Group #12014052 — subgroups

Subgroups — #12014052

Order 1280.

340 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 91 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	#10599853	C₂	1	2	yes	maximal, complemented
2	#10599851	C₂	1	2	yes	maximal, complemented
3	#10598855	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598851	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₀	—	2	4	no	—
7	#64984	—	2	1	no	—
8	#64984	—	2	1	no	—
9	#64984	C₂²	1	2	yes	—
10	#64984	C₂²	1	1	yes	characteristic
11	#64983	—	2	1	no	—
12	#64982	—	2	1	no	—
13	#64971	C₂²	1	1	yes	characteristic
14	C₄×C₅×C₁₆	C₂²	1	2	yes	—
15	C₅×C₈²	C₂²	1	1	yes	characteristic
16	#64729	—	2	4	no	—
17	#12525	—	5	1	no	maximal, Sylow
18	C₂×D₄₀	—	2	2	no	—
19	C₂×D₄₀	—	2	2	no	—
20	C₂×D₄₀	—	2	2	no	—
21	C₂×D₄₀	—	4	2	no	—
22	C₂×D₄₀	—	4	2	no	—
23	C₂×#677	—	4	2	no	—
24	#4020	—	2	1	no	—
25	#4020	—	2	2	no	—
26	#4015	—	2	1	no	—
27	C₂×C₅×C₁₆	D₄	1	4	yes	—
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×C₈	D₄	1	1	yes	characteristic
30	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
31	#3953	—	4	2	no	—
32	#3950	—	2	2	no	—
33	D₈₀	—	4	8	no	—
34	#2246	—	5	2	no	—
35	#2240	—	5	2	no	—
36	#1712	—	5	1	no	—
37	#1706	—	5	1	no	—
38	#1312	D₅	1	1	yes	characteristic, complemented
39	C₂×D₂₀	—	4	1	no	—
40	C₂×D₂₀	—	4	2	no	—
41	C₂×D₂₀	—	4	1	no	—
42	C₂×D₂₀	—	4	1	no	—
43	C₂×D₂₀	—	4	2	no	—
44	C₂×D₂₀	—	4	2	no	—
45	C₂×Dic₁₀	—	4	1	no	—
46	C₂×C₅×C₈	D₈	1	2	yes	—
47	C₂×C₅×C₈	D₈	1	2	yes	—
48	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
49	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
50	C₄⋊Dic₅	—	4	1	no	—
51	C₄⋊Dic₅	—	4	1	no	—
52	D₄₀	—	4	4	no	—
53	D₄₀	—	4	4	no	—
54	D₄₀	—	4	4	no	—
55	D₄₀	—	8	4	no	—
56	D₄₀	—	8	4	no	—
57	#677	—	8	4	no	—
58	C₅×C₁₆	—	2	4	no	—
59	C₂×D₁₆	—	10	4	no	—
60	#500	—	5	1	no	—
61	#493	—	5	1	no	—
62	#493	—	5	2	no	—
63	#493	—	10	1	no	—
64	#493	—	10	1	no	—
65	#492	—	10	1	no	—
66	#486	—	10	1	no	—
67	#357	—	10	4	no	—
68	C₄×C₁₆	C₂×D₅	1	2	yes	—
69	C₈²	C₂×D₅	1	1	yes	characteristic
70	C₂²×D₅	—	8	1	no	—
71	C₂²×D₅	—	8	2	no	—
72	C₂×C₄×C₅	#127	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
75	C₂×Dic₅	—	8	1	no	—
76	D₂₀	—	8	2	no	—
77	D₂₀	—	8	2	no	—
78	D₂₀	—	8	2	no	—
79	D₂₀	—	8	2	no	—
80	D₂₀	—	8	4	no	—
81	D₂₀	—	8	4	no	—
82	D₂₀	—	8	2	no	—
83	Dic₁₀	—	8	2	no	—
84	C₅×C₈	—	2	2	no	—
85	C₅×C₈	D₁₆	1	4	yes	complemented
86	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	—
87	C₂×SD₁₆	—	20	2	no	—
88	C₂×D₈	—	10	2	no	—
89	C₂×D₈	—	10	2	no	—
90	C₂×D₈	—	10	2	no	—
91	C₂×D₈	—	20	2	no	—
92	C₂×D₈	—	20	2	no	—
93	C₄⋊Q₈	—	10	1	no	—
94	#127	—	10	1	no	—
95	#127	—	10	2	no	—
96	D₁₆	—	20	8	no	—
97	C₂×C₁₆	D₂₀	1	4	yes	—
98	#107	—	10	2	no	—
99	#102	—	20	2	no	—
100	C₄×C₈	D₂₀	1	1	yes	characteristic
101	C₄×C₈	D₂₀	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
103	C₂²×C₅	#493	1	1	yes	characteristic
104	C₂×D₅	—	16	1	no	—
105	C₂×D₅	—	16	2	no	—
106	C₂×D₅	—	16	1	no	—
107	C₂×D₅	—	16	2	no	—
108	C₂×D₅	—	16	2	no	—
109	C₂×D₅	—	16	1	no	—
110	C₄×C₅	#496	1	2	yes	—
111	C₄×C₅	C₂×D₁₆	1	2	yes	—
112	C₄×C₅	#509	1	2	yes	—
113	Dic₅	—	16	1	no	—
114	C₂×Q₈	—	20	1	no	—
115	C₂×D₄	—	20	1	no	—
116	C₂×D₄	—	20	2	no	—
117	C₂×D₄	—	20	1	no	—
118	C₂×D₄	—	20	2	no	—
119	C₂×D₄	—	20	2	no	—
120	C₂×D₄	—	20	1	no	—
121	SD₁₆	—	40	4	no	—
122	D₈	—	20	4	no	—
123	D₈	—	20	4	no	—
124	D₈	—	20	4	no	—
125	D₈	—	40	4	no	—
126	D₈	—	40	4	no	—
127	C₂×C₈	D₄₀	1	2	yes	—
128	C₂×C₈	D₄₀	1	2	yes	—
129	C₂×C₈	C₂×D₂₀	1	2	yes	—
130	C₄⋊C₄	—	20	1	no	—
131	C₄⋊C₄	—	20	1	no	—
132	C₄²	C₂×D₂₀	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₂×C₅	#1721	1	1	yes	characteristic
135	C₂×C₅	#2246	1	1	yes	characteristic
136	C₂×C₅	#2250	1	1	yes	characteristic
137	D₅	—	32	1	no	—
138	D₅	—	32	2	no	—
139	C₂³	—	40	1	no	—
140	C₂³	—	40	2	no	—
141	Q₈	—	40	2	no	—
142	D₄	—	40	4	no	—
143	D₄	—	40	2	no	—
144	D₄	—	40	4	no	—
145	D₄	—	40	2	no	—
146	D₄	—	40	2	no	—
147	D₄	—	40	2	no	—
148	D₄	—	40	2	no	—
149	C₂×C₄	—	40	1	no	—
150	C₂×C₄	#4020	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	D₈₀	1	4	yes	complemented
155	C₈	#4054	1	4	yes	—
156	C₅	#12525	1	1	yes	characteristic, complemented, Sylow
157	C₂²	—	80	2	no	—
158	C₂²	—	80	1	no	—
159	C₂²	—	80	2	no	—
160	C₂²	—	80	1	no	—
161	C₂²	—	80	1	no	—
162	C₂²	—	80	2	no	—
163	C₂²	#64984	1	1	yes	characteristic
164	C₄	—	80	1	no	—
165	C₄	#65001	1	2	yes	—
166	C₄	C₂×D₈₀	1	2	yes	—
167	C₄	#65197	1	2	yes	—
168	C₂	—	160	2	no	—
169	C₂	—	160	1	no	—
170	C₂	#10598887	1	1	yes	characteristic
171	C₂	#10599853	1	1	yes	characteristic
172	C₂	#10599874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599853

C₂

yes

maximal, complemented

#10599851

C₂

yes

maximal, complemented

#10598855

C₂

yes

maximal, characteristic, complemented

#10598851

C₂

yes

maximal, characteristic, complemented

C₅×#1312

C₂

yes

maximal, characteristic, complemented

C₂×D₈₀

—

#64984

—

#64984

—

#64984

C₂²

yes

—

#64984

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

—

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

—

—

—

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

characteristic

#3953

—

#3950

—

D₈₀

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

D₄₀

—

D₄₀

—

D₄₀

—

D₄₀

—

D₄₀

—

#677

—

C₅×C₁₆

—

C₂×D₁₆

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₅

yes

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

D₁₆

yes

complemented

C₅×C₈

AGL(1,ℤ/8)

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

D₁₆

—

C₂×C₁₆

D₂₀

yes

—

#107

—

#102

—

100

C₄×C₈

D₂₀

yes

characteristic

101

C₄×C₈

D₂₀

yes

characteristic

102

C₄×C₈

C₂²×D₅

yes

characteristic

103

C₂²×C₅

#493

yes

characteristic

104

C₂×D₅

—

105

C₂×D₅

—

106

C₂×D₅

—

107

C₂×D₅

—

108

C₂×D₅

—

109

C₂×D₅

—

110

C₄×C₅

#496

yes

—

111

C₄×C₅

C₂×D₁₆

yes

—

112

C₄×C₅

#509

yes

—

113

Dic₅

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

SD₁₆

—

122

D₈

—

123

D₈

—

124

D₈

—

125

D₈

—

126

D₈

—

127

C₂×C₈

D₄₀

yes

—

128

C₂×C₈

D₄₀

yes

—

129

C₂×C₈

C₂×D₂₀

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄²

C₂×D₂₀

yes

characteristic

133

C₁₆

—

134

C₂×C₅

#1721

yes

characteristic

135

C₂×C₅

#2246

yes

characteristic

136

C₂×C₅

#2250

yes

characteristic

137

D₅

—

138

D₅

—

139

C₂³

—

140

C₂³

—

141

Q₈

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

147

D₄

—

148

D₄

—

149

C₂×C₄

—

150

C₂×C₄

#4020

yes

characteristic

151

C₂×C₄

C₂×D₄₀

yes

characteristic

152

C₂×C₄

C₂×D₄₀

yes

characteristic

153

C₈

—

154

C₈

D₈₀

yes

complemented

155

C₈

#4054

yes

—

156

C₅

#12525

yes

characteristic, complemented, Sylow

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#64984

yes

characteristic

164

C₄

—

165

C₄

#65001

yes

—

166

C₄

C₂×D₈₀

yes

—

167

C₄

#65197

yes

—

168

C₂

—

160

—

169

C₂

—

160

—

170

C₂

#10598887

yes

characteristic

171

C₂

#10599853

yes

characteristic

172

C₂

#10599874

yes

characteristic

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1312	C₂
socle	C₂²×C₅	#493
hypercenter	#1312	D₅
hypocenter	C₅	#12525
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#64984

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #10599853	C₂	semidirect
2. #10599851	C₂	semidirect
3. #10598855	C₂	semidirect
4. #10598851	C₂	semidirect
5. C₅×#1312	C₂	semidirect
38. #1312	D₅	semidirect
85. C₅×C₈	D₁₆	semidirect
154. C₈	D₈₀	semidirect
156. C₅	#12525	semidirect

H ≅ G/N

type

1. #10599853

C₂

semidirect

2. #10599851

C₂

semidirect

3. #10598855

C₂

semidirect

4. #10598851

C₂

semidirect

5. C₅×#1312

C₂

semidirect

38. #1312

D₅

semidirect

85. C₅×C₈

D₁₆

semidirect

154. C₈

D₈₀

semidirect

156. C₅

#12525

semidirect

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

subgroup	index
#10599853	2
#10599851	2
#10598855	2
#10598851	2
C₅×#1312	2
#12525	5

subgroup

index

Subgroups — #12014052

Conjugacy classes of subgroups

Complements

Maximal subgroups