Group #13163765 — subgroups

Subgroups — #13163765

Order 1728.

1428 conjugacy classes of subgroups; only the 795 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 795 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×C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#88855	C₂	1	2	yes	maximal
3	#11722189	C₂	1	4	yes	maximal, complemented
4	#10592824	—	3	13	no	maximal
5	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
6	C₃³×M₁₆	C₄	1	4	yes	—
7	C₂×C₃³×C₈	C₂²	1	2	yes	—
8	#88855	—	2	2	no	—
9	#88855	C₂²	1	4	yes	—
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
11	C₂×C₃²×M₁₆	D₃	1	13	yes	—
12	C₂×#3748	—	3	26	no	—
13	#63646	—	3	52	no	—
14	C₂³×C₃³	Q₈	1	1	yes	characteristic
15	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
16	C₂×C₃³×C₄	D₄	1	2	yes	—
17	C₂×C₃³×C₄	Q₈	1	1	yes	characteristic
18	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
19	C₃³×C₈	C₂×C₄	1	4	yes	—
20	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
21	#5125	—	9	13	no	—
22	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
23	C₃²×M₁₆	Dic₃	1	52	yes	—
24	C₂×C₃²×C₈	C₂×D₃	1	26	yes	—
25	#3748	—	3	52	no	—
26	#3748	—	6	26	no	—
27	C₂×C₃⋊(C₃⋊C₈)	—	6	26	no	—
28	C₂²×C₃³	—	4	1	no	—
29	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
30	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
31	C₃³×C₄	C₄⋊C₄	1	2	yes	—
32	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
33	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂×C₃×M₁₆	C₃⋊D₃	1	13	yes	—
35	C₂×#208	—	9	26	no	—
36	#819	—	9	52	no	—
37	C₂³×C₃²	Dic₆	1	13	yes	—
38	C₂×C₃²×C₄	Dic₆	1	13	yes	—
39	C₂×C₃²×C₄	D₁₂	1	26	yes	—
40	C₂×C₃²×C₄	C₂×Dic₃	1	26	yes	—
41	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
42	C₃²×C₈	C₂×Dic₃	1	52	yes	—
43	C₃⋊(C₃⋊C₈)	—	6	52	no	—
44	#430	—	27	1	no	Sylow
45	C₂×C₃³	—	2	2	no	—
46	C₂×C₃³	—	2	1	no	—
47	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
49	C₃×M₁₆	C₃⋊Dic₃	1	52	yes	—
50	C₂×C₃×C₈	C₂×C₃⋊D₃	1	26	yes	—
51	#208	—	9	52	no	—
52	#208	—	18	26	no	—
53	C₂×C₃⋊C₈	—	18	26	no	—
54	C₂²×C₃²	—	4	13	no	—
55	C₂²×C₃²	C₄⋊Dic₃	1	26	yes	—
56	C₂²×C₃²	C₂×Dic₆	1	13	yes	—
57	C₃²×C₄	C₄⋊Dic₃	1	26	yes	—
58	C₃²×C₄	C₂×D₁₂	1	13	yes	—
59	C₃²×C₄	C₂²×Dic₃	1	13	yes	—
60	C₂×M₁₆	—	27	2	no	—
61	C₂×M₁₆	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
62	#108	—	27	4	no	—
63	C₃³	#430	1	1	yes	characteristic, complemented, Sylow
64	C₂³×C₃	#634	1	13	yes	—
65	C₂×C₃×C₄	#634	1	13	yes	—
66	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	26	yes	—
67	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	26	yes	—
68	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
69	C₃×C₈	C₂×C₃⋊Dic₃	1	52	yes	—
70	C₃⋊C₈	—	18	52	no	—
71	C₂×C₃²	—	2	26	no	—
72	C₂×C₃²	—	2	13	no	—
73	C₂×C₃²	C₂×C₄⋊Dic₃	1	13	yes	—
74	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
75	M₁₆	—	27	4	no	—
76	M₁₆	—	54	2	no	—
77	M₁₆	C₃⋊(C₃⋊Dic₃)	1	4	yes	—
78	C₂×C₈	—	54	2	no	—
79	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
80	C₂²×C₃	—	4	13	no	—
81	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	26	yes	—
82	C₂²×C₃	C₂×#634	1	13	yes	—
83	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	26	yes	—
84	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
85	C₃×C₄	C₂²×C₃⋊Dic₃	1	13	yes	—
86	C₃²	#5125	1	13	yes	complemented
87	C₂³	#6308	1	1	yes	characteristic
88	C₂×C₄	#6308	1	1	yes	characteristic
89	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	2	yes	—
90	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	2	yes	—
91	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
92	C₈	—	54	4	no	—
93	C₈	C₂×C₃⋊(C₃⋊Dic₃)	1	4	yes	—
94	C₂×C₃	—	2	26	no	—
95	C₂×C₃	—	2	13	no	—
96	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	13	yes	—
97	C₂²	—	4	1	no	—
98	C₂²	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	2	yes	—
99	C₂²	C₂×#6308	1	1	yes	characteristic
100	C₄	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	2	yes	—
101	C₄	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
102	C₄	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
103	C₃	#10592824	1	13	yes	complemented
104	C₂	—	2	2	no	—
105	C₂	—	2	1	no	—
106	C₂	C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃³×M₁₆

C₂

yes

maximal, characteristic

C₂×#88855

C₂

yes

maximal

#11722189

C₂

yes

maximal, complemented

#10592824

—

maximal

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₃³×M₁₆

C₄

yes

—

C₂×C₃³×C₈

C₂²

yes

—

#88855

—

#88855

C₂²

yes

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

—

C₂×C₃²×M₁₆

D₃

yes

—

C₂×#3748

—

#63646

—

C₂³×C₃³

Q₈

yes

characteristic

C₂×C₃³×C₄

C₂×C₄

yes

—

C₂×C₃³×C₄

D₄

yes

—

C₂×C₃³×C₄

Q₈

yes

characteristic

C₂×C₃³×C₄

C₂³

yes

characteristic

C₃³×C₈

C₂×C₄

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#5125

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₃²×M₁₆

Dic₃

yes

—

C₂×C₃²×C₈

C₂×D₃

yes

—

#3748

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

—

C₂²×C₃³

C₄⋊C₄

yes

—

C₂²×C₃³

C₂×Q₈

yes

characteristic

C₃³×C₄

C₄⋊C₄

yes

—

C₃³×C₄

C₂²×C₄

yes

characteristic

C₃³×C₄

C₂×D₄

yes

characteristic

C₂×C₃×M₁₆

C₃⋊D₃

yes

—

C₂×#208

—

#819

—

C₂³×C₃²

Dic₆

yes

—

C₂×C₃²×C₄

Dic₆

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₃²×C₈

C₂×Dic₃

yes

—

C₃⋊(C₃⋊C₈)

—

#430

—

Sylow

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₃×M₁₆

C₃⋊Dic₃

yes

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

C₄⋊Dic₃

yes

—

C₂²×C₃²

C₂×Dic₆

yes

—

C₃²×C₄

C₄⋊Dic₃

yes

—

C₃²×C₄

C₂×D₁₂

yes

—

C₃²×C₄

C₂²×Dic₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₃⋊(C₃⋊D₃)

yes

characteristic

#108

—

C₃³

#430

yes

characteristic, complemented, Sylow

C₂³×C₃

#634

yes

—

C₂×C₃×C₄

#634

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₃×C₈

C₂×C₃⋊Dic₃

yes

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×C₄⋊Dic₃

yes

—

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

C₃⋊(C₃⋊Dic₃)

yes

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

—

C₂²×C₃

—

C₂²×C₃

C₄⋊(C₃⋊Dic₃)

yes

—

C₂²×C₃

C₂×#634

yes

—

C₃×C₄

C₄⋊(C₃⋊Dic₃)

yes

—

C₃×C₄

C₂×C₄⋊(C₃⋊D₃)

yes

—

C₃×C₄

C₂²×C₃⋊Dic₃

yes

—

C₃²

yes

C₂³

yes

C₂×C₄

yes

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

yes

—

C₂×C₄

C₂×C₃⋊(C₃⋊Dic₃)

yes

—

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

C₈

—

C₈

C₂×C₃⋊(C₃⋊Dic₃)

yes

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂×C₄⋊(C₃⋊Dic₃)

yes

—

C₂²

—

C₂²

C₄⋊(C₃⋊(C₃⋊Dic₃))

yes

—

C₂²

C₂×#6308

yes

characteristic

100

C₄

C₄⋊(C₃⋊(C₃⋊Dic₃))

yes

—

101

C₄

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

102

C₄

C₂²×C₃⋊(C₃⋊Dic₃)

yes

characteristic

103

C₃

#10592824

yes

complemented

104

C₂

—

105

C₂

—

106

C₂

C₂×C₄⋊(C₃⋊(C₃⋊Dic₃))

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊(C₃⋊(C₃⋊D₃))
commutator	C₃³×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×M₁₆	C₂
socle	C₂×C₃³	C₂×C₄⋊C₄
prime radical	C₂³×C₃³	Q₈
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₂×M₁₆	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#430
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

commutator

C₃³×C₄

C₂²×C₄

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₂×C₃³×M₁₆

C₂

socle

C₂×C₃³

C₂×C₄⋊C₄

prime radical

C₂³×C₃³

Q₈

cosocle

C₂×C₃³×C₄

C₂³

hypercenter

C₂×M₁₆

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#430

LCS

G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
3. #11722189	C₂	semidirect
63. C₃³	#430	semidirect
86. C₃²	#5125	semidirect
103. C₃	#10592824	semidirect

H ≅ G/N

type

3. #11722189

C₂

semidirect

63. C₃³

#430

semidirect

86. C₃²

#5125

semidirect

103. C₃

#10592824

semidirect

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

subgroup	index
C₂×C₃³×M₁₆	2
C₂×#88855	2
#11722189	2
#10592824	3

subgroup

index

C₂×C₃³×M₁₆

C₂×#88855

#11722189

#10592824

Subgroups — #13163765

Conjugacy classes of subgroups

Complements

Maximal subgroups