Group #44477718 — subgroups

Subgroups — C₂×C₁₃⋊C₉×D₆₅ #44477718

Order 30420.

136 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 55 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×C₁₃×C₁₃⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃⋊C₉×D₆₅	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₁₃×D₆₅	C₃	1	1	yes	maximal, characteristic
4	C₅×C₁₃×C₁₃⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×D₁₃×C₁₃⋊C₉	—	5	1	no	maximal, Hall
6	C₃×C₁₃×D₆₅	C₂×C₃	1	2	yes	—
7	C₂×C₃×C₅×C₁₃²	C₂×C₃	1	1	yes	characteristic
8	C₂×C₁₃×D₆₅	C₉	1	1	yes	characteristic, complemented, Hall
9	C₂×C₁₃×C₁₃⋊C₉	D₅	1	1	yes	characteristic, complemented
10	D₁₃×C₁₃⋊C₉	—	5	2	no	—
11	C₃×C₅×C₁₃²	C₂²×C₃	1	1	yes	characteristic
12	C₂×C₉×D₆₅	—	13	1	no	maximal
13	C₂×D₅×C₁₃⋊C₉	—	13	1	no	maximal
14	C₂×C₃×C₁₃×D₁₃	—	5	1	no	—
15	C₂×C₅×C₁₃²	C₂×C₉	1	1	yes	characteristic, complemented
16	C₁₃×D₆₅	C₂×C₉	1	2	yes	complemented
17	C₁₃×C₁₃⋊C₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
18	C₂×C₅×C₉×C₁₃	—	13	1	no	—
19	C₉×D₆₅	—	13	2	no	—
20	C₂×C₅×C₁₃⋊C₉	D₁₃	1	1	yes	characteristic, complemented
21	D₅×C₁₃⋊C₉	—	13	2	no	—
22	C₂×C₃×C₁₃²	C₃×D₅	1	1	yes	characteristic
23	C₃×C₁₃×D₁₃	—	5	2	no	—
24	C₅×C₁₃²	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×D₆₅	C₁₃⋊C₃	1	1	yes	characteristic
26	C₂×C₃×D₅×C₁₃	—	13	1	no	—
27	C₂×C₁₃×D₁₃	—	5	1	no	Hall
28	C₅×C₉×C₁₃	—	13	1	no	—
29	C₅×C₁₃⋊C₉	C₂×D₁₃	1	1	yes	characteristic, complemented
30	C₃×C₁₃²	C₂×C₃×D₅	1	1	yes	characteristic
31	C₂×C₉×D₁₃	—	65	1	no	—
32	C₂²×C₁₃⋊C₉	—	65	1	no	—
33	C₂×C₃×C₅×C₁₃	—	6	2	no	—
34	C₂×C₃×C₅×C₁₃	C₂×C₁₃⋊C₃	1	1	yes	characteristic
35	C₂×C₃×C₅×C₁₃	C₃×D₁₃	1	1	yes	characteristic
36	C₃×D₆₅	C₂×C₁₃⋊C₃	1	2	yes	—
37	C₃×D₅×C₁₃	—	13	2	no	—
38	C₂×C₁₃²	C₉×D₅	1	1	yes	characteristic, complemented
39	C₁₃×D₁₃	—	5	2	no	—
40	C₂×D₆₅	C₁₃⋊C₉	1	1	yes	characteristic, direct product factor
41	C₂×D₅×C₁₃	—	13	1	no	—
42	C₂×C₉×C₁₃	—	13	1	no	—
43	C₉×D₁₃	—	65	2	no	—
44	C₂×C₁₃⋊C₉	—	65	2	no	—
45	C₂×C₁₃⋊C₉	D₆₅	1	1	yes	characteristic, direct product factor
46	C₃×C₅×C₁₃	—	6	2	no	—
47	C₃×C₅×C₁₃	C₂²×C₁₃⋊C₃	1	1	yes	characteristic
48	C₃×C₅×C₁₃	C₂×C₃×D₁₃	1	1	yes	characteristic
49	C₂×C₉×D₅	—	169	1	no	Hall
50	C₁₃²	C₂×C₉×D₅	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₃×C₁₃	—	65	1	no	—
52	C₂×C₃×D₁₃	—	5	1	no	—
53	C₂×C₅×C₁₃	—	6	2	no	—
54	C₂×C₅×C₁₃	C₂×C₁₃⋊C₉	1	1	yes	characteristic, complemented
55	C₂×C₅×C₁₃	C₉×D₁₃	1	1	yes	characteristic, complemented
56	D₆₅	C₂×C₁₃⋊C₉	1	2	yes	direct product factor
57	D₅×C₁₃	—	13	2	no	—
58	C₉×C₁₃	—	13	1	no	—
59	C₁₃⋊C₉	C₂×D₆₅	1	1	yes	characteristic, direct product factor
60	C₂×C₅×C₉	—	13	1	no	—
61	C₉×D₅	—	169	2	no	—
62	C₂×C₃×C₁₃	—	6	2	no	—
63	C₂×C₃×C₁₃	—	65	2	no	—
64	C₂×C₃×C₁₃	D₅×C₁₃⋊C₃	1	1	yes	characteristic
65	C₂×C₃×C₁₃	C₃×D₆₅	1	1	yes	characteristic
66	C₃×D₁₃	—	5	2	no	—
67	C₅×C₁₃	—	6	2	no	—
68	C₅×C₁₃	C₂²×C₁₃⋊C₉	1	1	yes	characteristic, complemented
69	C₅×C₁₃	C₂×C₉×D₁₃	1	1	yes	characteristic, complemented
70	C₂×C₃×D₅	—	13	1	no	—
71	C₂²×C₁₃	—	65	1	no	—
72	C₂×D₁₃	—	5	1	no	—
73	C₅×C₉	—	13	1	no	Hall
74	C₃×C₁₃	—	6	2	no	—
75	C₃×C₁₃	C₂×D₅×C₁₃⋊C₃	1	1	yes	characteristic
76	C₃×C₁₃	C₂×C₃×D₆₅	1	1	yes	characteristic
77	C₂²×C₉	—	845	1	no	Hall
78	C₂×C₃×C₅	D₁₃×C₁₃⋊C₃	1	1	yes	characteristic
79	C₃×D₅	—	13	2	no	—
80	C₂×C₁₃	—	6	2	no	—
81	C₂×C₁₃	—	65	2	no	—
82	C₂×C₁₃	D₅×C₁₃⋊C₉	1	1	yes	characteristic, complemented
83	C₂×C₁₃	C₉×D₆₅	1	1	yes	characteristic, complemented
84	D₁₃	—	5	2	no	—
85	C₂×D₅	—	13	1	no	Hall
86	C₂×C₉	—	13	1	no	—
87	C₂×C₉	—	845	2	no	—
88	C₃×C₅	C₂×D₁₃×C₁₃⋊C₃	1	1	yes	characteristic
89	C₁₃	—	6	2	no	—
90	C₁₃	C₂×D₅×C₁₃⋊C₉	1	1	yes	characteristic, complemented
91	C₁₃	C₂×C₉×D₆₅	1	1	yes	characteristic, complemented
92	C₂²×C₃	—	65	1	no	—
93	C₂×C₅	D₁₃×C₁₃⋊C₉	1	1	yes	characteristic, complemented
94	D₅	—	13	2	no	—
95	C₉	—	13	1	no	Sylow
96	C₂×C₃	—	65	2	no	—
97	C₂×C₃	C₁₃⋊C₃×D₆₅	1	1	yes	characteristic
98	C₅	C₂×D₁₃×C₁₃⋊C₉	1	1	yes	characteristic, complemented, Sylow
99	C₂²	—	65	1	no	Sylow
100	C₃	C₂×C₁₃⋊C₃×D₆₅	1	1	yes	characteristic
101	C₂	—	65	2	no	—
102	C₂	C₁₃⋊C₉×D₆₅	1	1	yes	characteristic, direct product factor

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×C₁₃×C₁₃⋊C₉

C₂

yes

maximal, characteristic, complemented

C₁₃⋊C₉×D₆₅

C₂

yes

maximal, direct product factor

C₂×C₃×C₁₃×D₆₅

C₃

yes

maximal, characteristic

C₅×C₁₃×C₁₃⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₂×D₁₃×C₁₃⋊C₉

—

maximal, Hall

C₃×C₁₃×D₆₅

C₂×C₃

yes

—

C₂×C₃×C₅×C₁₃²

C₂×C₃

yes

characteristic

C₂×C₁₃×D₆₅

C₉

yes

characteristic, complemented, Hall

C₂×C₁₃×C₁₃⋊C₉

D₅

yes

characteristic, complemented

D₁₃×C₁₃⋊C₉

—

C₃×C₅×C₁₃²

C₂²×C₃

yes

characteristic

C₂×C₉×D₆₅

—

maximal

C₂×D₅×C₁₃⋊C₉

—

maximal

C₂×C₃×C₁₃×D₁₃

—

C₂×C₅×C₁₃²

C₂×C₉

yes

characteristic, complemented

C₁₃×D₆₅

C₂×C₉

yes

complemented

C₁₃×C₁₃⋊C₉

C₂×D₅

yes

characteristic, complemented, Hall

C₂×C₅×C₉×C₁₃

—

C₉×D₆₅

—

C₂×C₅×C₁₃⋊C₉

D₁₃

yes

characteristic, complemented

D₅×C₁₃⋊C₉

—

C₂×C₃×C₁₃²

C₃×D₅

yes

characteristic

C₃×C₁₃×D₁₃

—

C₅×C₁₃²

C₂²×C₉

yes

characteristic, complemented, Hall

C₂×C₃×D₆₅

C₁₃⋊C₃

yes

characteristic

C₂×C₃×D₅×C₁₃

—

C₂×C₁₃×D₁₃

—

Hall

C₅×C₉×C₁₃

—

C₅×C₁₃⋊C₉

C₂×D₁₃

yes

characteristic, complemented

C₃×C₁₃²

C₂×C₃×D₅

yes

characteristic

C₂×C₉×D₁₃

—

C₂²×C₁₃⋊C₉

—

C₂×C₃×C₅×C₁₃

—

C₂×C₃×C₅×C₁₃

C₂×C₁₃⋊C₃

yes

characteristic

C₂×C₃×C₅×C₁₃

C₃×D₁₃

yes

characteristic

C₃×D₆₅

C₂×C₁₃⋊C₃

yes

—

C₃×D₅×C₁₃

—

C₂×C₁₃²

C₉×D₅

yes

characteristic, complemented

C₁₃×D₁₃

—

C₂×D₆₅

C₁₃⋊C₉

yes

characteristic, direct product factor

C₂×D₅×C₁₃

—

C₂×C₉×C₁₃

—

C₉×D₁₃

—

C₂×C₁₃⋊C₉

—

C₂×C₁₃⋊C₉

D₆₅

yes

characteristic, direct product factor

C₃×C₅×C₁₃

—

C₃×C₅×C₁₃

C₂²×C₁₃⋊C₃

yes

characteristic

C₃×C₅×C₁₃

C₂×C₃×D₁₃

yes

characteristic

C₂×C₉×D₅

—

169

Hall

C₁₃²

C₂×C₉×D₅

yes

characteristic, complemented, Sylow

C₂²×C₃×C₁₃

—

C₂×C₃×D₁₃

—

C₂×C₅×C₁₃

—

C₂×C₅×C₁₃

C₂×C₁₃⋊C₉

yes

characteristic, complemented

C₂×C₅×C₁₃

C₉×D₁₃

yes

characteristic, complemented

D₆₅

C₂×C₁₃⋊C₉

yes

direct product factor

D₅×C₁₃

—

C₉×C₁₃

—

C₁₃⋊C₉

C₂×D₆₅

yes

characteristic, direct product factor

C₂×C₅×C₉

—

C₉×D₅

—

169

—

C₂×C₃×C₁₃

—

C₂×C₃×C₁₃

—

C₂×C₃×C₁₃

D₅×C₁₃⋊C₃

yes

characteristic

C₂×C₃×C₁₃

C₃×D₆₅

yes

characteristic

C₃×D₁₃

—

C₅×C₁₃

—

C₅×C₁₃

C₂²×C₁₃⋊C₉

yes

characteristic, complemented

C₅×C₁₃

C₂×C₉×D₁₃

yes

characteristic, complemented

C₂×C₃×D₅

—

C₂²×C₁₃

—

C₂×D₁₃

—

C₅×C₉

—

Hall

C₃×C₁₃

—

C₃×C₁₃

C₂×D₅×C₁₃⋊C₃

yes

characteristic

C₃×C₁₃

C₂×C₃×D₆₅

yes

characteristic

C₂²×C₉

—

845

Hall

C₂×C₃×C₅

D₁₃×C₁₃⋊C₃

yes

characteristic

C₃×D₅

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

D₅×C₁₃⋊C₉

yes

characteristic, complemented

C₂×C₁₃

C₉×D₆₅

yes

characteristic, complemented

D₁₃

—

C₂×D₅

—

Hall

C₂×C₉

—

C₂×C₉

—

845

—

C₃×C₅

C₂×D₁₃×C₁₃⋊C₃

yes

characteristic

C₁₃

—

C₁₃

C₂×D₅×C₁₃⋊C₉

yes

characteristic, complemented

C₁₃

C₂×C₉×D₆₅

yes

characteristic, complemented

C₂²×C₃

—

C₂×C₅

D₁₃×C₁₃⋊C₉

yes

characteristic, complemented

D₅

—

C₉

—

Sylow

C₂×C₃

—

C₂×C₃

C₁₃⋊C₃×D₆₅

yes

characteristic

C₅

C₂×D₁₃×C₁₃⋊C₉

yes

characteristic, complemented, Sylow

C₂²

—

Sylow

100

C₃

C₂×C₁₃⋊C₃×D₆₅

yes

characteristic

101

C₂

—

102

C₂

C₁₃⋊C₉×D₆₅

yes

characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	C₁₃⋊C₃×D₆₅
commutator, hypocenter	C₅×C₁₃²	C₂²×C₉
Frattini	C₃	C₂×C₁₃⋊C₃×D₆₅
Fitting, socle	C₂×C₃×C₅×C₁₃²	C₂×C₃
prime radical	C₂×C₃×C₁₃×D₆₅	C₃
cosocle	C₃×C₅×C₁₃²	C₂²×C₃
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₂×C₃

subgroup

G/H

center, hypercenter

C₂×C₃

C₁₃⋊C₃×D₆₅

commutator, hypocenter

C₅×C₁₃²

C₂²×C₉

Frattini

C₃

C₂×C₁₃⋊C₃×D₆₅

Fitting, socle

C₂×C₃×C₅×C₁₃²

C₂×C₃

prime radical

C₂×C₃×C₁₃×D₆₅

C₃

cosocle

C₃×C₅×C₁₃²

C₂²×C₃

LCS

G ▷ C₅×C₁₃²

UCS

C₁ ◁ C₂×C₃

N	H ≅ G/N	type
1. C₂×C₅×C₁₃×C₁₃⋊C₉	C₂	semidirect
2. C₁₃⋊C₉×D₆₅	C₂	direct (N × H)
4. C₅×C₁₃×C₁₃⋊C₉	C₂²	semidirect
8. C₂×C₁₃×D₆₅	C₉	semidirect
9. C₂×C₁₃×C₁₃⋊C₉	D₅	semidirect
15. C₂×C₅×C₁₃²	C₂×C₉	semidirect
16. C₁₃×D₆₅	C₂×C₉	semidirect
17. C₁₃×C₁₃⋊C₉	C₂×D₅	semidirect
20. C₂×C₅×C₁₃⋊C₉	D₁₃	semidirect
24. C₅×C₁₃²	C₂²×C₉	semidirect
29. C₅×C₁₃⋊C₉	C₂×D₁₃	semidirect
38. C₂×C₁₃²	C₉×D₅	semidirect
40. C₂×D₆₅	C₁₃⋊C₉	direct (N × H)
45. C₂×C₁₃⋊C₉	D₆₅	direct (N × H)
50. C₁₃²	C₂×C₉×D₅	semidirect
54. C₂×C₅×C₁₃	C₂×C₁₃⋊C₉	semidirect
55. C₂×C₅×C₁₃	C₉×D₁₃	semidirect
56. D₆₅	C₂×C₁₃⋊C₉	direct (N × H)
59. C₁₃⋊C₉	C₂×D₆₅	direct (N × H)
68. C₅×C₁₃	C₂²×C₁₃⋊C₉	semidirect
69. C₅×C₁₃	C₂×C₉×D₁₃	semidirect
82. C₂×C₁₃	D₅×C₁₃⋊C₉	semidirect
83. C₂×C₁₃	C₉×D₆₅	semidirect
90. C₁₃	C₂×D₅×C₁₃⋊C₉	semidirect
91. C₁₃	C₂×C₉×D₆₅	semidirect
93. C₂×C₅	D₁₃×C₁₃⋊C₉	semidirect
98. C₅	C₂×D₁₃×C₁₃⋊C₉	semidirect
102. C₂	C₁₃⋊C₉×D₆₅	direct (N × H)

H ≅ G/N

type

1. C₂×C₅×C₁₃×C₁₃⋊C₉

C₂

semidirect

2. C₁₃⋊C₉×D₆₅

C₂

direct (N × H)

4. C₅×C₁₃×C₁₃⋊C₉

C₂²

semidirect

8. C₂×C₁₃×D₆₅

C₉

semidirect

9. C₂×C₁₃×C₁₃⋊C₉

D₅

semidirect

15. C₂×C₅×C₁₃²

C₂×C₉

semidirect

16. C₁₃×D₆₅

C₂×C₉

semidirect

17. C₁₃×C₁₃⋊C₉

C₂×D₅

semidirect

20. C₂×C₅×C₁₃⋊C₉

D₁₃

semidirect

24. C₅×C₁₃²

C₂²×C₉

semidirect

29. C₅×C₁₃⋊C₉

C₂×D₁₃

semidirect

38. C₂×C₁₃²

C₉×D₅

semidirect

40. C₂×D₆₅

C₁₃⋊C₉

direct (N × H)

45. C₂×C₁₃⋊C₉

D₆₅

direct (N × H)

50. C₁₃²

C₂×C₉×D₅

semidirect

54. C₂×C₅×C₁₃

C₂×C₁₃⋊C₉

semidirect

55. C₂×C₅×C₁₃

C₉×D₁₃

semidirect

56. D₆₅

C₂×C₁₃⋊C₉

direct (N × H)

59. C₁₃⋊C₉

C₂×D₆₅

direct (N × H)

68. C₅×C₁₃

C₂²×C₁₃⋊C₉

semidirect

69. C₅×C₁₃

C₂×C₉×D₁₃

semidirect

82. C₂×C₁₃

D₅×C₁₃⋊C₉

semidirect

83. C₂×C₁₃

C₉×D₆₅

semidirect

90. C₁₃

C₂×D₅×C₁₃⋊C₉

semidirect

91. C₁₃

C₂×C₉×D₆₅

semidirect

93. C₂×C₅

D₁₃×C₁₃⋊C₉

semidirect

98. C₅

C₂×D₁₃×C₁₃⋊C₉

semidirect

102. C₂

C₁₃⋊C₉×D₆₅

direct (N × H)

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

subgroup	index
C₂×C₅×C₁₃×C₁₃⋊C₉	2
C₁₃⋊C₉×D₆₅	2
C₂×C₃×C₁₃×D₆₅	3
C₂×D₁₃×C₁₃⋊C₉	5
C₂×C₉×D₆₅	13
C₂×D₅×C₁₃⋊C₉	13

subgroup

index

C₂×C₅×C₁₃×C₁₃⋊C₉

C₁₃⋊C₉×D₆₅

C₂×C₃×C₁₃×D₆₅

C₂×D₁₃×C₁₃⋊C₉

C₂×C₉×D₆₅

C₂×D₅×C₁₃⋊C₉

Subgroups — C₂×C₁₃⋊C₉×D₆₅ #44477718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C13⋊C9×D65 #44477718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃⋊C₉×D₆₅ #44477718