Group #13108398 — subgroups

Subgroups — D₃×D₅×3₋¹⁺² #13108398

Order 1620.

204 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 70 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	3₋¹⁺²×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₃×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₅×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×D₃×D₅	C₃	1	1	yes	maximal, characteristic
5	C₂×D₅×3₋¹⁺²	—	3	1	no	maximal
6	D₃×C₉×D₅	C₃	1	3	yes	maximal, complemented
7	C₃×C₅×3₋¹⁺²	C₂²	1	1	yes	characteristic, complemented, Hall
8	C₂×D₃×3₋¹⁺²	—	5	1	no	maximal, Hall
9	C₃²×D₁₅	C₂×C₃	1	1	yes	characteristic
10	C₃²×C₅×D₃	C₂×C₃	1	1	yes	characteristic
11	C₃³×D₅	C₂×C₃	1	1	yes	characteristic
12	C₂×C₅×3₋¹⁺²	—	3	1	no	—
13	C₉×D₁₅	C₂×C₃	1	3	yes	complemented
14	C₅×D₃×C₉	C₂×C₃	1	3	yes	complemented
15	D₅×3₋¹⁺²	—	2	1	no	—
16	D₅×3₋¹⁺²	—	2	3	no	—
17	D₅×3₋¹⁺²	—	3	1	no	—
18	D₅×3₋¹⁺²	D₃	1	1	yes	characteristic, direct product factor
19	C₃×C₉×D₅	C₂×C₃	1	3	yes	complemented
20	C₂×C₃²×D₅	—	3	1	no	—
21	C₃×D₃×D₅	—	3	1	no	—
22	C₃×D₃×D₅	C₃²	1	1	yes	characteristic
23	C₂×C₉×D₅	—	3	3	no	—
24	C₂×C₃×3₋¹⁺²	—	5	1	no	—
25	D₃×3₋¹⁺²	—	5	1	no	—
26	D₃×3₋¹⁺²	D₅	1	1	yes	characteristic, direct product factor
27	C₃³×C₅	C₂²×C₃	1	1	yes	characteristic
28	C₅×3₋¹⁺²	—	2	1	no	—
29	C₅×3₋¹⁺²	—	2	3	no	—
30	C₅×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic, complemented
31	C₃×C₅×C₉	C₂²×C₃	1	3	yes	complemented
32	C₂×C₃²×D₃	—	5	1	no	—
33	C₂²×3₋¹⁺²	—	15	1	no	—
34	C₂×D₃×C₉	—	5	3	no	—
35	C₂×C₃²×C₅	—	3	1	no	—
36	C₃×D₁₅	—	3	1	no	—
37	C₃×D₁₅	C₂×C₃²	1	1	yes	characteristic
38	C₃×C₅×D₃	—	3	1	no	—
39	C₃×C₅×D₃	C₂×C₃²	1	1	yes	characteristic
40	C₃²×D₅	—	2	1	no	—
41	C₃²×D₅	—	3	1	no	—
42	C₃²×D₅	—	3	1	no	—
43	C₃²×D₅	—	6	1	no	—
44	C₃²×D₅	C₃×D₃	1	1	yes	characteristic
45	C₃²×D₅	C₂×C₃²	1	1	yes	characteristic
46	C₂×C₅×C₉	—	3	3	no	—
47	C₉×D₅	—	2	3	no	—
48	C₉×D₅	—	3	3	no	—
49	C₉×D₅	C₃×D₃	1	3	yes	complemented
50	C₃×3₋¹⁺²	C₂×D₅	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₃×D₅	—	3	1	no	—
52	C₂×C₃×D₅	—	9	1	no	—
53	D₃×D₅	3₋¹⁺²	1	1	yes	characteristic, direct product factor
54	C₂×C₃³	—	5	1	no	—
55	C₃²×D₃	—	5	1	no	—
56	C₃²×D₃	C₃×D₅	1	1	yes	characteristic
57	C₂×3₋¹⁺²	—	3	1	no	—
58	C₂×3₋¹⁺²	—	5	1	no	—
59	C₂×3₋¹⁺²	—	10	3	no	—
60	C₂×3₋¹⁺²	—	10	1	no	—
61	C₂×3₋¹⁺²	—	15	1	no	—
62	C₂×C₃×C₉	—	5	3	no	—
63	D₃×C₉	—	5	3	no	—
64	D₃×C₉	C₃×D₅	1	3	yes	complemented
65	C₃²×C₅	—	2	1	no	—
66	C₃²×C₅	—	3	1	no	—
67	C₃²×C₅	—	6	1	no	—
68	C₃²×C₅	C₂×C₃×D₃	1	1	yes	characteristic
69	C₃²×C₅	C₂²×C₃²	1	1	yes	characteristic
70	C₅×C₉	—	2	3	no	—
71	C₅×C₉	C₂×C₃×D₃	1	3	yes	complemented
72	C₂²×C₃²	—	15	1	no	—
73	C₂×C₃×D₃	—	5	1	no	—
74	C₂×C₃×D₃	—	15	1	no	—
75	C₂²×C₉	—	15	3	no	—
76	C₂×C₃×C₅	—	3	1	no	—
77	C₂×C₃×C₅	—	9	1	no	—
78	D₁₅	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
79	C₃×D₅	—	2	1	no	—
80	C₃×D₅	—	3	1	no	—
81	C₃×D₅	—	3	1	no	—
82	C₃×D₅	—	6	1	no	—
83	C₃×D₅	—	9	1	no	—
84	C₃×D₅	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
85	C₃×D₅	C₃²×D₃	1	1	yes	characteristic
86	C₅×D₃	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
87	C₃³	C₂×C₃×D₅	1	1	yes	characteristic
88	3₋¹⁺²	—	2	1	no	—
89	3₋¹⁺²	—	2	3	no	—
90	3₋¹⁺²	D₃×D₅	1	1	yes	characteristic, direct product factor
91	C₃×C₉	C₂×C₃×D₅	1	3	yes	complemented
92	C₂×D₅	—	3	1	no	Hall
93	C₂×C₃²	—	3	1	no	—
94	C₂×C₃²	—	5	1	no	—
95	C₂×C₃²	—	5	1	no	—
96	C₂×C₃²	—	10	1	no	—
97	C₂×C₃²	—	15	1	no	—
98	C₂×C₃²	—	15	1	no	—
99	C₂×C₃²	—	30	1	no	—
100	C₃×D₃	—	3	1	no	—
101	C₃×D₃	—	5	1	no	—
102	C₃×D₃	—	15	1	no	—
103	C₃×D₃	C₃²×D₅	1	1	yes	characteristic
104	C₂×C₉	—	3	3	no	—
105	C₂×C₉	—	5	3	no	—
106	C₂×C₉	—	10	3	no	—
107	C₂×C₉	—	15	3	no	—
108	C₃×C₅	—	2	1	no	—
109	C₃×C₅	—	3	1	no	—
110	C₃×C₅	—	6	1	no	—
111	C₃×C₅	C₂²×3₋¹⁺²	1	1	yes	characteristic, complemented
112	C₃×C₅	C₂×C₃²×D₃	1	1	yes	characteristic
113	C₂²×C₃	—	15	1	no	—
114	C₂²×C₃	—	45	1	no	—
115	C₂×D₃	—	5	1	no	—
116	C₂×C₅	—	3	1	no	—
117	D₅	—	3	1	no	—
118	D₅	D₃×3₋¹⁺²	1	1	yes	characteristic, direct product factor
119	C₃²	—	2	1	no	—
120	C₃²	—	3	1	no	—
121	C₃²	—	6	1	no	—
122	C₃²	C₃×D₃×D₅	1	1	yes	characteristic
123	C₃²	C₂×C₃²×D₅	1	1	yes	characteristic
124	C₉	—	2	3	no	—
125	C₉	C₃×D₃×D₅	1	3	yes	complemented
126	C₂×C₃	—	3	1	no	—
127	C₂×C₃	—	5	1	no	—
128	C₂×C₃	—	5	1	no	—
129	C₂×C₃	—	9	1	no	—
130	C₂×C₃	—	10	1	no	—
131	C₂×C₃	—	15	1	no	—
132	C₂×C₃	—	15	1	no	—
133	C₂×C₃	—	30	1	no	—
134	C₂×C₃	—	45	1	no	—
135	D₃	—	5	1	no	—
136	D₃	D₅×3₋¹⁺²	1	1	yes	characteristic, direct product factor
137	C₅	C₂×D₃×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
138	C₂²	—	15	1	no	Sylow
139	C₃	—	2	1	no	—
140	C₃	—	3	1	no	—
141	C₃	—	6	1	no	—
142	C₃	C₂×D₅×3₋¹⁺²	1	1	yes	characteristic, complemented
143	C₃	C₃²×D₃×D₅	1	1	yes	characteristic
144	C₂	—	3	1	no	—
145	C₂	—	5	1	no	—
146	C₂	—	15	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×D₁₅

C₂

yes

maximal, characteristic, complemented

C₅×D₃×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

C₃×D₅×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

C₃²×D₃×D₅

C₃

yes

maximal, characteristic

C₂×D₅×3₋¹⁺²

—

maximal

D₃×C₉×D₅

C₃

yes

maximal, complemented

C₃×C₅×3₋¹⁺²

C₂²

yes

characteristic, complemented, Hall

C₂×D₃×3₋¹⁺²

—

maximal, Hall

C₃²×D₁₅

C₂×C₃

yes

characteristic

C₃²×C₅×D₃

C₂×C₃

yes

characteristic

C₃³×D₅

C₂×C₃

yes

characteristic

C₂×C₅×3₋¹⁺²

—

C₉×D₁₅

C₂×C₃

yes

complemented

C₅×D₃×C₉

C₂×C₃

yes

complemented

D₅×3₋¹⁺²

—

D₅×3₋¹⁺²

—

D₅×3₋¹⁺²

—

D₅×3₋¹⁺²

D₃

yes

characteristic, direct product factor

C₃×C₉×D₅

C₂×C₃

yes

complemented

C₂×C₃²×D₅

—

C₃×D₃×D₅

—

C₃×D₃×D₅

C₃²

yes

characteristic

C₂×C₉×D₅

—

C₂×C₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

D₃×3₋¹⁺²

D₅

yes

characteristic, direct product factor

C₃³×C₅

C₂²×C₃

yes

characteristic

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

—

C₅×3₋¹⁺²

C₂×D₃

yes

characteristic, complemented

C₃×C₅×C₉

C₂²×C₃

yes

complemented

C₂×C₃²×D₃

—

C₂²×3₋¹⁺²

—

C₂×D₃×C₉

—

C₂×C₃²×C₅

—

C₃×D₁₅

—

C₃×D₁₅

C₂×C₃²

yes

characteristic

C₃×C₅×D₃

—

C₃×C₅×D₃

C₂×C₃²

yes

characteristic

C₃²×D₅

—

C₃²×D₅

—

C₃²×D₅

—

C₃²×D₅

—

C₃²×D₅

C₃×D₃

yes

characteristic

C₃²×D₅

C₂×C₃²

yes

characteristic

C₂×C₅×C₉

—

C₉×D₅

—

C₉×D₅

—

C₉×D₅

C₃×D₃

yes

complemented

C₃×3₋¹⁺²

C₂×D₅

yes

characteristic, complemented, Sylow

C₂×C₃×D₅

—

C₂×C₃×D₅

—

D₃×D₅

3₋¹⁺²

yes

characteristic, direct product factor

C₂×C₃³

—

C₃²×D₃

—

C₃²×D₃

C₃×D₅

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

D₃×C₉

—

D₃×C₉

C₃×D₅

yes

complemented

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

—

C₃²×C₅

C₂×C₃×D₃

yes

characteristic

C₃²×C₅

C₂²×C₃²

yes

characteristic

C₅×C₉

—

C₅×C₉

C₂×C₃×D₃

yes

complemented

C₂²×C₃²

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂²×C₉

—

C₂×C₃×C₅

—

C₂×C₃×C₅

—

D₁₅

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

C₃×D₅

—

C₃×D₅

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃×D₅

C₃²×D₃

yes

characteristic

C₅×D₃

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃³

C₂×C₃×D₅

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

D₃×D₅

yes

characteristic, direct product factor

C₃×C₉

C₂×C₃×D₅

yes

complemented

C₂×D₅

—

Hall

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

100

C₃×D₃

—

101

C₃×D₃

—

102

C₃×D₃

—

103

C₃×D₃

C₃²×D₅

yes

characteristic

104

C₂×C₉

—

105

C₂×C₉

—

106

C₂×C₉

—

107

C₂×C₉

—

108

C₃×C₅

—

109

C₃×C₅

—

110

C₃×C₅

—

111

C₃×C₅

C₂²×3₋¹⁺²

yes

characteristic, complemented

112

C₃×C₅

C₂×C₃²×D₃

yes

characteristic

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂×D₃

—

116

C₂×C₅

—

117

D₅

—

118

D₅

D₃×3₋¹⁺²

yes

characteristic, direct product factor

119

C₃²

—

120

C₃²

—

121

C₃²

—

122

C₃²

C₃×D₃×D₅

yes

characteristic

123

C₃²

C₂×C₃²×D₅

yes

characteristic

124

C₉

—

125

C₉

C₃×D₃×D₅

yes

complemented

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

D₃

—

136

D₃

D₅×3₋¹⁺²

yes

characteristic, direct product factor

137

C₅

C₂×D₃×3₋¹⁺²

yes

characteristic, complemented, Sylow

138

C₂²

—

Sylow

139

C₃

—

140

C₃

—

141

C₃

—

142

C₃

C₂×D₅×3₋¹⁺²

yes

characteristic, complemented

143

C₃

C₃²×D₃×D₅

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

—

	subgroup	G/H
center, Frattini	C₃	C₃²×D₃×D₅
commutator, socle, cosocle	C₃²×C₅	C₂²×C₃²
Fitting	C₃×C₅×3₋¹⁺²	C₂²
prime radical	C₃²×D₃×D₅	C₃
hypercenter	3₋¹⁺²	D₃×D₅
hypocenter	C₃×C₅	C₂²×3₋¹⁺²
LCS	G ▷ C₃²×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

subgroup

G/H

center, Frattini

C₃

C₃²×D₃×D₅

commutator, socle, cosocle

C₃²×C₅

C₂²×C₃²

Fitting

C₃×C₅×3₋¹⁺²

C₂²

prime radical

C₃²×D₃×D₅

C₃

hypercenter

3₋¹⁺²

D₃×D₅

hypocenter

C₃×C₅

C₂²×3₋¹⁺²

LCS

G ▷ C₃²×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₃ ◁ 3₋¹⁺²

N	H ≅ G/N	type
1. 3₋¹⁺²×D₁₅	C₂	semidirect
2. C₅×D₃×3₋¹⁺²	C₂	semidirect
3. C₃×D₅×3₋¹⁺²	C₂	semidirect
6. D₃×C₉×D₅	C₃	semidirect
7. C₃×C₅×3₋¹⁺²	C₂²	semidirect
13. C₉×D₁₅	C₂×C₃	semidirect
14. C₅×D₃×C₉	C₂×C₃	semidirect
18. D₅×3₋¹⁺²	D₃	direct (N × H)
19. C₃×C₉×D₅	C₂×C₃	semidirect
26. D₃×3₋¹⁺²	D₅	direct (N × H)
30. C₅×3₋¹⁺²	C₂×D₃	semidirect
31. C₃×C₅×C₉	C₂²×C₃	semidirect
49. C₉×D₅	C₃×D₃	semidirect
50. C₃×3₋¹⁺²	C₂×D₅	semidirect
53. D₃×D₅	3₋¹⁺²	direct (N × H)
64. D₃×C₉	C₃×D₅	semidirect
71. C₅×C₉	C₂×C₃×D₃	semidirect
78. D₁₅	C₂×3₋¹⁺²	semidirect
84. C₃×D₅	C₂×3₋¹⁺²	semidirect
86. C₅×D₃	C₂×3₋¹⁺²	semidirect
90. 3₋¹⁺²	D₃×D₅	direct (N × H)
91. C₃×C₉	C₂×C₃×D₅	semidirect
111. C₃×C₅	C₂²×3₋¹⁺²	semidirect
118. D₅	D₃×3₋¹⁺²	direct (N × H)
125. C₉	C₃×D₃×D₅	semidirect
136. D₃	D₅×3₋¹⁺²	direct (N × H)
137. C₅	C₂×D₃×3₋¹⁺²	semidirect
142. C₃	C₂×D₅×3₋¹⁺²	semidirect

H ≅ G/N

type

1. 3₋¹⁺²×D₁₅

C₂

semidirect

2. C₅×D₃×3₋¹⁺²

C₂

semidirect

3. C₃×D₅×3₋¹⁺²

C₂

semidirect

6. D₃×C₉×D₅

C₃

semidirect

7. C₃×C₅×3₋¹⁺²

C₂²

semidirect

13. C₉×D₁₅

C₂×C₃

semidirect

14. C₅×D₃×C₉

C₂×C₃

semidirect

18. D₅×3₋¹⁺²

D₃

direct (N × H)

19. C₃×C₉×D₅

C₂×C₃

semidirect

26. D₃×3₋¹⁺²

D₅

direct (N × H)

30. C₅×3₋¹⁺²

C₂×D₃

semidirect

31. C₃×C₅×C₉

C₂²×C₃

semidirect

49. C₉×D₅

C₃×D₃

semidirect

50. C₃×3₋¹⁺²

C₂×D₅

semidirect

53. D₃×D₅

3₋¹⁺²

direct (N × H)

64. D₃×C₉

C₃×D₅

semidirect

71. C₅×C₉

C₂×C₃×D₃

semidirect

78. D₁₅

C₂×3₋¹⁺²

semidirect

84. C₃×D₅

C₂×3₋¹⁺²

semidirect

86. C₅×D₃

C₂×3₋¹⁺²

semidirect

90. 3₋¹⁺²

D₃×D₅

direct (N × H)

91. C₃×C₉

C₂×C₃×D₅

semidirect

111. C₃×C₅

C₂²×3₋¹⁺²

semidirect

118. D₅

D₃×3₋¹⁺²

direct (N × H)

125. C₉

C₃×D₃×D₅

semidirect

136. D₃

D₅×3₋¹⁺²

direct (N × H)

137. C₅

C₂×D₃×3₋¹⁺²

semidirect

142. C₃

C₂×D₅×3₋¹⁺²

semidirect

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

subgroup	index
3₋¹⁺²×D₁₅	2
C₅×D₃×3₋¹⁺²	2
C₃×D₅×3₋¹⁺²	2
C₃²×D₃×D₅	3
C₂×D₅×3₋¹⁺²	3
D₃×C₉×D₅	3
C₂×D₃×3₋¹⁺²	5

subgroup

index

3₋¹⁺²×D₁₅

C₅×D₃×3₋¹⁺²

C₃×D₅×3₋¹⁺²

C₃²×D₃×D₅

C₂×D₅×3₋¹⁺²

D₃×C₉×D₅

C₂×D₃×3₋¹⁺²

Subgroups — D₃×D₅×3₋¹⁺² #13108398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×D5×3−1+2 #13108398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×D₅×3₋¹⁺² #13108398