Group #13102617 — subgroups

Subgroups — #13102617

Order 1600.

400 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 95 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	C₂×#87812	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717262	C₂	1	1	yes	maximal, characteristic
3	#11717229	C₂	1	1	yes	maximal, characteristic
4	#11717227	C₂	1	1	yes	maximal, characteristic
5	C₅²×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716691	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716689	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
9	#87900	C₂²	1	1	yes	characteristic
10	C₂×#6045	C₂²	1	1	yes	characteristic
11	C₅²×M₁₆	C₂²	1	1	yes	characteristic
12	C₅²×M₁₆	C₂²	1	1	yes	characteristic
13	#87812	—	2	1	no	—
14	#87812	—	2	1	no	—
15	#87812	—	2	1	no	—
16	#87811	—	2	1	no	—
17	#87804	C₂²	1	1	yes	characteristic
18	C₂×#6029	C₂²	1	1	yes	characteristic
19	#87802	—	2	1	no	—
20	#87800	—	2	1	no	—
21	#87799	—	2	1	no	—
22	C₈×C₅⋊D₅	—	2	1	no	—
23	#65044	—	5	6	no	maximal
24	C₅²×Q₈	—	2	1	no	—
25	C₅²×Q₈	—	2	1	no	—
26	C₂×C₄×C₅²	—	2	1	no	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
29	C₂×#1039	—	2	1	no	—
30	C₄⋊(C₅⋊D₅)	D₄	1	1	yes	characteristic
31	C₄×C₅⋊D₅	—	2	1	no	—
32	#6045	—	2	1	no	—
33	#6045	—	2	1	no	—
34	#6045	D₄	1	1	yes	characteristic
35	C₅²×C₈	—	2	1	no	—
36	C₅²×C₈	—	2	1	no	—
37	#6029	—	2	1	no	—
38	#6029	D₄	1	1	yes	characteristic
39	#6029	D₄	1	1	yes	characteristic
40	C₂×#689	—	5	6	no	—
41	#4088	—	5	6	no	—
42	#4055	—	5	6	no	—
43	#4053	—	5	6	no	—
44	C₅×#101	D₅	1	6	yes	complemented
45	#3956	—	5	6	no	—
46	#3954	—	5	6	no	—
47	C₂²×C₅²	#41	1	1	yes	characteristic
48	C₂×C₅⋊D₅	—	2	1	no	—
49	C₄×C₅²	—	4	1	no	—
50	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
51	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
52	#1039	—	2	1	no	—
53	#1039	—	4	1	no	—
54	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
55	#709	—	5	6	no	—
56	C₂×Dic₁₀	—	5	6	no	—
57	C₅×M₁₆	C₂×D₅	1	6	yes	—
58	C₅×M₁₆	C₂×D₅	1	6	yes	—
59	#689	—	10	6	no	—
60	#689	—	10	6	no	—
61	#689	—	10	6	no	—
62	#688	—	10	6	no	—
63	#681	—	5	6	no	—
64	C₂×#168	—	5	6	no	—
65	Dic₂₀	—	10	6	no	—
66	#677	—	10	6	no	—
67	#676	—	10	6	no	—
68	C₈×D₅	—	10	6	no	—
69	#473	—	25	1	no	Sylow
70	C₂×C₅²	—	2	1	no	—
71	C₂×C₅²	#121	1	1	yes	characteristic
72	C₅⋊D₅	—	4	1	no	—
73	C₅×Q₈	—	2	6	no	—
74	C₅×Q₈	—	2	6	no	—
75	C₂×C₄×C₅	—	2	6	no	—
76	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
77	C₂²⋊D₅	—	10	6	no	—
78	C₂×Dic₅	—	10	6	no	—
79	D₂₀	—	5	6	no	—
80	C₄×D₅	—	10	6	no	—
81	Dic₁₀	—	5	6	no	—
82	Dic₁₀	—	10	6	no	—
83	Dic₁₀	—	10	6	no	—
84	C₅×C₈	—	2	6	no	—
85	C₅×C₈	—	2	6	no	—
86	#168	—	5	6	no	—
87	#168	—	5	6	no	—
88	#168	—	10	6	no	—
89	#137	—	25	1	no	—
90	#137	—	25	1	no	—
91	C₂×Q₁₆	—	25	1	no	—
92	#131	—	25	1	no	—
93	#108	—	25	1	no	—
94	#101	—	25	1	no	—
95	#101	C₅⋊D₅	1	1	yes	characteristic, complemented
96	C₅²	#473	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₅	#709	1	6	yes	—
98	C₂×D₅	—	10	6	no	—
99	C₄×C₅	—	4	6	no	—
100	C₄×C₅	D₄×D₅	1	6	yes	—
101	C₄×C₅	D₄×D₅	1	6	yes	—
102	Dic₅	—	10	6	no	—
103	Dic₅	—	20	6	no	—
104	#41	—	25	1	no	—
105	C₂×Q₈	—	25	1	no	—
106	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
107	Q₁₆	—	50	1	no	—
108	Q₁₆	—	50	1	no	—
109	Q₁₆	—	50	1	no	—
110	Q₁₆	—	50	1	no	—
111	SD₁₆	—	50	1	no	—
112	SD₁₆	—	50	1	no	—
113	M₁₆	—	25	1	no	—
114	M₁₆	—	50	1	no	—
115	M₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
116	M₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
117	C₂×C₈	—	25	1	no	—
118	C₂×C₈	—	50	1	no	—
119	C₂×C₅	—	2	6	no	—
120	C₂×C₅	#4030	1	6	yes	—
121	D₅	—	20	6	no	—
122	Q₈	—	2	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	25	1	no	—
125	Q₈	—	50	1	no	—
126	Q₈	—	50	1	no	—
127	D₄	—	25	1	no	—
128	D₄	—	50	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	50	1	no	—
131	C₂×C₄	—	50	1	no	—
132	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	25	1	no	—
136	C₈	—	25	1	no	—
137	C₈	—	50	1	no	—
138	C₅	#65044	1	6	yes	complemented
139	C₂²	—	50	1	no	—
140	C₂²	#87900	1	1	yes	characteristic
141	C₄	—	4	1	no	—
142	C₄	—	50	1	no	—
143	C₄	—	100	1	no	—
144	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
145	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	100	1	no	—
148	C₂	#11717204	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87812

C₂

yes

maximal, characteristic, complemented

#11717262

C₂

yes

maximal, characteristic

#11717229

C₂

yes

maximal, characteristic

#11717227

C₂

yes

maximal, characteristic

C₅²×#101

C₂

yes

maximal, characteristic, complemented

#11716691

C₂

yes

maximal, characteristic, complemented

#11716689

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

#87900

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

—

—

C₈×C₅⋊D₅

—

#65044

—

maximal

C₅²×Q₈

—

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

—

C₄⋊(C₅⋊D₅)

D₄

yes

characteristic

C₄×C₅⋊D₅

—

#6045

—

#6045

—

#6045

D₄

yes

characteristic

C₅²×C₈

—

C₅²×C₈

—

#6029

—

#6029

D₄

yes

characteristic

#6029

D₄

yes

—

—

—

—

D₅

yes

complemented

#3956

—

#3954

—

C₂²×C₅²

#41

yes

characteristic

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

#1039

—

C₂×C₅×Q₈

C₂×D₅

yes

—

#709

—

C₂×Dic₁₀

—

C₅×M₁₆

C₂×D₅

yes

—

C₅×M₁₆

C₂×D₅

yes

—

—

—

—

—

—

—

Dic₂₀

—

#677

—

#676

—

C₈×D₅

—

#473

—

Sylow

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₅⋊D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

—

—

—

—

—

C₂×Q₁₆

—

—

—

—

C₅⋊D₅

yes

characteristic, complemented

C₅²

#473

yes

characteristic, complemented, Sylow

C₂²×C₅

#709

yes

—

C₂×D₅

—

C₄×C₅

—

100

C₄×C₅

D₄×D₅

yes

—

101

C₄×C₅

D₄×D₅

yes

—

102

Dic₅

—

103

Dic₅

—

104

#41

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

107

Q₁₆

—

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

—

111

SD₁₆

—

112

SD₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

C₂×C₅⋊D₅

yes

characteristic

116

M₁₆

C₂×C₅⋊D₅

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₅

—

120

C₂×C₅

#4030

yes

—

121

D₅

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₅

#65044

yes

complemented

139

C₂²

—

140

C₂²

#87900

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

100

—

144

C₄

D₄×C₅⋊D₅

yes

characteristic

145

C₄

D₄×C₅⋊D₅

yes

characteristic

146

C₂

—

147

C₂

—

100

—

148

C₂

#11717204

yes

characteristic

	subgroup	G/H
center	C₂	#11717204
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#101	C₂
socle	C₂×C₅²	#121
prime radical	#87900	C₂²
hypercenter	#101	C₅⋊D₅
hypocenter	C₅²	#473
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

#11717204

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#101

C₂

C₂×C₅²

C₂²

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
1. C₂×#87812	C₂	semidirect
5. C₅²×#101	C₂	semidirect
6. #11716691	C₂	semidirect
7. #11716689	C₂	semidirect
44. C₅×#101	D₅	semidirect
95. #101	C₅⋊D₅	semidirect
96. C₅²	#473	semidirect
138. C₅	#65044	semidirect

H ≅ G/N

type

1. C₂×#87812

C₂

semidirect

5. C₅²×#101

C₂

semidirect

6. #11716691

C₂

semidirect

7. #11716689

C₂

semidirect

44. C₅×#101

D₅

semidirect

95. #101

C₅⋊D₅

semidirect

96. C₅²

#473

semidirect

138. C₅

#65044

semidirect

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

subgroup	index
C₂×#87812	2
#11717262	2
#11717229	2
#11717227	2
C₅²×#101	2
#11716691	2
#11716689	2
#65044	5

subgroup

index

C₅²×#101

Subgroups — #13102617

Conjugacy classes of subgroups

Complements

Maximal subgroups