Group #45686443 — subgroups

Subgroups — C₅²×D₃×D₁₁₁ #45686443

Order 33300.

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

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₃×C₅²×D₃×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅²×D₁₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×C₃⋊D₁₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅²×D₁₁₁	—	3	1	no	maximal
5	C₅²×D₃×D₃₇	—	3	1	no	maximal
6	C₃²×C₅²×C₃₇	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₅×D₃×D₁₁₁	C₅	1	6	yes	maximal, direct product factor
8	C₅²×D₁₁₁	—	3	1	no	—
9	C₅²×D₁₁₁	—	3	1	no	—
10	C₅²×D₁₁₁	—	6	1	no	—
11	C₅²×D₁₁₁	D₃	1	1	yes	characteristic, direct product factor
12	C₅²×D₃×C₃₇	D₃	1	1	yes	characteristic, complemented
13	C₃×C₅²×D₃₇	—	3	1	no	—
14	C₂×C₃×C₅²×C₃₇	—	3	1	no	—
15	C₂×C₅²×D₃₇	—	9	1	no	Hall
16	C₅×C₃⋊D₁₁₁	C₂×C₅	1	6	yes	complemented
17	C₃×C₅×D₁₁₁	C₂×C₅	1	6	yes	complemented
18	C₃×C₅×D₃×C₃₇	C₂×C₅	1	6	yes	complemented
19	C₃×C₅²×C₃₇	—	2	1	no	—
20	C₃×C₅²×C₃₇	C₂×D₃	1	1	yes	characteristic, complemented
21	C₃×C₅²×C₃₇	C₂×D₃	1	1	yes	characteristic, complemented
22	C₂×C₅×D₁₁₁	—	3	6	no	—
23	C₅×D₃×D₃₇	—	3	6	no	—
24	C₂×C₅²×C₃₇	—	3	1	no	—
25	C₅²×D₃₇	—	3	1	no	—
26	C₅²×D₃₇	—	9	1	no	—
27	C₃²×C₅×C₃₇	C₂²×C₅	1	6	yes	complemented
28	D₃×D₁₁₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃×C₅×C₃₇	—	3	6	no	—
30	C₅×D₁₁₁	—	3	6	no	—
31	C₅×D₁₁₁	—	3	6	no	—
32	C₅×D₁₁₁	—	6	6	no	—
33	C₅×D₁₁₁	C₅×D₃	1	6	yes	direct product factor
34	C₅×D₃×C₃₇	C₅×D₃	1	6	yes	complemented
35	C₃×C₅×D₃₇	—	3	6	no	—
36	C₅²×C₃₇	D₃²	1	1	yes	characteristic, complemented, Hall
37	C₅²×D₃²	—	37	1	no	maximal, Hall
38	C₂×C₅×D₃₇	—	9	6	no	—
39	C₃⋊D₁₁₁	C₂×C₅²	1	1	yes	characteristic, complemented
40	C₃×D₁₁₁	C₂×C₅²	1	1	yes	characteristic, complemented
41	C₃×D₃×C₃₇	C₂×C₅²	1	1	yes	characteristic, complemented
42	C₃×C₅×C₃₇	—	2	6	no	—
43	C₃×C₅×C₃₇	C₂×C₅×D₃	1	6	yes	complemented
44	C₃×C₅×C₃₇	C₂×C₅×D₃	1	6	yes	complemented
45	C₅²×C₃⋊D₃	—	37	1	no	—
46	C₃×C₅²×D₃	—	37	1	no	—
47	C₃×C₅²×D₃	D₃₇	1	1	yes	characteristic, complemented
48	C₂×D₁₁₁	—	3	1	no	—
49	D₃×D₃₇	—	3	1	no	—
50	C₂×C₅×C₃₇	—	3	6	no	—
51	C₅×D₃₇	—	3	6	no	—
52	C₅×D₃₇	—	9	6	no	—
53	C₃²×C₃₇	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
54	C₂×C₅²×D₃	—	111	1	no	—
55	C₂×C₅²×D₃	—	111	1	no	—
56	C₃²×C₅²	C₂×D₃₇	1	1	yes	characteristic, complemented, Hall
57	C₂×C₃×C₃₇	—	3	1	no	—
58	D₁₁₁	—	3	1	no	—
59	D₁₁₁	—	3	1	no	—
60	D₁₁₁	—	6	1	no	—
61	D₁₁₁	C₅²×D₃	1	1	yes	characteristic, direct product factor
62	C₃×D₃₇	—	3	1	no	—
63	D₃×C₃₇	C₅²×D₃	1	1	yes	characteristic, complemented
64	C₅×C₃₇	C₅×D₃²	1	6	yes	complemented
65	C₅×D₃²	—	37	6	no	—
66	C₂×C₃×C₅²	—	3	1	no	—
67	C₂×C₃×C₅²	—	111	1	no	—
68	C₅²×D₃	—	37	1	no	—
69	C₅²×D₃	—	111	1	no	—
70	C₅²×D₃	—	111	1	no	—
71	C₅²×D₃	—	222	1	no	—
72	C₅²×D₃	D₁₁₁	1	1	yes	characteristic, direct product factor
73	C₂×D₃₇	—	9	1	no	Hall
74	C₃×C₃₇	—	2	1	no	—
75	C₃×C₃₇	C₂×C₅²×D₃	1	1	yes	characteristic, complemented
76	C₃×C₃₇	C₂×C₅²×D₃	1	1	yes	characteristic, complemented
77	C₂²×C₅²	—	333	1	no	Hall
78	C₅×C₃⋊D₃	—	37	6	no	—
79	C₃×C₅×D₃	—	37	6	no	—
80	C₃×C₅×D₃	C₅×D₃₇	1	6	yes	complemented
81	C₃×C₅²	—	2	1	no	—
82	C₃×C₅²	D₃×D₃₇	1	1	yes	characteristic, complemented
83	C₃×C₅²	C₂×D₁₁₁	1	1	yes	characteristic, complemented
84	C₂×C₃₇	—	3	1	no	—
85	D₃₇	—	3	1	no	—
86	D₃₇	—	9	1	no	—
87	C₂×C₅×D₃	—	111	6	no	—
88	C₂×C₅×D₃	—	111	6	no	—
89	C₂×C₅²	—	3	1	no	—
90	C₂×C₅²	—	333	1	no	—
91	C₂×C₅²	—	111	1	no	—
92	C₃²×C₅	C₂×C₅×D₃₇	1	6	yes	complemented
93	C₃₇	C₅²×D₃²	1	1	yes	characteristic, complemented, Sylow
94	D₃²	—	37	1	no	Hall
95	C₂×C₃×C₅	—	3	6	no	—
96	C₂×C₃×C₅	—	111	6	no	—
97	C₅×D₃	—	37	6	no	—
98	C₅×D₃	—	111	6	no	—
99	C₅×D₃	—	111	6	no	—
100	C₅×D₃	—	222	6	no	—
101	C₅×D₃	C₅×D₁₁₁	1	6	yes	direct product factor
102	C₅²	D₃×D₁₁₁	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²×C₅	—	333	6	no	—
104	C₃⋊D₃	—	37	1	no	—
105	C₃×D₃	—	37	1	no	—
106	C₃×D₃	C₅²×D₃₇	1	1	yes	characteristic, complemented
107	C₃×C₅	—	2	6	no	—
108	C₃×C₅	C₅×D₃×D₃₇	1	6	yes	complemented
109	C₃×C₅	C₂×C₅×D₁₁₁	1	6	yes	complemented
110	C₂×D₃	—	111	1	no	—
111	C₂×D₃	—	111	1	no	—
112	C₂×C₅	—	3	6	no	—
113	C₂×C₅	—	333	6	no	—
114	C₂×C₅	—	111	6	no	—
115	C₃²	C₂×C₅²×D₃₇	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₃	—	3	1	no	—
117	C₂×C₃	—	111	1	no	—
118	D₃	—	37	1	no	—
119	D₃	—	111	1	no	—
120	D₃	—	111	1	no	—
121	D₃	—	222	1	no	—
122	D₃	C₅²×D₁₁₁	1	1	yes	characteristic, direct product factor
123	C₅	C₅×D₃×D₁₁₁	1	6	yes	direct product factor
124	C₂²	—	333	1	no	Sylow
125	C₃	—	2	1	no	—
126	C₃	C₅²×D₃×D₃₇	1	1	yes	characteristic, complemented
127	C₃	C₂×C₅²×D₁₁₁	1	1	yes	characteristic, complemented
128	C₂	—	3	1	no	—
129	C₂	—	333	1	no	—
130	C₂	—	111	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₅²×D₃×C₃₇

C₂

yes

maximal, characteristic, complemented

C₃×C₅²×D₁₁₁

C₂

yes

maximal, characteristic, complemented

C₅²×C₃⋊D₁₁₁

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₁₁₁

—

maximal

C₅²×D₃×D₃₇

—

maximal

C₃²×C₅²×C₃₇

C₂²

yes

characteristic, complemented, Hall

C₅×D₃×D₁₁₁

C₅

yes

maximal, direct product factor

C₅²×D₁₁₁

—

C₅²×D₁₁₁

—

C₅²×D₁₁₁

—

C₅²×D₁₁₁

D₃

yes

characteristic, direct product factor

C₅²×D₃×C₃₇

D₃

yes

characteristic, complemented

C₃×C₅²×D₃₇

—

C₂×C₃×C₅²×C₃₇

—

C₂×C₅²×D₃₇

—

Hall

C₅×C₃⋊D₁₁₁

C₂×C₅

yes

complemented

C₃×C₅×D₁₁₁

C₂×C₅

yes

complemented

C₃×C₅×D₃×C₃₇

C₂×C₅

yes

complemented

C₃×C₅²×C₃₇

—

C₃×C₅²×C₃₇

C₂×D₃

yes

characteristic, complemented

C₃×C₅²×C₃₇

C₂×D₃

yes

characteristic, complemented

C₂×C₅×D₁₁₁

—

C₅×D₃×D₃₇

—

C₂×C₅²×C₃₇

—

C₅²×D₃₇

—

C₅²×D₃₇

—

C₃²×C₅×C₃₇

C₂²×C₅

yes

complemented

D₃×D₁₁₁

C₅²

yes

characteristic, direct product factor, Hall

C₂×C₃×C₅×C₃₇

—

C₅×D₁₁₁

—

C₅×D₁₁₁

—

C₅×D₁₁₁

—

C₅×D₁₁₁

C₅×D₃

yes

direct product factor

C₅×D₃×C₃₇

C₅×D₃

yes

complemented

C₃×C₅×D₃₇

—

C₅²×C₃₇

D₃²

yes

characteristic, complemented, Hall

C₅²×D₃²

—

maximal, Hall

C₂×C₅×D₃₇

—

C₃⋊D₁₁₁

C₂×C₅²

yes

characteristic, complemented

C₃×D₁₁₁

C₂×C₅²

yes

characteristic, complemented

C₃×D₃×C₃₇

C₂×C₅²

yes

characteristic, complemented

C₃×C₅×C₃₇

—

C₃×C₅×C₃₇

C₂×C₅×D₃

yes

complemented

C₃×C₅×C₃₇

C₂×C₅×D₃

yes

complemented

C₅²×C₃⋊D₃

—

C₃×C₅²×D₃

—

C₃×C₅²×D₃

D₃₇

yes

characteristic, complemented

C₂×D₁₁₁

—

D₃×D₃₇

—

C₂×C₅×C₃₇

—

C₅×D₃₇

—

C₅×D₃₇

—

C₃²×C₃₇

C₂²×C₅²

yes

characteristic, complemented, Hall

C₂×C₅²×D₃

—

111

—

C₂×C₅²×D₃

—

111

—

C₃²×C₅²

C₂×D₃₇

yes

characteristic, complemented, Hall

C₂×C₃×C₃₇

—

D₁₁₁

—

D₁₁₁

—

D₁₁₁

—

D₁₁₁

C₅²×D₃

yes

characteristic, direct product factor

C₃×D₃₇

—

D₃×C₃₇

C₅²×D₃

yes

characteristic, complemented

C₅×C₃₇

C₅×D₃²

yes

complemented

C₅×D₃²

—

C₂×C₃×C₅²

—

C₂×C₃×C₅²

—

111

—

C₅²×D₃

—

C₅²×D₃

—

111

—

C₅²×D₃

—

111

—

C₅²×D₃

—

222

—

C₅²×D₃

D₁₁₁

yes

characteristic, direct product factor

C₂×D₃₇

—

Hall

C₃×C₃₇

—

C₃×C₃₇

C₂×C₅²×D₃

yes

characteristic, complemented

C₃×C₃₇

C₂×C₅²×D₃

yes

characteristic, complemented

C₂²×C₅²

—

333

Hall

C₅×C₃⋊D₃

—

C₃×C₅×D₃

—

C₃×C₅×D₃

C₅×D₃₇

yes

complemented

C₃×C₅²

—

C₃×C₅²

D₃×D₃₇

yes

characteristic, complemented

C₃×C₅²

C₂×D₁₁₁

yes

characteristic, complemented

C₂×C₃₇

—

D₃₇

—

D₃₇

—

C₂×C₅×D₃

—

111

—

C₂×C₅×D₃

—

111

—

C₂×C₅²

—

C₂×C₅²

—

333

—

C₂×C₅²

—

111

—

C₃²×C₅

C₂×C₅×D₃₇

yes

complemented

C₃₇

C₅²×D₃²

yes

characteristic, complemented, Sylow

D₃²

—

Hall

C₂×C₃×C₅

—

C₂×C₃×C₅

—

111

—

C₅×D₃

—

C₅×D₃

—

111

—

C₅×D₃

—

111

—

100

C₅×D₃

—

222

—

101

C₅×D₃

C₅×D₁₁₁

yes

direct product factor

102

C₅²

D₃×D₁₁₁

yes

characteristic, direct product factor, Sylow

103

C₂²×C₅

—

333

—

104

C₃⋊D₃

—

105

C₃×D₃

—

106

C₃×D₃

C₅²×D₃₇

yes

characteristic, complemented

107

C₃×C₅

—

108

C₃×C₅

C₅×D₃×D₃₇

yes

complemented

109

C₃×C₅

C₂×C₅×D₁₁₁

yes

complemented

110

C₂×D₃

—

111

—

111

C₂×D₃

—

111

—

112

C₂×C₅

—

113

C₂×C₅

—

333

—

114

C₂×C₅

—

111

—

115

C₃²

C₂×C₅²×D₃₇

yes

characteristic, complemented, Sylow

116

C₂×C₃

—

117

C₂×C₃

—

111

—

118

D₃

—

119

D₃

—

111

—

120

D₃

—

111

—

121

D₃

—

222

—

122

D₃

C₅²×D₁₁₁

yes

characteristic, direct product factor

123

C₅

C₅×D₃×D₁₁₁

yes

direct product factor

124

C₂²

—

333

Sylow

125

C₃

—

126

C₃

C₅²×D₃×D₃₇

yes

characteristic, complemented

127

C₃

C₂×C₅²×D₁₁₁

yes

characteristic, complemented

128

C₂

—

129

C₂

—

333

—

130

C₂

—

111

—

	subgroup	G/H
center, hypercenter	C₅²	D₃×D₁₁₁
commutator, cosocle, hypocenter	C₃²×C₃₇	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₇	C₂²
LCS	G ▷ C₃²×C₃₇
UCS	C₁ ◁ C₅²

subgroup

G/H

center, hypercenter

C₅²

D₃×D₁₁₁

commutator, cosocle, hypocenter

C₃²×C₃₇

C₂²×C₅²

Frattini

C₁

Fitting, socle

C₃²×C₅²×C₃₇

C₂²

LCS

G ▷ C₃²×C₃₇

UCS

C₁ ◁ C₅²

N	H ≅ G/N	type
1. C₃×C₅²×D₃×C₃₇	C₂	semidirect
2. C₃×C₅²×D₁₁₁	C₂	semidirect
3. C₅²×C₃⋊D₁₁₁	C₂	semidirect
6. C₃²×C₅²×C₃₇	C₂²	semidirect
7. C₅×D₃×D₁₁₁	C₅	direct (N × H)
11. C₅²×D₁₁₁	D₃	direct (N × H)
12. C₅²×D₃×C₃₇	D₃	semidirect
16. C₅×C₃⋊D₁₁₁	C₂×C₅	semidirect
17. C₃×C₅×D₁₁₁	C₂×C₅	semidirect
18. C₃×C₅×D₃×C₃₇	C₂×C₅	semidirect
20. C₃×C₅²×C₃₇	C₂×D₃	semidirect
21. C₃×C₅²×C₃₇	C₂×D₃	semidirect
27. C₃²×C₅×C₃₇	C₂²×C₅	semidirect
28. D₃×D₁₁₁	C₅²	direct (N × H)
33. C₅×D₁₁₁	C₅×D₃	direct (N × H)
34. C₅×D₃×C₃₇	C₅×D₃	semidirect
36. C₅²×C₃₇	D₃²	semidirect
39. C₃⋊D₁₁₁	C₂×C₅²	semidirect
40. C₃×D₁₁₁	C₂×C₅²	semidirect
41. C₃×D₃×C₃₇	C₂×C₅²	semidirect
43. C₃×C₅×C₃₇	C₂×C₅×D₃	semidirect
44. C₃×C₅×C₃₇	C₂×C₅×D₃	semidirect
47. C₃×C₅²×D₃	D₃₇	semidirect
53. C₃²×C₃₇	C₂²×C₅²	semidirect
56. C₃²×C₅²	C₂×D₃₇	semidirect
61. D₁₁₁	C₅²×D₃	direct (N × H)
63. D₃×C₃₇	C₅²×D₃	semidirect
64. C₅×C₃₇	C₅×D₃²	semidirect
72. C₅²×D₃	D₁₁₁	direct (N × H)
75. C₃×C₃₇	C₂×C₅²×D₃	semidirect
76. C₃×C₃₇	C₂×C₅²×D₃	semidirect
80. C₃×C₅×D₃	C₅×D₃₇	semidirect
82. C₃×C₅²	D₃×D₃₇	semidirect
83. C₃×C₅²	C₂×D₁₁₁	semidirect
92. C₃²×C₅	C₂×C₅×D₃₇	semidirect
93. C₃₇	C₅²×D₃²	semidirect
101. C₅×D₃	C₅×D₁₁₁	direct (N × H)
102. C₅²	D₃×D₁₁₁	direct (N × H)
106. C₃×D₃	C₅²×D₃₇	semidirect
108. C₃×C₅	C₅×D₃×D₃₇	semidirect
109. C₃×C₅	C₂×C₅×D₁₁₁	semidirect
115. C₃²	C₂×C₅²×D₃₇	semidirect
122. D₃	C₅²×D₁₁₁	direct (N × H)
123. C₅	C₅×D₃×D₁₁₁	direct (N × H)
126. C₃	C₅²×D₃×D₃₇	semidirect
127. C₃	C₂×C₅²×D₁₁₁	semidirect

H ≅ G/N

type

1. C₃×C₅²×D₃×C₃₇

C₂

semidirect

2. C₃×C₅²×D₁₁₁

C₂

semidirect

3. C₅²×C₃⋊D₁₁₁

C₂

semidirect

6. C₃²×C₅²×C₃₇

C₂²

semidirect

7. C₅×D₃×D₁₁₁

C₅

direct (N × H)

11. C₅²×D₁₁₁

D₃

direct (N × H)

12. C₅²×D₃×C₃₇

D₃

semidirect

16. C₅×C₃⋊D₁₁₁

C₂×C₅

semidirect

17. C₃×C₅×D₁₁₁

C₂×C₅

semidirect

18. C₃×C₅×D₃×C₃₇

C₂×C₅

semidirect

20. C₃×C₅²×C₃₇

C₂×D₃

semidirect

21. C₃×C₅²×C₃₇

C₂×D₃

semidirect

27. C₃²×C₅×C₃₇

C₂²×C₅

semidirect

28. D₃×D₁₁₁

C₅²

direct (N × H)

33. C₅×D₁₁₁

C₅×D₃

direct (N × H)

34. C₅×D₃×C₃₇

C₅×D₃

semidirect

36. C₅²×C₃₇

D₃²

semidirect

39. C₃⋊D₁₁₁

C₂×C₅²

semidirect

40. C₃×D₁₁₁

C₂×C₅²

semidirect

41. C₃×D₃×C₃₇

C₂×C₅²

semidirect

43. C₃×C₅×C₃₇

C₂×C₅×D₃

semidirect

44. C₃×C₅×C₃₇

C₂×C₅×D₃

semidirect

47. C₃×C₅²×D₃

D₃₇

semidirect

53. C₃²×C₃₇

C₂²×C₅²

semidirect

56. C₃²×C₅²

C₂×D₃₇

semidirect

61. D₁₁₁

C₅²×D₃

direct (N × H)

63. D₃×C₃₇

C₅²×D₃

semidirect

64. C₅×C₃₇

C₅×D₃²

semidirect

72. C₅²×D₃

D₁₁₁

direct (N × H)

75. C₃×C₃₇

C₂×C₅²×D₃

semidirect

76. C₃×C₃₇

C₂×C₅²×D₃

semidirect

80. C₃×C₅×D₃

C₅×D₃₇

semidirect

82. C₃×C₅²

D₃×D₃₇

semidirect

83. C₃×C₅²

C₂×D₁₁₁

semidirect

92. C₃²×C₅

C₂×C₅×D₃₇

semidirect

93. C₃₇

C₅²×D₃²

semidirect

101. C₅×D₃

C₅×D₁₁₁

direct (N × H)

102. C₅²

D₃×D₁₁₁

direct (N × H)

106. C₃×D₃

C₅²×D₃₇

semidirect

108. C₃×C₅

C₅×D₃×D₃₇

semidirect

109. C₃×C₅

C₂×C₅×D₁₁₁

semidirect

115. C₃²

C₂×C₅²×D₃₇

semidirect

122. D₃

C₅²×D₁₁₁

direct (N × H)

123. C₅

C₅×D₃×D₁₁₁

direct (N × H)

126. C₃

C₅²×D₃×D₃₇

semidirect

127. C₃

C₂×C₅²×D₁₁₁

semidirect

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

subgroup	index
C₃×C₅²×D₃×C₃₇	2
C₃×C₅²×D₁₁₁	2
C₅²×C₃⋊D₁₁₁	2
C₂×C₅²×D₁₁₁	3
C₅²×D₃×D₃₇	3
C₅×D₃×D₁₁₁	5
C₅²×D₃²	37

subgroup

index

C₃×C₅²×D₃×C₃₇

C₃×C₅²×D₁₁₁

C₅²×C₃⋊D₁₁₁

C₂×C₅²×D₁₁₁

C₅²×D₃×D₃₇

C₅×D₃×D₁₁₁

C₅²×D₃²

Subgroups — C₅²×D₃×D₁₁₁ #45686443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C52×D3×D111 #45686443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₃×D₁₁₁ #45686443