Group #13132685 — subgroups

Subgroups — D₇×Dic₃₀ #13132685

Order 1680.

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

normal subgroups only

This group has 152 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	Dic₂₁₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₅×D₇	C₂	1	1	yes	maximal, characteristic
4	#11719760	C₂	1	2	yes	maximal, complemented
5	D₇×Dic₁₅	C₂	1	2	yes	maximal
6	D₇×Dic₁₀	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×D₇	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	Dic₁₀₅	C₂²	1	2	yes	—
10	C₇×Dic₁₅	C₂²	1	2	yes	—
11	C₃×C₅×Dic₇	C₂²	1	1	yes	characteristic
12	D₇×Dic₆	—	5	1	no	maximal, Hall
13	Dic₇₀	—	3	1	no	—
14	C₇×Dic₁₀	—	3	1	no	—
15	C₄×C₅×D₇	D₃	1	1	yes	characteristic
16	#63308	—	3	2	no	—
17	D₇×Dic₅	—	3	2	no	—
18	C₂×Dic₃₀	—	7	1	no	maximal, Hall
19	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
20	C₃×C₅×D₇	Q₈	1	2	yes	complemented
21	Dic₄₂	—	5	1	no	—
22	C₇×Dic₆	—	5	1	no	—
23	C₃×C₄×D₇	D₅	1	1	yes	characteristic
24	#4248	—	5	2	no	—
25	Dic₃×D₇	—	5	2	no	—
26	C₂×C₅×D₇	C₂×D₃	1	1	yes	characteristic
27	C₄×C₅×C₇	C₂×D₃	1	1	yes	characteristic
28	Dic₃₅	—	3	2	no	—
29	C₅×Dic₇	C₂×D₃	1	1	yes	characteristic
30	C₇×Dic₅	—	3	2	no	—
31	C₂×C₃×C₄×C₅	—	7	1	no	—
32	C₂×Dic₁₅	—	7	2	no	—
33	Dic₃₀	—	7	1	no	—
34	Dic₃₀	—	7	2	no	—
35	Dic₃₀	D₇	1	1	yes	characteristic, direct product factor
36	Q₈×D₇	—	15	1	no	Hall
37	C₃×C₅×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
38	C₂×C₃×D₇	C₂×D₅	1	1	yes	characteristic
39	C₃×C₄×C₇	C₂×D₅	1	1	yes	characteristic
40	Dic₂₁	—	5	2	no	—
41	C₃×Dic₇	C₂×D₅	1	1	yes	characteristic
42	C₇×Dic₃	—	5	2	no	—
43	C₂×Dic₁₀	—	21	1	no	Hall
44	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
45	C₅×D₇	Dic₆	1	2	yes	complemented
46	C₂²×C₃×C₅	—	7	1	no	—
47	C₃×C₄×C₅	—	7	1	no	—
48	C₃×C₄×C₅	C₂×D₇	1	1	yes	characteristic
49	Dic₁₅	—	7	2	no	—
50	Dic₁₅	C₂×D₇	1	2	yes	—
51	C₇×Q₈	—	15	1	no	—
52	C₄×D₇	—	15	2	no	—
53	C₄×D₇	D₁₅	1	1	yes	characteristic
54	Dic₁₄	—	15	1	no	—
55	Dic₁₄	—	15	2	no	—
56	C₂×Dic₆	—	35	1	no	Hall
57	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
58	C₃×D₇	Dic₁₀	1	2	yes	complemented
59	C₂×C₄×C₅	—	7	1	no	—
60	C₂×Dic₅	—	21	2	no	—
61	Dic₁₀	—	3	1	no	—
62	Dic₁₀	—	21	2	no	—
63	Dic₁₀	—	21	1	no	—
64	C₅×C₇	C₂×Dic₆	1	1	yes	characteristic, complemented, Hall
65	C₂×C₃×C₅	—	7	2	no	—
66	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
67	C₂×D₇	C₂×D₁₅	1	1	yes	characteristic
68	C₄×C₇	—	15	2	no	—
69	C₄×C₇	C₂×D₁₅	1	1	yes	characteristic
70	Dic₇	—	15	2	no	—
71	Dic₇	C₂×D₁₅	1	1	yes	characteristic
72	C₂×C₃×C₄	—	7	1	no	—
73	C₂×Dic₃	—	35	2	no	—
74	Dic₆	—	5	1	no	—
75	Dic₆	—	35	2	no	—
76	Dic₆	—	35	1	no	—
77	C₃×C₇	C₂×Dic₁₀	1	1	yes	characteristic, complemented, Hall
78	C₂²×C₅	—	7	1	no	—
79	C₄×C₅	—	7	1	no	—
80	C₄×C₅	D₃×D₇	1	1	yes	characteristic
81	Dic₅	—	3	2	no	—
82	Dic₅	—	21	2	no	—
83	C₂×Q₈	—	105	1	no	Sylow
84	C₃×C₅	Q₈×D₇	1	1	yes	characteristic, complemented, Hall
85	C₂×C₇	C₂²×D₁₅	1	1	yes	characteristic
86	D₇	Dic₃₀	1	2	yes	direct product factor
87	C₂²×C₃	—	7	1	no	—
88	C₃×C₄	—	7	1	no	—
89	C₃×C₄	D₅×D₇	1	1	yes	characteristic
90	Dic₃	—	5	2	no	—
91	Dic₃	—	35	2	no	—
92	C₂×C₅	—	7	2	no	—
93	C₂×C₅	C₂×D₃×D₇	1	1	yes	characteristic
94	Q₈	—	15	1	no	—
95	Q₈	—	105	2	no	—
96	Q₈	—	105	1	no	—
97	C₂×C₄	—	7	1	no	—
98	C₂×C₄	—	105	2	no	—
99	C₇	C₂×Dic₃₀	1	1	yes	characteristic, complemented, Sylow
100	C₂×C₃	—	7	2	no	—
101	C₂×C₃	C₂×D₅×D₇	1	1	yes	characteristic
102	C₅	D₇×Dic₆	1	1	yes	characteristic, complemented, Sylow
103	C₂²	—	7	1	no	—
104	C₄	—	7	1	no	—
105	C₄	—	15	2	no	—
106	C₄	—	105	2	no	—
107	C₄	D₇×D₁₅	1	1	yes	characteristic
108	C₃	D₇×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
109	C₂	—	7	2	no	—
110	C₂	C₂×D₇×D₁₅	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Dic₂₁₀

C₂

yes

maximal, characteristic, complemented

C₇×Dic₃₀

C₂

yes

maximal, characteristic, complemented

C₃×C₄×C₅×D₇

C₂

yes

maximal, characteristic

#11719760

C₂

yes

maximal, complemented

D₇×Dic₁₅

C₂

yes

maximal

D₇×Dic₁₀

—

maximal, Hall

C₂×C₃×C₅×D₇

C₂²

yes

characteristic

C₃×C₄×C₅×C₇

C₂²

yes

characteristic

Dic₁₀₅

C₂²

yes

—

C₇×Dic₁₅

C₂²

yes

—

C₃×C₅×Dic₇

C₂²

yes

characteristic

D₇×Dic₆

—

maximal, Hall

Dic₇₀

—

C₇×Dic₁₀

—

C₄×C₅×D₇

D₃

yes

characteristic

#63308

—

D₇×Dic₅

—

C₂×Dic₃₀

—

maximal, Hall

C₂×C₃×C₅×C₇

C₂³

yes

characteristic

C₃×C₅×D₇

Q₈

yes

complemented

Dic₄₂

—

C₇×Dic₆

—

C₃×C₄×D₇

D₅

yes

characteristic

#4248

—

Dic₃×D₇

—

C₂×C₅×D₇

C₂×D₃

yes

characteristic

C₄×C₅×C₇

C₂×D₃

yes

characteristic

Dic₃₅

—

C₅×Dic₇

C₂×D₃

yes

characteristic

C₇×Dic₅

—

C₂×C₃×C₄×C₅

—

C₂×Dic₁₅

—

Dic₃₀

—

Dic₃₀

—

Dic₃₀

D₇

yes

characteristic, direct product factor

Q₈×D₇

—

Hall

C₃×C₅×C₇

C₂×Q₈

yes

characteristic, complemented, Hall

C₂×C₃×D₇

C₂×D₅

yes

characteristic

C₃×C₄×C₇

C₂×D₅

yes

characteristic

Dic₂₁

—

C₃×Dic₇

C₂×D₅

yes

characteristic

C₇×Dic₃

—

C₂×Dic₁₀

—

Hall

C₂×C₅×C₇

C₂²×D₃

yes

characteristic

C₅×D₇

Dic₆

yes

complemented

C₂²×C₃×C₅

—

C₃×C₄×C₅

—

C₃×C₄×C₅

C₂×D₇

yes

characteristic

Dic₁₅

—

Dic₁₅

C₂×D₇

yes

—

C₇×Q₈

—

C₄×D₇

—

C₄×D₇

D₁₅

yes

characteristic

Dic₁₄

—

Dic₁₄

—

C₂×Dic₆

—

Hall

C₂×C₃×C₇

C₂²×D₅

yes

characteristic

C₃×D₇

Dic₁₀

yes

complemented

C₂×C₄×C₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₇

C₂×Dic₆

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

C₂×C₃×C₅

C₂²×D₇

yes

characteristic

C₂×D₇

C₂×D₁₅

yes

characteristic

C₄×C₇

—

C₄×C₇

C₂×D₁₅

yes

characteristic

Dic₇

—

Dic₇

C₂×D₁₅

yes

characteristic

C₂×C₃×C₄

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₇

C₂×Dic₁₀

yes

characteristic, complemented, Hall

C₂²×C₅

—

C₄×C₅

—

C₄×C₅

D₃×D₇

yes

characteristic

Dic₅

—

Dic₅

—

C₂×Q₈

—

105

Sylow

C₃×C₅

Q₈×D₇

yes

characteristic, complemented, Hall

C₂×C₇

C₂²×D₁₅

yes

characteristic

D₇

Dic₃₀

yes

direct product factor

C₂²×C₃

—

C₃×C₄

—

C₃×C₄

D₅×D₇

yes

characteristic

Dic₃

—

Dic₃

—

C₂×C₅

—

C₂×C₅

C₂×D₃×D₇

yes

characteristic

Q₈

—

Q₈

—

105

—

Q₈

—

105

—

C₂×C₄

—

C₂×C₄

—

105

—

C₇

C₂×Dic₃₀

yes

characteristic, complemented, Sylow

100

C₂×C₃

—

101

C₂×C₃

C₂×D₅×D₇

yes

characteristic

102

C₅

D₇×Dic₆

yes

characteristic, complemented, Sylow

103

C₂²

—

104

C₄

—

105

C₄

—

106

C₄

—

105

—

107

C₄

D₇×D₁₅

yes

characteristic

108

C₃

D₇×Dic₁₀

yes

characteristic, complemented, Sylow

109

C₂

—

110

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
prime radical	C₂×C₃×C₅×D₇	C₂²
hypercenter	C₄	D₇×D₁₅
hypocenter	C₃×C₅×C₇	C₂×Q₈
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

prime radical

C₂×C₃×C₅×D₇

C₂²

hypercenter

C₄

D₇×D₁₅

hypocenter

C₃×C₅×C₇

C₂×Q₈

LCS

G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇

UCS

C₁ ◁ C₂ ◁ C₄

N	H ≅ G/N	type
1. Dic₂₁₀	C₂	semidirect
2. C₇×Dic₃₀	C₂	semidirect
4. #11719760	C₂	semidirect
20. C₃×C₅×D₇	Q₈	semidirect
35. Dic₃₀	D₇	direct (N × H)
37. C₃×C₅×C₇	C₂×Q₈	semidirect
45. C₅×D₇	Dic₆	semidirect
58. C₃×D₇	Dic₁₀	semidirect
64. C₅×C₇	C₂×Dic₆	semidirect
77. C₃×C₇	C₂×Dic₁₀	semidirect
84. C₃×C₅	Q₈×D₇	semidirect
86. D₇	Dic₃₀	direct (N × H)
99. C₇	C₂×Dic₃₀	semidirect
102. C₅	D₇×Dic₆	semidirect
108. C₃	D₇×Dic₁₀	semidirect

H ≅ G/N

type

1. Dic₂₁₀

C₂

semidirect

2. C₇×Dic₃₀

C₂

semidirect

4. #11719760

C₂

semidirect

20. C₃×C₅×D₇

Q₈

semidirect

35. Dic₃₀

D₇

direct (N × H)

37. C₃×C₅×C₇

C₂×Q₈

semidirect

45. C₅×D₇

Dic₆

semidirect

58. C₃×D₇

Dic₁₀

semidirect

64. C₅×C₇

C₂×Dic₆

semidirect

77. C₃×C₇

C₂×Dic₁₀

semidirect

84. C₃×C₅

Q₈×D₇

semidirect

86. D₇

Dic₃₀

direct (N × H)

99. C₇

C₂×Dic₃₀

semidirect

102. C₅

D₇×Dic₆

semidirect

108. C₃

D₇×Dic₁₀

semidirect

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

subgroup	index
Dic₂₁₀	2
C₇×Dic₃₀	2
C₃×C₄×C₅×D₇	2
#11719760	2
D₇×Dic₁₅	2
D₇×Dic₁₀	3
D₇×Dic₆	5
C₂×Dic₃₀	7

subgroup

index

Dic₂₁₀

C₇×Dic₃₀

C₃×C₄×C₅×D₇

#11719760

D₇×Dic₁₅

D₇×Dic₁₀

D₇×Dic₆

C₂×Dic₃₀

Subgroups — D₇×Dic₃₀ #13132685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×Dic30 #13132685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×Dic₃₀ #13132685