Group #13046821 — subgroups

Subgroups — #13046821

Order 1296.

183 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 51 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	Q₈×C₃²⋊C₉	C₂	1	1	yes	maximal, characteristic
2	#10619654	C₂	1	1	yes	maximal, characteristic
3	#10619507	C₂	1	1	yes	maximal, characteristic
4	C₃×#3756	—	3	1	no	maximal
5	C₃×#3674	—	3	3	no	maximal
6	C₄×C₃²⋊C₉	—	2	1	no	—
7	C₄×C₃²⋊C₉	C₂²	1	1	yes	characteristic
8	#66328	—	2	1	no	—
9	C₃³×Q₈	D₃	1	1	yes	characteristic
10	C₃×#634	—	3	1	no	—
11	C₃×C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	C₃×Q₈×C₉	D₃	1	3	yes	—
13	C₃×Dic₁₈	—	3	3	no	—
14	C₃×C₉⋊C₈	—	3	3	no	—
15	C₂×C₃²⋊C₉	D₄	1	1	yes	characteristic
16	#3756	—	3	1	no	—
17	C₃×#216	—	9	3	no	—
18	C₃×#216	—	9	1	no	—
19	#3674	—	3	3	no	—
20	C₃³×C₄	—	2	1	no	—
21	C₃³×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃×C₃⋊Dic₃	—	6	1	no	—
23	C₃×C₄×C₉	—	2	3	no	—
24	C₃×C₄×C₉	C₂×D₃	1	3	yes	—
25	C₃×Dic₉	—	6	3	no	—
26	C₃²⋊C₉	Q₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₃²×Q₈	—	3	1	no	—
28	C₃²×Q₈	—	6	1	no	—
29	C₃²×Q₈	D₉	1	3	yes	—
30	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
31	#634	—	3	1	no	—
32	C₃×Dic₆	—	9	3	no	—
33	C₃×Dic₆	—	9	1	no	—
34	C₃⋊(C₃⋊C₈)	—	3	1	no	—
35	C₃×C₃⋊C₈	—	9	3	no	—
36	C₃×C₃⋊C₈	—	9	1	no	—
37	Q₈×C₉	—	3	3	no	—
38	Dic₁₈	—	3	3	no	—
39	C₉⋊C₈	—	3	3	no	—
40	C₂×C₃³	#67	1	1	yes	characteristic
41	C₂×C₃×C₉	#67	1	3	yes	—
42	C₃×Q₁₆	—	27	1	no	—
43	#216	—	9	3	no	—
44	#216	—	9	1	no	—
45	C₃²×C₄	—	2	1	no	—
46	C₃²×C₄	—	2	3	no	—
47	C₃²×C₄	—	3	1	no	—
48	C₃²×C₄	—	6	1	no	—
49	C₃²×C₄	—	6	1	no	—
50	C₃²×C₄	—	6	2	no	—
51	C₃²×C₄	C₂×D₉	1	3	yes	—
52	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
53	C₃⋊Dic₃	—	6	1	no	—
54	C₃×Dic₃	—	18	3	no	—
55	C₃×Dic₃	—	18	1	no	—
56	C₄×C₉	—	3	3	no	—
57	C₄×C₉	—	6	3	no	—
58	Dic₉	—	6	3	no	—
59	C₃³	#216	1	1	yes	characteristic
60	C₃×C₉	#216	1	3	yes	complemented
61	C₃×Q₈	—	2	1	no	—
62	C₃×Q₈	—	3	3	no	—
63	C₃×Q₈	—	27	1	no	—
64	C₃×Q₈	C₃⋊D₉	1	1	yes	characteristic
65	C₃×Q₈	#272	1	1	yes	characteristic
66	Dic₆	—	9	3	no	—
67	Dic₆	—	9	1	no	—
68	C₃×C₈	—	27	1	no	—
69	C₃⋊C₈	—	9	3	no	—
70	C₃⋊C₈	—	9	1	no	—
71	C₂×C₃²	—	3	1	no	—
72	C₂×C₃²	—	6	1	no	—
73	C₂×C₃²	#611	1	3	yes	—
74	C₂×C₃²	#638	1	1	yes	characteristic
75	C₂×C₉	—	3	3	no	—
76	Q₁₆	—	27	1	no	Sylow
77	C₃×C₄	—	2	2	no	—
78	C₃×C₄	—	2	1	no	—
79	C₃×C₄	—	2	1	no	—
80	C₃×C₄	—	2	1	no	—
81	C₃×C₄	—	3	3	no	—
82	C₃×C₄	—	6	3	no	—
83	C₃×C₄	—	54	1	no	—
84	C₃×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
85	C₃×C₄	C₂×#272	1	1	yes	characteristic
86	Dic₃	—	18	3	no	—
87	Dic₃	—	18	1	no	—
88	C₃²	—	3	1	no	—
89	C₃²	—	6	1	no	—
90	C₃²	#3674	1	3	yes	complemented
91	C₃²	#3756	1	1	yes	characteristic
92	C₉	—	3	3	no	—
93	Q₈	—	27	1	no	—
94	Q₈	#4181	1	1	yes	characteristic
95	C₈	—	27	1	no	—
96	C₂×C₃	—	2	1	no	—
97	C₂×C₃	—	3	3	no	—
98	C₂×C₃	#6233	1	1	yes	characteristic
99	C₂×C₃	#6235	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	54	1	no	—
102	C₄	C₂×#4181	1	1	yes	characteristic
103	C₃	—	2	1	no	—
104	C₃	—	3	3	no	—
105	C₃	#88547	1	1	yes	characteristic
106	C₃	#88550	1	1	yes	characteristic
107	C₂	#10619675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₃²⋊C₉

C₂

yes

maximal, characteristic

#10619654

C₂

yes

maximal, characteristic

#10619507

C₂

yes

maximal, characteristic

—

—

C₄×C₃²⋊C₉

—

C₄×C₃²⋊C₉

C₂²

yes

characteristic

#66328

—

C₃³×Q₈

D₃

yes

characteristic

C₃×#634

—

C₃×C₃⋊(C₃⋊C₈)

—

C₃×Q₈×C₉

D₃

yes

—

C₃×Dic₁₈

—

C₃×C₉⋊C₈

—

C₂×C₃²⋊C₉

D₄

yes

—

—

—

—

C₃³×C₄

—

C₃³×C₄

C₂×D₃

yes

characteristic

C₃×C₃⋊Dic₃

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×D₃

yes

—

C₃×Dic₉

—

C₃²⋊C₉

Q₁₆

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

D₉

yes

—

C₃²×Q₈

C₃⋊D₃

yes

characteristic

#634

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Dic₁₈

—

C₉⋊C₈

—

C₂×C₃³

#67

yes

characteristic

C₂×C₃×C₉

#67

yes

—

C₃×Q₁₆

—

#216

—

#216

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×D₉

yes

—

C₃²×C₄

C₂×C₃⋊D₃

yes

characteristic

C₃⋊Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

Dic₉

—

C₃³

yes

C₃×C₉

yes

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃⋊D₉

yes

characteristic

C₃×Q₈

#272

yes

characteristic

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

#611

yes

—

C₂×C₃²

#638

yes

characteristic

C₂×C₉

—

Q₁₆

—

Sylow

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₃⋊D₉

yes

characteristic

C₃×C₄

C₂×#272

yes

characteristic

Dic₃

—

Dic₃

—

C₃²

—

C₃²

—

C₃²

yes

C₃²

yes

C₉

—

Q₈

—

Q₈

#4181

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

yes

C₂×C₃

yes

100

C₄

—

101

C₄

—

102

C₄

C₂×#4181

yes

characteristic

103

C₃

—

104

C₃

—

105

C₃

#88547

yes

characteristic

106

C₃

#88550

yes

characteristic

107

C₂

#10619675

yes

characteristic

	subgroup	G/H
center	C₂×C₃	#6233
commutator, cosocle	C₄×C₃²⋊C₉	C₂²
Frattini	C₃²×C₄	C₂×C₃⋊D₃
Fitting	Q₈×C₃²⋊C₉	C₂
socle	C₂×C₃²	#638
prime radical	C₂×C₃³	#67
hypercenter	C₃×Q₈	C₃⋊D₉
hypocenter	C₃²⋊C₉	Q₁₆
LCS	G ▷ C₄×C₃²⋊C₉ ▷ C₂×C₃²⋊C₉ ▷ C₃²⋊C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

subgroup

G/H

center

C₂×C₃

#6233

commutator, cosocle

C₄×C₃²⋊C₉

C₂²

Frattini

C₃²×C₄

C₂×C₃⋊D₃

Fitting

Q₈×C₃²⋊C₉

C₂

C₂×C₃²

C₂×C₃³

C₃×Q₈

C₃⋊D₉

hypocenter

C₃²⋊C₉

Q₁₆

LCS

G ▷ C₄×C₃²⋊C₉ ▷ C₂×C₃²⋊C₉ ▷ C₃²⋊C₉

UCS

C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

N	H ≅ G/N	type
26. C₃²⋊C₉	Q₁₆	semidirect
60. C₃×C₉	#216	semidirect
90. C₃²	#3674	semidirect

H ≅ G/N

type

26. C₃²⋊C₉

Q₁₆

semidirect

60. C₃×C₉

#216

semidirect

90. C₃²

#3674

semidirect

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

subgroup	index
Q₈×C₃²⋊C₉	2
#10619654	2
#10619507	2
C₃×#3756	3
C₃×#3674	3

subgroup

index

Q₈×C₃²⋊C₉

Subgroups — #13046821

Conjugacy classes of subgroups

Complements

Maximal subgroups