Group #72269247 — subgroups

Subgroups — #72269247

Order 3200.

174 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 66 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	#13099493	C₂	1	1	yes	maximal, characteristic
2	C₅×#64683	C₂	1	2	yes	maximal
3	C₄×C₅²×C₈	C₂²	1	1	yes	characteristic
4	C₂×#87756	C₄	1	1	yes	characteristic
5	C₂×#87756	C₄	1	1	yes	characteristic
6	C₅×#3944	—	2	2	no	—
7	#10598058	—	5	2	no	maximal
8	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
9	C₂×C₅²×C₈	D₄	1	1	yes	characteristic
10	C₄²×C₅²	Q₈	1	1	yes	characteristic
11	#87756	—	2	1	no	—
12	#87756	C₈	1	2	yes	—
13	C₅×#672	—	4	2	no	—
14	C₅×#364	—	5	2	no	—
15	#64683	D₅	1	2	yes	—
16	#64681	—	5	2	no	—
17	#64681	—	10	2	no	—
18	C₂×C₄×C₅²	—	2	1	no	—
19	C₂×C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
20	C₅²×C₈	C₂×C₈	1	1	yes	characteristic
21	C₅²×C₈	M₁₆	1	1	yes	characteristic
22	C₅²×C₈	SD₁₆	1	2	yes	—
23	C₅×M₃₂	—	10	2	no	—
24	C₄×C₅×C₈	—	2	2	no	—
25	C₄×C₅×C₈	C₂×D₅	1	2	yes	—
26	#3944	—	2	2	no	—
27	C₂×#672	—	5	2	no	—
28	C₂×#672	—	5	2	no	—
29	C₂×#672	—	10	2	no	—
30	C₂×#672	—	10	2	no	—
31	#1369	—	25	1	no	Sylow
32	C₂²×C₅²	#108	1	1	yes	characteristic
33	C₄×C₅²	—	4	1	no	—
34	C₄×C₅²	C₄⋊C₈	1	1	yes	characteristic
35	C₄×C₅²	#106	1	1	yes	characteristic
36	C₂×C₅×C₈	—	2	2	no	—
37	C₂×C₅×C₈	—	2	2	no	—
38	C₂×C₅×C₈	C₄×D₅	1	2	yes	—
39	C₂×C₅×C₈	C₂²⋊D₅	1	2	yes	—
40	C₄²×C₅	—	2	2	no	—
41	C₄²×C₅	Dic₁₀	1	2	yes	—
42	C₅×C₁₆	—	20	2	no	—
43	#672	—	4	2	no	—
44	#672	—	5	4	no	—
45	#672	—	10	4	no	—
46	#672	—	10	4	no	—
47	#672	—	10	2	no	—
48	#364	—	5	2	no	—
49	#346	—	25	1	no	—
50	C₂×C₅²	—	2	1	no	—
51	C₂×C₅²	#334	1	1	yes	characteristic
52	C₂×C₄×C₅	—	2	4	no	—
53	C₂×C₄×C₅	—	2	2	no	—
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	#683	1	2	yes	—
56	C₅×C₈	—	2	2	no	—
57	C₅×C₈	—	2	2	no	—
58	C₅×C₈	—	2	4	no	—
59	C₅×C₈	C₈×D₅	1	2	yes	—
60	C₅×C₈	#676	1	2	yes	—
61	C₅×C₈	#687	1	2	yes	—
62	C₅×C₈	#688	1	2	yes	—
63	M₃₂	—	10	2	no	—
64	C₂×C₁₆	—	25	1	no	—
65	C₂×C₁₆	—	25	1	no	—
66	C₄×C₈	D₅²	1	1	yes	characteristic
67	C₅²	#1369	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₅	—	2	2	no	—
69	C₂²×C₅	#3954	1	2	yes	—
70	C₄×C₅	—	2	2	no	—
71	C₄×C₅	—	2	2	no	—
72	C₄×C₅	—	4	4	no	—
73	C₄×C₅	—	4	2	no	—
74	C₄×C₅	#3940	1	2	yes	—
75	C₄×C₅	#3946	1	2	yes	—
76	C₂×C₈	#6036	1	1	yes	characteristic
77	C₂×C₈	#6037	1	1	yes	characteristic
78	C₄²	#6039	1	1	yes	characteristic
79	C₁₆	—	20	2	no	—
80	C₁₆	—	25	2	no	—
81	C₁₆	—	50	1	no	—
82	C₂×C₅	—	2	4	no	—
83	C₂×C₅	—	2	2	no	—
84	C₂×C₅	—	2	2	no	—
85	C₂×C₅	#64700	1	2	yes	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	#87781	1	1	yes	characteristic
88	C₈	#87767	1	1	yes	characteristic
89	C₈	#87769	1	1	yes	characteristic
90	C₈	#87772	1	2	yes	—
91	C₅	—	2	2	no	—
92	C₅	#10598058	1	2	yes	complemented
93	C₂²	#11716619	1	1	yes	characteristic
94	C₄	—	4	1	no	—
95	C₄	#11716613	1	1	yes	characteristic
96	C₄	#11716615	1	1	yes	characteristic
97	C₂	—	2	1	no	—
98	C₂	#13099268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13099493

C₂

yes

maximal, characteristic

C₅×#64683

C₂

yes

maximal

C₄×C₅²×C₈

C₂²

yes

characteristic

C₂×#87756

C₄

yes

characteristic

C₂×#87756

C₄

yes

—

—

C₂×C₅²×C₈

C₂×C₄

yes

characteristic

C₂×C₅²×C₈

D₄

yes

characteristic

C₄²×C₅²

Q₈

yes

characteristic

#87756

—

#87756

C₈

yes

—

C₅×#672

—

C₅×#364

—

#64683

D₅

yes

—

#64681

—

#64681

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₄⋊C₄

yes

characteristic

C₅²×C₈

C₂×C₈

yes

characteristic

C₅²×C₈

M₁₆

yes

characteristic

C₅²×C₈

SD₁₆

yes

—

C₅×M₃₂

—

C₄×C₅×C₈

—

C₄×C₅×C₈

C₂×D₅

yes

—

—

—

—

—

—

—

C₂²×C₅²

yes

C₄×C₅²

—

C₄×C₅²

C₄⋊C₈

yes

characteristic

C₄×C₅²

#106

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄×D₅

yes

—

C₂×C₅×C₈

C₂²⋊D₅

yes

—

C₄²×C₅

—

C₄²×C₅

Dic₁₀

yes

—

C₅×C₁₆

—

—

—

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

#334

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#683

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₈×D₅

yes

—

C₅×C₈

#676

yes

—

C₅×C₈

#687

yes

—

C₅×C₈

#688

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

D₅²

yes

characteristic

C₅²

#1369

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

#3954

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#3940

yes

—

C₄×C₅

#3946

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#64700

yes

—

C₂×C₄

—

C₂×C₄

yes

C₈

yes

C₈

yes

C₈

yes

—

C₅

—

C₅

yes

C₂²

yes

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#13099268

yes

characteristic

	subgroup	G/H
center	C₄	#11716615
commutator	C₅²×C₈	C₂×C₈
Frattini, hypercenter	C₄×C₈	D₅²
Fitting, cosocle	C₄×C₅²×C₈	C₂²
socle	C₂×C₅²	#334
prime radical	C₂²×C₅²	#108
hypocenter	C₅²	#1369
LCS	G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₄

#11716615

commutator

C₅²×C₈

C₂×C₈

Frattini, hypercenter

C₄×C₈

D₅²

Fitting, cosocle

C₄×C₅²×C₈

C₂²

C₂×C₅²

C₂²×C₅²

C₅²

G ▷ C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
67. C₅²	#1369	semidirect
92. C₅	#10598058	semidirect

H ≅ G/N

type

67. C₅²

#1369

semidirect

92. C₅

#10598058

semidirect

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

subgroup	index
#13099493	2
C₅×#64683	2
#10598058	5

subgroup

index

#13099493

C₅×#64683

#10598058

Subgroups — #72269247

Conjugacy classes of subgroups

Complements

Maximal subgroups