Group #13101862 — subgroups

Subgroups — #13101862

Order 1600.

262 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11717268	C₂	1	1	yes	maximal, characteristic
2	C₅×#4097	C₂	1	2	yes	maximal
3	C₂×#87775	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716619	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716601	C₂	1	2	yes	maximal, complemented
6	C₅²×#41	C₂²	1	1	yes	characteristic
7	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
8	#87804	—	2	1	no	—
9	#87804	C₂²	1	1	yes	characteristic
10	C₂×#6029	—	2	1	no	—
11	C₂×#6029	C₂²	1	1	yes	characteristic
12	C₅×#689	—	2	2	no	—
13	C₅×#687	—	2	2	no	—
14	C₅×#681	C₂²	1	2	yes	—
15	#87775	—	2	1	no	—
16	#87775	—	2	1	no	—
17	#65493	—	5	2	no	maximal
18	C₅²×Q₈	D₄	1	1	yes	characteristic
19	C₅²×D₄	—	2	1	no	—
20	C₅²×D₄	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅²	—	2	1	no	—
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₂×C₅×Dic₅	—	2	2	no	—
24	C₅×Dic₁₀	—	2	2	no	—
25	C₅×Dic₁₀	—	2	2	no	—
26	#6029	—	2	1	no	—
27	#6029	D₄	1	1	yes	characteristic
28	#6029	D₄	1	1	yes	characteristic
29	C₅×#168	—	2	2	no	—
30	C₅×#137	—	5	2	no	—
31	#4097	D₅	1	2	yes	—
32	#4094	—	5	2	no	—
33	#4094	—	10	2	no	—
34	C₂×#689	—	5	2	no	—
35	#3968	—	5	2	no	—
36	#3956	—	5	2	no	—
37	#3954	—	5	2	no	—
38	C₂²×C₅²	—	2	1	no	—
39	C₂²×C₅²	#41	1	1	yes	characteristic
40	C₄×C₅²	—	2	1	no	—
41	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
42	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
43	C₅×Dic₅	—	4	2	no	—
44	C₅×#41	—	2	2	no	—
45	C₅×#41	C₂×D₅	1	2	yes	—
46	C₂×C₅×Q₈	—	5	2	no	—
47	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
48	C₅×Q₁₆	—	10	2	no	—
49	C₅×SD₁₆	—	10	2	no	—
50	C₅×M₁₆	—	5	2	no	—
51	#689	—	2	2	no	—
52	#689	—	10	2	no	—
53	#689	—	10	2	no	—
54	#687	—	2	2	no	—
55	#681	—	5	2	no	—
56	#681	—	10	2	no	—
57	#681	—	10	2	no	—
58	#681	—	10	4	no	—
59	#681	C₂×D₅	1	2	yes	—
60	C₂×#168	—	5	2	no	—
61	C₂×#168	—	10	2	no	—
62	C₂×#168	—	10	2	no	—
63	C₂×#168	—	10	4	no	—
64	#473	—	25	1	no	Sylow
65	C₂×C₅²	—	2	1	no	—
66	C₂×C₅²	—	4	1	no	—
67	C₂×C₅²	#121	1	1	yes	characteristic
68	C₅×Q₈	—	2	2	no	—
69	C₅×Q₈	—	10	2	no	—
70	C₅×Q₈	—	10	2	no	—
71	C₅×Q₈	C₂²⋊D₅	1	2	yes	—
72	C₅×D₄	—	2	4	no	—
73	C₅×D₄	—	2	2	no	—
74	C₅×D₄	—	2	2	no	—
75	C₅×D₄	C₂²⋊D₅	1	2	yes	—
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	10	2	no	—
80	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
81	C₂×Dic₅	—	2	2	no	—
82	Dic₁₀	—	2	2	no	—
83	Dic₁₀	—	2	2	no	—
84	C₅×C₈	—	10	2	no	—
85	#168	—	2	2	no	—
86	#168	—	5	2	no	—
87	#168	—	5	2	no	—
88	#168	—	10	2	no	—
89	#168	—	10	2	no	—
90	#168	—	10	4	no	—
91	#168	—	10	2	no	—
92	#137	—	5	2	no	—
93	C₂×Q₁₆	—	25	1	no	—
94	#131	—	25	1	no	—
95	#108	—	25	1	no	—
96	#101	—	25	2	no	—
97	C₅²	#473	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₅	—	2	4	no	—
99	C₂²×C₅	—	2	2	no	—
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	#711	1	2	yes	—
102	C₄×C₅	—	2	4	no	—
103	C₄×C₅	—	2	2	no	—
104	C₄×C₅	—	2	2	no	—
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	—	20	2	no	—
107	C₄×C₅	D₄×D₅	1	2	yes	—
108	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
109	Dic₅	—	4	2	no	—
110	#41	D₅²	1	1	yes	characteristic
111	C₂×Q₈	—	5	2	no	—
112	Q₁₆	—	10	2	no	—
113	Q₁₆	—	50	1	no	—
114	Q₁₆	—	50	1	no	—
115	SD₁₆	—	10	2	no	—
116	M₁₆	—	5	2	no	—
117	M₁₆	—	25	1	no	—
118	M₁₆	—	50	1	no	—
119	C₂×C₈	—	25	1	no	—
120	C₂×C₈	—	50	1	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	—	4	2	no	—
124	C₂×C₅	—	4	4	no	—
125	C₂×C₅	—	4	2	no	—
126	C₂×C₅	#4085	1	2	yes	—
127	Q₈	—	10	2	no	—
128	Q₈	—	10	2	no	—
129	Q₈	#6037	1	1	yes	characteristic
130	D₄	—	2	1	no	—
131	D₄	#6037	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	10	2	no	—
134	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
135	C₈	—	10	2	no	—
136	C₈	—	25	1	no	—
137	C₈	—	25	1	no	—
138	C₈	—	50	1	no	—
139	C₅	—	2	2	no	—
140	C₅	#65493	1	2	yes	complemented
141	C₂²	—	2	1	no	—
142	C₂²	#87880	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	20	2	no	—
145	C₄	C₂×#6037	1	1	yes	characteristic
146	C₄	#87886	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	4	1	no	—
149	C₂	#11717098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717268

C₂

yes

maximal, characteristic

C₅×#4097

C₂

yes

maximal

C₂×#87775

C₂

yes

maximal, characteristic, complemented

#11716619

C₂

yes

maximal, characteristic, complemented

#11716601

C₂

yes

maximal, complemented

C₅²×#41

C₂²

yes

characteristic

C₂×C₅×Dic₁₀

C₂²

yes

—

#87804

—

#87804

C₂²

yes

characteristic

C₂×#6029

—

C₂×#6029

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₅²×Q₈

D₄

yes

characteristic

C₅²×D₄

—

C₅²×D₄

D₄

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

—

#6029

—

#6029

D₄

yes

characteristic

#6029

D₄

yes

—

—

D₅

yes

—

—

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

#41

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

C₅×Dic₅

—

C₅×#41

—

C₅×#41

C₂×D₅

yes

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₅×Q₁₆

—

C₅×SD₁₆

—

C₅×M₁₆

—

—

—

—

—

—

—

—

—

C₂×D₅

yes

—

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

C₂²⋊D₅

yes

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

—

—

—

—

—

—

—

—

C₂×Q₁₆

—

#131

—

#108

—

#101

—

C₅²

#473

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

100

C₂²×C₅

—

101

C₂²×C₅

#711

yes

—

102

C₄×C₅

—

103

C₄×C₅

—

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

—

107

C₄×C₅

D₄×D₅

yes

—

108

C₄×C₅

C₂×C₂²⋊D₅

yes

—

109

Dic₅

—

110

#41

D₅²

yes

characteristic

111

C₂×Q₈

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

115

SD₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₅

—

122

C₂×C₅

—

123

C₂×C₅

—

124

C₂×C₅

—

125

C₂×C₅

—

126

C₂×C₅

#4085

yes

—

127

Q₈

—

128

Q₈

—

129

Q₈

#6037

yes

characteristic

130

D₄

—

131

D₄

#6037

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂×D₅²

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₅

—

140

C₅

#65493

yes

complemented

141

C₂²

—

142

C₂²

#87880

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

C₂×#6037

yes

characteristic

146

C₄

#87886

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#11717098

yes

characteristic

	subgroup	G/H
center	C₂	#11717098
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting, prime radical	C₅²×#41	C₂²
socle	C₂×C₅²	#121
hypercenter	#41	D₅²
hypocenter	C₅²	#473
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

subgroup

G/H

center

C₂

#11717098

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting, prime radical

C₅²×#41

C₂²

C₂×C₅²

D₅²

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

N	H ≅ G/N	type
3. C₂×#87775	C₂	semidirect
4. #11716619	C₂	semidirect
5. #11716601	C₂	semidirect
97. C₅²	#473	semidirect
140. C₅	#65493	semidirect

H ≅ G/N

type

3. C₂×#87775

C₂

semidirect

4. #11716619

C₂

semidirect

5. #11716601

C₂

semidirect

97. C₅²

#473

semidirect

140. C₅

#65493

semidirect

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

subgroup	index
#11717268	2
C₅×#4097	2
C₂×#87775	2
#11716619	2
#11716601	2
#65493	5

subgroup

index

Subgroups — #13101862

Conjugacy classes of subgroups

Complements

Maximal subgroups