Group #11722827 — subgroups

Subgroups — #11722827

Order 864.

322 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 70 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	#89042	C₂	1	1	yes	maximal, characteristic
2	C₃⋊D₃×C₃⋊C₈	C₂	1	3	yes	maximal, complemented
3	C₃×#3712	C₂	1	3	yes	maximal, complemented
4	#63805	—	3	3	no	maximal
5	C₃×C₄×C₃⋊D₃	C₂²	1	3	yes	—
6	#6296	C₄	1	1	yes	characteristic
7	#6295	C₄	1	3	yes	—
8	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
9	C₃²×C₃⋊C₈	C₂²	1	3	yes	—
10	#3798	—	3	3	no	—
11	C₈×C₃⋊D₃	—	3	3	no	—
12	C₃×ΓL(1,𝔽₂₅)	—	3	6	no	—
13	#3712	D₃	1	3	yes	complemented
14	D₃×C₃⋊C₈	—	3	6	no	—
15	D₃×C₃⋊C₈	—	6	3	no	—
16	C₂×C₃×C₃⋊D₃	C₂×C₄	1	3	yes	—
17	C₃³×C₄	C₂³	1	1	yes	characteristic
18	C₃×C₃⋊Dic₃	C₂×C₄	1	3	yes	—
19	#907	—	9	3	no	—
20	C₄×C₃⋊D₃	C₂×D₃	1	3	yes	—
21	C₃×C₄×D₃	—	3	6	no	—
22	C₃×C₄×D₃	—	6	3	no	—
23	#627	—	3	3	no	—
24	#626	—	3	6	no	—
25	#625	—	3	3	no	—
26	C₃²×C₈	—	3	3	no	—
27	C₃⋊(C₃⋊C₈)	—	3	3	no	—
28	C₃⋊(C₃⋊C₈)	—	6	3	no	—
29	C₃⋊(C₃⋊C₈)	—	12	1	no	—
30	C₃×C₃⋊C₈	—	2	3	no	—
31	C₃×C₃⋊C₈	C₂×D₃	1	6	yes	—
32	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
33	C₃×C₃⋊D₃	—	2	3	no	—
34	#235	—	9	3	no	—
35	C₃×M₁₆	—	9	3	no	—
36	C₂×C₃⋊C₈	—	9	3	no	—
37	ΓL(1,𝔽₂₅)	—	3	6	no	—
38	D₃×C₈	—	9	6	no	—
39	D₃×C₈	—	18	3	no	—
40	C₂×C₃⋊D₃	C₄×D₃	1	3	yes	—
41	C₂×C₃×D₃	—	3	6	no	—
42	C₂×C₃×D₃	—	6	3	no	—
43	C₃²×C₄	—	2	3	no	—
44	C₃²×C₄	—	4	1	no	—
45	C₃²×C₄	C₂²×D₃	1	3	yes	—
46	C₃⋊Dic₃	C₄×D₃	1	3	yes	—
47	C₃×Dic₃	—	3	6	no	—
48	C₃×Dic₃	—	6	3	no	—
49	#131	—	27	1	no	Sylow
50	C₃³	#131	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃×C₄	—	9	3	no	—
52	#67	—	9	6	no	—
53	D₁₂	—	9	3	no	—
54	C₄×D₃	—	3	6	no	—
55	C₄×D₃	—	6	3	no	—
56	Dic₆	—	9	3	no	—
57	C₃×C₈	—	3	6	no	—
58	C₃×C₈	—	6	3	no	—
59	C₃⋊C₈	—	9	3	no	—
60	C₃⋊C₈	—	18	3	no	—
61	C₃⋊C₈	—	36	1	no	—
62	C₃⋊C₈	D₃²	1	3	yes	—
63	C₂×C₃²	—	2	3	no	—
64	C₂×C₃²	—	4	1	no	—
65	C₂×C₃²	C₂×C₄×D₃	1	3	yes	—
66	C₃⋊D₃	—	2	3	no	—
67	C₃×D₃	—	6	6	no	—
68	C₃×D₃	—	6	6	no	—
69	#41	—	27	1	no	—
70	M₁₆	—	9	3	no	—
71	C₂×C₈	—	27	3	no	—
72	C₂²×C₃	—	9	3	no	—
73	C₂×D₃	—	3	6	no	—
74	C₂×D₃	—	6	3	no	—
75	C₃×C₄	—	2	3	no	—
76	C₃×C₄	—	4	1	no	—
77	C₃×C₄	—	9	3	no	—
78	C₃×C₄	C₂×D₃²	1	3	yes	—
79	Dic₃	—	3	6	no	—
80	Dic₃	—	6	3	no	—
81	C₃²	—	2	3	no	—
82	C₃²	—	4	1	no	—
83	C₃²	#907	1	3	yes	complemented
84	Q₈	—	27	1	no	—
85	D₄	—	27	3	no	—
86	C₂×C₄	—	9	3	no	—
87	C₈	—	3	3	no	—
88	C₈	—	27	1	no	—
89	C₂×C₃	—	2	3	no	—
90	C₂×C₃	—	4	1	no	—
91	C₂×C₃	—	18	3	no	—
92	C₂×C₃	C₄×D₃²	1	3	yes	—
93	D₃	—	6	6	no	—
94	D₃	—	6	6	no	—
95	C₂²	—	9	3	no	—
96	C₄	—	9	3	no	—
97	C₄	D₃³	1	1	yes	characteristic
98	C₃	—	2	3	no	—
99	C₃	—	4	1	no	—
100	C₃	#63805	1	3	yes	complemented
101	C₂	—	18	3	no	—
102	C₂	#88949	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#89042

C₂

yes

maximal, characteristic

C₃⋊D₃×C₃⋊C₈

C₂

yes

maximal, complemented

C₃×#3712

C₂

yes

maximal, complemented

#63805

—

maximal

C₃×C₄×C₃⋊D₃

C₂²

yes

—

#6296

C₄

yes

characteristic

#6295

C₄

yes

—

C₃⋊(C₃⋊(C₃⋊C₈))

C₂²

yes

characteristic

C₃²×C₃⋊C₈

C₂²

yes

—

#3798

—

C₈×C₃⋊D₃

—

C₃×ΓL(1,𝔽₂₅)

—

#3712

D₃

yes

complemented

D₃×C₃⋊C₈

—

D₃×C₃⋊C₈

—

C₂×C₃×C₃⋊D₃

C₂×C₄

yes

—

C₃³×C₄

C₂³

yes

characteristic

C₃×C₃⋊Dic₃

C₂×C₄

yes

—

#907

—

C₄×C₃⋊D₃

C₂×D₃

yes

—

C₃×C₄×D₃

—

C₃×C₄×D₃

—

#627

—

#626

—

#625

—

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

C₂×D₃

yes

—

C₂×C₃³

C₂²×C₄

yes

characteristic

C₃×C₃⋊D₃

—

#235

—

C₃×M₁₆

—

C₂×C₃⋊C₈

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

D₃×C₈

—

C₂×C₃⋊D₃

C₄×D₃

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂²×D₃

yes

—

C₃⋊Dic₃

C₄×D₃

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

#131

—

Sylow

C₃³

#131

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

#67

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

D₃²

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×C₄×D₃

yes

—

C₃⋊D₃

—

C₃×D₃

—

C₃×D₃

—

#41

—

M₁₆

—

C₂×C₈

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×D₃²

yes

—

Dic₃

—

Dic₃

—

C₃²

—

C₃²

—

C₃²

#907

yes

complemented

Q₈

—

D₄

—

C₂×C₄

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₄×D₃²

yes

—

D₃

—

D₃

—

C₂²

—

C₄

—

C₄

D₃³

yes

characteristic

C₃

—

C₃

—

100

C₃

#63805

yes

complemented

101

C₂

—

102

C₂

#88949

yes

characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃³
commutator, socle	C₂×C₃³	C₂²×C₄
Fitting, cosocle	C₃³×C₄	C₂³
prime radical	#89042	C₂
hypocenter	C₃³	#131
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄

subgroup

G/H

center, Frattini, hypercenter

C₄

D₃³

commutator, socle

C₂×C₃³

C₂²×C₄

Fitting, cosocle

C₃³×C₄

C₂³

C₂

C₃³

G ▷ C₂×C₃³ ▷ C₃³

C₁ ◁ C₄

N	H ≅ G/N	type
2. C₃⋊D₃×C₃⋊C₈	C₂	semidirect
3. C₃×#3712	C₂	semidirect
13. #3712	D₃	semidirect
50. C₃³	#131	semidirect
83. C₃²	#907	semidirect
100. C₃	#63805	semidirect

H ≅ G/N

type

2. C₃⋊D₃×C₃⋊C₈

C₂

semidirect

3. C₃×#3712

C₂

semidirect

13. #3712

D₃

semidirect

50. C₃³

#131

semidirect

83. C₃²

#907

semidirect

100. C₃

#63805

semidirect

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

subgroup	index
#89042	2
C₃⋊D₃×C₃⋊C₈	2
C₃×#3712	2
#63805	3

subgroup

index

#89042

C₃⋊D₃×C₃⋊C₈

C₃×#3712

#63805

Subgroups — #11722827

Conjugacy classes of subgroups

Complements

Maximal subgroups