Group #13135684 — subgroups

Subgroups — #10588230⋊C₃ #13135684

Order 1728.

208 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 76 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	C₂×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	#63371⋊C₃	C₂	1	2	yes	maximal, complemented
3	C₃×#4553	—	3	1	no	maximal
4	#10588230	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
6	M₁₆×3₋¹⁺²	—	2	2	no	—
7	C₂×C₈×3₋¹⁺²	C₄	1	1	yes	characteristic
8	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	3₋¹⁺²⋊C₁₆	—	2	2	no	—
10	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
11	C₃×#812	—	3	2	no	—
12	C₂×C₉×M₁₆	—	2	1	no	—
13	C₂×C₉×M₁₆	C₂×C₃	1	1	yes	characteristic
14	#63371	C₂×C₃	1	2	yes	complemented
15	C₂³×3₋¹⁺²	C₈	1	1	yes	characteristic
16	C₂×C₄×3₋¹⁺²	—	2	1	no	—
17	C₂×C₄×3₋¹⁺²	—	2	1	no	—
18	C₂×C₄×3₋¹⁺²	C₈	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
20	C₈×3₋¹⁺²	—	2	1	no	—
21	C₈×3₋¹⁺²	D₄	1	2	yes	—
22	C₃×#349	—	9	1	no	—
23	#4553	—	3	1	no	—
24	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
25	C₃²×M₁₆	—	2	2	no	—
26	C₂×C₃²×C₈	Dic₃	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
30	C₃×C₃⋊C₁₆	—	6	2	no	—
31	C₉×M₁₆	—	2	4	no	—
32	C₉×M₁₆	—	2	2	no	—
33	C₂×C₈×C₉	—	2	1	no	—
34	C₂×C₈×C₉	—	2	1	no	—
35	C₂×C₈×C₉	C₃×C₄	1	1	yes	characteristic
36	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₉⋊C₁₆	—	2	2	no	—
38	C₂²×3₋¹⁺²	—	2	1	no	—
39	C₂²×3₋¹⁺²	—	4	1	no	—
40	C₂²×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
41	C₄×3₋¹⁺²	—	2	1	no	—
42	C₄×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×3₋¹⁺²	M₁₆	1	1	yes	characteristic
44	C₂×C₃×M₁₆	—	3	1	no	—
45	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
46	C₃×M₃₂	—	9	2	no	—
47	#812	—	3	2	no	—
48	C₂³×C₃²	C₃⋊C₈	1	1	yes	characteristic
49	C₂×C₃²×C₄	—	2	1	no	—
50	C₂×C₃²×C₄	—	2	1	no	—
51	C₂×C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
52	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
53	C₂³×C₉	—	2	1	no	—
54	C₂³×C₉	C₃×C₈	1	1	yes	characteristic
55	C₃²×C₈	—	2	1	no	—
56	C₃²×C₈	#67	1	2	yes	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	2	no	—
59	C₂×C₄×C₉	—	2	1	no	—
60	C₂×C₄×C₉	—	2	1	no	—
61	C₂×C₄×C₉	—	2	1	no	—
62	C₂×C₄×C₉	—	2	1	no	—
63	C₂×C₄×C₉	C₃×C₈	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
65	C₈×C₉	—	2	2	no	—
66	C₈×C₉	—	2	1	no	—
67	C₈×C₉	—	2	2	no	—
68	C₈×C₉	C₃×D₄	1	2	yes	—
69	#349	—	9	1	no	Sylow
70	C₂×3₋¹⁺²	—	2	1	no	—
71	C₂×3₋¹⁺²	—	4	1	no	—
72	C₂×3₋¹⁺²	C₂²⋊C₈	1	1	yes	characteristic
73	C₂²×C₃×C₄	—	3	1	no	—
74	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
75	C₃×M₁₆	—	2	2	no	—
76	C₃×M₁₆	—	6	2	no	—
77	C₂×C₃×C₈	—	3	1	no	—
78	C₂×C₃×C₈	—	3	1	no	—
79	C₂×C₃×C₈	C₃×Dic₃	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
81	C₃×C₁₆	—	18	2	no	—
82	C₃⋊C₁₆	—	6	2	no	—
83	C₂²×C₃²	—	2	1	no	—
84	C₂²×C₃²	—	4	1	no	—
85	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
86	C₃²×C₄	—	2	1	no	—
87	C₃²×C₄	#208	1	1	yes	characteristic
88	C₃²×C₄	#217	1	1	yes	characteristic
89	C₂²×C₉	—	2	2	no	—
90	C₂²×C₉	—	2	1	no	—
91	C₂²×C₉	—	2	1	no	—
92	C₂²×C₉	—	4	1	no	—
93	C₂²×C₉	—	4	2	no	—
94	C₂²×C₉	C₂×C₃×C₈	1	1	yes	characteristic
95	C₄×C₉	—	2	2	no	—
96	C₄×C₉	—	2	1	no	—
97	C₄×C₉	—	2	1	no	—
98	C₄×C₉	—	2	1	no	—
99	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
100	C₄×C₉	C₃×M₁₆	1	1	yes	characteristic
101	C₂×M₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic
102	M₃₂	—	9	2	no	—
103	3₋¹⁺²	#349	1	1	yes	characteristic, complemented, Sylow
104	C₂³×C₃	—	3	1	no	—
105	C₂³×C₃	C₃×C₃⋊C₈	1	1	yes	characteristic
106	C₂×C₃×C₄	—	2	1	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	3	1	no	—
109	C₂×C₃×C₄	—	3	1	no	—
110	C₂×C₃×C₄	—	6	1	no	—
111	C₂×C₃×C₄	—	6	1	no	—
112	C₂×C₃×C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
114	C₃×C₈	—	2	1	no	—
115	C₃×C₈	—	3	2	no	—
116	C₃×C₈	—	6	1	no	—
117	C₃×C₈	C₃×#67	1	2	yes	—
118	C₂×C₃²	—	2	1	no	—
119	C₂×C₃²	—	4	1	no	—
120	C₂×C₃²	#830	1	1	yes	characteristic
121	C₂×C₉	—	2	2	no	—
122	C₂×C₉	—	2	1	no	—
123	C₂×C₉	—	2	1	no	—
124	C₂×C₉	—	4	2	no	—
125	C₂×C₉	—	4	1	no	—
126	C₂×C₉	C₃×C₂²⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	3₋¹⁺²⋊C₄	1	1	yes	characteristic
128	M₁₆	—	2	2	no	—
129	C₂×C₈	3₋¹⁺²⋊C₄	1	1	yes	characteristic
130	C₂×C₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
131	C₁₆	—	18	2	no	—
132	C₂²×C₃	—	2	1	no	—
133	C₂²×C₃	—	3	1	no	—
134	C₂²×C₃	—	4	1	no	—
135	C₂²×C₃	—	6	1	no	—
136	C₂²×C₃	—	12	1	no	—
137	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
138	C₃×C₄	—	2	1	no	—
139	C₃×C₄	—	3	1	no	—
140	C₃×C₄	—	3	1	no	—
141	C₃×C₄	—	6	1	no	—
142	C₃×C₄	C₃×#208	1	1	yes	characteristic
143	C₃×C₄	C₃×#217	1	1	yes	characteristic
144	C₃²	#4553	1	1	yes	characteristic
145	C₉	—	2	1	no	—
146	C₉	C₃×#349	1	1	yes	characteristic, complemented
147	C₂³	3₋¹⁺²⋊C₈	1	1	yes	characteristic
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	3₋¹⁺²⋊C₈	1	1	yes	characteristic
151	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	#611⋊C₃	1	2	yes	—
154	C₂×C₃	—	2	1	no	—
155	C₂×C₃	—	3	1	no	—
156	C₂×C₃	—	4	1	no	—
157	C₂×C₃	—	6	1	no	—
158	C₂×C₃	—	12	1	no	—
159	C₂×C₃	C₃×#830	1	1	yes	characteristic
160	C₂²	—	2	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	C₂×3₋¹⁺²⋊C₈	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	#3667⋊C₃	1	1	yes	characteristic
165	C₄	#3676⋊C₃	1	1	yes	characteristic
166	C₃	—	3	1	no	—
167	C₃	C₃×#4553	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	4	1	no	—
170	C₂	#63389⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic

#63371⋊C₃

C₂

yes

maximal, complemented

C₃×#4553

—

maximal

#10588230

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₄

yes

characteristic

M₁₆×3₋¹⁺²

—

C₂×C₈×3₋¹⁺²

C₄

yes

characteristic

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

3₋¹⁺²⋊C₁₆

—

C₂×C₃²×M₁₆

D₃

yes

characteristic

C₃×#812

—

C₂×C₉×M₁₆

—

C₂×C₉×M₁₆

C₂×C₃

yes

characteristic

#63371

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₈×3₋¹⁺²

—

C₈×3₋¹⁺²

D₄

yes

—

C₃×#349

—

#4553

—

C₂²×C₃²×C₄

Dic₃

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

Dic₃

yes

characteristic

C₂×C₃²×C₈

C₂×D₃

yes

characteristic

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₃×C₄

yes

characteristic

C₃×C₃⋊C₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₃×C₄

yes

characteristic

C₂×C₈×C₉

C₂²×C₃

yes

characteristic

C₉⋊C₁₆

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂×C₈

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

M₁₆

yes

characteristic

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₃×D₃

yes

characteristic

C₃×M₃₂

—

#812

—

C₂³×C₃²

C₃⋊C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃⋊C₈

yes

characteristic

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂³×C₉

—

C₂³×C₉

C₃×C₈

yes

characteristic

C₃²×C₈

—

C₃²×C₈

#67

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×C₈

yes

characteristic

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

C₃×D₄

yes

—

#349

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²⋊C₈

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃×Dic₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×Dic₃

yes

characteristic

C₂×C₃×C₈

C₂×C₃×D₃

yes

characteristic

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃⋊C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

yes

C₃²×C₄

yes

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×C₈

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×C₂²⋊C₄

yes

characteristic

100

C₄×C₉

C₃×M₁₆

yes

characteristic

101

C₂×M₁₆

AGL(1,ℤ/9)

yes

characteristic

102

M₃₂

—

103

3₋¹⁺²

#349

yes

characteristic, complemented, Sylow

104

C₂³×C₃

—

105

C₂³×C₃

C₃×C₃⋊C₈

yes

characteristic

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

C₃×C₃⋊C₈

yes

characteristic

113

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

114

C₃×C₈

—

115

C₃×C₈

—

116

C₃×C₈

—

117

C₃×C₈

C₃×#67

yes

—

118

C₂×C₃²

—

119

C₂×C₃²

—

120

C₂×C₃²

#830

yes

characteristic

121

C₂×C₉

—

122

C₂×C₉

—

123

C₂×C₉

—

124

C₂×C₉

—

125

C₂×C₉

—

126

C₂×C₉

C₃×C₂²⋊C₈

yes

characteristic

127

C₂²×C₄

3₋¹⁺²⋊C₄

yes

characteristic

128

M₁₆

—

129

C₂×C₈

3₋¹⁺²⋊C₄

yes

characteristic

130

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

characteristic

131

C₁₆

—

132

C₂²×C₃

—

133

C₂²×C₃

—

134

C₂²×C₃

—

135

C₂²×C₃

—

136

C₂²×C₃

—

137

C₂²×C₃

C₂×C₃×C₃⋊C₈

yes

characteristic

138

C₃×C₄

—

139

C₃×C₄

—

140

C₃×C₄

—

141

C₃×C₄

—

142

C₃×C₄

C₃×#208

yes

characteristic

143

C₃×C₄

C₃×#217

yes

characteristic

144

C₃²

#4553

yes

characteristic

145

C₉

—

146

C₉

C₃×#349

yes

characteristic, complemented

147

C₂³

3₋¹⁺²⋊C₈

yes

characteristic

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

3₋¹⁺²⋊C₈

yes

characteristic

151

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

characteristic

152

C₈

—

153

C₈

#611⋊C₃

yes

—

154

C₂×C₃

—

155

C₂×C₃

—

156

C₂×C₃

—

157

C₂×C₃

—

158

C₂×C₃

—

159

C₂×C₃

C₃×#830

yes

characteristic

160

C₂²

—

161

C₂²

—

162

C₂²

C₂×3₋¹⁺²⋊C₈

yes

characteristic

163

C₄

—

164

C₄

#3667⋊C₃

yes

characteristic

165

C₄

#3676⋊C₃

yes

characteristic

166

C₃

—

167

C₃

C₃×#4553

yes

characteristic

168

C₂

—

169

C₂

—

170

C₂

#63389⋊C₃

yes

characteristic

	subgroup	G/H
center	C₄	#3676⋊C₃
commutator	C₂²×C₉	C₂×C₃×C₈
Frattini	C₂×C₃×C₈	C₂×C₃×D₃
Fitting	C₂×M₁₆×3₋¹⁺²	C₂
socle	C₂×C₃	C₃×#830
prime radical	C₂³×C₃²	C₃⋊C₈
cosocle	C₂×C₈×C₉	C₂²×C₃
hypercenter	C₂×M₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#349
LCS	G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

#3676⋊C₃

commutator

C₂²×C₉

C₂×C₃×C₈

Frattini

C₂×C₃×C₈

C₂×C₃×D₃

Fitting

C₂×M₁₆×3₋¹⁺²

C₂

socle

C₂×C₃

C₃×#830

prime radical

C₂³×C₃²

C₃⋊C₈

cosocle

C₂×C₈×C₉

C₂²×C₃

C₂×M₁₆

C₉

G ▷ C₂²×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #63371⋊C₃	C₂	semidirect
4. #10588230	C₃	semidirect
14. #63371	C₂×C₃	semidirect
103. 3₋¹⁺²	#349	semidirect
146. C₉	C₃×#349	semidirect

H ≅ G/N

type

2. #63371⋊C₃

C₂

semidirect

4. #10588230

C₃

semidirect

14. #63371

C₂×C₃

semidirect

103. 3₋¹⁺²

#349

semidirect

146. C₉

C₃×#349

semidirect

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

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
#63371⋊C₃	2
C₃×#4553	3
#10588230	3

subgroup

index

C₂×M₁₆×3₋¹⁺²

#63371⋊C₃

C₃×#4553

#10588230

Subgroups — #10588230⋊C₃ #13135684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588230⋊C3 #13135684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588230⋊C₃ #13135684