Group #13060794 — subgroups

Subgroups — #576⋊ΓL(1,𝔽₈) #13060794

Order 1344.

146 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 32 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	#10621615	C₂	1	2	yes	maximal, complemented
2	#131⋊ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#576	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	#41⋊ΓL(1,𝔽₈)	—	2	2	no	—
5	#41⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	Q₁₆×ΓL(1,𝔽₈)	—	4	1	no	maximal
7	C₇×H(5,𝔽₂)	C₂×C₃	1	2	yes	—
8	C₇×AGL(1,ℤ/8)	—	3	2	no	—
9	C₇×#135	—	3	1	no	—
10	C₂×C₇×D₈	—	3	1	no	—
11	C₇×#131	C₂×C₃	1	1	yes	characteristic
12	#576⋊C₃	—	7	1	no	maximal, Hall
13	C₇⋊SL(2,𝔽₃)	D₄	1	1	yes	characteristic
14	Q₈×ΓL(1,𝔽₈)	—	4	2	no	—
15	C₈×ΓL(1,𝔽₈)	—	4	1	no	—
16	C₇×#41	—	2	2	no	—
17	C₇×#41	—	3	2	no	—
18	C₇×#41	C₂²×C₃	1	1	yes	characteristic
19	C₂×C₇×D₄	—	3	2	no	—
20	C₂×C₇×D₄	—	6	2	no	—
21	C₇×Q₁₆	A₄	1	1	yes	characteristic
22	C₇×SD₁₆	—	3	2	no	—
23	C₇×D₈	—	3	2	no	—
24	C₇×D₈	—	3	1	no	—
25	C₇×M₁₆	—	3	1	no	—
26	C₂×C₇×C₈	—	3	1	no	—
27	#994	—	7	2	no	—
28	#131⋊C₃	—	7	1	no	—
29	C₄×ΓL(1,𝔽₈)	—	4	1	no	—
30	C₄×ΓL(1,𝔽₈)	—	8	2	no	—
31	#576	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
32	C₂³×C₇	—	6	2	no	—
33	C₇×Q₈	C₃×D₄	1	1	yes	characteristic
34	C₇×Q₈	C₂×A₄	1	2	yes	—
35	C₇×D₄	—	3	1	no	—
36	C₇×D₄	—	3	2	no	—
37	C₇×D₄	—	6	2	no	—
38	C₇×D₄	—	6	2	no	—
39	C₂×C₄×C₇	—	3	1	no	—
40	C₂×C₄×C₇	—	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	#41⋊C₃	—	14	2	no	—
45	C₃×Q₁₆	—	28	1	no	—
46	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
47	H(5,𝔽₂)	C₂×ΓL(1,𝔽₈)	1	2	yes	—
48	AGL(1,ℤ/8)	—	3	2	no	—
49	#135	—	3	1	no	—
50	C₂×D₈	—	3	1	no	—
51	#131	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	C₂²×C₇	—	3	1	no	—
53	C₂²×C₇	—	6	2	no	—
54	C₂²×C₇	—	24	2	no	—
55	C₄×C₇	—	2	2	no	—
56	C₄×C₇	—	3	1	no	—
57	C₄×C₇	C₂²×A₄	1	1	yes	characteristic
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	—	2	2	no	—
63	#41	—	3	2	no	—
64	#41	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
65	C₂×D₄	—	3	2	no	—
66	C₂×D₄	—	6	2	no	—
67	Q₁₆	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
68	SD₁₆	—	3	2	no	—
69	D₈	—	3	2	no	—
70	D₈	—	3	1	no	—
71	M₁₆	—	3	1	no	—
72	C₂×C₈	—	3	1	no	—
73	C₂×C₇	—	6	1	no	—
74	C₂×C₇	—	12	2	no	—
75	C₂×C₇	D₄×A₄	1	1	yes	characteristic
76	C₃×C₄	—	28	1	no	—
77	C₃×C₄	—	56	2	no	—
78	C₂³	—	6	2	no	—
79	Q₈	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
80	Q₈	C₂×C₂²⋊ΓL(1,𝔽₈)	1	2	yes	—
81	D₄	—	3	1	no	—
82	D₄	—	3	2	no	—
83	D₄	—	6	2	no	—
84	D₄	—	6	2	no	—
85	C₂×C₄	—	3	1	no	—
86	C₂×C₄	—	6	2	no	—
87	C₈	—	3	1	no	—
88	C₈	C₂×C₂²⋊ΓL(1,𝔽₈)	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	—	6	2	no	—
93	C₂²	—	24	2	no	—
94	C₄	—	2	2	no	—
95	C₄	—	3	1	no	—
96	C₄	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
97	C₃	—	28	1	no	Sylow
98	C₂	—	6	1	no	—
99	C₂	—	12	2	no	—
100	C₂	D₄×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10621615

C₂

yes

maximal, complemented

#131⋊ΓL(1,𝔽₈)

C₂

yes

maximal, characteristic, complemented

C₇×#576

C₃

yes

maximal, characteristic, complemented, Hall

#41⋊ΓL(1,𝔽₈)

—

#41⋊ΓL(1,𝔽₈)

C₂²

yes

characteristic

Q₁₆×ΓL(1,𝔽₈)

—

maximal

C₇×H(5,𝔽₂)

C₂×C₃

yes

—

C₇×AGL(1,ℤ/8)

—

C₇×#135

—

C₂×C₇×D₈

—

C₇×#131

C₂×C₃

yes

—

D₄

yes

characteristic

Q₈×ΓL(1,𝔽₈)

—

C₈×ΓL(1,𝔽₈)

—

C₇×#41

—

C₇×#41

—

C₇×#41

C₂²×C₃

yes

characteristic

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₇×Q₁₆

A₄

yes

characteristic

C₇×SD₁₆

—

C₇×D₈

—

C₇×D₈

—

C₇×M₁₆

—

C₂×C₇×C₈

—

—

—

—

—

yes

characteristic, complemented, Sylow

C₂³×C₇

—

C₇×Q₈

C₃×D₄

yes

characteristic

C₇×Q₈

C₂×A₄

yes

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₇×C₈

—

C₇×C₈

C₂×A₄

yes

characteristic

#41⋊C₃

—

#41⋊C₃

—

C₃×Q₁₆

—

C₂×ΓL(1,𝔽₈)

—

H(5,𝔽₂)

C₂×ΓL(1,𝔽₈)

yes

—

AGL(1,ℤ/8)

—

#135

—

C₂×D₈

—

#131

C₂×ΓL(1,𝔽₈)

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²×A₄

yes

characteristic

C₃×Q₈

—

SL(2,𝔽₃)

—

C₃×C₈

—

—

—

—

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₂×D₄

—

C₂×D₄

—

Q₁₆

C₂²⋊ΓL(1,𝔽₈)

yes

characteristic

SD₁₆

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₇

—

C₂×C₇

—

C₂×C₇

D₄×A₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₂³

—

Q₈

D₄×ΓL(1,𝔽₈)

yes

characteristic

Q₈

C₂×C₂²⋊ΓL(1,𝔽₈)

yes

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

C₂×C₂²⋊ΓL(1,𝔽₈)

yes

characteristic

C₇

#576⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂²

—

C₂²

—

C₂²

—

C₄

—

C₄

—

C₄

C₂²×C₂²⋊ΓL(1,𝔽₈)

yes

characteristic

C₃

—

Sylow

C₂

—

C₂

—

100

C₂

D₄×C₂²⋊ΓL(1,𝔽₈)

yes

characteristic

	subgroup	G/H
center	C₂	D₄×C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle	C₇×#41	C₂²×C₃
Frattini	C₄	C₂²×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#576	C₃
socle	C₂×C₇	D₄×A₄
hypercenter	Q₁₆	C₂²⋊ΓL(1,𝔽₈)
hypocenter	C₇×Q₈	C₃×D₄
LCS	G ▷ C₇×#41 ▷ C₇×Q₈
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

subgroup

G/H

center

C₂

D₄×C₂²⋊ΓL(1,𝔽₈)

commutator, cosocle

C₇×#41

C₂²×C₃

Frattini

C₄

C₂²×C₂²⋊ΓL(1,𝔽₈)

Fitting

C₇×#576

C₃

socle

C₂×C₇

D₄×A₄

hypercenter

Q₁₆

C₂²⋊ΓL(1,𝔽₈)

hypocenter

C₇×Q₈

C₃×D₄

LCS

G ▷ C₇×#41 ▷ C₇×Q₈

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

N	H ≅ G/N	type
1. #10621615	C₂	semidirect
2. #131⋊ΓL(1,𝔽₈)	C₂	semidirect
3. C₇×#576	C₃	semidirect
31. #576	ΓL(1,𝔽₈)	semidirect
89. C₇	#576⋊C₃	semidirect

H ≅ G/N

type

1. #10621615

C₂

semidirect

2. #131⋊ΓL(1,𝔽₈)

C₂

semidirect

3. C₇×#576

C₃

semidirect

31. #576

ΓL(1,𝔽₈)

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
#10621615	2
#131⋊ΓL(1,𝔽₈)	2
C₇×#576	3
Q₁₆×ΓL(1,𝔽₈)	4
#576⋊C₃	7

subgroup

index

Q₁₆×ΓL(1,𝔽₈)

Subgroups — #576⋊ΓL(1,𝔽₈) #13060794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #576⋊ΓL(1,𝔽8) #13060794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #576⋊ΓL(1,𝔽₈) #13060794