Group #13143693 — subgroups

Subgroups — #13143693

Order 1728.

590 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic
2	#11720572	C₂	1	3	yes	maximal
3	#11720572	C₂	1	3	yes	maximal
4	#10592361	—	3	1	no	maximal
5	C₉⋊#401	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
7	C₂²×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₂²	1	3	yes	—
9	C₃²×#95	D₃	1	1	yes	characteristic
10	#63658	—	3	3	no	—
11	#63658	—	3	3	no	—
12	C₉×#95	D₃	1	3	yes	—
13	C₉⋊#95	—	3	9	no	—
14	C₉⋊#95	—	3	9	no	—
15	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
16	C₂×C₃×C₄×C₉	—	2	3	no	—
17	C₂×C₃×C₄×C₉	—	2	6	no	—
18	C₂×C₃⋊Dic₉	—	2	3	no	—
19	C₂×C₃⋊Dic₉	—	2	3	no	—
20	C₂×C₃⋊Dic₉	—	2	3	no	—
21	C₂×C₃⋊Dic₉	—	2	3	no	—
22	C₃⋊#401	—	9	3	no	—
23	C₃⋊#401	—	9	1	no	—
24	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
25	C₂²×C₃⋊Dic₃	—	3	1	no	—
26	C₂²×C₃⋊Dic₃	—	3	3	no	—
27	C₂²×C₄×C₉	C₂×D₃	1	9	yes	—
28	C₂²×Dic₉	—	3	3	no	—
29	C₂²×Dic₉	—	3	9	no	—
30	C₂²×C₃×C₉	#41	1	3	yes	—
31	C₂²×C₃×C₉	#41	1	3	yes	—
32	C₂²×C₃×C₉	#41	1	1	yes	characteristic
33	C₃×C₄×C₉	—	4	3	no	—
34	C₃⋊Dic₉	—	4	1	no	—
35	C₃⋊Dic₉	—	4	3	no	—
36	C₃×#95	D₉	1	3	yes	—
37	C₃×#95	C₃⋊D₃	1	1	yes	characteristic
38	C₃⋊#95	—	9	9	no	—
39	C₃⋊#95	—	9	9	no	—
40	C₃⋊#95	—	9	3	no	—
41	C₃⋊#95	—	9	3	no	—
42	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
43	C₂×C₃²×C₄	—	2	3	no	—
44	C₂×C₃²×C₄	—	2	6	no	—
45	C₂×C₃⋊Dic₃	—	6	3	no	—
46	C₂×C₃⋊Dic₃	—	6	3	no	—
47	C₂×C₃⋊Dic₃	—	6	3	no	—
48	C₂×C₃⋊Dic₃	—	6	3	no	—
49	C₂³×C₉	C₂²×D₃	1	3	yes	—
50	C₂×C₄×C₉	—	2	9	no	—
51	C₂×C₄×C₉	—	2	18	no	—
52	C₂×Dic₉	—	6	9	no	—
53	C₂×Dic₉	—	6	9	no	—
54	C₂×Dic₉	—	6	9	no	—
55	C₂×Dic₉	—	6	9	no	—
56	#401	—	27	1	no	Sylow
57	C₂×C₃×C₉	#126	1	1	yes	characteristic
58	C₂×C₃×C₉	#126	1	3	yes	—
59	C₂×C₃×C₉	#126	1	3	yes	—
60	C₂²×C₃×C₄	C₂×D₉	1	9	yes	—
61	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
62	C₂²×Dic₃	—	9	3	no	—
63	C₂²×Dic₃	—	9	9	no	—
64	C₂²×Dic₃	—	9	1	no	—
65	C₂²×Dic₃	—	9	3	no	—
66	C₂²×C₃²	#235	1	3	yes	—
67	C₂²×C₃²	#237	1	3	yes	—
68	C₂²×C₃²	#239	1	1	yes	characteristic
69	C₃²×C₄	—	4	3	no	—
70	C₃⋊Dic₃	—	12	1	no	—
71	C₃⋊Dic₃	—	12	3	no	—
72	C₂²×C₉	#235	1	9	yes	—
73	C₂²×C₉	#237	1	9	yes	—
74	C₂²×C₉	#239	1	3	yes	—
75	C₄×C₉	—	4	9	no	—
76	Dic₉	—	12	3	no	—
77	Dic₉	—	12	9	no	—
78	#95	—	27	3	no	—
79	#95	—	27	3	no	—
80	#95	C₃⋊D₉	1	1	yes	characteristic
81	C₃×C₉	#401	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₃	C₂²×D₉	1	3	yes	—
83	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
84	C₂×C₃×C₄	—	2	9	no	—
85	C₂×C₃×C₄	—	2	18	no	—
86	C₂×C₃×C₄	—	2	3	no	—
87	C₂×C₃×C₄	—	2	6	no	—
88	C₂×Dic₃	—	18	9	no	—
89	C₂×Dic₃	—	18	9	no	—
90	C₂×Dic₃	—	18	9	no	—
91	C₂×Dic₃	—	18	9	no	—
92	C₂×Dic₃	—	18	3	no	—
93	C₂×Dic₃	—	18	3	no	—
94	C₂×Dic₃	—	18	3	no	—
95	C₂×Dic₃	—	18	3	no	—
96	C₂×C₃²	#876	1	1	yes	characteristic
97	C₂×C₃²	#879	1	3	yes	—
98	C₂×C₃²	#898	1	3	yes	—
99	C₂×C₉	#876	1	3	yes	—
100	C₂×C₉	#879	1	9	yes	—
101	C₂×C₉	#898	1	9	yes	—
102	C₂²×C₄	—	27	1	no	—
103	C₂²×C₄	—	27	3	no	—
104	C₂²×C₄	C₂×C₃⋊D₉	1	3	yes	—
105	C₂²×C₃	#3697	1	9	yes	—
106	C₂²×C₃	#3699	1	9	yes	—
107	C₂²×C₃	#3701	1	3	yes	—
108	C₂²×C₃	#3828	1	3	yes	—
109	C₂²×C₃	#3830	1	3	yes	—
110	C₂²×C₃	#3832	1	1	yes	characteristic
111	C₃×C₄	—	4	3	no	—
112	C₃×C₄	—	4	9	no	—
113	Dic₃	—	36	1	no	—
114	Dic₃	—	36	3	no	—
115	Dic₃	—	36	3	no	—
116	Dic₃	—	36	9	no	—
117	C₃²	C₃⋊#401	1	1	yes	characteristic
118	C₉	C₃⋊#401	1	3	yes	complemented
119	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	54	3	no	—
123	C₂×C₄	—	54	3	no	—
124	C₂×C₄	—	54	3	no	—
125	C₂×C₄	—	54	3	no	—
126	C₂×C₃	#63438	1	3	yes	—
127	C₂×C₃	#63441	1	9	yes	—
128	C₂×C₃	#63460	1	9	yes	—
129	C₂×C₃	#64085	1	1	yes	characteristic
130	C₂×C₃	#64088	1	3	yes	—
131	C₂×C₃	#64107	1	3	yes	—
132	C₂²	#88736	1	3	yes	—
133	C₂²	#88742	1	3	yes	—
134	C₂²	#88746	1	1	yes	characteristic
135	C₄	—	4	3	no	—
136	C₄	—	108	3	no	—
137	C₄	—	108	1	no	—
138	C₃	C₉⋊#401	1	3	yes	complemented
139	C₃	#10592361	1	1	yes	characteristic
140	C₂	#11721713	1	1	yes	characteristic
141	C₂	#11721725	1	3	yes	—
142	C₂	#11721754	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#95

C₂

yes

maximal, characteristic

#11720572

C₂

yes

maximal

#11720572

C₂

yes

—

—

C₂²×C₃×C₄×C₉

C₂²

yes

—

C₂²×C₃⋊Dic₉

C₂²

yes

characteristic

C₂²×C₃⋊Dic₉

C₂²

yes

—

C₃²×#95

D₃

yes

characteristic

#63658

—

#63658

—

C₉×#95

D₃

yes

—

C₉⋊#95

—

C₉⋊#95

—

C₂³×C₃×C₉

C₂³

yes

characteristic

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

—

C₂×C₃⋊Dic₉

—

C₂×C₃⋊Dic₉

—

C₂×C₃⋊Dic₉

—

C₂×C₃⋊Dic₉

—

C₃⋊#401

—

C₃⋊#401

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃⋊Dic₃

—

C₂²×C₃⋊Dic₃

—

C₂²×C₄×C₉

C₂×D₃

yes

—

C₂²×Dic₉

—

C₂²×Dic₉

—

C₂²×C₃×C₉

#41

yes

—

C₂²×C₃×C₉

#41

yes

—

C₂²×C₃×C₉

#41

yes

characteristic

C₃×C₄×C₉

—

C₃⋊Dic₉

—

C₃⋊Dic₉

—

C₃×#95

D₉

yes

—

C₃×#95

C₃⋊D₃

yes

—

—

—

—

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

C₂²×D₃

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×C₃×C₉

#126

yes

—

C₂²×C₃×C₄

C₂×D₉

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

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

#235

yes

—

C₂²×C₉

#237

yes

—

C₂²×C₉

#239

yes

—

C₄×C₉

—

Dic₉

—

Dic₉

—

#95

—

#95

—

#95

C₃⋊D₉

yes

characteristic

C₃×C₉

#401

yes

characteristic, complemented, Sylow

C₂³×C₃

C₂²×D₉

yes

—

C₂³×C₃

C₂²×C₃⋊D₃

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

#876

yes

characteristic

C₂×C₃²

#879

yes

—

C₂×C₃²

#898

yes

—

C₂×C₉

#876

yes

—

100

C₂×C₉

#879

yes

—

101

C₂×C₉

#898

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂×C₃⋊D₉

yes

—

105

C₂²×C₃

#3697

yes

—

106

C₂²×C₃

#3699

yes

—

107

C₂²×C₃

#3701

yes

—

108

C₂²×C₃

#3828

yes

—

109

C₂²×C₃

#3830

yes

—

110

C₂²×C₃

#3832

yes

characteristic

111

C₃×C₄

—

112

C₃×C₄

—

113

Dic₃

—

114

Dic₃

—

115

Dic₃

—

116

Dic₃

—

117

C₃²

C₃⋊#401

yes

characteristic

118

C₉

C₃⋊#401

yes

complemented

119

C₂³

C₂²×C₃⋊D₉

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₃

#63438

yes

—

127

C₂×C₃

#63441

yes

—

128

C₂×C₃

#63460

yes

—

129

C₂×C₃

#64085

yes

characteristic

130

C₂×C₃

#64088

yes

—

131

C₂×C₃

#64107

yes

—

132

C₂²

#88736

yes

—

133

C₂²

#88742

yes

—

134

C₂²

#88746

yes

characteristic

135

C₄

—

136

C₄

—

108

—

137

C₄

—

108

—

138

C₃

C₉⋊#401

yes

complemented

139

C₃

#10592361

yes

characteristic

140

C₂

#11721713

yes

characteristic

141

C₂

#11721725

yes

—

142

C₂

#11721754

yes

—

	subgroup	G/H
center	C₂³	C₂²×C₃⋊D₉
commutator, cosocle	C₂³×C₃×C₉	C₂³
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#95	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	#95	C₃⋊D₉
hypocenter	C₃×C₉	#401
LCS	G ▷ C₂³×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center

C₂³

C₂²×C₃⋊D₉

commutator, cosocle

C₂³×C₃×C₉

C₂³

Frattini

C₂³×C₃

C₂²×C₃⋊D₃

Fitting

C₃×C₉×#95

C₂

socle, prime radical

C₂³×C₃²

C₂²×D₃

C₃⋊D₉

C₃×C₉

G ▷ C₂³×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
81. C₃×C₉	#401	semidirect
118. C₉	C₃⋊#401	semidirect
138. C₃	C₉⋊#401	semidirect

H ≅ G/N

type

81. C₃×C₉

#401

semidirect

118. C₉

C₃⋊#401

semidirect

138. C₃

C₉⋊#401

semidirect

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

subgroup	index
C₃×C₉×#95	2
#11720572	2
#10592361	3
C₉⋊#401	3

subgroup

index

C₃×C₉×#95

#11720572

#10592361

C₉⋊#401

Subgroups — #13143693

Conjugacy classes of subgroups

Complements

Maximal subgroups