Group #13025284 — subgroups

Subgroups — #13025284

Order 1280.

204 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 66 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	#10604084	C₂	1	2	yes	maximal, complemented
2	#10600151	C₂	1	1	yes	maximal, characteristic
3	#10598648	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598638	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598624	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	#64886	C₂²	1	1	yes	characteristic
12	#64879	C₂²	1	1	yes	characteristic
13	C₅×#368	C₄	1	1	yes	characteristic
14	#64722	C₂²	1	1	yes	characteristic
15	#64717	C₄	1	2	yes	—
16	#12068	—	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	2	yes	—
22	#3990	—	4	2	no	—
23	D₅⋊C₁₆	—	4	2	no	—
24	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₅×#108	C₂×C₄	1	2	yes	—
26	#3957	—	2	2	no	—
27	C₈×Dic₅	C₂³	1	1	yes	characteristic
28	C₂×#672	C₂×C₄	1	1	yes	characteristic
29	Dic₄₀	C₈	1	1	yes	characteristic, complemented
30	#3932	C₈	1	2	yes	complemented
31	D₈₀	C₈	1	1	yes	characteristic, complemented
32	#2179	—	5	1	no	—
33	#2171	—	5	2	no	—
34	#1369	—	5	1	no	—
35	#1336	—	5	2	no	—
36	#1313	—	5	1	no	—
37	#709	—	2	2	no	—
38	C₂×#170	—	4	1	no	—
39	C₅×M₁₆	—	2	2	no	—
40	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₅	#41	1	1	yes	characteristic
42	C₂×#168	C₂×D₄	1	1	yes	characteristic
43	Dic₂₀	C₂×C₈	1	2	yes	—
44	D₄₀	C₂×C₈	1	2	yes	—
45	#677	—	2	2	no	—
46	#676	—	4	2	no	—
47	C₈×D₅	—	4	2	no	—
48	#674	—	2	2	no	—
49	#674	—	4	2	no	—
50	C₅×C₁₆	C₂×C₈	1	1	yes	characteristic
51	C₅×C₁₆	C₂×C₈	1	1	yes	characteristic
52	#672	—	2	1	no	—
53	#508	—	5	1	no	—
54	#504	—	10	2	no	—
55	#443	—	5	2	no	—
56	#368	AGL(1,𝔽₅)	1	1	yes	characteristic
57	#364	—	5	2	no	—
58	#359	—	5	2	no	—
59	#350	—	10	2	no	—
60	#346	—	5	1	no	—
61	#346	—	5	1	no	—
62	#322	—	5	1	no	—
63	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
64	C₂²⋊D₅	—	4	2	no	—
65	C₂×Dic₅	—	2	1	no	—
66	D₂₀	—	2	2	no	—
67	C₄×D₅	—	4	2	no	—
68	Dic₁₀	—	2	2	no	—
69	#170	—	8	1	no	—
70	C₅×C₈	—	4	2	no	—
71	C₅×C₈	C₂²×C₈	1	1	yes	characteristic
72	C₅×C₈	#131	1	1	yes	characteristic
73	#168	AGL(1,ℤ/8)	1	2	yes	—
74	#135	—	5	2	no	—
75	#131	—	10	2	no	—
76	Q₃₂	—	5	1	no	—
77	SD₃₂	—	5	2	no	—
78	D₁₆	—	5	1	no	—
79	M₃₂	—	10	2	no	—
80	M₃₂	—	20	2	no	—
81	C₂×C₁₆	—	5	2	no	—
82	C₂×C₁₆	—	5	1	no	—
83	C₂×C₁₆	—	20	2	no	—
84	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
85	#108	C₂×AGL(1,𝔽₅)	1	2	yes	—
86	#104	—	10	2	no	—
87	C₄×C₈	—	5	1	no	—
88	C₄×C₈	—	10	1	no	—
89	C₂²×C₅	#444	1	1	yes	characteristic
90	C₂×D₅	—	4	2	no	—
91	C₄×C₅	C₈×D₄	1	1	yes	characteristic
92	C₄×C₅	#442	1	1	yes	characteristic
93	Dic₅	—	4	1	no	—
94	Dic₅	—	4	2	no	—
95	#41	—	10	2	no	—
96	Q₁₆	—	5	2	no	—
97	SD₁₆	—	10	2	no	—
98	D₈	—	5	2	no	—
99	M₁₆	—	2	2	no	—
100	M₁₆	—	20	2	no	—
101	C₂×C₈	—	5	1	no	—
102	C₂×C₈	—	20	2	no	—
103	C₂×C₈	—	20	1	no	—
104	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₄²	—	5	1	no	—
106	C₁₆	—	10	2	no	—
107	C₁₆	—	10	1	no	—
108	C₁₆	—	20	2	no	—
109	C₁₆	D₅⋊C₈	1	1	yes	characteristic
110	C₁₆	D₅⋊C₈	1	1	yes	characteristic
111	C₂×C₅	—	2	1	no	—
112	C₂×C₅	#1580	1	1	yes	characteristic
113	D₅	—	8	2	no	—
114	Q₈	—	10	2	no	—
115	D₄	—	10	2	no	—
116	D₄	—	20	2	no	—
117	C₂×C₄	—	10	1	no	—
118	C₂×C₄	—	20	2	no	—
119	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₈	—	4	2	no	—
121	C₈	—	5	2	no	—
122	C₈	—	40	1	no	—
123	C₈	C₂×D₅⋊C₈	1	1	yes	characteristic
124	C₈	#4133	1	1	yes	characteristic
125	C₅	#12068	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	20	2	no	—
127	C₂²	#65741	1	1	yes	characteristic
128	C₄	—	20	1	no	—
129	C₄	—	20	2	no	—
130	C₄	#65702	1	1	yes	characteristic
131	C₄	#65731	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	40	2	no	—
134	C₂	#10604062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604084

C₂

yes

maximal, complemented

#10600151

C₂

yes

maximal, characteristic

#10598648

C₂

yes

maximal, characteristic, complemented

#10598638

C₂

yes

maximal, characteristic, complemented

#10598624

C₂

yes

maximal, complemented

#65725

—

#65193

C₄

yes

characteristic

#65184

C₂²

yes

—

#64898

—

#64889

C₂²

yes

—

#64886

C₂²

yes

characteristic

#64879

C₂²

yes

characteristic

C₅×#368

C₄

yes

characteristic

#64722

C₂²

yes

characteristic

#64717

C₄

yes

—

—

—

C₂×C₄

yes

—

C₄×#170

—

#3998

—

C₂×#674

D₄

yes

—

#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

—

C₅×C₁₆

C₂×C₈

yes

characteristic

C₅×C₁₆

C₂×C₈

yes

—

—

—

—

yes

—

—

—

—

—

—

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

yes

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

#104

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

#444

yes

characteristic

C₂×D₅

—

C₄×C₅

C₈×D₄

yes

characteristic

C₄×C₅

#442

yes

characteristic

Dic₅

—

Dic₅

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

D₅⋊C₈

yes

characteristic

110

C₁₆

D₅⋊C₈

yes

characteristic

111

C₂×C₅

—

112

C₂×C₅

#1580

yes

characteristic

113

D₅

—

114

Q₈

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

C₂×D₅⋊C₈

yes

characteristic

124

C₈

#4133

yes

characteristic

125

C₅

#12068

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

#65741

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#65702

yes

characteristic

131

C₄

#65731

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#10604062

yes

characteristic

	subgroup	G/H
center	C₄	#65731
commutator	C₅×C₈	C₂²×C₈
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#368	C₄
socle	C₂×C₅	#1580
prime radical	#65193	C₄
cosocle	C₈×Dic₅	C₂³
hypercenter	#368	AGL(1,𝔽₅)
hypocenter	C₅	#12068
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

#65731

commutator

C₅×C₈

C₂²×C₈

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

C₄

C₂×C₅

C₄

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. #10604084	C₂	semidirect
3. #10598648	C₂	semidirect
4. #10598638	C₂	semidirect
5. #10598624	C₂	semidirect
29. Dic₄₀	C₈	semidirect
30. #3932	C₈	semidirect
31. D₈₀	C₈	semidirect
125. C₅	#12068	semidirect

H ≅ G/N

type

1. #10604084

C₂

semidirect

3. #10598648

C₂

semidirect

4. #10598638

C₂

semidirect

5. #10598624

C₂

semidirect

29. Dic₄₀

C₈

semidirect

30. #3932

C₈

semidirect

31. D₈₀

C₈

semidirect

125. C₅

#12068

semidirect

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

subgroup	index
#10604084	2
#10600151	2
#10598648	2
#10598638	2
#10598624	2
#12068	5

subgroup

index

Subgroups — #13025284

Conjugacy classes of subgroups

Complements

Maximal subgroups