Group #11955518 — subgroups

Subgroups — C₅×#32798 #11955518

Order 1280.

1074 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 238 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×#2867	C₂	1	1	yes	maximal, characteristic
2	C₅×#2863	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2041	C₂	1	6	yes	maximal, complemented
4	C₅×#2014	C₂	1	6	yes	maximal, complemented
5	C₅×#1839	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×SD₁₆	C₂²	1	4	yes	—
7	C₂×C₅×#127	—	2	3	no	—
8	C₂×C₅×#127	—	2	1	no	—
9	C₂×C₅×#127	C₂²	1	6	yes	—
10	C₂×C₅×#102	C₂²	1	12	yes	—
11	C₅×#391	C₂²	1	3	yes	—
12	C₅×#389	C₂²	1	3	yes	—
13	C₅×#387	—	2	3	no	—
14	C₅×#383	—	2	1	no	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₅×#336	C₂²	1	6	yes	—
17	#32798	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
19	C₂²×C₅×D₄	—	2	6	no	—
20	C₂²×C₅×D₄	—	2	6	no	—
21	C₂²×C₅×D₄	C₂³	1	4	yes	—
22	C₂×C₅×SD₁₆	—	4	12	no	—
23	C₂²×C₅×C₈	C₂³	1	4	yes	—
24	C₅×#127	—	4	12	no	—
25	C₅×#127	—	4	12	no	—
26	C₅×#127	—	4	4	no	—
27	C₂×C₅×C₄⋊C₄	—	2	3	no	—
28	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
29	C₂×C₄²×C₅	D₄	1	3	yes	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	C₂³	1	3	yes	—
32	C₅×#102	—	4	12	no	—
33	C₅×#95	—	2	3	no	—
34	C₅×#95	—	2	3	no	—
35	C₅×#95	—	2	3	no	—
36	#2867	C₂×C₅	1	1	yes	characteristic
37	#2863	C₂×C₅	1	1	yes	characteristic, complemented
38	#2041	C₂×C₅	1	6	yes	complemented
39	#2014	C₂×C₅	1	6	yes	complemented
40	#1839	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂⁴×C₅	—	2	4	no	—
42	C₂×C₅×Q₈	—	2	6	no	—
43	C₂×C₅×D₄	—	2	12	no	—
44	C₂×C₅×D₄	—	2	12	no	—
45	C₂×C₅×D₄	—	4	12	no	—
46	C₂×C₅×D₄	—	4	12	no	—
47	C₂×C₅×D₄	—	4	12	no	—
48	C₂×C₅×D₄	—	4	12	no	—
49	C₂×C₅×D₄	—	4	12	no	—
50	C₂×C₅×D₄	—	4	12	no	—
51	C₂²×C₄×C₅	—	2	3	no	—
52	C₂²×C₄×C₅	—	2	1	no	—
53	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
56	C₅×SD₁₆	—	4	16	no	—
57	C₂×C₅×C₈	—	2	12	no	—
58	C₅×C₄⋊C₄	—	2	6	no	—
59	C₅×C₄⋊C₄	—	4	6	no	—
60	C₄²×C₅	—	2	6	no	—
61	C₄²×C₅	—	2	6	no	—
62	C₄²×C₅	SD₁₆	1	4	yes	complemented
63	C₂²×SD₁₆	C₂²×C₅	1	4	yes	—
64	C₂×#127	—	2	3	no	—
65	C₂×#127	—	2	1	no	—
66	C₂×#127	C₂²×C₅	1	6	yes	—
67	C₂×#102	C₂²×C₅	1	12	yes	—
68	#391	C₂²×C₅	1	3	yes	—
69	#389	C₂²×C₅	1	3	yes	—
70	#387	—	2	3	no	—
71	#383	—	2	1	no	—
72	C₄³	C₂²×C₅	1	1	yes	characteristic
73	#336	C₂²×C₅	1	6	yes	—
74	C₂³×C₅	—	4	12	no	—
75	C₂³×C₅	—	4	4	no	—
76	C₂³×C₅	—	4	12	no	—
77	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
78	C₅×Q₈	—	4	4	no	—
79	C₅×D₄	—	4	4	no	—
80	C₅×D₄	—	4	12	no	—
81	C₅×D₄	—	8	24	no	—
82	C₅×D₄	—	8	24	no	—
83	C₂×C₄×C₅	—	2	3	no	—
84	C₂×C₄×C₅	—	2	3	no	—
85	C₂×C₄×C₅	—	2	3	no	—
86	C₂×C₄×C₅	—	2	3	no	—
87	C₂×C₄×C₅	—	4	3	no	—
88	C₂×C₄×C₅	—	4	3	no	—
89	C₂×C₄×C₅	—	4	6	no	—
90	C₂×C₄×C₅	C₂×SD₁₆	1	6	yes	—
91	C₂×C₄×C₅	AGL(1,ℤ/8)	1	6	yes	—
92	C₂×C₄×C₅	H(5,𝔽₂)	1	6	yes	—
93	C₅×C₈	—	4	4	no	—
94	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
95	C₂²×D₄	—	2	6	no	—
96	C₂²×D₄	—	2	6	no	—
97	C₂²×D₄	C₂³×C₅	1	4	yes	—
98	C₂×SD₁₆	—	4	12	no	—
99	C₂²×C₈	C₂³×C₅	1	4	yes	—
100	#127	—	4	12	no	—
101	#127	—	4	12	no	—
102	#127	—	4	4	no	—
103	C₂×C₄⋊C₄	—	2	3	no	—
104	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
105	C₂×C₄²	C₅×D₄	1	3	yes	—
106	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
107	C₂×C₄²	C₂³×C₅	1	3	yes	—
108	#102	—	4	12	no	—
109	#95	—	2	3	no	—
110	#95	—	2	3	no	—
111	#95	—	2	3	no	—
112	C₂²×C₅	—	8	4	no	—
113	C₂²×C₅	—	8	12	no	—
114	C₂²×C₅	—	8	12	no	—
115	C₂²×C₅	#535	1	3	yes	—
116	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
117	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	3	yes	—
118	C₄×C₅	—	2	6	no	—
119	C₄×C₅	—	2	6	no	—
120	C₄×C₅	—	8	1	no	—
121	C₄×C₅	—	8	3	no	—
122	C₄×C₅	#560	1	4	yes	—
123	C₂⁴	—	2	4	no	—
124	C₂×Q₈	—	2	6	no	—
125	C₂×D₄	—	2	12	no	—
126	C₂×D₄	—	2	12	no	—
127	C₂×D₄	—	4	12	no	—
128	C₂×D₄	—	4	12	no	—
129	C₂×D₄	—	4	12	no	—
130	C₂×D₄	—	4	12	no	—
131	C₂×D₄	—	4	12	no	—
132	C₂×D₄	—	4	12	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	3	no	—
135	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
136	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
137	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
138	SD₁₆	—	4	16	no	—
139	C₂×C₈	—	2	12	no	—
140	C₄⋊C₄	—	2	6	no	—
141	C₄⋊C₄	—	4	6	no	—
142	C₄²	—	2	6	no	—
143	C₄²	—	2	6	no	—
144	C₄²	C₅×SD₁₆	1	4	yes	complemented
145	C₂×C₅	—	16	4	no	—
146	C₂×C₅	#2818	1	1	yes	characteristic
147	C₂×C₅	#3208	1	3	yes	—
148	C₂×C₅	#3217	1	3	yes	—
149	C₂³	—	4	12	no	—
150	C₂³	—	4	4	no	—
151	C₂³	—	4	12	no	—
152	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
153	Q₈	—	4	4	no	—
154	D₄	—	4	4	no	—
155	D₄	—	4	12	no	—
156	D₄	—	8	24	no	—
157	D₄	—	8	24	no	—
158	C₂×C₄	—	2	3	no	—
159	C₂×C₄	—	2	3	no	—
160	C₂×C₄	—	2	3	no	—
161	C₂×C₄	—	2	3	no	—
162	C₂×C₄	—	4	3	no	—
163	C₂×C₄	—	4	3	no	—
164	C₂×C₄	—	4	6	no	—
165	C₂×C₄	C₂×C₅×SD₁₆	1	6	yes	—
166	C₂×C₄	C₅×AGL(1,ℤ/8)	1	6	yes	—
167	C₂×C₄	C₅×H(5,𝔽₂)	1	6	yes	—
168	C₈	—	4	4	no	—
169	C₅	#32798	1	1	yes	characteristic, direct product factor, Sylow
170	C₂²	—	8	4	no	—
171	C₂²	—	8	12	no	—
172	C₂²	—	8	12	no	—
173	C₂²	C₅×#535	1	3	yes	—
174	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
175	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	3	yes	—
176	C₄	—	2	6	no	—
177	C₄	—	2	6	no	—
178	C₄	—	8	1	no	—
179	C₄	—	8	3	no	—
180	C₄	C₅×#560	1	4	yes	—
181	C₂	—	16	4	no	—
182	C₂	C₅×#2818	1	1	yes	characteristic
183	C₂	C₅×#3208	1	3	yes	—
184	C₂	C₅×#3217	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2867

C₂

yes

maximal, characteristic

C₅×#2863

C₂

yes

maximal, characteristic, complemented

C₅×#2041

C₂

yes

maximal, complemented

C₅×#2014

C₂

yes

maximal, complemented

C₅×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×C₅×SD₁₆

C₂²

yes

—

C₂×C₅×#127

—

C₂×C₅×#127

—

C₂×C₅×#127

C₂²

yes

—

C₂×C₅×#102

C₂²

yes

—

C₅×#391

C₂²

yes

—

C₅×#389

C₂²

yes

—

C₅×#387

—

C₅×#383

—

C₄³×C₅

C₂²

yes

characteristic

C₅×#336

C₂²

yes

—

#32798

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

C₂³

yes

characteristic

C₂²×C₅×D₄

—

C₂²×C₅×D₄

—

C₂²×C₅×D₄

C₂³

yes

—

C₂×C₅×SD₁₆

—

C₂²×C₅×C₈

C₂³

yes

—

C₅×#127

—

C₅×#127

—

C₅×#127

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

C₂³

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄²×C₅

C₂³

yes

—

C₅×#102

—

C₅×#95

—

C₅×#95

—

C₅×#95

—

#2867

C₂×C₅

yes

characteristic

#2863

C₂×C₅

yes

characteristic, complemented

#2041

C₂×C₅

yes

complemented

#2014

C₂×C₅

yes

complemented

#1839

C₂×C₅

yes

characteristic, complemented

C₂⁴×C₅

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂⁴

yes

characteristic

C₅×SD₁₆

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

SD₁₆

yes

complemented

C₂²×SD₁₆

C₂²×C₅

yes

—

C₂×#127

—

C₂×#127

—

C₂×#127

C₂²×C₅

yes

—

C₂×#102

C₂²×C₅

yes

—

#391

C₂²×C₅

yes

—

#389

C₂²×C₅

yes

—

#387

—

#383

—

C₄³

C₂²×C₅

yes

characteristic

#336

C₂²×C₅

yes

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂²×D₄

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

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×SD₁₆

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

H(5,𝔽₂)

yes

—

C₅×C₈

—

C₂²×Q₈

C₂³×C₅

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂³×C₅

yes

—

C₂×SD₁₆

—

C₂²×C₈

C₂³×C₅

yes

—

100

#127

—

101

#127

—

102

#127

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂³×C₅

yes

—

105

C₂×C₄²

C₅×D₄

yes

—

106

C₂×C₄²

C₅×D₄

yes

characteristic

107

C₂×C₄²

C₂³×C₅

yes

—

108

#102

—

109

#95

—

110

#95

—

111

#95

—

112

C₂²×C₅

—

113

C₂²×C₅

—

114

C₂²×C₅

—

115

C₂²×C₅

#535

yes

—

116

C₂²×C₅

C₂²×SD₁₆

yes

characteristic

117

C₂²×C₅

C₂×AGL(1,ℤ/8)

yes

—

118

C₄×C₅

—

119

C₄×C₅

—

120

C₄×C₅

—

121

C₄×C₅

—

122

C₄×C₅

#560

yes

—

123

C₂⁴

—

124

C₂×Q₈

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×C₅×D₄

yes

—

136

C₂²×C₄

C₂×C₅×D₄

yes

—

137

C₂²×C₄

C₂⁴×C₅

yes

characteristic

138

SD₁₆

—

139

C₂×C₈

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄²

—

143

C₄²

—

144

C₄²

C₅×SD₁₆

yes

complemented

145

C₂×C₅

—

146

C₂×C₅

#2818

yes

characteristic

147

C₂×C₅

#3208

yes

—

148

C₂×C₅

#3217

yes

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

C₂²×C₅×D₄

yes

characteristic

153

Q₈

—

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

C₂×C₅×SD₁₆

yes

—

166

C₂×C₄

C₅×AGL(1,ℤ/8)

yes

—

167

C₂×C₄

C₅×H(5,𝔽₂)

yes

—

168

C₈

—

169

C₅

#32798

yes

characteristic, direct product factor, Sylow

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

C₅×#535

yes

—

174

C₂²

C₂²×C₅×SD₁₆

yes

characteristic

175

C₂²

C₂×C₅×AGL(1,ℤ/8)

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

C₅×#560

yes

—

181

C₂

—

182

C₂

C₅×#2818

yes

characteristic

183

C₂

C₅×#3208

yes

—

184

C₂

C₅×#3217

yes

—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₅×#2867	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

subgroup

G/H

center, socle

C₂³×C₅

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₅

C₂

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

N	H ≅ G/N	type
2. C₅×#2863	C₂	semidirect
3. C₅×#2041	C₂	semidirect
4. C₅×#2014	C₂	semidirect
5. C₅×#1839	C₂	semidirect
17. #32798	C₅	direct (N × H)
37. #2863	C₂×C₅	semidirect
38. #2041	C₂×C₅	semidirect
39. #2014	C₂×C₅	semidirect
40. #1839	C₂×C₅	semidirect
62. C₄²×C₅	SD₁₆	semidirect
144. C₄²	C₅×SD₁₆	semidirect
169. C₅	#32798	direct (N × H)

H ≅ G/N

type

2. C₅×#2863

C₂

semidirect

3. C₅×#2041

C₂

semidirect

4. C₅×#2014

C₂

semidirect

5. C₅×#1839

C₂

semidirect

17. #32798

C₅

direct (N × H)

37. #2863

C₂×C₅

semidirect

38. #2041

C₂×C₅

semidirect

39. #2014

C₂×C₅

semidirect

40. #1839

C₂×C₅

semidirect

62. C₄²×C₅

SD₁₆

semidirect

144. C₄²

C₅×SD₁₆

semidirect

169. C₅

#32798

direct (N × H)

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

subgroup	index
C₅×#2867	2
C₅×#2863	2
C₅×#2041	2
C₅×#2014	2
C₅×#1839	2
#32798	5

subgroup

index

Subgroups — C₅×#32798 #11955518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#32798 #11955518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#32798 #11955518