Group #13163820 — subgroups

Subgroups — #13163820

Order 1728.

1708 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1708 conjugacy classes of subgroups — too many to draw. Only its 411 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×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#11723985	C₂	1	2	yes	maximal
3	C₂×#88863	C₂	1	2	yes	maximal
4	C₂×#88853	C₂	1	1	yes	maximal, characteristic
5	C₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
6	#10592882	—	3	13	no	maximal
7	C₂×C₃³×Q₈	C₂²	1	2	yes	—
8	C₂×#6308	C₂²	1	2	yes	—
9	C₃³×Q₁₆	—	2	2	no	—
10	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
11	#88863	—	2	4	no	—
12	#88857	—	2	4	no	—
13	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
15	#88853	—	2	2	no	—
16	C₂×C₃²×Q₁₆	D₃	1	13	yes	—
17	#64153	—	3	26	no	—
18	C₂×#3756	—	3	26	no	—
19	C₂×#3746	—	3	13	no	—
20	C₈×C₃⋊Dic₃	—	3	13	no	—
21	C₃³×Q₈	—	2	2	no	—
22	C₃³×Q₈	—	2	2	no	—
23	C₂×C₃³×C₄	—	2	2	no	—
24	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
26	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
27	#6308	—	2	2	no	—
28	#6308	—	2	2	no	—
29	C₃³×C₈	D₄	1	2	yes	—
30	C₃⋊(C₃⋊(C₃⋊C₈))	D₄	1	2	yes	—
31	#5183	—	9	13	no	—
32	C₂×C₃²×Q₈	C₂×D₃	1	26	yes	—
33	C₂×#634	—	3	26	no	—
34	C₃²×Q₁₆	—	2	26	no	—
35	C₂×C₃²×C₈	C₂×D₃	1	13	yes	—
36	#3756	—	6	52	no	—
37	#3750	—	6	52	no	—
38	C₄×C₃⋊Dic₃	—	3	13	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
40	#3746	—	6	26	no	—
41	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
42	C₃³×C₄	—	4	2	no	—
43	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
44	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
45	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
46	C₃⋊(C₃⋊Dic₃)	Q₁₆	1	4	yes	complemented
47	C₂×C₃×Q₁₆	C₃⋊D₃	1	13	yes	—
48	#944	—	9	26	no	—
49	C₂×#216	—	9	26	no	—
50	C₂×Dic₁₂	—	9	13	no	—
51	C₈×Dic₃	—	9	13	no	—
52	C₃²×Q₈	—	2	26	no	—
53	C₃²×Q₈	—	2	26	no	—
54	C₂×C₃²×C₄	—	2	26	no	—
55	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
56	C₂×C₃⋊Dic₃	—	3	26	no	—
57	C₂×C₃⋊Dic₃	—	6	26	no	—
58	#634	—	6	26	no	—
59	#634	—	6	26	no	—
60	C₃²×C₈	#67	1	26	yes	—
61	C₃⋊(C₃⋊C₈)	—	3	26	no	—
62	#494	—	27	1	no	Sylow
63	C₂×C₃³	#127	1	1	yes	characteristic
64	C₂×C₃³	C₂×Q₁₆	1	2	yes	—
65	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	26	yes	—
66	C₂×Dic₆	—	9	26	no	—
67	C₃×Q₁₆	—	2	26	no	—
68	C₂×C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
69	#216	—	18	52	no	—
70	Dic₃⋊C₄	—	18	52	no	—
71	C₄×Dic₃	—	9	13	no	—
72	C₂×C₃⋊C₈	—	9	13	no	—
73	Dic₁₂	—	18	26	no	—
74	C₂²×C₃²	D₃×D₄	1	13	yes	—
75	C₃²×C₄	—	4	26	no	—
76	C₃²×C₄	D₃×D₄	1	13	yes	—
77	C₃²×C₄	C₂×#67	1	13	yes	—
78	C₃⋊Dic₃	—	3	52	no	—
79	C₃⋊Dic₃	—	12	26	no	—
80	C₂×Q₁₆	—	27	2	no	—
81	C₂×Q₁₆	—	27	1	no	—
82	C₂×Q₁₆	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
83	C₄⋊Q₈	—	27	2	no	—
84	C₄×C₈	—	27	1	no	—
85	C₃³	#494	1	1	yes	characteristic, complemented, Sylow
86	C₃×Q₈	—	2	26	no	—
87	C₃×Q₈	—	2	26	no	—
88	C₂×C₃×C₄	—	2	26	no	—
89	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
90	C₂×Dic₃	—	9	26	no	—
91	C₂×Dic₃	—	18	26	no	—
92	Dic₆	—	18	26	no	—
93	Dic₆	—	18	26	no	—
94	C₃×C₈	#638	1	26	yes	—
95	C₃⋊C₈	—	9	26	no	—
96	C₂×C₃²	D₃×Q₁₆	1	26	yes	—
97	C₂×C₃²	#940	1	13	yes	—
98	C₂×Q₈	—	27	2	no	—
99	C₂×Q₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	—	54	4	no	—
102	Q₁₆	—	54	2	no	—
103	C₂×C₈	—	27	1	no	—
104	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
105	C₄⋊C₄	—	54	4	no	—
106	C₄²	—	27	1	no	—
107	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
108	C₃×C₄	—	4	26	no	—
109	C₃×C₄	D₄×C₃⋊D₃	1	13	yes	—
110	C₃×C₄	C₂×#638	1	13	yes	—
111	Dic₃	—	9	52	no	—
112	Dic₃	—	36	26	no	—
113	C₃²	#5183	1	13	yes	complemented
114	Q₈	—	2	2	no	—
115	Q₈	—	2	2	no	—
116	Q₈	—	54	2	no	—
117	Q₈	—	54	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	27	2	no	—
120	C₂×C₄	—	54	2	no	—
121	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
122	C₈	—	27	2	no	—
123	C₈	#6312	1	2	yes	—
124	C₂×C₃	Q₁₆×C₃⋊D₃	1	26	yes	—
125	C₂×C₃	#64149	1	13	yes	—
126	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	27	4	no	—
129	C₄	—	108	2	no	—
130	C₄	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₄	C₂×#6312	1	1	yes	characteristic
132	C₃	#10592882	1	13	yes	complemented
133	C₂	Q₁₆×C₃⋊(C₃⋊D₃)	1	2	yes	—
134	C₂	#11723981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃³×Q₁₆

C₂

yes

maximal, characteristic

#11723985

C₂

yes

maximal

C₂×#88863

C₂

yes

maximal

C₂×#88853

C₂

yes

maximal, characteristic

C₈×C₃⋊(C₃⋊Dic₃)

C₂

yes

maximal, characteristic

#10592882

—

maximal

C₂×C₃³×Q₈

C₂²

yes

—

C₂×#6308

C₂²

yes

—

C₃³×Q₁₆

—

C₂×C₃³×C₈

C₂²

yes

characteristic

#88863

—

#88857

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

#88853

—

C₂×C₃²×Q₁₆

D₃

yes

—

#64153

—

C₂×#3756

—

C₂×#3746

—

C₈×C₃⋊Dic₃

—

C₃³×Q₈

—

C₃³×Q₈

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

D₄

yes

—

#6308

—

#6308

—

C₃³×C₈

D₄

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

D₄

yes

—

#5183

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×#634

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

#3756

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

#3746

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₂×D₄

yes

characteristic

C₃³×C₄

C₂×D₄

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

Q₁₆

yes

complemented

C₂×C₃×Q₁₆

C₃⋊D₃

yes

—

#944

—

C₂×#216

—

C₂×Dic₁₂

—

C₈×Dic₃

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

#67

yes

—

C₃⋊(C₃⋊C₈)

—

—

C₂×C₃³

yes

C₂×C₃³

C₂×Q₁₆

yes

—

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

#216

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

D₃×D₄

yes

—

C₃²×C₄

—

C₃²×C₄

D₃×D₄

yes

—

C₃²×C₄

C₂×#67

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₃⋊(C₃⋊D₃)

yes

characteristic

C₄⋊Q₈

—

C₄×C₈

—

C₃³

#494

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

#638

yes

—

C₃⋊C₈

—

C₂×C₃²

D₃×Q₁₆

yes

—

C₂×C₃²

#940

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×C₃⋊(C₃⋊D₃)

yes

—

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

—

107

C₂²×C₃

D₄×C₃⋊D₃

yes

—

108

C₃×C₄

—

109

C₃×C₄

D₄×C₃⋊D₃

yes

—

110

C₃×C₄

C₂×#638

yes

—

111

Dic₃

—

112

Dic₃

—

113

C₃²

#5183

yes

complemented

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

122

C₈

—

123

C₈

#6312

yes

—

124

C₂×C₃

Q₁₆×C₃⋊D₃

yes

—

125

C₂×C₃

#64149

yes

—

126

C₂²

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

108

—

130

C₄

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

131

C₄

C₂×#6312

yes

characteristic

132

C₃

#10592882

yes

complemented

133

C₂

Q₁₆×C₃⋊(C₃⋊D₃)

yes

—

134

C₂

#11723981

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×Q₁₆	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₂×Q₁₆	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#494
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×C₃⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₂×C₃³×Q₁₆

C₂

socle, prime radical

C₂²×C₃³

C₂×D₄

hypercenter

C₂×Q₁₆

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#494

LCS

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
46. C₃⋊(C₃⋊Dic₃)	Q₁₆	semidirect
85. C₃³	#494	semidirect
113. C₃²	#5183	semidirect
132. C₃	#10592882	semidirect

H ≅ G/N

type

46. C₃⋊(C₃⋊Dic₃)

Q₁₆

semidirect

85. C₃³

#494

semidirect

113. C₃²

#5183

semidirect

132. C₃

#10592882

semidirect

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

subgroup	index
C₂×C₃³×Q₁₆	2
#11723985	2
C₂×#88863	2
C₂×#88853	2
C₈×C₃⋊(C₃⋊Dic₃)	2
#10592882	3

subgroup

index

C₂×C₃³×Q₁₆

C₈×C₃⋊(C₃⋊Dic₃)

Subgroups — #13163820

Conjugacy classes of subgroups

Complements

Maximal subgroups