Group #11931509 — subgroups

Subgroups — C₅×#8789 #11931509

Order 1280.

342 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 166 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×#1842	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅×#1277	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊C₈	—	2	2	no	—
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
9	C₄³×C₅	C₂²	1	1	yes	characteristic
10	C₅×#336	—	2	2	no	—
11	C₅×#336	C₄	1	2	yes	—
12	C₅×#336	C₄	1	2	yes	—
13	#8789	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂²×C₅×C₈	—	2	2	no	—
15	C₂²×C₅×C₈	—	2	2	no	—
16	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	2	yes	—
20	C₂×C₄²×C₅	Q₈	1	2	yes	—
21	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
22	C₅×C₄⋊C₈	—	2	4	no	—
23	C₅×C₄⋊C₈	—	2	4	no	—
24	C₅×C₄⋊C₈	—	4	4	no	—
25	C₄×C₅×C₈	—	2	2	no	—
26	C₄×C₅×C₈	—	2	2	no	—
27	#1842	C₂×C₅	1	1	yes	characteristic
28	#1831	C₂×C₅	1	1	yes	characteristic
29	#1744	C₂×C₅	1	1	yes	characteristic
30	#1277	C₂×C₅	1	4	yes	—
31	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₅×C₈	—	2	4	no	—
37	C₂×C₅×C₈	—	2	4	no	—
38	C₂×C₅×C₈	—	2	4	no	—
39	C₂×C₅×C₈	—	4	2	no	—
40	C₂×C₅×C₈	—	4	2	no	—
41	C₂×C₅×C₈	—	4	2	no	—
42	C₂×C₅×C₈	—	4	4	no	—
43	C₂×C₅×C₈	—	4	2	no	—
44	C₄²×C₅	—	2	4	no	—
45	C₄²×C₅	—	2	2	no	—
46	C₄²×C₅	—	2	2	no	—
47	C₄²×C₅	—	2	2	no	—
48	C₄²×C₅	—	2	2	no	—
49	C₄²×C₅	#41	1	2	yes	—
50	C₄²×C₅	#41	1	2	yes	—
51	C₂×C₄⋊C₈	—	2	2	no	—
52	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
53	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
54	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
55	C₄³	C₂²×C₅	1	1	yes	characteristic
56	#336	—	2	2	no	—
57	#336	C₄×C₅	1	2	yes	—
58	#336	C₄×C₅	1	2	yes	—
59	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	#108	1	4	yes	—
67	C₂×C₄×C₅	#117	1	2	yes	—
68	C₂×C₄×C₅	#124	1	1	yes	characteristic
69	C₂×C₄×C₅	#124	1	1	yes	characteristic
70	C₂×C₄×C₅	#125	1	2	yes	—
71	C₂×C₄×C₅	#131	1	2	yes	—
72	C₂×C₄×C₅	#131	1	2	yes	—
73	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₅	#137	1	2	yes	—
75	C₅×C₈	—	4	4	no	—
76	C₅×C₈	—	8	2	no	—
77	C₅×C₈	—	8	2	no	—
78	C₂²×C₈	—	2	2	no	—
79	C₂²×C₈	—	2	2	no	—
80	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
81	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
82	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
83	C₂×C₄²	C₅×D₄	1	2	yes	—
84	C₂×C₄²	C₅×Q₈	1	2	yes	—
85	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
86	C₄⋊C₈	—	2	4	no	—
87	C₄⋊C₈	—	2	4	no	—
88	C₄⋊C₈	—	4	4	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₂²×C₅	#382	1	1	yes	characteristic
92	C₂²×C₅	#424	1	1	yes	characteristic
93	C₂²×C₅	#424	1	1	yes	characteristic
94	C₂²×C₅	#428	1	2	yes	—
95	C₂²×C₅	C₂×#108	1	2	yes	—
96	C₄×C₅	—	2	4	no	—
97	C₄×C₅	—	2	2	no	—
98	C₄×C₅	—	4	2	no	—
99	C₄×C₅	#433	1	4	yes	—
100	C₄×C₅	#489	1	2	yes	—
101	C₄×C₅	#490	1	1	yes	characteristic
102	C₄×C₅	#491	1	1	yes	characteristic
103	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
104	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
105	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	—	2	4	no	—
121	C₄²	C₅×#41	1	2	yes	—
122	C₄²	C₅×#41	1	2	yes	—
123	C₂×C₅	#1573	1	2	yes	—
124	C₂×C₅	#1574	1	2	yes	—
125	C₂×C₅	#1832	1	1	yes	characteristic
126	C₂×C₅	#1836	1	1	yes	characteristic
127	C₂×C₅	#1837	1	1	yes	characteristic
128	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₅×#108	1	4	yes	—
136	C₂×C₄	C₅×#117	1	2	yes	—
137	C₂×C₄	C₅×#124	1	1	yes	characteristic
138	C₂×C₄	C₅×#124	1	1	yes	characteristic
139	C₂×C₄	C₅×#125	1	2	yes	—
140	C₂×C₄	C₅×#131	1	2	yes	—
141	C₂×C₄	C₅×#131	1	2	yes	—
142	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄	C₅×#137	1	2	yes	—
144	C₈	—	4	4	no	—
145	C₈	—	8	2	no	—
146	C₈	—	8	2	no	—
147	C₅	#8789	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²	C₅×#382	1	1	yes	characteristic
149	C₂²	C₅×#424	1	1	yes	characteristic
150	C₂²	C₅×#424	1	1	yes	characteristic
151	C₂²	C₅×#428	1	2	yes	—
152	C₂²	C₂×C₅×#108	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	C₅×#433	1	4	yes	—
157	C₄	C₅×#489	1	2	yes	—
158	C₄	C₅×#490	1	1	yes	characteristic
159	C₄	C₅×#491	1	1	yes	characteristic
160	C₂	C₅×#1573	1	2	yes	—
161	C₂	C₅×#1574	1	2	yes	—
162	C₂	C₅×#1832	1	1	yes	characteristic
163	C₂	C₅×#1836	1	1	yes	characteristic
164	C₂	C₅×#1837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1842

C₂

yes

maximal, characteristic

C₅×#1831

C₂

yes

maximal, characteristic

C₅×#1744

C₂

yes

maximal, characteristic

C₅×#1277

C₂

yes

maximal

C₂×C₅×C₄⋊C₈

—

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

—

C₄³×C₅

C₂²

yes

characteristic

C₅×#336

—

C₅×#336

C₄

yes

—

C₅×#336

C₄

yes

—

#8789

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×C₈

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

C₂×C₄

yes

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

Q₈

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

—

#1842

C₂×C₅

yes

characteristic

#1831

C₂×C₅

yes

characteristic

#1744

C₂×C₅

yes

characteristic

#1277

C₂×C₅

yes

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₄⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×Q₈

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

#41

yes

—

C₄²×C₅

#41

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₅

yes

—

C₂×C₄⋊C₈

C₂²×C₅

yes

—

C₂×C₄×C₈

C₂²×C₅

yes

—

C₄³

C₂²×C₅

yes

characteristic

#336

—

#336

C₄×C₅

yes

—

#336

C₄×C₅

yes

—

C₂³×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#108

yes

—

C₂×C₄×C₅

#117

yes

—

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₅

yes

—

C₂×C₄²

C₂×C₄×C₅

yes

characteristic

C₂×C₄²

C₂×C₄×C₅

yes

characteristic

C₂×C₄²

C₅×D₄

yes

—

C₂×C₄²

C₅×Q₈

yes

—

C₂×C₄²

C₂³×C₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

C₂²×C₅

C₂×#108

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#433

yes

—

100

C₄×C₅

#489

yes

—

101

C₄×C₅

#490

yes

characteristic

102

C₄×C₅

#491

yes

characteristic

103

C₂²×C₄

C₅×C₄⋊C₄

yes

—

104

C₂²×C₄

C₅×C₄⋊C₄

yes

—

105

C₂²×C₄

C₂²×C₄×C₅

yes

characteristic

106

C₂²×C₄

C₂×C₅×D₄

yes

characteristic

107

C₂²×C₄

C₂×C₅×Q₈

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₅×#41

yes

—

122

C₄²

C₅×#41

yes

—

123

C₂×C₅

#1573

yes

—

124

C₂×C₅

#1574

yes

—

125

C₂×C₅

#1832

yes

characteristic

126

C₂×C₅

#1836

yes

characteristic

127

C₂×C₅

#1837

yes

characteristic

128

C₂³

C₂×C₅×C₄⋊C₄

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₅×#108

yes

—

136

C₂×C₄

C₅×#117

yes

—

137

C₂×C₄

C₅×#124

yes

characteristic

138

C₂×C₄

C₅×#124

yes

characteristic

139

C₂×C₄

C₅×#125

yes

—

140

C₂×C₄

C₅×#131

yes

—

141

C₂×C₄

C₅×#131

yes

—

142

C₂×C₄

C₅×AGL(1,ℤ/8)

yes

—

143

C₂×C₄

C₅×#137

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₅

#8789

yes

characteristic, direct product factor, Sylow

148

C₂²

C₅×#382

yes

characteristic

149

C₂²

C₅×#424

yes

characteristic

150

C₂²

C₅×#424

yes

characteristic

151

C₂²

C₅×#428

yes

—

152

C₂²

C₂×C₅×#108

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₅×#433

yes

—

157

C₄

C₅×#489

yes

—

158

C₄

C₅×#490

yes

characteristic

159

C₄

C₅×#491

yes

characteristic

160

C₂

C₅×#1573

yes

—

161

C₂

C₅×#1574

yes

—

162

C₂

C₅×#1832

yes

characteristic

163

C₂

C₅×#1836

yes

characteristic

164

C₂

C₅×#1837

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

subgroup

G/H

center

C₂²×C₄×C₅

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₅

Frattini, cosocle

C₂×C₄²

C₂³×C₅

socle, prime radical

C₂³×C₅

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

N	H ≅ G/N	type
13. #8789	C₅	direct (N × H)
147. C₅	#8789	direct (N × H)

H ≅ G/N

type

13. #8789

C₅

direct (N × H)

147. C₅

#8789

direct (N × H)

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

subgroup	index
C₅×#1842	2
C₅×#1831	2
C₅×#1744	2
C₅×#1277	2
#8789	5

subgroup

index

Subgroups — C₅×#8789 #11931509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#8789 #11931509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8789 #11931509