Group #13054958 — subgroups

Subgroups — ΓL(1,𝔽₈)×#427 #13054958

Order 1344.

228 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂²×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)×#117	C₂	1	2	yes	maximal
3	ΓL(1,𝔽₈)×#107	C₂	1	2	yes	maximal, complemented
4	ΓL(1,𝔽₈)×#106	C₂	1	2	yes	maximal, complemented
5	C₇×#427	C₃	1	1	yes	maximal, characteristic, complemented, Hall
6	C₂²×C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₂×C₈×ΓL(1,𝔽₈)	C₄	1	4	yes	complemented
8	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
9	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
11	C₂²⋊C₄×ΓL(1,𝔽₈)	—	2	2	no	—
12	C₄²×ΓL(1,𝔽₈)	—	2	2	no	—
13	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
14	C₇×#117	C₂×C₃	1	2	yes	—
15	C₇×#107	C₂×C₃	1	2	yes	complemented
16	C₇×#106	C₂×C₃	1	2	yes	complemented
17	C₃×#427	—	7	1	no	maximal, Hall
18	C₂³×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
19	C₂×C₄×ΓL(1,𝔽₈)	—	2	4	no	—
20	C₂×C₄×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
21	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
22	C₂×C₄×ΓL(1,𝔽₈)	Q₈	1	2	yes	—
23	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
24	C₈×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
25	C₈×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
27	C₂×C₇×C₈	C₃×C₄	1	4	yes	complemented
28	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
29	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
30	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
31	C₇×C₂²⋊C₄	—	2	2	no	—
32	C₄²×C₇	—	2	2	no	—
33	C₂²×C₃×C₈	—	7	1	no	—
34	C₃×#117	—	7	2	no	—
35	C₃×#107	—	7	2	no	—
36	C₃×#106	—	7	2	no	—
37	C₂²×ΓL(1,𝔽₈)	—	2	2	no	—
38	C₂²×ΓL(1,𝔽₈)	C₄⋊C₄	1	2	yes	—
39	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
40	C₄×ΓL(1,𝔽₈)	—	4	4	no	—
41	C₄×ΓL(1,𝔽₈)	C₄⋊C₄	1	2	yes	—
42	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
43	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic
44	#427	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇	—	2	4	no	—
47	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
48	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₃×Q₈	1	2	yes	—
50	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
51	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
52	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
53	C₂²×C₃×C₄	—	7	1	no	—
54	C₂×C₃×C₈	—	7	4	no	—
55	C₂×C₃×C₈	—	7	1	no	—
56	C₂×C₃×C₈	—	7	1	no	—
57	C₃×C₄⋊C₄	—	7	4	no	—
58	C₃×C₂²⋊C₄	—	14	2	no	—
59	C₃×C₄²	—	14	2	no	—
60	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
61	C₂×ΓL(1,𝔽₈)	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×ΓL(1,𝔽₈)	#135	1	2	yes	—
63	C₂²×C₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
64	#117	C₂×ΓL(1,𝔽₈)	1	2	yes	—
65	#107	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
66	#106	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	C₃×C₄⋊C₄	1	2	yes	—
69	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
70	C₄×C₇	—	4	4	no	—
71	C₄×C₇	C₃×C₄⋊C₄	1	2	yes	—
72	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
73	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
74	C₂³×C₃	—	7	1	no	—
75	C₂×C₃×C₄	—	7	2	no	—
76	C₂×C₃×C₄	—	7	1	no	—
77	C₂×C₃×C₄	—	7	2	no	—
78	C₂×C₃×C₄	—	7	1	no	—
79	C₂×C₃×C₄	—	14	4	no	—
80	C₃×C₈	—	7	2	no	—
81	C₃×C₈	—	7	2	no	—
82	ΓL(1,𝔽₈)	#427	1	1	yes	characteristic, direct product factor, Hall
83	C₂²×C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
84	C₂×C₈	C₄×ΓL(1,𝔽₈)	1	4	yes	complemented
85	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
86	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
87	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	4	yes	—
88	C₂²⋊C₄	—	2	2	no	—
89	C₄²	—	2	2	no	—
90	C₂×C₇	—	2	2	no	—
91	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₇	C₃×#135	1	2	yes	—
93	C₂²×C₃	—	7	2	no	—
94	C₂²×C₃	—	7	1	no	—
95	C₂²×C₃	—	14	2	no	—
96	C₃×C₄	—	7	2	no	—
97	C₃×C₄	—	7	1	no	—
98	C₃×C₄	—	7	1	no	—
99	C₃×C₄	—	28	4	no	—
100	C₂³	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₂×C₄×ΓL(1,𝔽₈)	1	2	yes	—
103	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
104	C₂×C₄	Q₈×ΓL(1,𝔽₈)	1	2	yes	—
105	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
106	C₈	C₂×C₄×ΓL(1,𝔽₈)	1	2	yes	—
107	C₈	C₂×C₄×ΓL(1,𝔽₈)	1	2	yes	—
108	C₇	C₃×#427	1	1	yes	characteristic, complemented, Sylow
109	C₂×C₃	—	7	1	no	—
110	C₂×C₃	—	7	2	no	—
111	C₂×C₃	—	14	2	no	—
112	C₂²	—	2	2	no	—
113	C₂²	C₄⋊C₄×ΓL(1,𝔽₈)	1	2	yes	—
114	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
115	C₄	—	4	4	no	—
116	C₄	C₄⋊C₄×ΓL(1,𝔽₈)	1	2	yes	—
117	C₄	C₂²×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
118	C₄	C₂×Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic
119	C₃	—	7	1	no	Sylow
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
122	C₂	ΓL(1,𝔽₈)×#135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×ΓL(1,𝔽₈)

C₂

yes

maximal, characteristic

ΓL(1,𝔽₈)×#117

C₂

yes

maximal

ΓL(1,𝔽₈)×#107

C₂

yes

maximal, complemented

ΓL(1,𝔽₈)×#106

C₂

yes

maximal, complemented

C₇×#427

C₃

yes

maximal, characteristic, complemented, Hall

C₂²×C₄×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₂×C₈×ΓL(1,𝔽₈)

C₄

yes

complemented

C₂×C₈×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₂×C₈×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₂²⋊C₄×ΓL(1,𝔽₈)

—

C₄²×ΓL(1,𝔽₈)

—

C₂²×C₇×C₈

C₂×C₃

yes

characteristic

C₇×#117

C₂×C₃

yes

—

C₇×#107

C₂×C₃

yes

complemented

C₇×#106

C₂×C₃

yes

complemented

C₃×#427

—

maximal, Hall

C₂³×ΓL(1,𝔽₈)

D₄

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

—

C₂×C₄×ΓL(1,𝔽₈)

C₂×C₄

yes

—

C₂×C₄×ΓL(1,𝔽₈)

D₄

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

Q₈

yes

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₈×ΓL(1,𝔽₈)

C₂×C₄

yes

—

C₈×ΓL(1,𝔽₈)

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂²×C₃

yes

characteristic

C₂×C₇×C₈

C₃×C₄

yes

complemented

C₂×C₇×C₈

C₂²×C₃

yes

characteristic

C₂×C₇×C₈

C₂²×C₃

yes

characteristic

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₂²×C₃×C₈

—

C₃×#117

—

C₃×#107

—

C₃×#106

—

C₂²×ΓL(1,𝔽₈)

—

C₂²×ΓL(1,𝔽₈)

C₄⋊C₄

yes

—

C₂²×ΓL(1,𝔽₈)

C₂×D₄

yes

characteristic

C₄×ΓL(1,𝔽₈)

—

C₄×ΓL(1,𝔽₈)

C₄⋊C₄

yes

—

C₄×ΓL(1,𝔽₈)

C₂²×C₄

yes

characteristic

C₄×ΓL(1,𝔽₈)

C₂×Q₈

yes

characteristic

#427

ΓL(1,𝔽₈)

yes

characteristic, direct product factor, Sylow

C₂³×C₇

C₃×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₃×C₄

yes

—

C₂×C₄×C₇

C₃×D₄

yes

characteristic

C₂×C₄×C₇

C₃×Q₈

yes

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₇×C₈

C₂×C₃×C₄

yes

—

C₇×C₈

C₂×C₃×C₄

yes

—

C₂²×C₃×C₄

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂×ΓL(1,𝔽₈)

—

C₂×ΓL(1,𝔽₈)

C₂×C₄⋊C₄

yes

characteristic

C₂×ΓL(1,𝔽₈)

#135

yes

—

C₂²×C₈

yes

yes

—

yes

yes

C₂²×C₇

—

C₂²×C₇

C₃×C₄⋊C₄

yes

—

C₂²×C₇

C₂×C₃×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

C₃×C₄⋊C₄

yes

—

C₄×C₇

C₂²×C₃×C₄

yes

characteristic

C₄×C₇

C₂×C₃×Q₈

yes

characteristic

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

ΓL(1,𝔽₈)

#427

yes

characteristic, direct product factor, Hall

C₂²×C₄

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₈

C₄×ΓL(1,𝔽₈)

yes

complemented

C₂×C₈

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₈

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₄⋊C₄

C₂²×ΓL(1,𝔽₈)

yes

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

C₂×C₃×C₄⋊C₄

yes

characteristic

C₂×C₇

C₃×#135

yes

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

100

C₂³

D₄×ΓL(1,𝔽₈)

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

C₂×C₄×ΓL(1,𝔽₈)

yes

—

103

C₂×C₄

D₄×ΓL(1,𝔽₈)

yes

characteristic

104

C₂×C₄

Q₈×ΓL(1,𝔽₈)

yes

—

105

C₂×C₄

C₂³×ΓL(1,𝔽₈)

yes

characteristic

106

C₈

C₂×C₄×ΓL(1,𝔽₈)

yes

—

107

C₈

C₂×C₄×ΓL(1,𝔽₈)

yes

—

108

C₇

C₃×#427

yes

characteristic, complemented, Sylow

109

C₂×C₃

—

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂²

—

113

C₂²

C₄⋊C₄×ΓL(1,𝔽₈)

yes

—

114

C₂²

C₂×D₄×ΓL(1,𝔽₈)

yes

characteristic

115

C₄

—

116

C₄

C₄⋊C₄×ΓL(1,𝔽₈)

yes

—

117

C₄

C₂²×C₄×ΓL(1,𝔽₈)

yes

characteristic

118

C₄

C₂×Q₈×ΓL(1,𝔽₈)

yes

characteristic

119

C₃

—

Sylow

120

C₂

—

121

C₂

C₂×C₄⋊C₄×ΓL(1,𝔽₈)

yes

characteristic

122

C₂

ΓL(1,𝔽₈)×#135

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#427	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂³×ΓL(1,𝔽₈)	D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#427	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#427
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×ΓL(1,𝔽₈)

commutator

C₄×C₇

C₂²×C₃×C₄

Frattini

C₂×C₄

C₂³×ΓL(1,𝔽₈)

Fitting

C₇×#427

C₃

socle

C₂²×C₇

C₂×C₃×D₄

prime radical

C₂³×ΓL(1,𝔽₈)

D₄

cosocle

C₂×C₄×C₇

C₂³×C₃

C₇

G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
3. ΓL(1,𝔽₈)×#107	C₂	semidirect
4. ΓL(1,𝔽₈)×#106	C₂	semidirect
5. C₇×#427	C₃	semidirect
7. C₂×C₈×ΓL(1,𝔽₈)	C₄	semidirect
15. C₇×#107	C₂×C₃	semidirect
16. C₇×#106	C₂×C₃	semidirect
27. C₂×C₇×C₈	C₃×C₄	semidirect
44. #427	ΓL(1,𝔽₈)	direct (N × H)
65. #107	C₂×ΓL(1,𝔽₈)	semidirect
66. #106	C₂×ΓL(1,𝔽₈)	semidirect
82. ΓL(1,𝔽₈)	#427	direct (N × H)
84. C₂×C₈	C₄×ΓL(1,𝔽₈)	semidirect
108. C₇	C₃×#427	semidirect

H ≅ G/N

type

3. ΓL(1,𝔽₈)×#107

C₂

semidirect

4. ΓL(1,𝔽₈)×#106

C₂

semidirect

5. C₇×#427

C₃

semidirect

7. C₂×C₈×ΓL(1,𝔽₈)

C₄

semidirect

15. C₇×#107

C₂×C₃

semidirect

16. C₇×#106

C₂×C₃

semidirect

27. C₂×C₇×C₈

C₃×C₄

semidirect

44. #427

ΓL(1,𝔽₈)

direct (N × H)

65. #107

C₂×ΓL(1,𝔽₈)

semidirect

66. #106

C₂×ΓL(1,𝔽₈)

semidirect

82. ΓL(1,𝔽₈)

#427

direct (N × H)

84. C₂×C₈

C₄×ΓL(1,𝔽₈)

semidirect

108. C₇

C₃×#427

semidirect

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

subgroup	index
C₂²×C₈×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×#117	2
ΓL(1,𝔽₈)×#107	2
ΓL(1,𝔽₈)×#106	2
C₇×#427	3
C₃×#427	7

subgroup

index

C₂²×C₈×ΓL(1,𝔽₈)

Subgroups — ΓL(1,𝔽₈)×#427 #13054958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽8)×#427 #13054958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#427 #13054958