Group #12014001 — subgroups

Subgroups — #12014001

Order 1280.

268 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 79 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	#10602175	C₂	1	2	yes	maximal, complemented
3	#10602174	C₂	1	2	yes	maximal
4	#10598853	C₂	1	1	yes	maximal, characteristic
5	#10597928	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#500	—	2	1	no	—
7	C₅×#500	C₂²	1	2	yes	—
8	C₅×#494	—	2	1	no	—
9	#64988	C₂²	1	1	yes	characteristic
10	#64984	C₂²	1	1	yes	characteristic
11	#64983	—	2	1	no	—
12	#64983	—	2	1	no	—
13	C₅×C₈²	C₂²	1	1	yes	characteristic
14	#64712	—	2	4	no	—
15	#64711	—	2	4	no	—
16	C₄×#672	C₂²	1	2	yes	—
17	#12512	—	5	1	no	maximal, Sylow
18	C₂×C₅×Q₁₆	—	4	2	no	—
19	C₅×C₄⋊Q₈	—	2	2	no	—
20	C₂×Dic₂₀	—	2	2	no	—
21	C₂×D₄₀	—	2	2	no	—
22	#4020	—	2	1	no	—
23	#4015	—	2	1	no	—
24	C₅×#107	—	2	2	no	—
25	C₅×#107	—	2	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	#3953	—	4	2	no	—
31	#3953	—	4	2	no	—
32	C₂×#672	D₄	1	4	yes	—
33	#2241	—	5	2	no	—
34	#2240	—	5	2	no	—
35	#1713	D₅	1	1	yes	characteristic, complemented
36	#1709	—	5	1	no	—
37	#1311	—	5	1	no	—
38	C₂×C₅×Q₈	—	4	2	no	—
39	C₂×D₂₀	—	4	1	no	—
40	C₂×D₂₀	—	4	1	no	—
41	C₂×D₂₀	—	4	1	no	—
42	C₂×Dic₁₀	—	4	1	no	—
43	C₅×Q₁₆	—	8	4	no	—
44	C₂×C₅×C₈	D₈	1	2	yes	—
45	C₂×C₅×C₈	D₈	1	2	yes	—
46	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
47	C₅×C₄⋊C₄	—	4	2	no	—
48	C₅×C₄⋊C₄	—	4	2	no	—
49	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
50	C₄⋊Dic₅	—	4	1	no	—
51	C₄⋊Dic₅	—	4	1	no	—
52	Dic₂₀	—	4	4	no	—
53	D₄₀	—	4	4	no	—
54	#672	—	2	4	no	—
55	#500	—	2	1	no	—
56	#500	C₂×D₅	1	2	yes	—
57	#494	—	2	1	no	—
58	#494	—	5	1	no	—
59	#493	—	5	1	no	—
60	#486	—	10	1	no	—
61	#486	—	10	1	no	—
62	#358	—	10	4	no	—
63	#357	—	10	4	no	—
64	C₄×C₁₆	—	5	2	no	—
65	C₈²	C₂×D₅	1	1	yes	characteristic
66	C₂²×D₅	—	8	1	no	—
67	C₅×Q₈	—	8	4	no	—
68	C₂×C₄×C₅	—	8	2	no	—
69	C₂×C₄×C₅	#127	1	1	yes	characteristic
70	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
72	C₂×Dic₅	—	8	1	no	—
73	D₂₀	—	8	2	no	—
74	D₂₀	—	8	2	no	—
75	D₂₀	—	8	2	no	—
76	Dic₁₀	—	8	2	no	—
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	—	2	2	no	—
79	C₅×C₈	#135	1	4	yes	—
80	C₂×Q₁₆	—	4	2	no	—
81	C₂×Q₁₆	—	10	2	no	—
82	C₂×D₈	—	10	2	no	—
83	C₄⋊Q₈	—	2	2	no	—
84	C₄⋊Q₈	—	10	1	no	—
85	#127	—	10	1	no	—
86	C₂×C₁₆	—	5	4	no	—
87	#107	—	2	2	no	—
88	#107	—	2	2	no	—
89	#107	—	4	2	no	—
90	#102	—	20	2	no	—
91	#102	—	20	2	no	—
92	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
94	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
95	C₂²×C₅	#493	1	1	yes	characteristic
96	C₂×D₅	—	16	1	no	—
97	C₂×D₅	—	16	1	no	—
98	C₂×D₅	—	16	1	no	—
99	C₄×C₅	—	16	2	no	—
100	C₄×C₅	#495	1	2	yes	—
101	C₄×C₅	#509	1	2	yes	—
102	C₄×C₅	#510	1	2	yes	—
103	Dic₅	—	16	1	no	—
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	—	20	1	no	—
106	C₂×D₄	—	20	1	no	—
107	C₂×D₄	—	20	1	no	—
108	C₂×D₄	—	20	1	no	—
109	Q₁₆	—	8	4	no	—
110	Q₁₆	—	20	4	no	—
111	D₈	—	20	4	no	—
112	C₂×C₈	#686	1	2	yes	—
113	C₂×C₈	#686	1	2	yes	—
114	C₂×C₈	D₄×D₅	1	2	yes	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	20	1	no	—
118	C₄⋊C₄	—	20	1	no	—
119	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
120	C₁₆	—	10	4	no	—
121	C₂×C₅	#1709	1	1	yes	characteristic
122	C₂×C₅	#2250	1	1	yes	characteristic
123	C₂×C₅	#2251	1	1	yes	characteristic
124	D₅	—	32	1	no	—
125	C₂³	—	40	1	no	—
126	Q₈	—	8	4	no	—
127	Q₈	—	40	2	no	—
128	D₄	—	40	2	no	—
129	D₄	—	40	2	no	—
130	D₄	—	40	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	40	1	no	—
133	C₂×C₄	C₂×#686	1	1	yes	characteristic
134	C₂×C₄	C₂×#686	1	1	yes	characteristic
135	C₂×C₄	#4086	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	#4065	1	4	yes	—
139	C₅	#12512	1	1	yes	characteristic, complemented, Sylow
140	C₂²	—	80	1	no	—
141	C₂²	—	80	1	no	—
142	C₂²	—	80	1	no	—
143	C₂²	#65362	1	1	yes	characteristic
144	C₄	—	16	2	no	—
145	C₄	—	80	1	no	—
146	C₄	#65436	1	2	yes	—
147	C₄	#65468	1	2	yes	—
148	C₄	#65480	1	2	yes	—
149	C₂	—	160	1	no	—
150	C₂	#10602085	1	1	yes	characteristic
151	C₂	#10602090	1	1	yes	characteristic
152	C₂	#10602107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1713

C₂

yes

maximal, characteristic, complemented

#10602175

C₂

yes

maximal, complemented

#10602174

C₂

yes

maximal

#10598853

C₂

yes

maximal, characteristic

#10597928

C₂

yes

maximal, characteristic, complemented

C₅×#500

—

C₅×#500

C₂²

yes

—

C₅×#494

—

#64988

C₂²

yes

characteristic

#64984

C₂²

yes

characteristic

#64983

—

#64983

—

C₅×C₈²

C₂²

yes

—

—

C₂²

yes

—

#12512

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

C₂×D₄₀

—

—

—

—

—

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

D₄

yes

—

#2241

—

#2240

—

#1713

D₅

yes

characteristic, complemented

#1709

—

#1311

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

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

—

Dic₂₀

—

D₄₀

—

#672

—

#500

—

#500

C₂×D₅

yes

—

—

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

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

—

C₅×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

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

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

100

C₄×C₅

#495

yes

—

101

C₄×C₅

#509

yes

—

102

C₄×C₅

#510

yes

—

103

Dic₅

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

Q₁₆

—

110

Q₁₆

—

111

D₈

—

112

C₂×C₈

#686

yes

—

113

C₂×C₈

#686

yes

—

114

C₂×C₈

D₄×D₅

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂×C₂²⋊D₅

yes

characteristic

120

C₁₆

—

121

C₂×C₅

#1709

yes

characteristic

122

C₂×C₅

#2250

yes

characteristic

123

C₂×C₅

#2251

yes

characteristic

124

D₅

—

125

C₂³

—

126

Q₈

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×#686

yes

characteristic

134

C₂×C₄

C₂×#686

yes

characteristic

135

C₂×C₄

#4086

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

#4065

yes

—

139

C₅

#12512

yes

characteristic, complemented, Sylow

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#65362

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

#65436

yes

—

147

C₄

#65468

yes

—

148

C₄

#65480

yes

—

149

C₂

—

160

—

150

C₂

#10602085

yes

characteristic

151

C₂

#10602090

yes

characteristic

152

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	#12512
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
2. #10602175	C₂	semidirect
5. #10597928	C₂	semidirect
35. #1713	D₅	semidirect
139. C₅	#12512	semidirect

H ≅ G/N

type

1. C₅×#1713

C₂

semidirect

2. #10602175

C₂

semidirect

5. #10597928

C₂

semidirect

35. #1713

D₅

semidirect

139. C₅

#12512

semidirect

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

subgroup	index
C₅×#1713	2
#10602175	2
#10602174	2
#10598853	2
#10597928	2
#12512	5

subgroup

index

Subgroups — #12014001

Conjugacy classes of subgroups

Complements

Maximal subgroups