Group #13101330 — subgroups

Subgroups — #13101330

Order 1600.

304 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₂²×#6029	C₂	1	1	yes	maximal, characteristic
2	C₅×#4072	C₂	1	2	yes	maximal
3	#11716617	C₂	1	2	yes	maximal, complemented
4	#11716615	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₂×#6029	C₄	1	4	yes	complemented
7	C₂×#6029	C₂²	1	1	yes	characteristic
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	C₅×C₂²⋊Dic₅	—	2	2	no	—
10	C₅×C₄⋊Dic₅	C₂²	1	4	yes	—
11	C₄×C₅×Dic₅	—	2	2	no	—
12	#65287	—	5	2	no	maximal
13	C₂³×C₅²	D₄	1	1	yes	characteristic
14	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	Q₈	1	2	yes	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×C₅×Dic₅	—	2	4	no	—
19	#6029	C₂×C₄	1	2	yes	—
20	#6029	C₂×C₄	1	2	yes	—
21	C₅×#117	—	5	2	no	—
22	#4072	D₅	1	2	yes	—
23	C₂²×#168	—	5	2	no	—
24	C₂²×#168	—	10	2	no	—
25	#3940	—	5	4	no	—
26	#3939	—	5	4	no	—
27	C₂²×C₅²	—	2	2	no	—
28	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
29	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
30	C₄×C₅²	C₄⋊C₄	1	2	yes	—
31	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
33	C₅×Dic₅	—	4	4	no	—
34	C₂²×C₄×C₅	—	2	2	no	—
35	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
36	C₅×C₄⋊C₄	—	5	4	no	—
37	C₅×C₂²⋊C₄	—	10	2	no	—
38	C₄²×C₅	—	10	2	no	—
39	C₂²⋊Dic₅	—	2	2	no	—
40	C₄⋊Dic₅	C₂×D₅	1	4	yes	—
41	C₄×Dic₅	—	2	2	no	—
42	C₂×#168	—	5	8	no	—
43	C₂×#168	—	5	2	no	—
44	C₂×#168	—	5	2	no	—
45	C₂×#168	—	10	8	no	—
46	C₂×#168	—	10	2	no	—
47	C₂×#168	—	10	2	no	—
48	#427	—	25	1	no	Sylow
49	C₂×C₅²	—	2	2	no	—
50	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅²	#135	1	2	yes	—
52	C₂³×C₅	—	2	2	no	—
53	C₂³×C₅	C₂²⋊D₅	1	2	yes	—
54	C₂×C₄×C₅	—	2	4	no	—
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	4	no	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	—	10	4	no	—
59	C₂×C₄×C₅	Dic₁₀	1	4	yes	—
60	C₂×C₄×C₅	C₄×D₅	1	4	yes	—
61	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
62	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
63	C₂×Dic₅	—	2	4	no	—
64	#168	—	5	4	no	—
65	#168	—	5	4	no	—
66	#168	—	10	4	no	—
67	#168	—	10	4	no	—
68	C₂²×C₈	—	25	1	no	—
69	#117	—	5	2	no	—
70	#107	—	25	2	no	—
71	#106	—	25	2	no	—
72	C₅²	#427	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₅	—	2	8	no	—
74	C₂²×C₅	—	2	4	no	—
75	C₂²×C₅	—	2	2	no	—
76	C₂²×C₅	—	2	4	no	—
77	C₂²×C₅	#683	1	4	yes	—
78	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
79	C₄×C₅	—	2	2	no	—
80	C₄×C₅	—	2	2	no	—
81	C₄×C₅	—	2	4	no	—
82	C₄×C₅	—	20	4	no	—
83	C₄×C₅	#683	1	4	yes	—
84	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
85	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
86	Dic₅	—	4	4	no	—
87	C₂²×C₄	D₅²	1	1	yes	characteristic
88	C₂×C₈	—	25	4	no	—
89	C₂×C₈	—	25	1	no	—
90	C₂×C₈	—	25	1	no	—
91	C₄⋊C₄	—	5	4	no	—
92	C₂²⋊C₄	—	10	2	no	—
93	C₄²	—	10	2	no	—
94	C₂×C₅	—	2	8	no	—
95	C₂×C₅	—	2	4	no	—
96	C₂×C₅	—	2	4	no	—
97	C₂×C₅	—	2	2	no	—
98	C₂×C₅	C₂×#683	1	2	yes	—
99	C₂×C₅	#4096	1	4	yes	—
100	C₂³	#6037	1	1	yes	characteristic
101	C₂×C₄	—	10	4	no	—
102	C₂×C₄	#6036	1	2	yes	—
103	C₂×C₄	#6037	1	1	yes	characteristic
104	C₂×C₄	#6039	1	2	yes	—
105	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
106	C₈	—	25	2	no	—
107	C₈	—	25	2	no	—
108	C₅	—	2	2	no	—
109	C₅	#65287	1	2	yes	complemented
110	C₂²	—	2	2	no	—
111	C₂²	#87781	1	2	yes	—
112	C₂²	C₂×#6037	1	1	yes	characteristic
113	C₄	—	20	4	no	—
114	C₄	#87781	1	2	yes	—
115	C₄	C₂×#6036	1	1	yes	characteristic
116	C₄	C₂×#6039	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#11716940	1	2	yes	—
119	C₂	C₂×#87781	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6029

C₂

yes

maximal, characteristic

C₅×#4072

C₂

yes

maximal

#11716617

C₂

yes

maximal, complemented

#11716615

C₂

yes

maximal, complemented

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂×#6029

C₄

yes

complemented

C₂×#6029

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₅×C₂²⋊Dic₅

—

C₅×C₄⋊Dic₅

C₂²

yes

—

C₄×C₅×Dic₅

—

#65287

—

maximal

C₂³×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

Q₈

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

#6029

C₂×C₄

yes

—

#6029

C₂×C₄

yes

—

C₅×#117

—

#4072

D₅

yes

—

C₂²×#168

—

C₂²×#168

—

#3940

—

#3939

—

C₂²×C₅²

—

C₂²×C₅²

C₄⋊C₄

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₄⋊C₄

yes

—

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

C₅×Dic₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

C₄×Dic₅

—

—

—

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#135

yes

—

C₂³×C₅

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

—

—

—

—

C₂²×C₈

—

#117

—

#107

—

#106

—

C₅²

#427

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#683

yes

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#683

yes

—

C₄×C₅

C₂×Dic₁₀

yes

—

C₄×C₅

C₂×C₄×D₅

yes

—

Dic₅

—

C₂²×C₄

D₅²

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

C₂×#683

yes

—

C₂×C₅

#4096

yes

—

100

C₂³

#6037

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

#6036

yes

—

103

C₂×C₄

#6037

yes

characteristic

104

C₂×C₄

#6039

yes

—

105

C₂×C₄

C₂×D₅²

yes

characteristic

106

C₈

—

107

C₈

—

108

C₅

—

109

C₅

#65287

yes

complemented

110

C₂²

—

111

C₂²

#87781

yes

—

112

C₂²

C₂×#6037

yes

characteristic

113

C₄

—

114

C₄

#87781

yes

—

115

C₄

C₂×#6036

yes

characteristic

116

C₄

C₂×#6039

yes

characteristic

117

C₂

—

118

C₂

#11716940

yes

—

119

C₂

C₂×#87781

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#6037
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₂²×C₄×C₅²	C₂²
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂³×C₅²	D₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#427
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

subgroup

G/H

center

C₂×C₄

#6037

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₂²×C₄×C₅²

C₂²

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂³×C₅²

D₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂²×C₄

D₅²

hypocenter

C₅²

#427

LCS

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄

N	H ≅ G/N	type
3. #11716617	C₂	semidirect
4. #11716615	C₂	semidirect
6. C₂×#6029	C₄	semidirect
72. C₅²	#427	semidirect
109. C₅	#65287	semidirect

H ≅ G/N

type

3. #11716617

C₂

semidirect

4. #11716615

C₂

semidirect

6. C₂×#6029

C₄

semidirect

72. C₅²

#427

semidirect

109. C₅

#65287

semidirect

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

subgroup	index
C₂²×#6029	2
C₅×#4072	2
#11716617	2
#11716615	2
#65287	5

subgroup

index

C₂²×#6029

Subgroups — #13101330

Conjugacy classes of subgroups

Complements

Maximal subgroups