Group #11955521 — subgroups

Subgroups — C₅×#32801 #11955521

Order 1280.

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

normal subgroups only

This group has 762 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×#2868	C₂	1	1	yes	maximal, characteristic
2	C₅×#2863	C₂	1	1	yes	maximal, characteristic
3	C₅×#2043	C₂	1	6	yes	maximal
4	C₅×#2016	C₂	1	6	yes	maximal
5	C₅×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
7	C₂×C₅×C₄⋊Q₈	C₂²	1	6	yes	—
8	C₂×C₅×#103	C₂²	1	12	yes	—
9	C₅×#391	C₂²	1	3	yes	—
10	C₅×#389	C₂²	1	3	yes	—
11	C₅×#387	—	2	3	no	—
12	C₅×#384	—	2	3	no	—
13	C₅×#384	—	2	1	no	—
14	C₅×#383	—	2	1	no	—
15	C₄³×C₅	C₂²	1	1	yes	characteristic
16	C₅×#336	C₂²	1	6	yes	—
17	#32801	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
19	C₂²×C₅×Q₈	C₂³	1	4	yes	—
20	C₂×C₅×Q₁₆	—	4	12	no	—
21	C₂²×C₅×C₈	C₂³	1	4	yes	—
22	C₅×C₄⋊Q₈	—	4	12	no	—
23	C₂×C₅×C₄⋊C₄	—	2	6	no	—
24	C₂×C₅×C₄⋊C₄	—	2	6	no	—
25	C₂×C₅×C₄⋊C₄	—	2	3	no	—
26	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
27	C₂×C₄²×C₅	D₄	1	3	yes	—
28	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂³	1	3	yes	—
30	C₅×#103	—	4	12	no	—
31	C₅×#95	—	2	3	no	—
32	C₅×#95	—	2	3	no	—
33	C₅×#95	—	2	3	no	—
34	#2868	C₂×C₅	1	1	yes	characteristic
35	#2863	C₂×C₅	1	1	yes	characteristic
36	#2043	C₂×C₅	1	6	yes	—
37	#2016	C₂×C₅	1	6	yes	—
38	#1839	C₂×C₅	1	1	yes	characteristic
39	C₂×C₅×Q₈	—	2	12	no	—
40	C₂×C₅×Q₈	—	2	12	no	—
41	C₂×C₅×Q₈	—	2	6	no	—
42	C₂²×C₄×C₅	—	2	4	no	—
43	C₂²×C₄×C₅	—	2	3	no	—
44	C₂²×C₄×C₅	—	2	1	no	—
45	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
48	C₅×Q₁₆	—	4	16	no	—
49	C₂×C₅×C₈	—	2	12	no	—
50	C₅×C₄⋊C₄	—	2	6	no	—
51	C₅×C₄⋊C₄	—	4	12	no	—
52	C₅×C₄⋊C₄	—	4	12	no	—
53	C₅×C₄⋊C₄	—	4	6	no	—
54	C₄²×C₅	—	2	6	no	—
55	C₄²×C₅	—	2	6	no	—
56	C₄²×C₅	Q₁₆	1	4	yes	complemented
57	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
58	C₂×C₄⋊Q₈	C₂²×C₅	1	6	yes	—
59	C₂×#103	C₂²×C₅	1	12	yes	—
60	#391	C₂²×C₅	1	3	yes	—
61	#389	C₂²×C₅	1	3	yes	—
62	#387	—	2	3	no	—
63	#384	—	2	3	no	—
64	#384	—	2	1	no	—
65	#383	—	2	1	no	—
66	C₄³	C₂²×C₅	1	1	yes	characteristic
67	#336	C₂²×C₅	1	6	yes	—
68	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
69	C₅×Q₈	—	4	4	no	—
70	C₅×Q₈	—	4	4	no	—
71	C₅×Q₈	—	4	12	no	—
72	C₂×C₄×C₅	—	2	3	no	—
73	C₂×C₄×C₅	—	2	3	no	—
74	C₂×C₄×C₅	—	2	3	no	—
75	C₂×C₄×C₅	—	2	3	no	—
76	C₂×C₄×C₅	—	4	3	no	—
77	C₂×C₄×C₅	—	4	12	no	—
78	C₂×C₄×C₅	—	4	3	no	—
79	C₂×C₄×C₅	—	4	6	no	—
80	C₂×C₄×C₅	C₂×Q₁₆	1	6	yes	—
81	C₂×C₄×C₅	#137	1	6	yes	—
82	C₂×C₄×C₅	H(5,𝔽₂)	1	6	yes	—
83	C₅×C₈	—	4	4	no	—
84	C₂²×Q₈	C₂³×C₅	1	4	yes	—
85	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
86	C₂×Q₁₆	—	4	12	no	—
87	C₂²×C₈	C₂³×C₅	1	4	yes	—
88	C₄⋊Q₈	—	4	12	no	—
89	C₂×C₄⋊C₄	—	2	6	no	—
90	C₂×C₄⋊C₄	—	2	6	no	—
91	C₂×C₄⋊C₄	—	2	3	no	—
92	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
93	C₂×C₄²	C₅×D₄	1	3	yes	—
94	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
95	C₂×C₄²	C₂³×C₅	1	3	yes	—
96	#103	—	4	12	no	—
97	#95	—	2	3	no	—
98	#95	—	2	3	no	—
99	#95	—	2	3	no	—
100	C₂²×C₅	#535	1	3	yes	—
101	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
102	C₂²×C₅	C₂×#137	1	3	yes	—
103	C₄×C₅	—	2	6	no	—
104	C₄×C₅	—	2	6	no	—
105	C₄×C₅	—	8	4	no	—
106	C₄×C₅	—	8	1	no	—
107	C₄×C₅	—	8	3	no	—
108	C₄×C₅	#560	1	4	yes	—
109	C₂×Q₈	—	2	12	no	—
110	C₂×Q₈	—	2	12	no	—
111	C₂×Q₈	—	2	6	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
116	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
117	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
118	Q₁₆	—	4	16	no	—
119	C₂×C₈	—	2	12	no	—
120	C₄⋊C₄	—	2	6	no	—
121	C₄⋊C₄	—	4	12	no	—
122	C₄⋊C₄	—	4	12	no	—
123	C₄⋊C₄	—	4	6	no	—
124	C₄²	—	2	6	no	—
125	C₄²	—	2	6	no	—
126	C₄²	C₅×Q₁₆	1	4	yes	complemented
127	C₂×C₅	#2818	1	1	yes	characteristic
128	C₂×C₅	#3209	1	3	yes	—
129	C₂×C₅	#3218	1	3	yes	—
130	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
131	Q₈	—	4	4	no	—
132	Q₈	—	4	4	no	—
133	Q₈	—	4	12	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	2	3	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	4	3	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	—	4	3	no	—
141	C₂×C₄	—	4	6	no	—
142	C₂×C₄	C₂×C₅×Q₁₆	1	6	yes	—
143	C₂×C₄	C₅×#137	1	6	yes	—
144	C₂×C₄	C₅×H(5,𝔽₂)	1	6	yes	—
145	C₈	—	4	4	no	—
146	C₅	#32801	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²	C₅×#535	1	3	yes	—
148	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
149	C₂²	C₂×C₅×#137	1	3	yes	—
150	C₄	—	2	6	no	—
151	C₄	—	2	6	no	—
152	C₄	—	8	4	no	—
153	C₄	—	8	1	no	—
154	C₄	—	8	3	no	—
155	C₄	C₅×#560	1	4	yes	—
156	C₂	C₅×#2818	1	1	yes	characteristic
157	C₂	C₅×#3209	1	3	yes	—
158	C₂	C₅×#3218	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2868

C₂

yes

maximal, characteristic

C₅×#2863

C₂

yes

maximal, characteristic

C₅×#2043

C₂

yes

maximal

C₅×#2016

C₂

yes

maximal

C₅×#1839

C₂

yes

maximal, characteristic

C₂²×C₅×Q₁₆

C₂²

yes

—

C₂×C₅×C₄⋊Q₈

C₂²

yes

—

C₂×C₅×#103

C₂²

yes

—

C₅×#391

C₂²

yes

—

C₅×#389

C₂²

yes

—

—

—

—

—

C₄³×C₅

C₂²

yes

characteristic

C₅×#336

C₂²

yes

—

#32801

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅×Q₈

C₂³

yes

characteristic

C₂²×C₅×Q₈

C₂³

yes

—

C₂×C₅×Q₁₆

—

C₂²×C₅×C₈

C₂³

yes

—

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

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×#103

—

C₅×#95

—

C₅×#95

—

C₅×#95

—

#2868

C₂×C₅

yes

characteristic

#2863

C₂×C₅

yes

characteristic

#2043

C₂×C₅

yes

—

#2016

C₂×C₅

yes

—

#1839

C₂×C₅

yes

characteristic

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

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×Q₁₆

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

Q₁₆

yes

complemented

C₂²×Q₁₆

C₂²×C₅

yes

—

C₂×C₄⋊Q₈

C₂²×C₅

yes

—

C₂×#103

C₂²×C₅

yes

—

#391

C₂²×C₅

yes

—

#389

C₂²×C₅

yes

—

—

—

—

—

C₄³

C₂²×C₅

yes

characteristic

#336

C₂²×C₅

yes

—

C₂³×C₅

C₂²×D₄

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

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×Q₁₆

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×C₄×C₅

H(5,𝔽₂)

yes

—

C₅×C₈

—

C₂²×Q₈

C₂³×C₅

yes

—

C₂²×Q₈

C₂³×C₅

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

C₂³×C₅

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³×C₅

yes

—

C₂×C₄²

C₅×D₄

yes

—

C₂×C₄²

C₅×D₄

yes

characteristic

C₂×C₄²

C₂³×C₅

yes

—

—

—

—

—

100

C₂²×C₅

#535

yes

—

101

C₂²×C₅

C₂²×Q₁₆

yes

characteristic

102

C₂²×C₅

C₂×#137

yes

—

103

C₄×C₅

—

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

—

107

C₄×C₅

—

108

C₄×C₅

#560

yes

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂×C₅×D₄

yes

—

116

C₂²×C₄

C₂×C₅×D₄

yes

—

117

C₂²×C₄

C₂⁴×C₅

yes

characteristic

118

Q₁₆

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₅×Q₁₆

yes

complemented

127

C₂×C₅

#2818

yes

characteristic

128

C₂×C₅

#3209

yes

—

129

C₂×C₅

#3218

yes

—

130

C₂³

C₂²×C₅×D₄

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

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

C₂×C₅×Q₁₆

yes

—

143

C₂×C₄

C₅×#137

yes

—

144

C₂×C₄

C₅×H(5,𝔽₂)

yes

—

145

C₈

—

146

C₅

#32801

yes

characteristic, direct product factor, Sylow

147

C₂²

C₅×#535

yes

—

148

C₂²

C₂²×C₅×Q₁₆

yes

characteristic

149

C₂²

C₂×C₅×#137

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

C₅×#560

yes

—

156

C₂

C₅×#2818

yes

characteristic

157

C₂

C₅×#3209

yes

—

158

C₂

C₅×#3218

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₅

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₅

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

N	H ≅ G/N	type
17. #32801	C₅	direct (N × H)
56. C₄²×C₅	Q₁₆	semidirect
126. C₄²	C₅×Q₁₆	semidirect
146. C₅	#32801	direct (N × H)

H ≅ G/N

type

17. #32801

C₅

direct (N × H)

56. C₄²×C₅

Q₁₆

semidirect

126. C₄²

C₅×Q₁₆

semidirect

146. C₅

#32801

direct (N × H)

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

subgroup	index
C₅×#2868	2
C₅×#2863	2
C₅×#2043	2
C₅×#2016	2
C₅×#1839	2
#32801	5

subgroup

index

Subgroups — C₅×#32801 #11955521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#32801 #11955521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#32801 #11955521