Group #11998691 — subgroups

Subgroups — #11998691

Order 1280.

370 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 111 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	#10610173	C₂	1	1	yes	maximal, characteristic
2	C₅×#1835	C₂	1	1	yes	maximal, characteristic
3	#10600519	C₂	1	1	yes	maximal, characteristic
4	#10600339	C₂	1	4	yes	maximal
5	#65804	C₂²	1	2	yes	—
6	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₅×#382	C₂²	1	2	yes	—
8	#65395	—	2	2	no	—
9	#65214	C₂²	1	2	yes	—
10	#64972	—	2	2	no	—
11	#64971	C₄	1	2	yes	complemented
12	#64969	C₄	1	2	yes	complemented
13	C₅×#340	—	2	2	no	—
14	#64674	—	2	2	no	—
15	#64672	—	2	8	no	—
16	#9759	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	2	yes	—
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
20	#4072	—	2	2	no	—
21	#4070	—	2	4	no	—
22	#4016	—	2	2	no	—
23	#4015	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₀	—	2	4	no	—
25	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₅×C₈	D₄	1	2	yes	—
28	C₅×#95	—	2	4	no	—
29	C₂²⋊#168	D₄	1	4	yes	—
30	#3950	—	2	4	no	—
31	#3949	—	2	4	no	—
32	C₄⋊#168	—	2	2	no	—
33	C₄×#168	—	2	2	no	—
34	#3160	—	5	1	no	—
35	#1835	D₅	1	1	yes	characteristic
36	#1518	—	5	4	no	—
37	#1452	—	5	1	no	—
38	C₂²×C₄×C₅	—	4	2	no	—
39	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀	—	2	4	no	—
42	C₂×C₅×C₈	—	2	4	no	—
43	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
44	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
45	C₅×C₄⋊C₄	—	4	4	no	—
46	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₅	#41	1	2	yes	—
49	C₂²⋊Dic₅	—	4	2	no	—
50	C₄⋊Dic₅	—	2	4	no	—
51	C₄⋊Dic₅	—	2	4	no	—
52	#683	—	4	4	no	—
53	C₄×Dic₅	—	4	2	no	—
54	C₂×#168	—	2	4	no	—
55	#533	—	5	2	no	—
56	#500	—	5	2	no	—
57	#499	—	10	2	no	—
58	#498	—	5	2	no	—
59	#431	—	5	2	no	—
60	#427	—	10	2	no	—
61	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
62	#382	C₂×D₅	1	2	yes	—
63	#340	—	2	2	no	—
64	#328	—	10	8	no	—
65	#322	—	10	2	no	—
66	C₂³×C₅	#127	1	1	yes	characteristic
67	C₂×C₄×C₅	—	4	2	no	—
68	C₂×C₄×C₅	—	8	4	no	—
69	C₂×C₄×C₅	—	8	2	no	—
70	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
72	C₂×C₄×C₅	#121	1	2	yes	—
73	C₂×C₄×C₅	#124	1	1	yes	characteristic
74	C₂×C₄×C₅	#137	1	4	yes	—
75	C₂×Dic₅	—	4	4	no	—
76	Dic₁₀	—	8	4	no	—
77	C₅×C₈	—	2	2	no	—
78	C₅×C₈	—	2	2	no	—
79	#168	—	4	4	no	—
80	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
81	C₄⋊Q₈	—	5	4	no	—
82	#125	—	10	2	no	—
83	C₂²⋊Q₈	—	10	4	no	—
84	C₄×Q₈	—	10	4	no	—
85	#117	—	10	2	no	—
86	C₂×C₄⋊C₄	—	2	2	no	—
87	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
88	#107	—	10	4	no	—
89	#106	—	10	4	no	—
90	C₄⋊C₈	—	10	2	no	—
91	C₂²⋊C₈	—	5	4	no	—
92	C₄×C₈	—	10	2	no	—
93	C₄×C₈	C₄×D₅	1	1	yes	characteristic
94	C₄×C₈	C₄×D₅	1	1	yes	characteristic
95	C₄×C₈	D₂₀	1	2	yes	—
96	#95	—	2	4	no	—
97	C₂²×C₅	—	2	2	no	—
98	C₂²×C₅	#390	1	1	yes	characteristic
99	C₂²×C₅	#497	1	2	yes	—
100	C₄×C₅	—	2	2	no	—
101	C₄×C₅	—	2	2	no	—
102	C₄×C₅	—	16	2	no	—
103	C₄×C₅	#419	1	2	yes	—
104	C₄×C₅	#491	1	2	yes	—
105	Dic₅	—	8	4	no	—
106	C₂×Q₈	—	10	4	no	—
107	C₂²×C₄	—	4	2	no	—
108	C₂²×C₄	D₄×D₅	1	2	yes	—
109	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	10	4	no	—
112	C₂×C₈	#685	1	2	yes	—
113	C₂×C₈	#685	1	2	yes	—
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	10	4	no	—
116	C₄⋊C₄	—	10	4	no	—
117	C₄⋊C₄	—	20	4	no	—
118	C₂²⋊C₄	—	20	2	no	—
119	C₄²	—	20	2	no	—
120	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
121	C₄²	C₂×D₂₀	1	1	yes	characteristic
122	C₄²	#713	1	2	yes	—
123	C₂×C₅	—	2	2	no	—
124	C₂×C₅	#1960	1	1	yes	characteristic
125	C₂×C₅	#1976	1	2	yes	—
126	C₂³	#4086	1	1	yes	characteristic
127	Q₈	—	40	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	—	20	4	no	—
132	C₂×C₄	#4039	1	2	yes	—
133	C₂×C₄	#4041	1	2	yes	—
134	C₂×C₄	C₂×#685	1	1	yes	characteristic
135	C₂×C₄	#4093	1	1	yes	characteristic
136	C₂×C₄	#4097	1	4	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	—	20	4	no	—
140	C₅	#9759	1	1	yes	characteristic, complemented, Sylow
141	C₂²	—	2	2	no	—
142	C₂²	#65277	1	1	yes	characteristic
143	C₂²	#65349	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	16	2	no	—
147	C₄	—	40	4	no	—
148	C₄	#65293	1	2	yes	—
149	C₄	#65358	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#10600345	1	1	yes	characteristic
152	C₂	#10601451	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10610173

C₂

yes

maximal, characteristic

C₅×#1835

C₂

yes

maximal, characteristic

#10600519

C₂

yes

maximal, characteristic

#10600339

C₂

yes

maximal

#65804

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#382

C₂²

yes

—

#65395

—

#65214

C₂²

yes

—

#64972

—

#64971

C₄

yes

complemented

#64969

C₄

yes

—

—

—

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₄×Dic₁₀

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

—

C₅×#95

—

C₂²⋊#168

D₄

yes

—

—

—

—

—

—

D₅

yes

characteristic

#1518

—

#1452

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

#41

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#382

C₂×D₅

yes

—

#340

—

#328

—

#322

—

C₂³×C₅

#127

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

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

#124

yes

characteristic

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅

yes

characteristic

C₄×C₈

C₄×D₅

yes

characteristic

C₄×C₈

D₂₀

yes

—

#95

—

C₂²×C₅

—

C₂²×C₅

#390

yes

characteristic

C₂²×C₅

#497

yes

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

#419

yes

—

104

C₄×C₅

#491

yes

—

105

Dic₅

—

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

D₄×D₅

yes

—

109

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

#685

yes

—

113

C₂×C₈

#685

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

C₂×C₄×D₅

yes

characteristic

121

C₄²

C₂×D₂₀

yes

characteristic

122

C₄²

#713

yes

—

123

C₂×C₅

—

124

C₂×C₅

#1960

yes

characteristic

125

C₂×C₅

#1976

yes

—

126

C₂³

#4086

yes

characteristic

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#4039

yes

—

133

C₂×C₄

#4041

yes

—

134

C₂×C₄

C₂×#685

yes

characteristic

135

C₂×C₄

#4093

yes

characteristic

136

C₂×C₄

#4097

yes

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₅

#9759

yes

characteristic, complemented, Sylow

141

C₂²

—

142

C₂²

#65277

yes

characteristic

143

C₂²

#65349

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#65293

yes

—

149

C₄

#65358

yes

—

150

C₂

—

151

C₂

#10600345

yes

characteristic

152

C₂

#10601451

yes

—

	subgroup	G/H
center	C₂²	#65277
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1835	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1835	D₅
hypocenter	C₅	#9759
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#65277

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
11. #64971	C₄	semidirect
12. #64969	C₄	semidirect
140. C₅	#9759	semidirect

H ≅ G/N

type

11. #64971

C₄

semidirect

12. #64969

C₄

semidirect

140. C₅

#9759

semidirect

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

subgroup	index
#10610173	2
C₅×#1835	2
#10600519	2
#10600339	2
#9759	5

subgroup

index

Subgroups — #11998691

Conjugacy classes of subgroups

Complements

Maximal subgroups