Group #13132955 — subgroups

Subgroups — C₂×C₅×A₄×D₇ #13132955

Order 1680.

156 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 42 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	C₂×C₅×C₇×A₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×A₄×D₇	C₂	1	2	yes	maximal, direct product factor
3	C₂³×C₅×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×C₃×C₅×D₇	—	4	1	no	maximal
5	C₅×C₇×A₄	C₂²	1	1	yes	characteristic, complemented
6	C₂×A₄×D₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₅×C₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂²×C₅×D₇	—	3	1	no	—
9	C₂²×C₅×D₇	—	3	1	no	—
10	C₂²×C₅×D₇	—	3	2	no	—
11	C₂²×C₅×D₇	C₂×C₃	1	2	yes	complemented
12	C₂²×C₅×A₄	—	7	1	no	maximal, Hall
13	C₂×C₃×C₅×C₇	—	4	1	no	—
14	C₃×C₅×D₇	—	4	2	no	—
15	C₂×C₇×A₄	C₂×C₅	1	1	yes	characteristic, complemented
16	A₄×D₇	C₂×C₅	1	2	yes	direct product factor
17	C₂²×C₅×C₇	—	3	1	no	—
18	C₂²×C₅×C₇	—	3	1	no	—
19	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic, complemented
20	C₂×C₅×D₇	—	3	2	no	—
21	C₂×C₅×D₇	—	3	1	no	—
22	C₂×C₅×D₇	—	3	2	no	—
23	C₂×C₅×D₇	—	3	2	no	—
24	C₂×C₅×D₇	—	3	2	no	—
25	C₂×C₅×D₇	A₄	1	1	yes	characteristic, direct product factor
26	C₂×C₅×A₄	—	7	2	no	—
27	C₂×C₅×A₄	D₇	1	1	yes	characteristic, direct product factor
28	C₂³×D₇	C₃×C₅	1	1	yes	characteristic, complemented, Hall
29	C₃×C₅×C₇	—	4	1	no	Hall
30	C₂×C₃×D₇	—	4	1	no	—
31	C₇×A₄	C₂²×C₅	1	1	yes	characteristic, complemented
32	C₂⁴×C₅	—	7	1	no	Hall
33	C₂×C₅×C₇	—	3	1	no	—
34	C₂×C₅×C₇	—	3	1	no	—
35	C₂×C₅×C₇	C₂×A₄	1	1	yes	characteristic, complemented
36	C₅×D₇	—	3	2	no	—
37	C₅×D₇	C₂×A₄	1	2	yes	direct product factor
38	C₂²×C₃×C₅	—	28	1	no	—
39	C₅×A₄	C₂×D₇	1	1	yes	characteristic, direct product factor
40	C₂³×C₇	C₂×C₃×C₅	1	1	yes	characteristic, complemented
41	C₂²×D₇	—	3	1	no	—
42	C₂²×D₇	—	3	1	no	—
43	C₂²×D₇	—	3	2	no	—
44	C₂²×D₇	C₂×C₃×C₅	1	2	yes	complemented
45	C₂²×A₄	—	7	1	no	Hall
46	C₂×C₃×C₇	—	4	1	no	—
47	C₃×D₇	—	4	2	no	—
48	C₂³×C₅	—	7	2	no	—
49	C₂³×C₅	—	21	1	no	—
50	C₂³×C₅	—	21	2	no	—
51	C₂³×C₅	—	21	1	no	—
52	C₂³×C₅	C₃×D₇	1	1	yes	characteristic, complemented
53	C₅×C₇	C₂²×A₄	1	1	yes	characteristic, complemented, Hall
54	C₂×C₃×C₅	—	4	1	no	—
55	C₂×C₃×C₅	—	28	2	no	—
56	C₂²×C₇	—	3	1	no	—
57	C₂²×C₇	—	3	1	no	—
58	C₂²×C₇	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
59	C₂×D₇	—	3	2	no	—
60	C₂×D₇	—	3	1	no	—
61	C₂×D₇	—	3	2	no	—
62	C₂×D₇	—	3	2	no	—
63	C₂×D₇	—	3	2	no	—
64	C₂×D₇	C₅×A₄	1	1	yes	characteristic, direct product factor
65	C₂×A₄	—	7	2	no	—
66	C₂×A₄	C₅×D₇	1	1	yes	characteristic, direct product factor
67	C₃×C₇	—	4	1	no	Hall
68	C₂²×C₅	—	3	1	no	—
69	C₂²×C₅	—	3	1	no	—
70	C₂²×C₅	—	7	1	no	—
71	C₂²×C₅	—	21	2	no	—
72	C₂²×C₅	—	21	2	no	—
73	C₂²×C₅	—	21	2	no	—
74	C₂²×C₅	—	21	2	no	—
75	C₂²×C₅	—	21	1	no	—
76	C₂²×C₅	C₂×C₃×D₇	1	1	yes	characteristic, complemented
77	C₂⁴	—	7	1	no	Sylow
78	C₃×C₅	—	4	1	no	Hall
79	C₂×C₇	—	3	1	no	—
80	C₂×C₇	—	3	1	no	—
81	C₂×C₇	C₂×C₅×A₄	1	1	yes	characteristic, complemented
82	D₇	—	3	2	no	—
83	D₇	C₂×C₅×A₄	1	2	yes	direct product factor
84	C₂²×C₃	—	28	1	no	—
85	A₄	C₂×C₅×D₇	1	1	yes	characteristic, direct product factor
86	C₂×C₅	—	3	1	no	—
87	C₂×C₅	—	3	1	no	—
88	C₂×C₅	—	7	2	no	—
89	C₂×C₅	—	21	2	no	—
90	C₂×C₅	A₄×D₇	1	1	yes	characteristic, direct product factor
91	C₂³	—	7	2	no	—
92	C₂³	—	21	1	no	—
93	C₂³	—	21	2	no	—
94	C₂³	—	21	1	no	—
95	C₂³	C₃×C₅×D₇	1	1	yes	characteristic, complemented
96	C₇	C₂²×C₅×A₄	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₃	—	4	1	no	—
98	C₂×C₃	—	28	2	no	—
99	C₅	C₂×A₄×D₇	1	1	yes	characteristic, direct product factor, Sylow
100	C₂²	—	3	1	no	—
101	C₂²	—	3	1	no	—
102	C₂²	—	7	1	no	—
103	C₂²	—	21	2	no	—
104	C₂²	—	21	2	no	—
105	C₂²	—	21	2	no	—
106	C₂²	—	21	2	no	—
107	C₂²	—	21	1	no	—
108	C₂²	C₂×C₃×C₅×D₇	1	1	yes	characteristic, complemented
109	C₃	—	4	1	no	Sylow
110	C₂	—	3	1	no	—
111	C₂	—	3	1	no	—
112	C₂	—	7	2	no	—
113	C₂	—	21	2	no	—
114	C₂	C₅×A₄×D₇	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×C₇×A₄

C₂

yes

maximal, characteristic, complemented

C₅×A₄×D₇

C₂

yes

maximal, direct product factor

C₂³×C₅×D₇

C₃

yes

maximal, characteristic, complemented, Hall

C₂×C₃×C₅×D₇

—

maximal

C₅×C₇×A₄

C₂²

yes

characteristic, complemented

C₂×A₄×D₇

C₅

yes

maximal, characteristic, direct product factor, Hall

C₂³×C₅×C₇

C₂×C₃

yes

characteristic, complemented

C₂²×C₅×D₇

—

C₂²×C₅×D₇

—

C₂²×C₅×D₇

—

C₂²×C₅×D₇

C₂×C₃

yes

complemented

C₂²×C₅×A₄

—

maximal, Hall

C₂×C₃×C₅×C₇

—

C₃×C₅×D₇

—

C₂×C₇×A₄

C₂×C₅

yes

characteristic, complemented

A₄×D₇

C₂×C₅

yes

direct product factor

C₂²×C₅×C₇

—

C₂²×C₅×C₇

—

C₂²×C₅×C₇

C₂²×C₃

yes

characteristic, complemented

C₂×C₅×D₇

—

C₂×C₅×D₇

—

C₂×C₅×D₇

—

C₂×C₅×D₇

—

C₂×C₅×D₇

—

C₂×C₅×D₇

A₄

yes

characteristic, direct product factor

C₂×C₅×A₄

—

C₂×C₅×A₄

D₇

yes

characteristic, direct product factor

C₂³×D₇

C₃×C₅

yes

characteristic, complemented, Hall

C₃×C₅×C₇

—

Hall

C₂×C₃×D₇

—

C₇×A₄

C₂²×C₅

yes

characteristic, complemented

C₂⁴×C₅

—

Hall

C₂×C₅×C₇

—

C₂×C₅×C₇

—

C₂×C₅×C₇

C₂×A₄

yes

characteristic, complemented

C₅×D₇

—

C₅×D₇

C₂×A₄

yes

direct product factor

C₂²×C₃×C₅

—

C₅×A₄

C₂×D₇

yes

characteristic, direct product factor

C₂³×C₇

C₂×C₃×C₅

yes

characteristic, complemented

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

C₂×C₃×C₅

yes

complemented

C₂²×A₄

—

Hall

C₂×C₃×C₇

—

C₃×D₇

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₃×D₇

yes

characteristic, complemented

C₅×C₇

C₂²×A₄

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂²×C₃×C₅

yes

characteristic, complemented

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₅×A₄

yes

characteristic, direct product factor

C₂×A₄

—

C₂×A₄

C₅×D₇

yes

characteristic, direct product factor

C₃×C₇

—

Hall

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₃×D₇

yes

characteristic, complemented

C₂⁴

—

Sylow

C₃×C₅

—

Hall

C₂×C₇

—

C₂×C₇

—

C₂×C₇

C₂×C₅×A₄

yes

characteristic, complemented

D₇

—

D₇

C₂×C₅×A₄

yes

direct product factor

C₂²×C₃

—

A₄

C₂×C₅×D₇

yes

characteristic, direct product factor

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

A₄×D₇

yes

characteristic, direct product factor

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₃×C₅×D₇

yes

characteristic, complemented

C₇

C₂²×C₅×A₄

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

—

C₅

C₂×A₄×D₇

yes

characteristic, direct product factor, Sylow

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂×C₃×C₅×D₇

yes

characteristic, complemented

109

C₃

—

Sylow

110

C₂

—

111

C₂

—

112

C₂

—

113

C₂

—

114

C₂

C₅×A₄×D₇

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	A₄×D₇
commutator, cosocle, hypocenter	C₂²×C₇	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂³×C₅×C₇	C₂×C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂×C₅

subgroup

G/H

center, hypercenter

C₂×C₅

A₄×D₇

commutator, cosocle, hypocenter

C₂²×C₇

C₂²×C₃×C₅

Frattini

C₁

Fitting, socle

C₂³×C₅×C₇

C₂×C₃

LCS

G ▷ C₂²×C₇

UCS

C₁ ◁ C₂×C₅

N	H ≅ G/N	type
1. C₂×C₅×C₇×A₄	C₂	semidirect
2. C₅×A₄×D₇	C₂	direct (N × H)
3. C₂³×C₅×D₇	C₃	semidirect
5. C₅×C₇×A₄	C₂²	semidirect
6. C₂×A₄×D₇	C₅	direct (N × H)
7. C₂³×C₅×C₇	C₂×C₃	semidirect
11. C₂²×C₅×D₇	C₂×C₃	semidirect
15. C₂×C₇×A₄	C₂×C₅	semidirect
16. A₄×D₇	C₂×C₅	direct (N × H)
19. C₂²×C₅×C₇	C₂²×C₃	semidirect
25. C₂×C₅×D₇	A₄	direct (N × H)
27. C₂×C₅×A₄	D₇	direct (N × H)
28. C₂³×D₇	C₃×C₅	semidirect
31. C₇×A₄	C₂²×C₅	semidirect
35. C₂×C₅×C₇	C₂×A₄	semidirect
37. C₅×D₇	C₂×A₄	direct (N × H)
39. C₅×A₄	C₂×D₇	direct (N × H)
40. C₂³×C₇	C₂×C₃×C₅	semidirect
44. C₂²×D₇	C₂×C₃×C₅	semidirect
52. C₂³×C₅	C₃×D₇	semidirect
53. C₅×C₇	C₂²×A₄	semidirect
58. C₂²×C₇	C₂²×C₃×C₅	semidirect
64. C₂×D₇	C₅×A₄	direct (N × H)
66. C₂×A₄	C₅×D₇	direct (N × H)
76. C₂²×C₅	C₂×C₃×D₇	semidirect
81. C₂×C₇	C₂×C₅×A₄	semidirect
83. D₇	C₂×C₅×A₄	direct (N × H)
85. A₄	C₂×C₅×D₇	direct (N × H)
90. C₂×C₅	A₄×D₇	direct (N × H)
95. C₂³	C₃×C₅×D₇	semidirect
96. C₇	C₂²×C₅×A₄	semidirect
99. C₅	C₂×A₄×D₇	direct (N × H)
108. C₂²	C₂×C₃×C₅×D₇	semidirect
114. C₂	C₅×A₄×D₇	direct (N × H)

H ≅ G/N

type

1. C₂×C₅×C₇×A₄

C₂

semidirect

2. C₅×A₄×D₇

C₂

direct (N × H)

3. C₂³×C₅×D₇

C₃

semidirect

5. C₅×C₇×A₄

C₂²

semidirect

6. C₂×A₄×D₇

C₅

direct (N × H)

7. C₂³×C₅×C₇

C₂×C₃

semidirect

11. C₂²×C₅×D₇

C₂×C₃

semidirect

15. C₂×C₇×A₄

C₂×C₅

semidirect

16. A₄×D₇

C₂×C₅

direct (N × H)

19. C₂²×C₅×C₇

C₂²×C₃

semidirect

25. C₂×C₅×D₇

A₄

direct (N × H)

27. C₂×C₅×A₄

D₇

direct (N × H)

28. C₂³×D₇

C₃×C₅

semidirect

31. C₇×A₄

C₂²×C₅

semidirect

35. C₂×C₅×C₇

C₂×A₄

semidirect

37. C₅×D₇

C₂×A₄

direct (N × H)

39. C₅×A₄

C₂×D₇

direct (N × H)

40. C₂³×C₇

C₂×C₃×C₅

semidirect

44. C₂²×D₇

C₂×C₃×C₅

semidirect

52. C₂³×C₅

C₃×D₇

semidirect

53. C₅×C₇

C₂²×A₄

semidirect

58. C₂²×C₇

C₂²×C₃×C₅

semidirect

64. C₂×D₇

C₅×A₄

direct (N × H)

66. C₂×A₄

C₅×D₇

direct (N × H)

76. C₂²×C₅

C₂×C₃×D₇

semidirect

81. C₂×C₇

C₂×C₅×A₄

semidirect

83. D₇

C₂×C₅×A₄

direct (N × H)

85. A₄

C₂×C₅×D₇

direct (N × H)

90. C₂×C₅

A₄×D₇

direct (N × H)

95. C₂³

C₃×C₅×D₇

semidirect

96. C₇

C₂²×C₅×A₄

semidirect

99. C₅

C₂×A₄×D₇

direct (N × H)

108. C₂²

C₂×C₃×C₅×D₇

semidirect

114. C₂

C₅×A₄×D₇

direct (N × H)

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

subgroup	index
C₂×C₅×C₇×A₄	2
C₅×A₄×D₇	2
C₂³×C₅×D₇	3
C₂×C₃×C₅×D₇	4
C₂×A₄×D₇	5
C₂²×C₅×A₄	7

subgroup

index

C₂×C₅×C₇×A₄

C₅×A₄×D₇

C₂³×C₅×D₇

C₂×C₃×C₅×D₇

C₂×A₄×D₇

C₂²×C₅×A₄

Subgroups — C₂×C₅×A₄×D₇ #13132955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C5×A4×D7 #13132955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×A₄×D₇ #13132955