Group #44552207 — subgroups

Subgroups — #44552207

Order 32128.

144 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 79 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	#30632104	C₂	1	2	yes	maximal, complemented
2	#30631508	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631471	C₂	1	2	yes	maximal, complemented
4	#30631463	C₂	1	1	yes	maximal, characteristic, complemented
5	#443×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548655	—	2	2	no	—
7	#22548655	C₂²	1	2	yes	—
8	C₂×C₂₅₁⋊C₁₆	—	2	2	no	—
9	C₂×C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#135×C₂₅₁	C₄	1	1	yes	characteristic
12	#131×C₂₅₁	C₂²	1	2	yes	—
13	#108×C₂₅₁	C₄	1	1	yes	characteristic
14	#104×C₂₅₁	C₄	1	2	yes	—
15	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂₅₁⋊C₁₆	—	2	2	no	—
17	C₂₅₁⋊C₁₆	—	2	1	no	—
18	C₂₅₁⋊C₁₆	D₄	1	2	yes	—
19	#41×C₂₅₁	C₂×C₄	1	2	yes	—
20	Q₁₆×C₂₅₁	C₈	1	1	yes	characteristic
21	SD₁₆×C₂₅₁	C₈	1	2	yes	—
22	D₈×C₂₅₁	C₈	1	1	yes	characteristic
23	M₁₆×C₂₅₁	—	2	2	no	—
24	M₁₆×C₂₅₁	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₅₁	—	2	2	no	—
26	C₂×C₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
28	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
29	Q₈×C₂₅₁	C₂×C₈	1	2	yes	—
30	D₄×C₂₅₁	—	2	2	no	—
31	D₄×C₂₅₁	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₂₅₁	—	2	2	no	—
33	C₂×C₄×C₂₅₁	—	2	1	no	—
34	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₂₅₁	—	2	2	no	—
36	C₈×C₂₅₁	C₂×C₈	1	1	yes	characteristic
37	C₈×C₂₅₁	C₂×C₈	1	1	yes	characteristic
38	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
39	C₈×C₂₅₁	#41	1	1	yes	characteristic
40	C₂²×C₂₅₁	—	2	2	no	—
41	C₂²×C₂₅₁	#131	1	1	yes	characteristic
42	C₄×C₂₅₁	—	2	2	no	—
43	C₄×C₂₅₁	—	4	1	no	—
44	C₄×C₂₅₁	C₄×D₄	1	1	yes	characteristic
45	C₄×C₂₅₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₂₅₁	—	2	1	no	—
47	C₂×C₂₅₁	—	4	2	no	—
48	C₂×C₂₅₁	C₈×D₄	1	1	yes	characteristic
49	C₂₅₁	#2171	1	1	yes	characteristic, complemented, Sylow
50	#2171	—	251	1	no	maximal, Sylow
51	#504	—	251	2	no	—
52	#443	D₂₅₁	1	1	yes	characteristic
53	#364	—	251	1	no	—
54	#350	—	251	2	no	—
55	#346	—	251	1	no	—
56	#135	Dic₂₅₁	1	1	yes	characteristic
57	#131	C₂×D₂₅₁	1	2	yes	—
58	M₃₂	—	502	2	no	—
59	M₃₂	—	251	2	no	—
60	C₂×C₁₆	—	502	2	no	—
61	C₂×C₁₆	—	251	1	no	—
62	C₂×C₁₆	—	251	1	no	—
63	#108	Dic₂₅₁	1	1	yes	characteristic
64	#104	Dic₂₅₁	1	2	yes	—
65	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
66	#41	C₂×Dic₂₅₁	1	2	yes	—
67	Q₁₆	C₂₅₁⋊C₈	1	1	yes	characteristic
68	SD₁₆	C₂₅₁⋊C₈	1	2	yes	—
69	D₈	C₂₅₁⋊C₈	1	1	yes	characteristic
70	M₁₆	—	2	2	no	—
71	M₁₆	C₂×Dic₂₅₁	1	2	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×Dic₂₅₁	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
75	C₄²	C₂×Dic₂₅₁	1	1	yes	characteristic
76	C₁₆	—	502	2	no	—
77	C₁₆	—	502	1	no	—
78	C₁₆	—	251	2	no	—
79	Q₈	C₂×C₂₅₁⋊C₈	1	2	yes	—
80	D₄	—	2	2	no	—
81	D₄	C₂×C₂₅₁⋊C₈	1	2	yes	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	C₂×C₂₅₁⋊C₈	1	1	yes	characteristic
87	C₈	C₂×C₂₅₁⋊C₈	1	1	yes	characteristic
88	C₈	D₄×D₂₅₁	1	1	yes	characteristic
89	C₈	#16860809	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#22548722	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	4	1	no	—
94	C₄	D₄×Dic₂₅₁	1	1	yes	characteristic
95	C₄	C₂²×C₂₅₁⋊C₈	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	4	2	no	—
98	C₂	D₄×C₂₅₁⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632104

C₂

yes

maximal, complemented

#30631508

C₂

yes

maximal, characteristic, complemented

#30631471

C₂

yes

maximal, complemented

#30631463

C₂

yes

maximal, characteristic, complemented

#443×C₂₅₁

C₂

yes

maximal, characteristic

#22548655

—

#22548655

C₂²

yes

—

C₂×C₂₅₁⋊C₁₆

—

C₂×C₂₅₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₂₅₁⋊C₁₆

C₂²

yes

characteristic

#135×C₂₅₁

C₄

yes

characteristic

#131×C₂₅₁

C₂²

yes

—

#108×C₂₅₁

C₄

yes

characteristic

#104×C₂₅₁

C₄

yes

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂₅₁⋊C₁₆

—

C₂₅₁⋊C₁₆

—

C₂₅₁⋊C₁₆

D₄

yes

—

#41×C₂₅₁

C₂×C₄

yes

—

Q₁₆×C₂₅₁

C₈

yes

characteristic

SD₁₆×C₂₅₁

C₈

yes

—

D₈×C₂₅₁

C₈

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

Q₈×C₂₅₁

C₂×C₈

yes

—

D₄×C₂₅₁

—

D₄×C₂₅₁

C₂×C₈

yes

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₈×C₂₅₁

—

C₈×C₂₅₁

C₂×C₈

yes

characteristic

C₈×C₂₅₁

C₂×C₈

yes

characteristic

C₈×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

#41

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#131

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

characteristic

C₄×C₂₅₁

C₂²×C₈

yes

characteristic

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₈×D₄

yes

characteristic

C₂₅₁

#2171

yes

characteristic, complemented, Sylow

#2171

—

251

maximal, Sylow

#504

—

251

—

#443

D₂₅₁

yes

characteristic

#364

—

251

—

#350

—

251

—

#346

—

251

—

#135

Dic₂₅₁

yes

characteristic

#131

C₂×D₂₅₁

yes

—

M₃₂

—

502

—

M₃₂

—

251

—

C₂×C₁₆

—

502

—

C₂×C₁₆

—

251

—

C₂×C₁₆

—

251

—

#108

Dic₂₅₁

yes

characteristic

#104

Dic₂₅₁

yes

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

#41

C₂×Dic₂₅₁

yes

—

Q₁₆

C₂₅₁⋊C₈

yes

characteristic

SD₁₆

C₂₅₁⋊C₈

yes

—

D₈

C₂₅₁⋊C₈

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₂₅₁

yes

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₅₁

yes

characteristic

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

C₄²

C₂×Dic₂₅₁

yes

characteristic

C₁₆

—

502

—

C₁₆

—

502

—

C₁₆

—

251

—

Q₈

C₂×C₂₅₁⋊C₈

yes

—

D₄

—

D₄

C₂×C₂₅₁⋊C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

C₈

—

C₈

C₂×C₂₅₁⋊C₈

yes

characteristic

C₈

C₂×C₂₅₁⋊C₈

yes

characteristic

C₈

D₄×D₂₅₁

yes

characteristic

C₈

#16860809

yes

characteristic

C₂²

—

C₂²

#22548722

yes

characteristic

C₄

—

C₄

—

C₄

D₄×Dic₂₅₁

yes

characteristic

C₄

C₂²×C₂₅₁⋊C₈

yes

characteristic

C₂

—

C₂

—

C₂

D₄×C₂₅₁⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#443×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₈×D₄
prime radical	#135×C₂₅₁	C₄
cosocle	C₂×C₈×C₂₅₁	C₂³
hypercenter	#443	D₂₅₁
hypocenter	C₂₅₁	#2171
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×D₂₅₁

commutator

C₄×C₂₅₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₂₅₁

Fitting

#443×C₂₅₁

C₂

socle

C₂×C₂₅₁

C₈×D₄

prime radical

#135×C₂₅₁

C₄

cosocle

C₂×C₈×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #30632104	C₂	semidirect
2. #30631508	C₂	semidirect
3. #30631471	C₂	semidirect
4. #30631463	C₂	semidirect
49. C₂₅₁	#2171	semidirect

H ≅ G/N

type

1. #30632104

C₂

semidirect

2. #30631508

C₂

semidirect

3. #30631471

C₂

semidirect

4. #30631463

C₂

semidirect

49. C₂₅₁

#2171

semidirect

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

subgroup	index
#30632104	2
#30631508	2
#30631471	2
#30631463	2
#443×C₂₅₁	2
#2171	251

subgroup

index

#443×C₂₅₁

251

Subgroups — #44552207

Conjugacy classes of subgroups

Complements

Maximal subgroups