Group #13100557 — subgroups

Subgroups — C₄×#87775 #13100557

Order 1600.

264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₄×C₅×Dic₁₀	C₂	1	2	yes	maximal
2	C₂×#87775	C₂	1	1	yes	maximal, characteristic
3	C₄×#6029	C₂	1	1	yes	maximal, characteristic
4	#11716617	C₂	1	1	yes	maximal, characteristic
5	#11716606	C₂	1	2	yes	maximal
6	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
7	C₄²×C₅²	C₂²	1	1	yes	characteristic
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	C₂×#6029	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₅×#683	—	2	2	no	—
12	C₄×C₅×Dic₅	—	2	2	no	—
13	#87775	C₄	1	4	yes	direct product factor
14	C₄×#689	—	5	2	no	maximal
15	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×C₅×Dic₅	—	2	2	no	—
19	C₂×C₅×Dic₅	—	2	2	no	—
20	C₅×Dic₁₀	—	2	2	no	—
21	C₅×Dic₁₀	C₂×C₄	1	4	yes	—
22	#6029	—	2	1	no	—
23	#6029	C₂×C₄	1	2	yes	—
24	C₄×C₅×Q₈	—	5	2	no	—
25	C₂×#689	—	5	2	no	—
26	C₄×Dic₁₀	D₅	1	2	yes	—
27	#3967	—	5	2	no	—
28	#3942	—	5	2	no	—
29	#3939	—	5	2	no	—
30	C₄×#168	—	5	2	no	—
31	C₄×#168	—	10	2	no	—
32	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅²	—	2	1	no	—
34	C₄×C₅²	Q₁₆	1	2	yes	complemented
35	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₅²	#41	1	1	yes	characteristic
37	C₅×Dic₅	—	2	4	no	—
38	C₅×Dic₅	—	4	2	no	—
39	C₂×C₅×Q₈	—	5	2	no	—
40	C₂×Dic₁₀	C₂×D₅	1	2	yes	—
41	C₅×C₄⋊C₄	—	5	2	no	—
42	C₅×C₄⋊C₄	—	10	2	no	—
43	C₄²×C₅	—	2	2	no	—
44	C₄²×C₅	—	10	2	no	—
45	C₄²×C₅	C₂×D₅	1	2	yes	—
46	#689	—	5	8	no	—
47	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
48	#683	—	2	2	no	—
49	C₄×Dic₅	—	2	2	no	—
50	C₂×#168	—	5	2	no	—
51	C₂×#168	—	5	2	no	—
52	C₂×#168	—	10	2	no	—
53	C₂×#168	—	10	2	no	—
54	C₄×Q₁₆	—	25	1	no	Sylow
55	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
56	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₅²	#135	1	1	yes	characteristic
58	C₅×Q₈	—	5	4	no	—
59	C₅×Q₈	—	10	2	no	—
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	10	2	no	—
64	C₂×C₄×C₅	—	10	2	no	—
65	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
66	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
67	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
68	C₂×Dic₅	—	2	2	no	—
69	C₂×Dic₅	—	2	2	no	—
70	Dic₁₀	—	2	2	no	—
71	Dic₁₀	C₄×D₅	1	4	yes	—
72	#168	—	5	4	no	—
73	#168	—	10	4	no	—
74	#168	—	10	4	no	—
75	#168	—	10	2	no	—
76	C₂×Q₁₆	—	25	1	no	—
77	C₄×Q₈	—	5	2	no	—
78	#107	—	25	1	no	—
79	#103	—	25	2	no	—
80	C₄×C₈	—	25	1	no	—
81	C₅²	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₅	—	2	2	no	—
83	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
84	C₄×C₅	—	2	4	no	—
85	C₄×C₅	—	2	2	no	—
86	C₄×C₅	—	2	2	no	—
87	C₄×C₅	—	2	2	no	—
88	C₄×C₅	—	2	4	no	—
89	C₄×C₅	—	10	4	no	—
90	C₄×C₅	—	20	2	no	—
91	C₄×C₅	#689	1	4	yes	complemented
92	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
93	C₄×C₅	#709	1	2	yes	—
94	Dic₅	—	2	4	no	—
95	Dic₅	—	4	2	no	—
96	C₂×Q₈	—	5	2	no	—
97	Q₁₆	—	25	4	no	—
98	C₂×C₈	—	25	1	no	—
99	C₂×C₈	—	25	1	no	—
100	C₄⋊C₄	—	5	2	no	—
101	C₄⋊C₄	—	10	2	no	—
102	C₄²	—	10	2	no	—
103	C₄²	D₅²	1	1	yes	characteristic
104	C₂×C₅	—	2	2	no	—
105	C₂×C₅	—	2	2	no	—
106	C₂×C₅	—	2	2	no	—
107	C₂×C₅	C₄×C₂²⋊D₅	1	2	yes	—
108	C₂×C₅	C₂×#689	1	2	yes	—
109	C₂×C₅	#4096	1	2	yes	—
110	Q₈	—	5	4	no	—
111	Q₈	—	10	2	no	—
112	C₂×C₄	—	10	2	no	—
113	C₂×C₄	—	10	2	no	—
114	C₂×C₄	#6037	1	1	yes	characteristic
115	C₂×C₄	#6037	1	1	yes	characteristic
116	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
117	C₈	—	25	2	no	—
118	C₈	—	50	1	no	—
119	C₅	—	2	2	no	—
120	C₅	C₄×#689	1	2	yes	complemented
121	C₂²	C₂×#6037	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	10	4	no	—
124	C₄	—	20	2	no	—
125	C₄	#87775	1	2	yes	direct product factor
126	C₄	#87873	1	1	yes	characteristic
127	C₄	C₄×D₅²	1	1	yes	characteristic
128	C₂	#11716940	1	1	yes	characteristic
129	C₂	C₂×#87775	1	1	yes	characteristic
130	C₂	C₄×#6037	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₅×Dic₁₀

C₂

yes

maximal

C₂×#87775

C₂

yes

maximal, characteristic

C₄×#6029

C₂

yes

maximal, characteristic

#11716617

C₂

yes

maximal, characteristic

#11716606

C₂

yes

maximal

C₂×C₅×Dic₁₀

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₅×#683

—

C₄×C₅×Dic₅

—

#87775

C₄

yes

direct product factor

C₄×#689

—

maximal

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×Dic₁₀

C₂×C₄

yes

—

#6029

—

#6029

C₂×C₄

yes

—

C₄×C₅×Q₈

—

C₂×#689

—

C₄×Dic₁₀

D₅

yes

—

—

—

—

—

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

Q₁₆

yes

complemented

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

#41

yes

characteristic

C₅×Dic₅

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

C₂×D₅

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

—

#689

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

C₄×Dic₅

—

—

—

—

—

C₄×Q₁₆

—

Sylow

C₂×C₅²

C₄×D₄

yes

characteristic

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₂×C₅²

#135

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

C₄×D₅

yes

—

—

—

—

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

#103

—

C₄×C₈

—

C₅²

C₄×Q₁₆

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#689

yes

complemented

C₄×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

#709

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

D₅²

yes

characteristic

104

C₂×C₅

—

105

C₂×C₅

—

106

C₂×C₅

—

107

C₂×C₅

C₄×C₂²⋊D₅

yes

—

108

C₂×C₅

C₂×#689

yes

—

109

C₂×C₅

#4096

yes

—

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#6037

yes

characteristic

115

C₂×C₄

#6037

yes

characteristic

116

C₂×C₄

C₂×D₅²

yes

characteristic

117

C₈

—

118

C₈

—

119

C₅

—

120

C₅

C₄×#689

yes

complemented

121

C₂²

C₂×#6037

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#87775

yes

direct product factor

126

C₄

#87873

yes

characteristic

127

C₄

C₄×D₅²

yes

characteristic

128

C₂

#11716940

yes

characteristic

129

C₂

C₂×#87775

yes

characteristic

130

C₂

C₄×#6037

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#6037
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₄²×C₅²	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₄²	D₅²
hypocenter	C₅²	C₄×Q₁₆
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

subgroup

G/H

center

C₂×C₄

#6037

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₄²×C₅²

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₄²

D₅²

hypocenter

C₅²

C₄×Q₁₆

LCS

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂×C₄ ◁ C₄²

N	H ≅ G/N	type
13. #87775	C₄	direct (N × H)
34. C₄×C₅²	Q₁₆	semidirect
81. C₅²	C₄×Q₁₆	semidirect
91. C₄×C₅	#689	semidirect
120. C₅	C₄×#689	semidirect
125. C₄	#87775	direct (N × H)

H ≅ G/N

type

13. #87775

C₄

direct (N × H)

34. C₄×C₅²

Q₁₆

semidirect

81. C₅²

C₄×Q₁₆

semidirect

91. C₄×C₅

#689

semidirect

120. C₅

C₄×#689

semidirect

125. C₄

#87775

direct (N × H)

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

subgroup	index
C₄×C₅×Dic₁₀	2
C₂×#87775	2
C₄×#6029	2
#11716617	2
#11716606	2
C₄×#689	5

subgroup

index

C₄×C₅×Dic₁₀

Subgroups — C₄×#87775 #13100557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#87775 #13100557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#87775 #13100557