Group #13014158 — subgroups

Subgroups — #13014158

Order 1280.

136 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 36 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	#10601309	C₂	1	1	yes	maximal, characteristic
2	#10598691	C₂	1	2	yes	maximal
3	C₅×#423	C₄	1	1	yes	characteristic
4	#65213	C₂²	1	1	yes	characteristic
5	C₂×C₄×#168	C₄	1	1	yes	characteristic
6	#64912	—	2	2	no	—
7	#64778	—	2	2	no	—
8	#7368	—	5	1	no	maximal, Sylow
9	C₂×C₅×M₁₆	—	2	1	no	—
10	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
11	C₂×#681	D₄	1	2	yes	—
12	C₂²×#168	—	2	1	no	—
13	#3998	—	4	2	no	—
14	C₅×C₄⋊C₈	—	4	1	no	—
15	C₄⋊#168	—	4	1	no	—
16	C₄×#168	—	2	1	no	—
17	C₄×#168	C₈	1	2	yes	—
18	#1837	—	5	1	no	—
19	#1327	—	5	2	no	—
20	C₂²×C₄×C₅	—	2	1	no	—
21	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
22	C₅×M₁₆	—	4	2	no	—
23	C₂×C₅×C₈	—	4	1	no	—
24	C₄²×C₅	—	2	1	no	—
25	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
26	C₄²×C₅	M₁₆	1	1	yes	characteristic
27	#681	—	4	2	no	—
28	C₂×#168	—	2	2	no	—
29	C₂×#168	—	2	2	no	—
30	C₂×#168	—	4	1	no	—
31	C₂×#168	—	4	1	no	—
32	#674	—	8	2	no	—
33	#423	—	5	1	no	—
34	#423	AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂×C₄×C₈	—	5	1	no	—
36	#349	—	10	2	no	—
37	#341	—	10	2	no	—
38	C₂³×C₅	#101	1	1	yes	characteristic
39	C₂×C₄×C₅	—	2	1	no	—
40	C₂×C₄×C₅	—	2	1	no	—
41	C₂×C₄×C₅	—	2	1	no	—
42	C₂×C₄×C₅	—	2	1	no	—
43	C₂×C₄×C₅	—	4	2	no	—
44	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅	#99	1	1	yes	characteristic
46	C₅×C₈	—	8	1	no	—
47	#168	—	4	2	no	—
48	#168	—	4	2	no	—
49	C₂×M₁₆	—	2	1	no	—
50	C₂×M₁₆	—	5	2	no	—
51	C₂²×C₈	—	10	1	no	—
52	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	M₃₂	—	20	2	no	—
54	C₄⋊C₈	—	4	1	no	—
55	C₄⋊C₈	—	20	1	no	—
56	C₄×C₈	—	5	2	no	—
57	C₄×C₈	—	10	1	no	—
58	C₂²×C₅	—	2	1	no	—
59	C₂²×C₅	—	4	1	no	—
60	C₂²×C₅	#324	1	1	yes	characteristic
61	C₄×C₅	—	2	2	no	—
62	C₄×C₅	—	2	1	no	—
63	C₄×C₅	—	4	1	no	—
64	C₄×C₅	#354	1	1	yes	characteristic
65	C₄×C₅	#355	1	1	yes	characteristic
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
68	M₁₆	—	4	2	no	—
69	M₁₆	—	20	2	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	—	10	2	no	—
72	C₂×C₈	—	10	2	no	—
73	C₂×C₈	—	20	1	no	—
74	C₂×C₈	—	20	1	no	—
75	C₄²	—	2	1	no	—
76	C₄²	D₅⋊C₈	1	1	yes	characteristic
77	C₄²	#700	1	1	yes	characteristic
78	C₁₆	—	40	2	no	—
79	C₂×C₅	—	2	1	no	—
80	C₂×C₅	—	4	1	no	—
81	C₂×C₅	#1324	1	1	yes	characteristic
82	C₂³	#4005	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	#4003	1	1	yes	characteristic
89	C₂×C₄	#4011	1	1	yes	characteristic
90	C₈	—	8	1	no	—
91	C₈	—	20	2	no	—
92	C₈	—	20	2	no	—
93	C₅	#7368	1	1	yes	characteristic, complemented, Sylow
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	#64914	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	4	1	no	—
100	C₄	#64923	1	1	yes	characteristic
101	C₄	#64928	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	4	1	no	—
104	C₂	#10598710	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10601309

C₂

yes

maximal, characteristic

#10598691

C₂

yes

maximal

C₅×#423

C₄

yes

characteristic

#65213

C₂²

yes

characteristic

C₂×C₄×#168

C₄

yes

—

—

—

C₂×C₅×M₁₆

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×#681

D₄

yes

—

C₂²×#168

—

#3998

—

C₅×C₄⋊C₈

—

C₄⋊#168

—

C₄×#168

—

C₄×#168

C₈

yes

—

#1837

—

#1327

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₅×M₁₆

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

C₂×C₈

yes

characteristic

C₄²×C₅

M₁₆

yes

—

—

—

—

—

—

—

yes

C₂×C₄×C₈

—

#349

—

#341

—

C₂³×C₅

#101

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅

#99

yes

characteristic

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

M₃₂

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#324

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₂²×C₄

—

C₂²×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

D₅⋊C₈

yes

characteristic

C₄²

#700

yes

characteristic

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₅

#7368

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#64914

yes

characteristic

C₄

—

C₄

—

C₄

—

100

C₄

#64923

yes

characteristic

101

C₄

#64928

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#10598710

yes

characteristic

	subgroup	G/H
center	C₄	#64928
commutator	C₄²×C₅	C₂×C₈
Frattini	C₂×C₄²	C₂×AGL(1,𝔽₅)
Fitting	C₅×#423	C₄
socle	C₂×C₅	#1324
prime radical	C₂³×C₅	#101
cosocle	#65213	C₂²
hypercenter	#423	AGL(1,𝔽₅)
hypocenter	C₅	#7368
LCS	G ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₄

#64928

commutator

C₄²×C₅

C₂×C₈

C₂×C₄²

C₄

C₂×C₅

C₂³×C₅

C₂²

C₅

G ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
93. C₅	#7368	semidirect

H ≅ G/N

type

93. C₅

#7368

semidirect

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

subgroup	index
#10601309	2
#10598691	2
#7368	5

subgroup

index

#10601309

#10598691

#7368

Subgroups — #13014158

Conjugacy classes of subgroups

Complements

Maximal subgroups