Group #13144513 — subgroups

Subgroups — #13144513

Order 1728.

450 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 151 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	#11721706	C₂	1	1	yes	maximal, characteristic
3	#11720527	C₂	1	4	yes	maximal
4	#11720516	C₂	1	1	yes	maximal, characteristic
5	#10592768	—	3	1	no	maximal
6	#10588746	—	3	3	no	maximal
7	C₂×#6226	C₂²	1	2	yes	—
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)	—	2	2	no	—
12	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
13	C₃²×#125	D₃	1	1	yes	characteristic
14	#64078	—	3	1	no	—
15	#63637	—	3	4	no	—
16	#63630	—	3	1	no	—
17	C₉×#125	D₃	1	3	yes	—
18	#63431	—	3	3	no	—
19	#63368	—	3	12	no	—
20	#63362	—	3	3	no	—
21	C₂×C₃×C₄×C₉	—	2	2	no	—
22	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊Dic₉	—	2	2	no	—
25	#6226	—	2	4	no	—
26	C₃⋊(C₉⋊C₈)	—	2	2	no	—
27	#5069	—	9	3	no	—
28	#5069	—	9	1	no	—
29	C₂×#634	—	3	2	no	—
30	C₃²×C₄⋊C₄	—	2	2	no	—
31	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
32	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
33	C₄⋊(C₃⋊Dic₃)	—	6	2	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
35	C₂×Dic₁₈	—	3	6	no	—
36	C₉×C₄⋊C₄	—	2	6	no	—
37	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	—
38	C₄²×C₉	C₂×D₃	1	3	yes	—
39	C₄⋊Dic₉	—	6	6	no	—
40	C₂×C₉⋊C₈	—	3	6	no	—
41	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
42	C₃×C₄×C₉	—	2	2	no	—
43	C₃×C₄×C₉	—	4	2	no	—
44	C₃×C₄×C₉	#41	1	2	yes	—
45	C₃⋊Dic₉	—	4	2	no	—
46	C₃×#125	D₉	1	3	yes	—
47	C₃×#125	C₃⋊D₃	1	1	yes	characteristic
48	#869	—	9	3	no	—
49	#869	—	9	1	no	—
50	#810	—	9	12	no	—
51	#810	—	9	4	no	—
52	#803	—	9	3	no	—
53	#803	—	9	1	no	—
54	C₂×C₃²×C₄	—	2	2	no	—
55	C₂×C₃²×C₄	#67	1	2	yes	—
56	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₂×C₃⋊Dic₃	—	6	2	no	—
58	#634	—	6	4	no	—
59	C₃⋊(C₃⋊C₈)	—	6	2	no	—
60	C₂×C₄×C₉	—	2	6	no	—
61	C₂×C₄×C₉	#67	1	6	yes	—
62	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
63	C₂×Dic₉	—	6	6	no	—
64	Dic₁₈	—	6	12	no	—
65	C₉⋊C₈	—	6	6	no	—
66	#491	—	27	1	no	Sylow
67	C₂×C₃×C₉	#124	1	1	yes	characteristic
68	C₂×C₃×C₉	#137	1	2	yes	—
69	C₂×Dic₆	—	9	6	no	—
70	C₂×Dic₆	—	9	2	no	—
71	C₃×C₄⋊C₄	—	2	6	no	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	—
74	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
75	C₃×C₄²	C₂×D₉	1	3	yes	—
76	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
77	C₄⋊Dic₃	—	18	6	no	—
78	C₄⋊Dic₃	—	18	2	no	—
79	C₂×C₃⋊C₈	—	9	6	no	—
80	C₂×C₃⋊C₈	—	9	2	no	—
81	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
82	C₃²×C₄	—	2	2	no	—
83	C₃²×C₄	—	4	2	no	—
84	C₃²×C₄	#239	1	2	yes	—
85	C₃⋊Dic₃	—	12	2	no	—
86	C₂²×C₉	C₂×#67	1	3	yes	—
87	C₄×C₉	—	2	6	no	—
88	C₄×C₉	—	4	6	no	—
89	C₄×C₉	#239	1	6	yes	—
90	Dic₉	—	12	6	no	—
91	C₄⋊Q₈	—	27	1	no	—
92	#125	C₃⋊D₉	1	1	yes	characteristic
93	#103	—	27	4	no	—
94	#97	—	27	1	no	—
95	C₃×C₉	#491	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₃×C₄	—	2	6	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	#611	1	6	yes	—
99	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
100	C₂×C₃×C₄	#638	1	2	yes	—
101	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
102	C₂×Dic₃	—	18	6	no	—
103	C₂×Dic₃	—	18	2	no	—
104	Dic₆	—	18	12	no	—
105	Dic₆	—	18	4	no	—
106	C₃⋊C₈	—	18	6	no	—
107	C₃⋊C₈	—	18	2	no	—
108	C₂×C₃²	#947	1	1	yes	characteristic
109	C₂×C₃²	#951	1	2	yes	—
110	C₂×C₉	#947	1	3	yes	—
111	C₂×C₉	#951	1	6	yes	—
112	C₂×Q₈	—	27	2	no	—
113	C₂×C₈	—	27	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	54	2	no	—
116	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	—
117	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
118	C₂²×C₃	C₂×#611	1	3	yes	—
119	C₂²×C₃	C₂×#638	1	1	yes	characteristic
120	C₃×C₄	—	2	6	no	—
121	C₃×C₄	—	2	2	no	—
122	C₃×C₄	—	4	6	no	—
123	C₃×C₄	—	4	2	no	—
124	C₃×C₄	#3701	1	6	yes	—
125	C₃×C₄	#3832	1	2	yes	—
126	Dic₃	—	36	6	no	—
127	Dic₃	—	36	2	no	—
128	C₃²	#5069	1	1	yes	characteristic
129	C₉	#5069	1	3	yes	complemented
130	Q₈	—	54	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	54	2	no	—
133	C₂×C₄	#6233	1	2	yes	—
134	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
135	C₈	—	54	2	no	—
136	C₂×C₃	#63509	1	3	yes	—
137	C₂×C₃	#63511	1	6	yes	—
138	C₂×C₃	#64156	1	1	yes	characteristic
139	C₂×C₃	#64160	1	2	yes	—
140	C₂²	C₂×#6233	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	—	108	2	no	—
144	C₄	#88746	1	2	yes	—
145	C₃	#10588746	1	3	yes	complemented
146	C₃	#10592768	1	1	yes	characteristic
147	C₂	#11721857	1	1	yes	characteristic
148	C₂	#11721866	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×#125

C₂

yes

maximal, characteristic

#11721706

C₂

yes

maximal, characteristic

#11720527

C₂

yes

maximal

#11720516

C₂

yes

maximal, characteristic

—

—

C₂²

yes

—

C₃×C₉×C₄⋊C₄

—

C₃×C₉×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

—

C₂×C₃⋊(C₉⋊C₈)

C₂²

yes

—

C₃²×#125

D₃

yes

—

—

—

D₃

yes

—

#63431

—

#63368

—

#63362

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

D₄

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃⋊Dic₉

—

#6226

—

C₃⋊(C₉⋊C₈)

—

#5069

—

#5069

—

C₂×#634

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₉

C₂×D₃

yes

—

C₄⋊Dic₉

—

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

C₂×D₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

#41

yes

—

C₃⋊Dic₉

—

C₃×#125

D₉

yes

—

C₃×#125

C₃⋊D₃

yes

—

—

—

—

—

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃⋊(C₃⋊C₈)

—

C₂×C₄×C₉

—

C₂×C₄×C₉

#67

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×Dic₉

—

Dic₁₈

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

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

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×#67

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#239

yes

—

C₃⋊Dic₃

—

C₂²×C₉

C₂×#67

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#239

yes

—

Dic₉

—

C₄⋊Q₈

—

#125

C₃⋊D₉

yes

characteristic

#103

—

#97

—

C₃×C₉

#491

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#611

yes

—

C₂×C₃×C₄

C₂²×D₉

yes

—

100

C₂×C₃×C₄

#638

yes

—

101

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

102

C₂×Dic₃

—

103

C₂×Dic₃

—

104

Dic₆

—

105

Dic₆

—

106

C₃⋊C₈

—

107

C₃⋊C₈

—

108

C₂×C₃²

#947

yes

characteristic

109

C₂×C₃²

#951

yes

—

110

C₂×C₉

#947

yes

—

111

C₂×C₉

#951

yes

—

112

C₂×Q₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

C₂×C₃⋊D₉

yes

—

117

C₄²

C₂×C₃⋊D₉

yes

characteristic

118

C₂²×C₃

C₂×#611

yes

—

119

C₂²×C₃

C₂×#638

yes

characteristic

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

#3701

yes

—

125

C₃×C₄

#3832

yes

—

126

Dic₃

—

127

Dic₃

—

128

C₃²

#5069

yes

characteristic

129

C₉

#5069

yes

complemented

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#6233

yes

—

134

C₂×C₄

C₂²×C₃⋊D₉

yes

characteristic

135

C₈

—

136

C₂×C₃

#63509

yes

—

137

C₂×C₃

#63511

yes

—

138

C₂×C₃

#64156

yes

characteristic

139

C₂×C₃

#64160

yes

—

140

C₂²

C₂×#6233

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

108

—

144

C₄

#88746

yes

—

145

C₃

#10588746

yes

complemented

146

C₃

#10592768

yes

characteristic

147

C₂

#11721857

yes

characteristic

148

C₂

#11721866

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	#491
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
95. C₃×C₉	#491	semidirect
129. C₉	#5069	semidirect
145. C₃	#10588746	semidirect

H ≅ G/N

type

95. C₃×C₉

#491

semidirect

129. C₉

#5069

semidirect

145. C₃

#10588746

semidirect

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

subgroup	index
C₃×C₉×#125	2
#11721706	2
#11720527	2
#11720516	2
#10592768	3
#10588746	3

subgroup

index

C₃×C₉×#125

Subgroups — #13144513

Conjugacy classes of subgroups

Complements

Maximal subgroups