Group #10614775 — subgroups

Subgroups — #10614775

Order 640.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#4015	C₂	1	1	yes	maximal, characteristic
3	#65272	C₂	1	4	yes	maximal
4	#65268	C₂	1	4	yes	maximal
5	#65266	C₂	1	2	yes	maximal
6	#65265	C₂	1	2	yes	maximal
7	#65221	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₀	C₂²	1	2	yes	—
9	C₅×#125	—	2	4	no	—
10	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×#683	C₂²	1	8	yes	—
18	C₂×C₄×Dic₅	C₂²	1	2	yes	—
19	#4015	—	2	4	no	—
20	#3963	C₂²	1	2	yes	—
21	#3963	C₂²	1	2	yes	—
22	#3963	C₂²	1	8	yes	—
23	#2660	—	5	1	no	maximal, Sylow
24	C₂²×C₄×C₅	C₂³	1	2	yes	—
25	C₂²×C₄×C₅	C₂³	1	2	yes	—
26	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×Dic₅	C₂³	1	4	yes	—
30	C₂²×Dic₅	C₂³	1	4	yes	—
31	C₂×Dic₁₀	—	2	4	no	—
32	C₂×Dic₁₀	—	4	4	no	—
33	C₅×C₄⋊C₄	—	2	4	no	—
34	C₅×C₄⋊C₄	—	2	4	no	—
35	C₅×C₄⋊C₄	—	2	2	no	—
36	C₅×C₄⋊C₄	—	2	2	no	—
37	C₄²×C₅	D₄	1	4	yes	—
38	C₄⋊Dic₅	—	2	4	no	—
39	C₄⋊Dic₅	—	2	4	no	—
40	#683	—	4	8	no	—
41	C₄×Dic₅	Q₈	1	8	yes	complemented
42	C₂×C₄⋊Q₈	—	5	1	no	—
43	C₂×#125	D₅	1	1	yes	characteristic
44	#398	—	5	4	no	—
45	#395	—	5	4	no	—
46	#384	—	5	2	no	—
47	#382	—	5	2	no	—
48	#376	—	5	1	no	—
49	C₂³×C₅	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₅	—	2	2	no	—
51	C₂×C₄×C₅	—	2	2	no	—
52	C₂×C₄×C₅	—	2	2	no	—
53	C₂×C₄×C₅	—	2	2	no	—
54	C₂×C₄×C₅	—	2	4	no	—
55	C₂×C₄×C₅	—	2	4	no	—
56	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
60	C₂×Dic₅	—	2	4	no	—
61	C₂×Dic₅	—	2	4	no	—
62	C₂×Dic₅	—	2	4	no	—
63	C₂×Dic₅	—	2	8	no	—
64	C₂×Dic₅	C₂×Q₈	1	8	yes	—
65	Dic₁₀	—	4	8	no	—
66	C₂²×Q₈	—	5	2	no	—
67	C₄⋊Q₈	—	10	4	no	—
68	#125	—	2	4	no	—
69	C₂×C₄⋊C₄	—	5	2	no	—
70	C₂×C₄⋊C₄	—	5	2	no	—
71	C₂×C₄⋊C₄	—	5	8	no	—
72	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
76	C₂×C₄²	—	5	2	no	—
77	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
78	#95	—	5	2	no	—
79	#95	—	5	2	no	—
80	#95	—	5	8	no	—
81	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
82	C₂²×C₅	C₂²×Q₈	1	2	yes	—
83	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
84	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
85	C₄×C₅	—	2	2	no	—
86	C₄×C₅	—	2	2	no	—
87	C₄×C₅	—	4	2	no	—
88	C₄×C₅	—	4	2	no	—
89	C₄×C₅	C₄⋊Q₈	1	4	yes	—
90	Dic₅	—	2	8	no	—
91	Dic₅	—	4	4	no	—
92	C₂×Q₈	—	10	4	no	—
93	C₂×Q₈	—	20	4	no	—
94	C₂²×C₄	—	5	4	no	—
95	C₂²×C₄	—	5	4	no	—
96	C₂²×C₄	C₂²×D₅	1	2	yes	—
97	C₂²×C₄	C₂²×D₅	1	2	yes	—
98	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	10	4	no	—
106	C₄⋊C₄	—	10	4	no	—
107	C₄⋊C₄	—	20	8	no	—
108	C₄²	—	5	8	no	—
109	C₄²	C₂²⋊D₅	1	4	yes	—
110	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₅	#535	1	1	yes	characteristic
112	C₂×C₅	#536	1	1	yes	characteristic
113	C₂×C₅	#544	1	4	yes	—
114	C₂³	C₂³×D₅	1	1	yes	characteristic
115	Q₈	—	20	8	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	5	8	no	—
123	C₂×C₄	—	10	4	no	—
124	C₂×C₄	—	10	4	no	—
125	C₂×C₄	—	10	8	no	—
126	C₂×C₄	—	10	4	no	—
127	C₂×C₄	Q₈×D₅	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
131	C₅	#2660	1	1	yes	characteristic, complemented, Sylow
132	C₂²	C₂×Q₈×D₅	1	2	yes	—
133	C₂²	#4149	1	2	yes	—
134	C₂²	#4150	1	2	yes	—
135	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	4	2	no	—
140	C₄	—	10	8	no	—
141	C₄	—	20	4	no	—
142	C₄	#4090	1	4	yes	—
143	C₂	#66023	1	4	yes	—
144	C₂	C₂×#4090	1	1	yes	characteristic
145	C₂	#66159	1	1	yes	characteristic
146	C₂	#66164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#125

C₂

yes

maximal, characteristic

C₂×#4015

C₂

yes

maximal, characteristic

#65272

C₂

yes

maximal

#65268

C₂

yes

maximal

#65266

C₂

yes

maximal

#65265

C₂

yes

maximal

#65221

C₂

yes

maximal, characteristic

C₂²×Dic₁₀

C₂²

yes

—

C₅×#125

—

C₂×C₅×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

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

—

#4015

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2660

—

maximal, Sylow

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

Q₈

yes

complemented

C₂×C₄⋊Q₈

—

C₂×#125

D₅

yes

—

—

—

—

—

C₂³×C₅

C₂⁴

yes

characteristic

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×D₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×Q₈

yes

—

Dic₁₀

—

C₂²×Q₈

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅

yes

characteristic

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

—

—

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂²×Q₈

yes

—

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄⋊Q₈

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₂²×C₄

C₂²×D₅

yes

characteristic

100

C₂²×C₄

C₂²×D₅

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

C₂²⋊D₅

yes

—

110

C₂×C₅

C₂×C₄⋊Q₈

yes

characteristic

111

C₂×C₅

#535

yes

characteristic

112

C₂×C₅

#536

yes

characteristic

113

C₂×C₅

#544

yes

—

114

C₂³

C₂³×D₅

yes

characteristic

115

Q₈

—

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

—

126

C₂×C₄

—

127

C₂×C₄

Q₈×D₅

yes

—

128

C₂×C₄

C₂×C₂²⋊D₅

yes

—

129

C₂×C₄

C₂×C₂²⋊D₅

yes

—

130

C₂×C₄

C₂×C₂²⋊D₅

yes

—

131

C₅

#2660

yes

characteristic, complemented, Sylow

132

C₂²

C₂×Q₈×D₅

yes

—

133

C₂²

#4149

yes

—

134

C₂²

#4150

yes

—

135

C₂²

C₂²×C₂²⋊D₅

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#4090

yes

—

143

C₂

#66023

yes

—

144

C₂

C₂×#4090

yes

characteristic

145

C₂

#66159

yes

characteristic

146

C₂

#66164

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₂×C₅×#125	C₂
hypercenter	C₂×#125	D₅
hypocenter	C₅	#2660
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, prime radical, cosocle

C₂³×C₅

C₂⁴

Fitting

C₂×C₅×#125

C₂

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
41. C₄×Dic₅	Q₈	semidirect
131. C₅	#2660	semidirect

H ≅ G/N

type

41. C₄×Dic₅

Q₈

semidirect

131. C₅

#2660

semidirect

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

subgroup	index
C₂×C₅×#125	2
C₂×#4015	2
#65272	2
#65268	2
#65266	2
#65265	2
#65221	2
#2660	5

subgroup

index

C₂×C₅×#125

Subgroups — #10614775

Conjugacy classes of subgroups

Complements

Maximal subgroups