Group #13018760 — subgroups

Subgroups — #13018760

Order 1280.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#10603821	C₂	1	2	yes	maximal
2	#10598861	C₂	1	1	yes	maximal, characteristic
3	#10598532	C₂	1	4	yes	maximal
4	#65683	C₂²	1	4	yes	—
5	#64974	C₂²	1	2	yes	—
6	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
7	#64884	—	2	2	no	—
8	#64883	—	2	2	no	—
9	C₅×#322	C₄	1	1	yes	characteristic
10	#64675	C₄	1	1	yes	characteristic
11	C₈×#168	C₄	1	2	yes	—
12	#8796	—	5	1	no	maximal, Sylow
13	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
14	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
15	C₄²×D₅	C₂³	1	1	yes	characteristic
16	#4003	—	2	4	no	—
17	#4002	—	2	4	no	—
18	C₄⋊#170	—	2	4	no	—
19	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	#3952	—	2	2	no	—
22	#3946	—	2	2	no	—
23	C₈×Dic₅	D₄	1	1	yes	characteristic
24	C₈×Dic₅	D₄	1	1	yes	characteristic
25	C₄×#168	C₂×C₄	1	2	yes	—
26	C₄×#168	C₂×C₄	1	1	yes	characteristic
27	#1568	—	5	2	no	—
28	#1450	—	5	1	no	—
29	#1287	—	5	4	no	—
30	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×D₅	#41	1	2	yes	—
32	C₂×#170	—	4	4	no	—
33	C₂×C₅×C₈	—	2	2	no	—
34	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
37	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
38	C₄×Dic₅	#41	1	2	yes	—
39	C₂×#168	—	2	2	no	—
40	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
42	C₈×D₅	—	2	2	no	—
43	C₈×D₅	—	2	2	no	—
44	#432	—	5	4	no	—
45	#431	—	5	2	no	—
46	C₂×C₄×C₈	—	5	1	no	—
47	#335	—	10	2	no	—
48	#334	—	10	2	no	—
49	#322	—	5	1	no	—
50	#322	AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₈²	—	5	2	no	—
52	C₂²×D₅	#125	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅	#117	1	2	yes	—
55	C₂×Dic₅	#124	1	2	yes	—
56	C₂×Dic₅	#125	1	1	yes	characteristic
57	C₄×D₅	—	2	1	no	—
58	C₄×D₅	—	2	1	no	—
59	C₄×D₅	—	4	2	no	—
60	#170	—	8	4	no	—
61	C₅×C₈	—	2	1	no	—
62	C₅×C₈	—	2	1	no	—
63	C₅×C₈	—	4	2	no	—
64	#168	—	4	2	no	—
65	#168	#108	1	2	yes	—
66	#168	#108	1	2	yes	—
67	C₂²×C₈	—	5	1	no	—
68	C₂²×C₈	—	5	1	no	—
69	C₂×C₄²	—	5	1	no	—
70	C₄⋊C₈	—	10	2	no	—
71	C₄⋊C₈	—	10	4	no	—
72	C₂²⋊C₈	—	10	2	no	—
73	C₂²⋊C₈	—	10	4	no	—
74	#97	—	10	4	no	—
75	C₄×C₈	—	5	1	no	—
76	C₄×C₈	—	5	1	no	—
77	C₄×C₈	—	5	1	no	—
78	C₄×C₈	—	5	2	no	—
79	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
80	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₂²×C₅	#382	1	1	yes	characteristic
82	C₂×D₅	—	2	2	no	—
83	C₂×D₅	—	2	1	no	—
84	C₄×C₅	—	2	2	no	—
85	C₄×C₅	C₂×#108	1	1	yes	characteristic
86	C₄×C₅	C₂×#108	1	1	yes	characteristic
87	Dic₅	—	2	2	no	—
88	Dic₅	—	2	1	no	—
89	C₂²×C₄	—	5	1	no	—
90	C₂²×C₄	—	5	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	5	1	no	—
93	C₂×C₈	—	5	1	no	—
94	C₂×C₈	—	10	2	no	—
95	C₂×C₈	—	10	2	no	—
96	C₂×C₈	—	10	2	no	—
97	C₂×C₈	—	20	4	no	—
98	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₄²	—	5	1	no	—
101	C₄²	—	5	2	no	—
102	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
103	C₂×C₅	#1833	1	2	yes	—
104	C₂×C₅	#1837	1	1	yes	characteristic
105	D₅	—	4	1	no	—
106	C₂³	—	5	1	no	—
107	C₂×C₄	—	5	2	no	—
108	C₂×C₄	—	5	1	no	—
109	C₂×C₄	—	10	1	no	—
110	C₂×C₄	—	10	1	no	—
111	C₂×C₄	—	20	2	no	—
112	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₂×C₄	#4130	1	2	yes	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	—	4	2	no	—
117	C₈	—	5	2	no	—
118	C₈	—	5	2	no	—
119	C₈	—	20	2	no	—
120	C₈	—	40	4	no	—
121	C₅	#8796	1	1	yes	characteristic, complemented, Sylow
122	C₂²	—	10	1	no	—
123	C₂²	—	10	2	no	—
124	C₂²	#65689	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	10	1	no	—
127	C₄	—	10	2	no	—
128	C₄	D₅⋊#108	1	1	yes	characteristic
129	C₄	D₅⋊#108	1	1	yes	characteristic
130	C₂	—	20	1	no	—
131	C₂	#10603857	1	1	yes	characteristic
132	C₂	#10603886	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603821

C₂

yes

maximal

#10598861

C₂

yes

maximal, characteristic

#10598532

C₂

yes

maximal

#65683

C₂²

yes

—

#64974

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

—

—

C₄

yes

characteristic

#64675

C₄

yes

characteristic

C₈×#168

C₄

yes

—

#8796

—

maximal, Sylow

C₂×C₈×D₅

Q₈

yes

characteristic

C₂×C₈×D₅

Q₈

yes

characteristic

C₄²×D₅

C₂³

yes

—

—

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

#3946

—

C₈×Dic₅

D₄

yes

characteristic

C₈×Dic₅

D₄

yes

characteristic

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

—

—

C₂×C₄×D₅

C₂×Q₈

yes

characteristic

C₂×C₄×D₅

#41

yes

—

C₂×#170

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

#41

yes

—

C₂×#168

—

C₂×#168

C₄⋊C₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

C₈×D₅

—

C₈×D₅

—

#432

—

#431

—

C₂×C₄×C₈

—

—

—

—

yes

C₈²

—

C₂²×D₅

#125

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#117

yes

—

C₂×Dic₅

#124

yes

—

C₂×Dic₅

#125

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

#108

yes

—

#168

#108

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₄×C₈

yes

C₂²×C₅

yes

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

103

C₂×C₅

#1833

yes

—

104

C₂×C₅

#1837

yes

characteristic

105

D₅

—

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

113

C₂×C₄

#4130

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₅

#8796

yes

characteristic, complemented, Sylow

122

C₂²

—

123

C₂²

—

124

C₂²

#65689

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

D₅⋊#108

yes

characteristic

129

C₄

D₅⋊#108

yes

characteristic

130

C₂

—

131

C₂

#10603857

yes

characteristic

132

C₂

#10603886

yes

—

	subgroup	G/H
center	C₂²	#65689
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#322	C₄
socle	C₂²×C₅	#382
prime radical	C₂²×D₅	#125
cosocle	C₄²×D₅	C₂³
hypercenter	#322	AGL(1,𝔽₅)
hypocenter	C₅	#8796
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#65689

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₂²×D₅

C₄²×D₅

C₂³

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
121. C₅	#8796	semidirect

H ≅ G/N

type

121. C₅

#8796

semidirect

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

subgroup	index
#10603821	2
#10598861	2
#10598532	2
#8796	5

subgroup

index

Subgroups — #13018760

Conjugacy classes of subgroups

Complements

Maximal subgroups