Group #44477694 — subgroups

Subgroups — D₅×(C₁₃⋊C₉)⋊D₁₃ #44477694

Order 30420.

200 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 42 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	C₅⋊((C₁₃⋊C₉)⋊D₁₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₅×C₁₃×C₁₃⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×(C₁₃⋊C₉)⋊D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₅×C₁₃⋊D₁₃	C₃	1	1	yes	maximal, characteristic
5	C₅×C₁₃×C₁₃⋊C₉	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₂×(C₁₃⋊C₉)⋊D₁₃	—	5	1	no	maximal, Hall
7	C₃×C₅⋊(C₁₃⋊D₁₃)	C₂×C₃	1	1	yes	characteristic
8	C₃×D₅×C₁₃²	C₂×C₃	1	1	yes	characteristic
9	C₃×C₅×C₁₃⋊D₁₃	C₂×C₃	1	1	yes	characteristic
10	D₅×C₁₃⋊D₁₃	C₉	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₃×C₁₃⋊C₉	—	5	1	no	—
12	(C₁₃⋊C₉)⋊D₁₃	—	5	1	no	—
13	(C₁₃⋊C₉)⋊D₁₃	D₅	1	1	yes	characteristic, direct product factor
14	C₃×C₅×C₁₃²	C₂²×C₃	1	1	yes	characteristic
15	C₉×D₅×D₁₃	—	13	1	no	maximal
16	D₅×D₁₃⋊C₉	—	13	1	no	maximal
17	C₂×C₃×C₁₃⋊D₁₃	—	5	1	no	—
18	C₅⋊(C₁₃⋊D₁₃)	C₂×C₉	1	1	yes	characteristic, complemented
19	D₅×C₁₃²	C₂×C₉	1	1	yes	characteristic, complemented
20	C₅×C₁₃⋊D₁₃	C₂×C₉	1	1	yes	characteristic, complemented
21	C₁₃×C₁₃⋊C₉	C₂×D₅	1	1	yes	characteristic, complemented, Hall
22	C₉×D₆₅	—	13	1	no	—
23	C₉×D₅×C₁₃	—	13	1	no	—
24	C₅×C₉×D₁₃	—	13	1	no	—
25	C₅⋊(D₁₃⋊C₉)	—	13	1	no	—
26	D₅×C₁₃⋊C₉	D₁₃	1	1	yes	characteristic, complemented
27	C₅×D₁₃⋊C₉	—	13	1	no	—
28	C₂×C₃×C₁₃²	—	5	1	no	—
29	C₃×C₁₃⋊D₁₃	—	5	1	no	—
30	C₃×C₁₃⋊D₁₃	C₃×D₅	1	1	yes	characteristic
31	C₅×C₁₃²	C₂²×C₉	1	1	yes	characteristic, complemented, Hall
32	C₃×D₅×D₁₃	—	13	1	no	—
33	C₃×D₅×D₁₃	—	13	1	no	—
34	C₃×D₅×D₁₃	—	39	4	no	—
35	C₂×C₁₃⋊D₁₃	—	5	1	no	Hall
36	C₅×C₉×C₁₃	—	13	1	no	—
37	C₅×C₁₃⋊C₉	C₂×D₁₃	1	1	yes	characteristic, complemented
38	C₃×C₁₃²	C₂×C₃×D₅	1	1	yes	characteristic
39	C₂×C₉×D₁₃	—	65	1	no	—
40	C₂×D₁₃⋊C₉	—	65	1	no	—
41	C₃×D₆₅	—	13	1	no	—
42	C₃×D₆₅	—	13	1	no	—
43	C₃×D₆₅	—	39	4	no	—
44	C₃×D₅×C₁₃	—	3	4	no	—
45	C₃×D₅×C₁₃	D₁₃⋊C₃	1	1	yes	characteristic
46	C₃×D₅×C₁₃	C₃×D₁₃	1	1	yes	characteristic
47	C₃×C₅×D₁₃	—	13	1	no	—
48	C₃×C₅×D₁₃	—	13	1	no	—
49	C₃×C₅×D₁₃	—	39	4	no	—
50	C₂×C₁₃²	—	5	1	no	—
51	C₁₃⋊D₁₃	—	5	1	no	—
52	C₁₃⋊D₁₃	C₉×D₅	1	1	yes	characteristic, complemented
53	D₅×D₁₃	—	13	1	no	—
54	D₅×D₁₃	—	13	1	no	—
55	D₅×D₁₃	—	39	4	no	—
56	C₂×C₉×C₁₃	—	65	1	no	—
57	C₉×D₁₃	—	13	1	no	—
58	C₉×D₁₃	—	65	1	no	—
59	C₂×C₁₃⋊C₉	—	5	1	no	—
60	D₁₃⋊C₉	—	13	1	no	—
61	D₁₃⋊C₉	—	65	1	no	—
62	C₃×C₅×C₁₃	—	3	4	no	—
63	C₃×C₅×C₁₃	C₂×D₁₃⋊C₃	1	1	yes	characteristic
64	C₃×C₅×C₁₃	C₂×C₃×D₁₃	1	1	yes	characteristic
65	C₂×C₉×D₅	—	169	1	no	Hall
66	C₁₃²	C₂×C₉×D₅	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₃×D₁₃	—	65	1	no	—
68	C₂×C₃×D₁₃	—	65	1	no	—
69	C₂×C₃×D₁₃	—	195	4	no	—
70	D₆₅	—	13	1	no	—
71	D₆₅	—	13	1	no	—
72	D₆₅	—	39	4	no	—
73	C₅×D₁₃	—	13	1	no	—
74	C₅×D₁₃	—	13	1	no	—
75	C₅×D₁₃	—	39	4	no	—
76	D₅×C₁₃	—	3	4	no	—
77	D₅×C₁₃	D₁₃⋊C₉	1	1	yes	characteristic, complemented
78	D₅×C₁₃	C₉×D₁₃	1	1	yes	characteristic, complemented
79	C₉×C₁₃	—	13	1	no	—
80	C₁₃⋊C₉	D₅×D₁₃	1	1	yes	characteristic, complemented
81	C₂×C₅×C₉	—	169	1	no	—
82	C₉×D₅	—	13	1	no	—
83	C₉×D₅	—	169	1	no	—
84	C₂×C₃×C₁₃	—	5	1	no	—
85	C₂×C₃×C₁₃	—	5	1	no	—
86	C₂×C₃×C₁₃	—	15	4	no	—
87	C₃×D₁₃	—	13	1	no	—
88	C₃×D₁₃	—	13	1	no	—
89	C₃×D₁₃	—	39	4	no	—
90	C₃×D₁₃	—	65	1	no	—
91	C₃×D₁₃	—	65	1	no	—
92	C₃×D₁₃	—	195	4	no	—
93	C₅×C₁₃	—	3	4	no	—
94	C₅×C₁₃	C₂×D₁₃⋊C₉	1	1	yes	characteristic, complemented
95	C₅×C₁₃	C₂×C₉×D₁₃	1	1	yes	characteristic, complemented
96	C₂×C₃×D₅	—	169	1	no	—
97	C₂×D₁₃	—	65	1	no	—
98	C₂×D₁₃	—	65	1	no	—
99	C₂×D₁₃	—	195	4	no	—
100	C₅×C₉	—	13	1	no	Hall
101	C₃×C₁₃	—	3	4	no	—
102	C₃×C₁₃	D₅×D₁₃⋊C₃	1	1	yes	characteristic
103	C₃×C₁₃	C₃×D₅×D₁₃	1	1	yes	characteristic
104	C₂²×C₉	—	845	1	no	Hall
105	C₂×C₃×C₅	—	169	1	no	—
106	C₃×D₅	—	169	1	no	—
107	C₃×D₅	(C₁₃⋊C₃)⋊D₁₃	1	1	yes	characteristic
108	C₂×C₁₃	—	5	1	no	—
109	C₂×C₁₃	—	5	1	no	—
110	C₂×C₁₃	—	15	4	no	—
111	D₁₃	—	13	1	no	—
112	D₁₃	—	13	1	no	—
113	D₁₃	—	39	4	no	—
114	D₁₃	—	65	1	no	—
115	D₁₃	—	65	1	no	—
116	D₁₃	—	195	4	no	—
117	C₂×D₅	—	169	1	no	Hall
118	C₂×C₉	—	65	1	no	—
119	C₂×C₉	—	845	1	no	—
120	C₂×C₉	—	169	1	no	—
121	C₃×C₅	C₂×(C₁₃⋊C₃)⋊D₁₃	1	1	yes	characteristic
122	C₁₃	—	3	4	no	—
123	C₁₃	D₅×D₁₃⋊C₉	1	1	yes	characteristic, complemented
124	C₁₃	C₉×D₅×D₁₃	1	1	yes	characteristic, complemented
125	C₂²×C₃	—	845	1	no	—
126	C₂×C₅	—	169	1	no	—
127	D₅	—	169	1	no	—
128	D₅	(C₁₃⋊C₉)⋊D₁₃	1	1	yes	characteristic, direct product factor
129	C₉	—	13	1	no	Sylow
130	C₂×C₃	—	5	1	no	—
131	C₂×C₃	—	845	1	no	—
132	C₂×C₃	—	169	1	no	—
133	C₅	C₂×(C₁₃⋊C₉)⋊D₁₃	1	1	yes	characteristic, complemented, Sylow
134	C₂²	—	845	1	no	Sylow
135	C₃	D₅×(C₁₃⋊C₃)⋊D₁₃	1	1	yes	characteristic
136	C₂	—	5	1	no	—
137	C₂	—	845	1	no	—
138	C₂	—	169	1	no	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅⋊((C₁₃⋊C₉)⋊D₁₃)

C₂

yes

maximal, characteristic, complemented

D₅×C₁₃×C₁₃⋊C₉

C₂

yes

maximal, characteristic, complemented

C₅×(C₁₃⋊C₉)⋊D₁₃

C₂

yes

maximal, characteristic, complemented

C₃×D₅×C₁₃⋊D₁₃

C₃

yes

maximal, characteristic

C₅×C₁₃×C₁₃⋊C₉

C₂²

yes

characteristic, complemented, Hall

C₂×(C₁₃⋊C₉)⋊D₁₃

—

maximal, Hall

C₃×C₅⋊(C₁₃⋊D₁₃)

C₂×C₃

yes

characteristic

C₃×D₅×C₁₃²

C₂×C₃

yes

characteristic

C₃×C₅×C₁₃⋊D₁₃

C₂×C₃

yes

characteristic

D₅×C₁₃⋊D₁₃

C₉

yes

characteristic, complemented, Hall

C₂×C₁₃×C₁₃⋊C₉

—

(C₁₃⋊C₉)⋊D₁₃

—

(C₁₃⋊C₉)⋊D₁₃

D₅

yes

characteristic, direct product factor

C₃×C₅×C₁₃²

C₂²×C₃

yes

characteristic

C₉×D₅×D₁₃

—

maximal

D₅×D₁₃⋊C₉

—

maximal

C₂×C₃×C₁₃⋊D₁₃

—

C₅⋊(C₁₃⋊D₁₃)

C₂×C₉

yes

characteristic, complemented

D₅×C₁₃²

C₂×C₉

yes

characteristic, complemented

C₅×C₁₃⋊D₁₃

C₂×C₉

yes

characteristic, complemented

C₁₃×C₁₃⋊C₉

C₂×D₅

yes

characteristic, complemented, Hall

C₉×D₆₅

—

C₉×D₅×C₁₃

—

C₅×C₉×D₁₃

—

C₅⋊(D₁₃⋊C₉)

—

D₅×C₁₃⋊C₉

D₁₃

yes

characteristic, complemented

C₅×D₁₃⋊C₉

—

C₂×C₃×C₁₃²

—

C₃×C₁₃⋊D₁₃

—

C₃×C₁₃⋊D₁₃

C₃×D₅

yes

characteristic

C₅×C₁₃²

C₂²×C₉

yes

characteristic, complemented, Hall

C₃×D₅×D₁₃

—

C₃×D₅×D₁₃

—

C₃×D₅×D₁₃

—

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

—

C₂×D₁₃⋊C₉

—

C₃×D₆₅

—

C₃×D₆₅

—

C₃×D₆₅

—

C₃×D₅×C₁₃

—

C₃×D₅×C₁₃

D₁₃⋊C₃

yes

characteristic

C₃×D₅×C₁₃

C₃×D₁₃

yes

characteristic

C₃×C₅×D₁₃

—

C₃×C₅×D₁₃

—

C₃×C₅×D₁₃

—

C₂×C₁₃²

—

C₁₃⋊D₁₃

—

C₁₃⋊D₁₃

C₉×D₅

yes

characteristic, complemented

D₅×D₁₃

—

D₅×D₁₃

—

D₅×D₁₃

—

C₂×C₉×C₁₃

—

C₉×D₁₃

—

C₉×D₁₃

—

C₂×C₁₃⋊C₉

—

D₁₃⋊C₉

—

D₁₃⋊C₉

—

C₃×C₅×C₁₃

—

C₃×C₅×C₁₃

C₂×D₁₃⋊C₃

yes

characteristic

C₃×C₅×C₁₃

C₂×C₃×D₁₃

yes

characteristic

C₂×C₉×D₅

—

169

Hall

C₁₃²

C₂×C₉×D₅

yes

characteristic, complemented, Sylow

C₂×C₃×D₁₃

—

C₂×C₃×D₁₃

—

C₂×C₃×D₁₃

—

195

—

D₆₅

—

D₆₅

—

D₆₅

—

C₅×D₁₃

—

C₅×D₁₃

—

C₅×D₁₃

—

D₅×C₁₃

—

D₅×C₁₃

D₁₃⋊C₉

yes

characteristic, complemented

D₅×C₁₃

C₉×D₁₃

yes

characteristic, complemented

C₉×C₁₃

—

C₁₃⋊C₉

D₅×D₁₃

yes

characteristic, complemented

C₂×C₅×C₉

—

169

—

C₉×D₅

—

C₉×D₅

—

169

—

C₂×C₃×C₁₃

—

C₂×C₃×C₁₃

—

C₂×C₃×C₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

C₃×D₁₃

—

195

—

C₅×C₁₃

—

C₅×C₁₃

C₂×D₁₃⋊C₉

yes

characteristic, complemented

C₅×C₁₃

C₂×C₉×D₁₃

yes

characteristic, complemented

C₂×C₃×D₅

—

169

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

195

—

100

C₅×C₉

—

Hall

101

C₃×C₁₃

—

102

C₃×C₁₃

D₅×D₁₃⋊C₃

yes

characteristic

103

C₃×C₁₃

C₃×D₅×D₁₃

yes

characteristic

104

C₂²×C₉

—

845

Hall

105

C₂×C₃×C₅

—

169

—

106

C₃×D₅

—

169

—

107

C₃×D₅

(C₁₃⋊C₃)⋊D₁₃

yes

characteristic

108

C₂×C₁₃

—

109

C₂×C₁₃

—

110

C₂×C₁₃

—

111

D₁₃

—

112

D₁₃

—

113

D₁₃

—

114

D₁₃

—

115

D₁₃

—

116

D₁₃

—

195

—

117

C₂×D₅

—

169

Hall

118

C₂×C₉

—

119

C₂×C₉

—

845

—

120

C₂×C₉

—

169

—

121

C₃×C₅

C₂×(C₁₃⋊C₃)⋊D₁₃

yes

characteristic

122

C₁₃

—

123

C₁₃

D₅×D₁₃⋊C₉

yes

characteristic, complemented

124

C₁₃

C₉×D₅×D₁₃

yes

characteristic, complemented

125

C₂²×C₃

—

845

—

126

C₂×C₅

—

169

—

127

D₅

—

169

—

128

D₅

(C₁₃⋊C₉)⋊D₁₃

yes

characteristic, direct product factor

129

C₉

—

Sylow

130

C₂×C₃

—

131

C₂×C₃

—

845

—

132

C₂×C₃

—

169

—

133

C₅

C₂×(C₁₃⋊C₉)⋊D₁₃

yes

characteristic, complemented, Sylow

134

C₂²

—

845

Sylow

135

C₃

D₅×(C₁₃⋊C₃)⋊D₁₃

yes

characteristic

136

C₂

—

137

C₂

—

845

—

138

C₂

—

169

—

	subgroup	G/H
center, Frattini, hypercenter	C₃	D₅×(C₁₃⋊C₃)⋊D₁₃
commutator, hypocenter	C₅×C₁₃²	C₂²×C₉
Fitting, socle, cosocle	C₃×C₅×C₁₃²	C₂²×C₃
prime radical	C₃×D₅×C₁₃⋊D₁₃	C₃
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₃

subgroup

G/H

center, Frattini, hypercenter

C₃

D₅×(C₁₃⋊C₃)⋊D₁₃

commutator, hypocenter

C₅×C₁₃²

C₂²×C₉

Fitting, socle, cosocle

C₃×C₅×C₁₃²

C₂²×C₃

prime radical

C₃×D₅×C₁₃⋊D₁₃

C₃

LCS

G ▷ C₅×C₁₃²

UCS

C₁ ◁ C₃

N	H ≅ G/N	type
1. C₅⋊((C₁₃⋊C₉)⋊D₁₃)	C₂	semidirect
2. D₅×C₁₃×C₁₃⋊C₉	C₂	semidirect
3. C₅×(C₁₃⋊C₉)⋊D₁₃	C₂	semidirect
5. C₅×C₁₃×C₁₃⋊C₉	C₂²	semidirect
10. D₅×C₁₃⋊D₁₃	C₉	semidirect
13. (C₁₃⋊C₉)⋊D₁₃	D₅	direct (N × H)
18. C₅⋊(C₁₃⋊D₁₃)	C₂×C₉	semidirect
19. D₅×C₁₃²	C₂×C₉	semidirect
20. C₅×C₁₃⋊D₁₃	C₂×C₉	semidirect
21. C₁₃×C₁₃⋊C₉	C₂×D₅	semidirect
26. D₅×C₁₃⋊C₉	D₁₃	semidirect
31. C₅×C₁₃²	C₂²×C₉	semidirect
37. C₅×C₁₃⋊C₉	C₂×D₁₃	semidirect
52. C₁₃⋊D₁₃	C₉×D₅	semidirect
66. C₁₃²	C₂×C₉×D₅	semidirect
77. D₅×C₁₃	D₁₃⋊C₉	semidirect
78. D₅×C₁₃	C₉×D₁₃	semidirect
80. C₁₃⋊C₉	D₅×D₁₃	semidirect
94. C₅×C₁₃	C₂×D₁₃⋊C₉	semidirect
95. C₅×C₁₃	C₂×C₉×D₁₃	semidirect
123. C₁₃	D₅×D₁₃⋊C₉	semidirect
124. C₁₃	C₉×D₅×D₁₃	semidirect
128. D₅	(C₁₃⋊C₉)⋊D₁₃	direct (N × H)
133. C₅	C₂×(C₁₃⋊C₉)⋊D₁₃	semidirect

H ≅ G/N

type

1. C₅⋊((C₁₃⋊C₉)⋊D₁₃)

C₂

semidirect

2. D₅×C₁₃×C₁₃⋊C₉

C₂

semidirect

3. C₅×(C₁₃⋊C₉)⋊D₁₃

C₂

semidirect

5. C₅×C₁₃×C₁₃⋊C₉

C₂²

semidirect

10. D₅×C₁₃⋊D₁₃

C₉

semidirect

13. (C₁₃⋊C₉)⋊D₁₃

D₅

direct (N × H)

18. C₅⋊(C₁₃⋊D₁₃)

C₂×C₉

semidirect

19. D₅×C₁₃²

C₂×C₉

semidirect

20. C₅×C₁₃⋊D₁₃

C₂×C₉

semidirect

21. C₁₃×C₁₃⋊C₉

C₂×D₅

semidirect

26. D₅×C₁₃⋊C₉

D₁₃

semidirect

31. C₅×C₁₃²

C₂²×C₉

semidirect

37. C₅×C₁₃⋊C₉

C₂×D₁₃

semidirect

52. C₁₃⋊D₁₃

C₉×D₅

semidirect

66. C₁₃²

C₂×C₉×D₅

semidirect

77. D₅×C₁₃

D₁₃⋊C₉

semidirect

78. D₅×C₁₃

C₉×D₁₃

semidirect

80. C₁₃⋊C₉

D₅×D₁₃

semidirect

94. C₅×C₁₃

C₂×D₁₃⋊C₉

semidirect

95. C₅×C₁₃

C₂×C₉×D₁₃

semidirect

123. C₁₃

D₅×D₁₃⋊C₉

semidirect

124. C₁₃

C₉×D₅×D₁₃

semidirect

128. D₅

(C₁₃⋊C₉)⋊D₁₃

direct (N × H)

133. C₅

C₂×(C₁₃⋊C₉)⋊D₁₃

semidirect

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

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

subgroup

index

C₅⋊((C₁₃⋊C₉)⋊D₁₃)

D₅×C₁₃×C₁₃⋊C₉

C₅×(C₁₃⋊C₉)⋊D₁₃

C₃×D₅×C₁₃⋊D₁₃

C₂×(C₁₃⋊C₉)⋊D₁₃

C₉×D₅×D₁₃

D₅×D₁₃⋊C₉

Subgroups — D₅×(C₁₃⋊C₉)⋊D₁₃ #44477694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5×(C13⋊C9)⋊D13 #44477694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×(C₁₃⋊C₉)⋊D₁₃ #44477694