Group #44550031 — subgroups

Subgroups — #44550031

Order 32128.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#30632064	C₂	1	2	yes	maximal, complemented
2	#30632064	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632060	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632034	C₂	1	2	yes	maximal, complemented
5	#322×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16860799	—	2	2	no	—
7	C₂×D₂₀₀₈	—	2	4	no	—
8	C₂×D₂₀₀₈	—	2	2	no	—
9	#22548715	C₂²	1	1	yes	characteristic
10	#22548712	C₂²	1	2	yes	—
11	#22548712	C₂²	1	1	yes	characteristic
12	#22548637	—	2	4	no	—
13	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₀₂	—	2	1	no	—
16	C₂×D₁₀₀₄	—	2	2	no	—
17	C₂×D₁₀₀₄	—	2	2	no	—
18	C₂×D₁₀₀₄	—	2	2	no	—
19	C₂×D₁₀₀₄	—	2	2	no	—
20	C₂×D₁₀₀₄	—	2	1	no	—
21	#16860799	—	4	4	no	—
22	D₂₀₀₈	—	4	8	no	—
23	D₂₀₀₈	—	4	4	no	—
24	C₄⋊Dic₂₅₁	—	2	2	no	—
25	C₂×C₈×C₂₅₁	D₄	1	4	yes	—
26	C₂×C₈×C₂₅₁	D₄	1	2	yes	—
27	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
28	C₂²×D₂₅₁	—	4	2	no	—
29	C₂²×D₂₅₁	—	4	1	no	—
30	Dic₅₀₂	—	4	2	no	—
31	D₁₀₀₄	—	4	4	no	—
32	D₁₀₀₄	—	4	2	no	—
33	D₁₀₀₄	—	4	4	no	—
34	D₁₀₀₄	—	4	4	no	—
35	D₁₀₀₄	—	4	4	no	—
36	C₂×Dic₂₅₁	—	4	1	no	—
37	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
39	C₈×C₂₅₁	—	2	4	no	—
40	C₈×C₂₅₁	D₈	1	4	yes	complemented
41	C₂²×C₂₅₁	#127	1	1	yes	characteristic
42	C₂×D₂₅₁	—	8	2	no	—
43	C₂×D₂₅₁	—	8	1	no	—
44	C₂×D₂₅₁	—	8	2	no	—
45	C₂×D₂₅₁	—	8	2	no	—
46	C₂×D₂₅₁	—	8	2	no	—
47	C₄×C₂₅₁	C₂×D₈	1	2	yes	—
48	C₄×C₂₅₁	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₂₅₁	—	8	1	no	—
50	C₂×C₂₅₁	#493	1	1	yes	characteristic
51	C₂×C₂₅₁	#496	1	2	yes	—
52	D₂₅₁	—	16	2	no	—
53	D₂₅₁	—	16	1	no	—
54	C₂₅₁	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	251	1	no	maximal, Sylow
56	#493	—	251	2	no	—
57	#493	—	251	1	no	—
58	#492	—	251	1	no	—
59	#486	—	251	2	no	—
60	#322	D₂₅₁	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	502	2	no	—
62	C₂×D₈	—	502	4	no	—
63	C₂×D₈	—	502	2	no	—
64	C₄⋊Q₈	—	251	1	no	—
65	#127	—	251	2	no	—
66	#127	—	251	1	no	—
67	#102	—	502	4	no	—
68	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
69	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
70	C₂×Q₈	—	502	1	no	—
71	C₂×D₄	—	502	2	no	—
72	C₂×D₄	—	502	2	no	—
73	C₂×D₄	—	502	2	no	—
74	C₂×D₄	—	502	2	no	—
75	C₂×D₄	—	502	1	no	—
76	SD₁₆	—	1004	4	no	—
77	D₈	—	1004	8	no	—
78	D₈	—	1004	4	no	—
79	C₂×C₈	D₁₀₀₄	1	4	yes	—
80	C₂×C₈	D₁₀₀₄	1	2	yes	—
81	C₄⋊C₄	—	502	2	no	—
82	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
83	C₂³	—	1004	2	no	—
84	C₂³	—	1004	1	no	—
85	Q₈	—	1004	2	no	—
86	D₄	—	1004	4	no	—
87	D₄	—	1004	4	no	—
88	D₄	—	1004	4	no	—
89	D₄	—	1004	4	no	—
90	D₄	—	1004	2	no	—
91	C₂×C₄	—	1004	1	no	—
92	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
93	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
94	C₈	—	2	4	no	—
95	C₈	D₂₀₀₈	1	4	yes	complemented
96	C₂²	—	2008	2	no	—
97	C₂²	—	2008	1	no	—
98	C₂²	—	2008	2	no	—
99	C₂²	—	2008	2	no	—
100	C₂²	—	2008	2	no	—
101	C₂²	#22548712	1	1	yes	characteristic
102	C₄	—	2008	1	no	—
103	C₄	C₂×D₂₀₀₈	1	2	yes	—
104	C₄	#22548743	1	4	yes	—
105	C₂	—	4016	2	no	—
106	C₂	—	4016	1	no	—
107	C₂	#30632064	1	1	yes	characteristic
108	C₂	#30632076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632064

C₂

yes

maximal, complemented

#30632064

C₂

yes

maximal, characteristic, complemented

#30632060

C₂

yes

maximal, characteristic, complemented

#30632034

C₂

yes

maximal, complemented

#322×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860799

—

C₂×D₂₀₀₈

—

C₂×D₂₀₀₈

—

#22548715

C₂²

yes

characteristic

#22548712

C₂²

yes

—

#22548712

C₂²

yes

characteristic

#22548637

—

C₄×C₈×C₂₅₁

C₂²

yes

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

#16860799

—

D₂₀₀₈

—

D₂₀₀₈

—

C₄⋊Dic₂₅₁

—

C₂×C₈×C₂₅₁

D₄

yes

—

C₂×C₈×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂×Dic₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₈×C₂₅₁

—

C₈×C₂₅₁

D₈

yes

complemented

C₂²×C₂₅₁

#127

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

C₂×D₈

yes

—

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

#493

yes

characteristic

C₂×C₂₅₁

#496

yes

—

D₂₅₁

—

D₂₅₁

—

C₂₅₁

#1721

yes

characteristic, complemented, Sylow

#1721

—

251

maximal, Sylow

#493

—

251

—

#493

—

251

—

#492

—

251

—

#486

—

251

—

#322

D₂₅₁

yes

characteristic, complemented

C₂×SD₁₆

—

502

—

C₂×D₈

—

502

—

C₂×D₈

—

502

—

C₄⋊Q₈

—

251

—

#127

—

251

—

#127

—

251

—

#102

—

502

—

C₄×C₈

C₂×D₂₅₁

yes

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

SD₁₆

—

1004

—

D₈

—

1004

—

D₈

—

1004

—

C₂×C₈

D₁₀₀₄

yes

—

C₂×C₈

D₁₀₀₄

yes

—

C₄⋊C₄

—

502

—

C₄²

C₂²×D₂₅₁

yes

characteristic

C₂³

—

1004

—

C₂³

—

1004

—

Q₈

—

1004

—

D₄

—

1004

—

D₄

—

1004

—

D₄

—

1004

—

D₄

—

1004

—

D₄

—

1004

—

C₂×C₄

—

1004

—

C₂×C₄

C₂×D₁₀₀₄

yes

—

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

C₈

—

C₈

D₂₀₀₈

yes

complemented

C₂²

—

2008

—

C₂²

—

2008

—

C₂²

—

2008

—

C₂²

—

2008

—

100

C₂²

—

2008

—

101

C₂²

#22548712

yes

characteristic

102

C₄

—

2008

—

103

C₄

C₂×D₂₀₀₈

yes

—

104

C₄

#22548743

yes

—

105

C₂

—

4016

—

106

C₂

—

4016

—

107

C₂

#30632064

yes

characteristic

108

C₂

#30632076

yes

—

	subgroup	G/H
center	C₂²	#22548712
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#322×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#127
hypercenter	#322	D₂₅₁
hypocenter	C₂₅₁	#1721
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#22548712

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#322×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #30632064	C₂	semidirect
2. #30632064	C₂	semidirect
3. #30632060	C₂	semidirect
4. #30632034	C₂	semidirect
5. #322×C₂₅₁	C₂	semidirect
40. C₈×C₂₅₁	D₈	semidirect
54. C₂₅₁	#1721	semidirect
60. #322	D₂₅₁	semidirect
95. C₈	D₂₀₀₈	semidirect

H ≅ G/N

type

1. #30632064

C₂

semidirect

2. #30632064

C₂

semidirect

3. #30632060

C₂

semidirect

4. #30632034

C₂

semidirect

5. #322×C₂₅₁

C₂

semidirect

40. C₈×C₂₅₁

D₈

semidirect

54. C₂₅₁

#1721

semidirect

60. #322

D₂₅₁

semidirect

95. C₈

D₂₀₀₈

semidirect

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

subgroup	index
#30632064	2
#30632060	2
#30632034	2
#322×C₂₅₁	2
#1721	251

subgroup

index

#322×C₂₅₁

251

Subgroups — #44550031

Conjugacy classes of subgroups

Complements

Maximal subgroups