Group #13137142 — subgroups

Subgroups — #13137142

Order 1728.

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

normal subgroups only

This group has 326 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	#11721707	C₂	1	1	yes	maximal, characteristic
2	C₉×#869	C₂	1	1	yes	maximal, characteristic
3	C₃×#63362	C₂	1	1	yes	maximal, characteristic
4	#11720298	C₂	1	4	yes	maximal
5	#10590289	—	3	1	no	maximal
6	#10588762	—	3	1	no	maximal
7	C₂×C₉×Dic₆	C₂²	1	2	yes	—
8	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
10	#88534	—	2	2	no	—
11	C₉×C₄⋊Dic₃	—	2	2	no	—
12	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
13	#64079	—	3	1	no	—
14	C₃×#869	D₃	1	1	yes	characteristic
15	C₃×#803	—	3	1	no	—
16	#63566	—	3	4	no	—
17	C₉×C₄⋊Q₈	—	3	1	no	—
18	#63432	—	3	1	no	—
19	#63432	—	6	1	no	—
20	#63395	—	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	C₂×C₉×Dic₃	—	2	2	no	—
26	C₉×Dic₆	—	2	4	no	—
27	C₃×C₉⋊C₈	—	2	2	no	—
28	#5082	—	9	1	no	—
29	#4703	—	9	1	no	—
30	C₂×C₃×Dic₆	C₂×D₃	1	2	yes	—
31	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
32	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
33	#3750	—	6	2	no	—
34	C₃×C₄⋊Dic₃	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	—	3	2	no	—
36	C₂×Q₈×C₉	—	3	2	no	—
37	C₉×C₄⋊C₄	—	6	2	no	—
38	C₄²×C₉	—	2	1	no	—
39	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
40	C₄⋊Dic₉	—	3	2	no	—
41	C₄⋊Dic₉	—	6	2	no	—
42	Dic₉⋊C₄	—	6	4	no	—
43	Dic₉⋊C₄	—	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×C₄⋊Q₈	—	3	1	no	—
51	#870	—	9	1	no	—
52	#870	—	9	1	no	—
53	#870	—	18	1	no	—
54	#869	D₉	1	1	yes	characteristic
55	C₃×#97	—	9	1	no	—
56	#835	—	9	4	no	—
57	#816	—	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	C₂×C₃×Dic₃	—	2	2	no	—
63	C₃×Dic₆	—	2	4	no	—
64	C₃×C₃⋊C₈	—	6	2	no	—
65	Q₈×C₉	—	6	4	no	—
66	C₂×C₄×C₉	—	2	2	no	—
67	C₂×C₄×C₉	—	2	1	no	—
68	C₂×C₄×C₉	—	6	2	no	—
69	C₂×C₄×C₉	D₁₂	1	2	yes	—
70	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
71	C₂×Dic₉	—	6	4	no	—
72	C₂×Dic₉	—	6	2	no	—
73	C₉⋊C₈	—	2	2	no	—
74	#491	—	27	1	no	Sylow
75	C₂×C₃×C₉	#124	1	1	yes	characteristic
76	C₂×C₃×C₉	#137	1	2	yes	—
77	C₂×C₃×Q₈	—	3	2	no	—
78	C₂×Dic₆	C₂×D₉	1	2	yes	—
79	C₂×C₃×C₈	—	9	2	no	—
80	C₃×C₄⋊C₄	—	6	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	—	2	2	no	—
85	C₄⋊Dic₃	—	9	2	no	—
86	C₄⋊Dic₃	—	9	2	no	—
87	C₄⋊Dic₃	—	18	2	no	—
88	Dic₃⋊C₄	—	18	4	no	—
89	Dic₃⋊C₄	—	18	2	no	—
90	Dic₃⋊C₄	—	18	2	no	—
91	C₂×C₃⋊C₈	—	3	2	no	—
92	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
93	C₃²×C₄	—	2	2	no	—
94	C₃²×C₄	#237	1	2	yes	—
95	C₃⋊Dic₃	—	12	2	no	—
96	C₃×Dic₃	—	4	2	no	—
97	C₂²×C₉	—	2	1	no	—
98	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	2	2	no	—
101	C₄×C₉	—	4	2	no	—
102	C₄×C₉	—	12	2	no	—
103	C₄×C₉	#235	1	2	yes	—
104	Dic₉	—	12	4	no	—
105	Dic₉	—	12	2	no	—
106	C₄⋊Q₈	—	3	1	no	—
107	#125	—	27	1	no	—
108	#103	—	27	4	no	—
109	#97	—	9	1	no	—
110	C₃×C₉	#491	1	1	yes	characteristic, complemented, Sylow
111	C₃×Q₈	—	6	4	no	—
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	—	2	4	no	—
124	C₃×C₈	—	18	2	no	—
125	C₃⋊C₈	—	6	2	no	—
126	C₂×C₃²	#936	1	1	yes	characteristic
127	C₂×C₃²	#942	1	2	yes	—
128	C₂×C₉	—	2	2	no	—
129	C₂×C₉	—	2	1	no	—
130	C₂×C₉	#875	1	1	yes	characteristic
131	C₂×C₉	#909	1	2	yes	—
132	C₂×Q₈	—	3	2	no	—
133	C₂×C₈	—	9	2	no	—
134	C₄⋊C₄	—	6	2	no	—
135	C₄⋊C₄	—	27	2	no	—
136	C₄⋊C₄	—	54	2	no	—
137	C₄²	D₃×D₉	1	1	yes	characteristic
138	C₂²×C₃	—	2	1	no	—
139	C₂²×C₃	C₂×#611	1	1	yes	characteristic
140	C₂²×C₃	C₂×#626	1	1	yes	characteristic
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	2	2	no	—
143	C₃×C₄	—	2	2	no	—
144	C₃×C₄	—	4	2	no	—
145	C₃×C₄	—	12	2	no	—
146	C₃×C₄	#3699	1	2	yes	—
147	C₃×C₄	#3795	1	2	yes	—
148	Dic₃	—	4	2	no	—
149	Dic₃	—	36	4	no	—
150	Dic₃	—	36	2	no	—
151	Dic₃	—	36	2	no	—
152	C₃²	#5082	1	1	yes	characteristic
153	C₉	—	2	1	no	—
154	C₉	#4703	1	1	yes	characteristic, complemented
155	Q₈	—	6	4	no	—
156	C₂×C₄	—	6	2	no	—
157	C₂×C₄	—	54	2	no	—
158	C₂×C₄	#6195	1	2	yes	—
159	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
160	C₈	—	18	2	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	2	no	—
163	C₂×C₃	#63498	1	1	yes	characteristic
164	C₂×C₃	#63505	1	2	yes	—
165	C₂×C₃	#63833	1	2	yes	—
166	C₂×C₃	#63860	1	1	yes	characteristic
167	C₂²	C₂×#6195	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	12	2	no	—
170	C₄	—	108	2	no	—
171	C₄	#88638	1	2	yes	—
172	C₃	—	2	1	no	—
173	C₃	#10588762	1	1	yes	characteristic, complemented
174	C₃	#10590289	1	1	yes	characteristic
175	C₂	#11720895	1	2	yes	—
176	C₂	#11720964	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721707

C₂

yes

maximal, characteristic

C₉×#869

C₂

yes

maximal, characteristic

C₃×#63362

C₂

yes

maximal, characteristic

#11720298

C₂

yes

—

—

C₂×C₉×Dic₆

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

C₂²

yes

—

#88534

—

C₉×C₄⋊Dic₃

—

C₂×C₃×C₉⋊C₈

C₂²

yes

—

#64079

—

C₃×#869

D₃

yes

characteristic

C₃×#803

—

#63566

—

C₉×C₄⋊Q₈

—

—

—

—

D₃

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

C₂×C₉×Dic₃

—

C₉×Dic₆

—

C₃×C₉⋊C₈

—

#5082

—

#4703

—

C₂×C₃×Dic₆

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₃×C₄⋊Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

Dic₉⋊C₄

—

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×C₄⋊Q₈

—

—

—

—

D₉

yes

characteristic

C₃×#97

—

#835

—

#816

—

#803

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

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

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

C₂×D₉

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₃×C₄²

D₃²

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#237

yes

—

C₃⋊Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×D₁₂

yes

characteristic

C₄×C₉

—

100

C₄×C₉

—

101

C₄×C₉

—

102

C₄×C₉

—

103

C₄×C₉

#235

yes

—

104

Dic₉

—

105

Dic₉

—

106

C₄⋊Q₈

—

107

#125

—

108

#103

—

109

#97

—

110

C₃×C₉

#491

yes

characteristic, complemented, Sylow

111

C₃×Q₈

—

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

C₃×C₈

—

125

C₃⋊C₈

—

126

C₂×C₃²

#936

yes

characteristic

127

C₂×C₃²

#942

yes

—

128

C₂×C₉

—

129

C₂×C₉

—

130

C₂×C₉

#875

yes

characteristic

131

C₂×C₉

#909

yes

—

132

C₂×Q₈

—

133

C₂×C₈

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

D₃×D₉

yes

characteristic

138

C₂²×C₃

—

139

C₂²×C₃

C₂×#611

yes

characteristic

140

C₂²×C₃

C₂×#626

yes

characteristic

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

C₃×C₄

#3699

yes

—

147

C₃×C₄

#3795

yes

—

148

Dic₃

—

149

Dic₃

—

150

Dic₃

—

151

Dic₃

—

152

C₃²

#5082

yes

characteristic

153

C₉

—

154

C₉

#4703

yes

characteristic, complemented

155

Q₈

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#6195

yes

—

159

C₂×C₄

C₂×D₃×D₉

yes

characteristic

160

C₈

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

#63498

yes

characteristic

164

C₂×C₃

#63505

yes

—

165

C₂×C₃

#63833

yes

—

166

C₂×C₃

#63860

yes

characteristic

167

C₂²

C₂×#6195

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

108

—

171

C₄

#88638

yes

—

172

C₃

—

173

C₃

#10588762

yes

characteristic, complemented

174

C₃

#10590289

yes

characteristic

175

C₂

#11720895

yes

—

176

C₂

#11720964

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
110. C₃×C₉	#491	semidirect
154. C₉	#4703	semidirect
173. C₃	#10588762	semidirect

H ≅ G/N

type

110. C₃×C₉

#491

semidirect

154. C₉

#4703

semidirect

173. C₃

#10588762

semidirect

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

subgroup	index
#11721707	2
C₉×#869	2
C₃×#63362	2
#11720298	2
#10590289	3
#10588762	3

subgroup

index

Subgroups — #13137142

Conjugacy classes of subgroups

Complements

Maximal subgroups