Group #13104867 — subgroups

Subgroups — #13104867

Order 1600.

272 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 66 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	#11717448	C₂	1	1	yes	maximal, characteristic
2	C₂×#87863	C₂	1	1	yes	maximal, characteristic
3	M₁₆×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
4	#11716889	C₂	1	2	yes	maximal, complemented
5	#11716888	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	#87867	—	2	1	no	—
8	C₂×#6034	—	2	1	no	—
9	#87863	—	2	1	no	—
10	#87863	C₂²	1	1	yes	characteristic
11	#87863	C₂²	1	2	yes	—
12	#87863	C₂²	1	1	yes	characteristic
13	#87862	—	2	1	no	—
14	C₅²×M₁₆	C₄	1	1	yes	characteristic
15	#87804	C₄	1	1	yes	characteristic
16	#87799	C₄	1	2	yes	—
17	C₈×C₅⋊D₅	C₂²	1	2	yes	—
18	#65729	—	5	2	no	maximal
19	C₂²×C₅⋊D₅	Q₈	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
21	C₂×#1039	D₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
24	C₄×C₅⋊D₅	Q₈	1	1	yes	characteristic
25	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
26	#6034	—	2	1	no	—
27	#6034	—	2	1	no	—
28	#6034	—	2	2	no	—
29	C₅²×C₈	C₂×C₄	1	2	yes	—
30	#6029	C₂×C₄	1	2	yes	—
31	D₅⋊M₁₆	—	5	2	no	—
32	C₂×#700	—	5	2	no	—
33	D₅×M₁₆	—	5	2	no	—
34	D₅×M₁₆	—	10	2	no	—
35	#3996	—	5	4	no	—
36	#3995	—	5	4	no	—
37	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₅⋊D₅	—	4	1	no	—
39	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₅⋊D₅	C₂×Q₈	1	1	yes	characteristic
41	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
43	#1039	C₄⋊C₄	1	1	yes	characteristic
44	#1039	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×D₅	—	5	2	no	—
46	C₂×C₄×D₅	—	10	2	no	—
47	#704	—	10	2	no	—
48	C₂×#170	—	10	2	no	—
49	#700	—	5	4	no	—
50	#700	—	5	2	no	—
51	#700	—	5	2	no	—
52	#700	—	10	2	no	—
53	D₅⋊C₈	—	10	2	no	—
54	C₅×M₁₆	—	2	2	no	—
55	C₅×M₁₆	AGL(1,𝔽₅)	1	2	yes	—
56	#681	—	5	2	no	—
57	#681	—	10	2	no	—
58	#676	—	5	4	no	—
59	#676	—	10	4	no	—
60	C₈×D₅	—	5	4	no	—
61	C₈×D₅	—	10	4	no	—
62	#430	—	25	1	no	Sylow
63	C₂×C₅²	—	2	1	no	—
64	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₅⋊D₅	—	2	1	no	—
66	C₅⋊D₅	—	2	1	no	—
67	C₂²×D₅	—	5	2	no	—
68	C₂²×D₅	—	10	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
71	C₂×Dic₅	—	5	2	no	—
72	C₂×Dic₅	—	10	2	no	—
73	C₄×D₅	—	5	2	no	—
74	C₄×D₅	—	5	2	no	—
75	C₄×D₅	—	5	2	no	—
76	C₄×D₅	—	5	2	no	—
77	C₄×D₅	—	10	2	no	—
78	C₄×D₅	—	10	2	no	—
79	C₄×D₅	—	10	2	no	—
80	C₄×D₅	—	10	2	no	—
81	#170	—	10	2	no	—
82	#170	—	10	2	no	—
83	#170	—	10	4	no	—
84	C₅×C₈	—	2	4	no	—
85	C₅×C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
86	#168	—	5	4	no	—
87	#168	—	10	4	no	—
88	C₂×M₁₆	—	25	1	no	—
89	C₂×M₁₆	—	25	1	no	—
90	C₂×M₁₆	—	25	1	no	—
91	#108	—	25	2	no	—
92	#108	—	25	2	no	—
93	C₅²	#430	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₅	—	2	2	no	—
95	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
96	C₂×D₅	—	5	2	no	—
97	C₂×D₅	—	5	2	no	—
98	C₂×D₅	—	10	2	no	—
99	C₂×D₅	—	10	2	no	—
100	C₂×D₅	—	20	2	no	—
101	C₂×D₅	—	20	4	no	—
102	C₄×C₅	—	2	2	no	—
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
105	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
106	Dic₅	—	5	2	no	—
107	Dic₅	—	5	2	no	—
108	Dic₅	—	10	2	no	—
109	Dic₅	—	10	2	no	—
110	C₂²×C₄	—	25	1	no	—
111	M₁₆	—	25	1	no	—
112	M₁₆	—	25	2	no	—
113	M₁₆	—	25	1	no	—
114	M₁₆	—	25	2	no	—
115	M₁₆	—	25	1	no	—
116	M₁₆	—	50	1	no	—
117	M₁₆	—	50	1	no	—
118	M₁₆	#1044	1	1	yes	characteristic
119	C₂×C₈	—	25	2	no	—
120	C₂×C₈	—	50	1	no	—
121	C₂×C₈	—	50	1	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	—	2	2	no	—
124	C₂×C₅	—	4	2	no	—
125	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
126	D₅	—	10	4	no	—
127	D₅	—	10	2	no	—
128	D₅	—	10	2	no	—
129	D₅	—	20	2	no	—
130	C₂³	—	25	1	no	—
131	C₂×C₄	—	25	1	no	—
132	C₂×C₄	—	25	1	no	—
133	C₂×C₄	—	25	1	no	—
134	C₂×C₄	—	25	1	no	—
135	C₂×C₄	—	25	1	no	—
136	C₂×C₄	C₂×#1044	1	1	yes	characteristic
137	C₈	—	25	2	no	—
138	C₈	—	50	1	no	—
139	C₈	—	50	1	no	—
140	C₈	—	50	2	no	—
141	C₈	C₂×#1044	1	2	yes	—
142	C₅	—	2	2	no	—
143	C₅	#65729	1	2	yes	complemented
144	C₂²	—	25	1	no	—
145	C₂²	—	25	1	no	—
146	C₂²	—	100	1	no	—
147	C₂²	C₄⋊#1044	1	1	yes	characteristic
148	C₄	—	25	1	no	—
149	C₄	—	25	1	no	—
150	C₄	C₄⋊#1044	1	1	yes	characteristic
151	C₄	C₂²×#1044	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	50	1	no	—
154	C₂	—	50	1	no	—
155	C₂	C₂×C₄⋊#1044	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717448

C₂

yes

maximal, characteristic

C₂×#87863

C₂

yes

maximal, characteristic

M₁₆×C₅⋊D₅

C₂

yes

maximal, characteristic

#11716889

C₂

yes

maximal, complemented

#11716888

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

—

—

—

C₂²

yes

characteristic

#87863

C₂²

yes

—

#87863

C₂²

yes

characteristic

#87862

—

C₅²×M₁₆

C₄

yes

characteristic

#87804

C₄

yes

characteristic

#87799

C₄

yes

—

C₈×C₅⋊D₅

C₂²

yes

—

#65729

—

maximal

C₂²×C₅⋊D₅

Q₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₄×C₅⋊D₅

Q₈

yes

characteristic

C₄×C₅⋊D₅

C₂³

yes

—

—

—

C₅²×C₈

C₂×C₄

yes

—

#6029

C₂×C₄

yes

—

D₅⋊M₁₆

—

C₂×#700

—

D₅×M₁₆

—

D₅×M₁₆

—

#3996

—

#3995

—

C₂²×C₅²

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

C₂×Q₈

yes

characteristic

C₄×C₅²

C₄⋊C₄

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

C₄⋊C₄

yes

characteristic

#1039

C₂×D₄

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

—

—

—

—

—

—

D₅⋊C₈

—

C₅×M₁₆

—

C₅×M₁₆

AGL(1,𝔽₅)

yes

—

—

—

—

—

C₈×D₅

—

C₈×D₅

—

#430

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

#170

—

#170

—

C₅×C₈

—

C₅×C₈

C₂×AGL(1,𝔽₅)

yes

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

#108

—

#108

—

C₅²

#430

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

100

C₂×D₅

—

101

C₂×D₅

—

102

C₄×C₅

—

103

C₄×C₅

—

104

C₄×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

105

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

106

Dic₅

—

107

Dic₅

—

108

Dic₅

—

109

Dic₅

—

110

C₂²×C₄

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

#1044

yes

characteristic

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₅

—

123

C₂×C₅

—

124

C₂×C₅

—

125

C₂×C₅

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

126

D₅

—

127

D₅

—

128

D₅

—

129

D₅

—

130

C₂³

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂×#1044

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

C₂×#1044

yes

—

142

C₅

—

143

C₅

#65729

yes

complemented

144

C₂²

—

145

C₂²

—

146

C₂²

—

100

—

147

C₂²

C₄⋊#1044

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

C₄⋊#1044

yes

characteristic

151

C₄

C₂²×#1044

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

—

155

C₂

C₂×C₄⋊#1044

yes

characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#1044
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₄	C₂²×#1044
Fitting	C₅²×M₁₆	C₄
socle	C₂×C₅²	C₂×C₄⋊C₄
prime radical	C₂²×C₅⋊D₅	Q₈
cosocle	C₄×C₅⋊D₅	C₂³
hypercenter	M₁₆	#1044
hypocenter	C₅²	#430
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

subgroup

G/H

center

C₂

C₂×C₄⋊#1044

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₄

C₂²×#1044

Fitting

C₅²×M₁₆

C₄

socle

C₂×C₅²

C₂×C₄⋊C₄

prime radical

C₂²×C₅⋊D₅

Q₈

cosocle

C₄×C₅⋊D₅

C₂³

M₁₆

C₅²

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

N	H ≅ G/N	type
4. #11716889	C₂	semidirect
5. #11716888	C₂	semidirect
93. C₅²	#430	semidirect
143. C₅	#65729	semidirect

H ≅ G/N

type

4. #11716889

C₂

semidirect

5. #11716888

C₂

semidirect

93. C₅²

#430

semidirect

143. C₅

#65729

semidirect

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

subgroup	index
#11717448	2
C₂×#87863	2
M₁₆×C₅⋊D₅	2
#11716889	2
#11716888	2
#65729	5

subgroup

index

M₁₆×C₅⋊D₅

Subgroups — #13104867

Conjugacy classes of subgroups

Complements

Maximal subgroups