Group #12012696 — subgroups

Subgroups — #12012696

Order 1280.

366 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 59 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	#10613539	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#1709	C₂	1	1	yes	maximal, characteristic, complemented
3	#10602088	C₂	1	2	yes	maximal, complemented
4	#10598285	C₂	1	1	yes	maximal, characteristic, complemented
5	#10597932	C₂	1	2	yes	maximal, complemented
6	#66100	—	2	2	no	—
7	C₅×#494	C₂²	1	1	yes	characteristic
8	C₅×#493	C₂²	1	1	yes	characteristic
9	C₅×#486	—	2	2	no	—
10	#65481	—	2	2	no	—
11	#65436	—	2	2	no	—
12	#65384	—	2	2	no	—
13	#64989	C₂²	1	2	yes	—
14	C₅×C₈²	C₂²	1	1	yes	characteristic
15	#64783	—	2	2	no	—
16	#64683	C₂²	1	2	yes	—
17	#12311	—	5	1	no	maximal, Sylow
18	#4147	—	2	2	no	—
19	C₂×C₅×Q₁₆	—	2	2	no	—
20	C₂×C₅×D₈	—	2	2	no	—
21	C₅×C₄⋊Q₈	D₄	1	2	yes	—
22	C₅×#127	—	2	1	no	—
23	#4088	—	2	4	no	—
24	#4065	—	4	2	no	—
25	#4064	—	4	4	no	—
26	D₅×Q₁₆	—	4	2	no	—
27	#4050	D₄	1	2	yes	—
28	#4047	—	2	2	no	—
29	C₅×#102	—	4	4	no	—
30	C₄×C₅×C₈	—	2	1	no	—
31	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
32	#3968	—	2	2	no	—
33	#3959	—	4	2	no	—
34	#3958	—	4	2	no	—
35	#3951	D₄	1	2	yes	—
36	#3944	—	2	2	no	—
37	#3937	—	2	2	no	—
38	#3293	—	5	1	no	—
39	#2213	—	5	2	no	—
40	#1709	D₅	1	1	yes	characteristic, complemented
41	#1358	—	5	1	no	—
42	#1335	—	5	2	no	—
43	C₂×C₅×Q₈	—	2	2	no	—
44	C₂×C₅×D₄	—	4	2	no	—
45	C₂×C₅×D₄	—	4	1	no	—
46	#713	—	4	4	no	—
47	Q₈×D₅	—	4	4	no	—
48	#709	—	2	2	no	—
49	#709	—	8	2	no	—
50	C₅×Q₁₆	—	4	2	no	—
51	C₅×Q₁₆	—	4	2	no	—
52	C₅×D₈	—	4	2	no	—
53	C₅×D₈	—	4	2	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
56	C₅×C₄⋊C₄	—	4	2	no	—
57	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
58	#689	—	4	4	no	—
59	#688	—	4	4	no	—
60	#681	—	2	2	no	—
61	Dic₂₀	D₈	1	2	yes	—
62	D₄₀	D₈	1	2	yes	—
63	#677	—	2	2	no	—
64	#676	—	4	2	no	—
65	C₈×D₅	—	4	2	no	—
66	#672	—	4	2	no	—
67	#578	—	10	2	no	—
68	#509	—	10	2	no	—
69	#494	C₂×D₅	1	1	yes	characteristic
70	#493	C₂×D₅	1	1	yes	characteristic
71	#486	—	2	2	no	—
72	#473	—	10	2	no	—
73	#456	—	10	2	no	—
74	#443	—	5	2	no	—
75	#364	—	5	2	no	—
76	#361	—	10	2	no	—
77	C₈²	C₂×D₅	1	1	yes	characteristic
78	C₂³×C₅	—	8	1	no	—
79	C₅×Q₈	—	4	4	no	—
80	C₅×D₄	—	8	4	no	—
81	C₅×D₄	—	8	2	no	—
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	4	2	no	—
84	C₂×C₄×C₅	#120	1	1	yes	characteristic
85	C₂²⋊D₅	—	4	2	no	—
86	D₂₀	—	2	2	no	—
87	D₂₀	—	8	2	no	—
88	C₄×D₅	—	4	2	no	—
89	C₄×D₅	—	8	4	no	—
90	Dic₁₀	—	2	2	no	—
91	Dic₁₀	—	8	2	no	—
92	C₅×C₈	—	4	2	no	—
93	C₅×C₈	C₂×D₈	1	2	yes	—
94	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
95	#168	—	4	2	no	—
96	2₋¹⁺⁴	—	10	2	no	—
97	#137	—	10	4	no	—
98	#137	—	20	4	no	—
99	#135	—	5	2	no	—
100	#135	—	20	2	no	—
101	C₂×Q₁₆	—	2	2	no	—
102	C₂×Q₁₆	—	20	2	no	—
103	C₂×D₈	—	2	2	no	—
104	#131	—	10	2	no	—
105	C₄⋊Q₈	C₂²⋊D₅	1	2	yes	—
106	#127	—	2	1	no	—
107	SD₃₂	—	20	2	no	—
108	D₁₆	—	20	2	no	—
109	M₃₂	—	10	2	no	—
110	#108	—	5	2	no	—
111	#104	—	10	2	no	—
112	#102	—	4	4	no	—
113	#101	—	10	2	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
116	C₂²×C₅	—	16	2	no	—
117	C₂²×C₅	—	16	1	no	—
118	C₂²×C₅	#453	1	1	yes	characteristic
119	C₂×D₅	—	4	2	no	—
120	C₄×C₅	—	2	2	no	—
121	C₄×C₅	—	8	2	no	—
122	C₄×C₅	#447	1	2	yes	—
123	Dic₅	—	4	2	no	—
124	#41	—	10	2	no	—
125	#41	—	20	4	no	—
126	#41	—	40	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	20	4	no	—
129	C₂×D₄	—	4	2	no	—
130	C₂×D₄	—	4	1	no	—
131	Q₁₆	—	4	2	no	—
132	Q₁₆	—	4	2	no	—
133	Q₁₆	—	5	2	no	—
134	Q₁₆	—	20	4	no	—
135	SD₁₆	—	10	2	no	—
136	SD₁₆	—	20	4	no	—
137	D₈	—	4	2	no	—
138	D₈	—	4	2	no	—
139	D₈	—	5	2	no	—
140	M₁₆	—	10	2	no	—
141	M₁₆	—	20	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	20	2	no	—
144	C₂×C₈	D₄×D₅	1	2	yes	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
147	C₁₆	—	20	2	no	—
148	C₂×C₅	—	2	1	no	—
149	C₂×C₅	—	32	1	no	—
150	C₂×C₅	#1619	1	1	yes	characteristic
151	D₅	—	8	2	no	—
152	C₂³	—	8	1	no	—
153	Q₈	—	4	4	no	—
154	Q₈	—	10	2	no	—
155	Q₈	—	40	2	no	—
156	D₄	—	8	4	no	—
157	D₄	—	8	2	no	—
158	D₄	—	10	2	no	—
159	D₄	—	20	2	no	—
160	D₄	—	40	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	20	2	no	—
164	C₂×C₄	—	40	4	no	—
165	C₂×C₄	#4083	1	1	yes	characteristic
166	C₈	—	4	2	no	—
167	C₈	—	20	2	no	—
168	C₈	D₅×D₈	1	2	yes	—
169	C₈	#4057	1	2	yes	—
170	C₅	#12311	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	16	2	no	—
172	C₂²	—	16	1	no	—
173	C₂²	—	20	2	no	—
174	C₂²	#65366	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	8	2	no	—
177	C₄	—	20	2	no	—
178	C₄	#65445	1	2	yes	—
179	C₂	—	2	1	no	—
180	C₂	—	32	1	no	—
181	C₂	—	40	2	no	—
182	C₂	#10602074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10613539

C₂

yes

maximal, characteristic, complemented

C₅×#1709

C₂

yes

maximal, characteristic, complemented

#10602088

C₂

yes

maximal, complemented

#10598285

C₂

yes

maximal, characteristic, complemented

#10597932

C₂

yes

maximal, complemented

#66100

—

C₅×#494

C₂²

yes

characteristic

C₅×#493

C₂²

yes

—

—

—

—

C₂²

yes

—

C₅×C₈²

C₂²

yes

characteristic

#64783

—

#64683

C₂²

yes

—

#12311

—

maximal, Sylow

#4147

—

C₂×C₅×Q₁₆

—

C₂×C₅×D₈

—

C₅×C₄⋊Q₈

D₄

yes

—

—

—

—

—

D₅×Q₁₆

—

#4050

D₄

yes

—

#4047

—

C₅×#102

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂³

yes

—

—

—

D₄

yes

—

—

—

—

—

D₅

yes

characteristic, complemented

#1358

—

#1335

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

#713

—

Q₈×D₅

—

#709

—

#709

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×D₈

—

C₅×D₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

—

—

—

Dic₂₀

D₈

yes

—

D₄₀

D₈

yes

—

#677

—

#676

—

C₈×D₅

—

—

—

—

C₂×D₅

yes

characteristic

#493

C₂×D₅

yes

—

—

—

—

—

—

C₈²

C₂×D₅

yes

characteristic

C₂³×C₅

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#120

yes

characteristic

C₂²⋊D₅

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

C₂×D₈

yes

—

C₅×C₈

AGL(1,ℤ/8)

yes

—

#168

—

2₋¹⁺⁴

—

#137

—

#137

—

#135

—

100

#135

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂×D₈

—

104

#131

—

105

C₄⋊Q₈

C₂²⋊D₅

yes

—

106

#127

—

107

SD₃₂

—

108

D₁₆

—

109

M₃₂

—

110

#108

—

111

#104

—

112

#102

—

113

#101

—

114

C₄×C₈

—

115

C₄×C₈

C₂²×D₅

yes

characteristic

116

C₂²×C₅

—

117

C₂²×C₅

—

118

C₂²×C₅

#453

yes

characteristic

119

C₂×D₅

—

120

C₄×C₅

—

121

C₄×C₅

—

122

C₄×C₅

#447

yes

—

123

Dic₅

—

124

#41

—

125

#41

—

126

#41

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×D₄

—

130

C₂×D₄

—

131

Q₁₆

—

132

Q₁₆

—

133

Q₁₆

—

134

Q₁₆

—

135

SD₁₆

—

136

SD₁₆

—

137

D₈

—

138

D₈

—

139

D₈

—

140

M₁₆

—

141

M₁₆

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

D₄×D₅

yes

—

145

C₄⋊C₄

—

146

C₄²

C₂×C₂²⋊D₅

yes

characteristic

147

C₁₆

—

148

C₂×C₅

—

149

C₂×C₅

—

150

C₂×C₅

#1619

yes

characteristic

151

D₅

—

152

C₂³

—

153

Q₈

—

154

Q₈

—

155

Q₈

—

156

D₄

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

#4083

yes

characteristic

166

C₈

—

167

C₈

—

168

C₈

D₅×D₈

yes

—

169

C₈

#4057

yes

—

170

C₅

#12311

yes

characteristic, complemented, Sylow

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

#65366

yes

characteristic

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#65445

yes

—

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

#10602074

yes

characteristic

	subgroup	G/H
center	C₂	#10602074
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1709	C₂
socle	C₂×C₅	#1619
hypercenter	#1709	D₅
hypocenter	C₅	#12311
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

subgroup

G/H

center

C₂

#10602074

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

N	H ≅ G/N	type
1. #10613539	C₂	semidirect
2. C₅×#1709	C₂	semidirect
3. #10602088	C₂	semidirect
4. #10598285	C₂	semidirect
5. #10597932	C₂	semidirect
40. #1709	D₅	semidirect
170. C₅	#12311	semidirect

H ≅ G/N

type

1. #10613539

C₂

semidirect

2. C₅×#1709

C₂

semidirect

3. #10602088

C₂

semidirect

4. #10598285

C₂

semidirect

5. #10597932

C₂

semidirect

40. #1709

D₅

semidirect

170. C₅

#12311

semidirect

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

subgroup	index
#10613539	2
C₅×#1709	2
#10602088	2
#10598285	2
#10597932	2
#12311	5

subgroup

index

Subgroups — #12012696

Conjugacy classes of subgroups

Complements

Maximal subgroups