Group #13132696 — subgroups

Subgroups — D₇×D₆₀ #13132696

Order 1680.

216 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 58 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×D₇×D₁₅	C₂	1	2	yes	maximal, complemented
2	D₄₂₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₅×D₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719759	C₂	1	2	yes	maximal, complemented
6	D₇×D₂₀	—	3	1	no	maximal, Hall
7	C₂×D₁₀₅	C₂²	1	2	yes	complemented
8	C₂×C₇×D₁₅	C₂²	1	2	yes	complemented
9	C₂×C₃×C₅×D₇	C₂²	1	1	yes	characteristic
10	D₇×D₁₅	—	2	4	no	—
11	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic, complemented
12	C₃×C₅×Dic₇	C₂²	1	1	yes	characteristic, complemented
13	D₇×D₁₂	—	5	1	no	maximal, Hall
14	C₂×D₅×D₇	—	3	2	no	—
15	D₁₄₀	—	3	1	no	—
16	C₇×D₂₀	—	3	1	no	—
17	C₄×C₅×D₇	D₃	1	1	yes	characteristic, complemented
18	#63307	—	3	2	no	—
19	C₂×D₆₀	—	7	1	no	maximal, Hall
20	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
21	D₁₀₅	—	2	2	no	—
22	C₇×D₁₅	—	2	2	no	—
23	C₃×C₅×D₇	D₄	1	2	yes	complemented
24	C₂×D₃×D₇	—	5	2	no	—
25	D₈₄	—	5	1	no	—
26	C₇×D₁₂	—	5	1	no	—
27	C₃×C₄×D₇	D₅	1	1	yes	characteristic, complemented
28	#4247	—	5	2	no	—
29	C₂×D₃₅	—	3	2	no	—
30	C₂×C₇×D₅	—	3	2	no	—
31	C₂×C₅×D₇	C₂×D₃	1	1	yes	characteristic
32	D₅×D₇	—	6	4	no	—
33	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic, complemented
34	C₅×Dic₇	C₂×D₃	1	1	yes	characteristic, complemented
35	C₂²×D₁₅	—	7	2	no	—
36	C₂×C₃×C₄×C₅	—	7	1	no	—
37	D₆₀	—	7	2	no	—
38	D₆₀	—	7	1	no	—
39	D₆₀	D₇	1	1	yes	characteristic, direct product factor
40	D₄×D₇	—	15	1	no	Hall
41	C₃×C₅×C₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
42	C₂×D₂₁	—	5	2	no	—
43	C₂×D₃×C₇	—	5	2	no	—
44	C₂×C₃×D₇	C₂×D₅	1	1	yes	characteristic
45	D₃×D₇	—	10	4	no	—
46	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic, complemented
47	C₃×Dic₇	C₂×D₅	1	1	yes	characteristic, complemented
48	C₂×D₂₀	—	21	1	no	Hall
49	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
50	D₃₅	—	6	2	no	—
51	C₅×D₇	D₁₂	1	2	yes	complemented
52	C₇×D₅	—	6	2	no	—
53	C₂²×C₃×C₅	—	7	1	no	—
54	C₂×D₁₅	—	7	2	no	—
55	C₂×D₁₅	—	14	4	no	—
56	C₂×D₁₅	C₂×D₇	1	2	yes	complemented
57	C₃×C₄×C₅	—	7	1	no	—
58	C₃×C₄×C₅	C₂×D₇	1	1	yes	characteristic, complemented
59	C₂²×D₇	—	15	2	no	—
60	C₇×D₄	—	15	1	no	—
61	C₂²⋊D₇	—	15	2	no	—
62	D₂₈	—	15	1	no	—
63	C₄×D₇	D₁₅	1	1	yes	characteristic, complemented
64	C₂×D₁₂	—	35	1	no	Hall
65	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
66	D₂₁	—	10	2	no	—
67	C₃×D₇	D₂₀	1	2	yes	complemented
68	D₃×C₇	—	10	2	no	—
69	C₂²×D₅	—	21	2	no	—
70	C₂×C₄×C₅	—	7	1	no	—
71	D₂₀	—	3	1	no	—
72	D₂₀	—	21	2	no	—
73	D₂₀	—	21	1	no	—
74	C₅×C₇	C₂×D₁₂	1	1	yes	characteristic, complemented, Hall
75	C₂×C₃×C₅	—	7	2	no	—
76	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
77	D₁₅	—	2	2	no	—
78	D₁₅	—	14	2	no	—
79	C₂²×C₇	—	15	2	no	—
80	C₂×D₇	—	15	2	no	—
81	C₂×D₇	—	30	4	no	—
82	C₂×D₇	C₂×D₁₅	1	1	yes	characteristic
83	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic, complemented
84	Dic₇	C₂×D₁₅	1	1	yes	characteristic, complemented
85	C₂²×D₃	—	35	2	no	—
86	C₂×C₃×C₄	—	7	1	no	—
87	D₁₂	—	5	1	no	—
88	D₁₂	—	35	2	no	—
89	D₁₂	—	35	1	no	—
90	C₃×C₇	C₂×D₂₀	1	1	yes	characteristic, complemented, Hall
91	C₂²×C₅	—	7	1	no	—
92	C₂×D₅	—	3	2	no	—
93	C₂×D₅	—	21	2	no	—
94	C₂×D₅	—	42	4	no	—
95	C₄×C₅	—	7	1	no	—
96	C₄×C₅	D₃×D₇	1	1	yes	characteristic, complemented
97	C₂×D₄	—	105	1	no	Sylow
98	C₃×C₅	D₄×D₇	1	1	yes	characteristic, complemented, Hall
99	C₂×C₇	—	30	2	no	—
100	C₂×C₇	C₂²×D₁₅	1	1	yes	characteristic
101	D₇	—	30	2	no	—
102	D₇	D₆₀	1	2	yes	direct product factor
103	C₂²×C₃	—	7	1	no	—
104	C₂×D₃	—	5	2	no	—
105	C₂×D₃	—	35	2	no	—
106	C₂×D₃	—	70	4	no	—
107	C₃×C₄	—	7	1	no	—
108	C₃×C₄	D₅×D₇	1	1	yes	characteristic, complemented
109	C₂×C₅	—	7	2	no	—
110	C₂×C₅	C₂×D₃×D₇	1	1	yes	characteristic
111	D₅	—	6	2	no	—
112	D₅	—	42	2	no	—
113	C₂³	—	105	2	no	—
114	D₄	—	15	1	no	—
115	D₄	—	105	2	no	—
116	D₄	—	105	1	no	—
117	C₂×C₄	—	7	1	no	—
118	C₇	C₂×D₆₀	1	1	yes	characteristic, complemented, Sylow
119	C₂×C₃	—	7	2	no	—
120	C₂×C₃	C₂×D₅×D₇	1	1	yes	characteristic
121	D₃	—	10	2	no	—
122	D₃	—	70	2	no	—
123	C₅	D₇×D₁₂	1	1	yes	characteristic, complemented, Sylow
124	C₂²	—	7	1	no	—
125	C₂²	—	15	2	no	—
126	C₂²	—	105	2	no	—
127	C₂²	—	210	4	no	—
128	C₄	—	7	1	no	—
129	C₄	D₇×D₁₅	1	1	yes	characteristic, complemented
130	C₃	D₇×D₂₀	1	1	yes	characteristic, complemented, Sylow
131	C₂	—	7	2	no	—
132	C₂	—	30	2	no	—
133	C₂	—	210	2	no	—
134	C₂	C₂×D₇×D₁₅	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₇×D₁₅

C₂

yes

maximal, complemented

D₄₂₀

C₂

yes

maximal, characteristic, complemented

C₇×D₆₀

C₂

yes

maximal, characteristic, complemented

C₃×C₄×C₅×D₇

C₂

yes

maximal, characteristic, complemented

#11719759

C₂

yes

maximal, complemented

D₇×D₂₀

—

maximal, Hall

C₂×D₁₀₅

C₂²

yes

complemented

C₂×C₇×D₁₅

C₂²

yes

complemented

C₂×C₃×C₅×D₇

C₂²

yes

characteristic

D₇×D₁₅

—

C₃×C₄×C₅×C₇

C₂²

yes

characteristic, complemented

C₃×C₅×Dic₇

C₂²

yes

characteristic, complemented

D₇×D₁₂

—

maximal, Hall

C₂×D₅×D₇

—

D₁₄₀

—

C₇×D₂₀

—

C₄×C₅×D₇

D₃

yes

characteristic, complemented

#63307

—

C₂×D₆₀

—

maximal, Hall

C₂×C₃×C₅×C₇

C₂³

yes

characteristic

D₁₀₅

—

C₇×D₁₅

—

C₃×C₅×D₇

D₄

yes

complemented

C₂×D₃×D₇

—

D₈₄

—

C₇×D₁₂

—

C₃×C₄×D₇

D₅

yes

characteristic, complemented

#4247

—

C₂×D₃₅

—

C₂×C₇×D₅

—

C₂×C₅×D₇

C₂×D₃

yes

characteristic

D₅×D₇

—

C₄×C₅×C₇

C₂×D₃

yes

characteristic, complemented

C₅×Dic₇

C₂×D₃

yes

characteristic, complemented

C₂²×D₁₅

—

C₂×C₃×C₄×C₅

—

D₆₀

—

D₆₀

—

D₆₀

D₇

yes

characteristic, direct product factor

D₄×D₇

—

Hall

C₃×C₅×C₇

C₂×D₄

yes

characteristic, complemented, Hall

C₂×D₂₁

—

C₂×D₃×C₇

—

C₂×C₃×D₇

C₂×D₅

yes

characteristic

D₃×D₇

—

C₃×C₄×C₇

C₂×D₅

yes

characteristic, complemented

C₃×Dic₇

C₂×D₅

yes

characteristic, complemented

C₂×D₂₀

—

Hall

C₂×C₅×C₇

C₂²×D₃

yes

characteristic

D₃₅

—

C₅×D₇

D₁₂

yes

complemented

C₇×D₅

—

C₂²×C₃×C₅

—

C₂×D₁₅

—

C₂×D₁₅

—

C₂×D₁₅

C₂×D₇

yes

complemented

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×D₇

yes

characteristic, complemented

C₂²×D₇

—

C₇×D₄

—

C₂²⋊D₇

—

D₂₈

—

C₄×D₇

D₁₅

yes

characteristic, complemented

C₂×D₁₂

—

Hall

C₂×C₃×C₇

C₂²×D₅

yes

characteristic

D₂₁

—

C₃×D₇

D₂₀

yes

complemented

D₃×C₇

—

C₂²×D₅

—

C₂×C₄×C₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₇

C₂×D₁₂

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

C₂×C₃×C₅

C₂²×D₇

yes

characteristic

D₁₅

—

D₁₅

—

C₂²×C₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂×D₁₅

yes

characteristic

C₄×C₇

C₂×D₁₅

yes

characteristic, complemented

Dic₇

C₂×D₁₅

yes

characteristic, complemented

C₂²×D₃

—

C₂×C₃×C₄

—

D₁₂

—

D₁₂

—

D₁₂

—

C₃×C₇

C₂×D₂₀

yes

characteristic, complemented, Hall

C₂²×C₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

D₃×D₇

yes

characteristic, complemented

C₂×D₄

—

105

Sylow

C₃×C₅

D₄×D₇

yes

characteristic, complemented, Hall

C₂×C₇

—

100

C₂×C₇

C₂²×D₁₅

yes

characteristic

101

D₇

—

102

D₇

D₆₀

yes

direct product factor

103

C₂²×C₃

—

104

C₂×D₃

—

105

C₂×D₃

—

106

C₂×D₃

—

107

C₃×C₄

—

108

C₃×C₄

D₅×D₇

yes

characteristic, complemented

109

C₂×C₅

—

110

C₂×C₅

C₂×D₃×D₇

yes

characteristic

111

D₅

—

112

D₅

—

113

C₂³

—

105

—

114

D₄

—

115

D₄

—

105

—

116

D₄

—

105

—

117

C₂×C₄

—

118

C₇

C₂×D₆₀

yes

characteristic, complemented, Sylow

119

C₂×C₃

—

120

C₂×C₃

C₂×D₅×D₇

yes

characteristic

121

D₃

—

122

D₃

—

123

C₅

D₇×D₁₂

yes

characteristic, complemented, Sylow

124

C₂²

—

125

C₂²

—

126

C₂²

—

105

—

127

C₂²

—

210

—

128

C₄

—

129

C₄

D₇×D₁₅

yes

characteristic, complemented

130

C₃

D₇×D₂₀

yes

characteristic, complemented, Sylow

131

C₂

—

132

C₂

—

133

C₂

—

210

—

134

C₂

C₂×D₇×D₁₅

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₇×D₁₅
commutator, socle, cosocle	C₂×C₃×C₅×C₇	C₂³
Fitting	C₃×C₄×C₅×C₇	C₂²
hypercenter	C₄	D₇×D₁₅
hypocenter	C₃×C₅×C₇	C₂×D₄
LCS	G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄

subgroup

G/H

center, Frattini

C₂

C₂×D₇×D₁₅

commutator, socle, cosocle

C₂×C₃×C₅×C₇

C₂³

Fitting

C₃×C₄×C₅×C₇

C₂²

hypercenter

C₄

D₇×D₁₅

hypocenter

C₃×C₅×C₇

C₂×D₄

LCS

G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇

UCS

C₁ ◁ C₂ ◁ C₄

N	H ≅ G/N	type
1. C₂×D₇×D₁₅	C₂	semidirect
2. D₄₂₀	C₂	semidirect
3. C₇×D₆₀	C₂	semidirect
4. C₃×C₄×C₅×D₇	C₂	semidirect
5. #11719759	C₂	semidirect
7. C₂×D₁₀₅	C₂²	semidirect
8. C₂×C₇×D₁₅	C₂²	semidirect
11. C₃×C₄×C₅×C₇	C₂²	semidirect
12. C₃×C₅×Dic₇	C₂²	semidirect
17. C₄×C₅×D₇	D₃	semidirect
23. C₃×C₅×D₇	D₄	semidirect
27. C₃×C₄×D₇	D₅	semidirect
33. C₄×C₅×C₇	C₂×D₃	semidirect
34. C₅×Dic₇	C₂×D₃	semidirect
39. D₆₀	D₇	direct (N × H)
41. C₃×C₅×C₇	C₂×D₄	semidirect
46. C₃×C₄×C₇	C₂×D₅	semidirect
47. C₃×Dic₇	C₂×D₅	semidirect
51. C₅×D₇	D₁₂	semidirect
56. C₂×D₁₅	C₂×D₇	semidirect
58. C₃×C₄×C₅	C₂×D₇	semidirect
63. C₄×D₇	D₁₅	semidirect
67. C₃×D₇	D₂₀	semidirect
74. C₅×C₇	C₂×D₁₂	semidirect
83. C₄×C₇	C₂×D₁₅	semidirect
84. Dic₇	C₂×D₁₅	semidirect
90. C₃×C₇	C₂×D₂₀	semidirect
96. C₄×C₅	D₃×D₇	semidirect
98. C₃×C₅	D₄×D₇	semidirect
102. D₇	D₆₀	direct (N × H)
108. C₃×C₄	D₅×D₇	semidirect
118. C₇	C₂×D₆₀	semidirect
123. C₅	D₇×D₁₂	semidirect
129. C₄	D₇×D₁₅	semidirect
130. C₃	D₇×D₂₀	semidirect

H ≅ G/N

type

1. C₂×D₇×D₁₅

C₂

semidirect

2. D₄₂₀

C₂

semidirect

3. C₇×D₆₀

C₂

semidirect

4. C₃×C₄×C₅×D₇

C₂

semidirect

5. #11719759

C₂

semidirect

7. C₂×D₁₀₅

C₂²

semidirect

8. C₂×C₇×D₁₅

C₂²

semidirect

11. C₃×C₄×C₅×C₇

C₂²

semidirect

12. C₃×C₅×Dic₇

C₂²

semidirect

17. C₄×C₅×D₇

D₃

semidirect

23. C₃×C₅×D₇

D₄

semidirect

27. C₃×C₄×D₇

D₅

semidirect

33. C₄×C₅×C₇

C₂×D₃

semidirect

34. C₅×Dic₇

C₂×D₃

semidirect

39. D₆₀

D₇

direct (N × H)

41. C₃×C₅×C₇

C₂×D₄

semidirect

46. C₃×C₄×C₇

C₂×D₅

semidirect

47. C₃×Dic₇

C₂×D₅

semidirect

51. C₅×D₇

D₁₂

semidirect

56. C₂×D₁₅

C₂×D₇

semidirect

58. C₃×C₄×C₅

C₂×D₇

semidirect

63. C₄×D₇

D₁₅

semidirect

67. C₃×D₇

D₂₀

semidirect

74. C₅×C₇

C₂×D₁₂

semidirect

83. C₄×C₇

C₂×D₁₅

semidirect

84. Dic₇

C₂×D₁₅

semidirect

90. C₃×C₇

C₂×D₂₀

semidirect

96. C₄×C₅

D₃×D₇

semidirect

98. C₃×C₅

D₄×D₇

semidirect

102. D₇

D₆₀

direct (N × H)

108. C₃×C₄

D₅×D₇

semidirect

118. C₇

C₂×D₆₀

semidirect

123. C₅

D₇×D₁₂

semidirect

129. C₄

D₇×D₁₅

semidirect

130. C₃

D₇×D₂₀

semidirect

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

subgroup	index
C₂×D₇×D₁₅	2
D₄₂₀	2
C₇×D₆₀	2
C₃×C₄×C₅×D₇	2
#11719759	2
D₇×D₂₀	3
D₇×D₁₂	5
C₂×D₆₀	7

subgroup

index

C₂×D₇×D₁₅

D₄₂₀

C₇×D₆₀

C₃×C₄×C₅×D₇

#11719759

D₇×D₂₀

D₇×D₁₂

C₂×D₆₀

Subgroups — D₇×D₆₀ #13132696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×D60 #13132696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×D₆₀ #13132696