Group #13135617 — subgroups

Subgroups — #10588217⋊C₃ #13135617

Order 1728.

248 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 66 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	3₋¹⁺²×#124	C₂	1	1	yes	maximal, characteristic
2	#63362⋊C₃	C₂	1	2	yes	maximal, complemented
3	C₃×#4540	—	3	1	no	maximal
4	#10588217	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×3₋¹⁺²	C₄	1	1	yes	characteristic
6	C₂×D₄×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₂²⋊C₄×3₋¹⁺²	—	2	2	no	—
8	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₂×3₋¹⁺²⋊C₈	—	2	2	no	—
10	C₃²×#124	D₃	1	1	yes	characteristic
11	C₃×#803	—	3	2	no	—
12	C₉×#124	—	2	1	no	—
13	C₉×#124	C₂×C₃	1	1	yes	characteristic
14	#63362	C₂×C₃	1	2	yes	complemented
15	C₂³×3₋¹⁺²	—	2	1	no	—
16	Q₈×3₋¹⁺²	—	4	1	no	—
17	D₄×3₋¹⁺²	—	4	1	no	—
18	C₂×C₄×3₋¹⁺²	—	2	1	no	—
19	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
21	3₋¹⁺²⋊C₈	—	2	4	no	—
22	C₃×#329	—	9	1	no	—
23	#4540	—	3	1	no	—
24	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
25	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
26	C₃²×C₂²⋊C₄	—	2	2	no	—
27	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₃⋊C₈	—	6	2	no	—
29	C₂×Q₈×C₉	—	2	1	no	—
30	C₂×Q₈×C₉	C₃×C₄	1	1	yes	characteristic
31	C₂×D₄×C₉	—	2	1	no	—
32	C₂×D₄×C₉	C₃×C₄	1	1	yes	characteristic
33	C₉×C₂²⋊C₄	—	2	4	no	—
34	C₉×C₂²⋊C₄	—	2	2	no	—
35	C₄²×C₉	—	2	1	no	—
36	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₉⋊C₈	—	2	2	no	—
38	C₂²×3₋¹⁺²	—	4	1	no	—
39	C₂²×3₋¹⁺²	—	4	2	no	—
40	C₂²×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×3₋¹⁺²	—	2	1	no	—
42	C₄×3₋¹⁺²	—	2	2	no	—
43	C₄×3₋¹⁺²	—	4	1	no	—
44	C₃×#124	—	3	1	no	—
45	C₃×#124	C₃×D₃	1	1	yes	characteristic
46	C₃×#97	—	9	2	no	—
47	#803	—	3	2	no	—
48	C₂³×C₃²	—	2	1	no	—
49	C₃²×Q₈	—	4	1	no	—
50	C₃²×D₄	—	4	1	no	—
51	C₂×C₃²×C₄	—	2	1	no	—
52	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
53	C₂×C₃²×C₄	#67	1	2	yes	—
54	C₂³×C₉	—	2	2	no	—
55	C₂³×C₉	—	2	1	no	—
56	C₃×C₃⋊C₈	—	6	4	no	—
57	Q₈×C₉	—	4	1	no	—
58	Q₈×C₉	—	4	2	no	—
59	D₄×C₉	—	4	1	no	—
60	D₄×C₉	—	4	2	no	—
61	C₂×C₄×C₉	—	2	2	no	—
62	C₂×C₄×C₉	—	2	1	no	—
63	C₂×C₄×C₉	—	2	1	no	—
64	C₂×C₄×C₉	—	2	2	no	—
65	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
67	C₉⋊C₈	—	2	4	no	—
68	#329	—	9	1	no	Sylow
69	C₂×3₋¹⁺²	—	8	1	no	—
70	C₂×3₋¹⁺²	#100	1	1	yes	characteristic
71	C₂×3₋¹⁺²	#104	1	2	yes	—
72	C₂×C₃×Q₈	—	3	1	no	—
73	C₂×C₃×Q₈	C₃×Dic₃	1	1	yes	characteristic
74	C₂×C₃×D₄	—	3	1	no	—
75	C₂×C₃×D₄	C₃×Dic₃	1	1	yes	characteristic
76	C₂×C₃×C₈	—	18	2	no	—
77	C₃×C₂²⋊C₄	—	2	2	no	—
78	C₃×C₂²⋊C₄	—	6	2	no	—
79	C₃×C₄²	—	3	1	no	—
80	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
81	C₂×C₃⋊C₈	—	6	2	no	—
82	C₂²×C₃²	—	4	1	no	—
83	C₂²×C₃²	—	4	2	no	—
84	C₂²×C₃²	#217	1	1	yes	characteristic
85	C₃²×C₄	—	2	1	no	—
86	C₃²×C₄	—	2	2	no	—
87	C₃²×C₄	—	4	1	no	—
88	C₂²×C₉	—	2	1	no	—
89	C₂²×C₉	—	4	2	no	—
90	C₂²×C₉	—	4	4	no	—
91	C₂²×C₉	—	4	1	no	—
92	C₂²×C₉	—	4	2	no	—
93	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₉	—	2	2	no	—
95	C₄×C₉	—	2	1	no	—
96	C₄×C₉	—	2	2	no	—
97	C₄×C₉	—	4	2	no	—
98	C₄×C₉	—	4	2	no	—
99	C₄×C₉	—	4	1	no	—
100	#124	AGL(1,ℤ/9)	1	1	yes	characteristic
101	#97	—	9	2	no	—
102	3₋¹⁺²	#329	1	1	yes	characteristic, complemented, Sylow
103	C₂³×C₃	—	2	1	no	—
104	C₂³×C₃	—	6	1	no	—
105	C₃×Q₈	—	4	1	no	—
106	C₃×Q₈	—	12	1	no	—
107	C₃×D₄	—	4	1	no	—
108	C₃×D₄	—	12	1	no	—
109	C₂×C₃×C₄	—	2	1	no	—
110	C₂×C₃×C₄	—	3	1	no	—
111	C₂×C₃×C₄	—	3	2	no	—
112	C₂×C₃×C₄	—	6	1	no	—
113	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
114	C₂×C₃×C₄	C₃×#67	1	2	yes	—
115	C₃×C₈	—	18	4	no	—
116	C₃⋊C₈	—	6	4	no	—
117	C₂×C₃²	—	8	1	no	—
118	C₂×C₃²	#833	1	1	yes	characteristic
119	C₂×C₃²	#837	1	2	yes	—
120	C₂×C₉	—	2	1	no	—
121	C₂×C₉	—	2	2	no	—
122	C₂×C₉	—	8	2	no	—
123	C₂×C₉	—	8	1	no	—
124	C₂×C₉	C₃×#100	1	1	yes	characteristic
125	C₂×C₉	C₃×#104	1	2	yes	—
126	C₂×Q₈	3₋¹⁺²⋊C₄	1	1	yes	characteristic
127	C₂×D₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
128	C₂×C₈	—	18	2	no	—
129	C₂²⋊C₄	—	2	2	no	—
130	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
131	C₂²×C₃	—	3	1	no	—
132	C₂²×C₃	—	4	1	no	—
133	C₂²×C₃	—	4	2	no	—
134	C₂²×C₃	—	12	1	no	—
135	C₂²×C₃	—	12	2	no	—
136	C₂²×C₃	C₃×#217	1	1	yes	characteristic
137	C₃×C₄	—	2	1	no	—
138	C₃×C₄	—	2	2	no	—
139	C₃×C₄	—	4	1	no	—
140	C₃×C₄	—	6	2	no	—
141	C₃×C₄	—	6	1	no	—
142	C₃×C₄	—	12	1	no	—
143	C₃²	#4540	1	1	yes	characteristic
144	C₉	—	2	1	no	—
145	C₉	C₃×#329	1	1	yes	characteristic, complemented
146	C₂³	—	2	1	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	4	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
151	C₂×C₄	#611⋊C₃	1	2	yes	—
152	C₈	—	18	4	no	—
153	C₂×C₃	—	3	1	no	—
154	C₂×C₃	—	3	2	no	—
155	C₂×C₃	—	8	1	no	—
156	C₂×C₃	—	24	1	no	—
157	C₂×C₃	C₃×#833	1	1	yes	characteristic
158	C₂×C₃	C₃×#837	1	2	yes	—
159	C₂²	—	4	1	no	—
160	C₂²	—	4	2	no	—
161	C₂²	#3676⋊C₃	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	1	no	—
165	C₃	—	3	1	no	—
166	C₃	C₃×#4540	1	1	yes	characteristic
167	C₂	—	8	1	no	—
168	C₂	#63391⋊C₃	1	1	yes	characteristic
169	C₂	#63396⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#124

C₂

yes

maximal, characteristic

#63362⋊C₃

C₂

yes

maximal, complemented

C₃×#4540

—

maximal

#10588217

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

C₄

yes

characteristic

C₂×D₄×3₋¹⁺²

C₄

yes

characteristic

C₂²⋊C₄×3₋¹⁺²

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₂×3₋¹⁺²⋊C₈

—

C₃²×#124

D₃

yes

—

—

C₂×C₃

yes

characteristic

#63362

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

D₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

—

3₋¹⁺²⋊C₈

—

C₃×#329

—

#4540

—

C₂×C₃²×Q₈

Dic₃

yes

characteristic

C₂×C₃²×D₄

Dic₃

yes

characteristic

C₃²×C₂²⋊C₄

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₃×C₄

yes

characteristic

C₂×D₄×C₉

—

C₂×D₄×C₉

C₃×C₄

yes

characteristic

C₉×C₂²⋊C₄

—

C₉×C₂²⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₈

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₃×#124

—

C₃×#124

C₃×D₃

yes

characteristic

C₃×#97

—

#803

—

C₂³×C₃²

—

C₃²×Q₈

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₂³×C₉

—

C₂³×C₉

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

—

C₉⋊C₈

—

#329

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#100

yes

characteristic

C₂×3₋¹⁺²

#104

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×Dic₃

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃×Dic₃

yes

characteristic

C₂×C₃×C₈

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#217

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₂²⋊C₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

100

yes

101

—

102

3₋¹⁺²

#329

yes

characteristic, complemented, Sylow

103

C₂³×C₃

—

104

C₂³×C₃

—

105

C₃×Q₈

—

106

C₃×Q₈

—

107

C₃×D₄

—

108

C₃×D₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

114

C₂×C₃×C₄

C₃×#67

yes

—

115

C₃×C₈

—

116

C₃⋊C₈

—

117

C₂×C₃²

—

118

C₂×C₃²

#833

yes

characteristic

119

C₂×C₃²

#837

yes

—

120

C₂×C₉

—

121

C₂×C₉

—

122

C₂×C₉

—

123

C₂×C₉

—

124

C₂×C₉

C₃×#100

yes

characteristic

125

C₂×C₉

C₃×#104

yes

—

126

C₂×Q₈

3₋¹⁺²⋊C₄

yes

characteristic

127

C₂×D₄

3₋¹⁺²⋊C₄

yes

characteristic

128

C₂×C₈

—

129

C₂²⋊C₄

—

130

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

131

C₂²×C₃

—

132

C₂²×C₃

—

133

C₂²×C₃

—

134

C₂²×C₃

—

135

C₂²×C₃

—

136

C₂²×C₃

C₃×#217

yes

characteristic

137

C₃×C₄

—

138

C₃×C₄

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

—

143

C₃²

#4540

yes

characteristic

144

C₉

—

145

C₉

C₃×#329

yes

characteristic, complemented

146

C₂³

—

147

Q₈

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

characteristic

151

C₂×C₄

#611⋊C₃

yes

—

152

C₈

—

153

C₂×C₃

—

154

C₂×C₃

—

155

C₂×C₃

—

156

C₂×C₃

—

157

C₂×C₃

C₃×#833

yes

characteristic

158

C₂×C₃

C₃×#837

yes

—

159

C₂²

—

160

C₂²

—

161

C₂²

#3676⋊C₃

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₃

—

166

C₃

C₃×#4540

yes

characteristic

167

C₂

—

168

C₂

#63391⋊C₃

yes

characteristic

169

C₂

#63396⋊C₃

yes

—

	subgroup	G/H
center	C₂²	#3676⋊C₃
commutator	C₂×C₄×C₉	C₂×C₃×C₄
Frattini	C₃×C₄²	C₂×C₃×D₃
Fitting	3₋¹⁺²×#124	C₂
socle	C₂²×C₃	C₃×#217
prime radical	C₂×C₃²×D₄	Dic₃
cosocle	C₄²×C₉	C₂²×C₃
hypercenter	#124	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#329
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

subgroup

G/H

center

C₂²

#3676⋊C₃

commutator

C₂×C₄×C₉

C₂×C₃×C₄

Frattini

C₃×C₄²

C₂×C₃×D₃

Fitting

3₋¹⁺²×#124

C₂

socle

C₂²×C₃

C₃×#217

prime radical

C₂×C₃²×D₄

Dic₃

cosocle

C₄²×C₉

C₂²×C₃

C₉

G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #124

N	H ≅ G/N	type
2. #63362⋊C₃	C₂	semidirect
4. #10588217	C₃	semidirect
14. #63362	C₂×C₃	semidirect
102. 3₋¹⁺²	#329	semidirect
145. C₉	C₃×#329	semidirect

H ≅ G/N

type

2. #63362⋊C₃

C₂

semidirect

4. #10588217

C₃

semidirect

14. #63362

C₂×C₃

semidirect

102. 3₋¹⁺²

#329

semidirect

145. C₉

C₃×#329

semidirect

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

subgroup	index
3₋¹⁺²×#124	2
#63362⋊C₃	2
C₃×#4540	3
#10588217	3

subgroup

index

3₋¹⁺²×#124

#63362⋊C₃

C₃×#4540

#10588217

Subgroups — #10588217⋊C₃ #13135617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588217⋊C3 #13135617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588217⋊C₃ #13135617