Group #13051614 — subgroups

Subgroups — #89485⋊C₃ #13051614

Order 1344.

220 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 64 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	#10620710	C₂	1	1	yes	maximal, characteristic
2	#6433⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395⋊C₃	C₂	1	4	yes	maximal, complemented
5	#89485	C₃	1	1	yes	maximal, characteristic, complemented, Hall
6	C₂×C₄⋊AGL(1,𝔽₇)	—	2	2	no	—
7	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	2	yes	—
8	C₂×C₈×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
9	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(ΓL(1,𝔽₈)⋊C₄)	C₂²	1	2	yes	—
11	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	—	2	2	no	—
12	#6437	C₂×C₃	1	1	yes	characteristic
13	#6433	C₂×C₃	1	1	yes	characteristic, complemented
14	C₇×#97	C₂×C₃	1	1	yes	characteristic, complemented
15	#6395	C₂×C₃	1	4	yes	complemented
16	C₃×#490	—	7	1	no	maximal, Hall
17	C₂²×AGL(1,𝔽₇)	—	2	2	no	—
18	C₂×C₄×ΓL(1,𝔽₈)	D₄	1	2	yes	—
19	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
20	C₂×ΓL(1,𝔽₈)⋊C₄	—	2	2	no	—
21	C₄⋊AGL(1,𝔽₇)	—	2	4	no	—
22	C₄⋊AGL(1,𝔽₇)	—	4	4	no	—
23	C₈×ΓL(1,𝔽₈)	—	2	2	no	—
24	C₂×D₂₈	—	2	2	no	—
25	C₂×D₂₈	C₂²×C₃	1	2	yes	—
26	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
27	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
28	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
29	Dic₇⋊C₄	—	2	2	no	—
30	C₃×#127	—	7	1	no	—
31	C₃×#125	—	7	1	no	—
32	C₃×#102	—	7	4	no	—
33	C₃×#97	—	7	1	no	—
34	C₂²×ΓL(1,𝔽₈)	C₂×D₄	1	1	yes	characteristic
35	C₂×AGL(1,𝔽₇)	—	4	2	no	—
36	C₂×AGL(1,𝔽₇)	—	4	4	no	—
37	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
38	C₄×ΓL(1,𝔽₈)	#41	1	2	yes	—
39	ΓL(1,𝔽₈)⋊C₄	—	4	2	no	—
40	#490	—	7	1	no	Sylow
41	C₂²×D₇	—	2	2	no	—
42	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
43	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
44	C₂×Dic₇	—	2	2	no	—
45	D₂₈	—	2	4	no	—
46	D₂₈	—	4	4	no	—
47	C₇×C₈	—	2	2	no	—
48	C₂×C₃×D₄	—	7	2	no	—
49	C₂×C₃×D₄	—	14	2	no	—
50	C₂×C₃×C₈	—	7	2	no	—
51	C₃×C₄⋊C₄	—	7	2	no	—
52	C₃×C₄⋊C₄	—	14	2	no	—
53	C₃×C₄²	—	7	1	no	—
54	C₂×ΓL(1,𝔽₈)	#124	1	1	yes	characteristic
55	C₂×ΓL(1,𝔽₈)	AGL(1,ℤ/8)	1	2	yes	—
56	AGL(1,𝔽₇)	—	8	2	no	—
57	#127	—	7	1	no	—
58	#125	—	7	1	no	—
59	#102	—	7	4	no	—
60	#97	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
61	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
62	C₂×D₇	—	4	2	no	—
63	C₂×D₇	—	4	4	no	—
64	C₄×C₇	—	2	2	no	—
65	C₄×C₇	C₃×#41	1	2	yes	—
66	Dic₇	—	4	2	no	—
67	C₂³×C₃	—	14	2	no	—
68	C₃×D₄	—	14	4	no	—
69	C₃×D₄	—	28	4	no	—
70	C₂×C₃×C₄	—	7	2	no	—
71	C₂×C₃×C₄	—	7	1	no	—
72	C₂×C₃×C₄	—	14	2	no	—
73	C₃×C₈	—	14	2	no	—
74	ΓL(1,𝔽₈)	#490	1	1	yes	characteristic, complemented, Hall
75	C₂×D₄	—	7	2	no	—
76	C₂×D₄	—	14	2	no	—
77	C₂×C₈	C₂×AGL(1,𝔽₇)	1	2	yes	—
78	C₄⋊C₄	—	7	2	no	—
79	C₄⋊C₄	—	14	2	no	—
80	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
81	C₂×C₇	C₃×#124	1	1	yes	characteristic
82	C₂×C₇	C₃×AGL(1,ℤ/8)	1	2	yes	—
83	D₇	—	8	2	no	—
84	C₂²×C₃	—	7	1	no	—
85	C₂²×C₃	—	28	2	no	—
86	C₂²×C₃	—	28	4	no	—
87	C₃×C₄	—	7	2	no	—
88	C₃×C₄	—	14	2	no	—
89	C₃×C₄	—	28	2	no	—
90	C₂³	—	14	2	no	—
91	D₄	—	14	4	no	—
92	D₄	—	28	4	no	—
93	C₂×C₄	—	14	2	no	—
94	C₂×C₄	C₄⋊AGL(1,𝔽₇)	1	2	yes	—
95	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₇	C₃×#490	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₃	—	7	1	no	—
99	C₂×C₃	—	7	2	no	—
100	C₂×C₃	—	56	2	no	—
101	C₂²	—	28	2	no	—
102	C₂²	—	28	4	no	—
103	C₂²	C₂×C₄⋊AGL(1,𝔽₇)	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	28	2	no	—
106	C₄	#1157⋊C₃	1	2	yes	—
107	C₃	—	7	1	no	Sylow
108	C₂	—	56	2	no	—
109	C₂	#6438⋊C₃	1	1	yes	characteristic
110	C₂	#6471⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10620710

C₂

yes

maximal, characteristic

#6433⋊C₃

C₂

yes

maximal, characteristic, complemented

ΓL(1,𝔽₈)×#97

C₂

yes

maximal, characteristic, complemented

#6395⋊C₃

C₂

yes

maximal, complemented

#89485

C₃

yes

maximal, characteristic, complemented, Hall

C₂×C₄⋊AGL(1,𝔽₇)

—

C₂×C₄⋊AGL(1,𝔽₇)

C₂²

yes

—

C₂×C₈×ΓL(1,𝔽₈)

C₂²

yes

—

C₄²×ΓL(1,𝔽₈)

C₂²

yes

characteristic

C₄⋊(ΓL(1,𝔽₈)⋊C₄)

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

—

#6437

C₂×C₃

yes

characteristic

#6433

C₂×C₃

yes

characteristic, complemented

C₇×#97

C₂×C₃

yes

characteristic, complemented

#6395

C₂×C₃

yes

complemented

C₃×#490

—

maximal, Hall

C₂²×AGL(1,𝔽₇)

—

C₂×C₄×ΓL(1,𝔽₈)

D₄

yes

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₂×ΓL(1,𝔽₈)⋊C₄

—

C₄⋊AGL(1,𝔽₇)

—

C₄⋊AGL(1,𝔽₇)

—

C₈×ΓL(1,𝔽₈)

—

C₂×D₂₈

—

C₂×D₂₈

C₂²×C₃

yes

—

C₂×C₇×C₈

C₂²×C₃

yes

—

C₄²×C₇

C₂²×C₃

yes

characteristic

C₄⋊Dic₇

C₂²×C₃

yes

—

Dic₇⋊C₄

—

C₃×#127

—

C₃×#125

—

C₃×#102

—

C₃×#97

—

C₂²×ΓL(1,𝔽₈)

C₂×D₄

yes

—

—

—

yes

—

ΓL(1,𝔽₈)⋊C₄

—

#490

—

Sylow

C₂²×D₇

—

C₂×C₄×C₇

C₃×D₄

yes

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₂×Dic₇

—

D₂₈

—

D₂₈

—

C₇×C₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

yes

yes

—

—

—

—

—

yes

characteristic, complemented

C₂²×C₇

C₂×C₃×D₄

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

C₃×#41

yes

—

Dic₇

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₃×C₈

—

ΓL(1,𝔽₈)

#490

yes

characteristic, complemented, Hall

C₂×D₄

—

C₂×D₄

—

C₂×C₈

C₂×AGL(1,𝔽₇)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

yes

C₂×C₇

yes

C₂×C₇

yes

—

D₇

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₄⋊AGL(1,𝔽₇)

yes

—

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

C₈

—

C₇

C₃×#490

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

100

C₂×C₃

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×C₄⋊AGL(1,𝔽₇)

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

#1157⋊C₃

yes

—

107

C₃

—

Sylow

108

C₂

—

109

C₂

#6438⋊C₃

yes

characteristic

110

C₂

#6471⋊C₃

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₇)
commutator, cosocle	C₂×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₇)
Fitting	C₇×#97	C₂×C₃
socle	C₂²×C₇	C₂×C₃×D₄
prime radical	#10620710	C₂
hypercenter	#97	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#490
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,𝔽₇)

commutator, cosocle

C₂×C₄×C₇

C₂³×C₃

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₇)

Fitting

C₇×#97

C₂×C₃

socle

C₂²×C₇

C₂×C₃×D₄

C₂

C₇

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
2. #6433⋊C₃	C₂	semidirect
3. ΓL(1,𝔽₈)×#97	C₂	semidirect
4. #6395⋊C₃	C₂	semidirect
5. #89485	C₃	semidirect
13. #6433	C₂×C₃	semidirect
14. C₇×#97	C₂×C₃	semidirect
15. #6395	C₂×C₃	semidirect
60. #97	AGL(1,𝔽₇)	semidirect
74. ΓL(1,𝔽₈)	#490	semidirect
97. C₇	C₃×#490	semidirect

H ≅ G/N

type

2. #6433⋊C₃

C₂

semidirect

3. ΓL(1,𝔽₈)×#97

C₂

semidirect

4. #6395⋊C₃

C₂

semidirect

5. #89485

C₃

semidirect

13. #6433

C₂×C₃

semidirect

14. C₇×#97

C₂×C₃

semidirect

15. #6395

C₂×C₃

semidirect

60. #97

AGL(1,𝔽₇)

semidirect

74. ΓL(1,𝔽₈)

#490

semidirect

97. C₇

C₃×#490

semidirect

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

subgroup	index
#10620710	2
#6433⋊C₃	2
ΓL(1,𝔽₈)×#97	2
#6395⋊C₃	2
#89485	3
C₃×#490	7

subgroup

index

Subgroups — #89485⋊C₃ #13051614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89485⋊C3 #13051614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #89485⋊C₃ #13051614