Group #72276538 — subgroups

Subgroups — ΓL(1,𝔽₈)×#1313 #72276538

Order 2688.

160 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 96 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	ΓL(1,𝔽₈)×#346	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)×#346	C₂	1	1	yes	maximal, characteristic
3	ΓL(1,𝔽₈)×#322	C₂	1	1	yes	maximal, characteristic
4	C₇×#1313	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×C₁₆×ΓL(1,𝔽₈)	C₄	1	2	yes	—
6	C₂×C₁₆×ΓL(1,𝔽₈)	C₄	1	2	yes	—
7	C₄×C₈×ΓL(1,𝔽₈)	C₄	1	2	yes	—
8	C₄×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
9	C₇×#346	C₂×C₃	1	1	yes	characteristic
10	C₇×#346	C₂×C₃	1	1	yes	characteristic
11	C₇×#322	C₂×C₃	1	1	yes	characteristic
12	C₃×#1313	—	7	1	no	maximal, Hall
13	C₂×C₈×ΓL(1,𝔽₈)	—	2	2	no	—
14	C₂×C₈×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
16	C₄²×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
17	C₁₆×ΓL(1,𝔽₈)	—	2	2	no	—
18	C₁₆×ΓL(1,𝔽₈)	C₈	1	4	yes	complemented
19	C₂×C₇×C₁₆	C₃×C₄	1	2	yes	—
20	C₂×C₇×C₁₆	C₃×C₄	1	2	yes	—
21	C₄×C₇×C₈	C₃×C₄	1	2	yes	—
22	C₄×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
23	C₃×#346	—	7	1	no	—
24	C₃×#346	—	7	1	no	—
25	C₃×#322	—	7	1	no	—
26	C₂×C₄×ΓL(1,𝔽₈)	C₄²	1	1	yes	characteristic
27	C₂×C₄×ΓL(1,𝔽₈)	M₁₆	1	2	yes	—
28	C₈×ΓL(1,𝔽₈)	—	4	2	no	—
29	C₈×ΓL(1,𝔽₈)	C₂×C₈	1	2	yes	—
30	C₈×ΓL(1,𝔽₈)	M₁₆	1	2	yes	—
31	#1313	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₇×C₈	—	2	2	no	—
33	C₂×C₇×C₈	C₂×C₃×C₄	1	1	yes	characteristic
34	C₂×C₇×C₈	C₂×C₃×C₄	1	1	yes	characteristic
35	C₄²×C₇	C₂×C₃×C₄	1	1	yes	characteristic
36	C₇×C₁₆	—	2	2	no	—
37	C₇×C₁₆	C₃×C₈	1	4	yes	complemented
38	C₂×C₃×C₁₆	—	7	2	no	—
39	C₂×C₃×C₁₆	—	7	2	no	—
40	C₃×C₄×C₈	—	7	2	no	—
41	C₃×C₄×C₈	—	7	1	no	—
42	C₂²×ΓL(1,𝔽₈)	#97	1	1	yes	characteristic
43	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
44	C₄×ΓL(1,𝔽₈)	C₄×C₈	1	1	yes	characteristic
45	C₄×ΓL(1,𝔽₈)	#97	1	1	yes	characteristic
46	#346	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
47	#346	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
48	#322	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
49	C₂×C₄×C₇	C₃×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₇	C₃×M₁₆	1	2	yes	—
51	C₇×C₈	—	4	2	no	—
52	C₇×C₈	C₂×C₃×C₈	1	2	yes	—
53	C₇×C₈	C₃×M₁₆	1	2	yes	—
54	C₂×C₃×C₈	—	7	1	no	—
55	C₂×C₃×C₈	—	7	1	no	—
56	C₂×C₃×C₈	—	14	2	no	—
57	C₃×C₄²	—	7	1	no	—
58	C₃×C₁₆	—	7	4	no	—
59	C₃×C₁₆	—	14	2	no	—
60	C₂×ΓL(1,𝔽₈)	#322	1	1	yes	characteristic
61	C₂×ΓL(1,𝔽₈)	#347	1	2	yes	—
62	C₂×C₁₆	C₄×ΓL(1,𝔽₈)	1	2	yes	—
63	C₂×C₁₆	C₄×ΓL(1,𝔽₈)	1	2	yes	—
64	C₄×C₈	C₄×ΓL(1,𝔽₈)	1	2	yes	—
65	C₄×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
66	C₂²×C₇	C₃×#97	1	1	yes	characteristic
67	C₄×C₇	—	2	2	no	—
68	C₄×C₇	C₃×C₄×C₈	1	1	yes	characteristic
69	C₄×C₇	C₃×#97	1	1	yes	characteristic
70	C₂×C₃×C₄	—	7	1	no	—
71	C₂×C₃×C₄	—	7	2	no	—
72	C₃×C₈	—	7	2	no	—
73	C₃×C₈	—	7	2	no	—
74	C₃×C₈	—	28	2	no	—
75	ΓL(1,𝔽₈)	#1313	1	1	yes	characteristic, direct product factor, Hall
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
78	C₂×C₈	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
79	C₄²	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
80	C₁₆	—	2	2	no	—
81	C₁₆	C₈×ΓL(1,𝔽₈)	1	4	yes	complemented
82	C₂×C₇	C₃×#322	1	1	yes	characteristic
83	C₂×C₇	C₃×#347	1	2	yes	—
84	C₂²×C₃	—	7	1	no	—
85	C₃×C₄	—	7	1	no	—
86	C₃×C₄	—	7	1	no	—
87	C₃×C₄	—	14	2	no	—
88	C₂×C₄	C₄²×ΓL(1,𝔽₈)	1	1	yes	characteristic
89	C₂×C₄	M₁₆×ΓL(1,𝔽₈)	1	2	yes	—
90	C₈	—	4	2	no	—
91	C₈	C₂×C₈×ΓL(1,𝔽₈)	1	2	yes	—
92	C₈	M₁₆×ΓL(1,𝔽₈)	1	2	yes	—
93	C₇	C₃×#1313	1	1	yes	characteristic, complemented, Sylow
94	C₂×C₃	—	7	1	no	—
95	C₂×C₃	—	7	2	no	—
96	C₂²	ΓL(1,𝔽₈)×#97	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	C₄×C₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
99	C₄	ΓL(1,𝔽₈)×#97	1	1	yes	characteristic
100	C₃	—	7	1	no	Sylow
101	C₂	ΓL(1,𝔽₈)×#322	1	1	yes	characteristic
102	C₂	ΓL(1,𝔽₈)×#347	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

ΓL(1,𝔽₈)×#346

C₂

yes

maximal, characteristic

ΓL(1,𝔽₈)×#346

C₂

yes

maximal, characteristic

ΓL(1,𝔽₈)×#322

C₂

yes

maximal, characteristic

C₇×#1313

C₃

yes

maximal, characteristic, complemented, Hall

C₂×C₁₆×ΓL(1,𝔽₈)

C₄

yes

—

C₂×C₁₆×ΓL(1,𝔽₈)

C₄

yes

—

C₄×C₈×ΓL(1,𝔽₈)

C₄

yes

—

C₄×C₈×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₇×#346

C₂×C₃

yes

characteristic

C₇×#346

C₂×C₃

yes

characteristic

C₇×#322

C₂×C₃

yes

characteristic

C₃×#1313

—

maximal, Hall

C₂×C₈×ΓL(1,𝔽₈)

—

C₂×C₈×ΓL(1,𝔽₈)

C₂×C₄

yes

characteristic

C₂×C₈×ΓL(1,𝔽₈)

C₂×C₄

yes

characteristic

C₄²×ΓL(1,𝔽₈)

C₂×C₄

yes

characteristic

C₁₆×ΓL(1,𝔽₈)

—

C₁₆×ΓL(1,𝔽₈)

C₈

yes

complemented

C₂×C₇×C₁₆

C₃×C₄

yes

—

C₂×C₇×C₁₆

C₃×C₄

yes

—

C₄×C₇×C₈

C₃×C₄

yes

—

C₄×C₇×C₈

C₂²×C₃

yes

—

—

—

C₂×C₄×ΓL(1,𝔽₈)

C₄²

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

M₁₆

yes

—

C₈×ΓL(1,𝔽₈)

—

C₈×ΓL(1,𝔽₈)

C₂×C₈

yes

—

C₈×ΓL(1,𝔽₈)

M₁₆

yes

—

#1313

ΓL(1,𝔽₈)

yes

characteristic, direct product factor, Sylow

C₂×C₇×C₈

—

C₂×C₇×C₈

C₂×C₃×C₄

yes

characteristic

C₂×C₇×C₈

C₂×C₃×C₄

yes

characteristic

C₄²×C₇

C₂×C₃×C₄

yes

characteristic

C₇×C₁₆

—

C₇×C₁₆

C₃×C₈

yes

complemented

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₂²×ΓL(1,𝔽₈)

yes

—

C₄×C₈

yes

yes

yes

yes

yes

C₂×C₄×C₇

C₃×C₄²

yes

characteristic

C₂×C₄×C₇

C₃×M₁₆

yes

—

C₇×C₈

—

C₇×C₈

C₂×C₃×C₈

yes

—

C₇×C₈

C₃×M₁₆

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₁₆

—

C₃×C₁₆

—

yes

yes

—

C₂×C₁₆

C₄×ΓL(1,𝔽₈)

yes

—

C₂×C₁₆

C₄×ΓL(1,𝔽₈)

yes

—

C₄×C₈

C₄×ΓL(1,𝔽₈)

yes

—

C₄×C₈

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₂²×C₇

C₃×#97

yes

characteristic

C₄×C₇

—

C₄×C₇

C₃×C₄×C₈

yes

characteristic

C₄×C₇

C₃×#97

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

ΓL(1,𝔽₈)

#1313

yes

characteristic, direct product factor, Hall

C₂×C₈

—

C₂×C₈

C₂×C₄×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₈

C₂×C₄×ΓL(1,𝔽₈)

yes

characteristic

C₄²

C₂×C₄×ΓL(1,𝔽₈)

yes

characteristic

C₁₆

—

C₁₆

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂×C₄

C₄²×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₄

M₁₆×ΓL(1,𝔽₈)

yes

—

C₈

—

C₈

C₂×C₈×ΓL(1,𝔽₈)

yes

—

C₈

M₁₆×ΓL(1,𝔽₈)

yes

—

C₇

C₃×#1313

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₂²

ΓL(1,𝔽₈)×#97

yes

characteristic

C₄

—

C₄

C₄×C₈×ΓL(1,𝔽₈)

yes

characteristic

C₄

ΓL(1,𝔽₈)×#97

yes

characteristic

100

C₃

—

Sylow

101

C₂

ΓL(1,𝔽₈)×#322

yes

characteristic

102

C₂

ΓL(1,𝔽₈)×#347

yes

—

	subgroup	G/H
center	C₂×C₄	C₄²×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₃×C₄×C₈
Frattini	C₄×C₈	C₂²×ΓL(1,𝔽₈)
Fitting	C₇×#1313	C₃
socle	C₂²×C₇	C₃×#97
prime radical	C₂²×ΓL(1,𝔽₈)	#97
cosocle	C₄×C₇×C₈	C₂²×C₃
hypercenter	#1313	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#1313
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #1313

subgroup

G/H

center

C₂×C₄

C₄²×ΓL(1,𝔽₈)

commutator

C₄×C₇

C₃×C₄×C₈

Frattini

C₄×C₈

C₂²×ΓL(1,𝔽₈)

Fitting

C₇×#1313

C₃

socle

C₂²×C₇

C₃×#97

prime radical

C₂²×ΓL(1,𝔽₈)

#97

cosocle

C₄×C₇×C₈

C₂²×C₃

C₇

G ▷ C₄×C₇ ▷ C₇

C₁ ◁ C₂×C₄ ◁ #1313

N	H ≅ G/N	type
4. C₇×#1313	C₃	semidirect
18. C₁₆×ΓL(1,𝔽₈)	C₈	semidirect
31. #1313	ΓL(1,𝔽₈)	direct (N × H)
37. C₇×C₁₆	C₃×C₈	semidirect
75. ΓL(1,𝔽₈)	#1313	direct (N × H)
81. C₁₆	C₈×ΓL(1,𝔽₈)	semidirect
93. C₇	C₃×#1313	semidirect

H ≅ G/N

type

4. C₇×#1313

C₃

semidirect

18. C₁₆×ΓL(1,𝔽₈)

C₈

semidirect

31. #1313

ΓL(1,𝔽₈)

direct (N × H)

37. C₇×C₁₆

C₃×C₈

semidirect

75. ΓL(1,𝔽₈)

#1313

direct (N × H)

81. C₁₆

C₈×ΓL(1,𝔽₈)

semidirect

93. C₇

C₃×#1313

semidirect

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

subgroup	index
ΓL(1,𝔽₈)×#346	2
ΓL(1,𝔽₈)×#322	2
C₇×#1313	3
C₃×#1313	7

subgroup

index

Subgroups — ΓL(1,𝔽₈)×#1313 #72276538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽8)×#1313 #72276538

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#1313 #72276538