Group #13163518 — subgroups

Subgroups — #13163518

Order 1728.

1204 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 1204 conjugacy classes of subgroups — too many to draw. Only its 407 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	#11723931	C₂	1	2	yes	maximal
2	C₃³×#107	C₂	1	1	yes	maximal, characteristic
3	#11722188	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
5	#11722178	C₂	1	2	yes	maximal
6	#10592577	—	3	13	no	maximal
7	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
8	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	2	no	—
10	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
11	#88857	—	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	#64099	—	3	26	no	—
15	C₃²×#107	D₃	1	13	yes	—
16	#63645	—	3	13	no	—
17	C₈×C₃⋊Dic₃	—	3	13	no	—
18	#63635	—	3	26	no	—
19	C₂×C₃³×C₄	—	2	2	no	—
20	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
21	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
22	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
23	C₂×C₃⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
24	C₃³×C₈	Q₈	1	2	yes	—
25	C₃⋊(C₃⋊(C₃⋊C₈))	Q₈	1	2	yes	—
26	#4878	—	9	13	no	—
27	C₂×C₃²×C₈	C₂×D₃	1	13	yes	—
28	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
29	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
30	C₄⋊(C₃⋊Dic₃)	—	6	26	no	—
31	#3750	—	6	26	no	—
32	C₄×C₃⋊Dic₃	—	3	13	no	—
33	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
34	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
35	C₃³×C₄	—	4	2	no	—
36	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
37	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
38	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
39	C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
40	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
41	#890	—	9	26	no	—
42	C₃×#107	C₃⋊D₃	1	13	yes	—
43	#818	—	9	13	no	—
44	C₈×Dic₃	—	9	13	no	—
45	#808	—	9	26	no	—
46	C₂×C₃²×C₄	—	2	26	no	—
47	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
48	C₂×C₃⋊Dic₃	—	3	13	no	—
49	C₂×C₃⋊Dic₃	—	3	13	no	—
50	C₂×C₃⋊Dic₃	—	6	26	no	—
51	C₃²×C₈	Dic₆	1	26	yes	—
52	C₃⋊(C₃⋊C₈)	—	3	26	no	—
53	#499	—	27	1	no	Sylow
54	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₃³	#135	1	1	yes	characteristic
56	C₂×C₃³	#135	1	1	yes	characteristic
57	C₂×C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
58	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
59	C₄⋊Dic₃	—	9	26	no	—
60	C₄⋊Dic₃	—	18	26	no	—
61	Dic₃⋊C₄	—	18	26	no	—
62	C₄×Dic₃	—	9	13	no	—
63	C₂×C₃⋊C₈	—	9	13	no	—
64	C₂²×C₃²	D₃×D₄	1	13	yes	—
65	C₃²×C₄	—	4	26	no	—
66	C₃²×C₄	C₂×Dic₆	1	13	yes	—
67	C₃²×C₄	D₃×Q₈	1	13	yes	—
68	C₃⋊Dic₃	—	6	13	no	—
69	C₃⋊Dic₃	—	6	13	no	—
70	C₃⋊Dic₃	—	12	26	no	—
71	#125	—	27	2	no	—
72	#107	—	27	1	no	—
73	#107	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
74	#106	—	27	2	no	—
75	C₄×C₈	—	27	1	no	—
76	C₃³	#499	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₃×C₄	—	2	26	no	—
78	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
79	C₂×Dic₃	—	9	13	no	—
80	C₂×Dic₃	—	9	13	no	—
81	C₂×Dic₃	—	18	26	no	—
82	C₃×C₈	#634	1	26	yes	—
83	C₃⋊C₈	—	9	26	no	—
84	C₂×C₃²	#888	1	13	yes	—
85	C₂×C₃²	#912	1	13	yes	—
86	C₂×C₃²	#919	1	13	yes	—
87	C₂×C₈	—	27	1	no	—
88	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
89	C₄⋊C₄	—	27	2	no	—
90	C₄⋊C₄	—	54	2	no	—
91	C₄⋊C₄	—	54	2	no	—
92	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
93	C₄²	—	27	1	no	—
94	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
95	C₃×C₄	—	4	26	no	—
96	C₃×C₄	C₂×#634	1	13	yes	—
97	C₃×C₄	Q₈×C₃⋊D₃	1	13	yes	—
98	Dic₃	—	18	13	no	—
99	Dic₃	—	18	13	no	—
100	Dic₃	—	36	26	no	—
101	C₃²	#4878	1	13	yes	complemented
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	27	1	no	—
104	C₂×C₄	—	27	1	no	—
105	C₂×C₄	—	54	2	no	—
106	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
107	C₈	—	27	2	no	—
108	C₈	#6308	1	2	yes	—
109	C₂×C₃	#64097	1	13	yes	—
110	C₂×C₃	#64121	1	13	yes	—
111	C₂×C₃	#64128	1	13	yes	—
112	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
113	C₄	—	4	2	no	—
114	C₄	—	54	1	no	—
115	C₄	—	54	1	no	—
116	C₄	—	108	2	no	—
117	C₄	C₂×#6308	1	1	yes	characteristic
118	C₄	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
119	C₃	#10592577	1	13	yes	complemented
120	C₂	#11723929	1	1	yes	characteristic
121	C₂	#11723953	1	1	yes	characteristic
122	C₂	#11723960	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11723931

C₂

yes

maximal

C₃³×#107

C₂

yes

maximal, characteristic

#11722188

C₂

yes

maximal, characteristic

C₈×C₃⋊(C₃⋊Dic₃)

C₂

yes

maximal, characteristic

#11722178

C₂

yes

maximal

#10592577

—

maximal

C₂×C₃³×C₈

C₂²

yes

characteristic

C₃³×C₄⋊C₄

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

#64099

—

C₃²×#107

D₃

yes

—

#63645

—

C₈×C₃⋊Dic₃

—

#63635

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

D₄

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

D₄

yes

characteristic

C₃³×C₈

Q₈

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

Q₈

yes

—

#4878

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

#890

—

C₃×#107

C₃⋊D₃

yes

—

#818

—

C₈×Dic₃

—

#808

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₃²×C₈

Dic₆

yes

—

C₃⋊(C₃⋊C₈)

—

#499

—

Sylow

C₂×C₃³

C₄⋊Q₈

yes

C₂×C₃³

yes

C₂×C₃³

yes

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

D₃×D₄

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃²×C₄

D₃×Q₈

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

#125

—

#107

—

#107

C₃⋊(C₃⋊D₃)

yes

characteristic

#106

—

C₄×C₈

—

C₃³

#499

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

#634

yes

—

C₃⋊C₈

—

C₂×C₃²

#888

yes

—

C₂×C₃²

#912

yes

—

C₂×C₃²

#919

yes

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

C₄²

—

C₂²×C₃

D₄×C₃⋊D₃

yes

—

C₃×C₄

—

C₃×C₄

C₂×#634

yes

—

C₃×C₄

Q₈×C₃⋊D₃

yes

—

Dic₃

—

Dic₃

—

100

Dic₃

—

101

C₃²

#4878

yes

complemented

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

107

C₈

—

108

C₈

#6308

yes

—

109

C₂×C₃

#64097

yes

—

110

C₂×C₃

#64121

yes

—

111

C₂×C₃

#64128

yes

—

112

C₂²

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

108

—

117

C₄

C₂×#6308

yes

characteristic

118

C₄

Q₈×C₃⋊(C₃⋊D₃)

yes

characteristic

119

C₃

#10592577

yes

complemented

120

C₂

#11723929

yes

characteristic

121

C₂

#11723953

yes

characteristic

122

C₂

#11723960

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	C₃³×#107	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	#107	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#499
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×C₃⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×#107

C₂

socle, prime radical

C₂²×C₃³

C₂×D₄

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
76. C₃³	#499	semidirect
101. C₃²	#4878	semidirect
119. C₃	#10592577	semidirect

H ≅ G/N

type

76. C₃³

#499

semidirect

101. C₃²

#4878

semidirect

119. C₃

#10592577

semidirect

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

subgroup	index
#11723931	2
C₃³×#107	2
#11722188	2
C₈×C₃⋊(C₃⋊Dic₃)	2
#11722178	2
#10592577	3

subgroup

index

C₃³×#107

C₈×C₃⋊(C₃⋊Dic₃)

Subgroups — #13163518

Conjugacy classes of subgroups

Complements

Maximal subgroups