Group #11998764 — subgroups

Subgroups — #11998764

Order 1280.

478 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 159 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×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	#10602566	C₂	1	4	yes	maximal
3	#10600585	C₂	1	1	yes	maximal, characteristic
4	#10600517	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×Q₁₆	—	2	2	no	—
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#494	—	2	1	no	—
8	C₅×#494	—	2	1	no	—
9	C₅×#494	C₄	1	4	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3951	—	2	2	no	—
12	#65214	C₂²	1	2	yes	—
13	#65213	C₂²	1	2	yes	—
14	#64767	—	2	8	no	—
15	#64675	—	2	1	no	—
16	C₈×#168	—	2	1	no	—
17	#9772	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	—	2	2	no	—
19	C₂×C₅×Q₁₆	—	2	4	no	—
20	C₂×C₅×Q₁₆	—	2	4	no	—
21	C₂×C₅×Q₁₆	—	4	4	no	—
22	C₂×C₅×Q₁₆	—	4	4	no	—
23	C₂²×C₅×C₈	D₄	1	2	yes	—
24	C₅×C₄⋊Q₈	—	2	4	no	—
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₅×C₄⋊C₄	—	2	4	no	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₂×#681	—	2	2	no	—
30	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₅×C₈	D₄	1	1	yes	characteristic
32	C₄×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²⋊#168	—	2	2	no	—
34	#3951	—	4	4	no	—
35	C₄⋊#168	—	2	4	no	—
36	C₄⋊#168	D₄	1	4	yes	—
37	C₄×#168	—	2	2	no	—
38	C₂×#494	D₅	1	1	yes	characteristic
39	#1848	—	5	1	no	—
40	#1546	—	5	4	no	—
41	#1450	—	5	1	no	—
42	C₂×C₅×Q₈	—	2	4	no	—
43	C₂×C₅×Q₈	—	4	2	no	—
44	C₂×C₅×Q₈	—	4	4	no	—
45	C₂×C₅×Q₈	—	4	4	no	—
46	C₂²×C₄×C₅	—	4	2	no	—
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	#41	1	2	yes	—
49	C₅×Q₁₆	—	4	8	no	—
50	C₅×Q₁₆	—	4	8	no	—
51	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
52	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
53	C₂×C₅×C₈	D₈	1	4	yes	—
54	C₂×C₅×C₈	SD₁₆	1	4	yes	—
55	C₅×C₄⋊C₄	—	2	8	no	—
56	C₅×C₄⋊C₄	—	4	4	no	—
57	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₅	C₂×D₄	1	2	yes	—
59	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
60	#681	—	4	4	no	—
61	C₂×#168	—	2	4	no	—
62	C₂×#168	—	4	2	no	—
63	C₂²×Q₁₆	—	2	2	no	—
64	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
65	#494	—	2	1	no	—
66	#494	—	2	1	no	—
67	#494	Dic₅	1	4	yes	—
68	#431	—	5	2	no	—
69	C₂×#108	—	10	2	no	—
70	#423	—	5	2	no	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	#332	—	10	8	no	—
73	#322	—	10	1	no	—
74	C₈²	—	10	1	no	—
75	C₂³×C₅	#124	1	1	yes	characteristic
76	C₅×Q₈	—	4	8	no	—
77	C₅×Q₈	—	8	2	no	—
78	C₅×Q₈	—	8	2	no	—
79	C₂×C₄×C₅	—	2	4	no	—
80	C₂×C₄×C₅	—	4	4	no	—
81	C₂×C₄×C₅	—	8	4	no	—
82	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅	#121	1	2	yes	—
85	C₂×C₄×C₅	#127	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
88	C₅×C₈	#102	1	4	yes	—
89	C₅×C₈	#102	1	4	yes	—
90	#168	—	4	4	no	—
91	#168	—	8	2	no	—
92	C₂²×Q₈	—	2	2	no	—
93	C₂×Q₁₆	—	2	4	no	—
94	C₂×Q₁₆	—	2	4	no	—
95	C₂×Q₁₆	—	4	4	no	—
96	C₂×Q₁₆	—	4	4	no	—
97	C₂×M₁₆	—	10	2	no	—
98	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
99	C₄⋊Q₈	—	2	4	no	—
100	C₄⋊Q₈	—	2	2	no	—
101	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
102	C₂×C₄⋊C₄	—	2	4	no	—
103	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
104	#108	—	20	4	no	—
105	C₄⋊C₈	—	5	4	no	—
106	C₄⋊C₈	—	10	4	no	—
107	C₂²⋊C₈	—	10	2	no	—
108	C₄×C₈	—	10	2	no	—
109	C₄×C₈	C₂×Dic₅	1	2	yes	—
110	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
112	C₂²×C₅	—	2	2	no	—
113	C₂²×C₅	#390	1	1	yes	characteristic
114	C₂²×C₅	#486	1	2	yes	—
115	C₄×C₅	—	2	2	no	—
116	C₄×C₅	—	8	4	no	—
117	C₄×C₅	C₂×#102	1	2	yes	—
118	C₄×C₅	#473	1	4	yes	—
119	C₄×C₅	#492	1	1	yes	characteristic
120	C₄×C₅	#493	1	1	yes	characteristic
121	C₂×Q₈	—	2	4	no	—
122	C₂×Q₈	—	4	2	no	—
123	C₂×Q₈	—	4	4	no	—
124	C₂×Q₈	—	4	4	no	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	#711	1	2	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
128	Q₁₆	—	4	8	no	—
129	Q₁₆	—	4	8	no	—
130	M₁₆	—	20	4	no	—
131	C₂×C₈	—	10	4	no	—
132	C₂×C₈	—	20	2	no	—
133	C₂×C₈	#686	1	4	yes	—
134	C₂×C₈	#687	1	4	yes	—
135	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
136	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
137	C₄⋊C₄	—	2	8	no	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄²	D₄×D₅	1	2	yes	—
140	C₄²	C₂²×Dic₅	1	1	yes	characteristic
141	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
142	C₂×C₅	—	2	2	no	—
143	C₂×C₅	#1957	1	1	yes	characteristic
144	C₂×C₅	#1979	1	2	yes	—
145	C₂³	#4082	1	1	yes	characteristic
146	Q₈	—	4	8	no	—
147	Q₈	—	8	2	no	—
148	Q₈	—	8	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	8	4	no	—
152	C₂×C₄	C₂×#686	1	2	yes	—
153	C₂×C₄	C₂×#687	1	2	yes	—
154	C₂×C₄	D₄×Dic₅	1	2	yes	—
155	C₂×C₄	#4084	1	2	yes	—
156	C₂×C₄	#4086	1	1	yes	characteristic
157	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
158	C₈	—	20	4	no	—
159	C₈	—	40	2	no	—
160	C₈	#3964	1	4	yes	—
161	C₈	#3964	1	4	yes	—
162	C₅	#9772	1	1	yes	characteristic, complemented, Sylow
163	C₂²	—	2	2	no	—
164	C₂²	#65359	1	2	yes	—
165	C₂²	#65510	1	1	yes	characteristic
166	C₄	—	2	2	no	—
167	C₄	—	8	4	no	—
168	C₄	#65362	1	1	yes	characteristic
169	C₄	#65365	1	1	yes	characteristic
170	C₄	#65481	1	4	yes	—
171	C₄	C₂×#3964	1	2	yes	—
172	C₂	—	2	2	no	—
173	C₂	#10602557	1	1	yes	characteristic
174	C₂	#10602862	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#494

C₂

yes

maximal, characteristic

#10602566

C₂

yes

maximal

#10600585

C₂

yes

maximal, characteristic

#10600517

C₂

yes

maximal, characteristic

C₂²×C₅×Q₁₆

—

C₂×C₅×C₄⋊Q₈

C₂²

yes

—

C₅×#494

—

C₅×#494

—

C₅×#494

C₄

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3951

—

#65214

C₂²

yes

—

#65213

C₂²

yes

—

—

—

—

—

C₂²×C₅×Q₈

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂²×C₅×C₈

D₄

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×#681

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₂²⋊#168

—

#3951

—

C₄⋊#168

—

C₄⋊#168

D₄

yes

—

C₄×#168

—

C₂×#494

D₅

yes

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

—

—

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₅

yes

—

#494

—

#494

—

#494

Dic₅

yes

—

#431

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#332

—

#322

—

C₈²

—

C₂³×C₅

#124

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

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

C₂×D₈

yes

—

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₅×C₈

#102

yes

—

C₅×C₈

#102

yes

—

#168

—

#168

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

C₂×Dic₅

yes

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄²

C₂²×D₅

yes

characteristic

104

#108

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

C₂×Dic₅

yes

—

110

C₄×C₈

C₂²⋊D₅

yes

characteristic

111

C₄×C₈

C₂²⋊D₅

yes

characteristic

112

C₂²×C₅

—

113

C₂²×C₅

#390

yes

characteristic

114

C₂²×C₅

#486

yes

—

115

C₄×C₅

—

116

C₄×C₅

—

117

C₄×C₅

C₂×#102

yes

—

118

C₄×C₅

#473

yes

—

119

C₄×C₅

#492

yes

characteristic

120

C₄×C₅

#493

yes

characteristic

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

125

C₂²×C₄

—

126

C₂²×C₄

#711

yes

—

127

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

128

Q₁₆

—

129

Q₁₆

—

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

#686

yes

—

134

C₂×C₈

#687

yes

—

135

C₂×C₈

C₂²⋊Dic₅

yes

—

136

C₂×C₈

C₂²⋊Dic₅

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄²

D₄×D₅

yes

—

140

C₄²

C₂²×Dic₅

yes

characteristic

141

C₄²

C₂×C₂²⋊D₅

yes

characteristic

142

C₂×C₅

—

143

C₂×C₅

#1957

yes

characteristic

144

C₂×C₅

#1979

yes

—

145

C₂³

#4082

yes

characteristic

146

Q₈

—

147

Q₈

—

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

C₂×#686

yes

—

153

C₂×C₄

C₂×#687

yes

—

154

C₂×C₄

D₄×Dic₅

yes

—

155

C₂×C₄

#4084

yes

—

156

C₂×C₄

#4086

yes

characteristic

157

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

158

C₈

—

159

C₈

—

160

C₈

#3964

yes

—

161

C₈

#3964

yes

—

162

C₅

#9772

yes

characteristic, complemented, Sylow

163

C₂²

—

164

C₂²

#65359

yes

—

165

C₂²

#65510

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

#65362

yes

characteristic

169

C₄

#65365

yes

characteristic

170

C₄

#65481

yes

—

171

C₄

C₂×#3964

yes

—

172

C₂

—

173

C₂

#10602557

yes

characteristic

174

C₂

#10602862

yes

—

	subgroup	G/H
center	C₂²	#65510
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#494	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#494	D₅
hypocenter	C₅	#9772
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#65510

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₂×C₅×#494

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
162. C₅	#9772	semidirect

H ≅ G/N

type

162. C₅

#9772

semidirect

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

subgroup	index
C₂×C₅×#494	2
#10602566	2
#10600585	2
#10600517	2
#9772	5

subgroup

index

C₂×C₅×#494

Subgroups — #11998764

Conjugacy classes of subgroups

Complements

Maximal subgroups