Group #13165067 — subgroups

Subgroups — #13165067

Order 1728.

202 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 34 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	C₂×#88926	C₂	1	1	yes	maximal, characteristic
2	#11724242	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#64218	C₂	1	1	yes	maximal, characteristic, complemented
4	#10593266	—	3	1	no	maximal
5	C₂×#6289	C₄	1	1	yes	characteristic
6	C₂×C₃⋊#622	C₄	1	1	yes	characteristic
7	C₂×C₃×#622	C₂²	1	1	yes	characteristic
8	#88926	—	2	1	no	—
9	#88926	—	2	1	no	—
10	C₃⋊#3771	—	2	1	no	—
11	C₃×#3771	—	2	1	no	—
12	C₂×#3788	—	3	1	no	—
13	#64218	—	3	1	no	—
14	#64218	D₃	1	1	yes	characteristic, complemented
15	C₂²×C₃⋊(C₃⋊D₃)	C₈	1	1	yes	characteristic
16	C₂×C₃×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃²×Dic₃	C₈	1	1	yes	characteristic
18	#6289	—	2	1	no	—
19	#6289	—	2	1	no	—
20	C₃⋊#622	—	2	1	no	—
21	C₃×#622	D₄	1	1	yes	characteristic
22	C₃×#622	D₄	1	1	yes	characteristic
23	#4514	—	9	1	no	maximal
24	C₂×C₄×C₃⋊D₃	—	3	1	no	—
25	C₂×#624	—	12	1	no	—
26	C₂×#622	—	3	1	no	—
27	C₂×#622	C₂×D₃	1	1	yes	characteristic
28	#3788	—	6	1	no	—
29	#3788	—	6	1	no	—
30	#3771	—	2	1	no	—
31	#3771	—	6	1	no	—
32	C₂²×C₃³	C₂×C₈	1	1	yes	characteristic
33	C₂×C₃⋊(C₃⋊D₃)	—	2	1	no	—
34	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	—
35	C₃×C₃⋊Dic₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₃×C₃⋊Dic₃	M₁₆	1	1	yes	characteristic
37	C₃²×Dic₃	—	2	1	no	—
38	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
39	C₃×M₃₂	—	9	1	no	—
40	#812	—	9	1	no	—
41	C₂²×C₃⋊D₃	—	3	1	no	—
42	C₂²×C₃⋊D₃	—	12	1	no	—
43	C₂²×C₃⋊D₃	—	24	1	no	—
44	C₂×C₃²×C₄	—	3	1	no	—
45	C₂×C₃⋊Dic₃	C₄×D₃	1	1	yes	characteristic
46	C₄×C₃⋊D₃	—	6	1	no	—
47	C₄×C₃⋊D₃	—	6	1	no	—
48	C₂×C₃×Dic₃	—	4	1	no	—
49	C₂×C₃×Dic₃	—	12	1	no	—
50	#624	—	12	2	no	—
51	#624	—	12	2	no	—
52	#622	—	6	1	no	—
53	#622	D₁₂	1	1	yes	characteristic
54	#622	#67	1	1	yes	characteristic
55	#349	—	27	1	no	Sylow
56	C₂×C₃³	—	2	1	no	—
57	C₂×C₃³	C₂²⋊C₈	1	1	yes	characteristic
58	C₃⋊(C₃⋊D₃)	—	4	1	no	—
59	C₂×C₄×D₃	—	9	1	no	—
60	C₂×C₄×D₃	—	36	1	no	—
61	C₂×C₃×C₈	—	9	1	no	—
62	C₂×C₃⋊C₈	—	9	1	no	—
63	ΓL(1,𝔽₂₅)	—	18	1	no	—
64	ΓL(1,𝔽₂₅)	—	18	1	no	—
65	C₃×C₁₆	—	18	1	no	—
66	C₃⋊C₁₆	—	18	1	no	—
67	C₂²×C₃²	—	4	1	no	—
68	C₂²×C₃²	—	8	1	no	—
69	C₂²×C₃²	D₃×C₈	1	1	yes	characteristic
70	C₂×C₃⋊D₃	—	6	1	no	—
71	C₂×C₃⋊D₃	—	12	2	no	—
72	C₂×C₃⋊D₃	—	12	1	no	—
73	C₂×C₃⋊D₃	—	12	4	no	—
74	C₂×C₃⋊D₃	—	24	4	no	—
75	C₂×C₃⋊D₃	—	24	2	no	—
76	C₃²×C₄	—	6	1	no	—
77	C₃⋊Dic₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
78	C₃⋊Dic₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
79	C₃×Dic₃	—	4	2	no	—
80	C₃×Dic₃	—	12	1	no	—
81	C₃×Dic₃	—	12	1	no	—
82	C₂×M₁₆	—	27	1	no	—
83	M₃₂	—	9	1	no	—
84	M₃₂	—	27	1	no	—
85	C₃³	#349	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₃	—	9	1	no	—
87	C₂²×D₃	—	36	1	no	—
88	C₂²×D₃	—	72	1	no	—
89	C₂×C₃×C₄	—	9	1	no	—
90	C₂×C₃×C₄	—	12	1	no	—
91	C₂×Dic₃	—	12	1	no	—
92	C₂×Dic₃	AGL(1,𝔽₉)	1	1	yes	characteristic
93	C₄×D₃	—	18	1	no	—
94	C₄×D₃	—	18	1	no	—
95	C₄×D₃	—	36	2	no	—
96	C₄×D₃	—	36	2	no	—
97	C₃×C₈	—	9	1	no	—
98	C₃×C₈	—	9	1	no	—
99	C₃⋊C₈	—	18	1	no	—
100	C₂×C₃²	—	2	1	no	—
101	C₂×C₃²	—	4	2	no	—
102	C₂×C₃²	—	4	1	no	—
103	C₂×C₃²	—	8	2	no	—
104	C₂×C₃²	—	8	1	no	—
105	C₂×C₃²	(C₂×D₃)⋊C₈	1	1	yes	characteristic
106	C₃⋊D₃	—	12	1	no	—
107	C₃⋊D₃	—	12	4	no	—
108	C₃⋊D₃	—	24	4	no	—
109	C₂²×C₄	—	27	1	no	—
110	M₁₆	—	54	1	no	—
111	M₁₆	—	54	1	no	—
112	C₂×C₈	—	9	1	no	—
113	C₂×C₈	—	27	1	no	—
114	C₁₆	—	18	1	no	—
115	C₁₆	—	54	1	no	—
116	C₂²×C₃	—	4	1	no	—
117	C₂²×C₃	—	8	1	no	—
118	C₂²×C₃	C₂×AGL(1,𝔽₉)	1	1	yes	characteristic
119	C₂×D₃	—	18	1	no	—
120	C₂×D₃	—	36	2	no	—
121	C₂×D₃	—	36	4	no	—
122	C₂×D₃	—	36	1	no	—
123	C₂×D₃	—	72	2	no	—
124	C₂×D₃	—	72	4	no	—
125	C₃×C₄	—	9	1	no	—
126	C₃×C₄	—	9	1	no	—
127	C₃×C₄	—	12	2	no	—
128	Dic₃	—	2	1	no	—
129	Dic₃	—	12	1	no	—
130	Dic₃	—	12	1	no	—
131	C₃²	—	4	1	no	—
132	C₃²	—	8	1	no	—
133	C₃²	#4514	1	1	yes	characteristic, complemented
134	C₂³	—	27	1	no	—
135	C₂×C₄	—	3	1	no	—
136	C₂×C₄	—	9	1	no	—
137	C₂×C₄	—	54	1	no	—
138	C₂×C₄	—	54	1	no	—
139	C₈	—	9	1	no	—
140	C₈	—	9	1	no	—
141	C₈	—	54	1	no	—
142	C₂×C₃	—	2	1	no	—
143	C₂×C₃	—	4	2	no	—
144	C₂×C₃	—	4	1	no	—
145	C₂×C₃	—	8	2	no	—
146	C₂×C₃	—	8	1	no	—
147	C₂×C₃	C₂²⋊AGL(1,𝔽₉)	1	1	yes	characteristic
148	D₃	—	36	4	no	—
149	D₃	—	36	1	no	—
150	D₃	—	72	4	no	—
151	C₂²	—	54	1	no	—
152	C₂²	—	108	1	no	—
153	C₂²	D₃×AGL(1,𝔽₉)	1	1	yes	characteristic
154	C₄	—	6	1	no	—
155	C₄	—	9	1	no	—
156	C₄	—	9	1	no	—
157	C₃	—	4	1	no	—
158	C₃	—	8	1	no	—
159	C₃	#10593266	1	1	yes	characteristic, complemented
160	C₂	—	2	1	no	—
161	C₂	—	108	1	no	—
162	C₂	(C₂×D₃)⋊AGL(1,𝔽₉)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88926

C₂

yes

maximal, characteristic

#11724242

C₂

yes

maximal, characteristic, complemented

C₃×#64218

C₂

yes

maximal, characteristic, complemented

#10593266

—

maximal

C₂×#6289

C₄

yes

characteristic

C₂×C₃⋊#622

C₄

yes

characteristic

C₂×C₃×#622

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂²×C₃⋊(C₃⋊D₃)

C₈

yes

characteristic

C₂×C₃×C₃⋊Dic₃

C₂×C₄

yes

characteristic

C₂×C₃²×Dic₃

C₈

yes

—

—

—

D₄

yes

characteristic

C₃×#622

D₄

yes

characteristic

#4514

—

maximal

C₂×C₄×C₃⋊D₃

—

C₂×#624

—

C₂×#622

—

C₂×#622

C₂×D₃

yes

—

—

—

—

C₂²×C₃³

C₂×C₈

yes

characteristic

C₂×C₃⋊(C₃⋊D₃)

—

C₂×C₃⋊(C₃⋊D₃)

—

C₃×C₃⋊Dic₃

C₂²⋊C₄

yes

characteristic

C₃×C₃⋊Dic₃

M₁₆

yes

characteristic

C₃²×Dic₃

—

C₂×ΓL(1,𝔽₂₅)

—

C₃×M₃₂

—

#812

—

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

—

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

—

C₂×C₃⋊Dic₃

C₄×D₃

yes

characteristic

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

—

—

—

D₁₂

yes

yes

—

C₂×C₃³

—

C₂×C₃³

C₂²⋊C₈

yes

characteristic

C₃⋊(C₃⋊D₃)

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

ΓL(1,𝔽₂₅)

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

D₃×C₈

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃⋊Dic₃

ΓL(1,𝔽₂₅)

yes

characteristic

C₃⋊Dic₃

(C₂×D₃)⋊C₄

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂×M₁₆

—

M₃₂

—

M₃₂

—

C₃³

#349

yes

characteristic, complemented, Sylow

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

AGL(1,𝔽₉)

yes

characteristic

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

—

103

C₂×C₃²

—

104

C₂×C₃²

—

105

C₂×C₃²

(C₂×D₃)⋊C₈

yes

characteristic

106

C₃⋊D₃

—

107

C₃⋊D₃

—

108

C₃⋊D₃

—

109

C₂²×C₄

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₁₆

—

115

C₁₆

—

116

C₂²×C₃

—

117

C₂²×C₃

—

118

C₂²×C₃

C₂×AGL(1,𝔽₉)

yes

characteristic

119

C₂×D₃

—

120

C₂×D₃

—

121

C₂×D₃

—

122

C₂×D₃

—

123

C₂×D₃

—

124

C₂×D₃

—

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

Dic₃

—

129

Dic₃

—

130

Dic₃

—

131

C₃²

—

132

C₃²

—

133

C₃²

#4514

yes

characteristic, complemented

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

—

146

C₂×C₃

—

147

C₂×C₃

C₂²⋊AGL(1,𝔽₉)

yes

characteristic

148

D₃

—

149

D₃

—

150

D₃

—

151

C₂²

—

152

C₂²

—

108

—

153

C₂²

D₃×AGL(1,𝔽₉)

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₃

—

158

C₃

—

159

C₃

#10593266

yes

characteristic, complemented

160

C₂

—

161

C₂

—

108

—

162

C₂

(C₂×D₃)⋊AGL(1,𝔽₉)

yes

characteristic

	subgroup	G/H
center	C₂	(C₂×D₃)⋊AGL(1,𝔽₉)
commutator, Fitting	C₂²×C₃³	C₂×C₈
Frattini, hypercenter	C₂²	D₃×AGL(1,𝔽₉)
socle	C₂×C₃³	C₂²⋊C₈
prime radical	C₂²×C₃⋊(C₃⋊D₃)	C₈
cosocle	C₂×C₃×#622	C₂²
hypocenter	C₃³	#349
LCS	G ▷ C₂²×C₃³ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂²

subgroup

G/H

center

C₂

(C₂×D₃)⋊AGL(1,𝔽₉)

commutator, Fitting

C₂²×C₃³

C₂×C₈

Frattini, hypercenter

C₂²

D₃×AGL(1,𝔽₉)

socle

C₂×C₃³

C₂²⋊C₈

prime radical

C₂²×C₃⋊(C₃⋊D₃)

C₈

cosocle

C₂×C₃×#622

C₂²

hypocenter

C₃³

#349

LCS

G ▷ C₂²×C₃³ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂ ◁ C₂²

N	H ≅ G/N	type
2. #11724242	C₂	semidirect
3. C₃×#64218	C₂	semidirect
14. #64218	D₃	semidirect
85. C₃³	#349	semidirect
133. C₃²	#4514	semidirect
159. C₃	#10593266	semidirect

H ≅ G/N

type

2. #11724242

C₂

semidirect

3. C₃×#64218

C₂

semidirect

14. #64218

D₃

semidirect

85. C₃³

#349

semidirect

133. C₃²

#4514

semidirect

159. C₃

#10593266

semidirect

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

subgroup	index
C₂×#88926	2
#11724242	2
C₃×#64218	2
#10593266	3
#4514	9

subgroup

index

Subgroups — #13165067

Conjugacy classes of subgroups

Complements

Maximal subgroups