Group #13146098 — subgroups

Subgroups — #13146098

Order 1728.

1260 conjugacy classes of subgroups; only the 623 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 623 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	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#88855	C₂	1	2	yes	maximal
3	#10589418	—	3	13	no	maximal
4	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
5	C₂×C₃³×C₈	—	2	2	no	—
6	C₂×C₃³×C₈	C₄	1	2	yes	—
7	#88855	—	2	2	no	—
8	#88855	C₄	1	4	yes	—
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	2	no	—
10	C₂²×C₃²×C₈	D₃	1	13	yes	—
11	C₂×#3748	—	3	26	no	—
12	C₂³×C₃³	Q₈	1	1	yes	characteristic
13	C₂×C₃³×C₄	C₂×C₄	1	2	yes	—
14	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
16	C₂×C₃³×C₄	D₄	1	2	yes	—
17	C₃³×C₈	—	2	2	no	—
18	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	4	no	—
19	#4552	—	9	13	no	—
20	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
21	C₂×C₃²×C₈	—	2	26	no	—
22	C₂×C₃²×C₈	Dic₃	1	26	yes	—
23	#3748	—	3	52	no	—
24	#3748	—	6	26	no	—
25	C₂×C₃⋊(C₃⋊C₈)	—	6	26	no	—
26	C₂²×C₃³	—	2	2	no	—
27	C₂²×C₃³	C₄⋊C₄	1	2	yes	—
28	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
29	C₃³×C₄	C₄²	1	1	yes	characteristic
30	C₃³×C₄	C₂²⋊C₄	1	2	yes	—
31	C₃³×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₃⋊D₃	1	13	yes	—
33	C₂×#208	—	9	26	no	—
34	C₂³×C₃²	Dic₆	1	13	yes	—
35	C₂×C₃²×C₄	C₄×D₃	1	26	yes	—
36	C₂×C₃²×C₄	D₁₂	1	13	yes	—
37	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
38	C₂×C₃²×C₄	#67	1	26	yes	—
39	C₃²×C₈	—	2	26	no	—
40	C₃⋊(C₃⋊C₈)	—	6	52	no	—
41	#341	—	27	1	no	Sylow
42	C₂×C₃³	—	2	2	no	—
43	C₂×C₃³	#95	1	1	yes	characteristic
44	C₂×C₃³	#108	1	2	yes	—
45	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
46	C₂×C₃×C₈	—	2	26	no	—
47	C₂×C₃×C₈	C₃⋊Dic₃	1	26	yes	—
48	#208	—	9	52	no	—
49	#208	—	18	26	no	—
50	C₂×C₃⋊C₈	—	18	26	no	—
51	C₂²×C₃²	—	2	26	no	—
52	C₂²×C₃²	Dic₃⋊C₄	1	26	yes	—
53	C₂²×C₃²	C₄⋊Dic₃	1	13	yes	—
54	C₃²×C₄	C₄×Dic₃	1	13	yes	—
55	C₃²×C₄	(C₂×D₃)⋊C₄	1	26	yes	—
56	C₃²×C₄	#217	1	13	yes	—
57	C₂×M₁₆	—	27	2	no	—
58	C₂²×C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
59	C₃³	#341	1	1	yes	characteristic, complemented, Sylow
60	C₂³×C₃	#634	1	13	yes	—
61	C₂×C₃×C₄	C₄×C₃⋊D₃	1	26	yes	—
62	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
63	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
64	C₂×C₃×C₄	#638	1	26	yes	—
65	C₃×C₈	—	2	26	no	—
66	C₃⋊C₈	—	18	52	no	—
67	C₂×C₃²	—	2	26	no	—
68	C₂×C₃²	#819	1	26	yes	—
69	C₂×C₃²	C₃⋊#95	1	13	yes	—
70	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
71	M₁₆	—	27	4	no	—
72	M₁₆	—	54	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	54	2	no	—
75	C₂×C₈	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
76	C₂²×C₃	—	2	26	no	—
77	C₂²×C₃	#3750	1	26	yes	—
78	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	13	yes	—
79	C₃×C₄	C₄×C₃⋊Dic₃	1	13	yes	—
80	C₃×C₄	#3752	1	26	yes	—
81	C₃×C₄	#3757	1	13	yes	—
82	C₃²	#4552	1	13	yes	complemented
83	C₂³	#6308	1	1	yes	characteristic
84	C₂×C₄	C₄×C₃⋊(C₃⋊D₃)	1	2	yes	—
85	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
86	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
87	C₂×C₄	#6312	1	2	yes	—
88	C₈	—	2	2	no	—
89	C₈	—	54	4	no	—
90	C₂×C₃	—	2	26	no	—
91	C₂×C₃	#63646	1	26	yes	—
92	C₂×C₃	#63658	1	13	yes	—
93	C₂²	—	2	2	no	—
94	C₂²	#88857	1	2	yes	—
95	C₂²	C₄⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
96	C₄	C₄×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
97	C₄	#88859	1	2	yes	—
98	C₄	#88864	1	1	yes	characteristic
99	C₃	#10589418	1	13	yes	complemented
100	C₂	—	2	2	no	—
101	C₂	#11722189	1	2	yes	—
102	C₂	#11722201	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃³×C₈

C₂

yes

maximal, characteristic

C₂×#88855

C₂

yes

maximal

#10589418

—

maximal

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₂×C₃³×C₈

—

C₂×C₃³×C₈

C₄

yes

—

#88855

—

#88855

C₄

yes

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

—

C₂²×C₃²×C₈

D₃

yes

—

C₂×#3748

—

C₂³×C₃³

Q₈

yes

characteristic

C₂×C₃³×C₄

C₂×C₄

yes

—

C₂×C₃³×C₄

C₂×C₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

—

C₃³×C₈

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#4552

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

Dic₃

yes

—

#3748

—

#3748

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

—

C₂²×C₃³

C₄⋊C₄

yes

—

C₂²×C₃³

C₄⋊C₄

yes

characteristic

C₃³×C₄

C₄²

yes

characteristic

C₃³×C₄

C₂²⋊C₄

yes

—

C₃³×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₃×C₈

C₃⋊D₃

yes

—

C₂×#208

—

C₂³×C₃²

Dic₆

yes

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

#341

—

Sylow

C₂×C₃³

—

C₂×C₃³

#95

yes

characteristic

C₂×C₃³

#108

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃⋊Dic₃

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

Dic₃⋊C₄

yes

—

C₂²×C₃²

C₄⋊Dic₃

yes

—

C₃²×C₄

C₄×Dic₃

yes

—

C₃²×C₄

(C₂×D₃)⋊C₄

yes

—

C₃²×C₄

#217

yes

—

C₂×M₁₆

—

C₂²×C₈

C₃⋊(C₃⋊D₃)

yes

characteristic

C₃³

#341

yes

characteristic, complemented, Sylow

C₂³×C₃

#634

yes

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

—

C₂×C₃×C₄

#638

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#819

yes

—

C₂×C₃²

C₃⋊#95

yes

—

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₃⋊(C₃⋊Dic₃)

yes

—

C₂²×C₃

—

C₂²×C₃

#3750

yes

—

C₂²×C₃

C₄⋊(C₃⋊Dic₃)

yes

—

C₃×C₄

C₄×C₃⋊Dic₃

yes

—

C₃×C₄

#3752

yes

—

C₃×C₄

#3757

yes

—

C₃²

yes

C₂³

yes

C₂×C₄

C₄×C₃⋊(C₃⋊D₃)

yes

—

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

C₂×C₄

C₂×C₃⋊(C₃⋊Dic₃)

yes

characteristic

C₂×C₄

#6312

yes

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

#63646

yes

—

C₂×C₃

#63658

yes

—

C₂²

—

C₂²

#88857

yes

—

C₂²

C₄⋊(C₃⋊(C₃⋊Dic₃))

yes

characteristic

C₄

C₄×C₃⋊(C₃⋊Dic₃)

yes

characteristic

C₄

#88859

yes

—

C₄

yes

C₃

yes

100

C₂

—

101

C₂

#11722189

yes

—

102

C₂

#11722201

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))
commutator	C₃³×C₄	C₄²
Frattini	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₄⋊C₄
prime radical	C₂³×C₃³	Q₈
cosocle	C₂²×C₃³×C₄	C₂²
hypercenter	C₂²×C₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#341
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

subgroup

G/H

center

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

commutator

C₃³×C₄

C₄²

Frattini

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

Fitting

C₂²×C₃³×C₈

C₂

socle

C₂²×C₃³

C₄⋊C₄

prime radical

C₂³×C₃³

Q₈

cosocle

C₂²×C₃³×C₄

C₂²

hypercenter

C₂²×C₈

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#341

LCS

G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

N	H ≅ G/N	type
59. C₃³	#341	semidirect
82. C₃²	#4552	semidirect
99. C₃	#10589418	semidirect

H ≅ G/N

type

59. C₃³

#341

semidirect

82. C₃²

#4552

semidirect

99. C₃

#10589418

semidirect

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

subgroup	index
C₂²×C₃³×C₈	2
C₂×#88855	2
#10589418	3

subgroup

index

C₂²×C₃³×C₈

C₂×#88855

#10589418

Subgroups — #13146098

Conjugacy classes of subgroups

Complements

Maximal subgroups