Group #13144512 — subgroups

Subgroups — #13144512

Order 1728.

430 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 155 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×#125	C₂	1	1	yes	maximal, characteristic
2	#11721707	C₂	1	1	yes	maximal, characteristic
3	#11720526	C₂	1	2	yes	maximal
4	#11720525	C₂	1	2	yes	maximal
5	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#10592764	—	3	1	no	maximal
7	#10588744	—	3	3	no	maximal
8	C₃×C₉×C₄⋊C₄	—	2	2	no	—
9	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
12	#88534	—	2	2	no	—
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₃²×#125	D₃	1	1	yes	characteristic
16	#64079	—	3	1	no	—
17	#63635	—	3	2	no	—
18	#63634	—	3	2	no	—
19	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
20	C₉×#125	D₃	1	3	yes	—
21	#63432	—	3	3	no	—
22	#63367	—	3	6	no	—
23	#63366	—	3	6	no	—
24	C₄×C₉⋊C₈	—	3	3	no	—
25	C₂×C₃×C₄×C₉	—	2	2	no	—
26	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₃⋊Dic₉	—	2	2	no	—
29	C₃⋊(C₉⋊C₈)	Q₈	1	2	yes	—
30	C₃⋊(C₉⋊C₈)	Q₈	1	2	yes	—
31	#5065	—	9	3	no	—
32	#5065	—	9	1	no	—
33	C₃²×C₄⋊C₄	—	2	2	no	—
34	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
35	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
36	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
37	#3750	—	6	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
40	C₉×C₄⋊C₄	—	2	6	no	—
41	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	—
42	C₄²×C₉	C₂×D₃	1	3	yes	—
43	C₄⋊Dic₉	—	3	6	no	—
44	Dic₉⋊C₄	—	6	6	no	—
45	C₂×C₉⋊C₈	—	3	3	no	—
46	C₂×C₉⋊C₈	—	3	3	no	—
47	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄×C₉	—	2	2	no	—
49	C₃×C₄×C₉	—	4	2	no	—
50	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
51	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
52	C₃⋊Dic₉	—	4	2	no	—
53	C₃×#125	D₉	1	3	yes	—
54	C₃×#125	C₃⋊D₃	1	1	yes	characteristic
55	#870	—	9	3	no	—
56	#870	—	9	1	no	—
57	#808	—	9	6	no	—
58	#808	—	9	2	no	—
59	#807	—	9	6	no	—
60	#807	—	9	2	no	—
61	C₄×C₃⋊C₈	—	9	3	no	—
62	C₄×C₃⋊C₈	—	9	1	no	—
63	C₂×C₃²×C₄	—	2	2	no	—
64	C₂×C₃²×C₄	#67	1	2	yes	—
65	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
66	C₂×C₃⋊Dic₃	—	6	2	no	—
67	C₃⋊(C₃⋊C₈)	—	3	2	no	—
68	C₃⋊(C₃⋊C₈)	—	3	2	no	—
69	C₂×C₄×C₉	—	2	6	no	—
70	C₂×C₄×C₉	#67	1	6	yes	—
71	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
72	C₂×Dic₉	—	6	6	no	—
73	C₉⋊C₈	—	3	6	no	—
74	C₉⋊C₈	—	3	6	no	—
75	#499	—	27	1	no	Sylow
76	C₂×C₃×C₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₃×C₉	#135	1	2	yes	—
78	C₃×C₄⋊C₄	—	2	6	no	—
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	—
81	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
82	C₃×C₄²	C₂×D₉	1	3	yes	—
83	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
84	C₄⋊Dic₃	—	9	6	no	—
85	C₄⋊Dic₃	—	9	2	no	—
86	Dic₃⋊C₄	—	18	6	no	—
87	Dic₃⋊C₄	—	18	2	no	—
88	C₂×C₃⋊C₈	—	9	3	no	—
89	C₂×C₃⋊C₈	—	9	3	no	—
90	C₂×C₃⋊C₈	—	9	1	no	—
91	C₂×C₃⋊C₈	—	9	1	no	—
92	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
93	C₃²×C₄	—	2	2	no	—
94	C₃²×C₄	—	4	2	no	—
95	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
96	C₃²×C₄	D₃×Q₈	1	1	yes	characteristic
97	C₃⋊Dic₃	—	12	2	no	—
98	C₂²×C₉	C₂×#67	1	3	yes	—
99	C₄×C₉	—	2	6	no	—
100	C₄×C₉	—	4	6	no	—
101	C₄×C₉	D₃×Q₈	1	3	yes	—
102	C₄×C₉	D₃×Q₈	1	3	yes	—
103	Dic₉	—	12	6	no	—
104	#125	—	27	1	no	—
105	#125	C₃⋊D₉	1	1	yes	characteristic
106	#107	—	27	2	no	—
107	#106	—	27	2	no	—
108	C₄×C₈	—	27	1	no	—
109	C₃×C₉	#499	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃×C₄	—	2	6	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	#611	1	6	yes	—
113	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
114	C₂×C₃×C₄	#638	1	2	yes	—
115	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
116	C₂×Dic₃	—	18	6	no	—
117	C₂×Dic₃	—	18	2	no	—
118	C₃⋊C₈	—	9	6	no	—
119	C₃⋊C₈	—	9	6	no	—
120	C₃⋊C₈	—	9	2	no	—
121	C₃⋊C₈	—	9	2	no	—
122	C₂×C₃²	#944	1	1	yes	characteristic
123	C₂×C₃²	#950	1	2	yes	—
124	C₂×C₉	#944	1	3	yes	—
125	C₂×C₉	#950	1	6	yes	—
126	C₂×C₈	—	27	1	no	—
127	C₂×C₈	—	27	1	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	27	2	no	—
130	C₄⋊C₄	—	54	2	no	—
131	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	—
132	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
133	C₂²×C₃	C₂×#611	1	3	yes	—
134	C₂²×C₃	C₂×#638	1	1	yes	characteristic
135	C₃×C₄	—	2	6	no	—
136	C₃×C₄	—	2	2	no	—
137	C₃×C₄	—	4	6	no	—
138	C₃×C₄	—	4	2	no	—
139	C₃×C₄	Q₈×D₉	1	3	yes	—
140	C₃×C₄	Q₈×D₉	1	3	yes	—
141	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
142	C₃×C₄	Q₈×C₃⋊D₃	1	1	yes	characteristic
143	Dic₃	—	36	6	no	—
144	Dic₃	—	36	2	no	—
145	C₃²	#5065	1	1	yes	characteristic
146	C₉	#5065	1	3	yes	complemented
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	54	2	no	—
149	C₂×C₄	#6233	1	2	yes	—
150	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
151	C₈	—	27	2	no	—
152	C₈	—	27	2	no	—
153	C₂×C₃	#63506	1	3	yes	—
154	C₂×C₃	#63513	1	6	yes	—
155	C₂×C₃	#64153	1	1	yes	characteristic
156	C₂×C₃	#64159	1	2	yes	—
157	C₂²	C₂×#6233	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	—	108	2	no	—
161	C₄	Q₈×C₃⋊D₉	1	1	yes	characteristic
162	C₄	Q₈×C₃⋊D₉	1	1	yes	characteristic
163	C₃	#10588744	1	3	yes	complemented
164	C₃	#10592764	1	1	yes	characteristic
165	C₂	#11721854	1	1	yes	characteristic
166	C₂	#11721868	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#125

C₂

yes

maximal, characteristic

#11721707

C₂

yes

maximal, characteristic

#11720526

C₂

yes

maximal

#11720525

C₂

yes

maximal

C₄×C₃⋊(C₉⋊C₈)

C₂

yes

maximal, characteristic

—

—

C₃×C₉×C₄⋊C₄

—

C₃×C₉×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

C₂²

yes

—

#88534

—

C₂×C₃⋊(C₉⋊C₈)

C₂²

yes

characteristic

C₂×C₃⋊(C₉⋊C₈)

C₂²

yes

characteristic

C₃²×#125

D₃

yes

—

—

—

C₄×C₃⋊(C₃⋊C₈)

—

C₉×#125

D₃

yes

—

#63432

—

#63367

—

#63366

—

C₄×C₉⋊C₈

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

C₃⋊(C₉⋊C₈)

Q₈

yes

—

C₃⋊(C₉⋊C₈)

Q₈

yes

—

#5065

—

#5065

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₉

C₂×D₃

yes

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×Q₈

yes

characteristic

C₃×C₄×C₉

C₂×Q₈

yes

characteristic

C₃⋊Dic₉

—

C₃×#125

D₉

yes

—

C₃×#125

C₃⋊D₃

yes

—

—

—

—

—

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×Dic₉

—

C₉⋊C₈

—

C₉⋊C₈

—

#499

—

Sylow

C₂×C₃×C₉

C₄⋊Q₈

yes

characteristic

C₂×C₃×C₉

#135

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×D₉

yes

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×D₉

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

D₃×Q₈

yes

characteristic

C₃²×C₄

D₃×Q₈

yes

characteristic

C₃⋊Dic₃

—

C₂²×C₉

C₂×#67

yes

—

C₄×C₉

—

100

C₄×C₉

—

101

C₄×C₉

D₃×Q₈

yes

—

102

C₄×C₉

D₃×Q₈

yes

—

103

Dic₉

—

104

#125

—

105

#125

C₃⋊D₉

yes

characteristic

106

#107

—

107

#106

—

108

C₄×C₈

—

109

C₃×C₉

#499

yes

characteristic, complemented, Sylow

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

#611

yes

—

113

C₂×C₃×C₄

C₂²×D₉

yes

—

114

C₂×C₃×C₄

#638

yes

—

115

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

116

C₂×Dic₃

—

117

C₂×Dic₃

—

118

C₃⋊C₈

—

119

C₃⋊C₈

—

120

C₃⋊C₈

—

121

C₃⋊C₈

—

122

C₂×C₃²

#944

yes

characteristic

123

C₂×C₃²

#950

yes

—

124

C₂×C₉

#944

yes

—

125

C₂×C₉

#950

yes

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

C₂×C₃⋊D₉

yes

—

132

C₄²

C₂×C₃⋊D₉

yes

characteristic

133

C₂²×C₃

C₂×#611

yes

—

134

C₂²×C₃

C₂×#638

yes

characteristic

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

—

139

C₃×C₄

Q₈×D₉

yes

—

140

C₃×C₄

Q₈×D₉

yes

—

141

C₃×C₄

Q₈×C₃⋊D₃

yes

characteristic

142

C₃×C₄

Q₈×C₃⋊D₃

yes

characteristic

143

Dic₃

—

144

Dic₃

—

145

C₃²

#5065

yes

characteristic

146

C₉

#5065

yes

complemented

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

#6233

yes

—

150

C₂×C₄

C₂²×C₃⋊D₉

yes

characteristic

151

C₈

—

152

C₈

—

153

C₂×C₃

#63506

yes

—

154

C₂×C₃

#63513

yes

—

155

C₂×C₃

#64153

yes

characteristic

156

C₂×C₃

#64159

yes

—

157

C₂²

C₂×#6233

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

108

—

161

C₄

Q₈×C₃⋊D₉

yes

characteristic

162

C₄

Q₈×C₃⋊D₉

yes

characteristic

163

C₃

#10588744

yes

complemented

164

C₃

#10592764

yes

characteristic

165

C₂

#11721854

yes

characteristic

166

C₂

#11721868

yes

—

	subgroup	G/H
center	C₂²	C₂×#6233
commutator, cosocle	C₂×C₃×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#125	C₂
socle, prime radical	C₂²×C₃²	C₂×#67
hypercenter	#125	C₃⋊D₉
hypocenter	C₃×C₉	#499
LCS	G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×#6233

commutator, cosocle

C₂×C₃×C₄×C₉

C₂³

Frattini

C₂×C₃×C₄

C₂²×C₃⋊D₃

Fitting

C₃×C₉×#125

C₂

socle, prime radical

C₂²×C₃²

C₂×#67

C₃⋊D₉

C₃×C₉

G ▷ C₂×C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
109. C₃×C₉	#499	semidirect
146. C₉	#5065	semidirect
163. C₃	#10588744	semidirect

H ≅ G/N

type

109. C₃×C₉

#499

semidirect

146. C₉

#5065

semidirect

163. C₃

#10588744

semidirect

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

subgroup	index
C₃×C₉×#125	2
#11721707	2
#11720526	2
#11720525	2
C₄×C₃⋊(C₉⋊C₈)	2
#10592764	3
#10588744	3

subgroup

index

C₃×C₉×#125

C₄×C₃⋊(C₉⋊C₈)

Subgroups — #13144512

Conjugacy classes of subgroups

Complements

Maximal subgroups