Group #13102760 — subgroups

Subgroups — #13102760

Order 1600.

416 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 109 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	#11717229	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717228	C₂	1	1	yes	maximal, characteristic
3	C₂×#87800	C₂	1	1	yes	maximal, characteristic, complemented
4	#11717221	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅²×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716691	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716690	C₂	1	1	yes	maximal, characteristic, complemented
8	#87900	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₅⋊D₅)	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	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
14	#87804	C₂²	1	1	yes	characteristic
15	#87802	—	2	1	no	—
16	#87801	—	2	1	no	—
17	#87800	—	2	1	no	—
18	#87800	—	2	1	no	—
19	#87800	—	2	1	no	—
20	#87800	—	2	1	no	—
21	#87799	—	2	1	no	—
22	C₈×C₅⋊D₅	—	2	1	no	—
23	#65187	—	5	6	no	maximal
24	C₂²×C₅⋊D₅	—	2	1	no	—
25	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
26	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
27	C₂×#1039	—	2	1	no	—
28	C₄⋊(C₅⋊D₅)	—	2	1	no	—
29	C₄⋊(C₅⋊D₅)	—	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	C₅²×C₈	D₄	1	1	yes	characteristic
38	C₅²×C₈	D₄	1	1	yes	characteristic
39	#6029	—	2	1	no	—
40	#4055	—	5	6	no	—
41	#4054	—	5	6	no	—
42	C₂×#677	—	5	6	no	—
43	#4047	—	5	6	no	—
44	C₅×#108	D₅	1	6	yes	complemented
45	#3956	—	5	6	no	—
46	#3955	—	5	6	no	—
47	C₂²×C₅²	#41	1	1	yes	characteristic
48	C₂×C₅⋊D₅	—	2	1	no	—
49	C₂×C₅⋊D₅	—	4	1	no	—
50	C₂×C₅⋊D₅	—	8	1	no	—
51	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
52	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
53	#1039	—	2	1	no	—
54	#1039	—	4	1	no	—
55	#709	—	5	6	no	—
56	C₂×D₂₀	—	5	6	no	—
57	C₂×Dic₁₀	—	5	6	no	—
58	C₅×M₁₆	C₂×D₅	1	6	yes	—
59	C₅×M₁₆	C₂×D₅	1	6	yes	—
60	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
61	#681	—	5	6	no	—
62	Dic₂₀	—	10	6	no	—
63	D₄₀	—	10	6	no	—
64	#677	—	10	6	no	—
65	#677	—	10	6	no	—
66	#677	—	10	6	no	—
67	#677	—	10	6	no	—
68	#676	—	10	6	no	—
69	C₈×D₅	—	10	6	no	—
70	#471	—	25	1	no	Sylow
71	C₂×C₅²	—	2	1	no	—
72	C₂×C₅²	#121	1	1	yes	characteristic
73	C₅⋊D₅	—	4	1	no	—
74	C₅⋊D₅	—	8	1	no	—
75	C₂²×D₅	—	10	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	D₂₀	—	10	6	no	—
81	D₂₀	—	10	6	no	—
82	C₄×D₅	—	10	6	no	—
83	Dic₁₀	—	5	6	no	—
84	Dic₁₀	—	10	6	no	—
85	Dic₁₀	—	10	6	no	—
86	C₅×C₈	—	2	6	no	—
87	C₅×C₈	—	2	6	no	—
88	C₅×C₈	D₂₀	1	6	yes	—
89	C₅×C₈	D₂₀	1	6	yes	—
90	#168	—	10	6	no	—
91	#137	—	25	1	no	—
92	AGL(1,ℤ/8)	—	25	1	no	—
93	C₂×SD₁₆	—	25	1	no	—
94	#131	—	25	1	no	—
95	#108	C₅⋊D₅	1	1	yes	characteristic, complemented
96	#101	—	25	1	no	—
97	#100	—	25	1	no	—
98	C₅²	#471	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₅	#713	1	6	yes	—
100	C₂×D₅	—	10	6	no	—
101	C₂×D₅	—	20	6	no	—
102	C₂×D₅	—	40	6	no	—
103	C₄×C₅	C₂×D₂₀	1	6	yes	—
104	C₄×C₅	D₄×D₅	1	6	yes	—
105	Dic₅	—	10	6	no	—
106	Dic₅	—	20	6	no	—
107	#41	—	25	1	no	—
108	C₂×Q₈	—	25	1	no	—
109	C₂×D₄	—	25	1	no	—
110	Q₁₆	—	50	1	no	—
111	SD₁₆	—	50	1	no	—
112	SD₁₆	—	50	1	no	—
113	SD₁₆	—	50	1	no	—
114	SD₁₆	—	50	1	no	—
115	D₈	—	50	1	no	—
116	M₁₆	—	25	1	no	—
117	M₁₆	—	50	1	no	—
118	M₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
119	M₁₆	C₂×C₅⋊D₅	1	1	yes	characteristic
120	C₂×C₈	—	50	1	no	—
121	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
122	C₂×C₅	—	2	6	no	—
123	C₂×C₅	#4041	1	6	yes	—
124	D₅	—	20	6	no	—
125	D₅	—	40	6	no	—
126	C₂³	—	50	1	no	—
127	Q₈	—	25	1	no	—
128	Q₈	—	50	1	no	—
129	Q₈	—	50	1	no	—
130	D₄	—	25	1	no	—
131	D₄	—	50	1	no	—
132	D₄	—	50	1	no	—
133	D₄	—	50	1	no	—
134	C₂×C₄	—	50	1	no	—
135	C₂×C₄	—	50	1	no	—
136	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	50	1	no	—
140	C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
141	C₈	C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
142	C₅	#65187	1	6	yes	complemented
143	C₂²	—	50	1	no	—
144	C₂²	—	100	1	no	—
145	C₂²	—	200	1	no	—
146	C₂²	#87904	1	1	yes	characteristic
147	C₄	—	50	1	no	—
148	C₄	—	100	1	no	—
149	C₄	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
150	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	100	1	no	—
153	C₂	—	200	1	no	—
154	C₂	#11717215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717229

C₂

yes

maximal, characteristic, complemented

#11717228

C₂

yes

maximal, characteristic

C₂×#87800

C₂

yes

maximal, characteristic, complemented

#11717221

C₂

yes

maximal, characteristic, complemented

C₅²×#108

C₂

yes

maximal, characteristic, complemented

#11716691

C₂

yes

maximal, characteristic, complemented

#11716690

C₂

yes

maximal, characteristic, complemented

#87900

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

#87804

C₂²

yes

—

—

—

—

—

—

—

C₈×C₅⋊D₅

—

#65187

—

maximal

C₂²×C₅⋊D₅

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

D₄

yes

characteristic

C₄×C₅⋊D₅

—

#6045

—

#6045

—

#6045

D₄

yes

characteristic

C₅²×C₈

—

C₅²×C₈

—

C₅²×C₈

D₄

yes

characteristic

C₅²×C₈

D₄

yes

—

—

—

—

—

D₅

yes

complemented

#3956

—

#3955

—

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

—

—

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₅×M₁₆

C₂×D₅

yes

—

C₅×M₁₆

C₂×D₅

yes

—

C₂×C₅×C₈

C₂×D₅

yes

—

#681

—

Dic₂₀

—

D₄₀

—

—

—

—

—

—

C₈×D₅

—

#471

—

Sylow

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

D₂₀

yes

—

C₅×C₈

D₂₀

yes

—

#168

—

#137

—

AGL(1,ℤ/8)

—

C₂×SD₁₆

—

#131

—

#108

C₅⋊D₅

yes

characteristic, complemented

#101

—

#100

—

C₅²

#471

yes

characteristic, complemented, Sylow

C₂²×C₅

#713

yes

—

100

C₂×D₅

—

101

C₂×D₅

—

102

C₂×D₅

—

103

C₄×C₅

C₂×D₂₀

yes

—

104

C₄×C₅

D₄×D₅

yes

—

105

Dic₅

—

106

Dic₅

—

107

#41

—

108

C₂×Q₈

—

109

C₂×D₄

—

110

Q₁₆

—

111

SD₁₆

—

112

SD₁₆

—

113

SD₁₆

—

114

SD₁₆

—

115

D₈

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

C₂×C₅⋊D₅

yes

characteristic

119

M₁₆

C₂×C₅⋊D₅

yes

characteristic

120

C₂×C₈

—

121

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

122

C₂×C₅

—

123

C₂×C₅

#4041

yes

—

124

D₅

—

125

D₅

—

126

C₂³

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

C₄⋊(C₅⋊D₅)

yes

characteristic

141

C₈

C₄⋊(C₅⋊D₅)

yes

characteristic

142

C₅

#65187

yes

complemented

143

C₂²

—

144

C₂²

—

100

—

145

C₂²

—

200

—

146

C₂²

#87904

yes

characteristic

147

C₄

—

148

C₄

—

100

—

149

C₄

C₂×C₄⋊(C₅⋊D₅)

yes

characteristic

150

C₄

D₄×C₅⋊D₅

yes

characteristic

151

C₂

—

152

C₂

—

100

—

153

C₂

—

200

—

154

C₂

#11717215

yes

characteristic

	subgroup	G/H
center	C₂	#11717215
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#108	C₂
socle	C₂×C₅²	#121
prime radical	#11717228	C₂
hypercenter	#108	C₅⋊D₅
hypocenter	C₅²	#471
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#11717215

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#108

C₂

C₂×C₅²

C₂

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
1. #11717229	C₂	semidirect
3. C₂×#87800	C₂	semidirect
4. #11717221	C₂	semidirect
5. C₅²×#108	C₂	semidirect
6. #11716691	C₂	semidirect
7. #11716690	C₂	semidirect
44. C₅×#108	D₅	semidirect
95. #108	C₅⋊D₅	semidirect
98. C₅²	#471	semidirect
142. C₅	#65187	semidirect

H ≅ G/N

type

1. #11717229

C₂

semidirect

3. C₂×#87800

C₂

semidirect

4. #11717221

C₂

semidirect

5. C₅²×#108

C₂

semidirect

6. #11716691

C₂

semidirect

7. #11716690

C₂

semidirect

44. C₅×#108

D₅

semidirect

95. #108

C₅⋊D₅

semidirect

98. C₅²

#471

semidirect

142. C₅

#65187

semidirect

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

subgroup	index
#11717229	2
#11717228	2
C₂×#87800	2
#11717221	2
C₅²×#108	2
#11716691	2
#11716690	2
#65187	5

subgroup

index

C₅²×#108

Subgroups — #13102760

Conjugacy classes of subgroups

Complements

Maximal subgroups