Group #13046827 — subgroups

Subgroups — #13046827

Order 1296.

126 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 35 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×#731	C₂	1	1	yes	maximal, characteristic
2	#10619663	C₂	1	1	yes	maximal, characteristic
3	#10619510	C₂	1	1	yes	maximal, characteristic
4	#3674⋊C₃	—	3	2	no	maximal
5	#3756⋊C₃	—	3	1	no	maximal
6	C₃×#3674	—	3	1	no	maximal
7	C₄×#731	—	2	1	no	—
8	C₄×#731	C₂²	1	1	yes	characteristic
9	#66331	—	2	1	no	—
10	Q₈×3₋¹⁺²	D₃	1	2	yes	—
11	Q₈×H(3,𝔽₃)	D₃	1	1	yes	characteristic
12	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
13	3₋¹⁺²⋊Q₈	—	3	2	no	—
14	#634⋊C₃	—	3	1	no	—
15	C₃×Dic₁₈	—	3	1	no	—
16	3₋¹⁺²⋊C₈	—	3	2	no	—
17	(C₃⋊(C₃⋊C₈))⋊C₃	—	3	1	no	—
18	C₃×C₉⋊C₈	—	3	1	no	—
19	C₂×#731	D₄	1	1	yes	characteristic
20	#3756	—	3	1	no	—
21	C₃×#216	—	9	1	no	—
22	#3674	—	3	1	no	—
23	#3674	—	3	2	no	—
24	C₄×3₋¹⁺²	—	2	2	no	—
25	C₄×3₋¹⁺²	C₂×D₃	1	2	yes	—
26	C₄×H(3,𝔽₃)	—	2	1	no	—
27	C₄×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
28	C₃×C₄×C₉	—	2	1	no	—
29	C₃×C₄×C₉	C₂×D₃	1	1	yes	characteristic
30	3₋¹⁺²⋊C₄	—	6	2	no	—
31	(C₃⋊Dic₃)⋊C₃	—	6	1	no	—
32	C₃×Dic₉	—	6	1	no	—
33	#731	Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₃²×Q₈	—	3	1	no	—
35	C₃²×Q₈	C₃⋊D₃	1	1	yes	characteristic
36	#634	—	3	1	no	—
37	C₃×Dic₆	—	9	1	no	—
38	C₃⋊(C₃⋊C₈)	—	3	1	no	—
39	C₃×C₃⋊C₈	—	9	1	no	—
40	Q₈×C₉	—	3	1	no	—
41	Q₈×C₉	—	3	2	no	—
42	Dic₁₈	—	3	1	no	—
43	Dic₁₈	—	3	2	no	—
44	C₉⋊C₈	—	3	1	no	—
45	C₉⋊C₈	—	3	2	no	—
46	C₂×3₋¹⁺²	#67	1	2	yes	—
47	C₂×H(3,𝔽₃)	#67	1	1	yes	characteristic
48	C₂×C₃×C₉	#67	1	1	yes	characteristic
49	C₃×Q₁₆	—	27	1	no	—
50	#216	—	9	1	no	—
51	#216	—	27	1	no	—
52	C₃²×C₄	—	2	1	no	—
53	C₃²×C₄	—	3	1	no	—
54	C₃²×C₄	—	6	1	no	—
55	C₃²×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
56	C₃⋊Dic₃	—	6	1	no	—
57	C₃×Dic₃	—	18	1	no	—
58	C₄×C₉	—	3	1	no	—
59	C₄×C₉	—	3	2	no	—
60	C₄×C₉	—	6	1	no	—
61	C₄×C₉	—	6	2	no	—
62	Dic₉	—	6	1	no	—
63	Dic₉	—	6	2	no	—
64	3₋¹⁺²	#216	1	2	yes	complemented
65	H(3,𝔽₃)	#216	1	1	yes	characteristic
66	C₃×C₉	#216	1	1	yes	characteristic, complemented
67	C₃×Q₈	—	3	1	no	—
68	C₃×Q₈	—	9	1	no	—
69	C₃×Q₈	—	27	1	no	—
70	C₃×Q₈	#272	1	1	yes	characteristic
71	Dic₆	—	9	1	no	—
72	Dic₆	—	27	1	no	—
73	C₃×C₈	—	27	1	no	—
74	C₃⋊C₈	—	9	1	no	—
75	C₃⋊C₈	—	27	1	no	—
76	C₂×C₃²	—	3	1	no	—
77	C₂×C₃²	#638	1	1	yes	characteristic
78	C₂×C₉	—	3	1	no	—
79	C₂×C₉	—	3	2	no	—
80	Q₁₆	—	27	1	no	Sylow
81	C₃×C₄	—	2	1	no	—
82	C₃×C₄	—	3	1	no	—
83	C₃×C₄	—	6	1	no	—
84	C₃×C₄	—	9	1	no	—
85	C₃×C₄	—	18	1	no	—
86	C₃×C₄	—	54	1	no	—
87	C₃×C₄	C₂×#272	1	1	yes	characteristic
88	Dic₃	—	18	1	no	—
89	Dic₃	—	54	1	no	—
90	C₃²	—	3	1	no	—
91	C₃²	#3756	1	1	yes	characteristic
92	C₉	—	3	1	no	—
93	C₉	—	3	2	no	—
94	Q₈	—	27	1	no	—
95	Q₈	#4184	1	1	yes	characteristic
96	C₈	—	27	1	no	—
97	C₂×C₃	—	3	1	no	—
98	C₂×C₃	—	9	1	no	—
99	C₂×C₃	#6235	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	54	1	no	—
102	C₄	C₂×#4184	1	1	yes	characteristic
103	C₃	—	3	1	no	—
104	C₃	—	9	1	no	—
105	C₃	#88550	1	1	yes	characteristic
106	C₂	#10619681	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#731

C₂

yes

maximal, characteristic

#10619663

C₂

yes

maximal, characteristic

#10619510

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

characteristic

#66331

—

Q₈×3₋¹⁺²

D₃

yes

—

Q₈×H(3,𝔽₃)

D₃

yes

characteristic

C₃×Q₈×C₉

D₃

yes

characteristic

3₋¹⁺²⋊Q₈

—

#634⋊C₃

—

C₃×Dic₁₈

—

3₋¹⁺²⋊C₈

—

(C₃⋊(C₃⋊C₈))⋊C₃

—

C₃×C₉⋊C₈

—

C₂×#731

D₄

yes

—

—

—

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×D₃

yes

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×D₃

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×D₃

yes

characteristic

3₋¹⁺²⋊C₄

—

(C₃⋊Dic₃)⋊C₃

—

C₃×Dic₉

—

#731

Q₁₆

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

C₃⋊D₃

yes

characteristic

#634

—

C₃×Dic₆

—

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

Dic₁₈

—

Dic₁₈

—

C₉⋊C₈

—

C₉⋊C₈

—

C₂×3₋¹⁺²

#67

yes

—

C₂×H(3,𝔽₃)

yes

C₂×C₃×C₉

yes

C₃×Q₁₆

—

#216

—

#216

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×C₃⋊D₃

yes

characteristic

C₃⋊Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

Dic₉

—

Dic₉

—

3₋¹⁺²

yes

yes

C₃×C₉

yes

characteristic, complemented

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

#272

yes

characteristic

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#638

yes

characteristic

C₂×C₉

—

C₂×C₉

—

Q₁₆

—

Sylow

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×#272

yes

characteristic

Dic₃

—

Dic₃

—

C₃²

—

C₃²

#3756

yes

characteristic

C₉

—

C₉

—

Q₈

—

Q₈

#4184

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#6235

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

C₂×#4184

yes

characteristic

103

C₃

—

104

C₃

—

105

C₃

#88550

yes

characteristic

106

C₂

#10619681

yes

characteristic

	subgroup	G/H
center	C₂	#10619681
commutator, cosocle	C₄×#731	C₂²
Frattini	C₃²×C₄	C₂×C₃⋊D₃
Fitting	Q₈×#731	C₂
socle	C₂×C₃	#6235
prime radical	C₂×H(3,𝔽₃)	#67
hypercenter	Q₈	#4184
hypocenter	#731	Q₁₆
LCS	G ▷ C₄×#731 ▷ C₂×#731 ▷ #731
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

subgroup

G/H

C₂

C₂²

C₃²×C₄

C₂×C₃⋊D₃

C₂

C₂×C₃

C₂×H(3,𝔽₃)

Q₈

Q₁₆

G ▷ C₄×#731 ▷ C₂×#731 ▷ #731

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₈

N	H ≅ G/N	type
33. #731	Q₁₆	semidirect
64. 3₋¹⁺²	#216	semidirect
66. C₃×C₉	#216	semidirect

H ≅ G/N

type

33. #731

Q₁₆

semidirect

64. 3₋¹⁺²

#216

semidirect

66. C₃×C₉

#216

semidirect

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

subgroup	index
Q₈×#731	2
#10619663	2
#10619510	2
#3674⋊C₃	3
#3756⋊C₃	3
C₃×#3674	3

subgroup

index

Subgroups — #13046827

Conjugacy classes of subgroups

Complements

Maximal subgroups