Group #13165121 — subgroups

Subgroups — #13165121

Order 1728.

210 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 31 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×#88941	C₂	1	1	yes	maximal, characteristic
2	#11724242	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#63748	C₂	1	1	yes	maximal, characteristic, complemented
4	#10593168	—	3	1	no	maximal
5	C₂×#6295	—	2	1	no	—
6	C₃×#3792	—	2	1	no	—
7	C₂×C₃×#622	C₂²	1	1	yes	characteristic
8	#88941	—	2	1	no	—
9	#88941	C₄	1	2	yes	—
10	C₃⋊#3771	—	2	1	no	—
11	C₂×#3774	—	3	1	no	—
12	#64218	—	3	1	no	—
13	#63748	D₃	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₃⋊D₃	—	4	1	no	—
15	C₂×C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
16	#6295	—	2	2	no	—
17	#6295	—	4	1	no	—
18	C₃×#622	—	4	1	no	—
19	C₃×#622	C₂×C₄	1	1	yes	characteristic
20	C₃×#622	D₄	1	1	yes	characteristic
21	#4495	—	9	1	no	maximal
22	C₂×#626	—	12	1	no	—
23	C₂×#625	—	6	1	no	—
24	#3792	—	2	1	no	—
25	C₂×#622	C₂×D₃	1	1	yes	characteristic
26	#3774	—	3	2	no	—
27	#3774	—	6	1	no	—
28	#3771	—	6	1	no	—
29	C₂²×C₃³	SD₁₆	1	1	yes	characteristic
30	C₂×C₃×C₃⋊D₃	—	4	2	no	—
31	C₂×C₃×C₃⋊D₃	—	8	2	no	—
32	C₃×C₃⋊Dic₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₃⋊Dic₃	D₈	1	1	yes	characteristic
34	C₂×D₂₄	—	9	1	no	—
35	C₃×#108	—	9	1	no	—
36	#812	—	9	1	no	—
37	C₂²×C₃⋊D₃	—	4	1	no	—
38	C₂²×C₃×D₃	—	12	1	no	—
39	C₂²×C₃×D₃	—	12	1	no	—
40	C₂²×C₃×D₃	—	24	1	no	—
41	C₂×C₃⋊Dic₃	#67	1	1	yes	characteristic
42	C₂×C₃×Dic₃	—	12	1	no	—
43	#626	—	12	2	no	—
44	#626	—	12	2	no	—
45	#625	—	6	2	no	—
46	#625	—	12	1	no	—
47	#622	—	4	1	no	—
48	#622	C₄×D₃	1	1	yes	characteristic
49	#622	D₁₂	1	1	yes	characteristic
50	#361	—	27	1	no	Sylow
51	C₂×C₃³	—	2	1	no	—
52	C₂×C₃³	#102	1	1	yes	characteristic
53	C₃×C₃⋊D₃	—	8	2	no	—
54	C₂×#67	—	36	1	no	—
55	C₂×D₁₂	—	18	1	no	—
56	C₃×M₁₆	—	18	1	no	—
57	C₂×C₃×C₈	—	9	1	no	—
58	D₂₄	—	9	2	no	—
59	D₂₄	—	18	1	no	—
60	C₃⋊C₁₆	—	18	1	no	—
61	C₂²×C₃²	—	4	1	no	—
62	C₂²×C₃²	—	8	1	no	—
63	C₂²×C₃²	#215	1	1	yes	characteristic
64	C₂×C₃⋊D₃	—	4	2	no	—
65	C₂×C₃⋊D₃	—	8	2	no	—
66	C₂×C₃×D₃	—	12	2	no	—
67	C₂×C₃×D₃	—	12	2	no	—
68	C₂×C₃×D₃	—	24	2	no	—
69	C₂×C₃×D₃	—	24	2	no	—
70	C₂×C₃×D₃	—	24	2	no	—
71	C₂×C₃×D₃	—	24	4	no	—
72	C₃⋊Dic₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
73	C₃⋊Dic₃	#213	1	1	yes	characteristic
74	C₃×Dic₃	—	12	1	no	—
75	C₃×Dic₃	—	12	1	no	—
76	C₂×D₈	—	27	1	no	—
77	M₃₂	—	27	1	no	—
78	#108	—	9	1	no	—
79	C₃³	#361	1	1	yes	characteristic, complemented, Sylow
80	C₂³×C₃	—	36	1	no	—
81	C₂²×D₃	—	12	1	no	—
82	C₂²×D₃	—	12	1	no	—
83	C₂²×D₃	—	24	1	no	—
84	C₂×C₃×C₄	—	9	1	no	—
85	#67	—	36	2	no	—
86	#67	—	36	2	no	—
87	C₂×Dic₃	—	12	1	no	—
88	D₁₂	—	18	2	no	—
89	D₁₂	—	36	1	no	—
90	C₃×C₈	—	9	1	no	—
91	C₃×C₈	—	9	1	no	—
92	C₃×C₈	—	36	1	no	—
93	C₂×C₃²	—	2	1	no	—
94	C₂×C₃²	—	4	2	no	—
95	C₂×C₃²	—	4	1	no	—
96	C₂×C₃²	—	8	2	no	—
97	C₂×C₃²	—	8	1	no	—
98	C₂×C₃²	#809	1	1	yes	characteristic
99	C₃⋊D₃	—	8	2	no	—
100	C₃×D₃	—	24	2	no	—
101	C₃×D₃	—	24	4	no	—
102	C₃×D₃	—	24	2	no	—
103	C₂×D₄	—	54	1	no	—
104	D₈	—	27	2	no	—
105	D₈	—	54	1	no	—
106	M₁₆	—	18	1	no	—
107	C₂×C₈	—	9	1	no	—
108	C₁₆	—	54	1	no	—
109	C₂²×C₃	—	4	1	no	—
110	C₂²×C₃	—	8	1	no	—
111	C₂²×C₃	—	36	2	no	—
112	C₂²×C₃	—	72	2	no	—
113	C₂²×C₃	AΓL(1,𝔽₉)	1	1	yes	characteristic
114	C₂×D₃	—	12	2	no	—
115	C₂×D₃	—	12	2	no	—
116	C₂×D₃	—	24	2	no	—
117	C₂×D₃	—	24	2	no	—
118	C₂×D₃	—	24	2	no	—
119	C₂×D₃	—	24	4	no	—
120	C₃×C₄	—	9	1	no	—
121	C₃×C₄	—	9	1	no	—
122	Dic₃	—	12	1	no	—
123	Dic₃	—	12	1	no	—
124	C₃²	—	4	1	no	—
125	C₃²	—	8	1	no	—
126	C₃²	#4495	1	1	yes	characteristic, complemented
127	C₂³	—	36	1	no	—
128	D₄	—	54	2	no	—
129	D₄	—	108	1	no	—
130	C₂×C₄	—	9	1	no	—
131	C₈	—	9	1	no	—
132	C₈	—	9	1	no	—
133	C₈	—	36	1	no	—
134	C₂×C₃	—	2	1	no	—
135	C₂×C₃	—	4	2	no	—
136	C₂×C₃	—	4	1	no	—
137	C₂×C₃	—	8	2	no	—
138	C₂×C₃	—	8	1	no	—
139	C₂×C₃	—	72	2	no	—
140	C₂×C₃	GO⁺(4,𝔽₂)⋊C₄	1	1	yes	characteristic
141	D₃	—	24	2	no	—
142	D₃	—	24	4	no	—
143	D₃	—	24	2	no	—
144	C₂²	—	36	2	no	—
145	C₂²	—	72	2	no	—
146	C₂²	#89092	1	1	yes	characteristic
147	C₄	—	9	1	no	—
148	C₄	—	9	1	no	—
149	C₃	—	4	1	no	—
150	C₃	—	8	1	no	—
151	C₃	#10593168	1	1	yes	characteristic, complemented
152	C₂	—	2	1	no	—
153	C₂	—	72	2	no	—
154	C₂	#11724101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88941

C₂

yes

maximal, characteristic

#11724242

C₂

yes

maximal, characteristic, complemented

C₃×#63748

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂×C₃×#622

C₂²

yes

characteristic

#88941

—

#88941

C₄

yes

—

—

—

—

D₃

yes

characteristic, complemented

C₂²×C₃×C₃⋊D₃

—

C₂×C₃×C₃⋊Dic₃

D₄

yes

—

—

—

C₂×C₄

yes

characteristic

C₃×#622

D₄

yes

—

—

—

—

C₂×D₃

yes

—

—

—

C₂²×C₃³

SD₁₆

yes

characteristic

C₂×C₃×C₃⋊D₃

—

C₂×C₃×C₃⋊D₃

—

C₃×C₃⋊Dic₃

C₂²⋊C₄

yes

characteristic

C₃×C₃⋊Dic₃

D₈

yes

characteristic

C₂×D₂₄

—

C₃×#108

—

#812

—

C₂²×C₃⋊D₃

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂×C₃⋊Dic₃

#67

yes

characteristic

C₂×C₃×Dic₃

—

—

—

—

—

—

C₄×D₃

yes

characteristic

#622

D₁₂

yes

characteristic

#361

—

Sylow

C₂×C₃³

—

C₂×C₃³

#102

yes

characteristic

C₃×C₃⋊D₃

—

C₂×#67

—

C₂×D₁₂

—

C₃×M₁₆

—

C₂×C₃×C₈

—

D₂₄

—

D₂₄

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

#215

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃⋊Dic₃

(C₂×D₃)⋊C₄

yes

characteristic

C₃⋊Dic₃

#213

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₂×D₈

—

M₃₂

—

#108

—

C₃³

#361

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

#67

—

#67

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#809

yes

characteristic

C₃⋊D₃

—

100

C₃×D₃

—

101

C₃×D₃

—

102

C₃×D₃

—

103

C₂×D₄

—

104

D₈

—

105

D₈

—

106

M₁₆

—

107

C₂×C₈

—

108

C₁₆

—

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂²×C₃

—

113

C₂²×C₃

AΓL(1,𝔽₉)

yes

characteristic

114

C₂×D₃

—

115

C₂×D₃

—

116

C₂×D₃

—

117

C₂×D₃

—

118

C₂×D₃

—

119

C₂×D₃

—

120

C₃×C₄

—

121

C₃×C₄

—

122

Dic₃

—

123

Dic₃

—

124

C₃²

—

125

C₃²

—

126

C₃²

#4495

yes

characteristic, complemented

127

C₂³

—

128

D₄

—

129

D₄

—

108

—

130

C₂×C₄

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

GO⁺(4,𝔽₂)⋊C₄

yes

characteristic

141

D₃

—

142

D₃

—

143

D₃

—

144

C₂²

—

145

C₂²

—

146

C₂²

#89092

yes

characteristic

147

C₄

—

148

C₄

—

149

C₃

—

150

C₃

—

151

C₃

#10593168

yes

characteristic, complemented

152

C₂

—

153

C₂

—

154

C₂

#11724101

yes

characteristic

	subgroup	G/H
center	C₂	#11724101
commutator	C₃×#622	C₂×C₄
Frattini, hypercenter	C₂²	#89092
Fitting	C₂²×C₃³	SD₁₆
socle	C₂×C₃³	#102
prime radical	C₂×#88941	C₂
cosocle	C₂×C₃×#622	C₂²
hypocenter	C₃³	#361
LCS	G ▷ C₃×#622 ▷ C₃×C₃⋊Dic₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂²

subgroup

G/H

C₂

C₂×C₄

Frattini, hypercenter

C₂²

#89092

Fitting

C₂²×C₃³

SD₁₆

C₂×C₃³

C₂

C₂×C₃×#622

C₂²

hypocenter

C₃³

#361

LCS

G ▷ C₃×#622 ▷ C₃×C₃⋊Dic₃ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂ ◁ C₂²

N	H ≅ G/N	type
2. #11724242	C₂	semidirect
3. C₃×#63748	C₂	semidirect
13. #63748	D₃	semidirect
79. C₃³	#361	semidirect
126. C₃²	#4495	semidirect
151. C₃	#10593168	semidirect

H ≅ G/N

type

2. #11724242

C₂

semidirect

3. C₃×#63748

C₂

semidirect

13. #63748

D₃

semidirect

79. C₃³

#361

semidirect

126. C₃²

#4495

semidirect

151. C₃

#10593168

semidirect

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

subgroup	index
C₂×#88941	2
#11724242	2
C₃×#63748	2
#10593168	3
#4495	9

subgroup

index

Subgroups — #13165121

Conjugacy classes of subgroups

Complements

Maximal subgroups