Group #11998787 — subgroups

Subgroups — #11998787

Order 1280.

390 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 127 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×#498	C₂	1	1	yes	maximal, characteristic
2	#10602568	C₂	1	2	yes	maximal
3	#10602567	C₂	1	2	yes	maximal
4	#10600584	C₂	1	1	yes	maximal, characteristic
5	#10600519	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#498	—	2	2	no	—
8	C₅×#498	C₄	1	4	yes	—
9	C₂×C₅×#106	—	2	2	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	#65214	C₂²	1	2	yes	—
12	#65213	C₂²	1	1	yes	characteristic
13	#65213	C₂²	1	1	yes	characteristic
14	#64772	—	2	2	no	—
15	#64767	—	2	4	no	—
16	#64766	—	2	4	no	—
17	#64674	—	2	2	no	—
18	#9776	—	5	1	no	maximal, Sylow
19	C₂²×C₅×Q₈	—	2	2	no	—
20	C₂²×C₅×C₈	D₄	1	2	yes	—
21	C₅×C₄⋊Q₈	—	2	2	no	—
22	C₅×C₄⋊Q₈	—	2	2	no	—
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₅×C₄⋊C₄	—	2	2	no	—
26	C₂×C₅×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
28	C₂×#681	—	2	1	no	—
29	C₂×#681	—	2	1	no	—
30	C₅×#106	—	2	8	no	—
31	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₅×C₈	D₄	1	2	yes	—
33	C₂²⋊#168	—	2	2	no	—
34	C₄⋊#168	—	2	2	no	—
35	C₄⋊#168	—	2	2	no	—
36	C₄⋊#168	D₄	1	4	yes	—
37	C₄×#168	—	2	2	no	—
38	C₂×#498	D₅	1	1	yes	characteristic
39	#1837	—	5	1	no	—
40	#1549	—	5	2	no	—
41	#1548	—	5	2	no	—
42	#1452	—	5	1	no	—
43	C₂×C₅×Q₈	—	2	4	no	—
44	C₂×C₅×Q₈	—	4	2	no	—
45	C₂×C₅×Q₈	—	8	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	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	—	2	2	no	—
51	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
52	C₅×C₄⋊C₄	—	2	4	no	—
53	C₅×C₄⋊C₄	—	2	4	no	—
54	C₅×C₄⋊C₄	—	4	2	no	—
55	C₅×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₅	C₂×D₄	1	2	yes	—
58	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
59	#681	—	4	2	no	—
60	#681	—	4	2	no	—
61	C₂×#168	—	2	4	no	—
62	C₂×#168	—	4	1	no	—
63	C₂×#168	—	4	1	no	—
64	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
65	#498	—	2	2	no	—
66	#498	Dic₅	1	4	yes	—
67	#431	—	5	2	no	—
68	C₂×#106	—	2	2	no	—
69	#423	—	5	1	no	—
70	#423	—	5	1	no	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	#341	—	10	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	C₅×Q₈	—	8	4	no	—
78	C₅×Q₈	—	8	4	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	#137	1	2	yes	—
87	C₂×C₄×C₅	#137	1	2	yes	—
88	C₅×C₈	—	2	4	no	—
89	#168	—	4	4	no	—
90	#168	—	8	1	no	—
91	#168	—	8	1	no	—
92	C₂²×Q₈	—	2	2	no	—
93	C₂×M₁₆	—	10	1	no	—
94	C₂×M₁₆	—	10	1	no	—
95	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
96	C₄⋊Q₈	—	2	2	no	—
97	C₄⋊Q₈	—	2	2	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
103	#106	—	2	8	no	—
104	C₄⋊C₈	—	5	4	no	—
105	C₄⋊C₈	—	10	2	no	—
106	C₄⋊C₈	—	10	2	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	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	—	8	4	no	—
116	C₄×C₅	#419	1	2	yes	—
117	C₄×C₅	#471	1	4	yes	—
118	C₄×C₅	#497	1	1	yes	characteristic
119	C₄×C₅	#497	1	1	yes	characteristic
120	C₂×Q₈	—	2	4	no	—
121	C₂×Q₈	—	4	2	no	—
122	C₂×Q₈	—	8	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	#711	1	2	yes	—
125	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
126	M₁₆	—	20	2	no	—
127	M₁₆	—	20	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	10	4	no	—
131	C₂×C₈	—	20	1	no	—
132	C₂×C₈	—	20	1	no	—
133	C₂×C₈	C₂²⋊Dic₅	1	4	yes	—
134	C₄⋊C₄	—	2	4	no	—
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	2	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄²	D₄×D₅	1	2	yes	—
139	C₄²	C₂²×Dic₅	1	1	yes	characteristic
140	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
141	C₂×C₅	—	2	2	no	—
142	C₂×C₅	#1960	1	1	yes	characteristic
143	C₂×C₅	#1977	1	2	yes	—
144	C₂³	#4082	1	1	yes	characteristic
145	Q₈	—	8	4	no	—
146	Q₈	—	8	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	D₄×Dic₅	1	2	yes	—
151	C₂×C₄	#4084	1	2	yes	—
152	C₂×C₄	#4086	1	1	yes	characteristic
153	C₂×C₄	#4088	1	2	yes	—
154	C₂×C₄	#4088	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₈	—	20	4	no	—
158	C₈	—	40	1	no	—
159	C₈	—	40	1	no	—
160	C₅	#9776	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	2	2	no	—
162	C₂²	#65371	1	2	yes	—
163	C₂²	#65510	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	8	4	no	—
166	C₄	#65375	1	1	yes	characteristic
167	C₄	#65375	1	1	yes	characteristic
168	C₄	#65466	1	4	yes	—
169	C₄	#65514	1	2	yes	—
170	C₂	—	2	2	no	—
171	C₂	#10602579	1	1	yes	characteristic
172	C₂	#10602845	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#498

C₂

yes

maximal, characteristic

#10602568

C₂

yes

maximal

#10602567

C₂

yes

maximal

#10600584

C₂

yes

maximal, characteristic

#10600519

C₂

yes

maximal, characteristic

C₂×C₅×C₄⋊Q₈

C₂²

yes

—

C₅×#498

—

C₅×#498

C₄

yes

—

C₂×C₅×#106

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

#65214

C₂²

yes

—

#65213

C₂²

yes

characteristic

#65213

C₂²

yes

—

—

—

—

—

C₂²×C₅×Q₈

—

C₂²×C₅×C₈

D₄

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₂²⋊#168

—

C₄⋊#168

—

C₄⋊#168

—

C₄⋊#168

D₄

yes

—

C₄×#168

—

C₂×#498

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

—

—

—

—

—

C₂×C₄⋊Q₈

C₂×D₅

yes

—

#498

—

#498

Dic₅

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

—

C₂³×C₅

#124

yes

characteristic

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

#137

yes

—

C₂×C₄×C₅

#137

yes

—

C₅×C₈

—

#168

—

#168

—

#168

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₅

yes

—

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

C₂×Dic₅

yes

—

110

C₄×C₈

C₂²⋊D₅

yes

—

111

C₂²×C₅

—

112

C₂²×C₅

#390

yes

characteristic

113

C₂²×C₅

#491

yes

—

114

C₄×C₅

—

115

C₄×C₅

—

116

C₄×C₅

#419

yes

—

117

C₄×C₅

#471

yes

—

118

C₄×C₅

#497

yes

characteristic

119

C₄×C₅

#497

yes

characteristic

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

#711

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

C₂²⋊Dic₅

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

D₄×D₅

yes

—

139

C₄²

C₂²×Dic₅

yes

characteristic

140

C₄²

C₂×C₂²⋊D₅

yes

characteristic

141

C₂×C₅

—

142

C₂×C₅

#1960

yes

characteristic

143

C₂×C₅

#1977

yes

—

144

C₂³

#4082

yes

characteristic

145

Q₈

—

146

Q₈

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

D₄×Dic₅

yes

—

151

C₂×C₄

#4084

yes

—

152

C₂×C₄

#4086

yes

characteristic

153

C₂×C₄

#4088

yes

—

154

C₂×C₄

#4088

yes

—

155

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₅

#9776

yes

characteristic, complemented, Sylow

161

C₂²

—

162

C₂²

#65371

yes

—

163

C₂²

#65510

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

#65375

yes

characteristic

167

C₄

#65375

yes

characteristic

168

C₄

#65466

yes

—

169

C₄

#65514

yes

—

170

C₂

—

171

C₂

#10602579

yes

characteristic

172

C₂

#10602845

yes

—

	subgroup	G/H
center	C₂²	#65510
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#498	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#498	D₅
hypocenter	C₅	#9776
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#65510

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₂×C₅×#498

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

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×C₅×#498	2
#10602568	2
#10602567	2
#10600584	2
#10600519	2
#9776	5

subgroup

index

C₂×C₅×#498

Subgroups — #11998787

Conjugacy classes of subgroups

Complements

Maximal subgroups