Group #13137056 — subgroups

Subgroups — #13137056

Order 1728.

338 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 89 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	#11721706	C₂	1	1	yes	maximal, characteristic
2	C₉×#870	C₂	1	1	yes	maximal, characteristic
3	C₃×#63362	C₂	1	1	yes	maximal, characteristic
4	#11720269	C₂	1	4	yes	maximal
5	#10590295	—	3	1	no	maximal
6	#10588746	—	3	1	no	maximal
7	C₂×#6226	C₂²	1	2	yes	—
8	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊Dic₉)	—	2	2	no	—
10	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₉×Dic₃⋊C₄	—	2	2	no	—
12	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
13	#64078	—	3	1	no	—
14	C₃×#870	D₃	1	1	yes	characteristic
15	C₃×#803	—	3	1	no	—
16	#63567	—	3	4	no	—
17	C₉×#125	—	3	1	no	—
18	#63431	—	3	1	no	—
19	#63431	—	6	1	no	—
20	#63368	—	3	4	no	—
21	#63362	D₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊Dic₉	—	2	2	no	—
25	#6226	—	2	4	no	—
26	C₂×C₉×Dic₃	—	2	2	no	—
27	C₃×C₉⋊C₈	—	2	2	no	—
28	#5069	—	9	1	no	—
29	#4703	—	9	1	no	—
30	C₂×#634	—	3	2	no	—
31	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
32	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
33	C₃×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
34	C₃×Dic₃⋊C₄	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	—	3	2	no	—
36	C₂×Dic₁₈	—	3	2	no	—
37	C₂×Dic₁₈	—	6	2	no	—
38	C₉×C₄⋊C₄	—	3	2	no	—
39	C₉×C₄⋊C₄	—	6	2	no	—
40	C₄²×C₉	—	2	1	no	—
41	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
42	C₄⋊Dic₉	—	6	4	no	—
43	C₄⋊Dic₉	—	6	2	no	—
44	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
45	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
46	C₃×C₄×C₉	—	2	2	no	—
47	C₃×C₄×C₉	#41	1	2	yes	—
48	C₃⋊Dic₉	—	4	2	no	—
49	C₉×Dic₃	—	4	2	no	—
50	C₃×#125	—	3	1	no	—
51	#870	D₉	1	1	yes	characteristic
52	#869	—	9	1	no	—
53	#869	—	9	1	no	—
54	#869	—	18	1	no	—
55	C₃×#97	—	9	1	no	—
56	#816	—	9	4	no	—
57	#810	—	9	4	no	—
58	#803	—	3	1	no	—
59	C₂×C₃²×C₄	#67	1	2	yes	—
60	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
61	C₂×C₃⋊Dic₃	—	6	2	no	—
62	#634	—	6	4	no	—
63	C₂×C₃×Dic₃	—	2	2	no	—
64	C₃×C₃⋊C₈	—	6	2	no	—
65	C₂×C₄×C₉	—	2	2	no	—
66	C₂×C₄×C₉	—	2	1	no	—
67	C₂×C₄×C₉	—	6	2	no	—
68	C₂×C₄×C₉	D₁₂	1	2	yes	—
69	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
70	C₂×Dic₉	—	6	4	no	—
71	C₂×Dic₉	—	6	2	no	—
72	Dic₁₈	—	6	4	no	—
73	Dic₁₈	—	12	4	no	—
74	C₉⋊C₈	—	2	2	no	—
75	#491	—	27	1	no	Sylow
76	C₂×C₃×C₉	#124	1	1	yes	characteristic
77	C₂×C₃×C₉	#137	1	2	yes	—
78	C₂×Dic₆	—	9	2	no	—
79	C₂×Dic₆	—	9	2	no	—
80	C₂×Dic₆	—	18	2	no	—
81	C₂×C₃×C₈	—	9	2	no	—
82	C₃×C₄⋊C₄	—	3	2	no	—
83	C₃×C₄⋊C₄	—	6	2	no	—
84	C₃×C₄²	—	2	1	no	—
85	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
86	C₃×C₄²	D₃²	1	1	yes	characteristic
87	C₄⋊Dic₃	—	18	4	no	—
88	C₄⋊Dic₃	—	18	2	no	—
89	C₄⋊Dic₃	—	18	2	no	—
90	C₄⋊Dic₃	C₂×D₉	1	2	yes	—
91	Dic₃⋊C₄	—	2	2	no	—
92	C₂×C₃⋊C₈	—	3	2	no	—
93	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
94	C₃²×C₄	—	2	2	no	—
95	C₃²×C₄	#239	1	2	yes	—
96	C₃⋊Dic₃	—	12	2	no	—
97	C₃×Dic₃	—	4	2	no	—
98	C₂²×C₉	—	2	1	no	—
99	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
100	C₄×C₉	—	2	4	no	—
101	C₄×C₉	—	2	2	no	—
102	C₄×C₉	—	2	2	no	—
103	C₄×C₉	—	12	2	no	—
104	C₄×C₉	#235	1	2	yes	—
105	Dic₉	—	12	4	no	—
106	Dic₉	—	12	2	no	—
107	C₄⋊Q₈	—	27	1	no	—
108	#125	—	3	1	no	—
109	#103	—	27	4	no	—
110	#97	—	9	1	no	—
111	C₃×C₉	#491	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	6	2	no	—
115	C₂×C₃×C₄	#611	1	2	yes	—
116	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
117	C₂×C₃×C₄	#626	1	2	yes	—
118	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
119	C₂×Dic₃	—	2	2	no	—
120	C₂×Dic₃	—	18	4	no	—
121	C₂×Dic₃	—	18	2	no	—
122	C₂×Dic₃	—	18	2	no	—
123	Dic₆	—	18	4	no	—
124	Dic₆	—	18	4	no	—
125	Dic₆	—	36	4	no	—
126	C₃×C₈	—	18	2	no	—
127	C₃⋊C₈	—	6	2	no	—
128	C₂×C₃²	#947	1	1	yes	characteristic
129	C₂×C₃²	#951	1	2	yes	—
130	C₂×C₉	—	2	2	no	—
131	C₂×C₉	—	2	1	no	—
132	C₂×C₉	#875	1	1	yes	characteristic
133	C₂×C₉	#909	1	2	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×D₉	1	1	yes	characteristic
140	C₂²×C₃	—	2	1	no	—
141	C₂²×C₃	C₂×#611	1	1	yes	characteristic
142	C₂²×C₃	C₂×#626	1	1	yes	characteristic
143	C₃×C₄	—	2	4	no	—
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	2	2	no	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	12	2	no	—
148	C₃×C₄	#3701	1	2	yes	—
149	C₃×C₄	#3799	1	2	yes	—
150	Dic₃	—	4	2	no	—
151	Dic₃	—	36	4	no	—
152	Dic₃	—	36	2	no	—
153	Dic₃	—	36	2	no	—
154	C₃²	#5069	1	1	yes	characteristic
155	C₉	—	2	1	no	—
156	C₉	#4703	1	1	yes	characteristic, complemented
157	Q₈	—	54	4	no	—
158	C₂×C₄	—	6	2	no	—
159	C₂×C₄	—	54	2	no	—
160	C₂×C₄	#6195	1	2	yes	—
161	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
162	C₈	—	18	2	no	—
163	C₂×C₃	—	2	2	no	—
164	C₂×C₃	—	2	1	no	—
165	C₂×C₃	#63509	1	1	yes	characteristic
166	C₂×C₃	#63511	1	2	yes	—
167	C₂×C₃	#63831	1	2	yes	—
168	C₂×C₃	#63864	1	1	yes	characteristic
169	C₂²	C₂×#6195	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	12	2	no	—
172	C₄	—	108	2	no	—
173	C₄	#88635	1	2	yes	—
174	C₃	—	2	1	no	—
175	C₃	#10588746	1	1	yes	characteristic, complemented
176	C₃	#10590295	1	1	yes	characteristic
177	C₂	#11720891	1	2	yes	—
178	C₂	#11720953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721706

C₂

yes

maximal, characteristic

C₉×#870

C₂

yes

maximal, characteristic

C₃×#63362

C₂

yes

maximal, characteristic

#11720269

C₂

yes

—

—

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

—

C₉×C₄⋊Dic₃

C₂²

yes

—

C₉×Dic₃⋊C₄

—

C₂×C₃×C₉⋊C₈

C₂²

yes

—

#64078

—

C₃×#870

D₃

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

#6226

—

C₂×C₉×Dic₃

—

C₃×C₉⋊C₈

—

#5069

—

#4703

—

C₂×#634

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₃×C₄⋊Dic₃

C₂×D₃

yes

—

C₃×Dic₃⋊C₄

—

C₂×C₃×C₃⋊C₈

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

C₂×D₃

yes

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

#41

yes

—

C₃⋊Dic₉

—

C₉×Dic₃

—

C₃×#125

—

#870

D₉

yes

—

—

—

C₃×#97

—

#816

—

#810

—

#803

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₂×C₃×Dic₃

—

C₃×C₃⋊C₈

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

D₁₂

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₃×C₄²

D₃²

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×D₉

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#239

yes

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×D₁₂

yes

characteristic

100

C₄×C₉

—

101

C₄×C₉

—

102

C₄×C₉

—

103

C₄×C₉

—

104

C₄×C₉

#235

yes

—

105

Dic₉

—

106

Dic₉

—

107

C₄⋊Q₈

—

108

#125

—

109

#103

—

110

#97

—

111

C₃×C₉

#491

yes

characteristic, complemented, Sylow

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

#611

yes

—

116

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

117

C₂×C₃×C₄

#626

yes

—

118

C₂×C₃×C₄

C₂×D₃²

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

#947

yes

characteristic

129

C₂×C₃²

#951

yes

—

130

C₂×C₉

—

131

C₂×C₉

—

132

C₂×C₉

#875

yes

characteristic

133

C₂×C₉

#909

yes

—

134

C₂×Q₈

—

135

C₂×C₈

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

D₃×D₉

yes

characteristic

140

C₂²×C₃

—

141

C₂²×C₃

C₂×#611

yes

characteristic

142

C₂²×C₃

C₂×#626

yes

characteristic

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

—

147

C₃×C₄

—

148

C₃×C₄

#3701

yes

—

149

C₃×C₄

#3799

yes

—

150

Dic₃

—

151

Dic₃

—

152

Dic₃

—

153

Dic₃

—

154

C₃²

#5069

yes

characteristic

155

C₉

—

156

C₉

#4703

yes

characteristic, complemented

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

#6195

yes

—

161

C₂×C₄

C₂×D₃×D₉

yes

characteristic

162

C₈

—

163

C₂×C₃

—

164

C₂×C₃

—

165

C₂×C₃

#63509

yes

characteristic

166

C₂×C₃

#63511

yes

—

167

C₂×C₃

#63831

yes

—

168

C₂×C₃

#63864

yes

characteristic

169

C₂²

C₂×#6195

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

108

—

173

C₄

#88635

yes

—

174

C₃

—

175

C₃

#10588746

yes

characteristic, complemented

176

C₃

#10590295

yes

characteristic

177

C₂

#11720891

yes

—

178

C₂

#11720953

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6195
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₃×C₄²×C₉	C₂²
socle, prime radical	C₂²×C₃²	C₂×#67
hypercenter	C₄²	D₃×D₉
hypocenter	C₃×C₉	#491
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#6195

commutator, cosocle

C₂×C₃×C₄×C₉

C₂³

Frattini

C₂×C₃×C₄

C₂×D₃²

Fitting

C₃×C₄²×C₉

C₂²

socle, prime radical

C₂²×C₃²

C₂×#67

hypercenter

C₄²

D₃×D₉

hypocenter

C₃×C₉

#491

LCS

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
111. C₃×C₉	#491	semidirect
156. C₉	#4703	semidirect
175. C₃	#10588746	semidirect

H ≅ G/N

type

111. C₃×C₉

#491

semidirect

156. C₉

#4703

semidirect

175. C₃

#10588746

semidirect

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

subgroup	index
#11721706	2
C₉×#870	2
C₃×#63362	2
#11720269	2
#10590295	3
#10588746	3

subgroup

index

Subgroups — #13137056

Conjugacy classes of subgroups

Complements

Maximal subgroups