Group #11993052 — subgroups

Subgroups — #11993052

Order 1280.

402 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 187 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	C₂×#64700	C₂	1	4	yes	maximal
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	#10600181	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₄⋊C₈	—	2	2	no	—
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₄³×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#168	—	2	2	no	—
9	C₂×C₄⋊#168	C₂²	1	2	yes	—
10	C₂×C₄×#168	C₂²	1	2	yes	—
11	#64744	C₄	1	2	yes	—
12	#64744	C₄	1	2	yes	—
13	#64700	—	2	8	no	—
14	#8809	—	5	1	no	maximal, Sylow
15	C₂²×C₅×C₈	—	2	2	no	—
16	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₅	D₄	1	2	yes	—
19	C₂×C₄²×C₅	Q₈	1	2	yes	—
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	C₂²×#168	—	2	2	no	—
22	C₂²×#168	C₂×C₄	1	4	yes	—
23	C₅×C₄⋊C₈	—	2	4	no	—
24	C₅×C₄⋊C₈	—	4	4	no	—
25	C₄⋊#168	—	2	4	no	—
26	C₄⋊#168	—	4	4	no	—
27	C₄×#168	D₄	1	4	yes	—
28	C₄×#168	Q₈	1	4	yes	—
29	#1842	—	5	1	no	—
30	#1842	D₅	1	1	yes	characteristic
31	#1744	—	5	1	no	—
32	C₂×#334	—	5	4	no	—
33	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₅×C₈	—	4	2	no	—
39	C₂×C₅×C₈	—	4	4	no	—
40	C₄²×C₅	—	2	2	no	—
41	C₄²×C₅	—	2	2	no	—
42	C₄²×C₅	—	4	2	no	—
43	C₄²×C₅	SD₁₆	1	4	yes	—
44	C₄²×C₅	SD₁₆	1	4	yes	—
45	C₄²×C₅	C₂×D₄	1	2	yes	—
46	C₄²×C₅	C₂×Q₈	1	2	yes	—
47	C₂×#168	—	2	4	no	—
48	C₂×#168	—	2	4	no	—
49	C₂×#168	—	4	2	no	—
50	C₂×#168	—	4	4	no	—
51	C₂×#168	C₄⋊C₄	1	8	yes	—
52	C₂×C₄⋊C₈	—	2	2	no	—
53	C₂×C₄⋊C₈	—	5	2	no	—
54	C₂×C₄⋊C₈	—	10	2	no	—
55	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
56	C₂×C₄×C₈	—	5	2	no	—
57	C₄³	C₂×D₅	1	1	yes	characteristic
58	#336	—	5	2	no	—
59	#336	—	5	2	no	—
60	#334	—	10	8	no	—
61	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
67	C₂×C₄×C₅	#106	1	4	yes	—
68	C₂×C₄×C₅	#106	1	4	yes	—
69	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
70	C₂×C₄×C₅	#127	1	1	yes	characteristic
71	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
72	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅	#131	1	2	yes	—
74	C₂×C₄×C₅	#131	1	2	yes	—
75	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
77	C₅×C₈	—	8	2	no	—
78	#168	—	2	8	no	—
79	#168	—	8	2	no	—
80	C₂²×C₈	—	2	2	no	—
81	C₂²×C₈	—	5	4	no	—
82	C₂²×C₈	—	10	2	no	—
83	C₂×C₄²	Dic₁₀	1	2	yes	—
84	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
85	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
86	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
87	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
88	C₄⋊C₈	—	2	4	no	—
89	C₄⋊C₈	—	4	4	no	—
90	C₄⋊C₈	—	10	4	no	—
91	C₄⋊C₈	—	20	4	no	—
92	C₄×C₈	—	5	4	no	—
93	C₄×C₈	—	5	4	no	—
94	C₂²×C₅	#384	1	1	yes	characteristic
95	C₂²×C₅	#424	1	1	yes	characteristic
96	C₂²×C₅	#424	1	1	yes	characteristic
97	C₂²×C₅	C₂×#106	1	2	yes	—
98	C₂²×C₅	#430	1	2	yes	—
99	C₄×C₅	—	2	4	no	—
100	C₄×C₅	—	2	4	no	—
101	C₄×C₅	—	2	2	no	—
102	C₄×C₅	#424	1	4	yes	—
103	C₄×C₅	#492	1	1	yes	characteristic
104	C₄×C₅	#492	1	1	yes	characteristic
105	C₄×C₅	#498	1	2	yes	—
106	C₂²×C₄	#683	1	2	yes	—
107	C₂²×C₄	#683	1	2	yes	—
108	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
111	C₂×C₈	—	4	4	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	5	8	no	—
114	C₂×C₈	—	10	4	no	—
115	C₂×C₈	—	10	4	no	—
116	C₂×C₈	—	20	2	no	—
117	C₂×C₈	—	20	4	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	—	4	2	no	—
121	C₄²	#687	1	4	yes	—
122	C₄²	#688	1	4	yes	—
123	C₄²	D₄×D₅	1	2	yes	—
124	C₄²	Q₈×D₅	1	2	yes	—
125	C₂×C₅	#1570	1	2	yes	—
126	C₂×C₅	#1570	1	2	yes	—
127	C₂×C₅	#1843	1	1	yes	characteristic
128	C₂×C₅	#1844	1	1	yes	characteristic
129	C₂×C₅	#1849	1	1	yes	characteristic
130	C₂³	C₂×#683	1	1	yes	characteristic
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	—	2	2	no	—
136	C₂×C₄	#3940	1	4	yes	—
137	C₂×C₄	#3940	1	4	yes	—
138	C₂×C₄	#4034	1	2	yes	—
139	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
140	C₂×C₄	#4047	1	2	yes	—
141	C₂×C₄	#4053	1	2	yes	—
142	C₂×C₄	C₂×#687	1	2	yes	—
143	C₂×C₄	#4086	1	1	yes	characteristic
144	C₂×C₄	C₂×#688	1	2	yes	—
145	C₂×C₄	#4090	1	1	yes	characteristic
146	C₈	—	8	2	no	—
147	C₈	—	10	8	no	—
148	C₈	—	40	2	no	—
149	C₅	#8809	1	1	yes	characteristic, complemented, Sylow
150	C₂²	C₂×#3940	1	2	yes	—
151	C₂²	#65265	1	1	yes	characteristic
152	C₂²	#65391	1	1	yes	characteristic
153	C₂²	#65408	1	1	yes	characteristic
154	C₂²	#65413	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	2	no	—
157	C₄	—	2	4	no	—
158	C₄	#65128	1	4	yes	—
159	C₄	#65365	1	1	yes	characteristic
160	C₄	#65368	1	2	yes	—
161	C₄	#65374	1	1	yes	characteristic
162	C₂	#10600369	1	1	yes	characteristic
163	C₂	#10601214	1	2	yes	—
164	C₂	#10601239	1	1	yes	characteristic
165	C₂	#10601286	1	2	yes	—
166	C₂	#10601312	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1842

C₂

yes

maximal, characteristic

C₂×#64700

C₂

yes

maximal

#10600187

C₂

yes

maximal, characteristic

#10600181

C₂

yes

maximal, characteristic

C₂×C₅×C₄⋊C₈

—

C₂×C₅×C₄⋊C₈

C₂²

yes

—

C₄³×C₅

C₂²

yes

characteristic

C₂×C₄⋊#168

—

C₂×C₄⋊#168

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

—

#64744

C₄

yes

—

#64744

C₄

yes

—

#64700

—

#8809

—

maximal, Sylow

C₂²×C₅×C₈

—

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

C₂²×#168

—

C₂²×#168

C₂×C₄

yes

—

C₅×C₄⋊C₈

—

C₅×C₄⋊C₈

—

C₄⋊#168

—

C₄⋊#168

—

C₄×#168

D₄

yes

—

C₄×#168

Q₈

yes

—

#1842

—

#1842

D₅

yes

characteristic

#1744

—

C₂×#334

—

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

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

SD₁₆

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×Q₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₅

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₅

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

#106

yes

—

C₂×C₄×C₅

#106

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₅×C₈

—

#168

—

#168

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₁₀

yes

—

C₂×C₄²

C₄×D₅

yes

characteristic

C₂×C₄²

C₄×D₅

yes

characteristic

C₂×C₄²

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

yes

—

C₂²×C₅

#430

yes

—

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

#424

yes

—

103

C₄×C₅

#492

yes

characteristic

104

C₄×C₅

#492

yes

characteristic

105

C₄×C₅

#498

yes

—

106

C₂²×C₄

#683

yes

—

107

C₂²×C₄

#683

yes

—

108

C₂²×C₄

C₂×Dic₁₀

yes

characteristic

109

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

110

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

#687

yes

—

122

C₄²

#688

yes

—

123

C₄²

D₄×D₅

yes

—

124

C₄²

Q₈×D₅

yes

—

125

C₂×C₅

#1570

yes

—

126

C₂×C₅

#1570

yes

—

127

C₂×C₅

#1843

yes

characteristic

128

C₂×C₅

#1844

yes

characteristic

129

C₂×C₅

#1849

yes

characteristic

130

C₂³

C₂×#683

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#3940

yes

—

137

C₂×C₄

#3940

yes

—

138

C₂×C₄

#4034

yes

—

139

C₂×C₄

D₅×C₄⋊C₄

yes

—

140

C₂×C₄

#4047

yes

—

141

C₂×C₄

#4053

yes

—

142

C₂×C₄

C₂×#687

yes

—

143

C₂×C₄

#4086

yes

characteristic

144

C₂×C₄

C₂×#688

yes

—

145

C₂×C₄

#4090

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₅

#8809

yes

characteristic, complemented, Sylow

150

C₂²

C₂×#3940

yes

—

151

C₂²

#65265

yes

characteristic

152

C₂²

#65391

yes

characteristic

153

C₂²

#65408

yes

characteristic

154

C₂²

#65413

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#65128

yes

—

159

C₄

#65365

yes

characteristic

160

C₄

#65368

yes

—

161

C₄

#65374

yes

characteristic

162

C₂

#10600369

yes

characteristic

163

C₂

#10601214

yes

—

164

C₂

#10601239

yes

characteristic

165

C₂

#10601286

yes

—

166

C₂

#10601312

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1842	C₂
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1842	D₅
hypocenter	C₅	#8809
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₅

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#1842

C₂

socle, prime radical

C₂³×C₅

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
149. C₅	#8809	semidirect

H ≅ G/N

type

149. C₅

#8809

semidirect

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

subgroup	index
C₅×#1842	2
C₂×#64700	2
#10600187	2
#10600181	2
#8809	5

subgroup

index

Subgroups — #11993052

Conjugacy classes of subgroups

Complements

Maximal subgroups