Group #13102834 — subgroups

Subgroups — #13102834

Order 1600.

616 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 259 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×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#1039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#1039	C₂	1	1	yes	maximal, characteristic
4	#11716698	C₂	1	4	yes	maximal
5	C₂²×C₄×C₅²	C₂²	1	2	yes	—
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂²×#1039	C₂²	1	2	yes	—
8	C₂²×#1039	C₂²	1	2	yes	—
9	C₅²×C₄⋊C₄	—	2	2	no	—
10	C₄⋊#1039	—	2	2	no	—
11	C₄×#1039	C₄	1	4	yes	complemented
12	#65266	—	5	6	no	maximal
13	C₂³×C₅²	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₅²	—	2	2	no	—
15	C₂×C₄×C₅²	—	2	4	no	—
16	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅²	D₄	1	2	yes	—
18	C₂×C₄×C₅²	Q₈	1	2	yes	—
19	C₂×#1039	—	2	2	no	—
20	C₂×#1039	—	2	2	no	—
21	C₂×#1039	—	2	2	no	—
22	C₂×#1039	—	2	4	no	—
23	C₂×#1039	C₂×C₄	1	4	yes	—
24	C₂×C₅×C₄⋊C₄	D₅	1	6	yes	—
25	C₂×C₄⋊Dic₅	—	5	6	no	—
26	C₂×C₄×Dic₅	—	5	6	no	—
27	#3963	—	5	24	no	—
28	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₅²	#41	1	2	yes	—
32	C₂²×C₅²	#41	1	2	yes	—
33	C₄×C₅²	—	4	2	no	—
34	C₄×C₅²	C₄⋊C₄	1	4	yes	—
35	#1039	—	2	4	no	—
36	#1039	—	4	2	no	—
37	C₂²×C₄×C₅	C₂×D₅	1	12	yes	—
38	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
39	C₂²×Dic₅	—	5	12	no	—
40	C₂²×Dic₅	—	5	12	no	—
41	C₅×C₄⋊C₄	—	2	12	no	—
42	C₄⋊Dic₅	—	10	12	no	—
43	C₄×Dic₅	—	5	24	no	—
44	#382	—	25	1	no	Sylow
45	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅²	#117	1	2	yes	—
47	C₂×C₅²	#124	1	1	yes	characteristic
48	C₂×C₅²	#124	1	1	yes	characteristic
49	C₂×C₅²	#125	1	2	yes	—
50	C₂³×C₅	C₂²×D₅	1	6	yes	—
51	C₂×C₄×C₅	—	2	24	no	—
52	C₂×C₄×C₅	—	2	12	no	—
53	C₂×C₄×C₅	Dic₁₀	1	12	yes	—
54	C₂×C₄×C₅	C₄×D₅	1	12	yes	—
55	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
56	C₂×Dic₅	—	5	24	no	—
57	C₂×Dic₅	—	10	12	no	—
58	C₂×Dic₅	—	10	12	no	—
59	C₂×Dic₅	—	10	24	no	—
60	C₂×Dic₅	—	10	12	no	—
61	C₂×C₄⋊C₄	—	25	1	no	—
62	C₂×C₄⋊C₄	C₅⋊D₅	1	1	yes	characteristic
63	C₂×C₄²	—	25	1	no	—
64	#95	—	25	4	no	—
65	C₅²	#382	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₅	C₂×Dic₁₀	1	6	yes	—
67	C₂²×C₅	C₂×C₄×D₅	1	6	yes	—
68	C₂²×C₅	#711	1	12	yes	—
69	C₂²×C₅	#713	1	12	yes	—
70	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
71	C₄×C₅	—	4	12	no	—
72	C₄×C₅	#683	1	24	yes	—
73	Dic₅	—	10	24	no	—
74	Dic₅	—	20	12	no	—
75	C₂²×C₄	—	25	2	no	—
76	C₂²×C₄	—	25	2	no	—
77	C₂²×C₄	C₂×C₅⋊D₅	1	2	yes	—
78	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	50	2	no	—
81	C₄²	—	25	4	no	—
82	C₂×C₅	#4036	1	12	yes	—
83	C₂×C₅	#4038	1	12	yes	—
84	C₂×C₅	C₂×#683	1	6	yes	—
85	C₂×C₅	#4082	1	6	yes	—
86	C₂×C₅	#4093	1	6	yes	—
87	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	25	4	no	—
91	C₂×C₄	—	50	2	no	—
92	C₂×C₄	—	50	2	no	—
93	C₂×C₄	—	50	4	no	—
94	C₂×C₄	—	50	2	no	—
95	C₂×C₄	#6045	1	2	yes	—
96	C₂×C₄	C₄×C₅⋊D₅	1	2	yes	—
97	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
98	C₅	#65266	1	6	yes	complemented
99	C₂²	C₂×#6045	1	1	yes	characteristic
100	C₂²	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
101	C₂²	#87902	1	2	yes	—
102	C₂²	#87904	1	2	yes	—
103	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
104	C₄	—	4	2	no	—
105	C₄	—	50	4	no	—
106	C₄	—	100	2	no	—
107	C₄	#87806	1	4	yes	—
108	C₂	#11717210	1	2	yes	—
109	C₂	#11717212	1	2	yes	—
110	C₂	C₂×#87806	1	1	yes	characteristic
111	C₂	#11717256	1	1	yes	characteristic
112	C₂	#11717267	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅²×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×C₄⋊#1039

C₂

yes

maximal, characteristic

C₂×C₄×#1039

C₂

yes

maximal, characteristic

#11716698

C₂

yes

maximal

C₂²×C₄×C₅²

C₂²

yes

—

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂²×#1039

C₂²

yes

—

C₂²×#1039

C₂²

yes

—

C₅²×C₄⋊C₄

—

C₄⋊#1039

—

C₄×#1039

C₄

yes

complemented

#65266

—

maximal

C₂³×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₅×C₄⋊C₄

D₅

yes

—

C₂×C₄⋊Dic₅

—

C₂×C₄×Dic₅

—

#3963

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂²×C₅²

C₂×Q₈

yes

characteristic

C₂²×C₅²

#41

yes

—

C₂²×C₅²

#41

yes

—

C₄×C₅²

—

C₄×C₅²

C₄⋊C₄

yes

—

#1039

—

#1039

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₄⋊Dic₅

—

C₄×Dic₅

—

#382

—

Sylow

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#117

yes

—

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₅²

yes

—

C₂³×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₅⋊D₅

yes

characteristic

C₂×C₄²

—

#95

—

C₅²

#382

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×Dic₁₀

yes

—

C₂²×C₅

C₂×C₄×D₅

yes

—

C₂²×C₅

#711

yes

—

C₂²×C₅

#713

yes

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

#683

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₅⋊D₅

yes

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂×C₅

#4036

yes

—

C₂×C₅

#4038

yes

—

C₂×C₅

C₂×#683

yes

—

C₂×C₅

#4082

yes

—

C₂×C₅

#4093

yes

—

C₂³

C₂²×C₅⋊D₅

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6045

yes

—

C₂×C₄

C₄×C₅⋊D₅

yes

—

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

C₅

yes

C₂²

yes

100

C₂²

C₂×C₄×C₅⋊D₅

yes

characteristic

101

C₂²

#87902

yes

—

102

C₂²

#87904

yes

—

103

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

100

—

107

C₄

#87806

yes

—

108

C₂

#11717210

yes

—

109

C₂

#11717212

yes

—

110

C₂

C₂×#87806

yes

characteristic

111

C₂

#11717256

yes

characteristic

112

C₂

#11717267

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₅²×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₅²	C₂³
hypercenter	C₂×C₄⋊C₄	C₅⋊D₅
hypocenter	C₅²	#382
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×C₅⋊D₅

commutator

C₂²×C₅²

C₂²×C₄

Fitting

C₂×C₅²×C₄⋊C₄

C₂

socle, prime radical, cosocle

C₂³×C₅²

C₂³

hypercenter

C₂×C₄⋊C₄

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
11. C₄×#1039	C₄	semidirect
65. C₅²	#382	semidirect
98. C₅	#65266	semidirect

H ≅ G/N

type

11. C₄×#1039

C₄

semidirect

65. C₅²

#382

semidirect

98. C₅

#65266

semidirect

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

subgroup	index
C₂×C₅²×C₄⋊C₄	2
C₂×C₄⋊#1039	2
C₂×C₄×#1039	2
#11716698	2
#65266	5

subgroup

index

C₂×C₅²×C₄⋊C₄

C₂×C₄⋊#1039

C₂×C₄×#1039

#11716698

#65266

Subgroups — #13102834

Conjugacy classes of subgroups

Complements

Maximal subgroups