Group #11998803 — subgroups

Subgroups — #11998803

Order 1280.

558 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 119 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×#64984	C₂	1	1	yes	maximal, characteristic
2	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	#10601295	C₂	1	4	yes	maximal, complemented
4	#10600519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₀	—	2	2	no	—
6	C₂×#4020	C₂²	1	2	yes	—
7	C₅×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	#65214	C₂²	1	2	yes	—
10	#64984	—	2	2	no	—
11	#64984	C₄	1	4	yes	—
12	C₅×#341	—	2	2	no	—
13	#64705	—	2	8	no	—
14	#64674	—	2	2	no	—
15	#9780	—	5	1	no	maximal, Sylow
16	C₂²×D₂₀	—	2	2	no	—
17	C₂²×D₂₀	—	2	4	no	—
18	C₂×C₅×M₁₆	—	2	2	no	—
19	C₂²×C₅×C₈	D₄	1	2	yes	—
20	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
21	C₂×D₄₀	—	2	8	no	—
22	C₂×D₄₀	—	4	8	no	—
23	#4020	—	2	4	no	—
24	#4020	—	2	2	no	—
25	#4020	C₂×C₄	1	4	yes	—
26	C₅×C₄⋊C₈	—	2	4	no	—
27	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₅×C₈	D₄	1	2	yes	—
29	C₂²⋊#168	—	2	2	no	—
30	C₄⋊#168	D₄	1	4	yes	—
31	C₄×#168	—	2	2	no	—
32	C₂×#493	—	5	1	no	—
33	#1837	D₅	1	1	yes	characteristic, complemented
34	#1545	—	5	4	no	—
35	#1452	—	5	1	no	—
36	C₂³×D₅	—	4	2	no	—
37	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₅	#41	1	2	yes	—
39	C₂×D₂₀	—	2	4	no	—
40	C₂×D₂₀	—	2	8	no	—
41	C₂×D₂₀	—	4	8	no	—
42	C₂×D₂₀	—	4	4	no	—
43	C₂×D₂₀	—	4	2	no	—
44	C₂×D₂₀	—	8	8	no	—
45	C₅×M₁₆	—	4	4	no	—
46	C₂×C₅×C₈	—	2	4	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
49	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅	C₂×D₄	1	2	yes	—
52	C₂×#168	—	2	4	no	—
53	D₄₀	—	4	16	no	—
54	C₂²×D₈	—	10	2	no	—
55	C₂×#127	—	5	2	no	—
56	#493	—	5	4	no	—
57	#493	—	10	2	no	—
58	#431	—	5	2	no	—
59	#423	C₂×D₅	1	2	yes	—
60	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
61	#341	—	2	2	no	—
62	#331	—	10	8	no	—
63	#322	—	10	2	no	—
64	C₂³×C₅	#124	1	1	yes	characteristic
65	C₂²×D₅	—	4	4	no	—
66	C₂²×D₅	—	8	8	no	—
67	C₂²×D₅	—	8	4	no	—
68	C₂×C₄×C₅	—	2	4	no	—
69	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₅	#121	1	2	yes	—
72	C₂×C₄×C₅	#127	1	1	yes	characteristic
73	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
74	D₂₀	—	4	8	no	—
75	D₂₀	—	8	8	no	—
76	D₂₀	—	8	8	no	—
77	D₂₀	—	8	4	no	—
78	C₅×C₈	—	2	4	no	—
79	C₅×C₈	—	8	2	no	—
80	#168	—	4	4	no	—
81	C₂²×D₄	—	10	4	no	—
82	C₂²×D₄	—	10	2	no	—
83	C₂×D₈	—	10	8	no	—
84	C₂×D₈	—	20	8	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
87	#127	—	5	4	no	—
88	#127	—	10	2	no	—
89	#127	—	10	4	no	—
90	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	5	4	no	—
93	C₂²⋊C₈	—	10	2	no	—
94	C₄×C₈	—	10	2	no	—
95	C₄×C₈	C₄×D₅	1	2	yes	—
96	C₄×C₈	D₂₀	1	2	yes	—
97	C₂²×C₅	—	2	2	no	—
98	C₂²×C₅	#390	1	1	yes	characteristic
99	C₂²×C₅	#490	1	2	yes	—
100	C₂×D₅	—	8	4	no	—
101	C₂×D₅	—	8	8	no	—
102	C₂×D₅	—	16	8	no	—
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	#418	1	2	yes	—
105	C₄×C₅	#472	1	4	yes	—
106	C₄×C₅	#496	1	2	yes	—
107	C₂⁴	—	20	2	no	—
108	C₂×D₄	—	10	8	no	—
109	C₂×D₄	—	10	4	no	—
110	C₂×D₄	—	20	4	no	—
111	C₂×D₄	—	20	2	no	—
112	C₂×D₄	—	20	8	no	—
113	C₂×D₄	—	40	8	no	—
114	C₂²×C₄	#713	1	2	yes	—
115	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
116	D₈	—	20	16	no	—
117	M₁₆	—	4	4	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	10	4	no	—
121	C₂×C₈	#685	1	4	yes	—
122	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
123	C₄²	C₂×D₂₀	1	1	yes	characteristic
124	C₄²	D₄×D₅	1	2	yes	—
125	C₂×C₅	—	2	2	no	—
126	C₂×C₅	#1961	1	1	yes	characteristic
127	C₂×C₅	#1978	1	2	yes	—
128	D₅	—	16	4	no	—
129	C₂³	—	20	4	no	—
130	C₂³	—	40	8	no	—
131	C₂³	—	40	4	no	—
132	C₂³	#4093	1	1	yes	characteristic
133	D₄	—	20	8	no	—
134	D₄	—	40	8	no	—
135	D₄	—	40	4	no	—
136	D₄	—	40	8	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	#4039	1	2	yes	—
139	C₂×C₄	#4041	1	2	yes	—
140	C₂×C₄	C₂×#685	1	1	yes	characteristic
141	C₂×C₄	#4086	1	1	yes	characteristic
142	C₂×C₄	#4095	1	4	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	8	2	no	—
145	C₈	—	20	4	no	—
146	C₅	#9780	1	1	yes	characteristic, complemented, Sylow
147	C₂²	—	2	2	no	—
148	C₂²	—	40	8	no	—
149	C₂²	—	40	4	no	—
150	C₂²	—	80	8	no	—
151	C₂²	#65277	1	1	yes	characteristic
152	C₂²	#65356	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#65186	1	4	yes	—
155	C₄	#65290	1	2	yes	—
156	C₄	#65347	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	80	4	no	—
159	C₂	#10600346	1	1	yes	characteristic
160	C₂	#10601502	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64984

C₂

yes

maximal, characteristic

C₅×#1837

C₂

yes

maximal, characteristic, complemented

#10601295

C₂

yes

maximal, complemented

#10600519

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₀

—

C₂×#4020

C₂²

yes

—

C₅×#423

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

#65214

C₂²

yes

—

#64984

—

#64984

C₄

yes

—

—

—

—

—

C₂²×D₂₀

—

C₂²×D₂₀

—

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

#4020

—

#4020

C₂×C₄

yes

—

C₅×C₄⋊C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₂²⋊#168

—

C₄⋊#168

D₄

yes

—

C₄×#168

—

C₂×#493

—

#1837

D₅

yes

characteristic, complemented

#1545

—

#1452

—

C₂³×D₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₂×#168

—

D₄₀

—

C₂²×D₈

—

—

—

—

—

C₂×D₅

yes

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₂³×C₅

#124

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

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

AGL(1,ℤ/8)

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₅

yes

—

#127

—

#127

—

#127

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅

yes

—

C₄×C₈

D₂₀

yes

—

C₂²×C₅

—

C₂²×C₅

#390

yes

characteristic

C₂²×C₅

#490

yes

—

100

C₂×D₅

—

101

C₂×D₅

—

102

C₂×D₅

—

103

C₄×C₅

—

104

C₄×C₅

#418

yes

—

105

C₄×C₅

#472

yes

—

106

C₄×C₅

#496

yes

—

107

C₂⁴

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

#713

yes

—

115

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

116

D₈

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

#685

yes

—

122

C₄²

C₂×C₄×D₅

yes

characteristic

123

C₄²

C₂×D₂₀

yes

characteristic

124

C₄²

D₄×D₅

yes

—

125

C₂×C₅

—

126

C₂×C₅

#1961

yes

characteristic

127

C₂×C₅

#1978

yes

—

128

D₅

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

#4093

yes

characteristic

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

#4039

yes

—

139

C₂×C₄

#4041

yes

—

140

C₂×C₄

C₂×#685

yes

characteristic

141

C₂×C₄

#4086

yes

characteristic

142

C₂×C₄

#4095

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₅

#9780

yes

characteristic, complemented, Sylow

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#65277

yes

characteristic

152

C₂²

#65356

yes

—

153

C₄

—

154

C₄

#65186

yes

—

155

C₄

#65290

yes

—

156

C₄

#65347

yes

—

157

C₂

—

158

C₂

—

159

C₂

#10600346

yes

characteristic

160

C₂

#10601502

yes

—

	subgroup	G/H
center	C₂²	#65277
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1837	C₂
socle	C₂²×C₅	#390
prime radical	C₂×#64984	C₂
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1837	D₅
hypocenter	C₅	#9780
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂²

#65277

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
2. C₅×#1837	C₂	semidirect
3. #10601295	C₂	semidirect
4. #10600519	C₂	semidirect
33. #1837	D₅	semidirect
146. C₅	#9780	semidirect

H ≅ G/N

type

2. C₅×#1837

C₂

semidirect

3. #10601295

C₂

semidirect

4. #10600519

C₂

semidirect

33. #1837

D₅

semidirect

146. C₅

#9780

semidirect

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

subgroup	index
C₂×#64984	2
C₅×#1837	2
#10601295	2
#10600519	2
#9780	5

subgroup

index

Subgroups — #11998803

Conjugacy classes of subgroups

Complements

Maximal subgroups