Group #45686262 — subgroups

Subgroups — (C₅⋊D₅)⋊C₃×D₁₁₁ #45686262

Order 33300.

128 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 27 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	C₅²⋊C₃×D₁₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊#19149853	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃₇×(C₅⋊D₅)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅⋊D₅×D₁₁₁	C₃	1	1	yes	maximal, characteristic, complemented
5	D₃₇×(C₅⋊D₅)⋊C₃	—	3	1	no	maximal
6	C₃×C₃₇×C₅²⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₃⋊(C₅⋊D₁₈₅)	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅²×D₁₁₁	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×C₃₇×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
10	D₃₇×C₅²⋊C₃	—	3	1	no	—
11	#19149853	—	3	1	no	—
12	C₃₇×(C₅⋊D₅)⋊C₃	—	2	1	no	—
13	C₃₇×(C₅⋊D₅)⋊C₃	D₃	1	1	yes	characteristic, complemented
14	C₅⋊D₅×D₃₇	—	3	1	no	Hall
15	C₃₇×C₅²⋊C₃	—	2	1	no	—
16	C₃₇×C₅²⋊C₃	C₂×D₃	1	1	yes	characteristic, complemented
17	C₃×C₅²×C₃₇	C₂²×C₃	1	1	yes	characteristic, complemented
18	D₅×D₁₁₁	—	15	2	no	—
19	C₅⋊D₁₈₅	—	3	1	no	—
20	C₃₇×C₅⋊D₅	C₃×D₃	1	1	yes	characteristic, complemented
21	C₅²×D₃₇	—	3	1	no	—
22	C₂×C₃×D₁₁₁	—	25	1	no	maximal, Hall
23	D₅₅₅	—	15	2	no	—
24	C₅×D₁₁₁	—	3	2	no	—
25	C₃×D₅×C₃₇	—	15	2	no	—
26	C₅²×C₃₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
27	D₃×(C₅⋊D₅)⋊C₃	—	37	1	no	maximal, Hall
28	D₅×D₃₇	—	45	2	no	—
29	C₂×C₃²×C₃₇	—	25	1	no	—
30	C₃×D₁₁₁	—	25	1	no	—
31	C₃×D₁₁₁	—	25	1	no	—
32	C₃×C₅×C₃₇	—	3	2	no	—
33	C₃⋊((C₅⋊D₅)⋊C₃)	—	37	1	no	—
34	D₃×C₅²⋊C₃	—	37	1	no	—
35	C₃×(C₅⋊D₅)⋊C₃	D₃₇	1	1	yes	characteristic, complemented
36	C₂×D₁₁₁	—	25	1	no	—
37	C₂×C₃×D₃₇	—	75	1	no	—
38	D₁₈₅	—	45	2	no	—
39	C₅×D₃₇	—	9	2	no	—
40	D₅×C₃₇	—	15	2	no	—
41	C₃²×C₃₇	—	25	1	no	Hall
42	D₃×C₅⋊D₅	—	37	1	no	—
43	C₂×(C₅⋊D₅)⋊C₃	—	111	1	no	—
44	C₃×C₅²⋊C₃	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
45	C₂×C₃×C₃₇	—	25	1	no	—
46	C₂×C₃×C₃₇	—	25	1	no	—
47	C₂×C₃×C₃₇	—	50	1	no	—
48	D₁₁₁	—	25	1	no	—
49	D₁₁₁	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, direct product factor
50	C₃×D₃₇	—	75	1	no	—
51	C₃×D₃₇	—	75	1	no	—
52	C₅×C₃₇	—	3	2	no	—
53	C₃⋊(C₅⋊D₅)	—	37	1	no	—
54	C₅²×D₃	—	37	1	no	—
55	C₃×C₅⋊D₅	C₃×D₃₇	1	1	yes	characteristic, complemented
56	C₂×C₅²⋊C₃	—	111	1	no	—
57	(C₅⋊D₅)⋊C₃	—	2	1	no	—
58	(C₅⋊D₅)⋊C₃	—	111	1	no	—
59	(C₅⋊D₅)⋊C₃	D₁₁₁	1	1	yes	characteristic, direct product factor
60	C₂×D₃₇	—	75	1	no	Hall
61	C₃×C₃₇	—	25	1	no	—
62	C₃×C₃₇	—	50	1	no	—
63	C₃×C₃₇	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
64	C₂×C₅⋊D₅	—	111	1	no	Hall
65	C₃×C₅²	C₂×C₃×D₃₇	1	1	yes	characteristic, complemented
66	C₅²⋊C₃	—	2	1	no	—
67	C₅²⋊C₃	C₂×D₁₁₁	1	1	yes	characteristic, complemented
68	C₂×C₃₇	—	25	1	no	—
69	D₃₇	—	3	1	no	—
70	D₃₇	—	75	1	no	—
71	D₃×D₅	—	555	2	no	—
72	C₂×C₅²	—	111	1	no	—
73	C₅⋊D₅	—	111	1	no	—
74	C₅⋊D₅	C₃×D₁₁₁	1	1	yes	characteristic, complemented
75	C₃₇	D₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₃×D₃	—	925	1	no	Hall
77	D₁₅	—	555	2	no	—
78	C₃×D₅	—	15	2	no	—
79	C₅×D₃	—	111	2	no	—
80	C₅²	C₂×C₃×D₁₁₁	1	1	yes	characteristic, complemented, Sylow
81	C₂×D₅	—	1665	2	no	—
82	C₂×C₃²	—	25	1	no	—
83	C₃×D₃	—	925	1	no	—
84	C₃×D₃	—	925	1	no	—
85	C₃×C₅	—	3	2	no	—
86	C₂²×C₃	—	2775	1	no	—
87	C₂×D₃	—	925	1	no	—
88	C₂×C₅	—	333	2	no	—
89	D₅	—	15	2	no	—
90	D₅	—	1665	2	no	—
91	C₃²	—	25	1	no	Sylow
92	C₂×C₃	—	25	1	no	—
93	C₂×C₃	—	25	1	no	—
94	C₂×C₃	—	50	1	no	—
95	C₂×C₃	—	2775	1	no	—
96	C₂×C₃	—	2775	1	no	—
97	D₃	—	37	1	no	—
98	D₃	—	925	1	no	—
99	C₅	—	3	2	no	—
100	C₂²	—	2775	1	no	Sylow
101	C₃	—	25	1	no	—
102	C₃	—	50	1	no	—
103	C₃	D₃₇×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
104	C₂	—	25	1	no	—
105	C₂	—	111	1	no	—
106	C₂	—	2775	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²⋊C₃×D₁₁₁

C₂

yes

maximal, characteristic, complemented

C₃⋊#19149853

C₂

yes

maximal, characteristic, complemented

C₃×C₃₇×(C₅⋊D₅)⋊C₃

C₂

yes

maximal, characteristic, complemented

C₅⋊D₅×D₁₁₁

C₃

yes

maximal, characteristic, complemented

D₃₇×(C₅⋊D₅)⋊C₃

—

maximal

C₃×C₃₇×C₅²⋊C₃

C₂²

yes

characteristic, complemented, Hall

C₃⋊(C₅⋊D₁₈₅)

C₂×C₃

yes

characteristic, complemented

C₅²×D₁₁₁

C₂×C₃

yes

characteristic, complemented

C₃×C₃₇×C₅⋊D₅

C₂×C₃

yes

characteristic, complemented

D₃₇×C₅²⋊C₃

—

#19149853

—

C₃₇×(C₅⋊D₅)⋊C₃

—

C₃₇×(C₅⋊D₅)⋊C₃

D₃

yes

characteristic, complemented

C₅⋊D₅×D₃₇

—

Hall

C₃₇×C₅²⋊C₃

—

C₃₇×C₅²⋊C₃

C₂×D₃

yes

characteristic, complemented

C₃×C₅²×C₃₇

C₂²×C₃

yes

characteristic, complemented

D₅×D₁₁₁

—

C₅⋊D₁₈₅

—

C₃₇×C₅⋊D₅

C₃×D₃

yes

characteristic, complemented

C₅²×D₃₇

—

C₂×C₃×D₁₁₁

—

maximal, Hall

D₅₅₅

—

C₅×D₁₁₁

—

C₃×D₅×C₃₇

—

C₅²×C₃₇

C₂×C₃×D₃

yes

characteristic, complemented, Hall

D₃×(C₅⋊D₅)⋊C₃

—

maximal, Hall

D₅×D₃₇

—

C₂×C₃²×C₃₇

—

C₃×D₁₁₁

—

C₃×D₁₁₁

—

C₃×C₅×C₃₇

—

C₃⋊((C₅⋊D₅)⋊C₃)

—

D₃×C₅²⋊C₃

—

C₃×(C₅⋊D₅)⋊C₃

D₃₇

yes

characteristic, complemented

C₂×D₁₁₁

—

C₂×C₃×D₃₇

—

D₁₈₅

—

C₅×D₃₇

—

D₅×C₃₇

—

C₃²×C₃₇

—

Hall

D₃×C₅⋊D₅

—

C₂×(C₅⋊D₅)⋊C₃

—

111

—

C₃×C₅²⋊C₃

C₂×D₃₇

yes

characteristic, complemented, Hall

C₂×C₃×C₃₇

—

C₂×C₃×C₃₇

—

C₂×C₃×C₃₇

—

D₁₁₁

—

D₁₁₁

(C₅⋊D₅)⋊C₃

yes

characteristic, direct product factor

C₃×D₃₇

—

C₃×D₃₇

—

C₅×C₃₇

—

C₃⋊(C₅⋊D₅)

—

C₅²×D₃

—

C₃×C₅⋊D₅

C₃×D₃₇

yes

characteristic, complemented

C₂×C₅²⋊C₃

—

111

—

(C₅⋊D₅)⋊C₃

—

(C₅⋊D₅)⋊C₃

—

111

—

(C₅⋊D₅)⋊C₃

D₁₁₁

yes

characteristic, direct product factor

C₂×D₃₇

—

Hall

C₃×C₃₇

—

C₃×C₃₇

—

C₃×C₃₇

C₂×(C₅⋊D₅)⋊C₃

yes

characteristic, complemented

C₂×C₅⋊D₅

—

111

Hall

C₃×C₅²

C₂×C₃×D₃₇

yes

characteristic, complemented

C₅²⋊C₃

—

C₅²⋊C₃

C₂×D₁₁₁

yes

characteristic, complemented

C₂×C₃₇

—

D₃₇

—

D₃₇

—

D₃×D₅

—

555

—

C₂×C₅²

—

111

—

C₅⋊D₅

—

111

—

C₅⋊D₅

C₃×D₁₁₁

yes

characteristic, complemented

C₃₇

D₃×(C₅⋊D₅)⋊C₃

yes

characteristic, complemented, Sylow

C₂×C₃×D₃

—

925

Hall

D₁₅

—

555

—

C₃×D₅

—

C₅×D₃

—

111

—

C₅²

C₂×C₃×D₁₁₁

yes

characteristic, complemented, Sylow

C₂×D₅

—

1665

—

C₂×C₃²

—

C₃×D₃

—

925

—

C₃×D₃

—

925

—

C₃×C₅

—

C₂²×C₃

—

2775

—

C₂×D₃

—

925

—

C₂×C₅

—

333

—

D₅

—

D₅

—

1665

—

C₃²

—

Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

2775

—

C₂×C₃

—

2775

—

D₃

—

D₃

—

925

—

C₅

—

100

C₂²

—

2775

Sylow

101

C₃

—

102

C₃

—

103

C₃

D₃₇×(C₅⋊D₅)⋊C₃

yes

characteristic, complemented

104

C₂

—

105

C₂

—

111

—

106

C₂

—

2775

—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₃₇	C₂²×C₃
LCS	G ▷ C₃×C₅²×C₃₇
UCS	C₁

subgroup

G/H

center, Frattini, hypercenter

C₁

commutator, Fitting, socle, cosocle, hypocenter

C₃×C₅²×C₃₇

C₂²×C₃

LCS

G ▷ C₃×C₅²×C₃₇

UCS

C₁

N	H ≅ G/N	type
1. C₅²⋊C₃×D₁₁₁	C₂	semidirect
2. C₃⋊#19149853	C₂	semidirect
3. C₃×C₃₇×(C₅⋊D₅)⋊C₃	C₂	semidirect
4. C₅⋊D₅×D₁₁₁	C₃	semidirect
6. C₃×C₃₇×C₅²⋊C₃	C₂²	semidirect
7. C₃⋊(C₅⋊D₁₈₅)	C₂×C₃	semidirect
8. C₅²×D₁₁₁	C₂×C₃	semidirect
9. C₃×C₃₇×C₅⋊D₅	C₂×C₃	semidirect
13. C₃₇×(C₅⋊D₅)⋊C₃	D₃	semidirect
16. C₃₇×C₅²⋊C₃	C₂×D₃	semidirect
17. C₃×C₅²×C₃₇	C₂²×C₃	semidirect
20. C₃₇×C₅⋊D₅	C₃×D₃	semidirect
26. C₅²×C₃₇	C₂×C₃×D₃	semidirect
35. C₃×(C₅⋊D₅)⋊C₃	D₃₇	semidirect
44. C₃×C₅²⋊C₃	C₂×D₃₇	semidirect
49. D₁₁₁	(C₅⋊D₅)⋊C₃	direct (N × H)
55. C₃×C₅⋊D₅	C₃×D₃₇	semidirect
59. (C₅⋊D₅)⋊C₃	D₁₁₁	direct (N × H)
63. C₃×C₃₇	C₂×(C₅⋊D₅)⋊C₃	semidirect
65. C₃×C₅²	C₂×C₃×D₃₇	semidirect
67. C₅²⋊C₃	C₂×D₁₁₁	semidirect
74. C₅⋊D₅	C₃×D₁₁₁	semidirect
75. C₃₇	D₃×(C₅⋊D₅)⋊C₃	semidirect
80. C₅²	C₂×C₃×D₁₁₁	semidirect
103. C₃	D₃₇×(C₅⋊D₅)⋊C₃	semidirect

H ≅ G/N

type

1. C₅²⋊C₃×D₁₁₁

C₂

semidirect

2. C₃⋊#19149853

C₂

semidirect

3. C₃×C₃₇×(C₅⋊D₅)⋊C₃

C₂

semidirect

4. C₅⋊D₅×D₁₁₁

C₃

semidirect

6. C₃×C₃₇×C₅²⋊C₃

C₂²

semidirect

7. C₃⋊(C₅⋊D₁₈₅)

C₂×C₃

semidirect

8. C₅²×D₁₁₁

C₂×C₃

semidirect

9. C₃×C₃₇×C₅⋊D₅

C₂×C₃

semidirect

13. C₃₇×(C₅⋊D₅)⋊C₃

D₃

semidirect

16. C₃₇×C₅²⋊C₃

C₂×D₃

semidirect

17. C₃×C₅²×C₃₇

C₂²×C₃

semidirect

20. C₃₇×C₅⋊D₅

C₃×D₃

semidirect

26. C₅²×C₃₇

C₂×C₃×D₃

semidirect

35. C₃×(C₅⋊D₅)⋊C₃

D₃₇

semidirect

44. C₃×C₅²⋊C₃

C₂×D₃₇

semidirect

49. D₁₁₁

(C₅⋊D₅)⋊C₃

direct (N × H)

55. C₃×C₅⋊D₅

C₃×D₃₇

semidirect

59. (C₅⋊D₅)⋊C₃

D₁₁₁

direct (N × H)

63. C₃×C₃₇

C₂×(C₅⋊D₅)⋊C₃

semidirect

65. C₃×C₅²

C₂×C₃×D₃₇

semidirect

67. C₅²⋊C₃

C₂×D₁₁₁

semidirect

74. C₅⋊D₅

C₃×D₁₁₁

semidirect

75. C₃₇

D₃×(C₅⋊D₅)⋊C₃

semidirect

80. C₅²

C₂×C₃×D₁₁₁

semidirect

103. C₃

D₃₇×(C₅⋊D₅)⋊C₃

semidirect

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

subgroup	index
C₅²⋊C₃×D₁₁₁	2
C₃⋊#19149853	2
C₃×C₃₇×(C₅⋊D₅)⋊C₃	2
C₅⋊D₅×D₁₁₁	3
D₃₇×(C₅⋊D₅)⋊C₃	3
C₂×C₃×D₁₁₁	25
D₃×(C₅⋊D₅)⋊C₃	37

subgroup

index

C₅²⋊C₃×D₁₁₁

C₃⋊#19149853

C₃×C₃₇×(C₅⋊D₅)⋊C₃

C₅⋊D₅×D₁₁₁

D₃₇×(C₅⋊D₅)⋊C₃

C₂×C₃×D₁₁₁

D₃×(C₅⋊D₅)⋊C₃

Subgroups — (C₅⋊D₅)⋊C₃×D₁₁₁ #45686262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C5⋊D5)⋊C3×D111 #45686262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₅⋊D₅)⋊C₃×D₁₁₁ #45686262