Group #13132699 — subgroups

Subgroups — C₂×D₇×Dic₁₅ #13132699

Order 1680.

216 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₂²×C₃×C₅×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₀₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	D₇×Dic₁₅	C₂	1	4	yes	maximal, direct product factor
5	C₂×D₇×Dic₅	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₅×D₇	C₄	1	4	yes	complemented
8	C₂×C₃×C₅×D₇	C₂²	1	2	yes	—
9	Dic₁₀₅	C₂²	1	2	yes	complemented
10	C₇×Dic₁₅	C₂²	1	2	yes	complemented
11	C₂×Dic₃×D₇	—	5	1	no	maximal, Hall
12	C₂²×C₅×D₇	D₃	1	1	yes	characteristic
13	C₂×Dic₃₅	—	3	1	no	—
14	C₂×C₇×Dic₅	—	3	1	no	—
15	D₇×Dic₅	—	3	4	no	—
16	C₂²×Dic₁₅	—	7	1	no	maximal, Hall
17	C₂×C₃×C₅×C₇	C₂×C₄	1	2	yes	complemented
18	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
19	C₃×C₅×D₇	C₂×C₄	1	4	yes	complemented
20	C₂²×C₃×D₇	D₅	1	1	yes	characteristic
21	C₂×Dic₂₁	—	5	1	no	—
22	C₂×C₇×Dic₃	—	5	1	no	—
23	Dic₃×D₇	—	5	4	no	—
24	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
25	C₂×C₅×D₇	Dic₃	1	4	yes	complemented
26	C₂×C₅×D₇	C₂×D₃	1	2	yes	—
27	Dic₃₅	—	3	2	no	—
28	C₇×Dic₅	—	3	2	no	—
29	C₂³×C₃×C₅	—	7	1	no	—
30	C₂×Dic₁₅	—	7	1	no	—
31	C₂×Dic₁₅	—	7	4	no	—
32	C₂×Dic₁₅	D₇	1	1	yes	characteristic, direct product factor
33	C₂×C₄×D₇	—	15	1	no	Hall
34	C₃×C₅×C₇	C₂²×C₄	1	1	yes	characteristic, complemented, Hall
35	C₂²×C₃×C₇	C₂×D₅	1	1	yes	characteristic
36	C₂×C₃×D₇	Dic₅	1	4	yes	complemented
37	C₂×C₃×D₇	C₂×D₅	1	2	yes	—
38	Dic₂₁	—	5	2	no	—
39	C₇×Dic₃	—	5	2	no	—
40	C₂²×Dic₅	—	21	1	no	Hall
41	C₂×C₅×C₇	C₂×Dic₃	1	2	yes	complemented
42	C₂×C₅×C₇	C₂²×D₃	1	1	yes	characteristic
43	C₅×D₇	C₂×Dic₃	1	4	yes	complemented
44	C₂²×C₃×C₅	—	7	4	no	—
45	C₂²×C₃×C₅	—	7	2	no	—
46	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
47	Dic₁₅	—	7	2	no	—
48	Dic₁₅	C₂×D₇	1	2	yes	direct product factor
49	C₂²×D₇	D₁₅	1	1	yes	characteristic
50	C₂×C₄×C₇	—	15	1	no	—
51	C₂×Dic₇	—	15	1	no	—
52	C₄×D₇	—	15	4	no	—
53	C₂²×Dic₃	—	35	1	no	Hall
54	C₂×C₃×C₇	C₂×Dic₅	1	2	yes	complemented
55	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
56	C₃×D₇	C₂×Dic₅	1	4	yes	complemented
57	C₂³×C₅	—	7	1	no	—
58	C₂×Dic₅	—	3	1	no	—
59	C₂×Dic₅	—	21	4	no	—
60	C₂×Dic₅	—	21	1	no	—
61	C₅×C₇	C₂²×Dic₃	1	1	yes	characteristic, complemented, Hall
62	C₂×C₃×C₅	—	7	4	no	—
63	C₂×C₃×C₅	C₄×D₇	1	2	yes	complemented
64	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
65	C₂²×C₇	C₂×D₁₅	1	1	yes	characteristic
66	C₂×D₇	Dic₁₅	1	4	yes	direct product factor
67	C₂×D₇	C₂×D₁₅	1	2	yes	—
68	C₄×C₇	—	15	2	no	—
69	Dic₇	—	15	2	no	—
70	C₂³×C₃	—	7	1	no	—
71	C₂×Dic₃	—	5	1	no	—
72	C₂×Dic₃	—	35	4	no	—
73	C₂×Dic₃	—	35	1	no	—
74	C₃×C₇	C₂²×Dic₅	1	1	yes	characteristic, complemented, Hall
75	C₂²×C₅	—	7	4	no	—
76	C₂²×C₅	—	7	2	no	—
77	C₂²×C₅	D₃×D₇	1	1	yes	characteristic
78	Dic₅	—	3	2	no	—
79	Dic₅	—	21	2	no	—
80	C₂²×C₄	—	105	1	no	Sylow
81	C₃×C₅	C₂×C₄×D₇	1	1	yes	characteristic, complemented, Hall
82	C₂×C₇	C₂×Dic₁₅	1	2	yes	complemented
83	C₂×C₇	C₂²×D₁₅	1	1	yes	characteristic
84	D₇	C₂×Dic₁₅	1	4	yes	direct product factor
85	C₂²×C₃	—	7	4	no	—
86	C₂²×C₃	—	7	2	no	—
87	C₂²×C₃	D₅×D₇	1	1	yes	characteristic
88	Dic₃	—	5	2	no	—
89	Dic₃	—	35	2	no	—
90	C₂×C₅	—	7	4	no	—
91	C₂×C₅	Dic₃×D₇	1	2	yes	complemented
92	C₂×C₅	C₂×D₃×D₇	1	1	yes	characteristic
93	C₂³	—	7	1	no	—
94	C₂×C₄	—	15	1	no	—
95	C₂×C₄	—	105	4	no	—
96	C₂×C₄	—	105	1	no	—
97	C₇	C₂²×Dic₁₅	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₃	—	7	4	no	—
99	C₂×C₃	D₇×Dic₅	1	2	yes	complemented
100	C₂×C₃	C₂×D₅×D₇	1	1	yes	characteristic
101	C₅	C₂×Dic₃×D₇	1	1	yes	characteristic, complemented, Sylow
102	C₂²	—	7	4	no	—
103	C₂²	—	7	2	no	—
104	C₂²	D₇×D₁₅	1	1	yes	characteristic
105	C₄	—	15	2	no	—
106	C₄	—	105	2	no	—
107	C₃	C₂×D₇×Dic₅	1	1	yes	characteristic, complemented, Sylow
108	C₂	—	7	4	no	—
109	C₂	D₇×Dic₁₅	1	2	yes	direct product factor
110	C₂	C₂×D₇×D₁₅	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃×C₅×D₇

C₂

yes

maximal, characteristic

C₂×Dic₁₀₅

C₂

yes

maximal, characteristic, complemented

C₂×C₇×Dic₁₅

C₂

yes

maximal, characteristic, complemented

D₇×Dic₁₅

C₂

yes

maximal, direct product factor

C₂×D₇×Dic₅

—

maximal, Hall

C₂²×C₃×C₅×C₇

C₂²

yes

characteristic

C₂×C₃×C₅×D₇

C₄

yes

complemented

C₂×C₃×C₅×D₇

C₂²

yes

—

Dic₁₀₅

C₂²

yes

complemented

C₇×Dic₁₅

C₂²

yes

complemented

C₂×Dic₃×D₇

—

maximal, Hall

C₂²×C₅×D₇

D₃

yes

characteristic

C₂×Dic₃₅

—

C₂×C₇×Dic₅

—

D₇×Dic₅

—

C₂²×Dic₁₅

—

maximal, Hall

C₂×C₃×C₅×C₇

C₂×C₄

yes

complemented

C₂×C₃×C₅×C₇

C₂³

yes

characteristic

C₃×C₅×D₇

C₂×C₄

yes

complemented

C₂²×C₃×D₇

D₅

yes

characteristic

C₂×Dic₂₁

—

C₂×C₇×Dic₃

—

Dic₃×D₇

—

C₂²×C₅×C₇

C₂×D₃

yes

characteristic

C₂×C₅×D₇

Dic₃

yes

complemented

C₂×C₅×D₇

C₂×D₃

yes

—

Dic₃₅

—

C₇×Dic₅

—

C₂³×C₃×C₅

—

C₂×Dic₁₅

—

C₂×Dic₁₅

—

C₂×Dic₁₅

D₇

yes

characteristic, direct product factor

C₂×C₄×D₇

—

Hall

C₃×C₅×C₇

C₂²×C₄

yes

characteristic, complemented, Hall

C₂²×C₃×C₇

C₂×D₅

yes

characteristic

C₂×C₃×D₇

Dic₅

yes

complemented

C₂×C₃×D₇

C₂×D₅

yes

—

Dic₂₁

—

C₇×Dic₃

—

C₂²×Dic₅

—

Hall

C₂×C₅×C₇

C₂×Dic₃

yes

complemented

C₂×C₅×C₇

C₂²×D₃

yes

characteristic

C₅×D₇

C₂×Dic₃

yes

complemented

C₂²×C₃×C₅

—

C₂²×C₃×C₅

—

C₂²×C₃×C₅

C₂×D₇

yes

characteristic

Dic₁₅

—

Dic₁₅

C₂×D₇

yes

direct product factor

C₂²×D₇

D₁₅

yes

characteristic

C₂×C₄×C₇

—

C₂×Dic₇

—

C₄×D₇

—

C₂²×Dic₃

—

Hall

C₂×C₃×C₇

C₂×Dic₅

yes

complemented

C₂×C₃×C₇

C₂²×D₅

yes

characteristic

C₃×D₇

C₂×Dic₅

yes

complemented

C₂³×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₇

C₂²×Dic₃

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

C₂×C₃×C₅

C₄×D₇

yes

complemented

C₂×C₃×C₅

C₂²×D₇

yes

characteristic

C₂²×C₇

C₂×D₁₅

yes

characteristic

C₂×D₇

Dic₁₅

yes

direct product factor

C₂×D₇

C₂×D₁₅

yes

—

C₄×C₇

—

Dic₇

—

C₂³×C₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₇

C₂²×Dic₅

yes

characteristic, complemented, Hall

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

D₃×D₇

yes

characteristic

Dic₅

—

Dic₅

—

C₂²×C₄

—

105

Sylow

C₃×C₅

C₂×C₄×D₇

yes

characteristic, complemented, Hall

C₂×C₇

C₂×Dic₁₅

yes

complemented

C₂×C₇

C₂²×D₁₅

yes

characteristic

D₇

C₂×Dic₁₅

yes

direct product factor

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

D₅×D₇

yes

characteristic

Dic₃

—

Dic₃

—

C₂×C₅

—

C₂×C₅

Dic₃×D₇

yes

complemented

C₂×C₅

C₂×D₃×D₇

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

105

—

C₂×C₄

—

105

—

C₇

C₂²×Dic₁₅

yes

characteristic, complemented, Sylow

C₂×C₃

—

C₂×C₃

D₇×Dic₅

yes

complemented

100

C₂×C₃

C₂×D₅×D₇

yes

characteristic

101

C₅

C₂×Dic₃×D₇

yes

characteristic, complemented, Sylow

102

C₂²

—

103

C₂²

—

104

C₂²

D₇×D₁₅

yes

characteristic

105

C₄

—

106

C₄

—

105

—

107

C₃

C₂×D₇×Dic₅

yes

characteristic, complemented, Sylow

108

C₂

—

109

C₂

D₇×Dic₁₅

yes

direct product factor

110

C₂

C₂×D₇×D₁₅

yes

characteristic

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

subgroup

G/H

center, hypercenter

C₂²

D₇×D₁₅

commutator, hypocenter

C₃×C₅×C₇

C₂²×C₄

Frattini

C₂

C₂×D₇×D₁₅

Fitting, socle

C₂²×C₃×C₅×C₇

C₂²

prime radical

C₂²×C₃×C₅×D₇

C₂

cosocle

C₂×C₃×C₅×C₇

C₂³

LCS

G ▷ C₃×C₅×C₇

UCS

C₁ ◁ C₂²

N	H ≅ G/N	type
2. C₂×Dic₁₀₅	C₂	semidirect
3. C₂×C₇×Dic₁₅	C₂	semidirect
4. D₇×Dic₁₅	C₂	direct (N × H)
7. C₂×C₃×C₅×D₇	C₄	semidirect
9. Dic₁₀₅	C₂²	semidirect
10. C₇×Dic₁₅	C₂²	semidirect
17. C₂×C₃×C₅×C₇	C₂×C₄	semidirect
19. C₃×C₅×D₇	C₂×C₄	semidirect
25. C₂×C₅×D₇	Dic₃	semidirect
32. C₂×Dic₁₅	D₇	direct (N × H)
34. C₃×C₅×C₇	C₂²×C₄	semidirect
36. C₂×C₃×D₇	Dic₅	semidirect
41. C₂×C₅×C₇	C₂×Dic₃	semidirect
43. C₅×D₇	C₂×Dic₃	semidirect
48. Dic₁₅	C₂×D₇	direct (N × H)
54. C₂×C₃×C₇	C₂×Dic₅	semidirect
56. C₃×D₇	C₂×Dic₅	semidirect
61. C₅×C₇	C₂²×Dic₃	semidirect
63. C₂×C₃×C₅	C₄×D₇	semidirect
66. C₂×D₇	Dic₁₅	direct (N × H)
74. C₃×C₇	C₂²×Dic₅	semidirect
81. C₃×C₅	C₂×C₄×D₇	semidirect
82. C₂×C₇	C₂×Dic₁₅	semidirect
84. D₇	C₂×Dic₁₅	direct (N × H)
91. C₂×C₅	Dic₃×D₇	semidirect
97. C₇	C₂²×Dic₁₅	semidirect
99. C₂×C₃	D₇×Dic₅	semidirect
101. C₅	C₂×Dic₃×D₇	semidirect
107. C₃	C₂×D₇×Dic₅	semidirect
109. C₂	D₇×Dic₁₅	direct (N × H)

H ≅ G/N

type

2. C₂×Dic₁₀₅

C₂

semidirect

3. C₂×C₇×Dic₁₅

C₂

semidirect

4. D₇×Dic₁₅

C₂

direct (N × H)

7. C₂×C₃×C₅×D₇

C₄

semidirect

9. Dic₁₀₅

C₂²

semidirect

10. C₇×Dic₁₅

C₂²

semidirect

17. C₂×C₃×C₅×C₇

C₂×C₄

semidirect

19. C₃×C₅×D₇

C₂×C₄

semidirect

25. C₂×C₅×D₇

Dic₃

semidirect

32. C₂×Dic₁₅

D₇

direct (N × H)

34. C₃×C₅×C₇

C₂²×C₄

semidirect

36. C₂×C₃×D₇

Dic₅

semidirect

41. C₂×C₅×C₇

C₂×Dic₃

semidirect

43. C₅×D₇

C₂×Dic₃

semidirect

48. Dic₁₅

C₂×D₇

direct (N × H)

54. C₂×C₃×C₇

C₂×Dic₅

semidirect

56. C₃×D₇

C₂×Dic₅

semidirect

61. C₅×C₇

C₂²×Dic₃

semidirect

63. C₂×C₃×C₅

C₄×D₇

semidirect

66. C₂×D₇

Dic₁₅

direct (N × H)

74. C₃×C₇

C₂²×Dic₅

semidirect

81. C₃×C₅

C₂×C₄×D₇

semidirect

82. C₂×C₇

C₂×Dic₁₅

semidirect

84. D₇

C₂×Dic₁₅

direct (N × H)

91. C₂×C₅

Dic₃×D₇

semidirect

97. C₇

C₂²×Dic₁₅

semidirect

99. C₂×C₃

D₇×Dic₅

semidirect

101. C₅

C₂×Dic₃×D₇

semidirect

107. C₃

C₂×D₇×Dic₅

semidirect

109. C₂

D₇×Dic₁₅

direct (N × H)

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

subgroup	index
C₂²×C₃×C₅×D₇	2
C₂×Dic₁₀₅	2
C₂×C₇×Dic₁₅	2
D₇×Dic₁₅	2
C₂×D₇×Dic₅	3
C₂×Dic₃×D₇	5
C₂²×Dic₁₅	7

subgroup

index

C₂²×C₃×C₅×D₇

C₂×Dic₁₀₅

C₂×C₇×Dic₁₅

D₇×Dic₁₅

C₂×D₇×Dic₅

C₂×Dic₃×D₇

C₂²×Dic₁₅

Subgroups — C₂×D₇×Dic₁₅ #13132699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D7×Dic15 #13132699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₇×Dic₁₅ #13132699