Group #50304116 — subgroups

Subgroups — #50304116

Order 38220.

144 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 33 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×#3861	C₂	1	1	yes	maximal, characteristic
2	C₇×C₇⋊#63118	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₁₃×#3861	C₄	1	1	yes	characteristic, complemented, Hall
4	#21439627	—	5	1	no	maximal, Hall
5	C₅×C₇²×D₁₃	C₂×C₃	1	1	yes	characteristic
6	ΓL(1,𝔽₈)×#63118	—	7	1	no	maximal
7	ΓL(1,𝔽₈)⋊#63118	—	7	1	no	maximal
8	D₁₃×#3861	D₅	1	1	yes	characteristic
9	C₅×C₇²×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	#15668736	—	13	1	no	maximal, Hall
11	C₅×ΓL(1,𝔽₈)×D₁₃	—	7	1	no	—
12	C₅×ΓL(1,𝔽₈)×D₁₃	—	7	1	no	—
13	C₅×ΓL(1,𝔽₈)×D₁₃	—	14	3	no	—
14	C₇×C₇⋊#261	—	5	1	no	Hall
15	C₁₃×#3861	Dic₅	1	1	yes	characteristic, complemented, Hall
16	C₇⋊#63118	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
17	C₇×#63118	—	7	1	no	—
18	C₂×C₅×#3861	—	13	1	no	—
19	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
20	C₅×C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
21	C₅×C₁₃×ΓL(1,𝔽₈)	—	14	3	no	—
22	C₇²×D₁₃	C₃×D₅	1	1	yes	characteristic
23	ΓL(1,𝔽₈)⋊#261	—	35	1	no	—
24	ΓL(1,𝔽₈)×#261	—	35	1	no	—
25	C₇×Dic₃₅	—	13	1	no	Hall
26	C₅×C₇×D₁₃	—	2	3	no	—
27	C₅×C₇×D₁₃	AGL(1,𝔽₇)	1	1	yes	characteristic
28	C₅×C₇×D₁₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₃×#63118	—	49	1	no	Hall
30	C₅×#3861	#261	1	1	yes	characteristic, complemented, Hall
31	C₇²×C₁₃	C₃×Dic₅	1	1	yes	characteristic, complemented, Hall
32	#10596873	—	65	1	no	Hall
33	ΓL(1,𝔽₈)×D₁₃	—	7	1	no	—
34	ΓL(1,𝔽₈)×D₁₃	—	7	1	no	—
35	ΓL(1,𝔽₈)×D₁₃	—	14	3	no	—
36	C₂×C₅×C₇²	—	13	1	no	—
37	C₅×C₇×C₁₃	—	2	3	no	—
38	C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented
39	C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
40	ΓL(1,𝔽₈)⋊Dic₅	—	91	1	no	—
41	Dic₅×ΓL(1,𝔽₈)	—	91	1	no	—
42	C₃×C₅×D₁₃	—	49	1	no	—
43	C₇⋊#261	—	5	1	no	—
44	C₇×#261	—	35	1	no	—
45	C₂×#3861	—	13	1	no	—
46	C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
47	C₁₃×ΓL(1,𝔽₈)	—	7	1	no	—
48	C₁₃×ΓL(1,𝔽₈)	—	14	3	no	—
49	#63118	—	7	1	no	Hall
50	C₅×C₇²	C₃×#261	1	1	yes	characteristic, complemented, Hall
51	C₂×C₅×ΓL(1,𝔽₈)	—	91	1	no	—
52	C₂×C₅×ΓL(1,𝔽₈)	—	91	1	no	—
53	C₂×C₅×ΓL(1,𝔽₈)	—	182	3	no	—
54	C₇×Dic₇	—	65	1	no	Hall
55	C₃×C₅×C₁₃	—	49	1	no	Hall
56	C₇×D₁₃	—	2	3	no	—
57	C₇×D₁₃	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
58	C₇×D₁₃	C₅⋊AGL(1,𝔽₇)	1	1	yes	characteristic
59	C₃×#261	—	245	1	no	Hall
60	#3861	#63118	1	1	yes	characteristic, complemented, Hall
61	Dic₃₅	—	13	1	no	—
62	C₇×Dic₅	—	91	1	no	—
63	C₅×D₁₃	#64423	1	1	yes	characteristic
64	C₅×ΓL(1,𝔽₈)	—	7	1	no	—
65	C₅×ΓL(1,𝔽₈)	—	7	1	no	—
66	C₅×ΓL(1,𝔽₈)	—	14	3	no	—
67	C₂×C₇²	—	13	1	no	—
68	C₇×C₁₃	—	2	3	no	—
69	C₇×C₁₃	Dic₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
70	C₇×C₁₃	ΓL(1,𝔽₈)⋊Dic₅	1	1	yes	characteristic, complemented
71	C₄×ΓL(1,𝔽₈)	—	455	1	no	—
72	ΓL(1,𝔽₈)⋊C₄	—	455	1	no	—
73	C₃×D₁₃	—	49	1	no	—
74	C₂×C₅×C₇	—	13	1	no	—
75	C₂×C₅×C₇	—	13	1	no	—
76	C₂×C₅×C₇	—	26	3	no	—
77	C₅×C₁₃	#10596873	1	1	yes	characteristic, complemented, Hall
78	C₃×Dic₅	—	637	1	no	Hall
79	#261	—	35	1	no	Hall
80	C₇²	C₃×#63118	1	1	yes	characteristic, complemented, Sylow
81	C₂×ΓL(1,𝔽₈)	—	91	1	no	—
82	C₂×ΓL(1,𝔽₈)	—	91	1	no	—
83	C₂×ΓL(1,𝔽₈)	—	182	3	no	—
84	C₃×C₁₃	—	49	1	no	Hall
85	C₅×C₇	—	2	3	no	—
86	C₅×C₇	ΓL(1,𝔽₈)×#261	1	1	yes	characteristic, complemented
87	C₅×C₇	ΓL(1,𝔽₈)⋊#261	1	1	yes	characteristic, complemented
88	C₂×C₃×C₅	—	637	1	no	—
89	C₄×C₇	—	455	1	no	—
90	Dic₇	—	65	1	no	—
91	D₁₃	C₅⋊#64423	1	1	yes	characteristic
92	ΓL(1,𝔽₈)	—	7	1	no	—
93	ΓL(1,𝔽₈)	—	7	1	no	—
94	ΓL(1,𝔽₈)	—	14	3	no	—
95	Dic₅	—	91	1	no	Hall
96	C₃×C₅	—	49	1	no	Hall
97	C₂×C₇	—	13	1	no	—
98	C₂×C₇	—	13	1	no	—
99	C₂×C₇	—	26	3	no	—
100	C₁₃	#15668736	1	1	yes	characteristic, complemented, Sylow
101	C₃×C₄	—	3185	1	no	Hall
102	C₂×C₅	—	13	1	no	—
103	C₇	—	2	3	no	—
104	C₇	ΓL(1,𝔽₈)⋊#63118	1	1	yes	characteristic, complemented
105	C₇	ΓL(1,𝔽₈)×#63118	1	1	yes	characteristic, complemented
106	C₂×C₃	—	637	1	no	—
107	C₅	#21439627	1	1	yes	characteristic, complemented, Sylow
108	C₄	—	455	1	no	Sylow
109	C₃	—	49	1	no	Sylow
110	C₂	—	13	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×D₁₃×#3861

C₂

yes

maximal, characteristic

C₇×C₇⋊#63118

C₃

yes

maximal, characteristic, complemented, Hall

C₅×C₁₃×#3861

C₄

yes

characteristic, complemented, Hall

#21439627

—

maximal, Hall

C₅×C₇²×D₁₃

C₂×C₃

yes

—

—

D₁₃×#3861

D₅

yes

characteristic

C₅×C₇²×C₁₃

C₃×C₄

yes

characteristic, complemented, Hall

#15668736

—

maximal, Hall

C₅×ΓL(1,𝔽₈)×D₁₃

—

C₅×ΓL(1,𝔽₈)×D₁₃

—

C₅×ΓL(1,𝔽₈)×D₁₃

—

C₇×C₇⋊#261

—

Hall

C₁₃×#3861

Dic₅

yes

characteristic, complemented, Hall

C₇⋊#63118

ΓL(1,𝔽₈)

yes

characteristic, complemented

C₇×#63118

—

C₂×C₅×#3861

—

C₅×C₁₃×ΓL(1,𝔽₈)

—

C₅×C₁₃×ΓL(1,𝔽₈)

—

C₅×C₁₃×ΓL(1,𝔽₈)

—

C₇²×D₁₃

C₃×D₅

yes

characteristic

ΓL(1,𝔽₈)⋊#261

—

ΓL(1,𝔽₈)×#261

—

C₇×Dic₃₅

—

Hall

C₅×C₇×D₁₃

—

C₅×C₇×D₁₃

yes

C₅×C₇×D₁₃

yes

—

yes

characteristic, complemented, Hall

C₇²×C₁₃

C₃×Dic₅

yes

characteristic, complemented, Hall

#10596873

—

Hall

ΓL(1,𝔽₈)×D₁₃

—

ΓL(1,𝔽₈)×D₁₃

—

ΓL(1,𝔽₈)×D₁₃

—

C₂×C₅×C₇²

—

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)⋊Dic₅

—

Dic₅×ΓL(1,𝔽₈)

—

C₃×C₅×D₁₃

—

C₇⋊#261

—

C₇×#261

—

C₂×#3861

—

C₁₃×ΓL(1,𝔽₈)

—

C₁₃×ΓL(1,𝔽₈)

—

C₁₃×ΓL(1,𝔽₈)

—

#63118

—

Hall

C₅×C₇²

C₃×#261

yes

characteristic, complemented, Hall

C₂×C₅×ΓL(1,𝔽₈)

—

C₂×C₅×ΓL(1,𝔽₈)

—

C₂×C₅×ΓL(1,𝔽₈)

—

182

—

C₇×Dic₇

—

Hall

C₃×C₅×C₁₃

—

Hall

C₇×D₁₃

—

C₇×D₁₃

yes

C₇×D₁₃

yes

—

245

Hall

#3861

#63118

yes

characteristic, complemented, Hall

Dic₃₅

—

C₇×Dic₅

—

C₅×D₁₃

yes

—

—

—

C₂×C₇²

—

C₇×C₁₃

—

C₇×C₁₃

Dic₅×ΓL(1,𝔽₈)

yes

characteristic, complemented

C₇×C₁₃

ΓL(1,𝔽₈)⋊Dic₅

yes

characteristic, complemented

C₄×ΓL(1,𝔽₈)

—

455

—

ΓL(1,𝔽₈)⋊C₄

—

455

—

C₃×D₁₃

—

C₂×C₅×C₇

—

C₂×C₅×C₇

—

C₂×C₅×C₇

—

C₅×C₁₃

#10596873

yes

characteristic, complemented, Hall

C₃×Dic₅

—

637

—

C₇²

yes

characteristic, complemented, Sylow

C₂×ΓL(1,𝔽₈)

—

C₂×ΓL(1,𝔽₈)

—

C₂×ΓL(1,𝔽₈)

—

182

—

C₃×C₁₃

—

Hall

C₅×C₇

—

C₅×C₇

ΓL(1,𝔽₈)×#261

yes

characteristic, complemented

C₅×C₇

ΓL(1,𝔽₈)⋊#261

yes

characteristic, complemented

C₂×C₃×C₅

—

637

—

C₄×C₇

—

455

—

Dic₇

—

D₁₃

yes

—

—

—

Dic₅

—

Hall

C₃×C₅

—

Hall

C₂×C₇

—

C₂×C₇

—

C₂×C₇

—

100

C₁₃

#15668736

yes

characteristic, complemented, Sylow

101

C₃×C₄

—

3185

Hall

102

C₂×C₅

—

103

C₇

—

104

C₇

ΓL(1,𝔽₈)⋊#63118

yes

characteristic, complemented

105

C₇

ΓL(1,𝔽₈)×#63118

yes

characteristic, complemented

106

C₂×C₃

—

637

—

107

C₅

#21439627

yes

characteristic, complemented, Sylow

108

C₄

—

455

Sylow

109

C₃

—

Sylow

110

C₂

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇²×C₁₃	C₃×C₄
prime radical	C₅×D₁₃×#3861	C₂
cosocle	C₅×C₇²×D₁₃	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×#3861

C₂

cosocle

C₅×C₇²×D₁₃

C₂×C₃

LCS

G ▷ C₅×C₇²×C₁₃

UCS

C₁

N	H ≅ G/N	type
2. C₇×C₇⋊#63118	C₃	semidirect
3. C₅×C₁₃×#3861	C₄	semidirect
9. C₅×C₇²×C₁₃	C₃×C₄	semidirect
15. C₁₃×#3861	Dic₅	semidirect
16. C₇⋊#63118	ΓL(1,𝔽₈)	semidirect
30. C₅×#3861	#261	semidirect
31. C₇²×C₁₃	C₃×Dic₅	semidirect
38. C₅×C₇×C₁₃	ΓL(1,𝔽₈)⋊C₄	semidirect
39. C₅×C₇×C₁₃	C₄×ΓL(1,𝔽₈)	semidirect
50. C₅×C₇²	C₃×#261	semidirect
60. #3861	#63118	semidirect
69. C₇×C₁₃	Dic₅×ΓL(1,𝔽₈)	semidirect
70. C₇×C₁₃	ΓL(1,𝔽₈)⋊Dic₅	semidirect
77. C₅×C₁₃	#10596873	semidirect
80. C₇²	C₃×#63118	semidirect
86. C₅×C₇	ΓL(1,𝔽₈)×#261	semidirect
87. C₅×C₇	ΓL(1,𝔽₈)⋊#261	semidirect
100. C₁₃	#15668736	semidirect
104. C₇	ΓL(1,𝔽₈)⋊#63118	semidirect
105. C₇	ΓL(1,𝔽₈)×#63118	semidirect
107. C₅	#21439627	semidirect

H ≅ G/N

type

2. C₇×C₇⋊#63118

C₃

semidirect

3. C₅×C₁₃×#3861

C₄

semidirect

9. C₅×C₇²×C₁₃

C₃×C₄

semidirect

15. C₁₃×#3861

Dic₅

semidirect

16. C₇⋊#63118

ΓL(1,𝔽₈)

semidirect

30. C₅×#3861

#261

semidirect

31. C₇²×C₁₃

C₃×Dic₅

semidirect

38. C₅×C₇×C₁₃

ΓL(1,𝔽₈)⋊C₄

semidirect

39. C₅×C₇×C₁₃

C₄×ΓL(1,𝔽₈)

semidirect

50. C₅×C₇²

C₃×#261

semidirect

60. #3861

#63118

semidirect

69. C₇×C₁₃

Dic₅×ΓL(1,𝔽₈)

semidirect

70. C₇×C₁₃

ΓL(1,𝔽₈)⋊Dic₅

semidirect

77. C₅×C₁₃

#10596873

semidirect

80. C₇²

C₃×#63118

semidirect

86. C₅×C₇

ΓL(1,𝔽₈)×#261

semidirect

87. C₅×C₇

ΓL(1,𝔽₈)⋊#261

semidirect

100. C₁₃

#15668736

semidirect

104. C₇

ΓL(1,𝔽₈)⋊#63118

semidirect

105. C₇

ΓL(1,𝔽₈)×#63118

semidirect

107. C₅

#21439627

semidirect

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

subgroup	index
C₅×D₁₃×#3861	2
C₇×C₇⋊#63118	3
#21439627	5
ΓL(1,𝔽₈)×#63118	7
ΓL(1,𝔽₈)⋊#63118	7
#15668736	13

subgroup

index

C₅×D₁₃×#3861

C₇×C₇⋊#63118

Subgroups — #50304116

Conjugacy classes of subgroups

Complements

Maximal subgroups