Group #13163758 — subgroups

Subgroups — #13163758

Order 1728.

1764 conjugacy classes of subgroups; only the 639 normal subgroups are shown.

normal subgroups only

This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 639 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×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
3	C₂×#88855	C₂	1	1	yes	maximal, characteristic
4	#11722184	C₂	1	4	yes	maximal, complemented
5	#10592817	—	3	13	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊(C₃⋊Dic₃)	C₄	1	2	yes	—
8	C₃³×M₁₆	—	2	2	no	—
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₄×C₃⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
11	C₄×C₃⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
12	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
13	#88855	—	2	2	no	—
14	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
15	C₂×C₃²×M₁₆	D₃	1	13	yes	—
16	C₂×C₄×C₃⋊Dic₃	—	3	13	no	—
17	C₂×#3748	—	3	13	no	—
18	#63641	—	3	52	no	—
19	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃³×C₄	D₄	1	2	yes	—
22	C₂×C₃³×C₄	Q₈	1	2	yes	—
23	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
25	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
26	C₂×C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
27	C₃³×C₈	—	2	2	no	—
28	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
29	#5118	—	9	13	no	—
30	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
31	C₂²×C₃⋊Dic₃	—	3	26	no	—
32	C₃²×M₁₆	—	2	26	no	—
33	C₂×C₃²×C₈	C₂×D₃	1	26	yes	—
34	C₄×C₃⋊Dic₃	—	3	13	no	—
35	C₄×C₃⋊Dic₃	—	3	13	no	—
36	C₄×C₃⋊Dic₃	—	3	26	no	—
37	#3748	—	6	26	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	3	26	no	—
39	C₂²×C₃³	—	2	2	no	—
40	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
41	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
42	C₃³×C₄	C₄⋊C₄	1	2	yes	—
43	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
44	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
46	C₃⋊(C₃⋊Dic₃)	M₁₆	1	4	yes	complemented
47	C₂×C₃×M₁₆	C₃⋊D₃	1	13	yes	—
48	C₂×C₄×Dic₃	—	9	13	no	—
49	C₂×#208	—	9	13	no	—
50	Dic₃⋊C₈	—	9	52	no	—
51	C₂³×C₃²	C₄×D₃	1	13	yes	—
52	C₂×C₃²×C₄	Dic₆	1	26	yes	—
53	C₂×C₃²×C₄	C₄×D₃	1	13	yes	—
54	C₂×C₃²×C₄	#67	1	26	yes	—
55	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
56	C₂×C₃⋊Dic₃	—	3	26	no	—
57	C₂×C₃⋊Dic₃	—	3	26	no	—
58	C₂×C₃⋊Dic₃	—	6	52	no	—
59	C₃²×C₈	—	2	26	no	—
60	C₃⋊(C₃⋊C₈)	—	6	26	no	—
61	#423	—	27	1	no	Sylow
62	C₂×C₃³	—	2	2	no	—
63	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃³	C₂×M₁₆	1	2	yes	—
65	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
66	C₂²×Dic₃	—	9	26	no	—
67	C₃×M₁₆	—	2	26	no	—
68	C₂×C₃×C₈	C₂×C₃⋊D₃	1	26	yes	—
69	C₄×Dic₃	—	9	13	no	—
70	C₄×Dic₃	—	9	13	no	—
71	C₄×Dic₃	—	9	26	no	—
72	#208	—	18	26	no	—
73	C₂×C₃⋊C₈	—	9	26	no	—
74	C₂²×C₃²	—	2	26	no	—
75	C₂²×C₃²	Dic₃⋊C₄	1	26	yes	—
76	C₂²×C₃²	C₂×C₄×D₃	1	13	yes	—
77	C₃²×C₄	Dic₃⋊C₄	1	26	yes	—
78	C₃²×C₄	C₂×Dic₆	1	13	yes	—
79	C₃²×C₄	C₂×#67	1	13	yes	—
80	C₃⋊Dic₃	—	3	52	no	—
81	C₃⋊Dic₃	—	6	26	no	—
82	C₂×M₁₆	—	27	1	no	—
83	C₂×M₁₆	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
84	C₂×C₄²	—	27	1	no	—
85	C₄⋊C₈	—	27	4	no	—
86	C₃³	#423	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₃	C₄×C₃⋊D₃	1	13	yes	—
88	C₂×C₃×C₄	#634	1	26	yes	—
89	C₂×C₃×C₄	C₄×C₃⋊D₃	1	13	yes	—
90	C₂×C₃×C₄	#638	1	26	yes	—
91	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
92	C₂×Dic₃	—	9	26	no	—
93	C₂×Dic₃	—	9	26	no	—
94	C₂×Dic₃	—	18	52	no	—
95	C₃×C₈	—	2	26	no	—
96	C₃⋊C₈	—	18	26	no	—
97	C₂×C₃²	—	2	26	no	—
98	C₂×C₃²	D₃×M₁₆	1	26	yes	—
99	C₂×C₃²	C₂×Dic₃⋊C₄	1	13	yes	—
100	C₂²×C₄	—	27	2	no	—
101	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	54	2	no	—
104	C₂×C₈	—	27	2	no	—
105	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
106	C₄²	—	27	1	no	—
107	C₄²	—	27	1	no	—
108	C₄²	—	27	2	no	—
109	C₂²×C₃	—	2	26	no	—
110	C₂²×C₃	#3750	1	26	yes	—
111	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	13	yes	—
112	C₃×C₄	#3750	1	26	yes	—
113	C₃×C₄	C₂×#634	1	13	yes	—
114	C₃×C₄	C₂×#638	1	13	yes	—
115	Dic₃	—	9	52	no	—
116	Dic₃	—	18	26	no	—
117	C₃²	#5118	1	13	yes	complemented
118	C₂³	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
119	C₂×C₄	—	27	2	no	—
120	C₂×C₄	—	27	2	no	—
121	C₂×C₄	—	54	4	no	—
122	C₂×C₄	#6308	1	2	yes	—
123	C₂×C₄	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
124	C₂×C₄	#6312	1	2	yes	—
125	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	54	2	no	—
128	C₂×C₃	—	2	26	no	—
129	C₂×C₃	M₁₆×C₃⋊D₃	1	26	yes	—
130	C₂×C₃	C₂×#3750	1	13	yes	—
131	C₂²	—	2	2	no	—
132	C₂²	#88857	1	2	yes	—
133	C₂²	C₂×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
134	C₄	—	27	4	no	—
135	C₄	—	54	2	no	—
136	C₄	#88857	1	2	yes	—
137	C₄	C₂×#6308	1	1	yes	characteristic
138	C₄	C₂×#6312	1	1	yes	characteristic
139	C₃	#10592817	1	13	yes	complemented
140	C₂	—	2	2	no	—
141	C₂	M₁₆×C₃⋊(C₃⋊D₃)	1	2	yes	—
142	C₂	C₂×#88857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃³×M₁₆

C₂

yes

maximal, characteristic

C₂×C₄×C₃⋊(C₃⋊Dic₃)

C₂

yes

maximal, characteristic

C₂×#88855

C₂

yes

maximal, characteristic

#11722184

C₂

yes

maximal, complemented

#10592817

—

maximal

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₂²×C₃⋊(C₃⋊Dic₃)

C₄

yes

—

C₃³×M₁₆

—

C₂×C₃³×C₈

C₂²

yes

—

C₄×C₃⋊(C₃⋊Dic₃)

C₄

yes

characteristic

C₄×C₃⋊(C₃⋊Dic₃)

C₄

yes

characteristic

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

—

#88855

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₂×C₃²×M₁₆

D₃

yes

—

C₂×C₄×C₃⋊Dic₃

—

C₂×#3748

—

#63641

—

C₂³×C₃³

C₂×C₄

yes

characteristic

C₂×C₃³×C₄

C₂×C₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃³×C₄

Q₈

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂×C₄

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂×C₄

yes

—

C₃³×C₈

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#5118

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃⋊Dic₃

—

C₃²×M₁₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₄×C₃⋊Dic₃

—

C₄×C₃⋊Dic₃

—

C₄×C₃⋊Dic₃

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

—

C₂²×C₃³

C₄⋊C₄

yes

—

C₂²×C₃³

C₂²×C₄

yes

characteristic

C₃³×C₄

C₄⋊C₄

yes

—

C₃³×C₄

C₂×D₄

yes

characteristic

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

M₁₆

yes

complemented

C₂×C₃×M₁₆

C₃⋊D₃

yes

—

C₂×C₄×Dic₃

—

C₂×#208

—

Dic₃⋊C₈

—

C₂³×C₃²

C₄×D₃

yes

—

C₂×C₃²×C₄

Dic₆

yes

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#423

—

Sylow

C₂×C₃³

—

C₂×C₃³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃³

C₂×M₁₆

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

Dic₃⋊C₄

yes

—

C₂²×C₃²

C₂×C₄×D₃

yes

—

C₃²×C₄

Dic₃⋊C₄

yes

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃²×C₄

C₂×#67

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×M₁₆

—

C₂×M₁₆

C₃⋊(C₃⋊D₃)

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₃³

#423

yes

characteristic, complemented, Sylow

C₂³×C₃

C₄×C₃⋊D₃

yes

—

C₂×C₃×C₄

#634

yes

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

—

C₂×C₃×C₄

#638

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

D₃×M₁₆

yes

—

C₂×C₃²

C₂×Dic₃⋊C₄

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂²×C₃

—

110

C₂²×C₃

#3750

yes

—

111

C₂²×C₃

C₂×C₄×C₃⋊D₃

yes

—

112

C₃×C₄

#3750

yes

—

113

C₃×C₄

C₂×#634

yes

—

114

C₃×C₄

C₂×#638

yes

—

115

Dic₃

—

116

Dic₃

—

117

C₃²

#5118

yes

complemented

118

C₂³

C₄×C₃⋊(C₃⋊D₃)

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#6308

yes

—

123

C₂×C₄

C₄×C₃⋊(C₃⋊D₃)

yes

characteristic

124

C₂×C₄

#6312

yes

—

125

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂×C₃

—

129

C₂×C₃

M₁₆×C₃⋊D₃

yes

—

130

C₂×C₃

C₂×#3750

yes

—

131

C₂²

—

132

C₂²

#88857

yes

—

133

C₂²

C₂×C₄×C₃⋊(C₃⋊D₃)

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

#88857

yes

—

137

C₄

C₂×#6308

yes

characteristic

138

C₄

C₂×#6312

yes

characteristic

139

C₃

#10592817

yes

complemented

140

C₂

—

141

C₂

M₁₆×C₃⋊(C₃⋊D₃)

yes

—

142

C₂

C₂×#88857

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle	C₂²×C₃³	C₂²×C₄
Fitting	C₂×C₃³×M₁₆	C₂
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂×M₁₆	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#423
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

commutator, socle

C₂²×C₃³

C₂²×C₄

Fitting

C₂×C₃³×M₁₆

C₂

prime radical

C₂³×C₃³

C₂×C₄

cosocle

C₂×C₃³×C₄

C₂³

hypercenter

C₂×M₁₆

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂²×C₃³ ▷ C₃³

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
4. #11722184	C₂	semidirect
46. C₃⋊(C₃⋊Dic₃)	M₁₆	semidirect
86. C₃³	#423	semidirect
117. C₃²	#5118	semidirect
139. C₃	#10592817	semidirect

H ≅ G/N

type

4. #11722184

C₂

semidirect

46. C₃⋊(C₃⋊Dic₃)

M₁₆

semidirect

86. C₃³

#423

semidirect

117. C₃²

#5118

semidirect

139. C₃

#10592817

semidirect

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

subgroup	index
C₂×C₃³×M₁₆	2
C₂×C₄×C₃⋊(C₃⋊Dic₃)	2
C₂×#88855	2
#11722184	2
#10592817	3

subgroup

index

C₂×C₃³×M₁₆

C₂×C₄×C₃⋊(C₃⋊Dic₃)

C₂×#88855

#11722184

#10592817

Subgroups — #13163758

Conjugacy classes of subgroups

Complements

Maximal subgroups