Group #13141543 — subgroups

Subgroups — #10588383⋊C₃ #13141543

Order 1728.

290 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 92 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	#63432⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#63431⋊C₃	C₂	1	1	yes	maximal, characteristic
3	3₋¹⁺²×#97	C₂	1	1	yes	maximal, characteristic
4	#63376⋊C₃	C₂	1	4	yes	maximal
5	C₃×#4703	—	3	1	no	maximal
6	#10588383	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂×3₋¹⁺²⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
9	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₄⋊(3₋¹⁺²⋊C₄)	—	2	2	no	—
11	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	2	yes	—
12	(3₋¹⁺²⋊C₄)⋊C₄	—	2	2	no	—
13	C₃×#870	—	3	1	no	—
14	C₃×#869	—	3	1	no	—
15	C₃²×#97	D₃	1	1	yes	characteristic
16	C₃×#816	—	3	4	no	—
17	#63432	C₂×C₃	1	1	yes	characteristic
18	#63431	C₂×C₃	1	1	yes	characteristic
19	C₉×#97	—	2	1	no	—
20	C₉×#97	C₂×C₃	1	1	yes	characteristic
21	#63376	C₂×C₃	1	4	yes	—
22	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
23	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
24	C₂×3₋¹⁺²⋊C₄	—	2	2	no	—
25	C₂×3₋¹⁺²⋊C₄	—	2	2	no	—
26	3₋¹⁺²⋊Q₈	—	2	4	no	—
27	C₈×3₋¹⁺²	—	2	2	no	—
28	C₃×#491	—	9	1	no	—
29	#4703	—	3	1	no	—
30	C₂×C₃×Dic₆	—	3	2	no	—
31	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
32	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
33	C₃×C₄⋊Dic₃	—	3	2	no	—
34	C₃×C₄⋊Dic₃	—	6	2	no	—
35	C₃×Dic₃⋊C₄	—	6	2	no	—
36	C₂×Dic₁₈	C₂²×C₃	1	2	yes	—
37	C₂×C₈×C₉	—	2	2	no	—
38	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
39	C₄²×C₉	—	2	1	no	—
40	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
41	C₄⋊Dic₉	—	2	2	no	—
42	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
43	Dic₉⋊C₄	—	2	2	no	—
44	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
45	C₄×3₋¹⁺²	—	2	2	no	—
46	C₄×3₋¹⁺²	#41	1	2	yes	—
47	3₋¹⁺²⋊C₄	—	4	2	no	—
48	3₋¹⁺²⋊C₄	—	4	2	no	—
49	C₃×C₄⋊Q₈	—	9	1	no	—
50	C₃×#125	—	9	1	no	—
51	#870	—	3	1	no	—
52	#869	—	3	1	no	—
53	C₃×#103	—	9	4	no	—
54	C₃×#97	—	3	1	no	—
55	C₃×#97	C₃×D₃	1	1	yes	characteristic
56	#816	—	3	4	no	—
57	C₂×C₃²×C₄	D₁₂	1	2	yes	—
58	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
59	C₂×C₃×Dic₃	—	6	2	no	—
60	C₂×C₃×Dic₃	—	6	2	no	—
61	C₃×Dic₆	—	6	4	no	—
62	C₃²×C₈	—	2	2	no	—
63	C₂×C₄×C₉	—	2	2	no	—
64	C₂×C₄×C₉	—	2	1	no	—
65	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
66	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
67	C₂×Dic₉	—	2	2	no	—
68	C₂×Dic₉	—	2	2	no	—
69	Dic₁₈	—	2	4	no	—
70	C₈×C₉	—	2	4	no	—
71	C₈×C₉	—	2	2	no	—
72	#491	—	9	1	no	Sylow
73	C₂×3₋¹⁺²	#124	1	1	yes	characteristic
74	C₂×3₋¹⁺²	#137	1	2	yes	—
75	C₂×C₃×Q₈	—	9	2	no	—
76	C₂×Dic₆	—	3	2	no	—
77	C₂×C₃×C₈	—	3	2	no	—
78	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
79	C₃×C₄⋊C₄	—	9	2	no	—
80	C₃×C₄⋊C₄	—	18	2	no	—
81	C₃×C₄⋊C₄	—	18	2	no	—
82	C₃×C₄²	—	3	1	no	—
83	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
84	C₄⋊Dic₃	—	3	2	no	—
85	C₄⋊Dic₃	—	6	2	no	—
86	Dic₃⋊C₄	—	6	2	no	—
87	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
88	C₃²×C₄	—	2	2	no	—
89	C₃²×C₄	#235	1	2	yes	—
90	C₃×Dic₃	—	12	2	no	—
91	C₃×Dic₃	—	12	2	no	—
92	C₂²×C₉	—	2	1	no	—
93	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
94	C₄×C₉	—	2	2	no	—
95	C₄×C₉	—	2	2	no	—
96	C₄×C₉	—	4	2	no	—
97	C₄×C₉	C₃×#41	1	2	yes	—
98	Dic₉	—	4	2	no	—
99	Dic₉	—	4	2	no	—
100	C₄⋊Q₈	—	9	1	no	—
101	#125	—	9	1	no	—
102	#103	—	9	4	no	—
103	#97	AGL(1,ℤ/9)	1	1	yes	characteristic
104	3₋¹⁺²	#491	1	1	yes	characteristic, complemented, Sylow
105	C₃×Q₈	—	18	4	no	—
106	C₂×C₃×C₄	—	3	2	no	—
107	C₂×C₃×C₄	—	3	1	no	—
108	C₂×C₃×C₄	—	18	2	no	—
109	C₂×C₃×C₄	—	18	2	no	—
110	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
111	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
112	C₂×Dic₃	—	6	2	no	—
113	C₂×Dic₃	—	6	2	no	—
114	Dic₆	—	6	4	no	—
115	C₃×C₈	—	2	2	no	—
116	C₃×C₈	—	6	2	no	—
117	C₂×C₃²	#875	1	1	yes	characteristic
118	C₂×C₃²	#909	1	2	yes	—
119	C₂×C₉	—	2	2	no	—
120	C₂×C₉	—	2	1	no	—
121	C₂×C₉	C₃×#124	1	1	yes	characteristic
122	C₂×C₉	C₃×#137	1	2	yes	—
123	C₂×Q₈	—	9	2	no	—
124	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
125	C₄⋊C₄	—	9	2	no	—
126	C₄⋊C₄	—	18	2	no	—
127	C₄⋊C₄	—	18	2	no	—
128	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
129	C₂²×C₃	—	3	1	no	—
130	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
131	C₃×C₄	—	2	2	no	—
132	C₃×C₄	—	3	2	no	—
133	C₃×C₄	—	6	2	no	—
134	C₃×C₄	—	36	2	no	—
135	C₃×C₄	—	36	2	no	—
136	C₃×C₄	C₃×#235	1	2	yes	—
137	Dic₃	—	12	2	no	—
138	Dic₃	—	12	2	no	—
139	C₃²	#4703	1	1	yes	characteristic
140	C₉	—	2	1	no	—
141	C₉	C₃×#491	1	1	yes	characteristic, complemented
142	Q₈	—	18	4	no	—
143	C₂×C₄	—	18	2	no	—
144	C₂×C₄	—	18	2	no	—
145	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	2	yes	—
146	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₂×C₃	—	3	2	no	—
149	C₂×C₃	—	3	1	no	—
150	C₂×C₃	C₃×#875	1	1	yes	characteristic
151	C₂×C₃	C₃×#909	1	2	yes	—
152	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	36	2	no	—
155	C₄	—	36	2	no	—
156	C₄	#3697⋊C₃	1	2	yes	—
157	C₃	—	3	1	no	—
158	C₃	C₃×#4703	1	1	yes	characteristic
159	C₂	#63437⋊C₃	1	1	yes	characteristic
160	C₂	#63471⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#63432⋊C₃

C₂

yes

maximal, characteristic

#63431⋊C₃

C₂

yes

maximal, characteristic

3₋¹⁺²×#97

C₂

yes

maximal, characteristic

#63376⋊C₃

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×3₋¹⁺²⋊Q₈

C₂²

yes

—

C₂×C₈×3₋¹⁺²

C₂²

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

—

C₄⋊(3₋¹⁺²⋊C₄)

C₂²

yes

—

(3₋¹⁺²⋊C₄)⋊C₄

—

C₃×#870

—

C₃×#869

—

C₃²×#97

D₃

yes

characteristic

C₃×#816

—

#63432

C₂×C₃

yes

characteristic

#63431

C₂×C₃

yes

characteristic

C₉×#97

—

C₉×#97

C₂×C₃

yes

characteristic

#63376

C₂×C₃

yes

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊Q₈

—

C₈×3₋¹⁺²

—

C₃×#491

—

#4703

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₂×Dic₁₈

C₂²×C₃

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

C₂²×C₃

yes

—

Dic₉⋊C₄

—

C₂²×3₋¹⁺²

C₂×D₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

#41

yes

—

3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊C₄

—

C₃×C₄⋊Q₈

—

—

—

—

—

C₃×#97

—

C₃×#97

C₃×D₃

yes

characteristic

#816

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃²×C₈

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

—

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

yes

—

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂²×C₃²

C₂×D₁₂

yes

characteristic

C₃²×C₄

—

C₃²×C₄

#235

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×#41

yes

—

Dic₉

—

Dic₉

—

100

C₄⋊Q₈

—

101

#125

—

102

#103

—

103

#97

AGL(1,ℤ/9)

yes

characteristic

104

3₋¹⁺²

#491

yes

characteristic, complemented, Sylow

105

C₃×Q₈

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

C₃×D₁₂

yes

—

111

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

112

C₂×Dic₃

—

113

C₂×Dic₃

—

114

Dic₆

—

115

C₃×C₈

—

116

C₃×C₈

—

117

C₂×C₃²

#875

yes

characteristic

118

C₂×C₃²

#909

yes

—

119

C₂×C₉

—

120

C₂×C₉

—

121

C₂×C₉

C₃×#124

yes

characteristic

122

C₂×C₉

C₃×#137

yes

—

123

C₂×Q₈

—

124

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

129

C₂²×C₃

—

130

C₂²×C₃

C₂×C₃×D₁₂

yes

characteristic

131

C₃×C₄

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

C₃×#235

yes

—

137

Dic₃

—

138

Dic₃

—

139

C₃²

#4703

yes

characteristic

140

C₉

—

141

C₉

C₃×#491

yes

characteristic, complemented

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

—

146

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

147

C₈

—

148

C₂×C₃

—

149

C₂×C₃

—

150

C₂×C₃

C₃×#875

yes

characteristic

151

C₂×C₃

C₃×#909

yes

—

152

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#3697⋊C₃

yes

—

157

C₃

—

158

C₃

C₃×#4703

yes

characteristic

159

C₂

#63437⋊C₃

yes

characteristic

160

C₂

#63471⋊C₃

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,ℤ/9)
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	3₋¹⁺²×#97	C₂
socle	C₂²×C₃	C₂×C₃×D₁₂
prime radical	C₂²×C₃²	C₂×D₁₂
hypercenter	#97	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#491
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

commutator, cosocle

C₂×C₄×C₉

C₂³×C₃

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

3₋¹⁺²×#97

C₂

socle

C₂²×C₃

C₂×C₃×D₁₂

prime radical

C₂²×C₃²

C₂×D₁₂

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
6. #10588383	C₃	semidirect
104. 3₋¹⁺²	#491	semidirect
141. C₉	C₃×#491	semidirect

H ≅ G/N

type

6. #10588383

C₃

semidirect

104. 3₋¹⁺²

#491

semidirect

141. C₉

C₃×#491

semidirect

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

subgroup	index
#63432⋊C₃	2
#63431⋊C₃	2
3₋¹⁺²×#97	2
#63376⋊C₃	2
C₃×#4703	3
#10588383	3

subgroup

index

3₋¹⁺²×#97

Subgroups — #10588383⋊C₃ #13141543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588383⋊C3 #13141543

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588383⋊C₃ #13141543