Group #11998784 — subgroups

Subgroups — #11998784

Order 1280.

390 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₂×#64969	C₂	1	1	yes	maximal, characteristic
2	C₅×#1837	C₂	1	1	yes	maximal, characteristic
3	#10601301	C₂	1	2	yes	maximal
4	#10601300	C₂	1	2	yes	maximal
5	#10600519	C₂	1	1	yes	maximal, characteristic
6	C₂×#4015	C₂²	1	2	yes	—
7	C₅×#423	C₂²	1	1	yes	characteristic
8	C₅×#423	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#3949	—	2	2	no	—
11	#65214	C₂²	1	2	yes	—
12	#64969	—	2	2	no	—
13	#64969	C₄	1	4	yes	—
14	C₅×#341	—	2	2	no	—
15	#64706	—	2	4	no	—
16	#64701	—	2	4	no	—
17	#64674	—	2	2	no	—
18	#9776	—	5	1	no	maximal, Sylow
19	C₂²×Dic₁₀	—	2	2	no	—
20	C₂×C₅×M₁₆	—	2	1	no	—
21	C₂×C₅×M₁₆	—	2	1	no	—
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	#4015	—	2	2	no	—
27	#4015	—	2	2	no	—
28	#4015	—	2	2	no	—
29	#4015	C₂×C₄	1	4	yes	—
30	C₅×C₄⋊C₈	—	2	2	no	—
31	C₅×C₄⋊C₈	—	2	2	no	—
32	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₄×C₅×C₈	D₄	1	2	yes	—
34	C₂²⋊#168	—	2	2	no	—
35	#3949	—	2	8	no	—
36	C₄⋊#168	D₄	1	4	yes	—
37	C₄×#168	—	2	2	no	—
38	C₂×#498	—	5	1	no	—
39	#1837	D₅	1	1	yes	characteristic
40	#1549	—	5	2	no	—
41	#1548	—	5	2	no	—
42	#1452	—	5	1	no	—
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	#41	1	2	yes	—
45	C₂²×Dic₅	—	4	2	no	—
46	C₂×Dic₁₀	—	2	4	no	—
47	C₂×Dic₁₀	—	4	2	no	—
48	C₂×Dic₁₀	—	8	4	no	—
49	C₅×M₁₆	—	4	2	no	—
50	C₅×M₁₆	—	4	2	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	—	2	2	no	—
53	C₂×C₅×C₈	—	4	1	no	—
54	C₂×C₅×C₈	—	4	1	no	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅	C₂×D₄	1	2	yes	—
59	C₄⋊Dic₅	—	2	4	no	—
60	C₄⋊Dic₅	—	2	4	no	—
61	C₄⋊Dic₅	—	4	2	no	—
62	C₄⋊Dic₅	—	4	2	no	—
63	C₂×#168	—	2	4	no	—
64	C₂×C₄⋊Q₈	—	5	2	no	—
65	#498	—	5	4	no	—
66	#498	—	10	2	no	—
67	#431	—	5	2	no	—
68	C₂×#106	—	10	2	no	—
69	#423	C₂×D₅	1	1	yes	characteristic
70	#423	C₂×D₅	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	#341	—	2	2	no	—
73	#333	—	10	4	no	—
74	#332	—	10	4	no	—
75	#322	—	10	2	no	—
76	C₂³×C₅	#124	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	4	no	—
78	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₅	#121	1	2	yes	—
81	C₂×C₄×C₅	#127	1	1	yes	characteristic
82	C₂×C₄×C₅	#137	1	2	yes	—
83	C₂×C₄×C₅	#137	1	2	yes	—
84	C₂×Dic₅	—	4	4	no	—
85	C₂×Dic₅	—	8	4	no	—
86	Dic₁₀	—	8	4	no	—
87	Dic₁₀	—	8	4	no	—
88	C₅×C₈	—	2	4	no	—
89	C₅×C₈	—	8	1	no	—
90	C₅×C₈	—	8	1	no	—
91	#168	—	4	4	no	—
92	C₂²×Q₈	—	10	2	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
96	C₄⋊Q₈	—	5	4	no	—
97	C₄⋊Q₈	—	10	2	no	—
98	C₄⋊Q₈	—	10	2	no	—
99	C₄⋊Q₈	—	10	2	no	—
100	C₂×C₄⋊C₄	—	10	2	no	—
101	C₂×C₄⋊C₄	—	10	2	no	—
102	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
103	#106	—	10	8	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	5	4	no	—
107	C₂²⋊C₈	—	10	2	no	—
108	C₄×C₈	—	10	2	no	—
109	C₄×C₈	C₄×D₅	1	2	yes	—
110	C₄×C₈	D₂₀	1	2	yes	—
111	C₂²×C₅	—	2	2	no	—
112	C₂²×C₅	#390	1	1	yes	characteristic
113	C₂²×C₅	#491	1	2	yes	—
114	C₄×C₅	—	2	2	no	—
115	C₄×C₅	#419	1	2	yes	—
116	C₄×C₅	#471	1	4	yes	—
117	C₄×C₅	#497	1	1	yes	characteristic
118	C₄×C₅	#497	1	1	yes	characteristic
119	Dic₅	—	8	4	no	—
120	C₂×Q₈	—	10	4	no	—
121	C₂×Q₈	—	20	2	no	—
122	C₂×Q₈	—	40	4	no	—
123	C₂²×C₄	—	20	2	no	—
124	C₂²×C₄	#713	1	2	yes	—
125	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
126	M₁₆	—	4	2	no	—
127	M₁₆	—	4	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	10	4	no	—
133	C₂×C₈	#685	1	4	yes	—
134	C₄⋊C₄	—	10	4	no	—
135	C₄⋊C₄	—	10	4	no	—
136	C₄⋊C₄	—	20	2	no	—
137	C₄⋊C₄	—	20	2	no	—
138	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
139	C₄²	C₂×D₂₀	1	1	yes	characteristic
140	C₄²	D₄×D₅	1	2	yes	—
141	C₂×C₅	—	2	2	no	—
142	C₂×C₅	#1960	1	1	yes	characteristic
143	C₂×C₅	#1977	1	2	yes	—
144	C₂³	#4093	1	1	yes	characteristic
145	Q₈	—	40	4	no	—
146	Q₈	—	40	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	20	4	no	—
149	C₂×C₄	—	40	4	no	—
150	C₂×C₄	#4039	1	2	yes	—
151	C₂×C₄	#4041	1	2	yes	—
152	C₂×C₄	C₂×#685	1	1	yes	characteristic
153	C₂×C₄	#4086	1	1	yes	characteristic
154	C₂×C₄	#4097	1	2	yes	—
155	C₂×C₄	#4097	1	2	yes	—
156	C₈	—	2	4	no	—
157	C₈	—	8	1	no	—
158	C₈	—	8	1	no	—
159	C₈	—	20	4	no	—
160	C₅	#9776	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	2	2	no	—
162	C₂²	#65277	1	1	yes	characteristic
163	C₂²	#65358	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	40	4	no	—
166	C₄	#65187	1	4	yes	—
167	C₄	#65293	1	2	yes	—
168	C₄	#65349	1	1	yes	characteristic
169	C₄	#65349	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#10600345	1	1	yes	characteristic
172	C₂	#10601501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64969

C₂

yes

maximal, characteristic

C₅×#1837

C₂

yes

maximal, characteristic

#10601301

C₂

yes

maximal

#10601300

C₂

yes

maximal

#10600519

C₂

yes

maximal, characteristic

C₂×#4015

C₂²

yes

—

C₅×#423

C₂²

yes

characteristic

C₅×#423

C₂²

yes

characteristic

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3949

—

#65214

C₂²

yes

—

#64969

—

#64969

C₄

yes

—

—

—

—

—

—

C₂²×Dic₁₀

—

C₂×C₅×M₁₆

—

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

—

—

—

C₂×C₄

yes

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₂²⋊#168

—

#3949

—

C₄⋊#168

D₄

yes

—

C₄×#168

—

C₂×#498

—

#1837

D₅

yes

—

—

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

—

C₂×C₄⋊Q₈

—

—

—

—

—

C₂×D₅

yes

characteristic

#423

C₂×D₅

yes

characteristic

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

—

C₂³×C₅

#124

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

#137

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄²

C₂²×D₅

yes

characteristic

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₄×D₅

yes

—

110

C₄×C₈

D₂₀

yes

—

111

C₂²×C₅

—

112

C₂²×C₅

#390

yes

characteristic

113

C₂²×C₅

#491

yes

—

114

C₄×C₅

—

115

C₄×C₅

#419

yes

—

116

C₄×C₅

#471

yes

—

117

C₄×C₅

#497

yes

characteristic

118

C₄×C₅

#497

yes

characteristic

119

Dic₅

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

#713

yes

—

125

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

126

M₁₆

—

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

#685

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

C₂×C₄×D₅

yes

characteristic

139

C₄²

C₂×D₂₀

yes

characteristic

140

C₄²

D₄×D₅

yes

—

141

C₂×C₅

—

142

C₂×C₅

#1960

yes

characteristic

143

C₂×C₅

#1977

yes

—

144

C₂³

#4093

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#4039

yes

—

151

C₂×C₄

#4041

yes

—

152

C₂×C₄

C₂×#685

yes

characteristic

153

C₂×C₄

#4086

yes

characteristic

154

C₂×C₄

#4097

yes

—

155

C₂×C₄

#4097

yes

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₅

#9776

yes

characteristic, complemented, Sylow

161

C₂²

—

162

C₂²

#65277

yes

characteristic

163

C₂²

#65358

yes

—

164

C₄

—

165

C₄

—

166

C₄

#65187

yes

—

167

C₄

#65293

yes

—

168

C₄

#65349

yes

characteristic

169

C₄

#65349

yes

characteristic

170

C₂

—

171

C₂

#10600345

yes

characteristic

172

C₂

#10601501

yes

—

	subgroup	G/H
center	C₂²	#65277
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1837	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1837	D₅
hypocenter	C₅	#9776
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#65277

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
160. C₅	#9776	semidirect

H ≅ G/N

type

160. C₅

#9776

semidirect

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

subgroup	index
C₂×#64969	2
C₅×#1837	2
#10601301	2
#10601300	2
#10600519	2
#9776	5

subgroup

index

Subgroups — #11998784

Conjugacy classes of subgroups

Complements

Maximal subgroups