Group #10616235 — subgroups

Subgroups — C₅×#2640 #10616235

Order 640.

500 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 200 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#124	C₂	1	1	yes	maximal, characteristic
3	C₅×#397	C₂	1	4	yes	maximal, complemented
4	C₅×#393	C₂	1	4	yes	maximal, complemented
5	C₅×#391	C₂	1	2	yes	maximal, complemented
6	C₅×#390	C₂	1	2	yes	maximal
7	C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Q₈	C₂²	1	2	yes	—
10	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊Q₈	—	2	4	no	—
12	C₅×#124	—	2	4	no	—
13	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
16	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₂×C₄²×C₅	C₂²	1	2	yes	—
19	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₅×#95	C₂²	1	2	yes	—
21	C₅×#95	C₂²	1	2	yes	—
22	C₅×#95	C₂²	1	8	yes	—
23	#2640	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁴×C₅	C₂³	1	2	yes	—
25	C₂×C₅×Q₈	—	2	2	no	—
26	C₂×C₅×Q₈	—	2	4	no	—
27	C₂×C₅×Q₈	—	2	4	no	—
28	C₂×C₅×Q₈	—	4	4	no	—
29	C₂×C₅×D₄	—	2	2	no	—
30	C₂×C₅×D₄	—	2	4	no	—
31	C₂²×C₄×C₅	C₂³	1	4	yes	—
32	C₂²×C₄×C₅	C₂³	1	4	yes	—
33	C₂²×C₄×C₅	C₂³	1	2	yes	—
34	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₅×C₄⋊C₄	—	2	4	no	—
38	C₅×C₄⋊C₄	—	2	4	no	—
39	C₅×C₂²⋊C₄	—	2	4	no	—
40	C₅×C₂²⋊C₄	—	2	4	no	—
41	C₅×C₂²⋊C₄	—	4	8	no	—
42	C₄²×C₅	—	2	4	no	—
43	C₄²×C₅	D₄	1	4	yes	—
44	C₂×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂×#124	C₂×C₅	1	1	yes	characteristic
46	#397	C₂×C₅	1	4	yes	complemented
47	#393	C₂×C₅	1	4	yes	complemented
48	#391	C₂×C₅	1	2	yes	complemented
49	#390	C₂×C₅	1	2	yes	—
50	#376	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂³×C₅	—	2	2	no	—
52	C₂³×C₅	—	2	4	no	—
53	C₂³×C₅	—	2	4	no	—
54	C₂³×C₅	—	2	4	no	—
55	C₂³×C₅	C₂⁴	1	1	yes	characteristic
56	C₅×Q₈	—	4	8	no	—
57	C₅×Q₈	—	4	4	no	—
58	C₅×D₄	—	4	4	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	4	no	—
63	C₂×C₄×C₅	—	2	4	no	—
64	C₂×C₄×C₅	—	2	4	no	—
65	C₂×C₄×C₅	—	2	4	no	—
66	C₂×C₄×C₅	—	2	8	no	—
67	C₂×C₄×C₅	—	2	4	no	—
68	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₅	#41	1	4	yes	—
72	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
73	C₂²×Q₈	C₂²×C₅	1	2	yes	—
74	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	4	no	—
76	#124	—	2	4	no	—
77	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
79	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
80	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
81	C₂×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
82	C₂×C₄²	C₂²×C₅	1	2	yes	—
83	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
84	#95	C₂²×C₅	1	2	yes	—
85	#95	C₂²×C₅	1	2	yes	—
86	#95	C₂²×C₅	1	8	yes	—
87	C₂²×C₅	—	4	2	no	—
88	C₂²×C₅	—	4	2	no	—
89	C₂²×C₅	—	4	2	no	—
90	C₂²×C₅	—	4	8	no	—
91	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₅	C₂×#41	1	2	yes	—
93	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
94	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
95	C₄×C₅	—	2	2	no	—
96	C₄×C₅	—	2	2	no	—
97	C₄×C₅	—	4	2	no	—
98	C₄×C₅	—	4	4	no	—
99	C₄×C₅	—	4	4	no	—
100	C₄×C₅	#124	1	4	yes	—
101	C₂⁴	C₂³×C₅	1	2	yes	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	4	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	4	4	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	4	no	—
108	C₂²×C₄	C₂³×C₅	1	4	yes	—
109	C₂²×C₄	C₂³×C₅	1	2	yes	—
110	C₂²×C₄	C₂³×C₅	1	4	yes	—
111	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
112	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₂²⋊C₄	—	2	4	no	—
117	C₂²⋊C₄	—	2	4	no	—
118	C₂²⋊C₄	—	4	8	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₅×D₄	1	4	yes	—
121	C₂×C₅	—	8	2	no	—
122	C₂×C₅	C₂×#124	1	1	yes	characteristic
123	C₂×C₅	#535	1	1	yes	characteristic
124	C₂×C₅	#536	1	1	yes	characteristic
125	C₂×C₅	#542	1	4	yes	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	4	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	C₂⁴×C₅	1	1	yes	characteristic
131	Q₈	—	4	8	no	—
132	Q₈	—	4	4	no	—
133	D₄	—	4	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
144	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
145	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
146	C₂×C₄	C₅×#41	1	4	yes	—
147	C₅	#2640	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²	—	4	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	8	no	—
152	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
153	C₂²	C₂×C₅×#41	1	2	yes	—
154	C₂²	C₅×H(5,𝔽₂)	1	2	yes	—
155	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	4	4	no	—
161	C₄	C₅×#124	1	4	yes	—
162	C₂	—	8	2	no	—
163	C₂	C₂×C₅×#124	1	1	yes	characteristic
164	C₂	C₅×#535	1	1	yes	characteristic
165	C₂	C₅×#536	1	1	yes	characteristic
166	C₂	C₅×#542	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×C₅×#124

C₂

yes

maximal, characteristic

C₅×#397

C₂

yes

maximal, complemented

C₅×#393

C₂

yes

maximal, complemented

C₅×#391

C₂

yes

maximal, complemented

C₅×#390

C₂

yes

maximal

C₅×#376

C₂

yes

maximal, characteristic, complemented

C₂²×C₅×Q₈

C₂²

yes

characteristic

C₂²×C₅×Q₈

C₂²

yes

—

C₂²×C₅×D₄

C₂²

yes

characteristic

C₅×C₄⋊Q₈

—

C₅×#124

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

C₅×#95

C₂²

yes

—

#2640

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₅

C₂³

yes

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

C₂³

yes

—

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

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₅×C₂²⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

D₄

yes

—

C₂×C₄⋊Q₈

C₂×C₅

yes

characteristic, complemented

C₂×#124

C₂×C₅

yes

characteristic

#397

C₂×C₅

yes

complemented

#393

C₂×C₅

yes

complemented

#391

C₂×C₅

yes

complemented

#390

C₂×C₅

yes

—

#376

C₂×C₅

yes

characteristic, complemented

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂⁴

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×D₄

—

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×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

#41

yes

—

C₂²×Q₈

C₂²×C₅

yes

characteristic

C₂²×Q₈

C₂²×C₅

yes

—

C₂²×D₄

C₂²×C₅

yes

characteristic

C₄⋊Q₈

—

#124

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₂²⋊C₄

C₂²×C₅

yes

—

C₂×C₂²⋊C₄

C₂²×C₅

yes

—

C₂×C₂²⋊C₄

C₂²×C₅

yes

—

C₂×C₄²

C₂²×C₅

yes

—

C₂×C₄²

C₂²×C₅

yes

characteristic

#95

C₂²×C₅

yes

—

#95

C₂²×C₅

yes

—

#95

C₂²×C₅

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂×#41

yes

—

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

#124

yes

—

101

C₂⁴

C₂³×C₅

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

C₂³×C₅

yes

—

109

C₂²×C₄

C₂³×C₅

yes

—

110

C₂²×C₄

C₂³×C₅

yes

—

111

C₂²×C₄

C₂³×C₅

yes

characteristic

112

C₂²×C₄

C₂³×C₅

yes

characteristic

113

C₂²×C₄

C₂³×C₅

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

C₅×D₄

yes

—

121

C₂×C₅

—

122

C₂×C₅

C₂×#124

yes

characteristic

123

C₂×C₅

#535

yes

characteristic

124

C₂×C₅

#536

yes

characteristic

125

C₂×C₅

#542

yes

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

C₂⁴×C₅

yes

characteristic

131

Q₈

—

132

Q₈

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂×C₅×D₄

yes

—

144

C₂×C₄

C₂×C₅×D₄

yes

—

145

C₂×C₄

C₂×C₅×D₄

yes

—

146

C₂×C₄

C₅×#41

yes

—

147

C₅

#2640

yes

characteristic, direct product factor, Sylow

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂²×C₅×D₄

yes

characteristic

153

C₂²

C₂×C₅×#41

yes

—

154

C₂²

C₅×H(5,𝔽₂)

yes

—

155

C₂²

C₅×2₋¹⁺⁴

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₅×#124

yes

—

162

C₂

—

163

C₂

C₂×C₅×#124

yes

characteristic

164

C₂

C₅×#535

yes

characteristic

165

C₂

C₅×#536

yes

characteristic

166

C₂

C₅×#542

yes

—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

subgroup

G/H

center, socle

C₂³×C₅

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₅

prime radical

C₂²×C₅×D₄

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ G

N	H ≅ G/N	type
1. C₂×C₅×C₄⋊Q₈	C₂	semidirect
3. C₅×#397	C₂	semidirect
4. C₅×#393	C₂	semidirect
5. C₅×#391	C₂	semidirect
7. C₅×#376	C₂	semidirect
23. #2640	C₅	direct (N × H)
44. C₂×C₄⋊Q₈	C₂×C₅	semidirect
46. #397	C₂×C₅	semidirect
47. #393	C₂×C₅	semidirect
48. #391	C₂×C₅	semidirect
50. #376	C₂×C₅	semidirect
147. C₅	#2640	direct (N × H)

H ≅ G/N

type

1. C₂×C₅×C₄⋊Q₈

C₂

semidirect

3. C₅×#397

C₂

semidirect

4. C₅×#393

C₂

semidirect

5. C₅×#391

C₂

semidirect

7. C₅×#376

C₂

semidirect

23. #2640

C₅

direct (N × H)

44. C₂×C₄⋊Q₈

C₂×C₅

semidirect

46. #397

C₂×C₅

semidirect

47. #393

C₂×C₅

semidirect

48. #391

C₂×C₅

semidirect

50. #376

C₂×C₅

semidirect

147. C₅

#2640

direct (N × H)

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

subgroup	index
C₂×C₅×C₄⋊Q₈	2
C₂×C₅×#124	2
C₅×#397	2
C₅×#393	2
C₅×#391	2
C₅×#390	2
C₅×#376	2
#2640	5

subgroup

index

C₂×C₅×C₄⋊Q₈

C₂×C₅×#124

Subgroups — C₅×#2640 #10616235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#2640 #10616235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2640 #10616235