Group #13054972 — subgroups

Subgroups — ΓL(1,𝔽₈)×#441 #13054972

Order 1344.

216 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×Q₁₆×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal
3	ΓL(1,𝔽₈)×#106	C₂	1	1	yes	maximal, characteristic
4	ΓL(1,𝔽₈)×#103	C₂	1	2	yes	maximal
5	ΓL(1,𝔽₈)×#97	C₂	1	1	yes	maximal, characteristic
6	C₇×#441	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂×Q₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
8	Q₁₆×ΓL(1,𝔽₈)	C₄	1	4	yes	complemented
9	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
10	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×ΓL(1,𝔽₈)	—	2	2	no	—
12	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
13	C₄²×ΓL(1,𝔽₈)	—	2	2	no	—
14	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
15	C₂×C₇×Q₁₆	C₂×C₃	1	1	yes	characteristic
16	C₄×C₇×Q₈	C₂×C₃	1	2	yes	—
17	C₇×#106	C₂×C₃	1	1	yes	characteristic
18	C₇×#103	C₂×C₃	1	2	yes	—
19	C₇×#97	C₂×C₃	1	1	yes	characteristic
20	C₃×#441	—	7	1	no	maximal, Hall
21	Q₈×ΓL(1,𝔽₈)	—	2	2	no	—
22	Q₈×ΓL(1,𝔽₈)	C₂×C₄	1	4	yes	—
23	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
24	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
25	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	2	yes	—
26	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
27	C₈×ΓL(1,𝔽₈)	—	2	1	no	—
28	C₈×ΓL(1,𝔽₈)	C₂×C₄	1	2	yes	—
29	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
30	C₇×Q₁₆	C₃×C₄	1	4	yes	complemented
31	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
32	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
33	C₇×C₄⋊C₄	—	2	2	no	—
34	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
35	C₄²×C₇	—	2	2	no	—
36	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
37	C₂×C₃×Q₁₆	—	7	1	no	—
38	C₃×C₄×Q₈	—	7	2	no	—
39	C₃×#106	—	7	1	no	—
40	C₃×#103	—	7	2	no	—
41	C₃×#97	—	7	1	no	—
42	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
43	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
44	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
45	C₄×ΓL(1,𝔽₈)	—	4	2	no	—
46	C₄×ΓL(1,𝔽₈)	C₂²×C₄	1	1	yes	characteristic
47	C₄×ΓL(1,𝔽₈)	#41	1	1	yes	characteristic
48	#441	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
49	C₇×Q₈	—	2	2	no	—
50	C₇×Q₈	C₂×C₃×C₄	1	4	yes	—
51	C₂×C₄×C₇	—	2	2	no	—
52	C₂×C₄×C₇	—	2	2	no	—
53	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
54	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
55	C₇×C₈	—	2	1	no	—
56	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
57	C₂×C₃×Q₈	—	7	2	no	—
58	C₃×Q₁₆	—	7	4	no	—
59	C₂×C₃×C₈	—	7	1	no	—
60	C₂×C₃×C₈	—	7	1	no	—
61	C₃×C₄⋊C₄	—	7	2	no	—
62	C₃×C₄⋊C₄	—	14	2	no	—
63	C₃×C₄²	—	7	1	no	—
64	C₃×C₄²	—	14	2	no	—
65	C₂×ΓL(1,𝔽₈)	C₄×D₄	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₈)	#137	1	2	yes	—
67	C₂×Q₁₆	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
68	C₄×Q₈	C₂×ΓL(1,𝔽₈)	1	2	yes	—
69	#106	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
70	#103	C₂×ΓL(1,𝔽₈)	1	2	yes	—
71	#97	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
72	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
73	C₄×C₇	—	2	4	no	—
74	C₄×C₇	—	2	2	no	—
75	C₄×C₇	—	4	2	no	—
76	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
77	C₄×C₇	C₃×#41	1	1	yes	characteristic
78	C₃×Q₈	—	7	4	no	—
79	C₃×Q₈	—	14	2	no	—
80	C₂×C₃×C₄	—	7	2	no	—
81	C₂×C₃×C₄	—	7	1	no	—
82	C₂×C₃×C₄	—	14	2	no	—
83	C₂×C₃×C₄	—	14	2	no	—
84	C₃×C₈	—	7	2	no	—
85	C₃×C₈	—	14	1	no	—
86	ΓL(1,𝔽₈)	#441	1	1	yes	characteristic, direct product factor, Hall
87	C₂×Q₈	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
88	Q₁₆	C₄×ΓL(1,𝔽₈)	1	4	yes	complemented
89	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
90	C₂×C₈	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
93	C₄²	—	2	2	no	—
94	C₄²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
95	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
96	C₂×C₇	C₃×#137	1	2	yes	—
97	C₂²×C₃	—	7	1	no	—
98	C₃×C₄	—	7	1	no	—
99	C₃×C₄	—	7	1	no	—
100	C₃×C₄	—	14	2	no	—
101	C₃×C₄	—	14	4	no	—
102	C₃×C₄	—	28	2	no	—
103	Q₈	—	2	2	no	—
104	Q₈	C₂×C₄×ΓL(1,𝔽₈)	1	4	yes	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	D₄×ΓL(1,𝔽₈)	1	2	yes	—
108	C₂×C₄	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	C₂×C₄×ΓL(1,𝔽₈)	1	2	yes	—
111	C₇	C₃×#441	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₃	—	7	1	no	—
113	C₂×C₃	—	7	2	no	—
114	C₂²	C₂×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
115	C₄	—	2	4	no	—
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	C₂²×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
119	C₄	#41×ΓL(1,𝔽₈)	1	1	yes	characteristic
120	C₃	—	7	1	no	Sylow
121	C₂	C₄×D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
122	C₂	ΓL(1,𝔽₈)×#137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×ΓL(1,𝔽₈)

C₂

yes

maximal, characteristic

C₄×Q₈×ΓL(1,𝔽₈)

C₂

yes

maximal

ΓL(1,𝔽₈)×#106

C₂

yes

maximal, characteristic

ΓL(1,𝔽₈)×#103

C₂

yes

maximal

ΓL(1,𝔽₈)×#97

C₂

yes

maximal, characteristic

C₇×#441

C₃

yes

maximal, characteristic, complemented, Hall

C₂×Q₈×ΓL(1,𝔽₈)

C₂²

yes

—

Q₁₆×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)

—

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄²×ΓL(1,𝔽₈)

—

C₄²×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₂×C₇×Q₁₆

C₂×C₃

yes

characteristic

C₄×C₇×Q₈

C₂×C₃

yes

—

C₇×#106

C₂×C₃

yes

characteristic

C₇×#103

C₂×C₃

yes

—

C₇×#97

C₂×C₃

yes

—

—

C₂×C₄

yes

—

C₂×C₄×ΓL(1,𝔽₈)

—

C₂×C₄×ΓL(1,𝔽₈)

—

C₂×C₄×ΓL(1,𝔽₈)

D₄

yes

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₈×ΓL(1,𝔽₈)

—

C₈×ΓL(1,𝔽₈)

C₂×C₄

yes

—

C₂×C₇×Q₈

C₂²×C₃

yes

—

C₇×Q₁₆

C₃×C₄

yes

complemented

C₂×C₇×C₈

C₂²×C₃

yes

characteristic

C₂×C₇×C₈

C₂²×C₃

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₇

—

C₄²×C₇

C₂²×C₃

yes

characteristic

C₂×C₃×Q₁₆

—

C₃×C₄×Q₈

—

C₃×#106

—

C₃×#103

—

C₃×#97

—

C₂²×ΓL(1,𝔽₈)

C₂×D₄

yes

—

—

—

C₂²×C₄

yes

yes

yes

characteristic, direct product factor, Sylow

C₇×Q₈

—

C₇×Q₈

C₂×C₃×C₄

yes

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₃×D₄

yes

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₇×C₈

—

C₇×C₈

C₂×C₃×C₄

yes

—

C₂×C₃×Q₈

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₂×ΓL(1,𝔽₈)

C₄×D₄

yes

characteristic

C₂×ΓL(1,𝔽₈)

#137

yes

—

C₂×Q₁₆

C₂×ΓL(1,𝔽₈)

yes

characteristic

C₄×Q₈

C₂×ΓL(1,𝔽₈)

yes

—

yes

yes

—

#97

C₂×ΓL(1,𝔽₈)

yes

characteristic

C₂²×C₇

C₂×C₃×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²×C₃×C₄

yes

characteristic

C₄×C₇

C₃×#41

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

ΓL(1,𝔽₈)

#441

yes

characteristic, direct product factor, Hall

C₂×Q₈

C₂²×ΓL(1,𝔽₈)

yes

—

Q₁₆

C₄×ΓL(1,𝔽₈)

yes

complemented

C₂×C₈

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₈

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂²×ΓL(1,𝔽₈)

yes

—

C₄²

—

C₄²

C₂²×ΓL(1,𝔽₈)

yes

characteristic

C₂×C₇

C₃×C₄×D₄

yes

characteristic

C₂×C₇

C₃×#137

yes

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

Q₈

—

104

Q₈

C₂×C₄×ΓL(1,𝔽₈)

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×ΓL(1,𝔽₈)

yes

—

108

C₂×C₄

C₂³×ΓL(1,𝔽₈)

yes

characteristic

109

C₈

—

110

C₈

C₂×C₄×ΓL(1,𝔽₈)

yes

—

111

C₇

C₃×#441

yes

characteristic, complemented, Sylow

112

C₂×C₃

—

113

C₂×C₃

—

114

C₂²

C₂×D₄×ΓL(1,𝔽₈)

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂²×C₄×ΓL(1,𝔽₈)

yes

characteristic

119

C₄

#41×ΓL(1,𝔽₈)

yes

characteristic

120

C₃

—

Sylow

121

C₂

C₄×D₄×ΓL(1,𝔽₈)

yes

characteristic

122

C₂

ΓL(1,𝔽₈)×#137

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×ΓL(1,𝔽₈)
commutator	C₄×C₇	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#441	C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	#441	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#441
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

subgroup

G/H

center

C₂²

C₂×D₄×ΓL(1,𝔽₈)

commutator

C₄×C₇

C₂²×C₃×C₄

Frattini

C₂×C₄

C₂³×ΓL(1,𝔽₈)

Fitting

C₇×#441

C₃

socle

C₂²×C₇

C₂×C₃×D₄

prime radical

C₂²×ΓL(1,𝔽₈)

C₂×D₄

cosocle

C₂×C₄×C₇

C₂³×C₃

C₇

G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #441

N	H ≅ G/N	type
6. C₇×#441	C₃	semidirect
8. Q₁₆×ΓL(1,𝔽₈)	C₄	semidirect
30. C₇×Q₁₆	C₃×C₄	semidirect
48. #441	ΓL(1,𝔽₈)	direct (N × H)
86. ΓL(1,𝔽₈)	#441	direct (N × H)
88. Q₁₆	C₄×ΓL(1,𝔽₈)	semidirect
111. C₇	C₃×#441	semidirect

H ≅ G/N

type

6. C₇×#441

C₃

semidirect

8. Q₁₆×ΓL(1,𝔽₈)

C₄

semidirect

30. C₇×Q₁₆

C₃×C₄

semidirect

48. #441

ΓL(1,𝔽₈)

direct (N × H)

86. ΓL(1,𝔽₈)

#441

direct (N × H)

88. Q₁₆

C₄×ΓL(1,𝔽₈)

semidirect

111. C₇

C₃×#441

semidirect

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

subgroup	index
C₂×Q₁₆×ΓL(1,𝔽₈)	2
C₄×Q₈×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×#106	2
ΓL(1,𝔽₈)×#103	2
ΓL(1,𝔽₈)×#97	2
C₇×#441	3
C₃×#441	7

subgroup

index

C₂×Q₁₆×ΓL(1,𝔽₈)

C₄×Q₈×ΓL(1,𝔽₈)

Subgroups — ΓL(1,𝔽₈)×#441 #13054972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽8)×#441 #13054972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#441 #13054972