Group #13025197 — subgroups

Subgroups — #13025197

Order 1280.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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	#10604083	C₂	1	2	yes	maximal, complemented
2	#10600150	C₂	1	1	yes	maximal, characteristic
3	#10598650	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598636	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598623	C₂	1	2	yes	maximal, complemented
6	#65725	—	2	2	no	—
7	#65193	C₄	1	1	yes	characteristic
8	#65184	C₂²	1	2	yes	—
9	#64898	—	2	2	no	—
10	#64889	C₂²	1	2	yes	—
11	#64885	C₂²	1	1	yes	characteristic
12	C₈×#170	C₂²	1	1	yes	characteristic
13	C₅×#368	C₄	1	1	yes	characteristic
14	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
15	#64717	C₄	1	2	yes	—
16	#12047	—	5	1	no	maximal, Sylow
17	#4053	—	2	2	no	—
18	#4050	C₂×C₄	1	2	yes	—
19	C₄×#170	—	2	1	no	—
20	#3998	—	2	2	no	—
21	C₂×#674	D₄	1	1	yes	characteristic
22	C₂×#674	D₄	1	1	yes	characteristic
23	#3990	—	4	2	no	—
24	D₅⋊C₁₆	—	4	2	no	—
25	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₅×#108	C₂×C₄	1	2	yes	—
27	#3957	—	2	2	no	—
28	C₈×Dic₅	C₂³	1	1	yes	characteristic
29	C₂×#672	C₂×C₄	1	1	yes	characteristic
30	Dic₄₀	C₈	1	1	yes	characteristic, complemented
31	#3932	C₈	1	2	yes	complemented
32	D₈₀	C₈	1	1	yes	characteristic, complemented
33	#2178	—	5	1	no	—
34	#2170	—	5	2	no	—
35	#1373	—	5	1	no	—
36	#1335	—	5	2	no	—
37	#1311	—	5	1	no	—
38	#709	—	2	2	no	—
39	C₂×#170	—	2	2	no	—
40	C₅×M₁₆	—	2	2	no	—
41	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₅	#41	1	1	yes	characteristic
43	C₂×#168	C₂×D₄	1	1	yes	characteristic
44	Dic₂₀	C₂×C₈	1	2	yes	—
45	D₄₀	C₂×C₈	1	2	yes	—
46	#677	—	2	2	no	—
47	#676	—	4	2	no	—
48	C₈×D₅	—	4	2	no	—
49	#674	—	2	1	no	—
50	#674	—	4	2	no	—
51	#674	D₈	1	2	yes	—
52	C₅×C₁₆	C₂×C₈	1	1	yes	characteristic
53	C₅×C₁₆	C₂×C₈	1	1	yes	characteristic
54	#672	—	2	1	no	—
55	#508	—	5	1	no	—
56	#504	—	10	2	no	—
57	#443	—	5	2	no	—
58	#368	AGL(1,𝔽₅)	1	1	yes	characteristic
59	#364	—	5	2	no	—
60	#359	—	5	2	no	—
61	#350	—	10	2	no	—
62	C₄×C₁₆	—	5	1	no	—
63	C₄×C₁₆	—	5	1	no	—
64	C₈²	—	5	1	no	—
65	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
66	C₂²⋊D₅	—	4	2	no	—
67	C₂×Dic₅	—	2	1	no	—
68	D₂₀	—	2	2	no	—
69	C₄×D₅	—	4	2	no	—
70	Dic₁₀	—	2	2	no	—
71	#170	—	4	2	no	—
72	C₅×C₈	—	4	2	no	—
73	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
74	C₅×C₈	#131	1	1	yes	characteristic
75	#168	C₂×D₈	1	1	yes	characteristic
76	#168	#135	1	1	yes	characteristic
77	#135	—	5	2	no	—
78	#131	—	10	2	no	—
79	Q₃₂	—	5	1	no	—
80	SD₃₂	—	5	2	no	—
81	D₁₆	—	5	1	no	—
82	M₃₂	—	10	2	no	—
83	M₃₂	—	20	2	no	—
84	C₂×C₁₆	—	5	1	no	—
85	C₂×C₁₆	—	5	1	no	—
86	C₂×C₁₆	—	5	1	no	—
87	C₂×C₁₆	—	20	2	no	—
88	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
89	#108	C₂×AGL(1,𝔽₅)	1	2	yes	—
90	#104	—	10	2	no	—
91	C₄×C₈	—	5	1	no	—
92	C₄×C₈	—	10	1	no	—
93	C₂²×C₅	#443	1	1	yes	characteristic
94	C₂×D₅	—	4	2	no	—
95	C₄×C₅	C₈×D₄	1	1	yes	characteristic
96	C₄×C₅	C₄×D₈	1	1	yes	characteristic
97	Dic₅	—	2	2	no	—
98	Dic₅	—	4	2	no	—
99	#41	—	10	2	no	—
100	Q₁₆	—	5	2	no	—
101	SD₁₆	—	10	2	no	—
102	D₈	—	5	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	20	2	no	—
105	C₂×C₈	—	5	1	no	—
106	C₂×C₈	—	10	2	no	—
107	C₂×C₈	—	20	2	no	—
108	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
109	C₄²	—	5	1	no	—
110	C₁₆	—	5	2	no	—
111	C₁₆	—	10	1	no	—
112	C₁₆	—	10	1	no	—
113	C₁₆	—	20	2	no	—
114	C₁₆	D₅⋊C₈	1	1	yes	characteristic
115	C₁₆	D₅⋊C₈	1	1	yes	characteristic
116	C₂×C₅	—	2	1	no	—
117	C₂×C₅	C₈×D₈	1	1	yes	characteristic
118	D₅	—	8	2	no	—
119	Q₈	—	10	2	no	—
120	D₄	—	10	2	no	—
121	D₄	—	20	2	no	—
122	C₂×C₄	—	10	1	no	—
123	C₂×C₄	—	20	2	no	—
124	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₈	—	5	1	no	—
127	C₈	—	5	1	no	—
128	C₈	—	20	2	no	—
129	C₈	C₂×D₅⋊C₈	1	1	yes	characteristic
130	C₈	#4133	1	1	yes	characteristic
131	C₅	#12047	1	1	yes	characteristic, complemented, Sylow
132	C₂²	—	20	2	no	—
133	C₂²	#65740	1	1	yes	characteristic
134	C₄	—	10	2	no	—
135	C₄	—	20	2	no	—
136	C₄	#65702	1	1	yes	characteristic
137	C₄	D₈×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	40	2	no	—
140	C₂	#10604059	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604083

C₂

yes

maximal, complemented

#10600150

C₂

yes

maximal, characteristic

#10598650

C₂

yes

maximal, characteristic, complemented

#10598636

C₂

yes

maximal, characteristic, complemented

#10598623

C₂

yes

maximal, complemented

#65725

—

#65193

C₄

yes

characteristic

#65184

C₂²

yes

—

#64898

—

#64889

C₂²

yes

—

#64885

C₂²

yes

characteristic

C₈×#170

C₂²

yes

characteristic

C₅×#368

C₄

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

#64717

C₄

yes

—

—

—

C₂×C₄

yes

—

C₄×#170

—

#3998

—

C₂×#674

D₄

yes

characteristic

C₂×#674

D₄

yes

characteristic

#3990

—

D₅⋊C₁₆

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₅×#108

C₂×C₄

yes

—

#3957

—

C₈×Dic₅

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

Dic₄₀

C₈

yes

characteristic, complemented

#3932

C₈

yes

complemented

D₈₀

C₈

yes

characteristic, complemented

—

—

—

—

—

—

—

C₅×M₁₆

—

C₂×C₅×C₈

C₂²×C₄

yes

C₄×Dic₅

yes

C₂×D₄

yes

characteristic

Dic₂₀

C₂×C₈

yes

—

D₄₀

C₂×C₈

yes

—

#677

—

#676

—

C₈×D₅

—

#674

—

#674

—

#674

D₈

yes

—

C₅×C₁₆

C₂×C₈

yes

characteristic

C₅×C₁₆

C₂×C₈

yes

—

—

—

—

yes

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

#170

—

C₅×C₈

—

C₅×C₈

C₂²×C₈

yes

C₅×C₈

yes

C₂×D₈

yes

yes

—

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

#104

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

#443

yes

characteristic

C₂×D₅

—

C₄×C₅

C₈×D₄

yes

characteristic

C₄×C₅

C₄×D₈

yes

characteristic

Dic₅

—

Dic₅

—

#41

—

100

Q₁₆

—

101

SD₁₆

—

102

D₈

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

D₅⋊C₈

yes

characteristic

115

C₁₆

D₅⋊C₈

yes

characteristic

116

C₂×C₅

—

117

C₂×C₅

C₈×D₈

yes

characteristic

118

D₅

—

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

C₂×D₅⋊C₈

yes

characteristic

130

C₈

#4133

yes

characteristic

131

C₅

#12047

yes

characteristic, complemented, Sylow

132

C₂²

—

133

C₂²

#65740

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

#65702

yes

characteristic

137

C₄

D₈×AGL(1,𝔽₅)

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#10604059

yes

characteristic

	subgroup	G/H
center	C₄	D₈×AGL(1,𝔽₅)
commutator	C₅×C₈	C₂²×C₈
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#368	C₄
socle	C₂×C₅	C₈×D₈
prime radical	#65193	C₄
cosocle	C₈×Dic₅	C₂³
hypercenter	#368	AGL(1,𝔽₅)
hypocenter	C₅	#12047
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×AGL(1,𝔽₅)

commutator

C₅×C₈

C₂²×C₈

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#368

C₄

socle

C₂×C₅

C₈×D₈

prime radical

#65193

C₄

cosocle

C₈×Dic₅

C₂³

C₅

G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #10604083	C₂	semidirect
3. #10598650	C₂	semidirect
4. #10598636	C₂	semidirect
5. #10598623	C₂	semidirect
30. Dic₄₀	C₈	semidirect
31. #3932	C₈	semidirect
32. D₈₀	C₈	semidirect
131. C₅	#12047	semidirect

H ≅ G/N

type

1. #10604083

C₂

semidirect

3. #10598650

C₂

semidirect

4. #10598636

C₂

semidirect

5. #10598623

C₂

semidirect

30. Dic₄₀

C₈

semidirect

31. #3932

C₈

semidirect

32. D₈₀

C₈

semidirect

131. C₅

#12047

semidirect

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

subgroup	index
#10604083	2
#10600150	2
#10598650	2
#10598636	2
#10598623	2
#12047	5

subgroup

index

Subgroups — #13025197

Conjugacy classes of subgroups

Complements

Maximal subgroups