Group #13137053 — subgroups

Subgroups — #13137053

Order 1728.

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

normal subgroups only

This group has 310 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	C₉×#870	C₂	1	1	yes	maximal, characteristic
2	C₃×#63431	C₂	1	1	yes	maximal, characteristic
3	#11720516	C₂	1	1	yes	maximal, characteristic
4	#11720268	C₂	1	4	yes	maximal
5	#10590291	—	3	1	no	maximal
6	#10588746	—	3	1	no	maximal
7	C₂×C₃×Dic₁₈	C₂²	1	2	yes	—
8	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
10	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₉×Dic₃⋊C₄	—	2	2	no	—
12	C₃×C₄⋊Dic₉	—	2	2	no	—
13	C₃×#870	D₃	1	1	yes	characteristic
14	C₃×#869	—	3	1	no	—
15	#63630	—	3	1	no	—
16	#63565	—	3	4	no	—
17	C₉×#125	—	3	1	no	—
18	#63431	D₃	1	1	yes	characteristic
19	#63368	—	3	4	no	—
20	#63362	—	3	1	no	—
21	#63362	—	6	1	no	—
22	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₉×Dic₃	—	2	2	no	—
25	C₂×C₃×Dic₉	—	2	2	no	—
26	C₃×Dic₁₈	—	2	4	no	—
27	C₃⋊(C₉⋊C₈)	—	2	2	no	—
28	#5082	—	9	1	no	—
29	#5069	—	9	1	no	—
30	C₂×C₃×Dic₆	—	3	2	no	—
31	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
32	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
33	C₃×C₄⋊Dic₃	—	6	2	no	—
34	C₃×C₄⋊Dic₃	C₂×D₃	1	2	yes	—
35	C₃×Dic₃⋊C₄	—	2	2	no	—
36	C₂×Dic₁₈	C₂×D₃	1	2	yes	—
37	C₉×C₄⋊C₄	—	3	2	no	—
38	C₉×C₄⋊C₄	—	6	2	no	—
39	C₄²×C₉	—	2	1	no	—
40	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
41	C₄⋊Dic₉	—	2	2	no	—
42	C₂×C₉⋊C₈	—	3	2	no	—
43	C₂×C₉⋊C₈	—	6	2	no	—
44	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
45	C₃×C₄×C₉	—	2	2	no	—
46	C₃×C₄×C₉	#41	1	2	yes	—
47	C₉×Dic₃	—	4	2	no	—
48	C₃×Dic₉	—	4	2	no	—
49	C₃×C₄⋊Q₈	—	9	1	no	—
50	C₃×#125	—	3	1	no	—
51	#870	D₉	1	1	yes	characteristic
52	#869	—	3	1	no	—
53	#835	—	9	4	no	—
54	#810	—	9	4	no	—
55	#803	—	9	1	no	—
56	#803	—	9	1	no	—
57	#803	—	18	1	no	—
58	C₂×C₃²×C₄	#67	1	2	yes	—
59	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂×C₃×Dic₃	—	2	2	no	—
61	C₂×C₃×Dic₃	—	6	2	no	—
62	C₃×Dic₆	—	6	4	no	—
63	C₃⋊(C₃⋊C₈)	—	6	2	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	—	2	1	no	—
66	C₂×C₄×C₉	—	6	2	no	—
67	C₂×C₄×C₉	#67	1	2	yes	—
68	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
69	C₂×Dic₉	—	2	2	no	—
70	Dic₁₈	—	2	4	no	—
71	C₉⋊C₈	—	6	4	no	—
72	C₉⋊C₈	—	6	2	no	—
73	#491	—	27	1	no	Sylow
74	C₂×C₃×C₉	#124	1	1	yes	characteristic
75	C₂×C₃×C₉	#137	1	2	yes	—
76	C₂×C₃×Q₈	—	9	2	no	—
77	C₂×Dic₆	—	3	2	no	—
78	C₃×C₄⋊C₄	—	3	2	no	—
79	C₃×C₄⋊C₄	—	6	2	no	—
80	C₃×C₄⋊C₄	—	18	2	no	—
81	C₃×C₄²	—	2	1	no	—
82	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
83	C₃×C₄²	D₃²	1	1	yes	characteristic
84	C₄⋊Dic₃	—	6	2	no	—
85	C₄⋊Dic₃	C₂×D₉	1	2	yes	—
86	Dic₃⋊C₄	—	2	2	no	—
87	C₂×C₃⋊C₈	—	9	2	no	—
88	C₂×C₃⋊C₈	—	9	2	no	—
89	C₂×C₃⋊C₈	—	18	2	no	—
90	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
91	C₃²×C₄	—	2	2	no	—
92	C₃²×C₄	#239	1	2	yes	—
93	C₃×Dic₃	—	4	2	no	—
94	C₃×Dic₃	—	12	2	no	—
95	C₂²×C₉	—	2	1	no	—
96	C₂²×C₉	C₂×#67	1	1	yes	characteristic
97	C₄×C₉	—	2	2	no	—
98	C₄×C₉	—	2	2	no	—
99	C₄×C₉	—	4	2	no	—
100	C₄×C₉	—	12	2	no	—
101	C₄×C₉	#237	1	2	yes	—
102	Dic₉	—	4	2	no	—
103	C₄⋊Q₈	—	9	1	no	—
104	#125	—	3	1	no	—
105	#103	—	27	4	no	—
106	#97	—	27	1	no	—
107	C₃×C₉	#491	1	1	yes	characteristic, complemented, Sylow
108	C₃×Q₈	—	18	4	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	1	no	—
111	C₂×C₃×C₄	—	6	2	no	—
112	C₂×C₃×C₄	—	18	2	no	—
113	C₂×C₃×C₄	#611	1	2	yes	—
114	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
115	C₂×C₃×C₄	#625	1	2	yes	—
116	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
117	C₂×Dic₃	—	2	2	no	—
118	C₂×Dic₃	—	6	2	no	—
119	Dic₆	—	6	4	no	—
120	C₃⋊C₈	—	18	4	no	—
121	C₃⋊C₈	—	18	2	no	—
122	C₃⋊C₈	—	18	2	no	—
123	C₂×C₃²	#947	1	1	yes	characteristic
124	C₂×C₃²	#951	1	2	yes	—
125	C₂×C₉	—	2	2	no	—
126	C₂×C₉	—	2	1	no	—
127	C₂×C₉	#936	1	1	yes	characteristic
128	C₂×C₉	#942	1	2	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×D₉	1	1	yes	characteristic
135	C₂²×C₃	—	2	1	no	—
136	C₂²×C₃	C₂×#611	1	1	yes	characteristic
137	C₂²×C₃	C₂×#625	1	1	yes	characteristic
138	C₃×C₄	—	2	2	no	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	—	2	2	no	—
141	C₃×C₄	—	4	2	no	—
142	C₃×C₄	—	12	2	no	—
143	C₃×C₄	—	36	2	no	—
144	C₃×C₄	#3701	1	2	yes	—
145	C₃×C₄	#3796	1	2	yes	—
146	Dic₃	—	4	2	no	—
147	Dic₃	—	12	2	no	—
148	C₃²	#5069	1	1	yes	characteristic
149	C₉	—	2	1	no	—
150	C₉	#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	#6194	1	2	yes	—
155	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
156	C₈	—	54	2	no	—
157	C₂×C₃	—	2	2	no	—
158	C₂×C₃	—	2	1	no	—
159	C₂×C₃	#63509	1	1	yes	characteristic
160	C₂×C₃	#63511	1	2	yes	—
161	C₂×C₃	#63827	1	2	yes	—
162	C₂×C₃	#63863	1	1	yes	characteristic
163	C₂²	C₂×#6194	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	12	2	no	—
166	C₄	—	36	2	no	—
167	C₄	#88634	1	2	yes	—
168	C₃	—	2	1	no	—
169	C₃	#10588746	1	1	yes	characteristic, complemented
170	C₃	#10590291	1	1	yes	characteristic
171	C₂	#11720887	1	2	yes	—
172	C₂	#11720951	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#870

C₂

yes

maximal, characteristic

C₃×#63431

C₂

yes

maximal, characteristic

#11720516

C₂

yes

maximal, characteristic

#11720268

C₂

yes

—

—

C₂×C₃×Dic₁₈

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₂×C₃⋊(C₉⋊C₈)

C₂²

yes

—

C₉×C₄⋊Dic₃

C₂²

yes

—

C₉×Dic₃⋊C₄

—

C₃×C₄⋊Dic₉

—

C₃×#870

D₃

yes

—

—

—

—

D₃

yes

—

—

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₉×Dic₃

—

C₂×C₃×Dic₉

—

C₃×Dic₁₈

—

C₃⋊(C₉⋊C₈)

—

#5082

—

#5069

—

C₂×C₃×Dic₆

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

—

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

C₂×D₃

yes

—

C₃×Dic₃⋊C₄

—

C₂×Dic₁₈

C₂×D₃

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₄⋊Dic₉

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

#41

yes

—

C₉×Dic₃

—

C₃×Dic₉

—

C₃×C₄⋊Q₈

—

C₃×#125

—

#870

D₉

yes

—

—

—

—

—

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃⋊(C₃⋊C₈)

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×Dic₉

—

Dic₁₈

—

C₉⋊C₈

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₃×C₄²

D₃²

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×D₉

yes

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#239

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×#67

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

100

C₄×C₉

—

101

C₄×C₉

#237

yes

—

102

Dic₉

—

103

C₄⋊Q₈

—

104

#125

—

105

#103

—

106

#97

—

107

C₃×C₉

#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

#611

yes

—

114

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

115

C₂×C₃×C₄

#625

yes

—

116

C₂×C₃×C₄

C₂×D₃²

yes

characteristic

117

C₂×Dic₃

—

118

C₂×Dic₃

—

119

Dic₆

—

120

C₃⋊C₈

—

121

C₃⋊C₈

—

122

C₃⋊C₈

—

123

C₂×C₃²

#947

yes

characteristic

124

C₂×C₃²

#951

yes

—

125

C₂×C₉

—

126

C₂×C₉

—

127

C₂×C₉

#936

yes

characteristic

128

C₂×C₉

#942

yes

—

129

C₂×Q₈

—

130

C₂×C₈

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄²

D₃×D₉

yes

characteristic

135

C₂²×C₃

—

136

C₂²×C₃

C₂×#611

yes

characteristic

137

C₂²×C₃

C₂×#625

yes

characteristic

138

C₃×C₄

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

#3701

yes

—

145

C₃×C₄

#3796

yes

—

146

Dic₃

—

147

Dic₃

—

148

C₃²

#5069

yes

characteristic

149

C₉

—

150

C₉

#5082

yes

characteristic, complemented

151

Q₈

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

#6194

yes

—

155

C₂×C₄

C₂×D₃×D₉

yes

characteristic

156

C₈

—

157

C₂×C₃

—

158

C₂×C₃

—

159

C₂×C₃

#63509

yes

characteristic

160

C₂×C₃

#63511

yes

—

161

C₂×C₃

#63827

yes

—

162

C₂×C₃

#63863

yes

characteristic

163

C₂²

C₂×#6194

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#88634

yes

—

168

C₃

—

169

C₃

#10588746

yes

characteristic, complemented

170

C₃

#10590291

yes

characteristic

171

C₂

#11720887

yes

—

172

C₂

#11720951

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#6194
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×#6194

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
107. C₃×C₉	#491	semidirect
150. C₉	#5082	semidirect
169. C₃	#10588746	semidirect

H ≅ G/N

type

107. C₃×C₉

#491

semidirect

150. C₉

#5082

semidirect

169. C₃

#10588746

semidirect

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

subgroup	index
C₉×#870	2
C₃×#63431	2
#11720516	2
#11720268	2
#10590291	3
#10588746	3

subgroup

index

Subgroups — #13137053

Conjugacy classes of subgroups

Complements

Maximal subgroups