Group #13158854 — subgroups

Subgroups — #13158854

Order 1728.

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

normal subgroups only

This group has 700 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	C₃×#64078	C₂	1	1	yes	maximal, characteristic
2	C₃²×#870	C₂	1	1	yes	maximal, characteristic
3	#11722173	C₂	1	1	yes	maximal, characteristic
4	#11722039	C₂	1	4	yes	maximal
5	#10592768	—	3	1	no	maximal
6	#10590291	—	3	4	no	maximal
7	C₂×C₃×#634	C₂²	1	2	yes	—
8	C₃³×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
10	C₃×C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
11	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₃²×Dic₃⋊C₄	—	2	2	no	—
13	C₃²×#125	—	3	1	no	—
14	#64078	D₃	1	1	yes	characteristic
15	C₃×#870	D₃	1	4	yes	—
16	C₃×#869	—	3	4	no	—
17	#63637	—	3	4	no	—
18	#63630	—	3	1	no	—
19	#63630	—	3	4	no	—
20	#63630	—	6	4	no	—
21	#63565	—	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	C₃×#634	—	2	4	no	—
26	C₂×C₃²×Dic₃	—	2	2	no	—
27	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
28	#5082	—	9	1	no	—
29	#5069	—	9	4	no	—
30	C₂×#634	C₂×D₃	1	2	yes	—
31	C₂×C₃×Dic₆	—	3	8	no	—
32	C₃²×C₄⋊C₄	—	3	2	no	—
33	C₃²×C₄⋊C₄	—	6	2	no	—
34	C₃²×C₄²	—	2	4	no	—
35	C₃²×C₄²	C₂×D₃	1	4	yes	—
36	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	3	8	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	6	8	no	—
41	C₃×C₄⋊Dic₃	—	6	8	no	—
42	C₃×C₄⋊Dic₃	C₂×D₃	1	8	yes	—
43	C₃×Dic₃⋊C₄	—	2	8	no	—
44	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
45	C₃³×C₄	—	2	2	no	—
46	C₃³×C₄	#41	1	2	yes	—
47	C₃×C₃⋊Dic₃	—	4	2	no	—
48	C₃²×Dic₃	—	4	2	no	—
49	C₃×C₄⋊Q₈	—	9	1	no	—
50	C₃×#125	—	3	4	no	—
51	#870	C₃⋊D₃	1	1	yes	characteristic
52	#869	—	3	4	no	—
53	#835	—	9	4	no	—
54	#810	—	9	16	no	—
55	#803	—	9	4	no	—
56	#803	—	9	1	no	—
57	#803	—	18	4	no	—
58	C₂×C₃²×C₄	—	2	8	no	—
59	C₂×C₃²×C₄	—	2	4	no	—
60	C₂×C₃²×C₄	—	6	2	no	—
61	C₂×C₃²×C₄	#67	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	#634	—	2	4	no	—
67	C₂×C₃×Dic₃	—	2	8	no	—
68	C₂×C₃×Dic₃	—	6	8	no	—
69	C₃×Dic₆	—	6	16	no	—
70	C₃⋊(C₃⋊C₈)	—	6	16	no	—
71	C₃⋊(C₃⋊C₈)	—	6	2	no	—
72	C₃⋊(C₃⋊C₈)	—	6	8	no	—
73	#491	—	27	1	no	Sylow
74	C₂×C₃³	#124	1	1	yes	characteristic
75	C₂×C₃³	#137	1	2	yes	—
76	C₂×C₃×Q₈	—	9	2	no	—
77	C₂×Dic₆	—	3	8	no	—
78	C₃×C₄⋊C₄	—	3	8	no	—
79	C₃×C₄⋊C₄	—	6	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	C₂×C₃⋊D₃	1	2	yes	—
86	Dic₃⋊C₄	—	2	2	no	—
87	C₂×C₃⋊C₈	—	9	8	no	—
88	C₂×C₃⋊C₈	—	9	2	no	—
89	C₂×C₃⋊C₈	—	18	8	no	—
90	C₂²×C₃²	—	2	4	no	—
91	C₂²×C₃²	C₂×#67	1	4	yes	—
92	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
93	C₃²×C₄	—	2	2	no	—
94	C₃²×C₄	—	2	8	no	—
95	C₃²×C₄	—	2	8	no	—
96	C₃²×C₄	—	4	8	no	—
97	C₃²×C₄	—	12	2	no	—
98	C₃²×C₄	#237	1	2	yes	—
99	C₃²×C₄	#239	1	8	yes	—
100	C₃⋊Dic₃	—	4	2	no	—
101	C₃×Dic₃	—	4	8	no	—
102	C₃×Dic₃	—	12	8	no	—
103	C₄⋊Q₈	—	9	1	no	—
104	#125	—	3	1	no	—
105	#103	—	27	4	no	—
106	#97	—	27	1	no	—
107	C₃³	#491	1	1	yes	characteristic, complemented, Sylow
108	C₃×Q₈	—	18	4	no	—
109	C₂×C₃×C₄	—	2	8	no	—
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	6	8	no	—
112	C₂×C₃×C₄	—	18	2	no	—
113	C₂×C₃×C₄	#625	1	8	yes	—
114	C₂×C₃×C₄	#638	1	2	yes	—
115	C₂×C₃×C₄	C₂×D₃²	1	4	yes	—
116	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
117	C₂×Dic₃	—	2	2	no	—
118	C₂×Dic₃	—	6	8	no	—
119	Dic₆	—	6	16	no	—
120	C₃⋊C₈	—	18	16	no	—
121	C₃⋊C₈	—	18	8	no	—
122	C₃⋊C₈	—	18	2	no	—
123	C₂×C₃²	—	2	8	no	—
124	C₂×C₃²	—	2	4	no	—
125	C₂×C₃²	#936	1	1	yes	characteristic
126	C₂×C₃²	#942	1	2	yes	—
127	C₂×C₃²	#947	1	4	yes	—
128	C₂×C₃²	#951	1	8	yes	—
129	C₂×Q₈	—	9	2	no	—
130	C₂×C₈	—	27	2	no	—
131	C₄⋊C₄	—	3	2	no	—
132	C₄⋊C₄	—	6	2	no	—
133	C₄⋊C₄	—	18	2	no	—
134	C₄²	D₃×C₃⋊D₃	1	1	yes	characteristic
135	C₂²×C₃	—	2	4	no	—
136	C₂²×C₃	C₂×#625	1	4	yes	—
137	C₂²×C₃	C₂×#638	1	1	yes	characteristic
138	C₃×C₄	—	2	8	no	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	—	2	8	no	—
141	C₃×C₄	—	4	8	no	—
142	C₃×C₄	—	12	8	no	—
143	C₃×C₄	—	36	2	no	—
144	C₃×C₄	#3796	1	8	yes	—
145	C₃×C₄	#3832	1	2	yes	—
146	Dic₃	—	4	2	no	—
147	Dic₃	—	12	8	no	—
148	C₃²	—	2	4	no	—
149	C₃²	#5069	1	4	yes	complemented
150	C₃²	#5082	1	1	yes	characteristic, complemented
151	Q₈	—	18	4	no	—
152	C₂×C₄	—	6	2	no	—
153	C₂×C₄	—	18	2	no	—
154	C₂×C₄	#6290	1	2	yes	—
155	C₂×C₄	C₂×D₃×C₃⋊D₃	1	1	yes	characteristic
156	C₈	—	54	2	no	—
157	C₂×C₃	—	2	8	no	—
158	C₂×C₃	—	2	4	no	—
159	C₂×C₃	#63827	1	8	yes	—
160	C₂×C₃	#63863	1	4	yes	—
161	C₂×C₃	#64156	1	1	yes	characteristic
162	C₂×C₃	#64160	1	2	yes	—
163	C₂²	C₂×#6290	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	12	2	no	—
166	C₄	—	36	2	no	—
167	C₄	#89017	1	2	yes	—
168	C₃	—	2	4	no	—
169	C₃	#10590291	1	4	yes	complemented
170	C₃	#10592768	1	1	yes	characteristic, complemented
171	C₂	#11723373	1	2	yes	—
172	C₂	#11723436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#64078

C₂

yes

maximal, characteristic

C₃²×#870

C₂

yes

maximal, characteristic

#11722173

C₂

yes

maximal, characteristic

#11722039

C₂

yes

—

—

C₂×C₃×#634

C₂²

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₃×C₄⋊(C₃⋊Dic₃)

—

C₃²×C₄⋊Dic₃

C₂²

yes

—

C₃²×Dic₃⋊C₄

—

C₃²×#125

—

#64078

D₃

yes

characteristic

C₃×#870

D₃

yes

—

—

—

—

—

—

—

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃×C₃⋊Dic₃

—

C₃×#634

—

C₂×C₃²×Dic₃

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#5082

—

#5069

—

C₂×#634

C₂×D₃

yes

—

C₂×C₃×Dic₆

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

—

C₃²×C₄²

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

C₂×D₃

yes

—

C₃×Dic₃⋊C₄

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

#41

yes

—

C₃×C₃⋊Dic₃

—

C₃²×Dic₃

—

C₃×C₄⋊Q₈

—

C₃×#125

—

#870

C₃⋊D₃

yes

—

—

—

—

—

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

—

C₂×C₃³

yes

C₂×C₃³

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

D₃²

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₃⋊D₃

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₂×#67

yes

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#237

yes

—

C₃²×C₄

#239

yes

—

100

C₃⋊Dic₃

—

101

C₃×Dic₃

—

102

C₃×Dic₃

—

103

C₄⋊Q₈

—

104

#125

—

105

#103

—

106

#97

—

107

C₃³

#491

yes

characteristic, complemented, Sylow

108

C₃×Q₈

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

#625

yes

—

114

C₂×C₃×C₄

#638

yes

—

115

C₂×C₃×C₄

C₂×D₃²

yes

—

116

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

117

C₂×Dic₃

—

118

C₂×Dic₃

—

119

Dic₆

—

120

C₃⋊C₈

—

121

C₃⋊C₈

—

122

C₃⋊C₈

—

123

C₂×C₃²

—

124

C₂×C₃²

—

125

C₂×C₃²

#936

yes

characteristic

126

C₂×C₃²

#942

yes

—

127

C₂×C₃²

#947

yes

—

128

C₂×C₃²

#951

yes

—

129

C₂×Q₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

D₃×C₃⋊D₃

yes

characteristic

135

C₂²×C₃

—

136

C₂²×C₃

C₂×#625

yes

—

137

C₂²×C₃

C₂×#638

yes

characteristic

138

C₃×C₄

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

#3796

yes

—

145

C₃×C₄

#3832

yes

—

146

Dic₃

—

147

Dic₃

—

148

C₃²

—

149

C₃²

#5069

yes

complemented

150

C₃²

#5082

yes

characteristic, complemented

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#6290

yes

—

155

C₂×C₄

C₂×D₃×C₃⋊D₃

yes

characteristic

156

C₈

—

157

C₂×C₃

—

158

C₂×C₃

—

159

C₂×C₃

#63827

yes

—

160

C₂×C₃

#63863

yes

—

161

C₂×C₃

#64156

yes

characteristic

162

C₂×C₃

#64160

yes

—

163

C₂²

C₂×#6290

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#89017

yes

—

168

C₃

—

169

C₃

#10590291

yes

complemented

170

C₃

#10592768

yes

characteristic, complemented

171

C₂

#11723373

yes

—

172

C₂

#11723436

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6290
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×#6290

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
107. C₃³	#491	semidirect
149. C₃²	#5069	semidirect
150. C₃²	#5082	semidirect
169. C₃	#10590291	semidirect
170. C₃	#10592768	semidirect

H ≅ G/N

type

107. C₃³

#491

semidirect

149. C₃²

#5069

semidirect

150. C₃²

#5082

semidirect

169. C₃

#10590291

semidirect

170. C₃

#10592768

semidirect

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

subgroup	index
C₃×#64078	2
C₃²×#870	2
#11722173	2
#11722039	2
#10592768	3
#10590291	3

subgroup

index

C₃²×#870

Subgroups — #13158854

Conjugacy classes of subgroups

Complements

Maximal subgroups