Group #30655142 — subgroups

Subgroups — D₁₃×D₃₁₅ #30655142

Order 16380.

120 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 40 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	D₄₀₉₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×D₃₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₇×C₉×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₃×D₁₀₅	—	3	1	no	maximal
5	C₅×C₇×C₉×C₁₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₁₃×D₆₃	—	5	1	no	maximal, Hall
7	D₁₃₆₅	—	3	1	no	—
8	C₁₃×D₁₀₅	—	3	1	no	—
9	C₃×C₅×C₇×D₁₃	D₃	1	1	yes	characteristic
10	D₁₃×D₄₅	—	7	1	no	maximal, Hall
11	D₁₃×D₃₅	—	9	1	no	Hall
12	D₈₁₉	—	5	1	no	—
13	C₁₃×D₆₃	—	5	1	no	—
14	C₇×C₉×D₁₃	D₅	1	1	yes	characteristic, complemented
15	C₃×C₅×C₇×C₁₃	C₂×D₃	1	1	yes	characteristic
16	C₂×D₃₁₅	—	13	1	no	maximal, Hall
17	D₅₈₅	—	7	1	no	—
18	C₁₃×D₄₅	—	7	1	no	—
19	C₅×C₉×D₁₃	D₇	1	1	yes	characteristic, complemented
20	D₁₃×D₂₁	—	15	1	no	—
21	D₄₅₅	—	9	1	no	—
22	C₁₃×D₃₅	—	9	1	no	—
23	C₅×C₇×D₁₃	D₉	1	1	yes	characteristic, complemented
24	C₇×C₉×C₁₃	C₂×D₅	1	1	yes	characteristic, complemented, Hall
25	D₁₃×D₁₅	—	21	1	no	—
26	C₂×C₅×C₇×C₉	—	13	1	no	—
27	D₃₁₅	—	13	1	no	—
28	D₃₁₅	D₁₃	1	1	yes	characteristic, direct product factor
29	C₅×C₉×C₁₃	C₂×D₇	1	1	yes	characteristic, complemented, Hall
30	D₂₇₃	—	15	1	no	—
31	C₁₃×D₂₁	—	15	1	no	—
32	C₃×C₇×D₁₃	D₁₅	1	1	yes	characteristic
33	D₉×D₁₃	—	35	1	no	Hall
34	C₅×C₇×C₁₃	C₂×D₉	1	1	yes	characteristic, complemented, Hall
35	C₂×D₁₀₅	—	39	1	no	—
36	D₁₉₅	—	21	1	no	—
37	C₁₃×D₁₅	—	21	1	no	—
38	C₃×C₅×D₁₃	D₂₁	1	1	yes	characteristic
39	D₇×D₁₃	—	45	1	no	Hall
40	C₅×C₇×C₉	C₂×D₁₃	1	1	yes	characteristic, complemented, Hall
41	C₃×C₇×C₁₃	C₂×D₁₅	1	1	yes	characteristic
42	D₅×D₁₃	—	63	1	no	Hall
43	C₂×D₆₃	—	65	1	no	Hall
44	D₁₁₇	—	35	1	no	—
45	C₉×D₁₃	D₃₅	1	1	yes	characteristic, complemented
46	C₁₃×D₉	—	35	1	no	—
47	C₂×C₃×C₅×C₇	—	13	1	no	—
48	D₁₀₅	—	3	1	no	—
49	D₁₀₅	—	39	1	no	—
50	C₃×C₅×C₁₃	C₂×D₂₁	1	1	yes	characteristic
51	D₉₁	—	45	1	no	—
52	C₇×D₁₃	D₄₅	1	1	yes	characteristic, complemented
53	C₁₃×D₇	—	45	1	no	—
54	C₂×D₄₅	—	91	1	no	Hall
55	D₃×D₁₃	—	105	1	no	—
56	C₂×D₃₅	—	117	1	no	Hall
57	D₆₅	—	63	1	no	—
58	C₅×D₁₃	D₆₃	1	1	yes	characteristic, complemented
59	D₅×C₁₃	—	63	1	no	—
60	C₂×C₇×C₉	—	13	1	no	—
61	D₆₃	—	5	1	no	—
62	D₆₃	—	65	1	no	—
63	C₉×C₁₃	C₂×D₃₅	1	1	yes	characteristic, complemented, Hall
64	C₃×C₅×C₇	D₃×D₁₃	1	1	yes	characteristic
65	C₇×C₁₃	C₂×D₄₅	1	1	yes	characteristic, complemented, Hall
66	C₂×C₅×C₉	—	13	1	no	—
67	D₄₅	—	7	1	no	—
68	D₄₅	—	91	1	no	—
69	C₂×D₂₁	—	195	1	no	—
70	D₃₉	—	105	1	no	—
71	C₃×D₁₃	D₁₀₅	1	1	yes	characteristic
72	D₃×C₁₃	—	105	1	no	—
73	C₂×C₅×C₇	—	13	1	no	—
74	D₃₅	—	9	1	no	—
75	D₃₅	—	117	1	no	—
76	C₅×C₁₃	C₂×D₆₃	1	1	yes	characteristic, complemented, Hall
77	C₇×C₉	D₅×D₁₃	1	1	yes	characteristic, complemented, Hall
78	C₂×D₁₅	—	273	1	no	—
79	C₂×D₁₃	—	315	1	no	Hall
80	C₅×C₉	D₇×D₁₃	1	1	yes	characteristic, complemented, Hall
81	C₂×C₃×C₇	—	13	1	no	—
82	D₂₁	—	15	1	no	—
83	D₂₁	—	195	1	no	—
84	C₃×C₁₃	C₂×D₁₀₅	1	1	yes	characteristic
85	C₂×D₉	—	455	1	no	Hall
86	C₅×C₇	D₉×D₁₃	1	1	yes	characteristic, complemented, Hall
87	C₂×C₃×C₅	—	13	1	no	—
88	D₁₅	—	273	1	no	—
89	D₁₅	—	21	1	no	—
90	C₂×D₇	—	585	1	no	Hall
91	C₂×C₁₃	—	315	1	no	—
92	D₁₃	—	315	1	no	—
93	D₁₃	D₃₁₅	1	1	yes	characteristic, direct product factor
94	C₃×C₇	D₁₃×D₁₅	1	1	yes	characteristic
95	C₂×D₅	—	819	1	no	Hall
96	C₂×C₉	—	13	1	no	—
97	D₉	—	35	1	no	—
98	D₉	—	455	1	no	—
99	C₃×C₅	D₁₃×D₂₁	1	1	yes	characteristic
100	C₂×C₇	—	13	1	no	—
101	D₇	—	45	1	no	—
102	D₇	—	585	1	no	—
103	C₁₃	C₂×D₃₁₅	1	1	yes	characteristic, complemented, Sylow
104	C₂×D₃	—	1365	1	no	—
105	C₂×C₅	—	13	1	no	—
106	D₅	—	819	1	no	—
107	D₅	—	63	1	no	—
108	C₉	D₁₃×D₃₅	1	1	yes	characteristic, complemented, Sylow
109	C₇	D₁₃×D₄₅	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₃	—	13	1	no	—
111	D₃	—	1365	1	no	—
112	D₃	—	105	1	no	—
113	C₅	D₁₃×D₆₃	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	4095	1	no	Sylow
115	C₃	D₁₃×D₁₀₅	1	1	yes	characteristic
116	C₂	—	13	1	no	—
117	C₂	—	315	1	no	—
118	C₂	—	4095	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₄₀₉₅

C₂

yes

maximal, characteristic, complemented

C₁₃×D₃₁₅

C₂

yes

maximal, characteristic, complemented

C₅×C₇×C₉×D₁₃

C₂

yes

maximal, characteristic, complemented

D₁₃×D₁₀₅

—

maximal

C₅×C₇×C₉×C₁₃

C₂²

yes

characteristic, complemented, Hall

D₁₃×D₆₃

—

maximal, Hall

D₁₃₆₅

—

C₁₃×D₁₀₅

—

C₃×C₅×C₇×D₁₃

D₃

yes

characteristic

D₁₃×D₄₅

—

maximal, Hall

D₁₃×D₃₅

—

Hall

D₈₁₉

—

C₁₃×D₆₃

—

C₇×C₉×D₁₃

D₅

yes

characteristic, complemented

C₃×C₅×C₇×C₁₃

C₂×D₃

yes

characteristic

C₂×D₃₁₅

—

maximal, Hall

D₅₈₅

—

C₁₃×D₄₅

—

C₅×C₉×D₁₃

D₇

yes

characteristic, complemented

D₁₃×D₂₁

—

D₄₅₅

—

C₁₃×D₃₅

—

C₅×C₇×D₁₃

D₉

yes

characteristic, complemented

C₇×C₉×C₁₃

C₂×D₅

yes

characteristic, complemented, Hall

D₁₃×D₁₅

—

C₂×C₅×C₇×C₉

—

D₃₁₅

—

D₃₁₅

D₁₃

yes

characteristic, direct product factor

C₅×C₉×C₁₃

C₂×D₇

yes

characteristic, complemented, Hall

D₂₇₃

—

C₁₃×D₂₁

—

C₃×C₇×D₁₃

D₁₅

yes

characteristic

D₉×D₁₃

—

Hall

C₅×C₇×C₁₃

C₂×D₉

yes

characteristic, complemented, Hall

C₂×D₁₀₅

—

D₁₉₅

—

C₁₃×D₁₅

—

C₃×C₅×D₁₃

D₂₁

yes

characteristic

D₇×D₁₃

—

Hall

C₅×C₇×C₉

C₂×D₁₃

yes

characteristic, complemented, Hall

C₃×C₇×C₁₃

C₂×D₁₅

yes

characteristic

D₅×D₁₃

—

Hall

C₂×D₆₃

—

Hall

D₁₁₇

—

C₉×D₁₃

D₃₅

yes

characteristic, complemented

C₁₃×D₉

—

C₂×C₃×C₅×C₇

—

D₁₀₅

—

D₁₀₅

—

C₃×C₅×C₁₃

C₂×D₂₁

yes

characteristic

D₉₁

—

C₇×D₁₃

D₄₅

yes

characteristic, complemented

C₁₃×D₇

—

C₂×D₄₅

—

Hall

D₃×D₁₃

—

105

—

C₂×D₃₅

—

117

Hall

D₆₅

—

C₅×D₁₃

D₆₃

yes

characteristic, complemented

D₅×C₁₃

—

C₂×C₇×C₉

—

D₆₃

—

D₆₃

—

C₉×C₁₃

C₂×D₃₅

yes

characteristic, complemented, Hall

C₃×C₅×C₇

D₃×D₁₃

yes

characteristic

C₇×C₁₃

C₂×D₄₅

yes

characteristic, complemented, Hall

C₂×C₅×C₉

—

D₄₅

—

D₄₅

—

C₂×D₂₁

—

195

—

D₃₉

—

105

—

C₃×D₁₃

D₁₀₅

yes

characteristic

D₃×C₁₃

—

105

—

C₂×C₅×C₇

—

D₃₅

—

D₃₅

—

117

—

C₅×C₁₃

C₂×D₆₃

yes

characteristic, complemented, Hall

C₇×C₉

D₅×D₁₃

yes

characteristic, complemented, Hall

C₂×D₁₅

—

273

—

C₂×D₁₃

—

315

Hall

C₅×C₉

D₇×D₁₃

yes

characteristic, complemented, Hall

C₂×C₃×C₇

—

D₂₁

—

D₂₁

—

195

—

C₃×C₁₃

C₂×D₁₀₅

yes

characteristic

C₂×D₉

—

455

Hall

C₅×C₇

D₉×D₁₃

yes

characteristic, complemented, Hall

C₂×C₃×C₅

—

D₁₅

—

273

—

D₁₅

—

C₂×D₇

—

585

Hall

C₂×C₁₃

—

315

—

D₁₃

—

315

—

D₁₃

D₃₁₅

yes

characteristic, direct product factor

C₃×C₇

D₁₃×D₁₅

yes

characteristic

C₂×D₅

—

819

Hall

C₂×C₉

—

D₉

—

D₉

—

455

—

C₃×C₅

D₁₃×D₂₁

yes

characteristic

100

C₂×C₇

—

101

D₇

—

102

D₇

—

585

—

103

C₁₃

C₂×D₃₁₅

yes

characteristic, complemented, Sylow

104

C₂×D₃

—

1365

—

105

C₂×C₅

—

106

D₅

—

819

—

107

D₅

—

108

C₉

D₁₃×D₃₅

yes

characteristic, complemented, Sylow

109

C₇

D₁₃×D₄₅

yes

characteristic, complemented, Sylow

110

C₂×C₃

—

111

D₃

—

1365

—

112

D₃

—

105

—

113

C₅

D₁₃×D₆₃

yes

characteristic, complemented, Sylow

114

C₂²

—

4095

Sylow

115

C₃

D₁₃×D₁₀₅

yes

characteristic

116

C₂

—

117

C₂

—

315

—

118

C₂

—

4095

—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₇×C₉×C₁₃	C₂²
Frattini	C₃	D₁₃×D₁₀₅
socle	C₃×C₅×C₇×C₁₃	C₂×D₃
LCS	G ▷ C₅×C₇×C₉×C₁₃
UCS	C₁

subgroup

G/H

center, hypercenter

C₁

commutator, Fitting, cosocle, hypocenter

C₅×C₇×C₉×C₁₃

C₂²

Frattini

C₃

D₁₃×D₁₀₅

socle

C₃×C₅×C₇×C₁₃

C₂×D₃

LCS

G ▷ C₅×C₇×C₉×C₁₃

UCS

C₁

N	H ≅ G/N	type
1. D₄₀₉₅	C₂	semidirect
2. C₁₃×D₃₁₅	C₂	semidirect
3. C₅×C₇×C₉×D₁₃	C₂	semidirect
5. C₅×C₇×C₉×C₁₃	C₂²	semidirect
14. C₇×C₉×D₁₃	D₅	semidirect
19. C₅×C₉×D₁₃	D₇	semidirect
23. C₅×C₇×D₁₃	D₉	semidirect
24. C₇×C₉×C₁₃	C₂×D₅	semidirect
28. D₃₁₅	D₁₃	direct (N × H)
29. C₅×C₉×C₁₃	C₂×D₇	semidirect
34. C₅×C₇×C₁₃	C₂×D₉	semidirect
40. C₅×C₇×C₉	C₂×D₁₃	semidirect
45. C₉×D₁₃	D₃₅	semidirect
52. C₇×D₁₃	D₄₅	semidirect
58. C₅×D₁₃	D₆₃	semidirect
63. C₉×C₁₃	C₂×D₃₅	semidirect
65. C₇×C₁₃	C₂×D₄₅	semidirect
76. C₅×C₁₃	C₂×D₆₃	semidirect
77. C₇×C₉	D₅×D₁₃	semidirect
80. C₅×C₉	D₇×D₁₃	semidirect
86. C₅×C₇	D₉×D₁₃	semidirect
93. D₁₃	D₃₁₅	direct (N × H)
103. C₁₃	C₂×D₃₁₅	semidirect
108. C₉	D₁₃×D₃₅	semidirect
109. C₇	D₁₃×D₄₅	semidirect
113. C₅	D₁₃×D₆₃	semidirect

H ≅ G/N

type

1. D₄₀₉₅

C₂

semidirect

2. C₁₃×D₃₁₅

C₂

semidirect

3. C₅×C₇×C₉×D₁₃

C₂

semidirect

5. C₅×C₇×C₉×C₁₃

C₂²

semidirect

14. C₇×C₉×D₁₃

D₅

semidirect

19. C₅×C₉×D₁₃

D₇

semidirect

23. C₅×C₇×D₁₃

D₉

semidirect

24. C₇×C₉×C₁₃

C₂×D₅

semidirect

28. D₃₁₅

D₁₃

direct (N × H)

29. C₅×C₉×C₁₃

C₂×D₇

semidirect

34. C₅×C₇×C₁₃

C₂×D₉

semidirect

40. C₅×C₇×C₉

C₂×D₁₃

semidirect

45. C₉×D₁₃

D₃₅

semidirect

52. C₇×D₁₃

D₄₅

semidirect

58. C₅×D₁₃

D₆₃

semidirect

63. C₉×C₁₃

C₂×D₃₅

semidirect

65. C₇×C₁₃

C₂×D₄₅

semidirect

76. C₅×C₁₃

C₂×D₆₃

semidirect

77. C₇×C₉

D₅×D₁₃

semidirect

80. C₅×C₉

D₇×D₁₃

semidirect

86. C₅×C₇

D₉×D₁₃

semidirect

93. D₁₃

D₃₁₅

direct (N × H)

103. C₁₃

C₂×D₃₁₅

semidirect

108. C₉

D₁₃×D₃₅

semidirect

109. C₇

D₁₃×D₄₅

semidirect

113. C₅

D₁₃×D₆₃

semidirect

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

subgroup	index
D₄₀₉₅	2
C₁₃×D₃₁₅	2
C₅×C₇×C₉×D₁₃	2
D₁₃×D₁₀₅	3
D₁₃×D₆₃	5
D₁₃×D₄₅	7
C₂×D₃₁₅	13

subgroup

index

D₄₀₉₅

C₁₃×D₃₁₅

C₅×C₇×C₉×D₁₃

D₁₃×D₁₀₅

D₁₃×D₆₃

D₁₃×D₄₅

C₂×D₃₁₅

Subgroups — D₁₃×D₃₁₅ #30655142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D13×D315 #30655142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃×D₃₁₅ #30655142