Group #10606076 — subgroups

Subgroups — #10606076

Order 640.

360 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₅×#384	C₂	1	1	yes	maximal, characteristic
2	#65275	C₂	1	4	yes	maximal
3	#65274	C₂	1	2	yes	maximal
4	#65273	C₂	1	4	yes	maximal
5	#65262	C₂	1	2	yes	maximal
6	#65225	C₂	1	1	yes	maximal, characteristic
7	#65221	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₅	C₂²	1	4	yes	—
13	C₂×#683	C₂²	1	4	yes	—
14	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	#3963	C₂²	1	2	yes	—
17	#3963	C₂²	1	2	yes	—
18	#3963	C₂²	1	4	yes	—
19	#3963	C₂²	1	4	yes	—
20	#3963	C₂²	1	4	yes	—
21	#2647	—	5	1	no	maximal, Sylow
22	C₂²×C₄×C₅	C₂³	1	4	yes	—
23	C₂²×C₄×C₅	C₂³	1	2	yes	—
24	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅	C₂³	1	2	yes	—
27	C₂²×Dic₅	C₂³	1	4	yes	—
28	C₂²×Dic₅	C₂³	1	1	yes	characteristic
29	C₅×C₄⋊C₄	—	2	4	no	—
30	C₅×C₄⋊C₄	—	4	4	no	—
31	C₄²×C₅	Q₈	1	4	yes	—
32	C₄⋊Dic₅	—	2	4	no	—
33	C₄⋊Dic₅	—	4	4	no	—
34	#683	—	4	4	no	—
35	C₄×Dic₅	—	2	2	no	—
36	C₄×Dic₅	—	2	2	no	—
37	#400	—	5	4	no	—
38	#398	—	5	4	no	—
39	#389	—	5	2	no	—
40	#389	—	5	2	no	—
41	#384	D₅	1	1	yes	characteristic
42	#382	—	5	1	no	—
43	#376	—	5	1	no	—
44	C₂³×C₅	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₅	—	2	2	no	—
46	C₂×C₄×C₅	—	2	2	no	—
47	C₂×C₄×C₅	—	2	4	no	—
48	C₂×C₄×C₅	—	2	4	no	—
49	C₂×C₄×C₅	—	2	4	no	—
50	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
51	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₅	#41	1	2	yes	—
53	C₂×C₄×C₅	#41	1	2	yes	—
54	C₂×Dic₅	—	2	1	no	—
55	C₂×Dic₅	—	2	1	no	—
56	C₂×Dic₅	—	2	2	no	—
57	C₂×Dic₅	—	2	4	no	—
58	C₂×Dic₅	—	2	2	no	—
59	C₂×Dic₅	—	2	4	no	—
60	C₂×Dic₅	—	2	2	no	—
61	C₂×Dic₅	—	2	2	no	—
62	C₂×Dic₅	—	2	2	no	—
63	C₂×Dic₅	—	2	4	no	—
64	C₂×C₄⋊C₄	—	5	2	no	—
65	C₂×C₄⋊C₄	—	5	4	no	—
66	C₂×C₄⋊C₄	—	5	4	no	—
67	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
68	C₂×C₄⋊C₄	C₂×D₅	1	4	yes	—
69	C₂×C₄²	—	5	1	no	—
70	C₂×C₄²	—	5	1	no	—
71	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
72	#95	—	5	2	no	—
73	#95	—	5	2	no	—
74	#95	—	5	4	no	—
75	#95	—	5	4	no	—
76	#95	—	5	4	no	—
77	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
78	C₂²×C₅	C₂×#41	1	1	yes	characteristic
79	C₂²×C₅	C₂×#41	1	1	yes	characteristic
80	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
81	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
82	C₄×C₅	—	2	4	no	—
83	C₄×C₅	—	4	4	no	—
84	C₄×C₅	#125	1	4	yes	—
85	Dic₅	—	4	1	no	—
86	Dic₅	—	4	1	no	—
87	Dic₅	—	4	2	no	—
88	Dic₅	—	4	4	no	—
89	C₂²×C₄	—	5	1	no	—
90	C₂²×C₄	—	5	2	no	—
91	C₂²×C₄	—	5	4	no	—
92	C₂²×C₄	—	5	1	no	—
93	C₂²×C₄	C₂²×D₅	1	4	yes	—
94	C₂²×C₄	C₂²×D₅	1	2	yes	—
95	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	10	4	no	—
99	C₄⋊C₄	—	20	4	no	—
100	C₄⋊C₄	—	20	4	no	—
101	C₄²	—	10	2	no	—
102	C₄²	—	10	2	no	—
103	C₄²	Dic₁₀	1	4	yes	—
104	C₂×C₅	C₂×#125	1	1	yes	characteristic
105	C₂×C₅	#540	1	1	yes	characteristic
106	C₂×C₅	#541	1	1	yes	characteristic
107	C₂×C₅	#542	1	2	yes	—
108	C₂×C₅	#544	1	2	yes	—
109	C₂³	C₂³×D₅	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	10	1	no	—
116	C₂×C₄	—	10	1	no	—
117	C₂×C₄	—	10	2	no	—
118	C₂×C₄	—	10	4	no	—
119	C₂×C₄	—	10	2	no	—
120	C₂×C₄	—	10	2	no	—
121	C₂×C₄	—	10	4	no	—
122	C₂×C₄	—	10	2	no	—
123	C₂×C₄	—	10	2	no	—
124	C₂×C₄	—	10	4	no	—
125	C₂×C₄	C₂×Dic₁₀	1	4	yes	—
126	C₂×C₄	C₂×Dic₁₀	1	2	yes	—
127	C₂×C₄	#711	1	2	yes	—
128	C₂×C₄	#713	1	2	yes	—
129	C₅	#2647	1	1	yes	characteristic, complemented, Sylow
130	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
131	C₂²	C₂×#711	1	1	yes	characteristic
132	C₂²	#4144	1	2	yes	—
133	C₂²	C₂×#713	1	1	yes	characteristic
134	C₂²	#4147	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	4	4	no	—
137	C₄	—	20	1	no	—
138	C₄	—	20	1	no	—
139	C₄	—	20	2	no	—
140	C₄	—	20	4	no	—
141	C₄	#4036	1	4	yes	—
142	C₂	C₂×#4036	1	1	yes	characteristic
143	C₂	#65834	1	2	yes	—
144	C₂	#66053	1	1	yes	characteristic
145	C₂	#66065	1	1	yes	characteristic
146	C₂	#66067	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#384

C₂

yes

maximal, characteristic

#65275

C₂

yes

maximal

#65274

C₂

yes

maximal

#65273

C₂

yes

maximal

#65262

C₂

yes

maximal

#65225

C₂

yes

maximal, characteristic

#65221

C₂

yes

maximal, characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

characteristic

C₂×C₄×Dic₅

C₂²

yes

characteristic

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2647

—

maximal, Sylow

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

Q₈

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

—

—

D₅

yes

characteristic

#382

—

#376

—

C₂³×C₅

C₂⁴

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

—

—

—

—

—

C₂²×C₅

C₂²×Q₈

yes

characteristic

C₂²×C₅

C₂×#41

yes

characteristic

C₂²×C₅

C₂×#41

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#125

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

Dic₁₀

yes

—

104

C₂×C₅

C₂×#125

yes

characteristic

105

C₂×C₅

#540

yes

characteristic

106

C₂×C₅

#541

yes

characteristic

107

C₂×C₅

#542

yes

—

108

C₂×C₅

#544

yes

—

109

C₂³

C₂³×D₅

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×Dic₁₀

yes

—

126

C₂×C₄

C₂×Dic₁₀

yes

—

127

C₂×C₄

#711

yes

—

128

C₂×C₄

#713

yes

—

129

C₅

#2647

yes

characteristic, complemented, Sylow

130

C₂²

C₂²×Dic₁₀

yes

characteristic

131

C₂²

C₂×#711

yes

characteristic

132

C₂²

#4144

yes

—

133

C₂²

C₂×#713

yes

characteristic

134

C₂²

#4147

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#4036

yes

—

142

C₂

C₂×#4036

yes

characteristic

143

C₂

#65834

yes

—

144

C₂

#66053

yes

characteristic

145

C₂

#66065

yes

characteristic

146

C₂

#66067

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#384	C₂
hypercenter	#384	D₅
hypocenter	C₅	#2647
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #384

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 ◁ #384

N	H ≅ G/N	type
129. C₅	#2647	semidirect

H ≅ G/N

type

129. C₅

#2647

semidirect

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

subgroup	index
C₅×#384	2
#65275	2
#65274	2
#65273	2
#65262	2
#65225	2
#65221	2
#2647	5

subgroup

index

Subgroups — #10606076

Conjugacy classes of subgroups

Complements

Maximal subgroups