Group #13100722 — subgroups

Subgroups — #13100722

Order 1600.

334 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 62 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	#11717213	C₂	1	1	yes	maximal, characteristic
2	#11716985	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87774	C₂	1	2	yes	maximal
4	C₅×#3942	C₂	1	2	yes	maximal, complemented
5	#11716613	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅⋊D₅	—	2	1	no	—
8	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
9	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	#87808	—	2	1	no	—
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₂×C₅×#168	C₂²	1	2	yes	—
13	#87781	—	2	1	no	—
14	#87781	—	2	1	no	—
15	#87774	—	2	4	no	—
16	#65107	—	5	2	no	maximal
17	C₂²×C₅⋊D₅	—	2	1	no	—
18	C₂×C₄×C₅²	—	2	1	no	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	C₂×#1039	D₄	1	1	yes	characteristic
22	C₄⋊(C₅⋊D₅)	—	2	1	no	—
23	C₄⋊(C₅⋊D₅)	D₄	1	2	yes	—
24	C₄×C₅⋊D₅	—	4	2	no	—
25	C₂×C₅×Dic₅	—	2	2	no	—
26	C₅×Dic₁₀	—	2	2	no	—
27	C₅×Dic₁₀	—	2	2	no	—
28	C₅×#168	—	2	2	no	—
29	#4090	—	5	2	no	—
30	C₂×#688	—	5	2	no	—
31	C₂×#677	—	5	2	no	—
32	#4039	—	5	2	no	—
33	#4039	—	10	2	no	—
34	C₅×#103	—	5	2	no	—
35	#3946	—	5	2	no	—
36	#3942	D₅	1	2	yes	complemented
37	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
38	C₂×C₅⋊D₅	—	2	2	no	—
39	C₂×C₅⋊D₅	—	4	1	no	—
40	C₂×C₅⋊D₅	—	4	1	no	—
41	C₄×C₅²	—	4	1	no	—
42	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
43	C₄×C₅²	#41	1	1	yes	characteristic
44	#1039	—	2	1	no	—
45	#1039	SD₁₆	1	2	yes	complemented
46	C₅×Dic₅	—	4	2	no	—
47	C₂×C₅×Q₈	—	5	2	no	—
48	C₂×D₂₀	—	5	2	no	—
49	C₂×D₂₀	—	10	2	no	—
50	C₂×C₄×D₅	—	10	2	no	—
51	C₂×C₄×D₅	—	10	4	no	—
52	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
53	C₂×C₅×C₈	—	5	2	no	—
54	C₅×C₄⋊C₄	—	2	2	no	—
55	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
56	#688	—	10	4	no	—
57	#685	—	10	2	no	—
58	#685	—	10	4	no	—
59	#683	—	10	2	no	—
60	#683	—	10	2	no	—
61	C₄×Dic₅	—	5	2	no	—
62	C₄×Dic₅	—	10	2	no	—
63	C₂×#168	C₂×D₅	1	2	yes	—
64	#677	—	10	4	no	—
65	#461	—	25	1	no	Sylow
66	C₂×C₅²	#121	1	1	yes	characteristic
67	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₅²	#137	1	1	yes	characteristic
69	C₅⋊D₅	—	4	2	no	—
70	C₂²×D₅	—	10	4	no	—
71	C₂²×D₅	—	10	2	no	—
72	C₅×Q₈	—	10	2	no	—
73	C₅×Q₈	—	10	2	no	—
74	C₂×C₄×C₅	—	2	4	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	10	2	no	—
78	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
79	C₂×Dic₅	—	2	2	no	—
80	C₂×Dic₅	—	5	2	no	—
81	C₂×Dic₅	—	5	2	no	—
82	C₂×Dic₅	—	10	2	no	—
83	C₂×Dic₅	—	10	2	no	—
84	D₂₀	—	5	4	no	—
85	D₂₀	—	10	4	no	—
86	D₂₀	—	10	4	no	—
87	D₂₀	—	10	2	no	—
88	C₄×D₅	—	20	4	no	—
89	C₄×D₅	—	20	8	no	—
90	Dic₁₀	—	2	2	no	—
91	Dic₁₀	—	2	2	no	—
92	C₅×C₈	—	10	2	no	—
93	#168	—	2	2	no	—
94	C₂×SD₁₆	—	25	2	no	—
95	C₄⋊Q₈	—	25	1	no	—
96	C₄×D₄	—	25	1	no	—
97	C₄⋊C₈	—	25	1	no	—
98	#103	—	5	2	no	—
99	C₅²	#461	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	D₄×D₅	1	2	yes	—
102	C₂×D₅	—	10	8	no	—
103	C₂×D₅	—	10	4	no	—
104	C₂×D₅	—	20	4	no	—
105	C₂×D₅	—	20	4	no	—
106	C₂×D₅	—	20	2	no	—
107	C₂×D₅	—	20	2	no	—
108	C₄×C₅	—	2	2	no	—
109	C₄×C₅	—	2	2	no	—
110	C₄×C₅	—	4	4	no	—
111	C₄×C₅	—	4	2	no	—
112	C₄×C₅	—	20	2	no	—
113	C₄×C₅	#709	1	2	yes	—
114	C₄×C₅	D₄×D₅	1	2	yes	—
115	Dic₅	—	4	2	no	—
116	Dic₅	—	5	4	no	—
117	Dic₅	—	10	4	no	—
118	Dic₅	—	10	2	no	—
119	Dic₅	—	20	2	no	—
120	C₂×Q₈	—	5	2	no	—
121	C₂×D₄	—	25	1	no	—
122	C₂²×C₄	—	50	1	no	—
123	SD₁₆	—	50	4	no	—
124	C₂×C₈	—	5	2	no	—
125	C₄⋊C₄	—	50	1	no	—
126	C₄⋊C₄	—	50	1	no	—
127	C₄⋊C₄	D₅²	1	1	yes	characteristic
128	C₂²⋊C₄	—	50	1	no	—
129	C₄²	—	25	1	no	—
130	C₂×C₅	—	2	2	no	—
131	C₂×C₅	—	2	2	no	—
132	C₂×C₅	—	2	2	no	—
133	C₂×C₅	#4030	1	2	yes	—
134	C₂×C₅	D₅×SD₁₆	1	2	yes	—
135	C₂×C₅	#4064	1	2	yes	—
136	D₅	—	20	8	no	—
137	D₅	—	20	4	no	—
138	C₂³	—	50	1	no	—
139	Q₈	—	10	2	no	—
140	Q₈	—	10	2	no	—
141	D₄	—	25	2	no	—
142	D₄	—	50	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	10	2	no	—
145	C₂×C₄	—	25	1	no	—
146	C₂×C₄	—	25	1	no	—
147	C₂×C₄	—	100	2	no	—
148	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
149	C₈	—	10	2	no	—
150	C₅	—	2	2	no	—
151	C₅	#65107	1	2	yes	complemented
152	C₂²	—	50	2	no	—
153	C₂²	—	100	1	no	—
154	C₂²	—	100	1	no	—
155	C₂²	#87886	1	1	yes	characteristic
156	C₄	—	4	1	no	—
157	C₄	—	20	2	no	—
158	C₄	—	25	2	no	—
159	C₄	—	50	1	no	—
160	C₄	#87873	1	1	yes	characteristic
161	C₄	#87877	1	1	yes	characteristic
162	C₂	—	100	2	no	—
163	C₂	#11717030	1	1	yes	characteristic
164	C₂	#11717048	1	1	yes	characteristic
165	C₂	#11717063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717213

C₂

yes

maximal, characteristic

#11716985

C₂

yes

maximal, characteristic, complemented

C₂×#87774

C₂

yes

maximal

C₅×#3942

C₂

yes

maximal, complemented

#11716613

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×C₄×C₅⋊D₅

—

C₂×C₅×Dic₁₀

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

characteristic

#87808

—

C₄×#1039

C₂²

yes

characteristic

C₂×C₅×#168

C₂²

yes

—

—

—

—

—

C₂²×C₅⋊D₅

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

D₄

yes

—

C₄×C₅⋊D₅

—

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

—

—

—

—

—

—

—

—

—

D₅

yes

complemented

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

SD₁₆

yes

complemented

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

—

—

—

—

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄×D₄

—

C₄⋊C₈

—

#103

—

C₅²

#461

yes

characteristic, complemented, Sylow

100

C₂²×C₅

—

101

C₂²×C₅

D₄×D₅

yes

—

102

C₂×D₅

—

103

C₂×D₅

—

104

C₂×D₅

—

105

C₂×D₅

—

106

C₂×D₅

—

107

C₂×D₅

—

108

C₄×C₅

—

109

C₄×C₅

—

110

C₄×C₅

—

111

C₄×C₅

—

112

C₄×C₅

—

113

C₄×C₅

#709

yes

—

114

C₄×C₅

D₄×D₅

yes

—

115

Dic₅

—

116

Dic₅

—

117

Dic₅

—

118

Dic₅

—

119

Dic₅

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂²×C₄

—

123

SD₁₆

—

124

C₂×C₈

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

D₅²

yes

characteristic

128

C₂²⋊C₄

—

129

C₄²

—

130

C₂×C₅

—

131

C₂×C₅

—

132

C₂×C₅

—

133

C₂×C₅

#4030

yes

—

134

C₂×C₅

D₅×SD₁₆

yes

—

135

C₂×C₅

#4064

yes

—

136

D₅

—

137

D₅

—

138

C₂³

—

139

Q₈

—

140

Q₈

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

100

—

148

C₂×C₄

C₂×D₅²

yes

characteristic

149

C₈

—

150

C₅

—

151

C₅

#65107

yes

complemented

152

C₂²

—

153

C₂²

—

100

—

154

C₂²

—

100

—

155

C₂²

#87886

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#87873

yes

characteristic

161

C₄

#87877

yes

characteristic

162

C₂

—

100

—

163

C₂

#11717030

yes

characteristic

164

C₂

#11717048

yes

characteristic

165

C₂

#11717063

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₅²×C₄⋊C₄	C₂²
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	C₄⋊C₄	D₅²
hypocenter	C₅²	#461
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₅²×C₄⋊C₄

C₂²

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×C₄⋊(C₅⋊D₅)

C₂²

hypercenter

C₄⋊C₄

D₅²

hypocenter

C₅²

#461

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

N	H ≅ G/N	type
2. #11716985	C₂	semidirect
4. C₅×#3942	C₂	semidirect
5. #11716613	C₂	semidirect
36. #3942	D₅	semidirect
45. #1039	SD₁₆	semidirect
99. C₅²	#461	semidirect
151. C₅	#65107	semidirect

H ≅ G/N

type

2. #11716985

C₂

semidirect

4. C₅×#3942

C₂

semidirect

5. #11716613

C₂

semidirect

36. #3942

D₅

semidirect

45. #1039

SD₁₆

semidirect

99. C₅²

#461

semidirect

151. C₅

#65107

semidirect

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

subgroup	index
#11717213	2
#11716985	2
C₂×#87774	2
C₅×#3942	2
#11716613	2
#65107	5

subgroup

index

Subgroups — #13100722

Conjugacy classes of subgroups

Complements

Maximal subgroups