Group #72223055 — subgroups

Subgroups — #72223055

Order 3200.

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

normal subgroups only

This group has 968 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×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	#13102952	C₂	1	2	yes	maximal, complemented
3	#13102935	C₂	1	2	yes	maximal, complemented
4	#13102558	C₂	1	1	yes	maximal, characteristic, complemented
5	#13099489	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×SD₁₆	—	2	2	no	—
7	C₅²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₅²×#127	C₂²	1	1	yes	characteristic
9	C₂×#87809	—	2	4	no	—
10	C₂×#87801	—	2	2	no	—
11	#11717194	C₂²	1	2	yes	—
12	C₄×C₅²×C₈	C₂²	1	1	yes	characteristic
13	#11716676	—	2	4	no	—
14	C₄×#6029	C₂²	1	2	yes	—
15	#10602055	—	5	6	no	maximal
16	C₂×C₅²×Q₈	—	2	1	no	—
17	C₂×C₅²×D₄	—	2	2	no	—
18	C₂×C₅²×D₄	—	2	1	no	—
19	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
20	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
21	C₂×C₄⋊(C₅⋊D₅)	—	2	2	no	—
22	C₅²×SD₁₆	—	4	4	no	—
23	C₂×C₅²×C₈	D₄	1	2	yes	—
24	C₅²×C₄⋊C₄	—	2	2	no	—
25	C₄²×C₅²	C₂³	1	1	yes	characteristic
26	#87809	—	4	8	no	—
27	C₂×#6029	D₄	1	4	yes	—
28	#87801	—	4	4	no	—
29	C₅×#492	D₅	1	6	yes	complemented
30	#65377	—	5	12	no	—
31	#65362	—	5	12	no	—
32	#64984	—	5	6	no	—
33	#64675	—	5	6	no	—
34	C₂³×C₅²	—	4	1	no	—
35	C₂²×C₅⋊D₅	—	4	2	no	—
36	C₅²×Q₈	—	4	2	no	—
37	C₅²×D₄	—	4	2	no	—
38	C₅²×D₄	—	4	4	no	—
39	C₂×C₄×C₅²	—	4	1	no	—
40	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂×D₄	1	2	yes	—
42	C₄⋊(C₅⋊D₅)	—	4	4	no	—
43	C₄⋊(C₅⋊D₅)	—	4	4	no	—
44	C₄⋊(C₅⋊D₅)	—	4	4	no	—
45	C₅²×C₈	D₈	1	4	yes	complemented
46	#6029	—	2	4	no	—
47	C₂×C₅×SD₁₆	—	2	12	no	—
48	C₅×C₄⋊Q₈	C₂×D₅	1	6	yes	—
49	C₅×#127	C₂×D₅	1	6	yes	—
50	C₂×#686	—	10	24	no	—
51	C₂×D₄₀	—	10	12	no	—
52	#4020	—	5	12	no	—
53	C₄×C₅×C₈	C₂×D₅	1	6	yes	—
54	#3941	—	10	24	no	—
55	C₄×#168	—	5	12	no	—
56	#1721	—	25	1	no	Sylow
57	C₂²×C₅²	—	8	1	no	—
58	C₂²×C₅²	—	8	2	no	—
59	C₂²×C₅²	#127	1	1	yes	characteristic
60	C₂×C₅⋊D₅	—	8	2	no	—
61	C₂×C₅⋊D₅	—	8	2	no	—
62	C₂×C₅⋊D₅	—	8	2	no	—
63	C₄×C₅²	—	8	1	no	—
64	C₄×C₅²	C₂×D₈	1	2	yes	—
65	C₄×C₅²	AGL(1,ℤ/8)	1	4	yes	—
66	C₂×C₅×Q₈	—	2	6	no	—
67	C₂×C₅×D₄	—	2	12	no	—
68	C₂×C₅×D₄	—	2	6	no	—
69	C₂×D₂₀	—	10	12	no	—
70	C₂×D₂₀	—	10	12	no	—
71	C₂×D₂₀	—	10	12	no	—
72	C₅×SD₁₆	—	4	24	no	—
73	C₂×C₅×C₈	C₂²⋊D₅	1	12	yes	—
74	C₅×C₄⋊C₄	—	2	12	no	—
75	C₄²×C₅	C₂²×D₅	1	6	yes	—
76	#686	—	20	48	no	—
77	C₂×#168	—	5	24	no	—
78	D₄₀	—	20	24	no	—
79	#493	—	25	2	no	—
80	#493	—	25	1	no	—
81	#492	C₅⋊D₅	1	1	yes	characteristic, complemented
82	#486	—	25	2	no	—
83	#322	—	25	1	no	—
84	C₂×C₅²	—	16	1	no	—
85	C₂×C₅²	#493	1	1	yes	characteristic
86	C₂×C₅²	#496	1	2	yes	—
87	C₅⋊D₅	—	16	2	no	—
88	C₂³×C₅	—	4	6	no	—
89	C₂²×D₅	—	20	12	no	—
90	C₅×Q₈	—	4	12	no	—
91	C₅×D₄	—	4	24	no	—
92	C₅×D₄	—	4	12	no	—
93	C₂×C₄×C₅	—	4	6	no	—
94	C₂×C₄×C₅	D₄×D₅	1	12	yes	—
95	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
96	D₂₀	—	20	24	no	—
97	D₂₀	—	20	24	no	—
98	D₂₀	—	20	24	no	—
99	C₅×C₈	#686	1	24	yes	complemented
100	#168	—	10	24	no	—
101	C₂×SD₁₆	—	2	2	no	—
102	C₂×D₈	—	50	4	no	—
103	C₂×D₈	—	50	2	no	—
104	C₄⋊Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
105	#127	—	25	2	no	—
106	#127	C₂×C₅⋊D₅	1	1	yes	characteristic
107	#102	—	50	4	no	—
108	C₄×C₈	—	25	2	no	—
109	C₄×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
110	C₅²	#1721	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₅	—	8	6	no	—
112	C₂²×C₅	—	8	12	no	—
113	C₂²×C₅	#4086	1	6	yes	—
114	C₂×D₅	—	40	12	no	—
115	C₂×D₅	—	40	12	no	—
116	C₂×D₅	—	40	12	no	—
117	C₄×C₅	—	8	6	no	—
118	C₄×C₅	#4060	1	24	yes	—
119	C₄×C₅	C₂×#686	1	12	yes	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	—	50	2	no	—
124	C₂×D₄	—	50	2	no	—
125	C₂×D₄	—	50	2	no	—
126	SD₁₆	—	4	4	no	—
127	D₈	—	100	8	no	—
128	D₈	—	100	4	no	—
129	C₂×C₈	—	25	4	no	—
130	C₂×C₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄²	C₂²×C₅⋊D₅	1	1	yes	characteristic
133	C₂×C₅	—	16	6	no	—
134	C₂×C₅	#65362	1	6	yes	—
135	C₂×C₅	#65465	1	12	yes	—
136	D₅	—	80	12	no	—
137	C₂³	—	4	1	no	—
138	C₂³	—	100	2	no	—
139	Q₈	—	4	2	no	—
140	D₄	—	4	4	no	—
141	D₄	—	4	2	no	—
142	D₄	—	100	4	no	—
143	D₄	—	100	4	no	—
144	D₄	—	100	4	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	D₄×C₅⋊D₅	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
148	C₈	—	50	4	no	—
149	C₈	#87809	1	4	yes	complemented
150	C₅	#10602055	1	6	yes	complemented
151	C₂²	—	8	1	no	—
152	C₂²	—	8	2	no	—
153	C₂²	—	200	2	no	—
154	C₂²	—	200	2	no	—
155	C₂²	—	200	2	no	—
156	C₂²	#11717260	1	1	yes	characteristic
157	C₄	—	8	1	no	—
158	C₄	#11717234	1	4	yes	—
159	C₄	C₂×#87809	1	2	yes	—
160	C₂	—	16	1	no	—
161	C₂	—	400	2	no	—
162	C₂	#13102935	1	1	yes	characteristic
163	C₂	#13103030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#492

C₂

yes

maximal, characteristic, complemented

#13102952

C₂

yes

maximal, complemented

#13102935

C₂

yes

maximal, complemented

#13102558

C₂

yes

maximal, characteristic, complemented

#13099489

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×SD₁₆

—

C₅²×C₄⋊Q₈

C₂²

yes

characteristic

C₅²×#127

C₂²

yes

—

—

C₂²

yes

—

C₄×C₅²×C₈

C₂²

yes

characteristic

#11716676

—

C₄×#6029

C₂²

yes

—

#10602055

—

maximal

C₂×C₅²×Q₈

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

—

C₅²×SD₁₆

—

C₂×C₅²×C₈

D₄

yes

—

C₅²×C₄⋊C₄

—

C₄²×C₅²

C₂³

yes

characteristic

#87809

—

C₂×#6029

D₄

yes

—

#87801

—

C₅×#492

D₅

yes

—

—

—

—

C₂³×C₅²

—

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₅²×D₄

—

C₅²×D₄

—

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)

—

C₅²×C₈

D₈

yes

complemented

#6029

—

C₂×C₅×SD₁₆

—

C₅×C₄⋊Q₈

C₂×D₅

yes

—

C₅×#127

C₂×D₅

yes

—

C₂×#686

—

C₂×D₄₀

—

#4020

—

C₄×C₅×C₈

C₂×D₅

yes

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

#127

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×D₈

yes

—

C₄×C₅²

AGL(1,ℤ/8)

yes

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₅×SD₁₆

—

C₂×C₅×C₈

C₂²⋊D₅

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×D₅

yes

—

#686

—

C₂×#168

—

D₄₀

—

#493

—

#493

—

#492

C₅⋊D₅

yes

characteristic, complemented

#486

—

#322

—

C₂×C₅²

—

C₂×C₅²

#493

yes

characteristic

C₂×C₅²

#496

yes

—

C₅⋊D₅

—

C₂³×C₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

D₄×D₅

yes

—

C₂×C₄×C₅

C₂×C₂²⋊D₅

yes

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

#686

yes

complemented

100

#168

—

101

C₂×SD₁₆

—

102

C₂×D₈

—

103

C₂×D₈

—

104

C₄⋊Q₈

C₂×C₅⋊D₅

yes

characteristic

105

#127

—

106

#127

C₂×C₅⋊D₅

yes

characteristic

107

#102

—

108

C₄×C₈

—

109

C₄×C₈

C₂×C₅⋊D₅

yes

characteristic

110

C₅²

#1721

yes

characteristic, complemented, Sylow

111

C₂²×C₅

—

112

C₂²×C₅

—

113

C₂²×C₅

#4086

yes

—

114

C₂×D₅

—

115

C₂×D₅

—

116

C₂×D₅

—

117

C₄×C₅

—

118

C₄×C₅

#4060

yes

—

119

C₄×C₅

C₂×#686

yes

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

SD₁₆

—

127

D₈

—

100

—

128

D₈

—

100

—

129

C₂×C₈

—

130

C₂×C₈

C₂²⋊(C₅⋊D₅)

yes

—

131

C₄⋊C₄

—

132

C₄²

C₂²×C₅⋊D₅

yes

characteristic

133

C₂×C₅

—

134

C₂×C₅

#65362

yes

—

135

C₂×C₅

#65465

yes

—

136

D₅

—

137

C₂³

—

138

C₂³

—

100

—

139

Q₈

—

140

D₄

—

141

D₄

—

142

D₄

—

100

—

143

D₄

—

100

—

144

D₄

—

100

—

145

C₂×C₄

—

146

C₂×C₄

D₄×C₅⋊D₅

yes

—

147

C₂×C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

148

C₈

—

149

C₈

#87809

yes

complemented

150

C₅

#10602055

yes

complemented

151

C₂²

—

152

C₂²

—

153

C₂²

—

200

—

154

C₂²

—

200

—

155

C₂²

—

200

—

156

C₂²

#11717260

yes

characteristic

157

C₄

—

158

C₄

#11717234

yes

—

159

C₄

C₂×#87809

yes

—

160

C₂

—

161

C₂

—

400

—

162

C₂

#13102935

yes

characteristic

163

C₂

#13103030

yes

—

	subgroup	G/H
center	C₂²	#11717260
commutator, cosocle	C₄²×C₅²	C₂³
Frattini	C₄²	C₂²×C₅⋊D₅
Fitting	C₅²×#492	C₂
socle	C₂²×C₅²	#127
hypercenter	#492	C₅⋊D₅
hypocenter	C₅²	#1721
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#11717260

commutator, cosocle

C₄²×C₅²

C₂³

Frattini

C₄²

C₂²×C₅⋊D₅

Fitting

C₅²×#492

C₂

C₂²×C₅²

C₅⋊D₅

C₅²

G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₅²×#492	C₂	semidirect
2. #13102952	C₂	semidirect
3. #13102935	C₂	semidirect
4. #13102558	C₂	semidirect
5. #13099489	C₂	semidirect
29. C₅×#492	D₅	semidirect
45. C₅²×C₈	D₈	semidirect
81. #492	C₅⋊D₅	semidirect
99. C₅×C₈	#686	semidirect
110. C₅²	#1721	semidirect
149. C₈	#87809	semidirect
150. C₅	#10602055	semidirect

H ≅ G/N

type

1. C₅²×#492

C₂

semidirect

2. #13102952

C₂

semidirect

3. #13102935

C₂

semidirect

4. #13102558

C₂

semidirect

5. #13099489

C₂

semidirect

29. C₅×#492

D₅

semidirect

45. C₅²×C₈

D₈

semidirect

81. #492

C₅⋊D₅

semidirect

99. C₅×C₈

#686

semidirect

110. C₅²

#1721

semidirect

149. C₈

#87809

semidirect

150. C₅

#10602055

semidirect

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

subgroup	index
C₅²×#492	2
#13102952	2
#13102935	2
#13102558	2
#13099489	2
#10602055	5

subgroup

index

C₅²×#492

Subgroups — #72223055

Conjugacy classes of subgroups

Complements

Maximal subgroups