Group #13057660 — subgroups

Subgroups — #13057660

Order 1344.

146 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 26 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	#10621541	C₂	1	2	yes	maximal, complemented
2	#131⋊ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	#90444	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	#41⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
5	D₇⋊SL(2,𝔽₃)	—	2	2	no	—
6	D₅₆⋊C₃	—	4	1	no	maximal
7	#6554	C₂×C₃	1	2	yes	complemented
8	C₇×#131	C₂×C₃	1	1	yes	characteristic, complemented
9	#6472	—	3	2	no	—
10	C₂×Dic₂₈	—	3	1	no	—
11	#6467	—	3	1	no	—
12	#578⋊C₃	—	7	1	no	maximal, Hall
13	C₇⋊SL(2,𝔽₃)	D₄	1	1	yes	characteristic
14	C₄⋊AGL(1,𝔽₇)	—	4	2	no	—
15	C₈×ΓL(1,𝔽₈)	—	4	1	no	—
16	C₇×#41	C₂²×C₃	1	1	yes	characteristic
17	Q₈×D₇	—	2	2	no	—
18	#1159	—	6	2	no	—
19	#1157	—	3	2	no	—
20	C₂×Dic₁₄	—	3	2	no	—
21	C₇×M₁₆	—	3	1	no	—
22	C₂×C₇×C₈	—	3	1	no	—
23	Dic₂₈	—	3	2	no	—
24	Dic₂₈	—	3	1	no	—
25	D₅₆	A₄	1	1	yes	characteristic
26	#1132	—	3	2	no	—
27	2₋¹⁺⁴⋊C₃	—	7	2	no	—
28	#131⋊C₃	—	7	1	no	—
29	C₂×AGL(1,𝔽₇)	—	8	2	no	—
30	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
31	#578	—	7	1	no	Sylow
32	C₇×Q₈	C₃×D₄	1	1	yes	characteristic
33	C₇×D₄	—	3	1	no	—
34	C₂×C₄×C₇	—	3	1	no	—
35	C₂²⋊D₇	—	6	2	no	—
36	C₂×Dic₇	—	6	2	no	—
37	D₂₈	C₂×A₄	1	2	yes	—
38	C₄×D₇	—	6	2	no	—
39	Dic₁₄	—	3	2	no	—
40	Dic₁₄	—	6	2	no	—
41	C₇×C₈	—	3	1	no	—
42	C₇×C₈	C₂×A₄	1	1	yes	characteristic
43	#41⋊C₃	—	7	1	no	—
44	C₂×SL(2,𝔽₃)	—	14	2	no	—
45	C₃×D₈	—	28	1	no	—
46	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
47	AGL(1,𝔽₇)	—	16	2	no	—
48	2₋¹⁺⁴	—	7	2	no	—
49	#137	—	21	2	no	—
50	#135	—	21	1	no	—
51	C₂×Q₁₆	—	21	1	no	—
52	#131	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
53	C₂²×C₇	—	3	1	no	—
54	C₂×D₇	—	2	2	no	—
55	C₄×C₇	—	3	1	no	—
56	C₄×C₇	C₂²×A₄	1	1	yes	characteristic
57	Dic₇	—	6	2	no	—
58	C₃×D₄	—	28	2	no	—
59	SL(2,𝔽₃)	—	7	1	no	—
60	C₃×C₈	—	28	1	no	—
61	ΓL(1,𝔽₈)	—	4	1	no	Hall
62	#41	—	21	2	no	—
63	#41	—	42	2	no	—
64	#41	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
65	C₂×Q₈	—	14	2	no	—
66	C₂×Q₈	—	21	2	no	—
67	Q₁₆	—	21	2	no	—
68	Q₁₆	—	21	1	no	—
69	SD₁₆	—	21	2	no	—
70	D₈	—	7	1	no	—
71	M₁₆	—	3	1	no	—
72	C₂×C₈	—	3	1	no	—
73	C₂×C₇	—	6	1	no	—
74	C₂×C₇	D₄×A₄	1	1	yes	characteristic
75	D₇	—	4	2	no	—
76	C₂²×C₃	—	56	2	no	—
77	C₃×C₄	—	28	1	no	—
78	Q₈	—	21	2	no	—
79	Q₈	—	42	2	no	—
80	Q₈	C₄⋊AGL(1,𝔽₇)	1	1	yes	characteristic
81	D₄	—	3	1	no	—
82	D₄	—	7	2	no	—
83	D₄	—	42	2	no	—
84	C₂×C₄	—	3	1	no	—
85	C₂×C₄	—	42	2	no	—
86	C₂×C₄	—	42	2	no	—
87	C₈	—	3	1	no	—
88	C₈	D₇⋊A₄	1	1	yes	characteristic
89	C₇	#578⋊C₃	1	1	yes	characteristic, complemented, Sylow
90	C₂×C₃	—	28	1	no	—
91	C₂×C₃	—	112	2	no	—
92	C₂²	—	3	1	no	—
93	C₂²	—	14	2	no	—
94	C₄	—	3	1	no	—
95	C₄	—	42	2	no	—
96	C₄	C₂×D₇⋊A₄	1	1	yes	characteristic
97	C₃	—	28	1	no	Sylow
98	C₂	—	6	1	no	—
99	C₂	—	28	2	no	—
100	C₂	C₄⋊(D₇⋊A₄)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10621541

C₂

yes

maximal, complemented

#131⋊ΓL(1,𝔽₈)

C₂

yes

maximal, characteristic, complemented

#90444

C₃

yes

maximal, characteristic, complemented, Hall

#41⋊ΓL(1,𝔽₈)

C₂²

yes

characteristic

D₇⋊SL(2,𝔽₃)

—

D₅₆⋊C₃

—

maximal

#6554

C₂×C₃

yes

complemented

C₇×#131

C₂×C₃

yes

characteristic, complemented

#6472

—

C₂×Dic₂₈

—

—

—

D₄

yes

characteristic

C₄⋊AGL(1,𝔽₇)

—

C₈×ΓL(1,𝔽₈)

—

C₇×#41

C₂²×C₃

yes

characteristic

Q₈×D₇

—

#1159

—

#1157

—

C₂×Dic₁₄

—

C₇×M₁₆

—

C₂×C₇×C₈

—

Dic₂₈

—

Dic₂₈

—

D₅₆

A₄

yes

characteristic

#1132

—

2₋¹⁺⁴⋊C₃

—

—

—

—

—

C₇×Q₈

C₃×D₄

yes

characteristic

C₇×D₄

—

C₂×C₄×C₇

—

C₂²⋊D₇

—

C₂×Dic₇

—

D₂₈

C₂×A₄

yes

—

C₄×D₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

—

C₇×C₈

C₂×A₄

yes

characteristic

#41⋊C₃

—

C₂×SL(2,𝔽₃)

—

C₃×D₈

—

C₂×ΓL(1,𝔽₈)

—

AGL(1,𝔽₇)

—

2₋¹⁺⁴

—

#137

—

#135

—

C₂×Q₁₆

—

#131

AGL(1,𝔽₇)

yes

characteristic, complemented

C₂²×C₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₂²×A₄

yes

characteristic

Dic₇

—

C₃×D₄

—

SL(2,𝔽₃)

—

C₃×C₈

—

—

—

—

yes

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₇

—

C₂×C₇

D₄×A₄

yes

characteristic

D₇

—

C₂²×C₃

—

C₃×C₄

—

Q₈

—

Q₈

—

Q₈

C₄⋊AGL(1,𝔽₇)

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

D₇⋊A₄

yes

characteristic

C₇

#578⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

112

—

C₂²

—

C₂²

—

C₄

—

C₄

—

C₄

C₂×D₇⋊A₄

yes

characteristic

C₃

—

Sylow

C₂

—

C₂

—

100

C₂

C₄⋊(D₇⋊A₄)

yes

characteristic

	subgroup	G/H
center	C₂	C₄⋊(D₇⋊A₄)
commutator, cosocle	C₇×#41	C₂²×C₃
Frattini	C₄	C₂×D₇⋊A₄
Fitting	C₇×#131	C₂×C₃
socle	C₂×C₇	D₄×A₄
hypercenter	C₈	D₇⋊A₄
hypocenter	C₇×Q₈	C₃×D₄
LCS	G ▷ C₇×#41 ▷ C₇×Q₈
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

subgroup

G/H

center

C₂

C₄⋊(D₇⋊A₄)

commutator, cosocle

C₇×#41

C₂²×C₃

Frattini

C₄

C₂×D₇⋊A₄

Fitting

C₇×#131

C₂×C₃

socle

C₂×C₇

D₄×A₄

hypercenter

C₈

D₇⋊A₄

hypocenter

C₇×Q₈

C₃×D₄

LCS

G ▷ C₇×#41 ▷ C₇×Q₈

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈

N	H ≅ G/N	type
1. #10621541	C₂	semidirect
2. #131⋊ΓL(1,𝔽₈)	C₂	semidirect
3. #90444	C₃	semidirect
7. #6554	C₂×C₃	semidirect
8. C₇×#131	C₂×C₃	semidirect
52. #131	AGL(1,𝔽₇)	semidirect
89. C₇	#578⋊C₃	semidirect

H ≅ G/N

type

1. #10621541

C₂

semidirect

2. #131⋊ΓL(1,𝔽₈)

C₂

semidirect

3. #90444

C₃

semidirect

7. #6554

C₂×C₃

semidirect

8. C₇×#131

C₂×C₃

semidirect

52. #131

AGL(1,𝔽₇)

semidirect

89. C₇

#578⋊C₃

semidirect

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

subgroup	index
#10621541	2
#131⋊ΓL(1,𝔽₈)	2
#90444	3
D₅₆⋊C₃	4
#578⋊C₃	7

subgroup

index

D₅₆⋊C₃

Subgroups — #13057660

Conjugacy classes of subgroups

Complements

Maximal subgroups