Group #13103101 — subgroups

Subgroups — #13103101

Order 1600.

600 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 235 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	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#87804	C₂	1	2	yes	maximal, complemented
3	#11716703	C₂	1	2	yes	maximal, complemented
4	#11716700	C₂	1	2	yes	maximal, complemented
5	C₅²×#41	—	2	2	no	—
6	C₅²×#41	—	2	2	no	—
7	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅²×D₄	C₄	1	2	yes	—
9	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅²	C₄	1	2	yes	—
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	#87804	—	2	2	no	—
13	#87804	C₂²	1	4	yes	—
14	C₂×#6029	—	2	2	no	—
15	#65526	—	5	6	no	maximal
16	C₂³×C₅²	C₂×C₄	1	2	yes	—
17	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₅²×Q₈	—	2	2	no	—
19	C₅²×D₄	—	2	4	no	—
20	C₅²×D₄	—	2	2	no	—
21	C₂×C₄×C₅²	—	2	2	no	—
22	C₂×C₄×C₅²	—	2	2	no	—
23	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅²	D₄	1	2	yes	—
26	C₂×C₄×C₅²	D₄	1	2	yes	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	#6029	—	2	4	no	—
29	C₂×C₅×#41	D₅	1	6	yes	—
30	C₂×#681	—	5	12	no	—
31	#3968	—	5	12	no	—
32	#3965	—	5	12	no	—
33	C₂²×C₅²	—	2	2	no	—
34	C₂²×C₅²	—	4	1	no	—
35	C₂²×C₅²	—	4	2	no	—
36	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₅²	—	2	2	no	—
39	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₅²	C₂×D₄	1	2	yes	—
42	C₅×#41	—	2	12	no	—
43	C₅×#41	—	2	12	no	—
44	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
45	C₂×C₅×D₄	Dic₅	1	12	yes	—
46	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
47	C₂²×C₄×C₅	Dic₅	1	12	yes	—
48	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
49	#681	—	5	24	no	—
50	#681	—	10	12	no	—
51	C₂×#168	—	10	12	no	—
52	#413	—	25	1	no	Sylow
53	C₂×C₅²	—	2	1	no	—
54	C₂×C₅²	—	2	2	no	—
55	C₂×C₅²	—	4	2	no	—
56	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₅	C₂×Dic₅	1	12	yes	—
58	C₂³×C₅	C₂×Dic₅	1	6	yes	—
59	C₅×Q₈	—	2	12	no	—
60	C₅×D₄	—	2	24	no	—
61	C₅×D₄	—	2	12	no	—
62	C₂×C₄×C₅	—	2	12	no	—
63	C₂×C₄×C₅	—	2	12	no	—
64	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	—
65	C₂×C₄×C₅	C₂×Dic₅	1	6	yes	—
66	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
67	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
68	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
69	#168	—	10	24	no	—
70	C₂×#41	C₅⋊D₅	1	1	yes	characteristic
71	C₂×M₁₆	—	25	2	no	—
72	#101	—	25	2	no	—
73	#100	—	25	2	no	—
74	C₅²	#413	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₅	—	2	12	no	—
76	C₂²×C₅	—	4	6	no	—
77	C₂²×C₅	—	4	12	no	—
78	C₂²×C₅	C₂²⋊Dic₅	1	12	yes	—
79	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
80	C₄×C₅	—	2	12	no	—
81	C₄×C₅	C₂²⋊Dic₅	1	6	yes	—
82	C₄×C₅	C₂²⋊Dic₅	1	6	yes	—
83	C₄×C₅	C₂×C₂²⋊D₅	1	12	yes	—
84	#41	—	2	2	no	—
85	#41	—	2	2	no	—
86	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
87	C₂×D₄	#1039	1	2	yes	—
88	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
89	C₂²×C₄	#1039	1	2	yes	—
90	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
91	M₁₆	—	25	4	no	—
92	M₁₆	—	50	2	no	—
93	C₂×C₈	—	50	2	no	—
94	C₂×C₅	—	2	6	no	—
95	C₂×C₅	—	2	12	no	—
96	C₂×C₅	—	4	12	no	—
97	C₂×C₅	C₂×C₂²⋊Dic₅	1	6	yes	—
98	C₂³	C₂×#1039	1	2	yes	—
99	C₂³	C₂×#1039	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	D₄	—	2	4	no	—
102	D₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂×#1039	1	2	yes	—
106	C₂×C₄	C₂×#1039	1	1	yes	characteristic
107	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
108	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
109	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
110	C₈	—	50	4	no	—
111	C₅	#65526	1	6	yes	complemented
112	C₂²	—	2	2	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	4	2	no	—
115	C₂²	C₂²⋊#1039	1	2	yes	—
116	C₂²	C₂²×#1039	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	C₂²⋊#1039	1	1	yes	characteristic
119	C₄	C₂²⋊#1039	1	1	yes	characteristic
120	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
121	C₂	—	2	1	no	—
122	C₂	—	2	2	no	—
123	C₂	—	4	2	no	—
124	C₂	C₂×C₂²⋊#1039	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅²×#41

C₂

yes

maximal, characteristic

C₂×#87804

C₂

yes

maximal, complemented

#11716703

C₂

yes

maximal, complemented

#11716700

C₂

yes

maximal, complemented

C₅²×#41

—

C₅²×#41

—

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅²×D₄

C₄

yes

—

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂²×C₄×C₅²

C₄

yes

—

C₂²×C₄×C₅²

C₂²

yes

characteristic

#87804

—

#87804

C₂²

yes

—

C₂×#6029

—

#65526

—

maximal

C₂³×C₅²

C₂×C₄

yes

—

C₂³×C₅²

C₂×C₄

yes

characteristic

C₅²×Q₈

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

#6029

—

C₂×C₅×#41

D₅

yes

—

C₂×#681

—

#3968

—

#3965

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂²⋊C₄

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂²⋊C₄

yes

characteristic

C₄×C₅²

C₂²⋊C₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

—

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×D₄

Dic₅

yes

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂²×C₄×C₅

Dic₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₅

C₂×Dic₅

yes

—

C₂³×C₅

C₂×Dic₅

yes

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

#168

—

C₂×#41

C₅⋊D₅

yes

characteristic

C₂×M₁₆

—

#101

—

#100

—

C₅²

#413

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²⋊Dic₅

yes

—

C₂²×C₅

C₂²×Dic₅

yes

—

C₄×C₅

—

C₄×C₅

C₂²⋊Dic₅

yes

—

C₄×C₅

C₂²⋊Dic₅

yes

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

#41

—

#41

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

C₂×D₄

#1039

yes

—

C₂×D₄

C₂×C₅⋊D₅

yes

characteristic

C₂²×C₄

#1039

yes

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×C₂²⋊Dic₅

yes

—

C₂³

C₂×#1039

yes

—

C₂³

C₂×#1039

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×#1039

yes

—

106

C₂×C₄

C₂×#1039

yes

characteristic

107

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

108

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

109

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

110

C₈

—

111

C₅

#65526

yes

complemented

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²⋊#1039

yes

—

116

C₂²

C₂²×#1039

yes

characteristic

117

C₄

—

118

C₄

C₂²⋊#1039

yes

characteristic

119

C₄

C₂²⋊#1039

yes

characteristic

120

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

—

121

C₂

—

122

C₂

—

123

C₂

—

124

C₂

C₂×C₂²⋊#1039

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊#1039
commutator	C₂²×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting, prime radical	C₂×C₅²×#41	C₂
socle	C₂×C₅²	C₂×C₂²⋊C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×#41	C₅⋊D₅
hypocenter	C₅²	#413
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₄

C₂²⋊#1039

commutator

C₂²×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting, prime radical

C₂×C₅²×#41

C₂

socle

C₂×C₅²

C₂×C₂²⋊C₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂×#41

C₅⋊D₅

hypocenter

C₅²

#413

LCS

G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
2. C₂×#87804	C₂	semidirect
3. #11716703	C₂	semidirect
4. #11716700	C₂	semidirect
74. C₅²	#413	semidirect
111. C₅	#65526	semidirect

H ≅ G/N

type

2. C₂×#87804

C₂

semidirect

3. #11716703

C₂

semidirect

4. #11716700

C₂

semidirect

74. C₅²

#413

semidirect

111. C₅

#65526

semidirect

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

subgroup	index
C₂×C₅²×#41	2
C₂×#87804	2
#11716703	2
#11716700	2
#65526	5

subgroup

index

C₂×C₅²×#41

Subgroups — #13103101

Conjugacy classes of subgroups

Complements

Maximal subgroups