Group #13163511 — subgroups

Subgroups — #13163511

Order 1728.

1512 conjugacy classes of subgroups; only the 413 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 413 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×#88853	C₂	1	1	yes	maximal, characteristic
2	#11723928	C₂	1	2	yes	maximal
3	C₃³×#106	C₂	1	1	yes	maximal, characteristic
4	#11722185	C₂	1	1	yes	maximal, characteristic
5	#11722180	C₂	1	2	yes	maximal
6	#10592552	—	3	13	no	maximal
7	C₂×#6308	C₂²	1	2	yes	—
8	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
9	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
10	#88857	—	2	2	no	—
11	C₄×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
12	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	#88853	C₄	1	4	yes	complemented
15	C₂×#3746	—	3	13	no	—
16	#64096	—	3	26	no	—
17	C₃²×#106	D₃	1	13	yes	—
18	#63642	—	3	13	no	—
19	#63637	—	3	26	no	—
20	C₂×C₃³×C₄	—	2	2	no	—
21	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
23	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	2	yes	—
24	#6308	—	2	2	no	—
25	#6308	C₂×C₄	1	4	yes	—
26	C₃³×C₈	C₂×C₄	1	2	yes	—
27	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
28	#4853	—	9	13	no	—
29	C₂×#634	—	3	26	no	—
30	C₂×C₃²×C₈	C₂×D₃	1	13	yes	—
31	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
32	#3750	—	6	26	no	—
33	C₄×C₃⋊Dic₃	—	3	13	no	—
34	C₄×C₃⋊Dic₃	—	6	26	no	—
35	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
36	#3746	—	3	52	no	—
37	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
38	C₃³×C₄	—	4	2	no	—
39	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₃³×C₄	#41	1	1	yes	characteristic
41	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
42	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
43	C₂×Dic₁₂	—	9	13	no	—
44	#887	—	9	26	no	—
45	C₃×#106	C₃⋊D₃	1	13	yes	—
46	#815	—	9	13	no	—
47	#810	—	9	26	no	—
48	C₂×C₃²×C₄	—	2	26	no	—
49	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
50	C₂×C₃⋊Dic₃	—	3	26	no	—
51	C₂×C₃⋊Dic₃	—	6	26	no	—
52	#634	—	3	52	no	—
53	#634	—	6	26	no	—
54	C₃²×C₈	C₄×D₃	1	26	yes	—
55	C₃⋊(C₃⋊C₈)	—	6	13	no	—
56	#441	—	27	1	no	Sylow
57	C₂×C₃³	C₄×D₄	1	1	yes	characteristic
58	C₂×C₃³	#137	1	2	yes	—
59	C₂×Dic₆	—	9	26	no	—
60	C₂×C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
61	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
62	Dic₃⋊C₄	—	18	26	no	—
63	C₄×Dic₃	—	9	13	no	—
64	C₄×Dic₃	—	18	26	no	—
65	C₂×C₃⋊C₈	—	9	13	no	—
66	Dic₁₂	—	9	52	no	—
67	C₂²×C₃²	D₃×D₄	1	13	yes	—
68	C₃²×C₄	—	4	26	no	—
69	C₃²×C₄	C₂×C₄×D₃	1	13	yes	—
70	C₃²×C₄	#239	1	13	yes	—
71	C₃⋊Dic₃	—	6	52	no	—
72	C₃⋊Dic₃	—	6	26	no	—
73	C₂×Q₁₆	—	27	1	no	—
74	C₄×Q₈	—	27	2	no	—
75	#106	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
76	#103	—	27	2	no	—
77	#97	—	27	1	no	—
78	C₃³	#441	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₃×C₄	—	2	26	no	—
80	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
81	C₂×Dic₃	—	9	26	no	—
82	C₂×Dic₃	—	18	26	no	—
83	Dic₆	—	9	52	no	—
84	Dic₆	—	18	26	no	—
85	C₃×C₈	C₄×C₃⋊D₃	1	26	yes	—
86	C₃⋊C₈	—	18	13	no	—
87	C₂×C₃²	#893	1	13	yes	—
88	C₂×C₃²	#915	1	26	yes	—
89	C₂×Q₈	—	27	2	no	—
90	Q₁₆	—	27	4	no	—
91	C₂×C₈	—	27	1	no	—
92	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
93	C₄⋊C₄	—	54	2	no	—
94	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
95	C₄²	—	27	1	no	—
96	C₄²	—	54	2	no	—
97	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
98	C₃×C₄	—	4	26	no	—
99	C₃×C₄	C₂×C₄×C₃⋊D₃	1	13	yes	—
100	C₃×C₄	#3832	1	13	yes	—
101	Dic₃	—	18	52	no	—
102	Dic₃	—	18	26	no	—
103	C₃²	#4853	1	13	yes	complemented
104	Q₈	—	27	4	no	—
105	Q₈	—	54	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	27	2	no	—
108	C₂×C₄	—	54	2	no	—
109	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
110	C₈	—	54	1	no	—
111	C₈	C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
112	C₂×C₃	#64102	1	13	yes	—
113	C₂×C₃	#64124	1	26	yes	—
114	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
115	C₄	—	4	2	no	—
116	C₄	—	54	4	no	—
117	C₄	—	54	2	no	—
118	C₄	C₂×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
119	C₄	#89080	1	1	yes	characteristic
120	C₃	#10592552	1	13	yes	complemented
121	C₂	#11723934	1	1	yes	characteristic
122	C₂	#11723956	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#88853

C₂

yes

maximal, characteristic

#11723928

C₂

yes

maximal

C₃³×#106

C₂

yes

maximal, characteristic

#11722185

C₂

yes

maximal, characteristic

#11722180

C₂

yes

—

C₂²

yes

—

C₂×C₃³×C₈

C₂²

yes

characteristic

C₃³×C₄⋊C₄

C₂²

yes

—

#88857

—

C₄×C₃⋊(C₃⋊Dic₃)

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

#88853

C₄

yes

complemented

C₂×#3746

—

#64096

—

C₃²×#106

D₃

yes

—

#63642

—

#63637

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

D₄

yes

—

#6308

—

#6308

C₂×C₄

yes

—

C₃³×C₈

C₂×C₄

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#4853

—

C₂×#634

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

#3750

—

C₄×C₃⋊Dic₃

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

#3746

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₂²×C₄

yes

characteristic

C₃³×C₄

#41

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₂×Dic₁₂

—

#887

—

C₃×#106

C₃⋊D₃

yes

—

#815

—

#810

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

C₃²×C₈

C₄×D₃

yes

—

C₃⋊(C₃⋊C₈)

—

#441

—

Sylow

C₂×C₃³

C₄×D₄

yes

characteristic

C₂×C₃³

#137

yes

—

C₂×Dic₆

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

D₃×D₄

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×C₄×D₃

yes

—

C₃²×C₄

#239

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₄×Q₈

—

#106

C₃⋊(C₃⋊D₃)

yes

characteristic

#103

—

#97

—

C₃³

#441

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

C₄×C₃⋊D₃

yes

—

C₃⋊C₈

—

C₂×C₃²

#893

yes

—

C₂×C₃²

#915

yes

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

C₄²

—

C₄²

—

C₂²×C₃

D₄×C₃⋊D₃

yes

—

C₃×C₄

—

C₃×C₄

C₂×C₄×C₃⋊D₃

yes

—

100

C₃×C₄

#3832

yes

—

101

Dic₃

—

102

Dic₃

—

103

C₃²

#4853

yes

complemented

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

110

C₈

—

111

C₈

C₄×C₃⋊(C₃⋊D₃)

yes

—

112

C₂×C₃

#64102

yes

—

113

C₂×C₃

#64124

yes

—

114

C₂²

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×C₄×C₃⋊(C₃⋊D₃)

yes

characteristic

119

C₄

#89080

yes

characteristic

120

C₃

#10592552

yes

complemented

121

C₂

#11723934

yes

characteristic

122

C₂

#11723956

yes

—

	subgroup	G/H
center	C₂²	D₄×C₃⋊(C₃⋊D₃)
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×#106	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	#106	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#441
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×C₃⋊(C₃⋊D₃)

commutator

C₃³×C₄

C₂²×C₄

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×#106

C₂

socle, prime radical

C₂²×C₃³

C₂×D₄

cosocle

C₂×C₃³×C₄

C₂³

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
14. #88853	C₄	semidirect
78. C₃³	#441	semidirect
103. C₃²	#4853	semidirect
120. C₃	#10592552	semidirect

H ≅ G/N

type

14. #88853

C₄

semidirect

78. C₃³

#441

semidirect

103. C₃²

#4853

semidirect

120. C₃

#10592552

semidirect

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

subgroup	index
C₂×#88853	2
#11723928	2
C₃³×#106	2
#11722185	2
#11722180	2
#10592552	3

subgroup

index

C₃³×#106

Subgroups — #13163511

Conjugacy classes of subgroups

Complements

Maximal subgroups