Group #10608305 — subgroups

Subgroups — #10608305

Order 640.

456 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 157 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×#4090	C₂	1	2	yes	maximal
2	C₅×#391	C₂	1	1	yes	maximal, characteristic
3	#65272	C₂	1	2	yes	maximal
4	#65217	C₂	1	2	yes	maximal
5	#64945	C₂	1	4	yes	maximal
6	#64942	C₂	1	4	yes	maximal
7	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₀	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#4090	—	2	8	no	—
12	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂×#683	C₂²	1	4	yes	—
16	C₂×#683	C₂²	1	4	yes	—
17	C₂×C₄×Dic₅	C₂²	1	2	yes	—
18	C₅×#95	C₂²	1	4	yes	—
19	#3963	C₂²	1	2	yes	—
20	#3963	C₂²	1	4	yes	—
21	#3963	C₂²	1	4	yes	—
22	#2734	—	5	1	no	maximal, Sylow
23	C₂×C₅×Q₈	—	4	1	no	—
24	C₂×C₅×Q₈	—	4	1	no	—
25	C₂×C₅×Q₈	D₄	1	4	yes	—
26	C₂²×C₄×C₅	C₂³	1	2	yes	—
27	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅	C₂³	1	2	yes	—
29	C₂²×C₄×C₅	C₂³	1	2	yes	—
30	C₂²×Dic₅	C₂³	1	4	yes	—
31	C₂²×Dic₅	C₂³	1	4	yes	—
32	C₂×Dic₁₀	—	2	4	no	—
33	C₂×Dic₁₀	—	4	2	no	—
34	C₂×Dic₁₀	—	4	2	no	—
35	C₅×C₄⋊C₄	—	4	1	no	—
36	C₄²×C₅	—	4	1	no	—
37	C₄⋊Dic₅	—	4	2	no	—
38	C₄⋊Dic₅	—	4	2	no	—
39	C₄⋊Dic₅	Q₈	1	8	yes	complemented
40	#683	—	2	8	no	—
41	#683	—	2	8	no	—
42	C₄×Dic₅	—	2	4	no	—
43	C₂×C₄⋊Q₈	—	5	2	no	—
44	#398	—	5	2	no	—
45	#395	—	5	4	no	—
46	#391	D₅	1	1	yes	characteristic
47	#389	—	5	2	no	—
48	#387	—	5	4	no	—
49	C₂³×C₅	C₂⁴	1	1	yes	characteristic
50	C₅×Q₈	—	4	4	no	—
51	C₂×C₄×C₅	—	2	2	no	—
52	C₂×C₄×C₅	—	2	1	no	—
53	C₂×C₄×C₅	—	2	1	no	—
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	—	2	4	no	—
59	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
60	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
62	C₂×Dic₅	—	2	4	no	—
63	C₂×Dic₅	—	2	4	no	—
64	C₂×Dic₅	—	2	4	no	—
65	C₂×Dic₅	—	2	4	no	—
66	C₂×Dic₅	—	2	4	no	—
67	C₂×Dic₅	C₂×Q₈	1	8	yes	—
68	Dic₁₀	—	4	8	no	—
69	C₂²×Q₈	—	5	2	no	—
70	C₂²×Q₈	C₂×D₅	1	1	yes	characteristic
71	C₄⋊Q₈	—	10	8	no	—
72	C₂×C₄⋊C₄	—	5	2	no	—
73	C₂×C₄⋊C₄	—	5	4	no	—
74	C₂×C₄⋊C₄	—	5	4	no	—
75	C₂×C₄⋊C₄	—	5	2	no	—
76	C₂×C₄⋊C₄	—	5	2	no	—
77	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
78	C₂×C₄²	—	5	2	no	—
79	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
80	#95	—	5	2	no	—
81	#95	—	5	4	no	—
82	#95	—	5	4	no	—
83	#95	C₂×D₅	1	4	yes	—
84	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₅	C₂²×Q₈	1	2	yes	—
86	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
89	C₄×C₅	—	2	4	no	—
90	C₄×C₅	—	2	2	no	—
91	C₄×C₅	—	2	4	no	—
92	C₄×C₅	—	4	2	no	—
93	Dic₅	—	2	8	no	—
94	Dic₅	—	4	4	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	4	1	no	—
97	C₂×Q₈	—	10	4	no	—
98	C₂×Q₈	—	20	2	no	—
99	C₂×Q₈	—	20	2	no	—
100	C₂×Q₈	D₂₀	1	4	yes	—
101	C₂²×C₄	—	5	4	no	—
102	C₂²×C₄	—	5	4	no	—
103	C₂²×C₄	C₂²×D₅	1	2	yes	—
104	C₂²×C₄	C₂²×D₅	1	2	yes	—
105	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₅	1	2	yes	—
107	C₄⋊C₄	—	4	1	no	—
108	C₄⋊C₄	—	5	8	no	—
109	C₄⋊C₄	—	10	8	no	—
110	C₄⋊C₄	—	10	8	no	—
111	C₄⋊C₄	—	20	2	no	—
112	C₄⋊C₄	—	20	2	no	—
113	C₄²	—	4	1	no	—
114	C₄²	—	10	4	no	—
115	C₂×C₅	#534	1	1	yes	characteristic
116	C₂×C₅	#544	1	2	yes	—
117	C₂×C₅	D₄×Q₈	1	2	yes	—
118	C₂×C₅	Q₈²	1	1	yes	characteristic
119	C₂×C₅	#563	1	1	yes	characteristic
120	C₂³	C₂³×D₅	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	Q₈	—	20	8	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	5	8	no	—
132	C₂×C₄	—	10	4	no	—
133	C₂×C₄	—	10	4	no	—
134	C₂×C₄	—	10	4	no	—
135	C₂×C₄	—	10	4	no	—
136	C₂×C₄	—	10	4	no	—
137	C₂×C₄	C₂×D₂₀	1	4	yes	—
138	C₂×C₄	C₂×D₂₀	1	2	yes	—
139	C₂×C₄	Q₈×D₅	1	4	yes	—
140	C₅	#2734	1	1	yes	characteristic, complemented, Sylow
141	C₂²	C₂²×D₂₀	1	1	yes	characteristic
142	C₂²	#4144	1	1	yes	characteristic
143	C₂²	#4144	1	1	yes	characteristic
144	C₂²	C₂×Q₈×D₅	1	2	yes	—
145	C₂²	#4150	1	2	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	2	2	no	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	10	8	no	—
151	C₄	—	20	4	no	—
152	C₂	#65827	1	1	yes	characteristic
153	C₂	Q₈×D₂₀	1	2	yes	—
154	C₂	#65957	1	2	yes	—
155	C₂	#66006	1	1	yes	characteristic
156	C₂	#66056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4090

C₂

yes

maximal

C₅×#391

C₂

yes

maximal, characteristic

#65272

C₂

yes

maximal

#65217

C₂

yes

maximal

#64945

C₂

yes

maximal

#64942

C₂

yes

maximal

C₂²×C₅×Q₈

C₂²

yes

characteristic

C₂²×Dic₁₀

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₅

C₂²

yes

characteristic

#4090

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₅×#95

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2734

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

Q₈

yes

complemented

#683

—

#683

—

C₄×Dic₅

—

C₂×C₄⋊Q₈

—

#398

—

#395

—

#391

D₅

yes

characteristic

#389

—

#387

—

C₂³×C₅

C₂⁴

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×Q₈

yes

—

Dic₁₀

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

—

—

—

C₂×D₅

yes

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂²×Q₈

yes

—

C₂²×C₅

yes

C₂²×C₅

yes

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

D₂₀

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₅

yes

—

104

C₂²×C₄

C₂²×D₅

yes

—

105

C₂²×C₄

C₂²×D₅

yes

characteristic

106

C₂²×C₄

C₂²×D₅

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₂×C₅

#534

yes

characteristic

116

C₂×C₅

#544

yes

—

117

C₂×C₅

D₄×Q₈

yes

—

118

C₂×C₅

Q₈²

yes

characteristic

119

C₂×C₅

#563

yes

characteristic

120

C₂³

C₂³×D₅

yes

characteristic

121

Q₈

—

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×D₂₀

yes

—

138

C₂×C₄

C₂×D₂₀

yes

—

139

C₂×C₄

Q₈×D₅

yes

—

140

C₅

#2734

yes

characteristic, complemented, Sylow

141

C₂²

C₂²×D₂₀

yes

characteristic

142

C₂²

#4144

yes

characteristic

143

C₂²

#4144

yes

characteristic

144

C₂²

C₂×Q₈×D₅

yes

—

145

C₂²

#4150

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₂

#65827

yes

characteristic

153

C₂

Q₈×D₂₀

yes

—

154

C₂

#65957

yes

—

155

C₂

#66006

yes

characteristic

156

C₂

#66056

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#391	C₂
hypercenter	#391	D₅
hypocenter	C₅	#2734
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, prime radical, cosocle

C₂³×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
39. C₄⋊Dic₅	Q₈	semidirect
140. C₅	#2734	semidirect

H ≅ G/N

type

39. C₄⋊Dic₅

Q₈

semidirect

140. C₅

#2734

semidirect

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

subgroup	index
C₂×#4090	2
C₅×#391	2
#65272	2
#65217	2
#64945	2
#64942	2
#2734	5

subgroup

index

Subgroups — #10608305

Conjugacy classes of subgroups

Complements

Maximal subgroups