#947	1	4	yes	—
133	C₂×C₃²	#951	1	8	yes	—
134	C₂×Q₈	—	27	2	no	—
135	C₂×C₈	—	9	2	no	—
136	C₄⋊C₄	—	3	2	no	—
137	C₄⋊C₄	—	6	2	no	—
138	C₄⋊C₄	—	54	2	no	—
139	C₄²	D₃×C₃⋊D₃	1	1	yes	characteristic
140	C₂²×C₃	—	2	4	no	—
141	C₂²×C₃	C₂×#626	1	4	yes	—
142	C₂²×C₃	C₂×#638	1	1	yes	characteristic
143	C₃×C₄	—	2	16	no	—
144	C₃×C₄	—	2	8	no	—
145	C₃×C₄	—	2	2	no	—
146	C₃×C₄	—	2	8	no	—
147	C₃×C₄	—	12	8	no	—
148	C₃×C₄	#3799	1	8	yes	—
149	C₃×C₄	#3832	1	2	yes	—
150	Dic₃	—	4	2	no	—
151	Dic₃	—	36	16	no	—
152	Dic₃	—	36	8	no	—
153	Dic₃	—	36	2	no	—
154	C₃²	—	2	4	no	—
155	C₃²	#4703	1	1	yes	characteristic, complemented
156	C₃²	#5069	1	4	yes	complemented
157	Q₈	—	54	4	no	—
158	C₂×C₄	—	6	2	no	—
159	C₂×C₄	—	54	2	no	—
160	C₂×C₄	#6291	1	2	yes	—
161	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
162	C₈	—	18	2	no	—
163	C₂×C₃	—	2	8	no	—
164	C₂×C₃	—	2	4	no	—
165	C₂×C₃	#63831	1	8	yes	—
166	C₂×C₃	#63864	1	4	yes	—
167	C₂×C₃	#64156	1	1	yes	characteristic
168	C₂×C₃	#64160	1	2	yes	—
169	C₂²	C₂×#6291	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	12	2	no	—
172	C₄	—	108	2	no	—
173	C₄	#89018	1	2	yes	—
174	C₃	—	2	4	no	—
175	C₃	#10590295	1	4	yes	complemented
176	C₃	#10592768	1	1	yes	characteristic, complemented
177	C₂	#11723379	1	2	yes	—
178	C₂	#11723440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11723910

C₂

yes

maximal, characteristic

C₃²×#870

C₂

yes

maximal, characteristic

C₃×#63630

C₂

yes

maximal, characteristic

#11722041

C₂

yes

—

—

C₂²

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊Dic₃))

—

C₂×C₃×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃²×C₄⋊Dic₃

C₂²

yes

—

C₃²×Dic₃⋊C₄

—

C₃²×#125

—

—

—

—

D₃

yes

—

#63637

—

#63630

D₃

yes

characteristic

C₃×#803

—

#63567

—

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

#6308

—

C₂×C₃²×Dic₃

—

C₃×C₃⋊(C₃⋊C₈)

—

—

—

—

—

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃²×C₄²

C₂×D₃

yes

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

C₂×D₃

yes

—

C₃×C₄⋊Dic₃

C₂×D₃

yes

—

C₃×Dic₃⋊C₄

—

C₂×C₃×C₃⋊C₈

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

#41

yes

—

C₃⋊(C₃⋊Dic₃)

—

C₃²×Dic₃

—

C₃×#125

—

#870

C₃⋊D₃

yes

—

—

—

C₃×#97

—

#816

—

#810

—

#803

—

C₂×C₃²×C₄

—

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

—

#634

—

#634

—

#634

—

C₂×C₃×Dic₃

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

—

C₂×C₃³

yes

C₂×C₃³

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

D₃²

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₃⋊D₃

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂×D₁₂

yes

characteristic

C₂²×C₃²

C₂×#67

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

100

C₃²×C₄

—

101

C₃²×C₄

#235

yes

—

102

C₃²×C₄

#239

yes

—

103

C₃⋊Dic₃

—

104

C₃⋊Dic₃

—

105

C₃⋊Dic₃

—

106

C₃×Dic₃

—

107

C₄⋊Q₈

—

108

#125

—

109

#103

—

110

#97

—

111

C₃³

#491

yes

characteristic, complemented, Sylow

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

#626

yes

—

116

C₂×C₃×C₄

#638

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

Dic₆

—

125

Dic₆

—

126

C₃×C₈

—

127

C₃⋊C₈

—

128

C₂×C₃²

—

129

C₂×C₃²

—

130

C₂×C₃²

#875

yes

characteristic

131

C₂×C₃²

#909

yes

—

132

C₂×C₃²

#947

yes

—

133

C₂×C₃²

#951

yes

—

134

C₂×Q₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

D₃×C₃⋊D₃

yes

characteristic

140

C₂²×C₃

—

141

C₂²×C₃

C₂×#626

yes

—

142

C₂²×C₃

C₂×#638

yes

characteristic

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

#3799

yes

—

149

C₃×C₄

#3832

yes

—

150

Dic₃

—

151

Dic₃

—

152

Dic₃

—

153

Dic₃

—

154

C₃²

—

155

C₃²

#4703

yes

characteristic, complemented

156

C₃²

#5069

yes

complemented

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#6291

yes

—

161

C₂×C₄

C₂×D₃×C₃⋊D₃

yes

characteristic

162

C₈

—

163

C₂×C₃

—

164

C₂×C₃

—

165

C₂×C₃

#63831

yes

—

166

C₂×C₃

#63864

yes

—

167

C₂×C₃

#64156

yes

characteristic

168

C₂×C₃

#64160

yes

—

169

C₂²

C₂×#6291

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

108

—

173

C₄

#89018

yes

—

174

C₃

—

175

C₃

#10590295

yes

complemented

176

C₃

#10592768

yes

characteristic, complemented

177

C₂

#11723379

yes

—

178

C₂

#11723440

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6291
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×#6291

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
111. C₃³	#491	semidirect
155. C₃²	#4703	semidirect
156. C₃²	#5069	semidirect
175. C₃	#10590295	semidirect
176. C₃	#10592768	semidirect

H ≅ G/N

type

111. C₃³

#491

semidirect

155. C₃²

#4703

semidirect

156. C₃²

#5069

semidirect

175. C₃

#10590295

semidirect

176. C₃

#10592768

semidirect

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

subgroup	index
#11723910	2
C₃²×#870	2
C₃×#63630	2
#11722041	2
#10592768	3
#10590295	3

subgroup

index

C₃²×#870

Subgroups — #13158864

Conjugacy classes of subgroups

Complements

Maximal subgroups