Group #44549693 — subgroups

Subgroups — #44549693

Order 32128.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#30632097	C₂	1	2	yes	maximal
2	#30632057	C₂	1	1	yes	maximal, characteristic
3	#30631429	C₂	1	1	yes	maximal, characteristic
4	#30631429	C₂	1	2	yes	maximal
5	#564×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548715	C₂²	1	1	yes	characteristic
7	#22548648	—	2	2	no	—
8	#22548646	—	2	2	no	—
9	#22548638	—	2	2	no	—
10	#22548626	C₂²	1	1	yes	characteristic
11	#22548626	C₂²	1	2	yes	—
12	#125×C₂₅₁	—	2	2	no	—
13	#125×C₂₅₁	—	2	2	no	—
14	#125×C₂₅₁	—	2	2	no	—
15	#125×C₂₅₁	C₂²	1	1	yes	characteristic
16	#125×C₂₅₁	C₂²	1	2	yes	—
17	C₂×Dic₅₀₂	—	2	1	no	—
18	C₂×C₂₅₁⋊C₈	—	2	2	no	—
19	C₂×C₂₅₁⋊C₈	—	2	1	no	—
20	C₄⋊Dic₂₅₁	—	2	2	no	—
21	C₄⋊C₄×C₂₅₁	—	2	2	no	—
22	C₄⋊C₄×C₂₅₁	—	2	2	no	—
23	C₄⋊C₄×C₂₅₁	—	2	2	no	—
24	C₄⋊C₄×C₂₅₁	—	2	2	no	—
25	C₄⋊C₄×C₂₅₁	—	2	2	no	—
26	C₄⋊C₄×C₂₅₁	—	2	2	no	—
27	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
28	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₅₁	D₄	1	2	yes	—
30	C₄²×C₂₅₁	—	2	2	no	—
31	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
32	Dic₅₀₂	—	4	2	no	—
33	C₂×Dic₂₅₁	—	4	1	no	—
34	C₂₅₁⋊C₈	—	2	4	no	—
35	C₂₅₁⋊C₈	—	4	1	no	—
36	C₂×C₄×C₂₅₁	—	2	2	no	—
37	C₂×C₄×C₂₅₁	—	2	2	no	—
38	C₂×C₄×C₂₅₁	—	2	2	no	—
39	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₂₅₁	#120	1	1	yes	characteristic
42	C₄×C₂₅₁	—	2	1	no	—
43	C₄×C₂₅₁	—	2	2	no	—
44	C₄×C₂₅₁	—	4	2	no	—
45	C₄×C₂₅₁	—	4	2	no	—
46	C₄×C₂₅₁	—	4	2	no	—
47	Dic₂₅₁	—	8	1	no	—
48	C₂×C₂₅₁	#454	1	1	yes	characteristic
49	C₂×C₂₅₁	#456	1	2	yes	—
50	C₂₅₁	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	251	1	no	maximal, Sylow
52	#564	D₂₅₁	1	1	yes	characteristic
53	#501	—	251	2	no	—
54	#491	—	251	1	no	—
55	#330	—	251	2	no	—
56	#330	—	251	1	no	—
57	C₄⋊Q₈	—	251	1	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	—	2	2	no	—
61	#125	C₂×D₂₅₁	1	1	yes	characteristic
62	#125	C₂×D₂₅₁	1	2	yes	—
63	#107	—	502	2	no	—
64	#106	—	502	2	no	—
65	#103	—	502	2	no	—
66	#97	—	251	1	no	—
67	#97	—	251	2	no	—
68	C₂×Q₈	—	502	1	no	—
69	C₂×C₈	—	502	1	no	—
70	C₂×C₈	—	502	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	502	2	no	—
78	C₄⋊C₄	C₂²⋊D₂₅₁	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₂₅₁	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₂₅₁	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
83	Q₈	—	1004	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	1004	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
90	C₈	—	1004	1	no	—
91	C₈	—	502	4	no	—
92	C₂²	#22548684	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	2008	1	no	—
99	C₂	#30631860	1	1	yes	characteristic
100	C₂	#30631870	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632097

C₂

yes

maximal

#30632057

C₂

yes

maximal, characteristic

#30631429

C₂

yes

maximal, characteristic

#30631429

C₂

yes

maximal

#564×C₂₅₁

C₂

yes

maximal, characteristic

#22548715

C₂²

yes

—

—

—

C₂²

yes

characteristic

#22548626

C₂²

yes

—

#125×C₂₅₁

—

#125×C₂₅₁

—

#125×C₂₅₁

—

#125×C₂₅₁

C₂²

yes

characteristic

#125×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₄⋊C₄×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂²×C₂₅₁

#120

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#454

yes

characteristic

C₂×C₂₅₁

#456

yes

—

C₂₅₁

#1663

yes

characteristic, complemented, Sylow

#1663

—

251

maximal, Sylow

#564

D₂₅₁

yes

characteristic

#501

—

251

—

#491

—

251

—

#330

—

251

—

#330

—

251

—

C₄⋊Q₈

—

251

—

—

—

—

C₂×D₂₅₁

yes

characteristic

#125

C₂×D₂₅₁

yes

—

#107

—

502

—

#106

—

502

—

#103

—

502

—

#97

—

251

—

#97

—

251

—

C₂×Q₈

—

502

—

C₂×C₈

—

502

—

C₂×C₈

—

502

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

502

—

C₄⋊C₄

C₂²⋊D₂₅₁

yes

—

C₄⋊C₄

C₂²⋊D₂₅₁

yes

—

C₄⋊C₄

C₂²⋊D₂₅₁

yes

—

C₄²

—

C₄²

C₂²×D₂₅₁

yes

characteristic

Q₈

—

1004

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1004

—

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

C₈

—

1004

—

C₈

—

502

—

C₂²

#22548684

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

2008

—

C₂

#30631860

yes

characteristic

100

C₂

#30631870

yes

—

	subgroup	G/H
center	C₂²	#22548684
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#564×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#120
hypercenter	#564	D₂₅₁
hypocenter	C₂₅₁	#1663
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#22548684

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#564×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
50. C₂₅₁	#1663	semidirect

H ≅ G/N

type

50. C₂₅₁

#1663

semidirect

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

subgroup	index
#30632097	2
#30632057	2
#30631429	2
#564×C₂₅₁	2
#1663	251

subgroup

index

#564×C₂₅₁

251

Subgroups — #44549693

Conjugacy classes of subgroups

Complements

Maximal subgroups