Group #13141381 — subgroups

Subgroups — 3₋¹⁺²⋊#401 #13141381

Order 1728.

392 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 122 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×#95	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²⋊#95	C₂	1	3	yes	maximal
3	3₋¹⁺²⋊#95	C₂	1	3	yes	maximal
4	C₃×C₃⋊#401	—	3	1	no	maximal
5	C₉⋊#401	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
7	C₂²×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
8	C₂²×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
9	C₃²×#95	D₃	1	1	yes	characteristic
10	C₃×C₃⋊#95	—	3	3	no	—
11	C₃×C₃⋊#95	—	3	3	no	—
12	C₉×#95	—	2	1	no	—
13	C₉×#95	C₂×C₃	1	1	yes	characteristic
14	C₉⋊#95	C₂×C₃	1	3	yes	—
15	C₉⋊#95	C₂×C₃	1	3	yes	—
16	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
17	C₂×C₄×3₋¹⁺²	—	2	3	no	—
18	C₂×C₄×3₋¹⁺²	—	2	6	no	—
19	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
20	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
21	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
22	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
23	C₃×#401	—	9	1	no	—
24	C₃⋊#401	—	3	1	no	—
25	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
26	C₂²×C₃×Dic₃	—	3	1	no	—
27	C₂²×C₃×Dic₃	—	3	3	no	—
28	C₂²×C₄×C₉	—	2	3	no	—
29	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
30	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
31	C₂²×Dic₉	C₂²×C₃	1	3	yes	—
32	C₂²×3₋¹⁺²	#41	1	3	yes	—
33	C₂²×3₋¹⁺²	#41	1	3	yes	—
34	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
35	C₄×3₋¹⁺²	—	4	3	no	—
36	3₋¹⁺²⋊C₄	—	4	1	no	—
37	3₋¹⁺²⋊C₄	—	4	3	no	—
38	C₃×#95	—	3	1	no	—
39	C₃×#95	—	9	3	no	—
40	C₃×#95	—	9	3	no	—
41	C₃×#95	C₃×D₃	1	1	yes	characteristic
42	C₃⋊#95	—	3	3	no	—
43	C₃⋊#95	—	3	3	no	—
44	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
45	C₂×C₃²×C₄	—	2	3	no	—
46	C₂×C₃²×C₄	—	2	6	no	—
47	C₂×C₃×Dic₃	—	6	3	no	—
48	C₂×C₃×Dic₃	—	6	3	no	—
49	C₂×C₃×Dic₃	—	6	3	no	—
50	C₂×C₃×Dic₃	—	6	3	no	—
51	C₂³×C₉	—	2	1	no	—
52	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
53	C₂×C₄×C₉	—	2	6	no	—
54	C₂×C₄×C₉	—	2	3	no	—
55	C₂×C₄×C₉	—	2	6	no	—
56	C₂×C₄×C₉	—	4	6	no	—
57	C₂×Dic₉	—	2	3	no	—
58	C₂×Dic₉	—	2	3	no	—
59	C₂×Dic₉	—	2	3	no	—
60	C₂×Dic₉	—	2	3	no	—
61	#401	—	9	1	no	Sylow
62	C₂×3₋¹⁺²	#126	1	1	yes	characteristic
63	C₂×3₋¹⁺²	#126	1	3	yes	—
64	C₂×3₋¹⁺²	#126	1	3	yes	—
65	C₂²×C₃×C₄	—	3	3	no	—
66	C₂²×C₃×C₄	—	9	1	no	—
67	C₂²×C₃×C₄	—	9	3	no	—
68	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
69	C₂²×Dic₃	—	3	1	no	—
70	C₂²×Dic₃	—	3	3	no	—
71	C₂²×C₃²	#235	1	3	yes	—
72	C₂²×C₃²	#237	1	3	yes	—
73	C₂²×C₃²	#239	1	1	yes	characteristic
74	C₃²×C₄	—	4	3	no	—
75	C₃×Dic₃	—	12	1	no	—
76	C₃×Dic₃	—	12	3	no	—
77	C₂²×C₉	—	2	3	no	—
78	C₂²×C₉	—	2	3	no	—
79	C₂²×C₉	—	2	1	no	—
80	C₂²×C₉	C₃×#41	1	3	yes	—
81	C₂²×C₉	C₃×#41	1	3	yes	—
82	C₂²×C₉	C₃×#41	1	1	yes	characteristic
83	C₄×C₉	—	4	6	no	—
84	C₄×C₉	—	4	3	no	—
85	Dic₉	—	4	3	no	—
86	Dic₉	—	4	1	no	—
87	#95	—	9	3	no	—
88	#95	—	9	3	no	—
89	#95	AGL(1,ℤ/9)	1	1	yes	characteristic
90	3₋¹⁺²	#401	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₃	—	3	1	no	—
92	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
93	C₂×C₃×C₄	—	2	3	no	—
94	C₂×C₃×C₄	—	2	6	no	—
95	C₂×C₃×C₄	—	6	3	no	—
96	C₂×C₃×C₄	—	6	6	no	—
97	C₂×C₃×C₄	—	18	3	no	—
98	C₂×C₃×C₄	—	18	3	no	—
99	C₂×C₃×C₄	—	18	3	no	—
100	C₂×C₃×C₄	—	18	3	no	—
101	C₂×Dic₃	—	6	3	no	—
102	C₂×Dic₃	—	6	3	no	—
103	C₂×Dic₃	—	6	3	no	—
104	C₂×Dic₃	—	6	3	no	—
105	C₂×C₃²	#876	1	1	yes	characteristic
106	C₂×C₃²	#879	1	3	yes	—
107	C₂×C₃²	#898	1	3	yes	—
108	C₂×C₉	—	2	3	no	—
109	C₂×C₉	—	2	3	no	—
110	C₂×C₉	—	2	1	no	—
111	C₂×C₉	C₃×#126	1	3	yes	—
112	C₂×C₉	C₃×#126	1	3	yes	—
113	C₂×C₉	C₃×#126	1	1	yes	characteristic
114	C₂²×C₄	—	9	1	no	—
115	C₂²×C₄	—	9	3	no	—
116	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
117	C₂²×C₃	—	3	3	no	—
118	C₂²×C₃	—	3	3	no	—
119	C₂²×C₃	—	3	1	no	—
120	C₂²×C₃	C₃×#235	1	3	yes	—
121	C₂²×C₃	C₃×#237	1	3	yes	—
122	C₂²×C₃	C₃×#239	1	1	yes	characteristic
123	C₃×C₄	—	4	3	no	—
124	C₃×C₄	—	12	3	no	—
125	C₃×C₄	—	36	3	no	—
126	C₃×C₄	—	36	1	no	—
127	Dic₃	—	12	3	no	—
128	Dic₃	—	12	1	no	—
129	C₃²	C₃⋊#401	1	1	yes	characteristic
130	C₉	—	2	1	no	—
131	C₉	C₃×#401	1	1	yes	characteristic, complemented
132	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	6	no	—
135	C₂×C₄	—	18	3	no	—
136	C₂×C₄	—	18	3	no	—
137	C₂×C₄	—	18	3	no	—
138	C₂×C₄	—	18	3	no	—
139	C₂×C₃	—	3	3	no	—
140	C₂×C₃	—	3	1	no	—
141	C₂×C₃	—	3	3	no	—
142	C₂×C₃	C₃×#876	1	1	yes	characteristic
143	C₂×C₃	C₃×#879	1	3	yes	—
144	C₂×C₃	C₃×#898	1	3	yes	—
145	C₂²	#3697⋊C₃	1	3	yes	—
146	C₂²	#3699⋊C₃	1	3	yes	—
147	C₂²	#88724	1	1	yes	characteristic
148	C₄	—	4	3	no	—
149	C₄	—	36	3	no	—
150	C₄	—	36	1	no	—
151	C₃	—	3	1	no	—
152	C₃	C₃×C₃⋊#401	1	1	yes	characteristic
153	C₂	#11721396	1	1	yes	characteristic
154	C₂	#63441⋊C₃	1	3	yes	—
155	C₂	#63460⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#95

C₂

yes

maximal, characteristic

3₋¹⁺²⋊#95

C₂

yes

maximal

3₋¹⁺²⋊#95

C₂

yes

maximal

C₃×C₃⋊#401

—

maximal

C₉⋊#401

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

characteristic

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

C₃²×#95

D₃

yes

characteristic

C₃×C₃⋊#95

—

C₃×C₃⋊#95

—

C₉×#95

—

C₉×#95

C₂×C₃

yes

characteristic

C₉⋊#95

C₂×C₃

yes

—

C₉⋊#95

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₃×#401

—

C₃⋊#401

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₂²×C₃×Dic₃

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

characteristic

C₂²×Dic₉

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

#41

yes

—

C₂²×3₋¹⁺²

#41

yes

—

C₂²×3₋¹⁺²

#41

yes

characteristic

C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊C₄

—

C₃×#95

—

C₃×#95

—

C₃×#95

—

C₃×#95

C₃×D₃

yes

characteristic

C₃⋊#95

—

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

yes

—

C₂×3₋¹⁺²

#126

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×C₃²

#235

yes

—

C₂²×C₃²

#237

yes

—

C₂²×C₃²

#239

yes

characteristic

C₃²×C₄

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×#41

yes

—

C₂²×C₉

C₃×#41

yes

—

C₂²×C₉

C₃×#41

yes

characteristic

C₄×C₉

—

C₄×C₉

—

Dic₉

—

Dic₉

—

—

—

yes

3₋¹⁺²

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×Dic₃

—

102

C₂×Dic₃

—

103

C₂×Dic₃

—

104

C₂×Dic₃

—

105

C₂×C₃²

#876

yes

characteristic

106

C₂×C₃²

#879

yes

—

107

C₂×C₃²

#898

yes

—

108

C₂×C₉

—

109

C₂×C₉

—

110

C₂×C₉

—

111

C₂×C₉

C₃×#126

yes

—

112

C₂×C₉

C₃×#126

yes

—

113

C₂×C₉

C₃×#126

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

C₃×#235

yes

—

121

C₂²×C₃

C₃×#237

yes

—

122

C₂²×C₃

C₃×#239

yes

characteristic

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

—

127

Dic₃

—

128

Dic₃

—

129

C₃²

C₃⋊#401

yes

characteristic

130

C₉

—

131

C₉

C₃×#401

yes

characteristic, complemented

132

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

C₃×#876

yes

characteristic

143

C₂×C₃

C₃×#879

yes

—

144

C₂×C₃

C₃×#898

yes

—

145

C₂²

#3697⋊C₃

yes

—

146

C₂²

#3699⋊C₃

yes

—

147

C₂²

#88724

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₃

—

152

C₃

C₃×C₃⋊#401

yes

characteristic

153

C₂

#11721396

yes

characteristic

154

C₂

#63441⋊C₃

yes

—

155

C₂

#63460⋊C₃

yes

—

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#95	C₂
prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	#95	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#401
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator, cosocle

C₂³×C₉

C₂³×C₃

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

3₋¹⁺²×#95

C₂

prime radical

C₂³×C₃²

C₂²×D₃

C₉

G ▷ C₂³×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
5. C₉⋊#401	C₃	semidirect
90. 3₋¹⁺²	#401	semidirect
131. C₉	C₃×#401	semidirect

H ≅ G/N

type

5. C₉⋊#401

C₃

semidirect

90. 3₋¹⁺²

#401

semidirect

131. C₉

C₃×#401

semidirect

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

subgroup	index
3₋¹⁺²×#95	2
3₋¹⁺²⋊#95	2
C₃×C₃⋊#401	3
C₉⋊#401	3

subgroup

index

3₋¹⁺²×#95

3₋¹⁺²⋊#95

C₃×C₃⋊#401

C₉⋊#401

Subgroups — 3₋¹⁺²⋊#401 #13141381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2⋊#401 #13141381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²⋊#401 #13141381