Group #13051758 — subgroups

Subgroups — #89630⋊C₃ #13051758

Order 1344.

172 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 72 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	#6453⋊C₃	C₂	1	2	yes	maximal
2	ΓL(1,𝔽₈)×#106	C₂	1	1	yes	maximal, characteristic
3	#6391⋊C₃	C₂	1	1	yes	maximal, characteristic
4	C₈×ΓL(1,𝔽₈)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#6381⋊C₃	C₂	1	2	yes	maximal
6	#89630	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
9	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	—	2	2	no	—
10	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
11	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	—	2	2	no	—
12	C₄×ΓL(1,𝔽₈)⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×ΓL(1,𝔽₈)⋊C₈	C₂²	1	1	yes	characteristic
14	#6453	C₂×C₃	1	2	yes	—
15	C₇×#106	C₂×C₃	1	1	yes	characteristic
16	#6391	C₂×C₃	1	1	yes	characteristic
17	C₈×Dic₇	C₂×C₃	1	1	yes	characteristic
18	#6381	C₂×C₃	1	2	yes	—
19	C₃×#499	—	7	1	no	maximal, Hall
20	C₂×C₄×ΓL(1,𝔽₈)	—	2	2	no	—
21	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
22	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
23	C₂×ΓL(1,𝔽₈)⋊C₄	D₄	1	2	yes	—
24	C₈×ΓL(1,𝔽₈)	Q₈	1	2	yes	—
25	ΓL(1,𝔽₈)⋊C₈	Q₈	1	2	yes	—
26	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
27	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
28	C₄⋊Dic₇	—	2	2	no	—
29	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
30	Dic₇⋊C₄	—	2	2	no	—
31	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
32	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
33	C₃×#125	—	7	2	no	—
34	C₃×#107	—	7	2	no	—
35	C₃×#106	—	7	1	no	—
36	C₃×#106	—	7	1	no	—
37	C₃×C₄×C₈	—	7	1	no	—
38	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
39	C₄×ΓL(1,𝔽₈)	—	4	2	no	—
40	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic
41	C₄×ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic
42	ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
43	ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
44	#499	—	7	1	no	Sylow
45	C₂×C₄×C₇	—	2	2	no	—
46	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
47	C₂×Dic₇	—	2	2	no	—
48	C₂×Dic₇	C₃×D₄	1	2	yes	—
49	C₇×C₈	C₃×Q₈	1	2	yes	—
50	C₇⋊C₈	C₃×Q₈	1	2	yes	—
51	C₂×C₃×C₈	—	7	1	no	—
52	C₂×C₃×C₈	—	7	1	no	—
53	C₃×C₄⋊C₄	—	7	2	no	—
54	C₃×C₄⋊C₄	—	7	2	no	—
55	C₃×C₄⋊C₄	—	14	2	no	—
56	C₃×C₄⋊C₄	—	14	2	no	—
57	C₃×C₄²	—	7	1	no	—
58	C₂×ΓL(1,𝔽₈)	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×ΓL(1,𝔽₈)	#135	1	2	yes	—
60	#125	—	7	2	no	—
61	#107	—	7	2	no	—
62	#106	—	7	1	no	—
63	#106	AGL(1,𝔽₇)	1	1	yes	characteristic
64	C₄×C₈	—	7	1	no	—
65	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
66	C₄×C₇	—	4	2	no	—
67	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
68	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
69	Dic₇	—	2	2	no	—
70	Dic₇	—	4	2	no	—
71	C₂×C₃×C₄	—	7	2	no	—
72	C₂×C₃×C₄	—	7	1	no	—
73	C₂×C₃×C₄	—	14	2	no	—
74	C₂×C₃×C₄	—	14	2	no	—
75	C₃×C₈	—	7	2	no	—
76	C₃×C₈	—	7	2	no	—
77	ΓL(1,𝔽₈)	#499	1	1	yes	characteristic, complemented, Hall
78	C₂×C₈	—	7	1	no	—
79	C₂×C₈	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
80	C₄⋊C₄	—	7	2	no	—
81	C₄⋊C₄	—	14	2	no	—
82	C₄⋊C₄	—	14	2	no	—
83	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
84	C₄²	—	7	1	no	—
85	C₂×C₇	C₃×C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₇	C₃×#135	1	2	yes	—
87	C₂²×C₃	—	7	1	no	—
88	C₃×C₄	—	7	1	no	—
89	C₃×C₄	—	7	1	no	—
90	C₃×C₄	—	14	2	no	—
91	C₃×C₄	—	28	2	no	—
92	C₃×C₄	—	28	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	7	2	no	—
95	C₂×C₄	—	14	2	no	—
96	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
97	C₈	—	7	2	no	—
98	C₈	ΓL(1,𝔽₈)⋊Q₈	1	2	yes	—
99	C₇	C₃×#499	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₃	—	7	1	no	—
101	C₂×C₃	—	7	2	no	—
102	C₂²	D₄×AGL(1,𝔽₇)	1	1	yes	characteristic
103	C₄	—	4	2	no	—
104	C₄	—	14	2	no	—
105	C₄	—	28	2	no	—
106	C₄	C₂×ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic
107	C₄	Q₈×AGL(1,𝔽₇)	1	1	yes	characteristic
108	C₃	—	7	1	no	Sylow
109	C₂	#6451⋊C₃	1	1	yes	characteristic
110	C₂	#6479⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#6453⋊C₃

C₂

yes

maximal

ΓL(1,𝔽₈)×#106

C₂

yes

maximal, characteristic

#6391⋊C₃

C₂

yes

maximal, characteristic

C₈×ΓL(1,𝔽₈)⋊C₄

C₂

yes

maximal, characteristic

#6381⋊C₃

C₂

yes

maximal

#89630

C₃

yes

maximal, characteristic, complemented, Hall

C₂×C₈×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

—

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

—

C₄×ΓL(1,𝔽₈)⋊C₄

C₂²

yes

characteristic

C₂×ΓL(1,𝔽₈)⋊C₈

C₂²

yes

characteristic

#6453

C₂×C₃

yes

—

C₇×#106

C₂×C₃

yes

characteristic

#6391

C₂×C₃

yes

characteristic

C₈×Dic₇

C₂×C₃

yes

characteristic

#6381

C₂×C₃

yes

—

C₃×#499

—

maximal, Hall

C₂×C₄×ΓL(1,𝔽₈)

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₂×ΓL(1,𝔽₈)⋊C₄

—

C₂×ΓL(1,𝔽₈)⋊C₄

D₄

yes

—

C₈×ΓL(1,𝔽₈)

Q₈

yes

—

ΓL(1,𝔽₈)⋊C₈

Q₈

yes

—

C₂×C₇×C₈

C₂²×C₃

yes

characteristic

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂²×C₃

yes

—

Dic₇⋊C₄

—

C₄×Dic₇

C₂²×C₃

yes

characteristic

C₂×C₇⋊C₈

C₂²×C₃

yes

—

—

—

—

C₃×C₄×C₈

—

C₂²×ΓL(1,𝔽₈)

C₂×D₄

yes

characteristic

C₄×ΓL(1,𝔽₈)

—

C₄×ΓL(1,𝔽₈)

C₂×Q₈

yes

characteristic

C₄×ΓL(1,𝔽₈)

C₂×Q₈

yes

—

—

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

C₃×D₄

yes

—

C₇×C₈

C₃×Q₈

yes

—

C₇⋊C₈

C₃×Q₈

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₂×ΓL(1,𝔽₈)

C₄⋊Q₈

yes

characteristic

C₂×ΓL(1,𝔽₈)

#135

yes

—

—

—

—

yes

C₄×C₈

—

C₂²×C₇

C₂×C₃×D₄

yes

characteristic

C₄×C₇

—

C₄×C₇

C₂×C₃×Q₈

yes

characteristic

C₄×C₇

C₂×C₃×Q₈

yes

characteristic

Dic₇

—

Dic₇

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

C₃×C₈

—

ΓL(1,𝔽₈)

#499

yes

characteristic, complemented, Hall

C₂×C₈

—

C₂×C₈

C₂×AGL(1,𝔽₇)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₇)

yes

—

C₄²

—

C₂×C₇

C₃×C₄⋊Q₈

yes

characteristic

C₂×C₇

C₃×#135

yes

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

C₈

—

C₈

ΓL(1,𝔽₈)⋊Q₈

yes

—

C₇

C₃×#499

yes

characteristic, complemented, Sylow

100

C₂×C₃

—

101

C₂×C₃

—

102

C₂²

D₄×AGL(1,𝔽₇)

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

C₂×ΓL(1,𝔽₈)⋊Q₈

yes

characteristic

107

C₄

Q₈×AGL(1,𝔽₇)

yes

characteristic

108

C₃

—

Sylow

109

C₂

#6451⋊C₃

yes

characteristic

110

C₂

#6479⋊C₃

yes

—

	subgroup	G/H
center	C₂²	D₄×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₇)
Fitting	C₇×#106	C₂×C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	C₂²×ΓL(1,𝔽₈)	C₂×D₄
hypercenter	#106	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#499
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×AGL(1,𝔽₇)

commutator, cosocle

C₂×C₄×C₇

C₂³×C₃

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₇)

Fitting

C₇×#106

C₂×C₃

socle

C₂²×C₇

C₂×C₃×D₄

prime radical

C₂²×ΓL(1,𝔽₈)

C₂×D₄

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
6. #89630	C₃	semidirect
77. ΓL(1,𝔽₈)	#499	semidirect
99. C₇	C₃×#499	semidirect

H ≅ G/N

type

6. #89630

C₃

semidirect

77. ΓL(1,𝔽₈)

#499

semidirect

99. C₇

C₃×#499

semidirect

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

subgroup	index
#6453⋊C₃	2
ΓL(1,𝔽₈)×#106	2
#6391⋊C₃	2
C₈×ΓL(1,𝔽₈)⋊C₄	2
#6381⋊C₃	2
#89630	3
C₃×#499	7

subgroup

index

C₈×ΓL(1,𝔽₈)⋊C₄

Subgroups — #89630⋊C₃ #13051758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89630⋊C3 #13051758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89630⋊C₃ #13051758