Group #13102612 — subgroups

Subgroups — #13102612

Order 1600.

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

normal subgroups only

This group has 432 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	#11717252	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87809	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717228	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717227	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅²×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716690	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716689	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
9	#87900	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
11	C₅²×M₁₆	C₂²	1	1	yes	characteristic
12	C₅²×M₁₆	C₂²	1	1	yes	characteristic
13	#87810	—	2	1	no	—
14	#87809	—	2	1	no	—
15	#87809	—	2	1	no	—
16	#87809	—	2	1	no	—
17	#87804	C₂²	1	1	yes	characteristic
18	C₂×#6029	C₂²	1	1	yes	characteristic
19	#87801	—	2	1	no	—
20	#87800	—	2	1	no	—
21	#87799	—	2	1	no	—
22	C₈×C₅⋊D₅	—	2	1	no	—
23	#65038	—	5	6	no	maximal
24	C₂³×C₅²	—	2	1	no	—
25	C₂²×C₅⋊D₅	—	2	1	no	—
26	C₅²×D₄	—	2	1	no	—
27	C₅²×D₄	—	2	1	no	—
28	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
29	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
30	C₄⋊(C₅⋊D₅)	—	2	1	no	—
31	C₄⋊(C₅⋊D₅)	—	2	1	no	—
32	C₄⋊(C₅⋊D₅)	D₄	1	1	yes	characteristic
33	C₄×C₅⋊D₅	—	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	#4078	—	5	6	no	—
41	C₂×#686	—	5	6	no	—
42	#4054	—	5	6	no	—
43	#4053	—	5	6	no	—
44	C₅×#100	D₅	1	6	yes	complemented
45	#3955	—	5	6	no	—
46	#3954	—	5	6	no	—
47	C₂²×C₅²	—	4	1	no	—
48	C₂²×C₅²	—	8	1	no	—
49	C₂²×C₅²	#41	1	1	yes	characteristic
50	C₂×C₅⋊D₅	—	2	1	no	—
51	C₂×C₅⋊D₅	—	4	1	no	—
52	C₂×C₅⋊D₅	—	8	1	no	—
53	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
54	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
55	#1039	—	2	1	no	—
56	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
57	#709	—	5	6	no	—
58	C₂×D₂₀	—	5	6	no	—
59	C₅×M₁₆	C₂×D₅	1	6	yes	—
60	C₅×M₁₆	C₂×D₅	1	6	yes	—
61	#687	—	10	6	no	—
62	#686	—	10	6	no	—
63	#686	—	10	6	no	—
64	#686	—	10	6	no	—
65	#681	—	5	6	no	—
66	C₂×#168	—	5	6	no	—
67	D₄₀	—	10	6	no	—
68	#677	—	10	6	no	—
69	#676	—	10	6	no	—
70	C₈×D₅	—	10	6	no	—
71	#472	—	25	1	no	Sylow
72	C₂×C₅²	—	2	1	no	—
73	C₂×C₅²	—	8	1	no	—
74	C₂×C₅²	#121	1	1	yes	characteristic
75	C₅⋊D₅	—	4	1	no	—
76	C₅⋊D₅	—	8	1	no	—
77	C₂³×C₅	—	2	6	no	—
78	C₂²×D₅	—	10	6	no	—
79	C₅×D₄	—	2	6	no	—
80	C₅×D₄	—	2	6	no	—
81	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
82	C₂²⋊D₅	—	10	6	no	—
83	D₂₀	—	5	6	no	—
84	D₂₀	—	10	6	no	—
85	D₂₀	—	10	6	no	—
86	C₄×D₅	—	10	6	no	—
87	Dic₁₀	—	5	6	no	—
88	C₅×C₈	—	2	6	no	—
89	C₅×C₈	—	2	6	no	—
90	#168	—	5	6	no	—
91	#168	—	5	6	no	—
92	#168	—	10	6	no	—
93	AGL(1,ℤ/8)	—	25	1	no	—
94	AGL(1,ℤ/8)	—	25	1	no	—
95	C₂×D₈	—	25	1	no	—
96	#131	—	25	1	no	—
97	#108	—	25	1	no	—
98	#100	—	25	1	no	—
99	#100	C₅⋊D₅	1	1	yes	characteristic, complemented
100	C₅²	#472	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₅	—	4	6	no	—
102	C₂²×C₅	—	8	6	no	—
103	C₂²×C₅	#709	1	6	yes	—
104	C₂×D₅	—	10	6	no	—
105	C₂×D₅	—	20	6	no	—
106	C₂×D₅	—	40	6	no	—
107	C₄×C₅	D₄×D₅	1	6	yes	—
108	C₄×C₅	D₄×D₅	1	6	yes	—
109	Dic₅	—	10	6	no	—
110	#41	—	25	1	no	—
111	C₂×D₄	—	25	1	no	—
112	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
113	SD₁₆	—	50	1	no	—
114	SD₁₆	—	50	1	no	—
115	D₈	—	50	1	no	—
116	D₈	—	50	1	no	—
117	D₈	—	50	1	no	—
118	D₈	—	50	1	no	—
119	M₁₆	—	25	1	no	—
120	M₁₆	—	50	1	no	—
121	M₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
122	M₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
123	C₂×C₈	—	25	1	no	—
124	C₂×C₈	—	50	1	no	—
125	C₂×C₅	—	2	6	no	—
126	C₂×C₅	—	8	6	no	—
127	C₂×C₅	#4030	1	6	yes	—
128	D₅	—	20	6	no	—
129	D₅	—	40	6	no	—
130	C₂³	—	2	1	no	—
131	C₂³	—	50	1	no	—
132	Q₈	—	25	1	no	—
133	D₄	—	2	1	no	—
134	D₄	—	2	1	no	—
135	D₄	—	25	1	no	—
136	D₄	—	50	1	no	—
137	D₄	—	50	1	no	—
138	D₄	—	50	1	no	—
139	C₂×C₄	—	50	1	no	—
140	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	25	1	no	—
144	C₈	—	25	1	no	—
145	C₈	—	50	1	no	—
146	C₅	#65038	1	6	yes	complemented
147	C₂²	—	4	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	50	1	no	—
150	C₂²	—	100	1	no	—
151	C₂²	—	200	1	no	—
152	C₂²	#87900	1	1	yes	characteristic
153	C₄	—	50	1	no	—
154	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
155	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	8	1	no	—
158	C₂	—	100	1	no	—
159	C₂	—	200	1	no	—
160	C₂	#11717204	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717252

C₂

yes

maximal, characteristic, complemented

C₂×#87809

C₂

yes

maximal, characteristic, complemented

#11717228

C₂

yes

maximal, characteristic, complemented

#11717227

C₂

yes

maximal, characteristic, complemented

C₅²×#100

C₂

yes

maximal, characteristic, complemented

#11716690

C₂

yes

maximal, characteristic, complemented

#11716689

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

C₂²

yes

characteristic

#87900

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

—

—

C₈×C₅⋊D₅

—

#65038

—

maximal

C₂³×C₅²

—

C₂²×C₅⋊D₅

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

D₄

yes

characteristic

C₄×C₅⋊D₅

—

#6045

D₄

yes

characteristic

C₅²×C₈

—

C₅²×C₈

—

#6029

—

#6029

D₄

yes

characteristic

#6029

D₄

yes

—

—

—

—

D₅

yes

complemented

#3955

—

#3954

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

#41

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

—

C₂×C₅×D₄

C₂×D₅

yes

—

#709

—

C₂×D₂₀

—

C₅×M₁₆

C₂×D₅

yes

—

C₅×M₁₆

C₂×D₅

yes

—

—

—

—

—

—

—

D₄₀

—

#677

—

#676

—

C₈×D₅

—

#472

—

Sylow

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂³×C₅

—

C₂²×D₅

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

—

—

—

—

—

C₂×D₈

—

—

—

—

C₅⋊D₅

yes

characteristic, complemented

100

C₅²

#472

yes

characteristic, complemented, Sylow

101

C₂²×C₅

—

102

C₂²×C₅

—

103

C₂²×C₅

#709

yes

—

104

C₂×D₅

—

105

C₂×D₅

—

106

C₂×D₅

—

107

C₄×C₅

D₄×D₅

yes

—

108

C₄×C₅

D₄×D₅

yes

—

109

Dic₅

—

110

#41

—

111

C₂×D₄

—

112

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

113

SD₁₆

—

114

SD₁₆

—

115

D₈

—

116

D₈

—

117

D₈

—

118

D₈

—

119

M₁₆

—

120

M₁₆

—

121

M₁₆

C₂×C₅⋊D₅

yes

characteristic

122

M₁₆

C₂×C₅⋊D₅

yes

characteristic

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₅

—

126

C₂×C₅

—

127

C₂×C₅

#4030

yes

—

128

D₅

—

129

D₅

—

130

C₂³

—

131

C₂³

—

132

Q₈

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₅

#65038

yes

complemented

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

100

—

151

C₂²

—

200

—

152

C₂²

#87900

yes

characteristic

153

C₄

—

154

C₄

D₄×C₅⋊D₅

yes

characteristic

155

C₄

D₄×C₅⋊D₅

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

—

100

—

159

C₂

—

200

—

160

C₂

#11717204

yes

characteristic

	subgroup	G/H
center	C₂	#11717204
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#100	C₂
socle	C₂×C₅²	#121
hypercenter	#100	C₅⋊D₅
hypocenter	C₅²	#472
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

subgroup

G/H

center

C₂

#11717204

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#100

C₂

C₂×C₅²

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

N	H ≅ G/N	type
1. #11717252	C₂	semidirect
2. C₂×#87809	C₂	semidirect
3. #11717228	C₂	semidirect
4. #11717227	C₂	semidirect
5. C₅²×#100	C₂	semidirect
6. #11716690	C₂	semidirect
7. #11716689	C₂	semidirect
44. C₅×#100	D₅	semidirect
99. #100	C₅⋊D₅	semidirect
100. C₅²	#472	semidirect
146. C₅	#65038	semidirect

H ≅ G/N

type

1. #11717252

C₂

semidirect

2. C₂×#87809

C₂

semidirect

3. #11717228

C₂

semidirect

4. #11717227

C₂

semidirect

5. C₅²×#100

C₂

semidirect

6. #11716690

C₂

semidirect

7. #11716689

C₂

semidirect

44. C₅×#100

D₅

semidirect

99. #100

C₅⋊D₅

semidirect

100. C₅²

#472

semidirect

146. C₅

#65038

semidirect

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

subgroup	index
#11717252	2
C₂×#87809	2
#11717228	2
#11717227	2
C₅²×#100	2
#11716690	2
#11716689	2
#65038	5

subgroup

index

C₅²×#100

Subgroups — #13102612

Conjugacy classes of subgroups

Complements

Maximal subgroups