Group #11989712 — subgroups

Subgroups — D₅×#1309 #11989712

Order 1280.

658 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 179 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	D₅×#423	C₂	1	3	yes	maximal
2	C₅×#1309	C₂	1	1	yes	maximal, characteristic, complemented
3	#10598203	C₂	1	3	yes	maximal, complemented
4	C₂×D₅×M₁₆	C₄	1	6	yes	—
5	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
6	C₅×#423	C₂²	1	3	yes	—
7	#65407	C₄	1	6	yes	—
8	#65213	C₂²	1	3	yes	—
9	#65126	—	2	6	no	—
10	D₅×C₄⋊C₈	—	2	6	no	—
11	C₂×#1309	—	5	1	no	maximal, Sylow
12	C₂²×C₄×D₅	C₂×C₄	1	3	yes	—
13	C₂×C₅×M₁₆	C₂×C₄	1	6	yes	—
14	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
15	C₂×C₄×Dic₅	C₂×C₄	1	3	yes	—
16	C₂×#681	C₂×C₄	1	6	yes	—
17	D₅×M₁₆	—	4	12	no	—
18	C₂×#676	—	2	6	no	—
19	C₂×C₈×D₅	—	2	6	no	—
20	C₄²×D₅	D₄	1	3	yes	—
21	C₄²×D₅	D₄	1	3	yes	—
22	C₄²×D₅	Q₈	1	1	yes	characteristic
23	C₄²×D₅	Q₈	1	1	yes	characteristic
24	C₅×C₄⋊C₈	—	2	6	no	—
25	#3946	—	2	12	no	—
26	C₄⋊#168	—	2	6	no	—
27	C₂×#423	—	5	3	no	—
28	#1309	—	5	3	no	—
29	#1309	D₅	1	1	yes	characteristic, direct product factor
30	C₂³×D₅	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
32	C₂²×Dic₅	C₄²	1	3	yes	—
33	C₂×C₄×D₅	—	2	6	no	—
34	C₂×C₄×D₅	—	2	6	no	—
35	C₂×C₄×D₅	C₂²⋊C₄	1	3	yes	—
36	C₂×C₄×D₅	C₂²⋊C₄	1	3	yes	—
37	C₂×C₄×D₅	C₄⋊C₄	1	3	yes	—
38	C₂×C₄×D₅	C₄⋊C₄	1	3	yes	—
39	C₅×M₁₆	—	4	6	no	—
40	C₂×C₅×C₈	—	2	6	no	—
41	C₄²×C₅	C₂×D₄	1	3	yes	—
42	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
43	C₄×Dic₅	C₂²⋊C₄	1	6	yes	—
44	C₄×Dic₅	C₄⋊C₄	1	3	yes	—
45	C₄×Dic₅	C₄⋊C₄	1	3	yes	—
46	#681	—	4	6	no	—
47	C₂×#168	—	2	6	no	—
48	#676	—	4	12	no	—
49	C₈×D₅	—	4	12	no	—
50	C₂²×M₁₆	—	5	6	no	—
51	C₂²×C₄²	—	5	1	no	—
52	#423	—	5	6	no	—
53	#423	—	5	3	no	—
54	#423	—	10	6	no	—
55	#423	C₂×D₅	1	3	yes	—
56	C₂×C₄⋊C₈	—	10	6	no	—
57	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
58	C₂²×D₅	—	2	6	no	—
59	C₂²×D₅	#95	1	1	yes	characteristic
60	C₂²×D₅	#95	1	1	yes	characteristic
61	C₂×C₄×C₅	—	2	3	no	—
62	C₂×C₄×C₅	—	2	3	no	—
63	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
64	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×Dic₅	—	2	6	no	—
66	C₂×Dic₅	#95	1	3	yes	—
67	C₂×Dic₅	#95	1	3	yes	—
68	C₄×D₅	—	2	6	no	—
69	C₄×D₅	—	2	3	no	—
70	C₄×D₅	—	2	3	no	—
71	C₄×D₅	—	2	6	no	—
72	C₄×D₅	#100	1	6	yes	—
73	C₄×D₅	#101	1	6	yes	—
74	C₅×C₈	—	4	6	no	—
75	#168	—	4	6	no	—
76	C₂³×C₄	—	5	3	no	—
77	C₂×M₁₆	—	5	6	no	—
78	C₂×M₁₆	—	10	6	no	—
79	C₂×M₁₆	—	20	12	no	—
80	C₂×M₁₆	C₄×D₅	1	6	yes	—
81	C₂²×C₈	—	10	6	no	—
82	C₂×C₄²	—	5	3	no	—
83	C₂×C₄²	—	5	3	no	—
84	C₂×C₄²	—	5	3	no	—
85	C₂×C₄²	—	5	1	no	—
86	C₂×C₄²	—	5	1	no	—
87	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
88	C₄⋊C₈	—	2	6	no	—
89	C₄⋊C₈	—	10	12	no	—
90	C₄⋊C₈	—	10	6	no	—
91	C₂²×C₅	—	2	3	no	—
92	C₂²×C₅	C₂×#95	1	1	yes	characteristic
93	C₂×D₅	—	2	3	no	—
94	C₂×D₅	—	4	4	no	—
95	C₂×D₅	#343	1	6	yes	—
96	C₄×C₅	—	2	3	no	—
97	C₄×C₅	C₂×#100	1	3	yes	—
98	C₄×C₅	C₂×#101	1	3	yes	—
99	Dic₅	—	2	3	no	—
100	Dic₅	#343	1	6	yes	—
101	C₂⁴	—	5	1	no	—
102	C₂²×C₄	—	5	3	no	—
103	C₂²×C₄	—	5	3	no	—
104	C₂²×C₄	—	5	3	no	—
105	C₂²×C₄	—	5	3	no	—
106	C₂²×C₄	—	5	3	no	—
107	C₂²×C₄	—	10	6	no	—
108	C₂²×C₄	—	10	6	no	—
109	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
110	M₁₆	—	4	6	no	—
111	M₁₆	—	20	12	no	—
112	M₁₆	—	20	6	no	—
113	C₂×C₈	—	2	6	no	—
114	C₂×C₈	—	10	6	no	—
115	C₂×C₈	—	20	12	no	—
116	C₄²	—	5	6	no	—
117	C₄²	—	5	3	no	—
118	C₄²	—	5	3	no	—
119	C₄²	D₄×D₅	1	3	yes	—
120	C₄²	Q₈×D₅	1	1	yes	characteristic
121	C₂×C₅	—	4	1	no	—
122	C₂×C₅	C₂×#343	1	3	yes	—
123	D₅	—	4	1	no	—
124	D₅	#1309	1	4	yes	direct product factor
125	C₂³	—	5	1	no	—
126	C₂³	—	5	1	no	—
127	C₂³	—	10	6	no	—
128	C₂³	C₄²×D₅	1	1	yes	characteristic
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	5	3	no	—
132	C₂×C₄	—	5	6	no	—
133	C₂×C₄	—	5	6	no	—
134	C₂×C₄	—	5	3	no	—
135	C₂×C₄	—	10	6	no	—
136	C₂×C₄	—	10	6	no	—
137	C₂×C₄	—	10	6	no	—
138	C₂×C₄	—	10	3	no	—
139	C₂×C₄	—	10	3	no	—
140	C₂×C₄	D₅×C₂²⋊C₄	1	3	yes	—
141	C₂×C₄	D₅×C₄⋊C₄	1	3	yes	—
142	C₈	—	4	6	no	—
143	C₈	—	20	6	no	—
144	C₅	C₂×#1309	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	2	3	no	—
146	C₂²	—	5	6	no	—
147	C₂²	—	10	3	no	—
148	C₂²	—	20	4	no	—
149	C₂²	D₅×#95	1	1	yes	characteristic
150	C₄	—	2	3	no	—
151	C₄	—	5	6	no	—
152	C₄	—	10	3	no	—
153	C₄	D₅×#100	1	3	yes	—
154	C₄	D₅×#101	1	3	yes	—
155	C₂	—	4	1	no	—
156	C₂	—	5	4	no	—
157	C₂	—	20	1	no	—
158	C₂	D₅×#343	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₅×#423

C₂

yes

maximal

C₅×#1309

C₂

yes

maximal, characteristic, complemented

#10598203

C₂

yes

maximal, complemented

C₂×D₅×M₁₆

C₄

yes

—

C₂×C₄²×D₅

C₂²

yes

characteristic

C₅×#423

C₂²

yes

—

#65407

C₄

yes

—

#65213

C₂²

yes

—

#65126

—

D₅×C₄⋊C₈

—

C₂×#1309

—

maximal, Sylow

C₂²×C₄×D₅

C₂×C₄

yes

—

C₂×C₅×M₁₆

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄×Dic₅

C₂×C₄

yes

—

C₂×#681

C₂×C₄

yes

—

D₅×M₁₆

—

C₂×#676

—

C₂×C₈×D₅

—

C₄²×D₅

D₄

yes

—

C₄²×D₅

D₄

yes

—

C₄²×D₅

Q₈

yes

characteristic

C₄²×D₅

Q₈

yes

characteristic

C₅×C₄⋊C₈

—

—

—

—

—

D₅

yes

characteristic, direct product factor

C₂³×D₅

C₄²

yes

characteristic

C₂²×C₄×C₅

C₂²×C₄

yes

—

C₂²×Dic₅

C₄²

yes

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂²⋊C₄

yes

—

C₂×C₄×D₅

C₂²⋊C₄

yes

—

C₂×C₄×D₅

C₄⋊C₄

yes

—

C₂×C₄×D₅

C₄⋊C₄

yes

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×Q₈

yes

characteristic

C₄×Dic₅

C₂²⋊C₄

yes

—

C₄×Dic₅

C₄⋊C₄

yes

—

C₄×Dic₅

C₄⋊C₄

yes

—

#681

—

C₂×#168

—

#676

—

C₈×D₅

—

C₂²×M₁₆

—

C₂²×C₄²

—

—

—

—

C₂×D₅

yes

—

C₂×C₄⋊C₈

—

C₂³×C₅

C₂×C₄²

yes

characteristic

C₂²×D₅

—

C₂²×D₅

yes

C₂²×D₅

yes

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×Dic₅

—

C₂×Dic₅

#95

yes

—

C₂×Dic₅

#95

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

#100

yes

—

C₄×D₅

#101

yes

—

C₅×C₈

—

#168

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₅

yes

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₅

—

C₂²×C₅

C₂×#95

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

#343

yes

—

C₄×C₅

—

C₄×C₅

C₂×#100

yes

—

C₄×C₅

C₂×#101

yes

—

Dic₅

—

100

Dic₅

#343

yes

—

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₄×D₅

yes

—

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

D₄×D₅

yes

—

120

C₄²

Q₈×D₅

yes

characteristic

121

C₂×C₅

—

122

C₂×C₅

C₂×#343

yes

—

123

D₅

—

124

D₅

#1309

yes

direct product factor

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₄²×D₅

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

D₅×C₂²⋊C₄

yes

—

141

C₂×C₄

D₅×C₄⋊C₄

yes

—

142

C₈

—

143

C₈

—

144

C₅

C₂×#1309

yes

characteristic, complemented, Sylow

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

D₅×#95

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

D₅×#100

yes

—

154

C₄

D₅×#101

yes

—

155

C₂

—

156

C₂

—

157

C₂

—

158

C₂

D₅×#343

yes

—

	subgroup	G/H
center	C₂²	D₅×#95
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1309	C₂
socle	C₂²×C₅	C₂×#95
prime radical	C₂³×D₅	C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1309	D₅
hypocenter	C₅	C₂×#1309
LCS	G ▷ C₂³×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

D₅×#95

commutator

C₂³×C₅

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#1309

C₂

socle

C₂²×C₅

C₂×#95

prime radical

C₂³×D₅

C₄²

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂³×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
2. C₅×#1309	C₂	semidirect
3. #10598203	C₂	semidirect
29. #1309	D₅	direct (N × H)
124. D₅	#1309	direct (N × H)
144. C₅	C₂×#1309	semidirect

H ≅ G/N

type

2. C₅×#1309

C₂

semidirect

3. #10598203

C₂

semidirect

29. #1309

D₅

direct (N × H)

124. D₅

#1309

direct (N × H)

144. C₅

C₂×#1309

semidirect

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

subgroup	index
D₅×#423	2
C₅×#1309	2
#10598203	2
C₂×#1309	5

subgroup

index

Subgroups — D₅×#1309 #11989712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D5×#1309 #11989712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#1309 #11989712