Group #50303945 — subgroups

Subgroups — #50303945

Order 38220.

120 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 25 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	C₅⋊(D₁₃⋊#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	#21439603	—	5	1	no	maximal, Hall
5	C₇²×D₆₅	C₂×C₃	1	1	yes	characteristic
6	#19129727	—	7	1	no	maximal
7	#19129726	—	7	1	no	maximal
8	D₁₃⋊#3862	—	5	1	no	—
9	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	#15668766	—	13	1	no	maximal, Hall
11	C₅⋊#10587717	—	7	1	no	—
12	C₅⋊#10587716	—	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	—	7	1	no	—
16	C₇⋊#63121	—	7	1	no	—
17	C₇⋊#63121	—	21	2	no	—
18	D₅×#3862	—	13	1	no	—
19	C₅×#63270	—	7	1	no	—
20	C₅×#63271	—	7	1	no	—
21	C₇²×D₁₃	—	5	1	no	—
22	#11763985	—	35	1	no	—
23	#11763984	—	35	1	no	—
24	C₇⋊(C₇⋊AGL(1,𝔽₅))	—	13	1	no	Hall
25	C₇×D₆₅	—	3	2	no	—
26	C₇×D₆₅	AGL(1,𝔽₇)	1	1	yes	characteristic
27	C₇×D₆₅	AGL(1,𝔽₇)	1	1	yes	characteristic
28	#63121⋊C₃	—	49	1	no	Hall
29	C₅×#3862	—	13	1	no	Hall
30	C₇²×C₁₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
31	#10596875	—	65	1	no	Hall
32	#10587717	—	35	1	no	—
33	#10587716	—	35	1	no	—
34	C₇²×D₅	—	13	1	no	—
35	C₅×C₇×C₁₃	—	3	2	no	—
36	C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
37	C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
38	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	—	91	1	no	—
39	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	—	91	1	no	—
40	C₅⋊(D₁₃⋊C₃)	—	49	1	no	—
41	C₇⋊#261	—	35	1	no	—
42	C₇⋊#261	—	35	1	no	—
43	C₇⋊#261	—	105	2	no	—
44	C₂×#3862	—	65	1	no	—
45	#63271	—	7	1	no	—
46	#63270	—	7	1	no	—
47	#63121	—	49	1	no	Hall
48	C₅×C₇²	AGL(1,𝔽₁₃)	1	1	yes	characteristic, complemented, Hall
49	D₅×ΓL(1,𝔽₈)	—	91	1	no	—
50	D₅×ΓL(1,𝔽₈)	—	91	1	no	—
51	C₇⋊Dic₇	—	65	1	no	Hall
52	C₅×C₁₃⋊C₃	—	49	1	no	Hall
53	C₇×D₁₃	—	5	1	no	—
54	C₇×D₁₃	—	5	1	no	—
55	C₇×D₁₃	—	15	2	no	—
56	AGL(1,𝔽₁₃)	—	245	1	no	Hall
57	#3862	—	13	1	no	Hall
58	C₇⋊AGL(1,𝔽₅)	—	273	2	no	—
59	C₇⋊AGL(1,𝔽₅)	—	91	1	no	—
60	C₇⋊AGL(1,𝔽₅)	—	91	1	no	—
61	D₆₅	#64425	1	1	yes	characteristic
62	C₅×ΓL(1,𝔽₈)	—	91	1	no	—
63	C₅×ΓL(1,𝔽₈)	—	91	1	no	—
64	C₂×C₇²	—	65	1	no	—
65	C₇×C₁₃	—	3	2	no	—
66	C₇×C₁₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
67	C₇×C₁₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
68	ΓL(1,𝔽₈)⋊C₄	—	455	1	no	—
69	ΓL(1,𝔽₈)⋊C₄	—	455	1	no	—
70	D₁₃⋊C₃	—	245	1	no	—
71	C₇×D₅	—	13	1	no	—
72	C₇×D₅	—	13	1	no	—
73	C₇×D₅	—	39	2	no	—
74	C₅×C₁₃	#10596875	1	1	yes	characteristic, complemented, Hall
75	C₃×AGL(1,𝔽₅)	—	637	1	no	Hall
76	#261	—	245	1	no	Hall
77	C₇²	#63121⋊C₃	1	1	yes	characteristic, complemented, Sylow
78	C₂×ΓL(1,𝔽₈)	—	455	1	no	—
79	C₂×ΓL(1,𝔽₈)	—	455	1	no	—
80	C₁₃⋊C₃	—	49	1	no	Hall
81	C₅×C₇	—	3	2	no	—
82	C₅×C₇	#11763984	1	1	yes	characteristic, complemented
83	C₅×C₇	#11763985	1	1	yes	characteristic, complemented
84	C₃×D₅	—	637	1	no	—
85	Dic₇	—	1365	2	no	—
86	Dic₇	—	455	1	no	—
87	Dic₇	—	455	1	no	—
88	D₁₃	—	5	1	no	—
89	ΓL(1,𝔽₈)	—	91	1	no	—
90	ΓL(1,𝔽₈)	—	91	1	no	—
91	AGL(1,𝔽₅)	—	637	1	no	Hall
92	C₃×C₅	—	637	1	no	Hall
93	C₂×C₇	—	65	1	no	—
94	C₂×C₇	—	65	1	no	—
95	C₂×C₇	—	195	2	no	—
96	C₁₃	#15668766	1	1	yes	characteristic, complemented, Sylow
97	C₃×C₄	—	3185	1	no	Hall
98	D₅	—	13	1	no	—
99	C₇	—	3	2	no	—
100	C₇	#19129726	1	1	yes	characteristic, complemented
101	C₇	#19129727	1	1	yes	characteristic, complemented
102	C₂×C₃	—	3185	1	no	—
103	C₅	#21439603	1	1	yes	characteristic, complemented, Sylow
104	C₄	—	3185	1	no	Sylow
105	C₃	—	637	1	no	Sylow
106	C₂	—	65	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅⋊(D₁₃⋊#3862)

C₂

yes

maximal, characteristic

C₇⋊(C₇⋊#63121)

C₃

yes

maximal, characteristic, complemented, Hall

C₅×C₁₃⋊#3862

C₄

yes

characteristic, complemented, Hall

#21439603

—

maximal, Hall

C₇²×D₆₅

C₂×C₃

yes

—

—

D₁₃⋊#3862

—

C₅×C₇²×C₁₃

C₃×C₄

yes

characteristic, complemented, Hall

—

—

—

—

C₁₃⋊#3862

yes

characteristic, complemented, Hall

—

—

—

—

—

—

C₇²×D₁₃

—

—

—

—

C₇×D₆₅

—

C₇×D₆₅

yes

C₇×D₆₅

yes

—

—

C₇²×C₁₃

yes

characteristic, complemented, Hall

—

—

—

C₇²×D₅

—

C₅×C₇×C₁₃

—

C₅×C₇×C₁₃

ΓL(1,𝔽₈)⋊C₄

yes

characteristic, complemented

C₅×C₇×C₁₃

ΓL(1,𝔽₈)⋊C₄

yes

characteristic, complemented

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

—

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

—

C₅⋊(D₁₃⋊C₃)

—

C₇⋊#261

—

C₇⋊#261

—

C₇⋊#261

—

105

—

—

—

—

—

C₅×C₇²

AGL(1,𝔽₁₃)

yes

characteristic, complemented, Hall

D₅×ΓL(1,𝔽₈)

—

D₅×ΓL(1,𝔽₈)

—

C₇⋊Dic₇

—

Hall

C₅×C₁₃⋊C₃

—

Hall

C₇×D₁₃

—

C₇×D₁₃

—

C₇×D₁₃

—

AGL(1,𝔽₁₃)

—

245

—

—

273

—

C₇⋊AGL(1,𝔽₅)

—

C₇⋊AGL(1,𝔽₅)

—

D₆₅

yes

—

—

C₂×C₇²

—

C₇×C₁₃

—

C₇×C₁₃

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

yes

characteristic, complemented

C₇×C₁₃

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

yes

characteristic, complemented

ΓL(1,𝔽₈)⋊C₄

—

455

—

ΓL(1,𝔽₈)⋊C₄

—

455

—

D₁₃⋊C₃

—

245

—

C₇×D₅

—

C₇×D₅

—

C₇×D₅

—

C₅×C₁₃

#10596875

yes

characteristic, complemented, Hall

C₃×AGL(1,𝔽₅)

—

637

Hall

#261

—

245

Hall

C₇²

#63121⋊C₃

yes

characteristic, complemented, Sylow

C₂×ΓL(1,𝔽₈)

—

455

—

C₂×ΓL(1,𝔽₈)

—

455

—

C₁₃⋊C₃

—

Hall

C₅×C₇

—

C₅×C₇

#11763984

yes

characteristic, complemented

C₅×C₇

#11763985

yes

characteristic, complemented

C₃×D₅

—

637

—

Dic₇

—

1365

—

Dic₇

—

455

—

Dic₇

—

455

—

D₁₃

—

ΓL(1,𝔽₈)

—

ΓL(1,𝔽₈)

—

AGL(1,𝔽₅)

—

637

Hall

C₃×C₅

—

637

Hall

C₂×C₇

—

C₂×C₇

—

C₂×C₇

—

195

—

C₁₃

#15668766

yes

characteristic, complemented, Sylow

C₃×C₄

—

3185

Hall

D₅

—

C₇

—

100

C₇

#19129726

yes

characteristic, complemented

101

C₇

#19129727

yes

characteristic, complemented

102

C₂×C₃

—

3185

—

103

C₅

#21439603

yes

characteristic, complemented, Sylow

104

C₄

—

3185

Sylow

105

C₃

—

637

Sylow

106

C₂

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	C₅⋊(D₁₃⋊#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

C₅⋊(D₁₃⋊#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
30. C₇²×C₁₃	C₃×AGL(1,𝔽₅)	semidirect
36. C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	semidirect
37. C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	semidirect
48. C₅×C₇²	AGL(1,𝔽₁₃)	semidirect
66. C₇×C₁₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	semidirect
67. C₇×C₁₃	ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)	semidirect
74. C₅×C₁₃	#10596875	semidirect
77. C₇²	#63121⋊C₃	semidirect
82. C₅×C₇	#11763984	semidirect
83. C₅×C₇	#11763985	semidirect
96. C₁₃	#15668766	semidirect
100. C₇	#19129726	semidirect
101. C₇	#19129727	semidirect
103. C₅	#21439603	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

30. C₇²×C₁₃

C₃×AGL(1,𝔽₅)

semidirect

36. C₅×C₇×C₁₃

ΓL(1,𝔽₈)⋊C₄

semidirect

37. C₅×C₇×C₁₃

ΓL(1,𝔽₈)⋊C₄

semidirect

48. C₅×C₇²

AGL(1,𝔽₁₃)

semidirect

66. C₇×C₁₃

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

semidirect

67. C₇×C₁₃

ΓL(1,𝔽₈)⋊AGL(1,𝔽₅)

semidirect

74. C₅×C₁₃

#10596875

semidirect

77. C₇²

#63121⋊C₃

semidirect

82. C₅×C₇

#11763984

semidirect

83. C₅×C₇

#11763985

semidirect

96. C₁₃

#15668766

semidirect

100. C₇

#19129726

semidirect

101. C₇

#19129727

semidirect

103. C₅

#21439603

semidirect

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

subgroup	index
C₅⋊(D₁₃⋊#3862)	2
C₇⋊(C₇⋊#63121)	3
#21439603	5
#19129727	7
#19129726	7
#15668766	13

subgroup

index

C₅⋊(D₁₃⋊#3862)

Subgroups — #50303945

Conjugacy classes of subgroups

Complements

Maximal subgroups