Group #30685921 — subgroups

Subgroups — C₅²×D₃×D₅₇ #30685921

Order 17100.

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	C₅²×C₃⋊D₅₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅²×D₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅²×D₃×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅²×D₅₇	—	3	1	no	maximal
5	C₅²×D₃×D₁₉	—	3	1	no	maximal
6	C₃²×C₅²×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₅×D₃×D₅₇	C₅	1	6	yes	maximal, direct product factor
8	C₅²×D₅₇	—	3	1	no	—
9	C₅²×D₅₇	—	3	1	no	—
10	C₅²×D₅₇	—	6	1	no	—
11	C₅²×D₅₇	D₃	1	1	yes	characteristic, direct product factor
12	C₅²×D₃×C₁₉	D₃	1	1	yes	characteristic, complemented
13	C₃×C₅²×D₁₉	—	3	1	no	—
14	C₂×C₃×C₅²×C₁₉	—	3	1	no	—
15	C₂×C₅²×D₁₉	—	9	1	no	Hall
16	C₅×C₃⋊D₅₇	C₂×C₅	1	6	yes	complemented
17	C₃×C₅×D₅₇	C₂×C₅	1	6	yes	complemented
18	C₃×C₅×D₃×C₁₉	C₂×C₅	1	6	yes	complemented
19	C₃×C₅²×C₁₉	—	2	1	no	—
20	C₃×C₅²×C₁₉	C₂×D₃	1	1	yes	characteristic, complemented
21	C₃×C₅²×C₁₉	C₂×D₃	1	1	yes	characteristic, complemented
22	C₂×C₅×D₅₇	—	3	6	no	—
23	C₅×D₃×D₁₉	—	3	6	no	—
24	C₂×C₅²×C₁₉	—	3	1	no	—
25	C₅²×D₁₉	—	3	1	no	—
26	C₅²×D₁₉	—	9	1	no	—
27	C₅²×D₃²	—	19	1	no	maximal, Hall
28	C₃²×C₅×C₁₉	C₂²×C₅	1	6	yes	complemented
29	D₃×D₅₇	C₅²	1	1	yes	characteristic, direct product factor, Hall
30	C₂×C₃×C₅×C₁₉	—	3	6	no	—
31	C₅×D₅₇	—	3	6	no	—
32	C₅×D₅₇	—	3	6	no	—
33	C₅×D₅₇	—	6	6	no	—
34	C₅×D₅₇	C₅×D₃	1	6	yes	direct product factor
35	C₅×D₃×C₁₉	C₅×D₃	1	6	yes	complemented
36	C₃×C₅×D₁₉	—	3	6	no	—
37	C₅²×C₁₉	D₃²	1	1	yes	characteristic, complemented, Hall
38	C₅²×C₃⋊D₃	—	19	1	no	—
39	C₃×C₅²×D₃	—	19	1	no	—
40	C₃×C₅²×D₃	D₁₉	1	1	yes	characteristic, complemented
41	C₂×C₅×D₁₉	—	9	6	no	—
42	C₃⋊D₅₇	C₂×C₅²	1	1	yes	characteristic, complemented
43	C₃×D₅₇	C₂×C₅²	1	1	yes	characteristic, complemented
44	C₃×D₃×C₁₉	C₂×C₅²	1	1	yes	characteristic, complemented
45	C₂×C₅²×D₃	—	57	1	no	—
46	C₂×C₅²×D₃	—	57	1	no	—
47	C₃×C₅×C₁₉	—	2	6	no	—
48	C₃×C₅×C₁₉	C₂×C₅×D₃	1	6	yes	complemented
49	C₃×C₅×C₁₉	C₂×C₅×D₃	1	6	yes	complemented
50	C₂×D₅₇	—	3	1	no	—
51	D₃×D₁₉	—	3	1	no	—
52	C₃²×C₅²	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
53	C₂×C₅×C₁₉	—	3	6	no	—
54	C₅×D₁₉	—	3	6	no	—
55	C₅×D₁₉	—	9	6	no	—
56	C₅×D₃²	—	19	6	no	—
57	C₃²×C₁₉	C₂²×C₅²	1	1	yes	characteristic, complemented, Hall
58	C₂×C₃×C₅²	—	3	1	no	—
59	C₂×C₃×C₅²	—	57	1	no	—
60	C₅²×D₃	—	19	1	no	—
61	C₅²×D₃	—	57	1	no	—
62	C₅²×D₃	—	57	1	no	—
63	C₅²×D₃	—	114	1	no	—
64	C₅²×D₃	D₅₇	1	1	yes	characteristic, direct product factor
65	C₂×C₃×C₁₉	—	3	1	no	—
66	D₅₇	—	3	1	no	—
67	D₅₇	—	3	1	no	—
68	D₅₇	—	6	1	no	—
69	D₅₇	C₅²×D₃	1	1	yes	characteristic, direct product factor
70	C₃×D₁₉	—	3	1	no	—
71	D₃×C₁₉	C₅²×D₃	1	1	yes	characteristic, complemented
72	C₂²×C₅²	—	171	1	no	Hall
73	C₅×C₁₉	C₅×D₃²	1	6	yes	complemented
74	C₅×C₃⋊D₃	—	19	6	no	—
75	C₃×C₅×D₃	—	19	6	no	—
76	C₃×C₅×D₃	C₅×D₁₉	1	6	yes	complemented
77	C₂×D₁₉	—	9	1	no	Hall
78	C₃×C₅²	—	2	1	no	—
79	C₃×C₅²	D₃×D₁₉	1	1	yes	characteristic, complemented
80	C₃×C₅²	C₂×D₅₇	1	1	yes	characteristic, complemented
81	C₂×C₅×D₃	—	57	6	no	—
82	C₂×C₅×D₃	—	57	6	no	—
83	C₃×C₁₉	—	2	1	no	—
84	C₃×C₁₉	C₂×C₅²×D₃	1	1	yes	characteristic, complemented
85	C₃×C₁₉	C₂×C₅²×D₃	1	1	yes	characteristic, complemented
86	C₂×C₅²	—	3	1	no	—
87	C₂×C₅²	—	57	1	no	—
88	C₂×C₅²	—	171	1	no	—
89	C₃²×C₅	C₂×C₅×D₁₉	1	6	yes	complemented
90	C₂×C₁₉	—	3	1	no	—
91	D₁₉	—	3	1	no	—
92	D₁₉	—	9	1	no	—
93	D₃²	—	19	1	no	Hall
94	C₂×C₃×C₅	—	3	6	no	—
95	C₂×C₃×C₅	—	57	6	no	—
96	C₅×D₃	—	19	6	no	—
97	C₅×D₃	—	57	6	no	—
98	C₅×D₃	—	57	6	no	—
99	C₅×D₃	—	114	6	no	—
100	C₅×D₃	C₅×D₅₇	1	6	yes	direct product factor
101	C₅²	D₃×D₅₇	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₅	—	171	6	no	—
103	C₁₉	C₅²×D₃²	1	1	yes	characteristic, complemented, Sylow
104	C₃⋊D₃	—	19	1	no	—
105	C₃×D₃	—	19	1	no	—
106	C₃×D₃	C₅²×D₁₉	1	1	yes	characteristic, complemented
107	C₃×C₅	—	2	6	no	—
108	C₃×C₅	C₅×D₃×D₁₉	1	6	yes	complemented
109	C₃×C₅	C₂×C₅×D₅₇	1	6	yes	complemented
110	C₂×D₃	—	57	1	no	—
111	C₂×D₃	—	57	1	no	—
112	C₂×C₅	—	3	6	no	—
113	C₂×C₅	—	57	6	no	—
114	C₂×C₅	—	171	6	no	—
115	C₃²	C₂×C₅²×D₁₉	1	1	yes	characteristic, complemented, Sylow
116	C₂×C₃	—	3	1	no	—
117	C₂×C₃	—	57	1	no	—
118	D₃	—	19	1	no	—
119	D₃	—	57	1	no	—
120	D₃	—	57	1	no	—
121	D₃	—	114	1	no	—
122	D₃	C₅²×D₅₇	1	1	yes	characteristic, direct product factor
123	C₅	C₅×D₃×D₅₇	1	6	yes	direct product factor
124	C₂²	—	171	1	no	Sylow
125	C₃	—	2	1	no	—
126	C₃	C₅²×D₃×D₁₉	1	1	yes	characteristic, complemented
127	C₃	C₂×C₅²×D₅₇	1	1	yes	characteristic, complemented
128	C₂	—	3	1	no	—
129	C₂	—	57	1	no	—
130	C₂	—	171	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₃⋊D₅₇

C₂

yes

maximal, characteristic, complemented

C₃×C₅²×D₅₇

C₂

yes

maximal, characteristic, complemented

C₃×C₅²×D₃×C₁₉

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₅₇

—

maximal

C₅²×D₃×D₁₉

—

maximal

C₃²×C₅²×C₁₉

C₂²

yes

characteristic, complemented, Hall

C₅×D₃×D₅₇

C₅

yes

maximal, direct product factor

C₅²×D₅₇

—

C₅²×D₅₇

—

C₅²×D₅₇

—

C₅²×D₅₇

D₃

yes

characteristic, direct product factor

C₅²×D₃×C₁₉

D₃

yes

characteristic, complemented

C₃×C₅²×D₁₉

—

C₂×C₃×C₅²×C₁₉

—

C₂×C₅²×D₁₉

—

Hall

C₅×C₃⋊D₅₇

C₂×C₅

yes

complemented

C₃×C₅×D₅₇

C₂×C₅

yes

complemented

C₃×C₅×D₃×C₁₉

C₂×C₅

yes

complemented

C₃×C₅²×C₁₉

—

C₃×C₅²×C₁₉

C₂×D₃

yes

characteristic, complemented

C₃×C₅²×C₁₉

C₂×D₃

yes

characteristic, complemented

C₂×C₅×D₅₇

—

C₅×D₃×D₁₉

—

C₂×C₅²×C₁₉

—

C₅²×D₁₉

—

C₅²×D₁₉

—

C₅²×D₃²

—

maximal, Hall

C₃²×C₅×C₁₉

C₂²×C₅

yes

complemented

D₃×D₅₇

C₅²

yes

characteristic, direct product factor, Hall

C₂×C₃×C₅×C₁₉

—

C₅×D₅₇

—

C₅×D₅₇

—

C₅×D₅₇

—

C₅×D₅₇

C₅×D₃

yes

direct product factor

C₅×D₃×C₁₉

C₅×D₃

yes

complemented

C₃×C₅×D₁₉

—

C₅²×C₁₉

D₃²

yes

characteristic, complemented, Hall

C₅²×C₃⋊D₃

—

C₃×C₅²×D₃

—

C₃×C₅²×D₃

D₁₉

yes

characteristic, complemented

C₂×C₅×D₁₉

—

C₃⋊D₅₇

C₂×C₅²

yes

characteristic, complemented

C₃×D₅₇

C₂×C₅²

yes

characteristic, complemented

C₃×D₃×C₁₉

C₂×C₅²

yes

characteristic, complemented

C₂×C₅²×D₃

—

C₂×C₅²×D₃

—

C₃×C₅×C₁₉

—

C₃×C₅×C₁₉

C₂×C₅×D₃

yes

complemented

C₃×C₅×C₁₉

C₂×C₅×D₃

yes

complemented

C₂×D₅₇

—

D₃×D₁₉

—

C₃²×C₅²

C₂×D₁₉

yes

characteristic, complemented, Hall

C₂×C₅×C₁₉

—

C₅×D₁₉

—

C₅×D₁₉

—

C₅×D₃²

—

C₃²×C₁₉

C₂²×C₅²

yes

characteristic, complemented, Hall

C₂×C₃×C₅²

—

C₂×C₃×C₅²

—

C₅²×D₃

—

C₅²×D₃

—

C₅²×D₃

—

C₅²×D₃

—

114

—

C₅²×D₃

D₅₇

yes

characteristic, direct product factor

C₂×C₃×C₁₉

—

D₅₇

—

D₅₇

—

D₅₇

—

D₅₇

C₅²×D₃

yes

characteristic, direct product factor

C₃×D₁₉

—

D₃×C₁₉

C₅²×D₃

yes

characteristic, complemented

C₂²×C₅²

—

171

Hall

C₅×C₁₉

C₅×D₃²

yes

complemented

C₅×C₃⋊D₃

—

C₃×C₅×D₃

—

C₃×C₅×D₃

C₅×D₁₉

yes

complemented

C₂×D₁₉

—

Hall

C₃×C₅²

—

C₃×C₅²

D₃×D₁₉

yes

characteristic, complemented

C₃×C₅²

C₂×D₅₇

yes

characteristic, complemented

C₂×C₅×D₃

—

C₂×C₅×D₃

—

C₃×C₁₉

—

C₃×C₁₉

C₂×C₅²×D₃

yes

characteristic, complemented

C₃×C₁₉

C₂×C₅²×D₃

yes

characteristic, complemented

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

—

171

—

C₃²×C₅

C₂×C₅×D₁₉

yes

complemented

C₂×C₁₉

—

D₁₉

—

D₁₉

—

D₃²

—

Hall

C₂×C₃×C₅

—

C₂×C₃×C₅

—

C₅×D₃

—

C₅×D₃

—

C₅×D₃

—

C₅×D₃

—

114

—

100

C₅×D₃

C₅×D₅₇

yes

direct product factor

101

C₅²

D₃×D₅₇

yes

characteristic, direct product factor, Sylow

102

C₂²×C₅

—

171

—

103

C₁₉

C₅²×D₃²

yes

characteristic, complemented, Sylow

104

C₃⋊D₃

—

105

C₃×D₃

—

106

C₃×D₃

C₅²×D₁₉

yes

characteristic, complemented

107

C₃×C₅

—

108

C₃×C₅

C₅×D₃×D₁₉

yes

complemented

109

C₃×C₅

C₂×C₅×D₅₇

yes

complemented

110

C₂×D₃

—

111

C₂×D₃

—

112

C₂×C₅

—

113

C₂×C₅

—

114

C₂×C₅

—

171

—

115

C₃²

C₂×C₅²×D₁₉

yes

characteristic, complemented, Sylow

116

C₂×C₃

—

117

C₂×C₃

—

118

D₃

—

119

D₃

—

120

D₃

—

121

D₃

—

114

—

122

D₃

C₅²×D₅₇

yes

characteristic, direct product factor

123

C₅

C₅×D₃×D₅₇

yes

direct product factor

124

C₂²

—

171

Sylow

125

C₃

—

126

C₃

C₅²×D₃×D₁₉

yes

characteristic, complemented

127

C₃

C₂×C₅²×D₅₇

yes

characteristic, complemented

128

C₂

—

129

C₂

—

130

C₂

—

171

—

	subgroup	G/H
center, hypercenter	C₅²	D₃×D₅₇
commutator, cosocle, hypocenter	C₃²×C₁₉	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₉	C₂²
LCS	G ▷ C₃²×C₁₉
UCS	C₁ ◁ C₅²

subgroup

G/H

center, hypercenter

C₅²

D₃×D₅₇

commutator, cosocle, hypocenter

C₃²×C₁₉

C₂²×C₅²

Frattini

C₁

Fitting, socle

C₃²×C₅²×C₁₉

C₂²

LCS

G ▷ C₃²×C₁₉

UCS

C₁ ◁ C₅²

N	H ≅ G/N	type
1. C₅²×C₃⋊D₅₇	C₂	semidirect
2. C₃×C₅²×D₅₇	C₂	semidirect
3. C₃×C₅²×D₃×C₁₉	C₂	semidirect
6. C₃²×C₅²×C₁₉	C₂²	semidirect
7. C₅×D₃×D₅₇	C₅	direct (N × H)
11. C₅²×D₅₇	D₃	direct (N × H)
12. C₅²×D₃×C₁₉	D₃	semidirect
16. C₅×C₃⋊D₅₇	C₂×C₅	semidirect
17. C₃×C₅×D₅₇	C₂×C₅	semidirect
18. C₃×C₅×D₃×C₁₉	C₂×C₅	semidirect
20. C₃×C₅²×C₁₉	C₂×D₃	semidirect
21. C₃×C₅²×C₁₉	C₂×D₃	semidirect
28. C₃²×C₅×C₁₉	C₂²×C₅	semidirect
29. D₃×D₅₇	C₅²	direct (N × H)
34. C₅×D₅₇	C₅×D₃	direct (N × H)
35. C₅×D₃×C₁₉	C₅×D₃	semidirect
37. C₅²×C₁₉	D₃²	semidirect
40. C₃×C₅²×D₃	D₁₉	semidirect
42. C₃⋊D₅₇	C₂×C₅²	semidirect
43. C₃×D₅₇	C₂×C₅²	semidirect
44. C₃×D₃×C₁₉	C₂×C₅²	semidirect
48. C₃×C₅×C₁₉	C₂×C₅×D₃	semidirect
49. C₃×C₅×C₁₉	C₂×C₅×D₃	semidirect
52. C₃²×C₅²	C₂×D₁₉	semidirect
57. C₃²×C₁₉	C₂²×C₅²	semidirect
64. C₅²×D₃	D₅₇	direct (N × H)
69. D₅₇	C₅²×D₃	direct (N × H)
71. D₃×C₁₉	C₅²×D₃	semidirect
73. C₅×C₁₉	C₅×D₃²	semidirect
76. C₃×C₅×D₃	C₅×D₁₉	semidirect
79. C₃×C₅²	D₃×D₁₉	semidirect
80. C₃×C₅²	C₂×D₅₇	semidirect
84. C₃×C₁₉	C₂×C₅²×D₃	semidirect
85. C₃×C₁₉	C₂×C₅²×D₃	semidirect
89. C₃²×C₅	C₂×C₅×D₁₉	semidirect
100. C₅×D₃	C₅×D₅₇	direct (N × H)
101. C₅²	D₃×D₅₇	direct (N × H)
103. C₁₉	C₅²×D₃²	semidirect
106. C₃×D₃	C₅²×D₁₉	semidirect
108. C₃×C₅	C₅×D₃×D₁₉	semidirect
109. C₃×C₅	C₂×C₅×D₅₇	semidirect
115. C₃²	C₂×C₅²×D₁₉	semidirect
122. D₃	C₅²×D₅₇	direct (N × H)
123. C₅	C₅×D₃×D₅₇	direct (N × H)
126. C₃	C₅²×D₃×D₁₉	semidirect
127. C₃	C₂×C₅²×D₅₇	semidirect

H ≅ G/N

type

1. C₅²×C₃⋊D₅₇

C₂

semidirect

2. C₃×C₅²×D₅₇

C₂

semidirect

3. C₃×C₅²×D₃×C₁₉

C₂

semidirect

6. C₃²×C₅²×C₁₉

C₂²

semidirect

7. C₅×D₃×D₅₇

C₅

direct (N × H)

11. C₅²×D₅₇

D₃

direct (N × H)

12. C₅²×D₃×C₁₉

D₃

semidirect

16. C₅×C₃⋊D₅₇

C₂×C₅

semidirect

17. C₃×C₅×D₅₇

C₂×C₅

semidirect

18. C₃×C₅×D₃×C₁₉

C₂×C₅

semidirect

20. C₃×C₅²×C₁₉

C₂×D₃

semidirect

21. C₃×C₅²×C₁₉

C₂×D₃

semidirect

28. C₃²×C₅×C₁₉

C₂²×C₅

semidirect

29. D₃×D₅₇

C₅²

direct (N × H)

34. C₅×D₅₇

C₅×D₃

direct (N × H)

35. C₅×D₃×C₁₉

C₅×D₃

semidirect

37. C₅²×C₁₉

D₃²

semidirect

40. C₃×C₅²×D₃

D₁₉

semidirect

42. C₃⋊D₅₇

C₂×C₅²

semidirect

43. C₃×D₅₇

C₂×C₅²

semidirect

44. C₃×D₃×C₁₉

C₂×C₅²

semidirect

48. C₃×C₅×C₁₉

C₂×C₅×D₃

semidirect

49. C₃×C₅×C₁₉

C₂×C₅×D₃

semidirect

52. C₃²×C₅²

C₂×D₁₉

semidirect

57. C₃²×C₁₉

C₂²×C₅²

semidirect

64. C₅²×D₃

D₅₇

direct (N × H)

69. D₅₇

C₅²×D₃

direct (N × H)

71. D₃×C₁₉

C₅²×D₃

semidirect

73. C₅×C₁₉

C₅×D₃²

semidirect

76. C₃×C₅×D₃

C₅×D₁₉

semidirect

79. C₃×C₅²

D₃×D₁₉

semidirect

80. C₃×C₅²

C₂×D₅₇

semidirect

84. C₃×C₁₉

C₂×C₅²×D₃

semidirect

85. C₃×C₁₉

C₂×C₅²×D₃

semidirect

89. C₃²×C₅

C₂×C₅×D₁₉

semidirect

100. C₅×D₃

C₅×D₅₇

direct (N × H)

101. C₅²

D₃×D₅₇

direct (N × H)

103. C₁₉

C₅²×D₃²

semidirect

106. C₃×D₃

C₅²×D₁₉

semidirect

108. C₃×C₅

C₅×D₃×D₁₉

semidirect

109. C₃×C₅

C₂×C₅×D₅₇

semidirect

115. C₃²

C₂×C₅²×D₁₉

semidirect

122. D₃

C₅²×D₅₇

direct (N × H)

123. C₅

C₅×D₃×D₅₇

direct (N × H)

126. C₃

C₅²×D₃×D₁₉

semidirect

127. C₃

C₂×C₅²×D₅₇

semidirect

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

subgroup	index
C₅²×C₃⋊D₅₇	2
C₃×C₅²×D₅₇	2
C₃×C₅²×D₃×C₁₉	2
C₂×C₅²×D₅₇	3
C₅²×D₃×D₁₉	3
C₅×D₃×D₅₇	5
C₅²×D₃²	19

subgroup

index

C₅²×C₃⋊D₅₇

C₃×C₅²×D₅₇

C₃×C₅²×D₃×C₁₉

C₂×C₅²×D₅₇

C₅²×D₃×D₁₉

C₅×D₃×D₅₇

C₅²×D₃²

Subgroups — C₅²×D₃×D₅₇ #30685921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C52×D3×D57 #30685921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₃×D₅₇ #30685921