Group #72223036 — subgroups

Subgroups — #72223036

Order 3200.

880 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 215 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	C₅²×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	#13102952	C₂	1	2	yes	maximal, complemented
3	#13102949	C₂	1	2	yes	maximal, complemented
4	#13102558	C₂	1	1	yes	maximal, characteristic
5	#13099489	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅²×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#87811	—	2	4	no	—
8	C₂×#87801	—	2	2	no	—
9	#11717194	C₂²	1	1	yes	characteristic
10	#11717194	C₂²	1	1	yes	characteristic
11	C₅²×#106	—	2	2	no	—
12	C₄×C₅²×C₈	C₂²	1	1	yes	characteristic
13	#11716676	—	2	4	no	—
14	C₄×#6029	C₂²	1	2	yes	—
15	#10602129	—	5	6	no	maximal
16	C₂×C₅²×Q₈	—	2	2	no	—
17	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
18	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
19	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
20	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
21	C₂×C₅²×C₈	D₄	1	2	yes	—
22	C₅²×C₄⋊C₄	—	2	2	no	—
23	C₅²×C₄⋊C₄	—	2	2	no	—
24	C₄²×C₅²	C₂³	1	1	yes	characteristic
25	#87811	—	4	8	no	—
26	C₂×#6029	D₄	1	4	yes	—
27	#87801	—	4	4	no	—
28	C₅×#498	D₅	1	6	yes	complemented
29	#65377	—	5	12	no	—
30	#65374	—	5	12	no	—
31	#64984	—	5	6	no	—
32	#64675	—	5	6	no	—
33	C₂²×C₅⋊D₅	—	4	1	no	—
34	C₂²×C₅⋊D₅	—	4	1	no	—
35	C₅²×Q₈	—	4	4	no	—
36	C₂×C₄×C₅²	—	4	2	no	—
37	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅²	C₂×D₄	1	2	yes	—
39	C₄⋊(C₅⋊D₅)	—	4	2	no	—
40	C₄⋊(C₅⋊D₅)	—	4	2	no	—
41	C₄⋊(C₅⋊D₅)	—	4	4	no	—
42	C₄⋊(C₅⋊D₅)	—	4	4	no	—
43	C₅²×C₈	SD₁₆	1	4	yes	complemented
44	#6029	—	2	4	no	—
45	C₅×C₄⋊Q₈	C₂×D₅	1	12	yes	—
46	C₂×#688	—	10	24	no	—
47	C₂×D₄₀	—	10	12	no	—
48	#4020	—	5	6	no	—
49	#4020	—	5	6	no	—
50	C₅×#106	—	2	12	no	—
51	C₄×C₅×C₈	C₂×D₅	1	6	yes	—
52	#3941	—	10	24	no	—
53	C₄×#168	—	5	12	no	—
54	#1715	—	25	1	no	Sylow
55	C₂²×C₅²	#127	1	1	yes	characteristic
56	C₂×C₅⋊D₅	—	8	1	no	—
57	C₂×C₅⋊D₅	—	8	1	no	—
58	C₂×C₅⋊D₅	—	8	2	no	—
59	C₂×C₅⋊D₅	—	8	2	no	—
60	C₄×C₅²	—	8	2	no	—
61	C₄×C₅²	C₂×SD₁₆	1	2	yes	—
62	C₄×C₅²	AGL(1,ℤ/8)	1	4	yes	—
63	C₂×C₅×Q₈	—	2	12	no	—
64	C₂×D₂₀	—	10	12	no	—
65	C₂×D₂₀	—	10	12	no	—
66	C₂×D₂₀	—	10	6	no	—
67	C₂×D₂₀	—	10	6	no	—
68	C₂×C₅×C₈	C₂²⋊D₅	1	12	yes	—
69	C₅×C₄⋊C₄	—	2	12	no	—
70	C₅×C₄⋊C₄	—	2	12	no	—
71	C₄²×C₅	C₂²×D₅	1	6	yes	—
72	#688	—	20	48	no	—
73	C₂×#168	—	5	24	no	—
74	D₄₀	—	20	24	no	—
75	#498	C₅⋊D₅	1	1	yes	characteristic, complemented
76	#493	—	25	1	no	—
77	#492	—	25	2	no	—
78	#486	—	25	2	no	—
79	#322	—	25	1	no	—
80	C₂×C₅²	#492	1	1	yes	characteristic
81	C₂×C₅²	#496	1	2	yes	—
82	C₅⋊D₅	—	16	1	no	—
83	C₅⋊D₅	—	16	1	no	—
84	C₂²×D₅	—	20	6	no	—
85	C₂²×D₅	—	20	6	no	—
86	C₅×Q₈	—	4	24	no	—
87	C₂×C₄×C₅	—	4	12	no	—
88	C₂×C₄×C₅	D₄×D₅	1	12	yes	—
89	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
90	D₂₀	—	20	24	no	—
91	D₂₀	—	20	24	no	—
92	D₂₀	—	20	12	no	—
93	D₂₀	—	20	12	no	—
94	C₅×C₈	#688	1	24	yes	complemented
95	#168	—	10	24	no	—
96	C₂×SD₁₆	—	50	4	no	—
97	C₂×D₈	—	50	2	no	—
98	C₄⋊Q₈	C₂×C₅⋊D₅	1	2	yes	—
99	#127	—	25	1	no	—
100	#127	—	25	1	no	—
101	#106	—	2	2	no	—
102	#102	—	50	4	no	—
103	C₄×C₈	—	25	2	no	—
104	C₄×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
105	C₅²	#1715	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₅	#4086	1	6	yes	—
107	C₂×D₅	—	40	6	no	—
108	C₂×D₅	—	40	6	no	—
109	C₂×D₅	—	40	12	no	—
110	C₂×D₅	—	40	12	no	—
111	C₄×C₅	—	8	12	no	—
112	C₄×C₅	#4060	1	24	yes	—
113	C₄×C₅	C₂×#688	1	12	yes	—
114	C₂×Q₈	—	2	2	no	—
115	C₂×D₄	—	50	2	no	—
116	C₂×D₄	—	50	2	no	—
117	C₂×D₄	—	50	1	no	—
118	C₂×D₄	—	50	1	no	—
119	SD₁₆	—	100	8	no	—
120	D₈	—	100	4	no	—
121	C₂×C₈	—	25	4	no	—
122	C₂×C₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄²	C₂²×C₅⋊D₅	1	1	yes	characteristic
126	C₂×C₅	#65374	1	6	yes	—
127	C₂×C₅	#65465	1	12	yes	—
128	D₅	—	80	6	no	—
129	D₅	—	80	6	no	—
130	C₂³	—	100	1	no	—
131	C₂³	—	100	1	no	—
132	Q₈	—	4	4	no	—
133	D₄	—	100	4	no	—
134	D₄	—	100	4	no	—
135	D₄	—	100	2	no	—
136	D₄	—	100	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	D₄×C₅⋊D₅	1	2	yes	—
139	C₂×C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
140	C₈	—	50	4	no	—
141	C₈	#87811	1	4	yes	complemented
142	C₅	#10602129	1	6	yes	complemented
143	C₂²	—	200	1	no	—
144	C₂²	—	200	1	no	—
145	C₂²	—	200	2	no	—
146	C₂²	—	200	2	no	—
147	C₂²	#11717260	1	1	yes	characteristic
148	C₄	—	8	2	no	—
149	C₄	#11717234	1	4	yes	—
150	C₄	C₂×#87811	1	2	yes	—
151	C₂	—	400	1	no	—
152	C₂	—	400	1	no	—
153	C₂	#13102949	1	1	yes	characteristic
154	C₂	#13103030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#498

C₂

yes

maximal, characteristic, complemented

#13102952

C₂

yes

maximal, complemented

#13102949

C₂

yes

maximal, complemented

#13102558

C₂

yes

maximal, characteristic

#13099489

C₂

yes

maximal, characteristic, complemented

C₅²×C₄⋊Q₈

C₂²

yes

—

C₂×#87811

—

C₂×#87801

—

#11717194

C₂²

yes

characteristic

#11717194

C₂²

yes

characteristic

C₅²×#106

—

C₄×C₅²×C₈

C₂²

yes

characteristic

#11716676

—

C₄×#6029

C₂²

yes

—

#10602129

—

maximal

C₂×C₅²×Q₈

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₅²×C₈

D₄

yes

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

—

C₄²×C₅²

C₂³

yes

characteristic

#87811

—

C₂×#6029

D₄

yes

—

#87801

—

C₅×#498

D₅

yes

—

—

—

—

C₂²×C₅⋊D₅

—

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×D₄

yes

characteristic

C₂×C₄×C₅²

C₂×D₄

yes

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₅²×C₈

SD₁₆

yes

complemented

#6029

—

C₅×C₄⋊Q₈

C₂×D₅

yes

—

C₂×#688

—

C₂×D₄₀

—

#4020

—

#4020

—

C₅×#106

—

C₄×C₅×C₈

C₂×D₅

yes

—

—

—

—

C₂²×C₅²

yes

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×SD₁₆

yes

—

C₄×C₅²

AGL(1,ℤ/8)

yes

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₅×C₈

C₂²⋊D₅

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×D₅

yes

—

#688

—

C₂×#168

—

D₄₀

—

#498

C₅⋊D₅

yes

characteristic, complemented

—

—

—

—

C₂×C₅²

#492

yes

characteristic

C₂×C₅²

#496

yes

—

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

D₄×D₅

yes

—

C₂×C₄×C₅

C₂×C₂²⋊D₅

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

#688

yes

complemented

#168

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×C₅⋊D₅

yes

—

#127

—

100

#127

—

101

#106

—

102

#102

—

103

C₄×C₈

—

104

C₄×C₈

C₂×C₅⋊D₅

yes

characteristic

105

C₅²

#1715

yes

characteristic, complemented, Sylow

106

C₂²×C₅

#4086

yes

—

107

C₂×D₅

—

108

C₂×D₅

—

109

C₂×D₅

—

110

C₂×D₅

—

111

C₄×C₅

—

112

C₄×C₅

#4060

yes

—

113

C₄×C₅

C₂×#688

yes

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

SD₁₆

—

100

—

120

D₈

—

100

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊(C₅⋊D₅)

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

C₂²×C₅⋊D₅

yes

characteristic

126

C₂×C₅

#65374

yes

—

127

C₂×C₅

#65465

yes

—

128

D₅

—

129

D₅

—

130

C₂³

—

100

—

131

C₂³

—

100

—

132

Q₈

—

133

D₄

—

100

—

134

D₄

—

100

—

135

D₄

—

100

—

136

D₄

—

100

—

137

C₂×C₄

—

138

C₂×C₄

D₄×C₅⋊D₅

yes

—

139

C₂×C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

140

C₈

—

141

C₈

#87811

yes

complemented

142

C₅

#10602129

yes

complemented

143

C₂²

—

200

—

144

C₂²

—

200

—

145

C₂²

—

200

—

146

C₂²

—

200

—

147

C₂²

#11717260

yes

characteristic

148

C₄

—

149

C₄

#11717234

yes

—

150

C₄

C₂×#87811

yes

—

151

C₂

—

400

—

152

C₂

—

400

—

153

C₂

#13102949

yes

characteristic

154

C₂

#13103030

yes

—

	subgroup	G/H
center	C₂²	#11717260
commutator, cosocle	C₄²×C₅²	C₂³
Frattini	C₄²	C₂²×C₅⋊D₅
Fitting	C₅²×#498	C₂
socle	C₂²×C₅²	#127
prime radical	#13102558	C₂
hypercenter	#498	C₅⋊D₅
hypocenter	C₅²	#1715
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#11717260

commutator, cosocle

C₄²×C₅²

C₂³

Frattini

C₄²

C₂²×C₅⋊D₅

Fitting

C₅²×#498

C₂

C₂²×C₅²

C₂

C₅⋊D₅

C₅²

G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. C₅²×#498	C₂	semidirect
2. #13102952	C₂	semidirect
3. #13102949	C₂	semidirect
5. #13099489	C₂	semidirect
28. C₅×#498	D₅	semidirect
43. C₅²×C₈	SD₁₆	semidirect
75. #498	C₅⋊D₅	semidirect
94. C₅×C₈	#688	semidirect
105. C₅²	#1715	semidirect
141. C₈	#87811	semidirect
142. C₅	#10602129	semidirect

H ≅ G/N

type

1. C₅²×#498

C₂

semidirect

2. #13102952

C₂

semidirect

3. #13102949

C₂

semidirect

5. #13099489

C₂

semidirect

28. C₅×#498

D₅

semidirect

43. C₅²×C₈

SD₁₆

semidirect

75. #498

C₅⋊D₅

semidirect

94. C₅×C₈

#688

semidirect

105. C₅²

#1715

semidirect

141. C₈

#87811

semidirect

142. C₅

#10602129

semidirect

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

subgroup	index
C₅²×#498	2
#13102952	2
#13102949	2
#13102558	2
#13099489	2
#10602129	5

subgroup

index

C₅²×#498

Subgroups — #72223036

Conjugacy classes of subgroups

Complements

Maximal subgroups