Group #12088629 — subgroups

Subgroups — #12088629

Order 1280.

736 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 179 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	#10610271	C₂	1	3	yes	maximal, complemented
2	#10610257	C₂	1	3	yes	maximal
3	C₅×#1455	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598890	C₂	1	6	yes	maximal, complemented
5	#10598854	C₂	1	2	yes	maximal, complemented
6	C₂×#4020	C₂²	1	1	yes	characteristic
7	#65804	C₂²	1	3	yes	—
8	C₄×C₅×M₁₆	C₂²	1	3	yes	—
9	#65426	—	2	6	no	—
10	#65420	—	2	6	no	—
11	#65005	—	2	6	no	—
12	#65002	—	2	6	no	—
13	#64988	C₂²	1	3	yes	complemented
14	#64983	C₂²	1	6	yes	—
15	#64983	C₂²	1	6	yes	—
16	#64970	C₂²	1	6	yes	complemented
17	#64969	C₂²	1	3	yes	complemented
18	C₅×#322	C₂²	1	3	yes	complemented
19	C₅×C₈²	C₂²	1	1	yes	characteristic, complemented
20	#19753	—	5	1	no	maximal, Sylow
21	C₂²×D₂₀	—	2	3	no	—
22	C₂×C₅×M₁₆	D₄	1	6	yes	—
23	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
24	#4072	—	2	3	no	—
25	#4070	—	2	6	no	—
26	C₂×Dic₂₀	—	2	6	no	—
27	#4020	—	2	3	no	—
28	#4020	C₂³	1	2	yes	—
29	#4016	—	2	3	no	—
30	#4015	C₂³	1	6	yes	—
31	C₄×Dic₁₀	—	2	6	no	—
32	C₅×#97	D₄	1	6	yes	—
33	C₄×C₅×C₈	C₂³	1	3	yes	—
34	C₄×C₅×C₈	C₂³	1	3	yes	—
35	#3953	—	2	12	no	—
36	#3953	—	2	12	no	—
37	#3949	—	2	6	no	—
38	#3948	—	2	12	no	—
39	#3185	—	5	3	no	—
40	#3138	—	5	3	no	—
41	#1720	—	5	6	no	—
42	#1711	—	5	2	no	—
43	#1455	D₅	1	1	yes	characteristic, complemented
44	C₂³×D₅	—	4	1	no	—
45	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
46	C₂×D₂₀	—	2	6	no	—
47	C₂×D₂₀	—	4	12	no	—
48	C₂×D₂₀	—	4	3	no	—
49	C₂×Dic₁₀	—	2	6	no	—
50	C₅×M₁₆	—	4	6	no	—
51	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
52	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
53	C₄²×C₅	C₂×D₄	1	3	yes	—
54	C₄²×C₅	C₂⁴	1	1	yes	characteristic
55	C₂²⋊Dic₅	—	4	3	no	—
56	C₄⋊Dic₅	—	2	6	no	—
57	C₄⋊Dic₅	—	2	6	no	—
58	#683	—	4	6	no	—
59	C₄×Dic₅	—	4	3	no	—
60	Dic₂₀	—	4	12	no	—
61	#533	—	5	3	no	—
62	C₂×#127	—	5	1	no	—
63	#498	—	5	3	no	—
64	#494	—	5	3	no	—
65	#490	—	10	6	no	—
66	#487	—	5	6	no	—
67	#486	—	5	6	no	—
68	#486	—	5	6	no	—
69	#470	—	10	6	no	—
70	#441	—	10	6	no	—
71	#418	—	10	6	no	—
72	C₄×M₁₆	C₂×D₅	1	3	yes	—
73	#322	C₂×D₅	1	3	yes	complemented
74	C₈²	C₂×D₅	1	1	yes	characteristic, complemented
75	C₂³×C₅	#127	1	1	yes	characteristic
76	C₂²×D₅	—	4	2	no	—
77	C₂²×D₅	—	8	6	no	—
78	C₂×C₄×C₅	—	2	6	no	—
79	C₂×C₄×C₅	#127	1	3	yes	—
80	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
81	C₂×Dic₅	—	4	6	no	—
82	D₂₀	—	4	12	no	—
83	D₂₀	—	8	6	no	—
84	Dic₁₀	—	4	12	no	—
85	C₅×C₈	—	2	6	no	—
86	C₅×C₈	—	2	6	no	—
87	C₂²×D₄	—	10	3	no	—
88	C₂×Q₁₆	—	10	6	no	—
89	C₂×M₁₆	D₂₀	1	6	yes	—
90	C₄⋊Q₈	—	5	6	no	—
91	#127	—	5	2	no	—
92	#127	—	10	3	no	—
93	#125	—	10	3	no	—
94	C₂²⋊Q₈	—	10	6	no	—
95	C₄×Q₈	—	10	6	no	—
96	#117	—	10	3	no	—
97	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
98	#106	—	10	6	no	—
99	#103	—	10	12	no	—
100	#102	—	10	12	no	—
101	#102	—	10	12	no	—
102	#97	D₂₀	1	6	yes	—
103	C₄×C₈	C₂²×D₅	1	3	yes	—
104	C₄×C₈	C₂²×D₅	1	3	yes	—
105	C₂²×C₅	—	2	3	no	—
106	C₂²×C₅	C₂×#127	1	1	yes	characteristic
107	C₂×D₅	—	8	6	no	—
108	C₂×D₅	—	16	4	no	—
109	C₄×C₅	—	2	3	no	—
110	C₄×C₅	#575	1	6	yes	—
111	Dic₅	—	8	6	no	—
112	C₂⁴	—	20	1	no	—
113	C₂×Q₈	—	10	6	no	—
114	C₂×D₄	—	10	6	no	—
115	C₂×D₄	—	20	12	no	—
116	C₂×D₄	—	20	3	no	—
117	C₂²×C₄	C₂×D₂₀	1	3	yes	—
118	Q₁₆	—	20	12	no	—
119	M₁₆	—	4	6	no	—
120	C₂×C₈	C₂×D₂₀	1	6	yes	—
121	C₂×C₈	C₂×D₂₀	1	6	yes	—
122	C₄⋊C₄	—	10	6	no	—
123	C₄⋊C₄	—	10	6	no	—
124	C₄⋊C₄	—	20	6	no	—
125	C₂²⋊C₄	—	20	3	no	—
126	C₄²	—	20	3	no	—
127	C₄²	C₂×D₂₀	1	3	yes	—
128	C₄²	C₂³×D₅	1	1	yes	characteristic
129	C₂×C₅	—	4	1	no	—
130	C₂×C₅	#3153	1	3	yes	—
131	D₅	—	16	2	no	—
132	C₂³	—	20	2	no	—
133	C₂³	—	40	6	no	—
134	C₂³	#4020	1	1	yes	characteristic
135	Q₈	—	20	12	no	—
136	D₄	—	20	12	no	—
137	D₄	—	40	6	no	—
138	C₂×C₄	—	2	6	no	—
139	C₂×C₄	—	20	6	no	—
140	C₂×C₄	#4020	1	3	yes	—
141	C₂×C₄	C₂²×D₂₀	1	3	yes	—
142	C₈	—	2	6	no	—
143	C₈	—	2	6	no	—
144	C₅	#19753	1	1	yes	characteristic, complemented, Sylow
145	C₂²	—	2	3	no	—
146	C₂²	—	40	6	no	—
147	C₂²	—	80	4	no	—
148	C₂²	C₂×#4020	1	1	yes	characteristic
149	C₄	—	2	3	no	—
150	C₄	—	40	6	no	—
151	C₄	#66082	1	6	yes	—
152	C₂	—	4	1	no	—
153	C₂	—	80	2	no	—
154	C₂	#10610262	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10610271

C₂

yes

maximal, complemented

#10610257

C₂

yes

maximal

C₅×#1455

C₂

yes

maximal, characteristic, complemented

#10598890

C₂

yes

maximal, complemented

#10598854

C₂

yes

maximal, complemented

C₂×#4020

C₂²

yes

characteristic

#65804

C₂²

yes

—

C₄×C₅×M₁₆

C₂²

yes

—

—

—

—

—

C₂²

yes

complemented

#64983

C₂²

yes

—

#64983

C₂²

yes

—

#64970

C₂²

yes

complemented

#64969

C₂²

yes

complemented

C₅×#322

C₂²

yes

complemented

C₅×C₈²

C₂²

yes

characteristic, complemented

#19753

—

maximal, Sylow

C₂²×D₂₀

—

C₂×C₅×M₁₆

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

#4072

—

#4070

—

C₂×Dic₂₀

—

#4020

—

#4020

C₂³

yes

—

#4016

—

#4015

C₂³

yes

—

C₄×Dic₁₀

—

C₅×#97

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

—

—

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂³×D₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂⁴

yes

characteristic

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

Dic₂₀

—

—

—

—

—

—

—

—

—

—

—

—

C₄×M₁₆

C₂×D₅

yes

—

#322

C₂×D₅

yes

complemented

C₈²

C₂×D₅

yes

characteristic, complemented

C₂³×C₅

#127

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

—

C₂×C₄×C₅

C₂²×D₄

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₂²×D₄

—

C₂×Q₁₆

—

C₂×M₁₆

D₂₀

yes

—

C₄⋊Q₈

—

#127

—

#127

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²×D₅

yes

characteristic

#106

—

#103

—

100

#102

—

101

#102

—

102

#97

D₂₀

yes

—

103

C₄×C₈

C₂²×D₅

yes

—

104

C₄×C₈

C₂²×D₅

yes

—

105

C₂²×C₅

—

106

C₂²×C₅

C₂×#127

yes

characteristic

107

C₂×D₅

—

108

C₂×D₅

—

109

C₄×C₅

—

110

C₄×C₅

#575

yes

—

111

Dic₅

—

112

C₂⁴

—

113

C₂×Q₈

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

C₂×D₂₀

yes

—

118

Q₁₆

—

119

M₁₆

—

120

C₂×C₈

C₂×D₂₀

yes

—

121

C₂×C₈

C₂×D₂₀

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₄²

C₂×D₂₀

yes

—

128

C₄²

C₂³×D₅

yes

characteristic

129

C₂×C₅

—

130

C₂×C₅

#3153

yes

—

131

D₅

—

132

C₂³

—

133

C₂³

—

134

C₂³

#4020

yes

characteristic

135

Q₈

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#4020

yes

—

141

C₂×C₄

C₂²×D₂₀

yes

—

142

C₈

—

143

C₈

—

144

C₅

#19753

yes

characteristic, complemented, Sylow

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×#4020

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

#66082

yes

—

152

C₂

—

153

C₂

—

154

C₂

#10610262

yes

—

	subgroup	G/H
center	C₂²	C₂×#4020
commutator, cosocle	C₄²×C₅	C₂⁴
Frattini	C₄²	C₂³×D₅
Fitting	C₅×#1455	C₂
socle	C₂²×C₅	C₂×#127
prime radical	C₂×#4020	C₂²
hypercenter	#1455	D₅
hypocenter	C₅	#19753
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

subgroup

G/H

center

C₂²

C₂×#4020

commutator, cosocle

C₄²×C₅

C₂⁴

Frattini

C₄²

C₂³×D₅

C₂

C₂²×C₅

C₂²

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

N	H ≅ G/N	type
1. #10610271	C₂	semidirect
3. C₅×#1455	C₂	semidirect
4. #10598890	C₂	semidirect
5. #10598854	C₂	semidirect
13. #64988	C₂²	semidirect
16. #64970	C₂²	semidirect
17. #64969	C₂²	semidirect
18. C₅×#322	C₂²	semidirect
19. C₅×C₈²	C₂²	semidirect
43. #1455	D₅	semidirect
73. #322	C₂×D₅	semidirect
74. C₈²	C₂×D₅	semidirect
144. C₅	#19753	semidirect

H ≅ G/N

type

1. #10610271

C₂

semidirect

3. C₅×#1455

C₂

semidirect

4. #10598890

C₂

semidirect

5. #10598854

C₂

semidirect

13. #64988

C₂²

semidirect

16. #64970

C₂²

semidirect

17. #64969

C₂²

semidirect

18. C₅×#322

C₂²

semidirect

19. C₅×C₈²

C₂²

semidirect

43. #1455

D₅

semidirect

73. #322

C₂×D₅

semidirect

74. C₈²

C₂×D₅

semidirect

144. C₅

#19753

semidirect

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

subgroup	index
#10610271	2
#10610257	2
C₅×#1455	2
#10598890	2
#10598854	2
#19753	5

subgroup

index

Subgroups — #12088629

Conjugacy classes of subgroups

Complements

Maximal subgroups