Group #12014058 — subgroups

Subgroups — #12014058

Order 1280.

296 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 87 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×#1713	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602106	C₂	1	2	yes	maximal
3	#10602103	C₂	1	2	yes	maximal, complemented
4	#10598851	C₂	1	1	yes	maximal, characteristic
5	#10597929	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#500	—	2	1	no	—
7	C₅×#500	—	2	1	no	—
8	C₅×#494	C₂²	1	2	yes	—
9	C₂×#3960	—	2	4	no	—
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	C₅×C₈²	C₂²	1	1	yes	characteristic
15	#64784	—	2	4	no	—
16	C₄×#672	C₂²	1	2	yes	—
17	#12526	—	5	1	no	maximal, Sylow
18	C₂×C₅×Q₁₆	—	2	2	no	—
19	C₂×C₅×Q₁₆	—	2	2	no	—
20	C₅×C₄⋊Q₈	—	2	2	no	—
21	C₂×D₄₀	—	2	2	no	—
22	C₂×#677	—	4	2	no	—
23	#4020	—	2	1	no	—
24	#4015	—	2	1	no	—
25	C₅×#107	—	4	2	no	—
26	C₅×#107	—	4	2	no	—
27	C₄×C₅×C₈	D₄	1	1	yes	characteristic
28	C₄×C₅×C₈	D₄	1	1	yes	characteristic
29	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
30	#3960	—	4	8	no	—
31	#3953	—	4	2	no	—
32	#3950	—	2	2	no	—
33	C₂×#672	D₄	1	4	yes	—
34	#2248	—	5	2	no	—
35	#2241	—	5	2	no	—
36	#1713	D₅	1	1	yes	characteristic, complemented
37	#1706	—	5	1	no	—
38	#1312	—	5	1	no	—
39	C₂×C₅×Q₈	—	4	2	no	—
40	C₂×D₂₀	—	4	1	no	—
41	C₂×D₂₀	—	4	1	no	—
42	C₂×D₂₀	—	4	1	no	—
43	C₂×Dic₁₀	—	4	1	no	—
44	C₅×Q₁₆	—	4	4	no	—
45	C₅×Q₁₆	—	4	4	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	C₅×C₄⋊C₄	—	4	2	no	—
50	C₅×C₄⋊C₄	—	4	2	no	—
51	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
52	C₄⋊Dic₅	—	4	1	no	—
53	C₄⋊Dic₅	—	4	1	no	—
54	D₄₀	—	4	4	no	—
55	#677	—	8	4	no	—
56	#672	—	2	4	no	—
57	C₂×SD₃₂	—	10	4	no	—
58	#500	—	2	1	no	—
59	#500	—	2	1	no	—
60	#500	—	5	1	no	—
61	#494	C₂×D₅	1	2	yes	—
62	#493	—	5	1	no	—
63	#492	—	10	1	no	—
64	#486	—	10	1	no	—
65	#358	—	10	4	no	—
66	C₄×C₁₆	—	5	2	no	—
67	C₈²	C₂×D₅	1	1	yes	characteristic
68	C₂²×D₅	—	8	1	no	—
69	C₅×Q₈	—	8	2	no	—
70	C₅×Q₈	—	8	2	no	—
71	C₂×C₄×C₅	—	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	Dic₁₀	—	8	2	no	—
80	C₅×C₈	—	2	2	no	—
81	C₅×C₈	SD₃₂	1	4	yes	complemented
82	C₅×C₈	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×Q₁₆	—	2	2	no	—
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×SD₁₆	—	20	2	no	—
86	C₂×D₈	—	10	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	C₄⋊Q₈	—	10	1	no	—
89	#127	—	10	1	no	—
90	SD₃₂	—	20	8	no	—
91	C₂×C₁₆	—	5	4	no	—
92	#107	—	4	2	no	—
93	#107	—	4	2	no	—
94	#107	—	10	2	no	—
95	#102	—	20	2	no	—
96	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
97	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
99	C₂²×C₅	#493	1	1	yes	characteristic
100	C₂×D₅	—	16	1	no	—
101	C₂×D₅	—	16	1	no	—
102	C₂×D₅	—	16	1	no	—
103	C₄×C₅	—	16	2	no	—
104	C₄×C₅	#496	1	2	yes	—
105	C₄×C₅	C₂×SD₃₂	1	2	yes	—
106	C₄×C₅	#510	1	2	yes	—
107	Dic₅	—	16	1	no	—
108	C₂×Q₈	—	4	2	no	—
109	C₂×Q₈	—	20	1	no	—
110	C₂×D₄	—	20	1	no	—
111	C₂×D₄	—	20	1	no	—
112	C₂×D₄	—	20	1	no	—
113	Q₁₆	—	4	4	no	—
114	Q₁₆	—	4	4	no	—
115	SD₁₆	—	40	4	no	—
116	D₈	—	20	4	no	—
117	C₂×C₈	#686	1	2	yes	—
118	C₂×C₈	#686	1	2	yes	—
119	C₂×C₈	D₄×D₅	1	2	yes	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	20	1	no	—
123	C₄⋊C₄	—	20	1	no	—
124	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
125	C₁₆	—	10	4	no	—
126	C₂×C₅	#1721	1	1	yes	characteristic
127	C₂×C₅	#2248	1	1	yes	characteristic
128	C₂×C₅	#2251	1	1	yes	characteristic
129	D₅	—	32	1	no	—
130	C₂³	—	40	1	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	40	2	no	—
134	D₄	—	40	2	no	—
135	D₄	—	40	2	no	—
136	D₄	—	40	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	40	1	no	—
139	C₂×C₄	C₂×#686	1	1	yes	characteristic
140	C₂×C₄	C₂×#686	1	1	yes	characteristic
141	C₂×C₄	#4086	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	#3960	1	4	yes	complemented
144	C₈	#4057	1	4	yes	—
145	C₅	#12526	1	1	yes	characteristic, complemented, Sylow
146	C₂²	—	80	1	no	—
147	C₂²	—	80	1	no	—
148	C₂²	—	80	1	no	—
149	C₂²	#65362	1	1	yes	characteristic
150	C₄	—	16	2	no	—
151	C₄	—	80	1	no	—
152	C₄	#65443	1	2	yes	—
153	C₄	C₂×#3960	1	2	yes	—
154	C₄	#65468	1	2	yes	—
155	C₂	—	160	1	no	—
156	C₂	#10602073	1	1	yes	characteristic
157	C₂	#10602106	1	1	yes	characteristic
158	C₂	#10602107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1713

C₂

yes

maximal, characteristic, complemented

#10602106

C₂

yes

maximal

#10602103

C₂

yes

maximal, complemented

#10598851

C₂

yes

maximal, characteristic

#10597929

C₂

yes

maximal, characteristic, complemented

C₅×#500

—

C₅×#500

—

C₅×#494

C₂²

yes

—

C₂×#3960

—

#64984

C₂²

yes

—

—

C₂²

yes

characteristic

C₅×C₈²

C₂²

yes

characteristic

#64784

—

C₄×#672

C₂²

yes

—

#12526

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₂×D₄₀

—

—

—

—

—

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

—

D₄

yes

—

#2248

—

#2241

—

#1713

D₅

yes

characteristic, complemented

#1706

—

#1312

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

D₄₀

—

#677

—

#672

—

C₂×SD₃₂

—

—

—

—

C₂×D₅

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

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

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

SD₃₂

yes

complemented

C₅×C₈

AGL(1,ℤ/8)

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

100

C₂×D₅

—

101

C₂×D₅

—

102

C₂×D₅

—

103

C₄×C₅

—

104

C₄×C₅

#496

yes

—

105

C₄×C₅

C₂×SD₃₂

yes

—

106

C₄×C₅

#510

yes

—

107

Dic₅

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

Q₁₆

—

114

Q₁₆

—

115

SD₁₆

—

116

D₈

—

117

C₂×C₈

#686

yes

—

118

C₂×C₈

#686

yes

—

119

C₂×C₈

D₄×D₅

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

C₂×C₂²⋊D₅

yes

characteristic

125

C₁₆

—

126

C₂×C₅

#1721

yes

characteristic

127

C₂×C₅

#2248

yes

characteristic

128

C₂×C₅

#2251

yes

characteristic

129

D₅

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×#686

yes

characteristic

140

C₂×C₄

C₂×#686

yes

characteristic

141

C₂×C₄

#4086

yes

characteristic

142

C₈

—

143

C₈

#3960

yes

complemented

144

C₈

#4057

yes

—

145

C₅

#12526

yes

characteristic, complemented, Sylow

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#65362

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

#65443

yes

—

153

C₄

C₂×#3960

yes

—

154

C₄

#65468

yes

—

155

C₂

—

160

—

156

C₂

#10602073

yes

characteristic

157

C₂

#10602106

yes

characteristic

158

C₂

#10602107

yes

characteristic

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1713	C₂
socle	C₂²×C₅	#493
prime radical	#64984	C₂²
hypercenter	#1713	D₅
hypocenter	C₅	#12526
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#65362

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂²

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. C₅×#1713	C₂	semidirect
3. #10602103	C₂	semidirect
5. #10597929	C₂	semidirect
36. #1713	D₅	semidirect
81. C₅×C₈	SD₃₂	semidirect
143. C₈	#3960	semidirect
145. C₅	#12526	semidirect

H ≅ G/N

type

1. C₅×#1713

C₂

semidirect

3. #10602103

C₂

semidirect

5. #10597929

C₂

semidirect

36. #1713

D₅

semidirect

81. C₅×C₈

SD₃₂

semidirect

143. C₈

#3960

semidirect

145. C₅

#12526

semidirect

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

subgroup	index
C₅×#1713	2
#10602106	2
#10602103	2
#10598851	2
#10597929	2
#12526	5

subgroup

index

Subgroups — #12014058

Conjugacy classes of subgroups

Complements

Maximal subgroups