Group #50303932 — subgroups

Subgroups — #50303932

Order 38220.

104 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 28 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	D₆₅×#3862	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#63121	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₃×#3862	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439633	—	5	1	no	maximal, Hall
5	C₇²×D₆₅	C₂×C₃	1	1	yes	characteristic
6	ΓL(1,𝔽₈)×#63121	—	7	1	no	maximal
7	ΓL(1,𝔽₈)⋊#63121	—	7	1	no	maximal
8	D₁₃×#3862	—	5	1	no	—
9	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	#15668768	—	13	1	no	maximal, Hall
11	ΓL(1,𝔽₈)×D₆₅	—	7	1	no	—
12	ΓL(1,𝔽₈)×D₆₅	—	7	1	no	—
13	C₇×C₇⋊#261	—	5	1	no	Hall
14	C₁₃×#3862	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
15	C₇⋊#63121	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	C₇×#63121	—	7	1	no	—
17	D₅×#3862	—	13	1	no	—
18	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
19	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
20	C₇²×D₁₃	—	5	1	no	—
21	ΓL(1,𝔽₈)⋊#261	—	35	1	no	—
22	ΓL(1,𝔽₈)×#261	—	35	1	no	—
23	C₇×C₇⋊AGL(1,𝔽₅)	—	13	1	no	Hall
24	C₇×D₆₅	—	6	1	no	—
25	C₇×D₆₅	AGL(1,𝔽₇)	1	1	yes	characteristic
26	C₇×D₆₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
27	C₃×#63121	—	49	1	no	Hall
28	C₅×#3862	#261	1	1	yes	characteristic, complemented, Hall
29	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
30	#10596872	—	65	1	no	Hall
31	ΓL(1,𝔽₈)×D₁₃	—	35	1	no	—
32	ΓL(1,𝔽₈)×D₁₃	—	35	1	no	—
33	C₇²×D₅	—	13	1	no	—
34	C₅×C₇×C₁₃	—	6	1	no	—
35	C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
36	C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
37	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	—	91	1	no	—
38	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	91	1	no	—
39	C₃×D₆₅	—	49	1	no	—
40	C₇⋊#261	—	5	1	no	—
41	C₇×#261	—	35	1	no	—
42	C₂×#3862	—	65	1	no	—
43	C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
44	C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
45	#63121	—	7	1	no	Hall
46	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
47	D₅×ΓL(1,𝔽₈)	—	91	1	no	—
48	D₅×ΓL(1,𝔽₈)	—	91	1	no	—
49	C₇×Dic₇	—	65	1	no	Hall
50	C₃×C₅×C₁₃	—	49	1	no	Hall
51	C₇×D₁₃	—	5	1	no	—
52	C₇×D₁₃	—	5	1	no	—
53	C₇×D₁₃	—	30	1	no	—
54	C₃×#261	—	245	1	no	Hall
55	#3862	#63121	1	1	yes	characteristic, complemented, Hall
56	C₇⋊AGL(1,𝔽₅)	—	13	1	no	—
57	C₇×AGL(1,𝔽₅)	—	91	1	no	—
58	D₆₅	#64422	1	1	yes	characteristic
59	C₅×ΓL(1,𝔽₈)	—	7	1	no	—
60	C₅×ΓL(1,𝔽₈)	—	7	1	no	—
61	C₂×C₇²	—	65	1	no	—
62	C₇×C₁₃	—	6	1	no	—
63	C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
64	C₇×C₁₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
65	C₄×ΓL(1,𝔽₈)	—	455	1	no	—
66	ΓL(1,𝔽₈)⋊C₄	—	455	1	no	—
67	C₃×D₁₃	—	245	1	no	—
68	C₇×D₅	—	13	1	no	—
69	C₇×D₅	—	13	1	no	—
70	C₇×D₅	—	78	1	no	—
71	C₅×C₁₃	#10596872	1	1	yes	characteristic, complemented, Hall
72	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
73	#261	—	35	1	no	Hall
74	C₇²	C₃×#63121	1	1	yes	characteristic, complemented, Sylow
75	C₂×ΓL(1,𝔽₈)	—	455	1	no	—
76	C₂×ΓL(1,𝔽₈)	—	455	1	no	—
77	C₃×C₁₃	—	49	1	no	Hall
78	C₅×C₇	—	6	1	no	—
79	C₅×C₇	ΓL(1,𝔽₈)×#261	1	1	yes	characteristic, complemented
80	C₅×C₇	ΓL(1,𝔽₈)⋊#261	1	1	yes	characteristic, complemented
81	C₃×D₅	—	637	1	no	—
82	C₄×C₇	—	455	1	no	—
83	Dic₇	—	65	1	no	—
84	D₁₃	—	5	1	no	—
85	ΓL(1,𝔽₈)	—	7	1	no	—
86	ΓL(1,𝔽₈)	—	7	1	no	—
87	AGL(1,𝔽₅)	—	91	1	no	Hall
88	C₃×C₅	—	49	1	no	Hall
89	C₂×C₇	—	65	1	no	—
90	C₂×C₇	—	65	1	no	—
91	C₂×C₇	—	390	1	no	—
92	C₁₃	#15668768	1	1	yes	characteristic, complemented, Sylow
93	C₃×C₄	—	3185	1	no	Hall
94	D₅	—	13	1	no	—
95	C₇	—	6	1	no	—
96	C₇	ΓL(1,𝔽₈)⋊#63121	1	1	yes	characteristic, complemented
97	C₇	ΓL(1,𝔽₈)×#63121	1	1	yes	characteristic, complemented
98	C₂×C₃	—	3185	1	no	—
99	C₅	#21439633	1	1	yes	characteristic, complemented, Sylow
100	C₄	—	455	1	no	Sylow
101	C₃	—	49	1	no	Sylow
102	C₂	—	65	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₆₅×#3862

C₂

yes

maximal, characteristic

C₇×C₇⋊#63121

C₃

yes

maximal, characteristic, complemented, Hall

C₅×C₁₃×#3862

C₄

yes

characteristic, complemented, Hall

#21439633

—

maximal, Hall

C₇²×D₆₅

C₂×C₃

yes

—

—

D₁₃×#3862

—

C₅×C₇²×C₁₃

C₃×C₄

yes

characteristic, complemented, Hall

#15668768

—

maximal, Hall

ΓL(1,𝔽₈)×D₆₅

—

ΓL(1,𝔽₈)×D₆₅

—

C₇×C₇⋊#261

—

Hall

C₁₃×#3862

AGL(1,𝔽₅)

yes

characteristic, complemented, Hall

C₇⋊#63121

ΓL(1,𝔽₈)

yes

characteristic, complemented

C₇×#63121

—

D₅×#3862

—

C₅×C₁₃×ΓL(1,𝔽₈)

—

C₅×C₁₃×ΓL(1,𝔽₈)

—

C₇²×D₁₃

—

ΓL(1,𝔽₈)⋊#261

—

ΓL(1,𝔽₈)×#261

—

C₇×C₇⋊AGL(1,𝔽₅)

—

Hall

C₇×D₆₅

—

C₇×D₆₅

yes

C₇×D₆₅

yes

—

yes

characteristic, complemented, Hall

C₇²×C₁₃

C₃×AGL(1,𝔽₅)

yes

characteristic, complemented, Hall

#10596872

—

Hall

ΓL(1,𝔽₈)×D₁₃

—

ΓL(1,𝔽₈)×D₁₃

—

C₇²×D₅

—

C₅×C₇×C₁₃

—

C₅×C₇×C₁₃

ΓL(1,𝔽₈)⋊C₄

yes

characteristic, complemented

C₅×C₇×C₁₃

C₄×ΓL(1,𝔽₈)

yes

characteristic, complemented

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

—

AGL(1,𝔽₅)×ΓL(1,𝔽₈)

—

C₃×D₆₅

—

C₇⋊#261

—

C₇×#261

—

C₂×#3862

—

C₁₃×ΓL(1,𝔽₈)

—

C₁₃×ΓL(1,𝔽₈)

—

#63121

—

Hall

C₅×C₇²

C₃×#261

yes

characteristic, complemented, Hall

D₅×ΓL(1,𝔽₈)

—

D₅×ΓL(1,𝔽₈)

—

C₇×Dic₇

—

Hall

C₃×C₅×C₁₃

—

Hall

C₇×D₁₃

—

C₇×D₁₃

—

C₇×D₁₃

—

C₃×#261

—

245

Hall

#3862

#63121

yes

characteristic, complemented, Hall

C₇⋊AGL(1,𝔽₅)

—

C₇×AGL(1,𝔽₅)

—

D₆₅

yes

—

—

C₂×C₇²

—

C₇×C₁₃

—

C₇×C₁₃

AGL(1,𝔽₅)×ΓL(1,𝔽₈)

yes

characteristic, complemented

C₇×C₁₃

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

yes

characteristic, complemented

C₄×ΓL(1,𝔽₈)

—

455

—

ΓL(1,𝔽₈)⋊C₄

—

455

—

C₃×D₁₃

—

245

—

C₇×D₅

—

C₇×D₅

—

C₇×D₅

—

C₅×C₁₃

#10596872

yes

characteristic, complemented, Hall

C₃×AGL(1,𝔽₅)

—

637

—

C₇²

yes

characteristic, complemented, Sylow

C₂×ΓL(1,𝔽₈)

—

455

—

C₂×ΓL(1,𝔽₈)

—

455

—

C₃×C₁₃

—

Hall

C₅×C₇

—

C₅×C₇

ΓL(1,𝔽₈)×#261

yes

characteristic, complemented

C₅×C₇

ΓL(1,𝔽₈)⋊#261

yes

characteristic, complemented

C₃×D₅

—

637

—

C₄×C₇

—

455

—

Dic₇

—

D₁₃

—

—

—

—

C₃×C₅

—

Hall

C₂×C₇

—

C₂×C₇

—

C₂×C₇

—

390

—

C₁₃

#15668768

yes

characteristic, complemented, Sylow

C₃×C₄

—

3185

Hall

D₅

—

C₇

—

C₇

ΓL(1,𝔽₈)⋊#63121

yes

characteristic, complemented

C₇

ΓL(1,𝔽₈)×#63121

yes

characteristic, complemented

C₂×C₃

—

3185

—

C₅

#21439633

yes

characteristic, complemented, Sylow

100

C₄

—

455

Sylow

101

C₃

—

Sylow

102

C₂

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	D₆₅×#3862	C₂
cosocle	C₇²×D₆₅	C₂×C₃
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, hypocenter

C₅×C₇²×C₁₃

C₃×C₄

prime radical

D₆₅×#3862

C₂

cosocle

C₇²×D₆₅

C₂×C₃

LCS

G ▷ C₅×C₇²×C₁₃

UCS

C₁

N	H ≅ G/N	type
2. C₇×C₇⋊#63121	C₃	semidirect
3. C₅×C₁₃×#3862	C₄	semidirect
9. C₅×C₇²×C₁₃	C₃×C₄	semidirect
14. C₁₃×#3862	AGL(1,𝔽₅)	semidirect
15. C₇⋊#63121	ΓL(1,𝔽₈)	semidirect
28. C₅×#3862	#261	semidirect
29. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
35. C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	semidirect
36. C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
46. C₅×C₇²	C₃×#261	semidirect
55. #3862	#63121	semidirect
63. C₇×C₁₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
64. C₇×C₁₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	semidirect
71. C₅×C₁₃	#10596872	semidirect
74. C₇²	C₃×#63121	semidirect
79. C₅×C₇	ΓL(1,𝔽₈)×#261	semidirect
80. C₅×C₇	ΓL(1,𝔽₈)⋊#261	semidirect
92. C₁₃	#15668768	semidirect
96. C₇	ΓL(1,𝔽₈)⋊#63121	semidirect
97. C₇	ΓL(1,𝔽₈)×#63121	semidirect
99. C₅	#21439633	semidirect

H ≅ G/N

type

2. C₇×C₇⋊#63121

C₃

semidirect

3. C₅×C₁₃×#3862

C₄

semidirect

9. C₅×C₇²×C₁₃

C₃×C₄

semidirect

14. C₁₃×#3862

AGL(1,𝔽₅)

semidirect

15. C₇⋊#63121

ΓL(1,𝔽₈)

semidirect

28. C₅×#3862

#261

semidirect

29. C₇²×C₁₃

C₃×AGL(1,𝔽₅)

semidirect

35. C₅×C₇×C₁₃

ΓL(1,𝔽₈)⋊C₄

semidirect

36. C₅×C₇×C₁₃

C₄×ΓL(1,𝔽₈)

semidirect

46. C₅×C₇²

C₃×#261

semidirect

55. #3862

#63121

semidirect

63. C₇×C₁₃

AGL(1,𝔽₅)×ΓL(1,𝔽₈)

semidirect

64. C₇×C₁₃

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

semidirect

71. C₅×C₁₃

#10596872

semidirect

74. C₇²

C₃×#63121

semidirect

79. C₅×C₇

ΓL(1,𝔽₈)×#261

semidirect

80. C₅×C₇

ΓL(1,𝔽₈)⋊#261

semidirect

92. C₁₃

#15668768

semidirect

96. C₇

ΓL(1,𝔽₈)⋊#63121

semidirect

97. C₇

ΓL(1,𝔽₈)×#63121

semidirect

99. C₅

#21439633

semidirect

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

subgroup	index
D₆₅×#3862	2
C₇×C₇⋊#63121	3
#21439633	5
ΓL(1,𝔽₈)×#63121	7
ΓL(1,𝔽₈)⋊#63121	7
#15668768	13

subgroup

index

D₆₅×#3862

C₇×C₇⋊#63121

Subgroups — #50303932

Conjugacy classes of subgroups

Complements

Maximal subgroups