Group #11998800 — subgroups

Subgroups — #11998800

Order 1280.

454 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 123 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×#64988	C₂	1	1	yes	maximal, characteristic
2	C₅×#1452	C₂	1	1	yes	maximal, characteristic
3	#10601571	C₂	1	4	yes	maximal
4	#10600584	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀	—	2	2	no	—
6	C₂×#4015	C₂²	1	2	yes	—
7	C₅×#431	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	#65213	C₂²	1	2	yes	—
10	#64988	—	2	2	no	—
11	#64988	C₄	1	4	yes	—
12	C₅×#322	—	2	2	no	—
13	#64772	—	2	2	no	—
14	#64706	—	2	8	no	—
15	#9779	—	5	1	no	maximal, Sylow
16	C₂²×Dic₁₀	—	2	2	no	—
17	C₂²×C₅×C₈	D₄	1	2	yes	—
18	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅	—	2	4	no	—
20	C₂×#681	—	2	2	no	—
21	C₂×Dic₂₀	—	2	8	no	—
22	C₂×Dic₂₀	—	4	8	no	—
23	#4015	—	2	4	no	—
24	#4015	—	2	2	no	—
25	#4015	C₂×C₄	1	4	yes	—
26	C₅×C₄⋊C₈	D₄	1	4	yes	—
27	C₅×C₂²⋊C₈	—	2	2	no	—
28	C₄×C₅×C₈	—	2	2	no	—
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₅×C₈	D₄	1	2	yes	—
31	C₄⋊#168	—	2	4	no	—
32	C₂×#494	—	5	1	no	—
33	#1837	—	5	1	no	—
34	#1546	—	5	4	no	—
35	#1452	D₅	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₅	#41	1	2	yes	—
38	C₂²×Dic₅	—	4	2	no	—
39	C₂×Dic₁₀	—	2	4	no	—
40	C₂×Dic₁₀	—	4	8	no	—
41	C₂×Dic₁₀	—	4	2	no	—
42	C₂×C₅×C₈	—	2	4	no	—
43	C₂×C₅×C₈	—	2	4	no	—
44	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
45	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₅	C₂×D₄	1	2	yes	—
47	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₅	—	2	8	no	—
49	C₄⋊Dic₅	—	4	4	no	—
50	#681	—	4	4	no	—
51	C₂×#168	—	4	2	no	—
52	Dic₂₀	—	4	16	no	—
53	C₂²×Q₁₆	—	10	2	no	—
54	C₂×C₄⋊Q₈	—	5	2	no	—
55	#494	—	5	4	no	—
56	#494	—	10	2	no	—
57	#431	C₂×D₅	1	2	yes	—
58	#423	—	5	2	no	—
59	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
60	#341	—	10	2	no	—
61	#332	—	10	8	no	—
62	#322	—	2	2	no	—
63	C₂³×C₅	#124	1	1	yes	characteristic
64	C₂×C₄×C₅	—	2	4	no	—
65	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₅	#121	1	2	yes	—
68	C₂×C₄×C₅	#127	1	1	yes	characteristic
69	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
70	C₂×Dic₅	—	4	4	no	—
71	C₂×Dic₅	—	8	4	no	—
72	Dic₁₀	—	4	8	no	—
73	Dic₁₀	—	8	4	no	—
74	C₅×C₈	—	2	4	no	—
75	C₅×C₈	—	4	4	no	—
76	#168	—	8	2	no	—
77	C₂²×Q₈	—	10	2	no	—
78	C₂×Q₁₆	—	10	8	no	—
79	C₂×Q₁₆	—	20	8	no	—
80	C₂×M₁₆	—	10	2	no	—
81	C₂²×C₈	D₂₀	1	2	yes	—
82	C₄⋊Q₈	—	5	4	no	—
83	C₄⋊Q₈	—	10	4	no	—
84	C₄⋊Q₈	—	10	2	no	—
85	C₂×C₄⋊C₄	—	10	4	no	—
86	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
87	C₄⋊C₈	—	10	4	no	—
88	C₄⋊C₈	D₂₀	1	4	yes	—
89	C₂²⋊C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	C₄×D₅	1	2	yes	—
92	C₄×C₈	C₂²⋊D₅	1	2	yes	—
93	C₂²×C₅	—	2	2	no	—
94	C₂²×C₅	#390	1	1	yes	characteristic
95	C₂²×C₅	#490	1	2	yes	—
96	C₄×C₅	—	2	2	no	—
97	C₄×C₅	#418	1	2	yes	—
98	C₄×C₅	#473	1	4	yes	—
99	C₄×C₅	#496	1	2	yes	—
100	Dic₅	—	8	4	no	—
101	C₂×Q₈	—	10	4	no	—
102	C₂×Q₈	—	20	2	no	—
103	C₂×Q₈	—	20	8	no	—
104	C₂²×C₄	—	20	2	no	—
105	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
106	C₂²×C₄	#709	1	2	yes	—
107	Q₁₆	—	20	16	no	—
108	M₁₆	—	20	4	no	—
109	C₂×C₈	—	2	4	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	20	2	no	—
112	C₂×C₈	#685	1	4	yes	—
113	C₄⋊C₄	—	10	8	no	—
114	C₄⋊C₄	—	20	4	no	—
115	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
116	C₄²	C₂×D₂₀	1	2	yes	—
117	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
118	C₂×C₅	—	2	2	no	—
119	C₂×C₅	#1961	1	1	yes	characteristic
120	C₂×C₅	#1979	1	2	yes	—
121	C₂³	#4021	1	1	yes	characteristic
122	Q₈	—	20	8	no	—
123	Q₈	—	40	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	20	4	no	—
126	C₂×C₄	—	40	4	no	—
127	C₂×C₄	C₄×D₂₀	1	2	yes	—
128	C₂×C₄	#4020	1	1	yes	characteristic
129	C₂×C₄	#4054	1	4	yes	—
130	C₂×C₄	C₂×#685	1	1	yes	characteristic
131	C₂×C₄	#4076	1	2	yes	—
132	C₈	—	2	4	no	—
133	C₈	—	4	4	no	—
134	C₈	—	40	2	no	—
135	C₅	#9779	1	1	yes	characteristic, complemented, Sylow
136	C₂²	—	2	2	no	—
137	C₂²	#65002	1	2	yes	—
138	C₂²	#65228	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	40	4	no	—
141	C₄	#65001	1	2	yes	—
142	C₄	#65420	1	2	yes	—
143	C₄	#65432	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	#10600616	1	1	yes	characteristic
146	C₂	#10600697	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64988

C₂

yes

maximal, characteristic

C₅×#1452

C₂

yes

maximal, characteristic

#10601571

C₂

yes

maximal

#10600584

C₂

yes

maximal, characteristic

C₂²×Dic₂₀

—

C₂×#4015

C₂²

yes

—

C₅×#431

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

#65213

C₂²

yes

—

#64988

—

#64988

C₄

yes

—

—

—

—

—

C₂²×Dic₁₀

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×#681

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

#4015

—

#4015

C₂×C₄

yes

—

C₅×C₄⋊C₈

D₄

yes

—

C₅×C₂²⋊C₈

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

D₅

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#681

—

C₂×#168

—

Dic₂₀

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

#431

C₂×D₅

yes

—

#423

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₂³×C₅

#124

yes

characteristic

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

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₂₀

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

D₂₀

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₅

yes

—

C₄×C₈

C₂²⋊D₅

yes

—

C₂²×C₅

—

C₂²×C₅

#390

yes

characteristic

C₂²×C₅

#490

yes

—

C₄×C₅

—

C₄×C₅

#418

yes

—

C₄×C₅

#473

yes

—

C₄×C₅

#496

yes

—

100

Dic₅

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂×D₂₀

yes

characteristic

106

C₂²×C₄

#709

yes

—

107

Q₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

#685

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂×C₄×D₅

yes

characteristic

116

C₄²

C₂×D₂₀

yes

—

117

C₄²

C₂×C₂²⋊D₅

yes

characteristic

118

C₂×C₅

—

119

C₂×C₅

#1961

yes

characteristic

120

C₂×C₅

#1979

yes

—

121

C₂³

#4021

yes

characteristic

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₄×D₂₀

yes

—

128

C₂×C₄

#4020

yes

characteristic

129

C₂×C₄

#4054

yes

—

130

C₂×C₄

C₂×#685

yes

characteristic

131

C₂×C₄

#4076

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₅

#9779

yes

characteristic, complemented, Sylow

136

C₂²

—

137

C₂²

#65002

yes

—

138

C₂²

#65228

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#65001

yes

—

142

C₄

#65420

yes

—

143

C₄

#65432

yes

—

144

C₂

—

145

C₂

#10600616

yes

characteristic

146

C₂

#10600697

yes

—

	subgroup	G/H
center	C₂²	#65228
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1452	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1452	D₅
hypocenter	C₅	#9779
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#65228

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 ◁ #1452

N	H ≅ G/N	type
135. C₅	#9779	semidirect

H ≅ G/N

type

135. C₅

#9779

semidirect

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

subgroup	index
C₂×#64988	2
C₅×#1452	2
#10601571	2
#10600584	2
#9779	5

subgroup

index

Subgroups — #11998800

Conjugacy classes of subgroups

Complements

Maximal subgroups