Group #13135858 — subgroups

Subgroups — #13135858

Order 1728.

253 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 77 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	H(3,𝔽₃)×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₈×#1084	C₂	1	1	yes	maximal, characteristic
3	#11720545	C₂	1	1	yes	maximal, characteristic
4	C₃×#4511	—	3	4	no	maximal
5	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄×#1084	C₄	1	1	yes	characteristic
7	C₂×#6180	C₄	1	1	yes	characteristic
8	C₁₆×H(3,𝔽₃)	—	2	1	no	—
9	#88386	—	2	1	no	—
10	C₃²×M₃₂	D₃	1	4	yes	—
11	C₃×C₈×Dic₃	—	3	4	no	—
12	C₃×#812	—	3	4	no	—
13	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
14	C₂×#1084	—	2	1	no	—
15	C₈×H(3,𝔽₃)	D₄	1	1	yes	characteristic
16	C₈×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
17	#6180	C₈	1	2	yes	—
18	C₃×#364	—	9	1	no	—
19	#4511	—	3	4	no	—
20	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
21	C₃×C₄×Dic₃	—	3	4	no	—
22	C₂×C₃×C₃⋊C₈	—	3	4	no	—
23	C₃²×C₁₆	—	2	4	no	—
24	C₃×C₃⋊C₁₆	—	6	4	no	—
25	C₂²×H(3,𝔽₃)	M₁₆	1	1	yes	characteristic
26	C₄×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
27	C₄×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
28	#1084	—	2	2	no	—
29	C₃×M₃₂	—	3	4	no	—
30	C₃×M₃₂	—	9	1	no	—
31	C₃×M₃₂	C₃⋊D₃	1	1	yes	characteristic
32	C₃×C₄×C₈	—	9	1	no	—
33	C₈×Dic₃	—	3	4	no	—
34	#812	—	3	4	no	—
35	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
36	C₂×C₃×Dic₃	—	6	4	no	—
37	C₃²×C₈	Dic₆	1	4	yes	—
38	C₃²×C₈	#67	1	4	yes	—
39	C₃×C₃⋊C₈	—	3	8	no	—
40	#364	—	9	1	no	Sylow
41	C₂×H(3,𝔽₃)	—	2	1	no	—
42	C₂×H(3,𝔽₃)	C₄⋊C₈	1	1	yes	characteristic
43	C₂×C₃×C₈	—	3	4	no	—
44	C₂×C₃×C₈	—	9	1	no	—
45	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
46	C₃×C₄²	—	9	1	no	—
47	C₄×Dic₃	—	3	4	no	—
48	C₂×C₃⋊C₈	—	3	4	no	—
49	C₃×C₁₆	—	2	1	no	—
50	C₃×C₁₆	—	6	4	no	—
51	C₃×C₁₆	—	18	1	no	—
52	C₃⋊C₁₆	—	6	4	no	—
53	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	4	yes	—
54	C₃²×C₄	D₃×C₈	1	4	yes	—
55	C₃²×C₄	Dic₃⋊C₄	1	4	yes	—
56	C₃×Dic₃	—	6	8	no	—
57	M₃₂	—	9	1	no	—
58	M₃₂	#272	1	1	yes	characteristic
59	C₄×C₈	—	9	1	no	—
60	H(3,𝔽₃)	#364	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₃×C₄	—	3	4	no	—
62	C₂×C₃×C₄	—	18	1	no	—
63	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
64	C₂×Dic₃	—	6	4	no	—
65	C₃×C₈	—	3	4	no	—
66	C₃×C₈	—	3	4	no	—
67	C₃×C₈	—	9	2	no	—
68	C₃×C₈	#634	1	1	yes	characteristic
69	C₃×C₈	#638	1	1	yes	characteristic
70	C₃⋊C₈	—	3	8	no	—
71	C₂×C₃²	—	2	4	no	—
72	C₂×C₃²	Dic₃⋊C₈	1	4	yes	—
73	C₂×C₈	—	9	1	no	—
74	C₂×C₈	C₂×#272	1	1	yes	characteristic
75	C₄²	—	9	1	no	—
76	C₁₆	—	2	1	no	—
77	C₁₆	—	18	1	no	—
78	C₂²×C₃	—	3	4	no	—
79	C₂²×C₃	#3743	1	1	yes	characteristic
80	C₃×C₄	—	3	4	no	—
81	C₃×C₄	—	3	4	no	—
82	C₃×C₄	—	18	2	no	—
83	C₃×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
84	C₃×C₄	#3750	1	1	yes	characteristic
85	Dic₃	—	6	8	no	—
86	C₃²	#4511	1	4	yes	complemented
87	C₂×C₄	—	18	1	no	—
88	C₂×C₄	C₄×#272	1	1	yes	characteristic
89	C₈	—	9	2	no	—
90	C₈	#6229	1	1	yes	characteristic
91	C₈	#6235	1	1	yes	characteristic
92	C₂×C₃	—	2	1	no	—
93	C₂×C₃	—	3	4	no	—
94	C₂×C₃	—	6	4	no	—
95	C₂×C₃	#63641	1	1	yes	characteristic
96	C₂²	#88527	1	1	yes	characteristic
97	C₄	—	18	2	no	—
98	C₄	C₈×#272	1	1	yes	characteristic
99	C₄	#88540	1	1	yes	characteristic
100	C₃	—	3	4	no	—
101	C₃	#10589377	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	#11720547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×M₃₂

C₂

yes

maximal, characteristic

C₈×#1084

C₂

yes

maximal, characteristic

#11720545

C₂

yes

maximal, characteristic

C₃×#4511

—

maximal

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₄×#1084

C₄

yes

characteristic

C₂×#6180

C₄

yes

characteristic

C₁₆×H(3,𝔽₃)

—

#88386

—

C₃²×M₃₂

D₃

yes

—

C₃×C₈×Dic₃

—

C₃×#812

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×#1084

—

C₈×H(3,𝔽₃)

D₄

yes

characteristic

C₈×H(3,𝔽₃)

Q₈

yes

characteristic

#6180

C₈

yes

—

C₃×#364

—

#4511

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×C₄×Dic₃

—

C₂×C₃×C₃⋊C₈

—

C₃²×C₁₆

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

M₁₆

yes

characteristic

C₄×H(3,𝔽₃)

C₄⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

C₂×C₈

yes

characteristic

#1084

—

C₃×M₃₂

—

C₃×M₃₂

—

C₃×M₃₂

C₃⋊D₃

yes

characteristic

C₃×C₄×C₈

—

C₈×Dic₃

—

#812

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃×Dic₃

—

C₃²×C₈

Dic₆

yes

—

C₃²×C₈

#67

yes

—

C₃×C₃⋊C₈

—

#364

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₄⋊C₈

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃×C₄²

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

ΓL(1,𝔽₂₅)

yes

—

C₃²×C₄

D₃×C₈

yes

—

C₃²×C₄

Dic₃⋊C₄

yes

—

C₃×Dic₃

—

M₃₂

—

M₃₂

#272

yes

characteristic

C₄×C₈

—

H(3,𝔽₃)

#364

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

yes

C₃×C₈

yes

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

Dic₃⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

C₂×#272

yes

characteristic

C₄²

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

#3743

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₈×C₃⋊D₃

yes

characteristic

C₃×C₄

#3750

yes

characteristic

Dic₃

—

C₃²

#4511

yes

complemented

C₂×C₄

—

C₂×C₄

C₄×#272

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

yes

C₂²

yes

C₄

—

C₄

yes

C₄

yes

100

C₃

—

101

C₃

#10589377

yes

characteristic

102

C₂

—

103

C₂

#11720547

yes

characteristic

	subgroup	G/H
center	C₃×C₈	#638
commutator	C₄×H(3,𝔽₃)	C₂×C₈
Frattini	C₂×C₃×C₈	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×M₃₂	C₂
socle	C₂×C₃	#63641
prime radical	C₂²×H(3,𝔽₃)	M₁₆
cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
hypercenter	C₃×M₃₂	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#364
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

subgroup

G/H

center

C₃×C₈

#638

commutator

C₄×H(3,𝔽₃)

C₂×C₈

Frattini

C₂×C₃×C₈

C₂×C₃⋊D₃

Fitting

H(3,𝔽₃)×M₃₂

C₂

socle

C₂×C₃

#63641

prime radical

C₂²×H(3,𝔽₃)

M₁₆

cosocle

C₂×C₈×H(3,𝔽₃)

C₂²

hypercenter

C₃×M₃₂

C₃⋊D₃

G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈ ◁ C₃×M₃₂

N	H ≅ G/N	type
60. H(3,𝔽₃)	#364	semidirect
86. C₃²	#4511	semidirect

H ≅ G/N

type

60. H(3,𝔽₃)

#364

semidirect

86. C₃²

#4511

semidirect

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

subgroup	index
H(3,𝔽₃)×M₃₂	2
C₈×#1084	2
#11720545	2
C₃×#4511	3

subgroup

index

H(3,𝔽₃)×M₃₂

C₈×#1084

#11720545

C₃×#4511

Subgroups — #13135858

Conjugacy classes of subgroups

Complements

Maximal subgroups