Group #13057661 — subgroups

Subgroups — #13057661

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	#10621539	C₂	1	2	yes	maximal, complemented
2	#131⋊ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	#90443	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	#41⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
5	#66665	—	2	2	no	—
6	Dic₂₈⋊C₃	—	4	1	no	maximal
7	#6553	C₂×C₃	1	2	yes	—
8	C₇×#131	C₂×C₃	1	1	yes	characteristic
9	#6471	—	3	2	no	—
10	#6467	—	3	1	no	—
11	C₂×D₅₆	—	3	1	no	—
12	#576⋊C₃	—	7	1	no	maximal, Hall
13	C₇⋊SL(2,𝔽₃)	D₄	1	1	yes	characteristic
14	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
15	C₈×ΓL(1,𝔽₈)	—	4	1	no	—
16	C₇×#41	C₂²×C₃	1	1	yes	characteristic
17	#1161	—	2	2	no	—
18	D₄×D₇	—	6	2	no	—
19	#1157	—	3	2	no	—
20	C₂×D₂₈	—	3	2	no	—
21	C₇×M₁₆	—	3	1	no	—
22	C₂×C₇×C₈	—	3	1	no	—
23	Dic₂₈	A₄	1	1	yes	characteristic
24	D₅₆	—	3	2	no	—
25	D₅₆	—	3	1	no	—
26	#1132	—	3	2	no	—
27	#994	—	7	2	no	—
28	#131⋊C₃	—	7	1	no	—
29	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
30	ΓL(1,𝔽₈)⋊C₄	—	8	2	no	—
31	#576	—	7	1	no	Sylow
32	C₂²×D₇	—	6	2	no	—
33	C₇×Q₈	C₃×D₄	1	1	yes	characteristic
34	C₇×D₄	—	3	1	no	—
35	C₂×C₄×C₇	—	3	1	no	—
36	C₂²⋊D₇	—	6	2	no	—
37	D₂₈	—	3	2	no	—
38	D₂₈	—	6	2	no	—
39	C₄×D₇	—	6	2	no	—
40	Dic₁₄	C₂×A₄	1	2	yes	—
41	C₇×C₈	—	3	1	no	—
42	C₇×C₈	C₂×A₄	1	1	yes	characteristic
43	#41⋊C₃	—	7	1	no	—
44	#41⋊C₃	—	14	2	no	—
45	C₃×Q₁₆	—	28	1	no	—
46	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
47	H(5,𝔽₂)	—	7	2	no	—
48	AGL(1,ℤ/8)	—	21	2	no	—
49	#135	—	21	1	no	—
50	C₂×D₈	—	21	1	no	—
51	#131	AGL(1,𝔽₇)	1	1	yes	characteristic
52	C₂²×C₇	—	3	1	no	—
53	C₂×D₇	—	6	2	no	—
54	C₂×D₇	—	24	2	no	—
55	C₄×C₇	—	3	1	no	—
56	C₄×C₇	C₂²×A₄	1	1	yes	characteristic
57	Dic₇	—	2	2	no	—
58	C₃×Q₈	—	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	—	14	2	no	—
63	#41	—	21	2	no	—
64	#41	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
65	C₂×D₄	—	21	2	no	—
66	C₂×D₄	—	42	2	no	—
67	Q₁₆	—	7	1	no	—
68	SD₁₆	—	21	2	no	—
69	D₈	—	21	2	no	—
70	D₈	—	21	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	—	12	2	no	—
76	C₃×C₄	—	28	1	no	—
77	C₃×C₄	—	56	2	no	—
78	C₂³	—	42	2	no	—
79	Q₈	—	7	2	no	—
80	Q₈	C₄⋊AGL(1,𝔽₇)	1	1	yes	characteristic
81	D₄	—	3	1	no	—
82	D₄	—	21	2	no	—
83	D₄	—	42	2	no	—
84	D₄	—	42	2	no	—
85	C₂×C₄	—	3	1	no	—
86	C₂×C₄	—	42	2	no	—
87	C₈	—	3	1	no	—
88	C₈	D₇⋊A₄	1	1	yes	characteristic
89	C₇	#576⋊C₃	1	1	yes	characteristic, complemented, Sylow
90	C₂×C₃	—	28	1	no	—
91	C₂²	—	3	1	no	—
92	C₂²	—	42	2	no	—
93	C₂²	—	168	2	no	—
94	C₄	—	3	1	no	—
95	C₄	—	14	2	no	—
96	C₄	C₂×D₇⋊A₄	1	1	yes	characteristic
97	C₃	—	28	1	no	Sylow
98	C₂	—	6	1	no	—
99	C₂	—	84	2	no	—
100	C₂	C₄⋊(D₇⋊A₄)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10621539

C₂

yes

maximal, complemented

#131⋊ΓL(1,𝔽₈)

C₂

yes

maximal, characteristic, complemented

#90443

C₃

yes

maximal, characteristic, complemented, Hall

#41⋊ΓL(1,𝔽₈)

C₂²

yes

characteristic

#66665

—

Dic₂₈⋊C₃

—

maximal

#6553

C₂×C₃

yes

—

C₇×#131

C₂×C₃

yes

characteristic

#6471

—

#6467

—

C₂×D₅₆

—

#576⋊C₃

—

maximal, Hall

C₇⋊SL(2,𝔽₃)

D₄

yes

characteristic

ΓL(1,𝔽₈)⋊Q₈

—

C₈×ΓL(1,𝔽₈)

—

C₇×#41

C₂²×C₃

yes

characteristic

#1161

—

D₄×D₇

—

#1157

—

C₂×D₂₈

—

C₇×M₁₆

—

C₂×C₇×C₈

—

Dic₂₈

A₄

yes

characteristic

D₅₆

—

D₅₆

—

—

—

—

—

—

—

C₂²×D₇

—

C₇×Q₈

C₃×D₄

yes

characteristic

C₇×D₄

—

C₂×C₄×C₇

—

C₂²⋊D₇

—

D₂₈

—

D₂₈

—

C₄×D₇

—

Dic₁₄

C₂×A₄

yes

—

C₇×C₈

—

C₇×C₈

C₂×A₄

yes

characteristic

#41⋊C₃

—

#41⋊C₃

—

C₃×Q₁₆

—

—

—

—

—

C₂×D₈

—

#131

AGL(1,𝔽₇)

yes

characteristic

C₂²×C₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₂²×A₄

yes

characteristic

Dic₇

—

C₃×Q₈

—

SL(2,𝔽₃)

—

C₃×C₈

—

—

—

—

yes

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₇

—

C₂×C₇

D₄×A₄

yes

characteristic

D₇

—

C₃×C₄

—

C₃×C₄

—

C₂³

—

Q₈

—

Q₈

C₄⋊AGL(1,𝔽₇)

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

D₇⋊A₄

yes

characteristic

C₇

#576⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂²

—

C₂²

—

C₂²

—

168

—

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. #10621539	C₂	semidirect
2. #131⋊ΓL(1,𝔽₈)	C₂	semidirect
3. #90443	C₃	semidirect
89. C₇	#576⋊C₃	semidirect

H ≅ G/N

type

1. #10621539

C₂

semidirect

2. #131⋊ΓL(1,𝔽₈)

C₂

semidirect

3. #90443

C₃

semidirect

89. C₇

#576⋊C₃

semidirect

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

subgroup	index
#10621539	2
#131⋊ΓL(1,𝔽₈)	2
#90443	3
Dic₂₈⋊C₃	4
#576⋊C₃	7

subgroup

index

Dic₂₈⋊C₃

Subgroups — #13057661

Conjugacy classes of subgroups

Complements

Maximal subgroups