Group #13163639 — subgroups

Subgroups — #13163639

Order 1728.

1596 conjugacy classes of subgroups; only the 851 normal subgroups are shown.

normal subgroups only

This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 851 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	C₃³×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#88855	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	4	yes	maximal, complemented
5	#10592698	—	3	13	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₃³×C₄⋊C₄	C₄	1	2	yes	—
8	C₃³×C₂²⋊C₄	C₄	1	2	yes	—
9	C₃³×C₄²	C₂²	1	2	yes	—
10	#88855	—	2	2	no	—
11	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
12	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
13	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
14	C₃²×#117	D₃	1	13	yes	—
15	C₂×#3748	—	3	13	no	—
16	C₂²×C₃⋊(C₃⋊C₈)	—	3	13	no	—
17	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	52	no	—
18	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃³×C₄	C₂×C₄	1	4	yes	—
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	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
25	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
26	#4999	—	9	13	no	—
27	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
28	C₃²×C₄⋊C₄	Dic₃	1	26	yes	—
29	C₃²×C₂²⋊C₄	Dic₃	1	26	yes	—
30	C₃²×C₄²	C₂×D₃	1	26	yes	—
31	#3748	—	6	26	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	3	26	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	3	26	no	—
34	C₂×C₃⋊(C₃⋊C₈)	—	6	26	no	—
35	C₂²×C₃³	—	2	2	no	—
36	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
37	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
38	C₃³×C₄	—	2	4	no	—
39	C₃³×C₄	C₄⋊C₄	1	2	yes	—
40	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
41	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
42	C₃×#117	C₃⋊D₃	1	13	yes	—
43	C₂×#208	—	9	13	no	—
44	C₂²×C₃⋊C₈	—	9	13	no	—
45	C₄⋊(C₃⋊C₈)	—	9	52	no	—
46	C₂³×C₃²	C₂×Dic₃	1	13	yes	—
47	C₂×C₃²×C₄	Dic₆	1	26	yes	—
48	C₂×C₃²×C₄	D₁₂	1	26	yes	—
49	C₂×C₃²×C₄	C₂×Dic₃	1	52	yes	—
50	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
51	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
52	C₃⋊(C₃⋊C₈)	—	6	26	no	—
53	C₃⋊(C₃⋊C₈)	—	6	26	no	—
54	#424	—	27	1	no	Sylow
55	C₂×C₃³	—	2	2	no	—
56	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃³	#131	1	2	yes	—
58	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
59	C₃×C₄⋊C₄	C₃⋊Dic₃	1	26	yes	—
60	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	26	yes	—
61	C₃×C₄²	C₂×C₃⋊D₃	1	26	yes	—
62	#208	—	18	26	no	—
63	C₂×C₃⋊C₈	—	9	26	no	—
64	C₂×C₃⋊C₈	—	9	26	no	—
65	C₂×C₃⋊C₈	—	18	26	no	—
66	C₂²×C₃²	—	2	26	no	—
67	C₂²×C₃²	C₄⋊Dic₃	1	26	yes	—
68	C₂²×C₃²	C₂²×Dic₃	1	13	yes	—
69	C₃²×C₄	—	2	52	no	—
70	C₃²×C₄	C₄⋊Dic₃	1	26	yes	—
71	C₃²×C₄	C₂×Dic₆	1	13	yes	—
72	C₃²×C₄	C₂×D₁₂	1	13	yes	—
73	C₂×M₁₆	—	27	1	no	—
74	C₂²×C₈	—	27	1	no	—
75	#117	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
76	C₄⋊C₈	—	27	4	no	—
77	C₃³	#424	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₃	C₂×C₃⋊Dic₃	1	13	yes	—
79	C₂×C₃×C₄	#634	1	26	yes	—
80	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	26	yes	—
81	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	52	yes	—
82	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
83	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
84	C₃⋊C₈	—	18	26	no	—
85	C₃⋊C₈	—	18	26	no	—
86	C₂×C₃²	—	2	26	no	—
87	C₂×C₃²	C₂×C₄⋊Dic₃	1	13	yes	—
88	C₂×C₃²	#948	1	26	yes	—
89	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
90	M₁₆	—	54	2	no	—
91	C₂×C₈	—	27	2	no	—
92	C₂×C₈	—	27	2	no	—
93	C₂×C₈	—	54	2	no	—
94	C₄⋊C₄	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
95	C₂²⋊C₄	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
96	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
97	C₂²×C₃	—	2	26	no	—
98	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	26	yes	—
99	C₂²×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
100	C₃×C₄	—	2	52	no	—
101	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	26	yes	—
102	C₃×C₄	C₂×#634	1	13	yes	—
103	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
104	C₃²	#4999	1	13	yes	complemented
105	C₂³	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
106	C₂×C₄	#6308	1	2	yes	—
107	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	2	yes	—
108	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	4	yes	—
109	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
110	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
111	C₈	—	54	2	no	—
112	C₈	—	54	2	no	—
113	C₂×C₃	—	2	26	no	—
114	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	13	yes	—
115	C₂×C₃	#64157	1	26	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	2	yes	—
118	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	2	yes	—
121	C₄	C₂×#6308	1	1	yes	characteristic
122	C₄	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
123	C₃	#10592698	1	13	yes	complemented
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
126	C₂	#11723989	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×#117

C₂

yes

maximal, characteristic

C₂×#88855

C₂

yes

maximal, characteristic

C₂²×C₃⋊(C₃⋊(C₃⋊C₈))

C₂

yes

maximal, characteristic

C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))

C₂

yes

maximal, complemented

#10592698

—

maximal

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₃³×C₄⋊C₄

C₄

yes

—

C₃³×C₂²⋊C₄

C₄

yes

—

C₃³×C₄²

C₂²

yes

—

#88855

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

—

C₃²×#117

D₃

yes

—

C₂×#3748

—

C₂²×C₃⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₂³×C₃³

C₂×C₄

yes

characteristic

C₂×C₃³×C₄

C₂×C₄

yes

—

C₂×C₃³×C₄

C₂×C₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃³×C₄

Q₈

yes

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₃⋊(C₃⋊(C₃⋊C₈))

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#4999

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₃²×C₄⋊C₄

Dic₃

yes

—

C₃²×C₂²⋊C₄

Dic₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

—

C₂²×C₃³

C₄⋊C₄

yes

—

C₂²×C₃³

C₂²×C₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₄⋊C₄

yes

—

C₃³×C₄

C₂×D₄

yes

characteristic

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃×#117

C₃⋊D₃

yes

—

C₂×#208

—

C₂²×C₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₂³×C₃²

C₂×Dic₃

yes

—

C₂×C₃²×C₄

Dic₆

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

#424

—

Sylow

C₂×C₃³

—

C₂×C₃³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃³

#131

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₃×C₄⋊C₄

C₃⋊Dic₃

yes

—

C₃×C₂²⋊C₄

C₃⋊Dic₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₄⋊Dic₃

yes

—

C₂²×C₃²

C₂²×Dic₃

yes

—

C₃²×C₄

—

C₃²×C₄

C₄⋊Dic₃

yes

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃²×C₄

C₂×D₁₂

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#117

C₃⋊(C₃⋊D₃)

yes

characteristic

C₄⋊C₈

—

C₃³

#424

yes

characteristic, complemented, Sylow

C₂³×C₃

C₂×C₃⋊Dic₃

yes

—

C₂×C₃×C₄

#634

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

C₂×C₄⋊Dic₃

yes

—

C₂×C₃²

#948

yes

—

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₃⋊(C₃⋊Dic₃)

yes

—

C₂²⋊C₄

C₃⋊(C₃⋊Dic₃)

yes

—

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

—

C₂²×C₃

—

C₂²×C₃

C₄⋊(C₃⋊Dic₃)

yes

—

C₂²×C₃

C₂²×C₃⋊Dic₃

yes

—

100

C₃×C₄

—

101

C₃×C₄

C₄⋊(C₃⋊Dic₃)

yes

—

102

C₃×C₄

C₂×#634

yes

—

103

C₃×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

—

104

C₃²

#4999

yes

complemented

105

C₂³

C₂×C₃⋊(C₃⋊Dic₃)

yes

characteristic

106

C₂×C₄

#6308

yes

—

107

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

yes

—

108

C₂×C₄

C₂×C₃⋊(C₃⋊Dic₃)

yes

—

109

C₂×C₄

C₂×C₃⋊(C₃⋊Dic₃)

yes

characteristic

110

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂×C₃

—

114

C₂×C₃

C₂×C₄⋊(C₃⋊Dic₃)

yes

—

115

C₂×C₃

#64157

yes

—

116

C₂²

—

117

C₂²

C₄⋊(C₃⋊(C₃⋊Dic₃))

yes

—

118

C₂²

C₂²×C₃⋊(C₃⋊Dic₃)

yes

characteristic

119

C₄

—

120

C₄

C₄⋊(C₃⋊(C₃⋊Dic₃))

yes

—

121

C₄

C₂×#6308

yes

characteristic

122

C₄

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

123

C₃

#10592698

yes

complemented

124

C₂

—

125

C₂

C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))

yes

characteristic

126

C₂

#11723989

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle	C₂²×C₃³	C₂²×C₄
Fitting	C₃³×#117	C₂
prime radical	C₂³×C₃³	C₂×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	#117	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#424
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

commutator, socle

C₂²×C₃³

C₂²×C₄

Fitting

C₃³×#117

C₂

prime radical

C₂³×C₃³

C₂×C₄

cosocle

C₂×C₃³×C₄

C₂³

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂²×C₃³ ▷ C₃³

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
4. C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	semidirect
77. C₃³	#424	semidirect
104. C₃²	#4999	semidirect
123. C₃	#10592698	semidirect

H ≅ G/N

type

4. C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))

C₂

semidirect

77. C₃³

#424

semidirect

104. C₃²

#4999

semidirect

123. C₃

#10592698

semidirect

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

subgroup	index
C₃³×#117	2
C₂×#88855	2
C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	2
C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
#10592698	3

subgroup

index

C₃³×#117

C₂×#88855

C₂²×C₃⋊(C₃⋊(C₃⋊C₈))

C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))

#10592698

Subgroups — #13163639

Conjugacy classes of subgroups

Complements

Maximal subgroups