Group #50304103 — subgroups

Subgroups — D₅×#16838446 #50304103

Order 38220.

400 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 71 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	D₅×#14274263	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅⋊#16838446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#16838446	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅×C₇⋊D₉₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₅×#14274263	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×#16838446	—	5	1	no	maximal, Hall
7	C₅⋊(C₇⋊D₉₁)	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇²×D₅×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
9	C₅×C₇⋊D₉₁	C₂×C₃	1	1	yes	characteristic, complemented
10	D₅×#10587722	—	7	8	no	maximal
11	C₂×#14274263	—	5	1	no	—
12	#16838446	—	5	1	no	—
13	#16838446	D₅	1	1	yes	characteristic, direct product factor
14	C₅×C₇²×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
15	D₅×#64424	—	13	1	no	maximal, Hall
16	C₅⋊#10587722	—	7	8	no	—
17	D₅×#63270	—	7	8	no	—
18	C₅×#10587722	—	7	8	no	—
19	C₂×C₇⋊D₉₁	—	5	1	no	Hall
20	#14274263	C₂×D₅	1	1	yes	characteristic, complemented, Hall
21	D₅×D₉₁	—	7	8	no	—
22	D₅×#3861	—	13	1	no	—
23	C₅⋊#64424	—	13	1	no	—
24	C₅×#64424	—	13	1	no	—
25	C₅×#63270	—	7	8	no	—
26	C₂×C₇²×C₁₃	—	5	1	no	—
27	C₇⋊D₉₁	—	5	1	no	—
28	C₇⋊D₉₁	C₃×D₅	1	1	yes	characteristic, complemented
29	C₂×#10587722	—	35	8	no	—
30	D₅×C₇⋊D₇	—	13	1	no	Hall
31	D₄₅₅	—	7	8	no	—
32	C₇×D₅×C₁₃	AGL(1,𝔽₇)	1	8	yes	complemented
33	C₅×D₉₁	—	7	8	no	—
34	D₅×D₁₃⋊C₃	—	49	1	no	Hall
35	C₅×#3861	—	13	1	no	Hall
36	C₇²×C₁₃	C₂×C₃×D₅	1	1	yes	characteristic, complemented, Hall
37	C₂×#64424	—	65	1	no	Hall
38	C₂×#63270	—	35	8	no	—
39	#10587722	—	7	8	no	—
40	#10587722	—	35	8	no	—
41	C₅⋊(C₇⋊D₇)	—	13	1	no	—
42	C₇²×D₅	D₁₃⋊C₃	1	1	yes	characteristic, complemented
43	C₅×C₇⋊D₇	—	13	1	no	—
44	C₅×C₇×C₁₃	C₂×AGL(1,𝔽₇)	1	8	yes	complemented
45	D₅×AGL(1,𝔽₇)	—	91	8	no	—
46	C₅⋊(D₁₃⋊C₃)	—	49	1	no	—
47	D₅×C₁₃⋊C₃	—	49	1	no	—
48	C₅×D₁₃⋊C₃	—	49	1	no	—
49	C₂×D₉₁	—	35	8	no	—
50	C₂×#3861	—	65	1	no	—
51	#64424	—	13	1	no	—
52	#64424	—	65	1	no	—
53	#63270	—	7	8	no	—
54	D₅×D₁₃	—	49	1	no	Hall
55	C₅×C₇²	C₂×D₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
56	C₅⋊AGL(1,𝔽₇)	—	91	8	no	—
57	D₅×ΓL(1,𝔽₈)	—	91	8	no	—
58	C₅×AGL(1,𝔽₇)	—	91	8	no	—
59	C₂×C₇⋊D₇	—	65	1	no	Hall
60	C₅×C₁₃⋊C₃	—	49	1	no	Hall
61	C₂×C₇×C₁₃	—	5	8	no	—
62	D₉₁	—	7	8	no	—
63	D₉₁	—	35	8	no	—
64	C₂×D₁₃⋊C₃	—	245	1	no	Hall
65	#3861	—	13	1	no	Hall
66	D₅×D₇	—	91	8	no	—
67	D₆₅	—	49	1	no	—
68	C₅×D₁₃	—	49	1	no	—
69	D₅×C₁₃	#64424	1	1	yes	characteristic, complemented
70	C₅×ΓL(1,𝔽₈)	—	91	8	no	—
71	C₂×C₇²	—	5	1	no	—
72	C₇⋊D₇	—	13	1	no	—
73	C₇⋊D₇	—	65	1	no	—
74	C₇×C₁₃	D₅×AGL(1,𝔽₇)	1	8	yes	complemented
75	C₂×AGL(1,𝔽₇)	—	455	8	no	—
76	C₂×C₁₃⋊C₃	—	245	1	no	—
77	D₁₃⋊C₃	—	49	1	no	—
78	D₁₃⋊C₃	—	245	1	no	—
79	D₃₅	—	91	8	no	—
80	C₅×D₇	—	91	8	no	—
81	C₇×D₅	#10587722	1	8	yes	complemented
82	C₅×C₁₃	C₂×#64424	1	1	yes	characteristic, complemented, Hall
83	C₂×C₃×D₅	—	637	1	no	Hall
84	C₂×D₁₃	—	245	1	no	Hall
85	C₇²	D₅×D₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
86	C₂×ΓL(1,𝔽₈)	—	455	8	no	—
87	AGL(1,𝔽₇)	—	91	8	no	—
88	AGL(1,𝔽₇)	—	455	8	no	—
89	C₁₃⋊C₃	—	49	1	no	Hall
90	C₅×C₇	C₂×#10587722	1	8	yes	complemented
91	C₂×C₃×C₅	—	637	1	no	—
92	C₃×D₅	—	637	1	no	—
93	C₃×D₅	—	637	1	no	—
94	C₂×D₇	—	455	8	no	—
95	C₂×C₁₃	—	5	1	no	—
96	D₁₃	—	49	1	no	—
97	D₁₃	—	245	1	no	—
98	ΓL(1,𝔽₈)	—	91	8	no	—
99	C₂×D₅	—	637	1	no	Hall
100	C₃×C₅	—	637	1	no	Hall
101	C₂×C₇	—	5	8	no	—
102	D₇	—	91	8	no	—
103	D₇	—	455	8	no	—
104	C₁₃	D₅×#64424	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₃	—	3185	1	no	Hall
106	C₂×C₅	—	637	1	no	—
107	D₅	—	637	1	no	—
108	D₅	#16838446	1	1	yes	characteristic, direct product factor
109	C₇	D₅×#10587722	1	8	yes	complemented
110	C₂×C₃	—	3185	1	no	—
111	C₂×C₃	—	3185	1	no	—
112	C₂×C₃	—	637	1	no	—
113	C₅	C₂×#16838446	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	3185	1	no	Sylow
115	C₃	—	637	1	no	Sylow
116	C₂	—	5	1	no	—
117	C₂	—	3185	1	no	—
118	C₂	—	637	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₅×#14274263

C₂

yes

maximal, characteristic, complemented

C₅⋊#16838446

C₂

yes

maximal, characteristic, complemented

C₅×#16838446

C₂

yes

maximal, characteristic, complemented

D₅×C₇⋊D₉₁

C₃

yes

maximal, characteristic, complemented, Hall

C₅×#14274263

C₂²

yes

characteristic, complemented, Hall

C₂×#16838446

—

maximal, Hall

C₅⋊(C₇⋊D₉₁)

C₂×C₃

yes

characteristic, complemented

C₇²×D₅×C₁₃

C₂×C₃

yes

characteristic, complemented

C₅×C₇⋊D₉₁

C₂×C₃

yes

characteristic, complemented

—

—

—

D₅

yes

characteristic, direct product factor

C₅×C₇²×C₁₃

C₂²×C₃

yes

characteristic, complemented, Hall

—

—

—

—

C₂×C₇⋊D₉₁

—

Hall

#14274263

C₂×D₅

yes

characteristic, complemented, Hall

D₅×D₉₁

—

—

—

—

—

C₂×C₇²×C₁₃

—

C₇⋊D₉₁

—

C₇⋊D₉₁

C₃×D₅

yes

characteristic, complemented

C₂×#10587722

—

D₅×C₇⋊D₇

—

Hall

D₄₅₅

—

C₇×D₅×C₁₃

AGL(1,𝔽₇)

yes

complemented

C₅×D₉₁

—

D₅×D₁₃⋊C₃

—

Hall

C₅×#3861

—

Hall

C₇²×C₁₃

C₂×C₃×D₅

yes

characteristic, complemented, Hall

—

—

—

—

C₅⋊(C₇⋊D₇)

—

C₇²×D₅

D₁₃⋊C₃

yes

characteristic, complemented

C₅×C₇⋊D₇

—

C₅×C₇×C₁₃

C₂×AGL(1,𝔽₇)

yes

complemented

D₅×AGL(1,𝔽₇)

—

C₅⋊(D₁₃⋊C₃)

—

D₅×C₁₃⋊C₃

—

C₅×D₁₃⋊C₃

—

C₂×D₉₁

—

—

—

—

—

D₅×D₁₃

—

Hall

C₅×C₇²

C₂×D₁₃⋊C₃

yes

characteristic, complemented, Hall

C₅⋊AGL(1,𝔽₇)

—

D₅×ΓL(1,𝔽₈)

—

C₅×AGL(1,𝔽₇)

—

C₂×C₇⋊D₇

—

Hall

C₅×C₁₃⋊C₃

—

Hall

C₂×C₇×C₁₃

—

D₉₁

—

D₉₁

—

C₂×D₁₃⋊C₃

—

245

Hall

#3861

—

Hall

D₅×D₇

—

D₆₅

—

C₅×D₁₃

—

D₅×C₁₃

#64424

yes

characteristic, complemented

C₅×ΓL(1,𝔽₈)

—

C₂×C₇²

—

C₇⋊D₇

—

C₇⋊D₇

—

C₇×C₁₃

D₅×AGL(1,𝔽₇)

yes

complemented

C₂×AGL(1,𝔽₇)

—

455

—

C₂×C₁₃⋊C₃

—

245

—

D₁₃⋊C₃

—

D₁₃⋊C₃

—

245

—

D₃₅

—

C₅×D₇

—

C₇×D₅

#10587722

yes

complemented

C₅×C₁₃

C₂×#64424

yes

characteristic, complemented, Hall

C₂×C₃×D₅

—

637

Hall

C₂×D₁₃

—

245

Hall

C₇²

D₅×D₁₃⋊C₃

yes

characteristic, complemented, Sylow

C₂×ΓL(1,𝔽₈)

—

455

—

AGL(1,𝔽₇)

—

AGL(1,𝔽₇)

—

455

—

C₁₃⋊C₃

—

Hall

C₅×C₇

C₂×#10587722

yes

complemented

C₂×C₃×C₅

—

637

—

C₃×D₅

—

637

—

C₃×D₅

—

637

—

C₂×D₇

—

455

—

C₂×C₁₃

—

D₁₃

—

D₁₃

—

245

—

ΓL(1,𝔽₈)

—

C₂×D₅

—

637

Hall

100

C₃×C₅

—

637

Hall

101

C₂×C₇

—

102

D₇

—

103

D₇

—

455

—

104

C₁₃

D₅×#64424

yes

characteristic, complemented, Sylow

105

C₂²×C₃

—

3185

Hall

106

C₂×C₅

—

637

—

107

D₅

—

637

—

108

D₅

#16838446

yes

characteristic, direct product factor

109

C₇

D₅×#10587722

yes

complemented

110

C₂×C₃

—

3185

—

111

C₂×C₃

—

3185

—

112

C₂×C₃

—

637

—

113

C₅

C₂×#16838446

yes

characteristic, complemented, Sylow

114

C₂²

—

3185

Sylow

115

C₃

—

637

Sylow

116

C₂

—

117

C₂

—

3185

—

118

C₂

—

637

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇²×C₁₃	C₂²×C₃
LCS	G ▷ C₅×C₇²×C₁₃
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₅×C₇²×C₁₃

C₂²×C₃

LCS

G ▷ C₅×C₇²×C₁₃

UCS

C₁

N	H ≅ G/N	type
1. D₅×#14274263	C₂	semidirect
2. C₅⋊#16838446	C₂	semidirect
3. C₅×#16838446	C₂	semidirect
4. D₅×C₇⋊D₉₁	C₃	semidirect
5. C₅×#14274263	C₂²	semidirect
7. C₅⋊(C₇⋊D₉₁)	C₂×C₃	semidirect
8. C₇²×D₅×C₁₃	C₂×C₃	semidirect
9. C₅×C₇⋊D₉₁	C₂×C₃	semidirect
13. #16838446	D₅	direct (N × H)
14. C₅×C₇²×C₁₃	C₂²×C₃	semidirect
20. #14274263	C₂×D₅	semidirect
28. C₇⋊D₉₁	C₃×D₅	semidirect
32. C₇×D₅×C₁₃	AGL(1,𝔽₇)	semidirect
36. C₇²×C₁₃	C₂×C₃×D₅	semidirect
42. C₇²×D₅	D₁₃⋊C₃	semidirect
44. C₅×C₇×C₁₃	C₂×AGL(1,𝔽₇)	semidirect
55. C₅×C₇²	C₂×D₁₃⋊C₃	semidirect
69. D₅×C₁₃	#64424	semidirect
74. C₇×C₁₃	D₅×AGL(1,𝔽₇)	semidirect
81. C₇×D₅	#10587722	semidirect
82. C₅×C₁₃	C₂×#64424	semidirect
85. C₇²	D₅×D₁₃⋊C₃	semidirect
90. C₅×C₇	C₂×#10587722	semidirect
104. C₁₃	D₅×#64424	semidirect
108. D₅	#16838446	direct (N × H)
109. C₇	D₅×#10587722	semidirect
113. C₅	C₂×#16838446	semidirect

H ≅ G/N

type

1. D₅×#14274263

C₂

semidirect

2. C₅⋊#16838446

C₂

semidirect

3. C₅×#16838446

C₂

semidirect

4. D₅×C₇⋊D₉₁

C₃

semidirect

5. C₅×#14274263

C₂²

semidirect

7. C₅⋊(C₇⋊D₉₁)

C₂×C₃

semidirect

8. C₇²×D₅×C₁₃

C₂×C₃

semidirect

9. C₅×C₇⋊D₉₁

C₂×C₃

semidirect

13. #16838446

D₅

direct (N × H)

14. C₅×C₇²×C₁₃

C₂²×C₃

semidirect

20. #14274263

C₂×D₅

semidirect

28. C₇⋊D₉₁

C₃×D₅

semidirect

32. C₇×D₅×C₁₃

AGL(1,𝔽₇)

semidirect

36. C₇²×C₁₃

C₂×C₃×D₅

semidirect

42. C₇²×D₅

D₁₃⋊C₃

semidirect

44. C₅×C₇×C₁₃

C₂×AGL(1,𝔽₇)

semidirect

55. C₅×C₇²

C₂×D₁₃⋊C₃

semidirect

69. D₅×C₁₃

#64424

semidirect

74. C₇×C₁₃

D₅×AGL(1,𝔽₇)

semidirect

81. C₇×D₅

#10587722

semidirect

82. C₅×C₁₃

C₂×#64424

semidirect

85. C₇²

D₅×D₁₃⋊C₃

semidirect

90. C₅×C₇

C₂×#10587722

semidirect

104. C₁₃

D₅×#64424

semidirect

108. D₅

#16838446

direct (N × H)

109. C₇

D₅×#10587722

semidirect

113. C₅

C₂×#16838446

semidirect

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

subgroup	index
D₅×#14274263	2
C₅⋊#16838446	2
C₅×#16838446	2
D₅×C₇⋊D₉₁	3
C₂×#16838446	5
D₅×#10587722	7
D₅×#64424	13

subgroup

index

D₅×C₇⋊D₉₁

Subgroups — D₅×#16838446 #50304103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5×#16838446 #50304103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#16838446 #50304103