Group #13025386 — subgroups

Subgroups — #13025386

Order 1280.

226 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10604070	C₂	1	2	yes	maximal, complemented
2	#10600132	C₂	1	1	yes	maximal, characteristic
3	#10598642	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598638	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598607	C₂	1	2	yes	maximal, complemented
6	#65705	—	2	2	no	—
7	C₂×D₈₀	C₄	1	1	yes	characteristic
8	#65161	C₂²	1	2	yes	—
9	#64886	C₂²	1	1	yes	characteristic
10	#64881	C₂²	1	2	yes	—
11	#64879	C₂²	1	1	yes	characteristic
12	#64871	—	2	2	no	—
13	C₅×#367	C₄	1	1	yes	characteristic
14	#64722	C₂²	1	1	yes	characteristic
15	#64711	C₄	1	2	yes	—
16	#12089	—	5	1	no	maximal, Sylow
17	C₂×#700	—	4	2	no	—
18	C₂×D₄₀	C₂×C₄	1	2	yes	—
19	#4039	—	2	2	no	—
20	#4003	—	4	2	no	—
21	C₄⋊#170	—	2	2	no	—
22	C₄×#170	D₄	1	2	yes	—
23	C₂×#674	—	2	1	no	—
24	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₅×#107	C₂×C₄	1	2	yes	—
26	C₈×Dic₅	C₂³	1	1	yes	characteristic
27	C₂×#672	C₂×C₄	1	1	yes	characteristic
28	#3941	—	2	2	no	—
29	D₈₀	—	2	2	no	—
30	#2177	—	5	1	no	—
31	#1594	—	5	2	no	—
32	#1368	—	5	1	no	—
33	#1333	—	5	2	no	—
34	#1313	—	5	1	no	—
35	C₂×D₂₀	—	2	2	no	—
36	C₂×C₄×D₅	—	4	2	no	—
37	C₂×#170	—	2	2	no	—
38	C₂×#170	—	4	2	no	—
39	#700	—	8	4	no	—
40	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
41	C₅×C₄⋊C₄	—	2	2	no	—
42	#685	—	4	2	no	—
43	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
44	C₂×#168	#41	1	1	yes	characteristic
45	D₄₀	—	2	2	no	—
46	D₄₀	—	2	2	no	—
47	#674	—	4	1	no	—
48	C₅×C₁₆	M₁₆	1	2	yes	complemented
49	#672	—	2	1	no	—
50	C₂×D₁₆	—	5	1	no	—
51	C₄×D₈	—	5	2	no	—
52	#436	—	10	2	no	—
53	#367	AGL(1,𝔽₅)	1	1	yes	characteristic
54	#357	—	5	2	no	—
55	#346	—	5	1	no	—
56	#346	—	5	1	no	—
57	#334	—	5	2	no	—
58	D₄⋊C₈	—	10	2	no	—
59	#322	—	5	1	no	—
60	C₂²×D₅	—	4	2	no	—
61	C₂×C₄×C₅	—	4	2	no	—
62	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
63	C₂×Dic₅	AGL(1,ℤ/8)	1	2	yes	—
64	D₂₀	—	4	2	no	—
65	D₂₀	—	4	2	no	—
66	C₄×D₅	—	8	2	no	—
67	C₄×D₅	—	8	2	no	—
68	#170	—	4	2	no	—
69	#170	—	8	2	no	—
70	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
71	C₅×C₈	#131	1	1	yes	characteristic
72	#168	—	2	1	no	—
73	C₂×D₈	—	5	2	no	—
74	C₂×M₁₆	—	20	2	no	—
75	C₄×D₄	—	10	2	no	—
76	D₁₆	—	10	2	no	—
77	C₂×C₁₆	—	5	1	no	—
78	C₂×C₁₆	—	10	1	no	—
79	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
80	#107	C₂×AGL(1,𝔽₅)	1	2	yes	—
81	C₄⋊C₈	—	10	2	no	—
82	#102	—	10	2	no	—
83	C₂²⋊C₈	—	20	2	no	—
84	C₄×C₈	—	5	1	no	—
85	C₄×C₈	—	5	2	no	—
86	C₂²×C₅	#442	1	1	yes	characteristic
87	C₂×D₅	—	8	2	no	—
88	C₂×D₅	—	8	2	no	—
89	C₂×D₅	—	8	2	no	—
90	C₄×C₅	—	8	2	no	—
91	C₄×C₅	#436	1	1	yes	characteristic
92	C₄×C₅	#444	1	1	yes	characteristic
93	Dic₅	—	2	2	no	—
94	C₂×D₄	—	10	2	no	—
95	C₂²×C₄	—	20	2	no	—
96	D₈	—	10	2	no	—
97	D₈	—	10	2	no	—
98	M₁₆	—	40	4	no	—
99	C₂×C₈	—	5	1	no	—
100	C₂×C₈	—	10	2	no	—
101	C₂×C₈	—	20	2	no	—
102	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₄⋊C₄	—	2	2	no	—
104	C₂²⋊C₄	—	20	2	no	—
105	C₄²	—	5	1	no	—
106	C₁₆	—	10	1	no	—
107	C₁₆	—	20	1	no	—
108	C₁₆	#700	1	2	yes	complemented
109	C₂×C₅	#1594	1	1	yes	characteristic
110	C₂×C₅	AGL(1,ℤ/16)	1	2	yes	—
111	D₅	—	16	2	no	—
112	C₂³	—	20	2	no	—
113	D₄	—	20	2	no	—
114	D₄	—	20	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	5	2	no	—
117	C₂×C₄	—	40	2	no	—
118	C₂×C₄	—	40	2	no	—
119	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₈	—	10	1	no	—
121	C₈	—	20	2	no	—
122	C₈	—	40	2	no	—
123	C₈	C₂×#700	1	1	yes	characteristic
124	C₈	#4133	1	1	yes	characteristic
125	C₅	#12089	1	1	yes	characteristic, complemented, Sylow
126	C₂²	—	40	2	no	—
127	C₂²	—	40	2	no	—
128	C₂²	—	40	2	no	—
129	C₂²	#65731	1	1	yes	characteristic
130	C₄	—	8	2	no	—
131	C₄	—	10	2	no	—
132	C₄	#65705	1	1	yes	characteristic
133	C₄	#65741	1	1	yes	characteristic
134	C₂	—	80	2	no	—
135	C₂	#10604070	1	1	yes	characteristic
136	C₂	#10604133	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10604070

C₂

yes

maximal, complemented

#10600132

C₂

yes

maximal, characteristic

#10598642

C₂

yes

maximal, characteristic, complemented

#10598638

C₂

yes

maximal, characteristic, complemented

#10598607

C₂

yes

maximal, complemented

#65705

—

C₂×D₈₀

C₄

yes

characteristic

#65161

C₂²

yes

—

#64886

C₂²

yes

characteristic

#64881

C₂²

yes

—

#64879

C₂²

yes

characteristic

#64871

—

C₅×#367

C₄

yes

characteristic

#64722

C₂²

yes

characteristic

#64711

C₄

yes

—

#12089

—

maximal, Sylow

C₂×#700

—

C₂×D₄₀

C₂×C₄

yes

—

—

—

—

D₄

yes

—

C₂×#674

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₅×#107

C₂×C₄

yes

—

C₈×Dic₅

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

#3941

—

D₈₀

—

—

—

—

—

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₂×#170

—

C₂×#170

—

#700

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₅×C₄⋊C₄

—

#685

—

C₄×Dic₅

C₂×D₄

yes

yes

D₄₀

—

D₄₀

—

#674

—

C₅×C₁₆

M₁₆

yes

complemented

#672

—

C₂×D₁₆

—

C₄×D₈

—

—

yes

—

—

—

—

D₄⋊C₈

—

#322

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×Dic₅

AGL(1,ℤ/8)

yes

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

#170

—

#170

—

C₅×C₈

C₂×M₁₆

yes

C₅×C₈

yes

—

C₂×D₈

—

C₂×M₁₆

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

#442

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

#700

yes

complemented

109

C₂×C₅

#1594

yes

characteristic

110

C₂×C₅

AGL(1,ℤ/16)

yes

—

111

D₅

—

112

C₂³

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

C₂×#700

yes

characteristic

124

C₈

#4133

yes

characteristic

125

C₅

#12089

yes

characteristic, complemented, Sylow

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#65731

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

#65705

yes

characteristic

133

C₄

#65741

yes

characteristic

134

C₂

—

135

C₂

#10604070

yes

characteristic

136

C₂

#10604133

yes

—

	subgroup	G/H
center	C₂²	#65731
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#367	C₄
socle	C₂²×C₅	#442
prime radical	C₂×D₈₀	C₄
cosocle	C₈×Dic₅	C₂³
hypercenter	#367	AGL(1,𝔽₅)
hypocenter	C₅	#12089
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#65731

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₂×D₈₀

C₄

cosocle

C₈×Dic₅

C₂³

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
1. #10604070	C₂	semidirect
3. #10598642	C₂	semidirect
4. #10598638	C₂	semidirect
5. #10598607	C₂	semidirect
48. C₅×C₁₆	M₁₆	semidirect
108. C₁₆	#700	semidirect
125. C₅	#12089	semidirect

H ≅ G/N

type

1. #10604070

C₂

semidirect

3. #10598642

C₂

semidirect

4. #10598638

C₂

semidirect

5. #10598607

C₂

semidirect

48. C₅×C₁₆

M₁₆

semidirect

108. C₁₆

#700

semidirect

125. C₅

#12089

semidirect

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

subgroup	index
#10604070	2
#10600132	2
#10598642	2
#10598638	2
#10598607	2
#12089	5

subgroup

index

Subgroups — #13025386

Conjugacy classes of subgroups

Complements

Maximal subgroups