Group #13135630 — subgroups

Subgroups — #10588221⋊C₃ #13135630

Order 1728.

306 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 86 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	2	yes	maximal, complemented
3	C₃×#4544	—	3	1	no	maximal
4	#10588221	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×3₋¹⁺²	—	2	2	no	—
6	C₂×D₄×3₋¹⁺²	C₄	1	2	yes	—
7	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×3₋¹⁺²⋊C₈	—	2	2	no	—
9	C₃²×#127	D₃	1	1	yes	characteristic
10	C₃×C₄⋊(C₃⋊C₈)	—	3	2	no	—
11	C₉×#127	—	2	1	no	—
12	C₉×#127	C₂×C₃	1	1	yes	characteristic
13	C₄⋊(C₉⋊C₈)	C₂×C₃	1	2	yes	complemented
14	C₂³×3₋¹⁺²	—	2	2	no	—
15	D₄×3₋¹⁺²	—	4	2	no	—
16	D₄×3₋¹⁺²	—	4	2	no	—
17	D₄×3₋¹⁺²	—	4	2	no	—
18	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
20	3₋¹⁺²⋊C₈	—	4	2	no	—
21	C₃×#331	—	9	1	no	—
22	#4544	—	3	1	no	—
23	C₂×C₃²×D₄	—	2	2	no	—
24	C₂×C₃²×D₄	Dic₃	1	2	yes	—
25	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
26	C₂×C₃×C₃⋊C₈	—	6	2	no	—
27	C₂×D₄×C₉	—	2	4	no	—
28	C₂×D₄×C₉	—	2	2	no	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	C₃×C₄	1	2	yes	—
31	C₄²×C₉	—	2	1	no	—
32	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
33	C₂×C₉⋊C₈	—	2	2	no	—
34	C₂²×3₋¹⁺²	—	4	2	no	—
35	C₂²×3₋¹⁺²	—	4	4	no	—
36	C₂²×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×3₋¹⁺²	—	2	1	no	—
38	C₄×3₋¹⁺²	D₈	1	2	yes	—
39	C₄×3₋¹⁺²	SD₁₆	1	2	yes	—
40	C₃×#127	—	3	1	no	—
41	C₃×#127	C₃×D₃	1	1	yes	characteristic
42	C₃×C₄⋊C₈	—	9	2	no	—
43	C₄⋊(C₃⋊C₈)	—	3	2	no	—
44	C₂³×C₃²	—	2	2	no	—
45	C₃²×D₄	—	4	2	no	—
46	C₃²×D₄	—	4	2	no	—
47	C₃²×D₄	—	4	2	no	—
48	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
49	C₂×C₃²×C₄	#67	1	2	yes	—
50	C₂³×C₉	—	2	4	no	—
51	C₂³×C₉	—	2	2	no	—
52	C₃×C₃⋊C₈	—	12	2	no	—
53	D₄×C₉	—	4	2	no	—
54	D₄×C₉	—	4	4	no	—
55	D₄×C₉	—	4	4	no	—
56	D₄×C₉	—	4	4	no	—
57	D₄×C₉	—	4	2	no	—
58	D₄×C₉	—	4	2	no	—
59	C₂×C₄×C₉	—	2	1	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
63	C₉⋊C₈	—	4	2	no	—
64	#331	—	9	1	no	Sylow
65	C₂×3₋¹⁺²	—	8	2	no	—
66	C₂×3₋¹⁺²	#100	1	1	yes	characteristic
67	C₂×3₋¹⁺²	#102	1	2	yes	—
68	C₂×C₃×D₄	—	2	2	no	—
69	C₂×C₃×D₄	—	3	2	no	—
70	C₂×C₃×D₄	—	6	2	no	—
71	C₂×C₃×D₄	C₃×Dic₃	1	2	yes	—
72	C₂×C₃×C₈	—	18	2	no	—
73	C₃×C₄²	—	3	1	no	—
74	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
75	C₂×C₃⋊C₈	—	6	2	no	—
76	C₂²×C₃²	—	4	4	no	—
77	C₂²×C₃²	—	4	2	no	—
78	C₂²×C₃²	#217	1	1	yes	characteristic
79	C₃²×C₄	—	2	1	no	—
80	C₃²×C₄	#213	1	2	yes	—
81	C₃²×C₄	#214	1	2	yes	—
82	C₂²×C₉	—	2	1	no	—
83	C₂²×C₉	—	4	8	no	—
84	C₂²×C₉	—	4	4	no	—
85	C₂²×C₉	—	4	2	no	—
86	C₂²×C₉	—	4	4	no	—
87	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₉	—	2	2	no	—
89	C₄×C₉	—	2	1	no	—
90	C₄×C₉	—	2	2	no	—
91	C₄×C₉	—	2	2	no	—
92	C₄×C₉	C₃×D₈	1	2	yes	—
93	C₄×C₉	C₃×SD₁₆	1	2	yes	—
94	#127	AGL(1,ℤ/9)	1	1	yes	characteristic
95	C₄⋊C₈	—	9	2	no	—
96	3₋¹⁺²	#331	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₃	—	2	2	no	—
98	C₂³×C₃	—	6	2	no	—
99	C₃×D₄	—	4	2	no	—
100	C₃×D₄	—	4	2	no	—
101	C₃×D₄	—	4	2	no	—
102	C₃×D₄	—	12	2	no	—
103	C₃×D₄	—	12	2	no	—
104	C₃×D₄	—	12	2	no	—
105	C₂×C₃×C₄	—	3	1	no	—
106	C₂×C₃×C₄	—	3	2	no	—
107	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₄	C₃×#67	1	2	yes	—
109	C₃×C₈	—	36	2	no	—
110	C₃⋊C₈	—	12	2	no	—
111	C₂×C₃²	—	8	2	no	—
112	C₂×C₃²	#832	1	2	yes	—
113	C₂×C₃²	#833	1	1	yes	characteristic
114	C₂×C₉	—	2	2	no	—
115	C₂×C₉	—	2	1	no	—
116	C₂×C₉	—	8	4	no	—
117	C₂×C₉	—	8	2	no	—
118	C₂×C₉	C₃×#100	1	1	yes	characteristic
119	C₂×C₉	C₃×#102	1	2	yes	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	3₋¹⁺²⋊C₄	1	2	yes	—
122	C₂×C₈	—	18	2	no	—
123	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
124	C₂²×C₃	—	3	1	no	—
125	C₂²×C₃	—	4	4	no	—
126	C₂²×C₃	—	4	2	no	—
127	C₂²×C₃	—	12	4	no	—
128	C₂²×C₃	—	12	2	no	—
129	C₂²×C₃	C₃×#217	1	1	yes	characteristic
130	C₃×C₄	—	2	1	no	—
131	C₃×C₄	—	3	2	no	—
132	C₃×C₄	—	3	2	no	—
133	C₃×C₄	—	6	1	no	—
134	C₃×C₄	C₃×#213	1	2	yes	—
135	C₃×C₄	C₃×#214	1	2	yes	—
136	C₃²	#4544	1	1	yes	characteristic
137	C₉	—	2	1	no	—
138	C₉	C₃×#331	1	1	yes	characteristic, complemented
139	C₂³	—	2	2	no	—
140	D₄	—	4	2	no	—
141	D₄	—	4	2	no	—
142	D₄	—	4	2	no	—
143	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#611⋊C₃	1	2	yes	—
145	C₈	—	36	2	no	—
146	C₂×C₃	—	3	1	no	—
147	C₂×C₃	—	3	2	no	—
148	C₂×C₃	—	8	2	no	—
149	C₂×C₃	—	24	2	no	—
150	C₂×C₃	C₃×#832	1	2	yes	—
151	C₂×C₃	C₃×#833	1	1	yes	characteristic
152	C₂²	—	4	2	no	—
153	C₂²	—	4	4	no	—
154	C₂²	#3676⋊C₃	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	#3672⋊C₃	1	2	yes	—
157	C₄	#3673⋊C₃	1	2	yes	—
158	C₃	—	3	1	no	—
159	C₃	C₃×#4544	1	1	yes	characteristic
160	C₂	—	8	2	no	—
161	C₂	#63391⋊C₃	1	1	yes	characteristic
162	C₂	#63392⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#127

C₂

yes

maximal, characteristic

C₄⋊(3₋¹⁺²⋊C₈)

C₂

yes

maximal, complemented

C₃×#4544

—

maximal

#10588221

C₃

yes

maximal, characteristic, complemented

C₂×D₄×3₋¹⁺²

—

C₂×D₄×3₋¹⁺²

C₄

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₂×3₋¹⁺²⋊C₈

—

C₃²×#127

D₃

yes

characteristic

C₃×C₄⋊(C₃⋊C₈)

—

C₉×#127

—

C₉×#127

C₂×C₃

yes

characteristic

C₄⋊(C₉⋊C₈)

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

—

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

—

3₋¹⁺²⋊C₈

—

C₃×#331

—

#4544

—

C₂×C₃²×D₄

—

C₂×C₃²×D₄

Dic₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

—

C₂×D₄×C₉

C₃×C₄

yes

—

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

D₈

yes

—

C₄×3₋¹⁺²

SD₁₆

yes

—

C₃×#127

—

C₃×#127

C₃×D₃

yes

characteristic

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₂³×C₉

—

C₂³×C₉

—

C₃×C₃⋊C₈

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

—

C₉⋊C₈

—

#331

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#100

yes

characteristic

C₂×3₋¹⁺²

#102

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃×Dic₃

yes

—

C₂×C₃×C₈

—

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

#213

yes

—

C₃²×C₄

#214

yes

—

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

C₃×D₈

yes

—

C₄×C₉

C₃×SD₁₆

yes

—

#127

AGL(1,ℤ/9)

yes

characteristic

C₄⋊C₈

—

3₋¹⁺²

#331

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

100

C₃×D₄

—

101

C₃×D₄

—

102

C₃×D₄

—

103

C₃×D₄

—

104

C₃×D₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

108

C₂×C₃×C₄

C₃×#67

yes

—

109

C₃×C₈

—

110

C₃⋊C₈

—

111

C₂×C₃²

—

112

C₂×C₃²

#832

yes

—

113

C₂×C₃²

#833

yes

characteristic

114

C₂×C₉

—

115

C₂×C₉

—

116

C₂×C₉

—

117

C₂×C₉

—

118

C₂×C₉

C₃×#100

yes

characteristic

119

C₂×C₉

C₃×#102

yes

—

120

C₂×D₄

—

121

C₂×D₄

3₋¹⁺²⋊C₄

yes

—

122

C₂×C₈

—

123

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

124

C₂²×C₃

—

125

C₂²×C₃

—

126

C₂²×C₃

—

127

C₂²×C₃

—

128

C₂²×C₃

—

129

C₂²×C₃

C₃×#217

yes

characteristic

130

C₃×C₄

—

131

C₃×C₄

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

C₃×#213

yes

—

135

C₃×C₄

C₃×#214

yes

—

136

C₃²

#4544

yes

characteristic

137

C₉

—

138

C₉

C₃×#331

yes

characteristic, complemented

139

C₂³

—

140

D₄

—

141

D₄

—

142

D₄

—

143

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

characteristic

144

C₂×C₄

#611⋊C₃

yes

—

145

C₈

—

146

C₂×C₃

—

147

C₂×C₃

—

148

C₂×C₃

—

149

C₂×C₃

—

150

C₂×C₃

C₃×#832

yes

—

151

C₂×C₃

C₃×#833

yes

characteristic

152

C₂²

—

153

C₂²

—

154

C₂²

#3676⋊C₃

yes

characteristic

155

C₄

—

156

C₄

#3672⋊C₃

yes

—

157

C₄

#3673⋊C₃

yes

—

158

C₃

—

159

C₃

C₃×#4544

yes

characteristic

160

C₂

—

161

C₂

#63391⋊C₃

yes

characteristic

162

C₂

#63392⋊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×#127	C₂
socle	C₂²×C₃	C₃×#217
prime radical	C₃²×#127	D₃
cosocle	C₄²×C₉	C₂²×C₃
hypercenter	#127	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#331
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

#3676⋊C₃

commutator

C₂×C₄×C₉

C₂×C₃×C₄

Frattini

C₃×C₄²

C₂×C₃×D₃

Fitting

3₋¹⁺²×#127

C₂

socle

C₂²×C₃

C₃×#217

prime radical

C₃²×#127

D₃

cosocle

C₄²×C₉

C₂²×C₃

C₉

G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. C₄⋊(3₋¹⁺²⋊C₈)	C₂	semidirect
4. #10588221	C₃	semidirect
13. C₄⋊(C₉⋊C₈)	C₂×C₃	semidirect
96. 3₋¹⁺²	#331	semidirect
138. C₉	C₃×#331	semidirect

H ≅ G/N

type

2. C₄⋊(3₋¹⁺²⋊C₈)

C₂

semidirect

4. #10588221

C₃

semidirect

13. C₄⋊(C₉⋊C₈)

C₂×C₃

semidirect

96. 3₋¹⁺²

#331

semidirect

138. C₉

C₃×#331

semidirect

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

subgroup	index
3₋¹⁺²×#127	2
C₄⋊(3₋¹⁺²⋊C₈)	2
C₃×#4544	3
#10588221	3

subgroup

index

3₋¹⁺²×#127

C₄⋊(3₋¹⁺²⋊C₈)

C₃×#4544

#10588221

Subgroups — #10588221⋊C₃ #13135630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588221⋊C3 #13135630

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588221⋊C₃ #13135630