Group #13163498 — subgroups

Subgroups — #13163498

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×#106	C₂	1	1	yes	maximal, characteristic
3	#11722187	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
5	#11722177	C₂	1	2	yes	maximal
6	#10592557	—	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×#106	D₃	1	13	yes	—
16	#63644	—	3	13	no	—
17	C₈×C₃⋊Dic₃	—	3	13	no	—
18	#63634	—	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	2	yes	—
23	C₃³×C₈	Q₈	1	2	yes	—
24	C₃⋊(C₃⋊(C₃⋊C₈))	Q₈	1	2	yes	—
25	#4858	—	9	13	no	—
26	C₂×C₃²×C₈	C₂×D₃	1	13	yes	—
27	C₃²×C₄⋊C₄	C₂×D₃	1	26	yes	—
28	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
29	C₄⋊(C₃⋊Dic₃)	—	6	26	no	—
30	#3750	—	6	26	no	—
31	C₄×C₃⋊Dic₃	—	3	13	no	—
32	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
33	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
34	C₃³×C₄	—	4	2	no	—
35	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
36	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
37	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
38	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
39	#890	—	9	26	no	—
40	C₃×#106	C₃⋊D₃	1	13	yes	—
41	#817	—	9	13	no	—
42	C₈×Dic₃	—	9	13	no	—
43	#807	—	9	26	no	—
44	C₂×C₃²×C₄	—	2	26	no	—
45	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
46	C₂×C₃⋊Dic₃	—	3	26	no	—
47	C₂×C₃⋊Dic₃	—	6	26	no	—
48	C₃²×C₈	Dic₆	1	26	yes	—
49	C₃⋊(C₃⋊C₈)	—	3	26	no	—
50	#499	—	27	1	no	Sylow
51	C₂×C₃³	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₃³	#135	1	2	yes	—
53	C₂×C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
54	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	26	yes	—
55	C₄⋊Dic₃	—	9	26	no	—
56	C₄⋊Dic₃	—	18	26	no	—
57	Dic₃⋊C₄	—	18	26	no	—
58	C₄×Dic₃	—	9	13	no	—
59	C₂×C₃⋊C₈	—	9	13	no	—
60	C₂²×C₃²	D₃×D₄	1	13	yes	—
61	C₃²×C₄	—	4	26	no	—
62	C₃²×C₄	C₂×Dic₆	1	13	yes	—
63	C₃²×C₄	D₃×Q₈	1	13	yes	—
64	C₃⋊Dic₃	—	6	26	no	—
65	C₃⋊Dic₃	—	12	26	no	—
66	#125	—	27	2	no	—
67	#107	—	27	2	no	—
68	#106	—	27	1	no	—
69	#106	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
70	C₄×C₈	—	27	1	no	—
71	C₃³	#499	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₃×C₄	—	2	26	no	—
73	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
74	C₂×Dic₃	—	9	26	no	—
75	C₂×Dic₃	—	18	26	no	—
76	C₃×C₈	#634	1	26	yes	—
77	C₃⋊C₈	—	9	26	no	—
78	C₂×C₃²	#888	1	13	yes	—
79	C₂×C₃²	#916	1	26	yes	—
80	C₂×C₈	—	27	1	no	—
81	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
82	C₄⋊C₄	—	27	2	no	—
83	C₄⋊C₄	—	54	2	no	—
84	C₄⋊C₄	—	54	2	no	—
85	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
86	C₄²	—	27	1	no	—
87	C₂²×C₃	D₄×C₃⋊D₃	1	13	yes	—
88	C₃×C₄	—	4	26	no	—
89	C₃×C₄	C₂×#634	1	13	yes	—
90	C₃×C₄	Q₈×C₃⋊D₃	1	13	yes	—
91	Dic₃	—	18	26	no	—
92	Dic₃	—	36	26	no	—
93	C₃²	#4858	1	13	yes	complemented
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	27	2	no	—
96	C₂×C₄	—	54	2	no	—
97	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
98	C₈	—	27	2	no	—
99	C₈	#6308	1	2	yes	—
100	C₂×C₃	#64097	1	13	yes	—
101	C₂×C₃	#64125	1	26	yes	—
102	C₂²	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
103	C₄	—	4	2	no	—
104	C₄	—	54	2	no	—
105	C₄	—	108	2	no	—
106	C₄	C₂×#6308	1	1	yes	characteristic
107	C₄	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
108	C₃	#10592557	1	13	yes	complemented
109	C₂	#11723929	1	1	yes	characteristic
110	C₂	#11723957	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11723931

C₂

yes

maximal

C₃³×#106

C₂

yes

maximal, characteristic

#11722187

C₂

yes

maximal, characteristic

C₈×C₃⋊(C₃⋊Dic₃)

C₂

yes

maximal, characteristic

#11722177

C₂

yes

maximal

#10592557

—

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×#106

D₃

yes

—

#63644

—

C₈×C₃⋊Dic₃

—

#63634

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

D₄

yes

—

C₃³×C₈

Q₈

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

Q₈

yes

—

#4858

—

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)

—

#890

—

C₃×#106

C₃⋊D₃

yes

—

#817

—

C₈×Dic₃

—

#807

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₃²×C₈

Dic₆

yes

—

C₃⋊(C₃⋊C₈)

—

#499

—

Sylow

C₂×C₃³

C₄⋊Q₈

yes

characteristic

C₂×C₃³

#135

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⋊(C₃⋊D₃)

yes

characteristic

C₄×C₈

—

C₃³

#499

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

#634

yes

—

C₃⋊C₈

—

C₂×C₃²

#888

yes

—

C₂×C₃²

#916

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

—

C₃²

#4858

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

C₈

—

C₈

#6308

yes

—

100

C₂×C₃

#64097

yes

—

101

C₂×C₃

#64125

yes

—

102

C₂²

D₄×C₃⋊(C₃⋊D₃)

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

108

—

106

C₄

C₂×#6308

yes

characteristic

107

C₄

Q₈×C₃⋊(C₃⋊D₃)

yes

characteristic

108

C₃

#10592557

yes

complemented

109

C₂

#11723929

yes

characteristic

110

C₂

#11723957

yes

—

	subgroup	G/H
center	C₂²	D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×#106	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	#106	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#499
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×C₃⋊(C₃⋊D₃)

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×#106

C₂

socle, prime radical

C₂²×C₃³

C₂×D₄

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
71. C₃³	#499	semidirect
93. C₃²	#4858	semidirect
108. C₃	#10592557	semidirect

H ≅ G/N

type

71. C₃³

#499

semidirect

93. C₃²

#4858

semidirect

108. C₃

#10592557

semidirect

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

subgroup	index
#11723931	2
C₃³×#106	2
#11722187	2
C₈×C₃⋊(C₃⋊Dic₃)	2
#11722177	2
#10592557	3

subgroup

index

C₃³×#106

C₈×C₃⋊(C₃⋊Dic₃)

Subgroups — #13163498

Conjugacy classes of subgroups

Complements

Maximal subgroups