Group #44552416 — subgroups

Subgroups — #44552416

Order 32128.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×Dic₂₀₀₈	C₂	1	2	yes	maximal
2	#30632067	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₁₀₀₄	C₂	1	1	yes	maximal, characteristic
4	#30631492	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₀₄	—	2	1	no	—
7	C₂×Dic₁₀₀₄	C₂²	1	2	yes	—
8	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
9	#22548715	—	2	1	no	—
10	C₄×Dic₅₀₂	—	2	1	no	—
11	Dic₂₀₀₈	—	2	4	no	—
12	#22548649	C₂²	1	1	yes	characteristic
13	#22548640	—	2	1	no	—
14	C₂×C₁₆×C₂₅₁	C₂²	1	2	yes	—
15	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₅₀₂	—	2	1	no	—
17	C₂×Dic₅₀₂	—	2	2	no	—
18	Dic₁₀₀₄	—	2	2	no	—
19	Dic₁₀₀₄	—	2	1	no	—
20	Dic₁₀₀₄	—	2	2	no	—
21	Dic₁₀₀₄	—	4	2	no	—
22	Dic₁₀₀₄	D₄	1	2	yes	—
23	C₄⋊Dic₂₅₁	—	2	1	no	—
24	C₄⋊Dic₂₅₁	—	4	1	no	—
25	C₄⋊Dic₂₅₁	—	4	1	no	—
26	Dic₂₅₁⋊C₄	—	4	1	no	—
27	C₄×Dic₂₅₁	—	4	1	no	—
28	C₂×C₈×C₂₅₁	D₄	1	1	yes	characteristic
29	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
30	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
31	C₁₆×C₂₅₁	—	2	2	no	—
32	Dic₅₀₂	—	2	2	no	—
33	Dic₅₀₂	—	4	1	no	—
34	Dic₅₀₂	—	4	2	no	—
35	Dic₅₀₂	—	4	2	no	—
36	C₂×Dic₂₅₁	—	4	1	no	—
37	C₂×Dic₂₅₁	—	4	1	no	—
38	C₂×Dic₂₅₁	—	4	2	no	—
39	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
40	C₂×C₄×C₂₅₁	D₈	1	1	yes	characteristic
41	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
42	C₈×C₂₅₁	—	2	1	no	—
43	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₅₁	#41	1	1	yes	characteristic
45	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
46	C₄×C₂₅₁	—	2	1	no	—
47	C₄×C₂₅₁	Q₃₂	1	2	yes	complemented
48	C₄×C₂₅₁	#121	1	1	yes	characteristic
49	C₄×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₂₅₁	—	4	2	no	—
51	Dic₂₅₁	—	8	2	no	—
52	Dic₂₅₁	—	8	1	no	—
53	C₂×C₂₅₁	#459	1	1	yes	characteristic
54	C₂×C₂₅₁	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₂₅₁	#510	1	1	yes	characteristic
56	C₂₅₁	#2209	1	1	yes	characteristic, complemented, Sylow
57	#2209	—	251	1	no	maximal, Sylow
58	C₂×Q₃₂	—	251	2	no	—
59	#494	—	251	1	no	—
60	C₄×Q₁₆	—	251	1	no	—
61	C₄⋊C₁₆	D₂₅₁	1	1	yes	characteristic
62	#358	—	251	2	no	—
63	C₂×Q₁₆	—	502	1	no	—
64	C₂×Q₁₆	—	251	2	no	—
65	C₂×Q₁₆	—	251	1	no	—
66	C₄⋊Q₈	—	502	1	no	—
67	C₄×Q₈	—	502	1	no	—
68	Q₃₂	—	502	4	no	—
69	C₂×C₁₆	C₂×D₂₅₁	1	2	yes	—
70	#107	—	251	1	no	—
71	#103	—	502	1	no	—
72	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
73	C₂×Q₈	—	502	2	no	—
74	C₂×Q₈	—	502	1	no	—
75	Q₁₆	—	1004	2	no	—
76	Q₁₆	—	502	2	no	—
77	Q₁₆	—	502	2	no	—
78	Q₁₆	—	502	1	no	—
79	Q₁₆	—	251	2	no	—
80	C₂×C₈	D₁₀₀₄	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
82	C₄⋊C₄	—	1004	1	no	—
83	C₄⋊C₄	—	1004	1	no	—
84	C₄⋊C₄	—	1004	1	no	—
85	C₄⋊C₄	—	502	1	no	—
86	C₄²	—	1004	1	no	—
87	C₄²	D₁₀₀₄	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	1004	2	no	—
90	Q₈	—	1004	2	no	—
91	Q₈	—	1004	1	no	—
92	Q₈	—	502	2	no	—
93	C₂×C₄	—	1004	1	no	—
94	C₂×C₄	—	1004	2	no	—
95	C₂×C₄	—	1004	1	no	—
96	C₂×C₄	D₂₀₀₈	1	1	yes	characteristic
97	C₂×C₄	D₂₀₀₈	1	1	yes	characteristic
98	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₂₅₁	1	1	yes	characteristic
101	C₈	#16860811	1	1	yes	characteristic
102	C₂²	C₂×D₂₀₀₈	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2008	1	no	—
105	C₄	—	2008	2	no	—
106	C₄	—	1004	2	no	—
107	C₄	Dic₂₀₀₈	1	2	yes	complemented
108	C₄	#22548690	1	1	yes	characteristic
109	C₄	#22548743	1	1	yes	characteristic
110	C₂	#30631883	1	1	yes	characteristic
111	C₂	C₂×Dic₂₀₀₈	1	1	yes	characteristic
112	C₂	#30632130	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₀₀₈

C₂

yes

maximal

#30632067

C₂

yes

maximal, characteristic

C₄×Dic₁₀₀₄

C₂

yes

maximal, characteristic

#30631492

C₂

yes

maximal

C₄⋊C₁₆×C₂₅₁

C₂

yes

maximal, characteristic

C₂×Dic₁₀₀₄

—

C₂×Dic₁₀₀₄

C₂²

yes

—

C₂×Dic₁₀₀₄

C₂²

yes

characteristic

#22548715

—

C₄×Dic₅₀₂

—

Dic₂₀₀₈

—

#22548649

C₂²

yes

characteristic

#22548640

—

C₂×C₁₆×C₂₅₁

C₂²

yes

—

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

Dic₁₀₀₄

—

Dic₁₀₀₄

—

Dic₁₀₀₄

—

Dic₁₀₀₄

—

Dic₁₀₀₄

D₄

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₂×C₈×C₂₅₁

D₄

yes

characteristic

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₁₆×C₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×C₄×C₂₅₁

D₈

yes

characteristic

C₂×C₄×C₂₅₁

D₈

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

—

C₈×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

#41

yes

characteristic

C₂²×C₂₅₁

C₂×D₈

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

Q₃₂

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#459

yes

characteristic

C₂×C₂₅₁

C₂×Q₃₂

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#2209

—

251

maximal, Sylow

C₂×Q₃₂

—

251

—

#494

—

251

—

C₄×Q₁₆

—

251

—

C₄⋊C₁₆

D₂₅₁

yes

characteristic

#358

—

251

—

C₂×Q₁₆

—

502

—

C₂×Q₁₆

—

251

—

C₂×Q₁₆

—

251

—

C₄⋊Q₈

—

502

—

C₄×Q₈

—

502

—

Q₃₂

—

502

—

C₂×C₁₆

C₂×D₂₅₁

yes

—

#107

—

251

—

#103

—

502

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

Q₁₆

—

1004

—

Q₁₆

—

502

—

Q₁₆

—

502

—

Q₁₆

—

502

—

Q₁₆

—

251

—

C₂×C₈

D₁₀₀₄

yes

characteristic

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

502

—

C₄²

—

1004

—

C₄²

D₁₀₀₄

yes

characteristic

C₁₆

—

Q₈

—

1004

—

Q₈

—

1004

—

Q₈

—

1004

—

Q₈

—

502

—

C₂×C₄

—

1004

—

C₂×C₄

—

1004

—

C₂×C₄

—

1004

—

C₂×C₄

D₂₀₀₈

yes

characteristic

C₂×C₄

D₂₀₀₈

yes

characteristic

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

C₈

—

100

C₈

D₄×D₂₅₁

yes

characteristic

101

C₈

#16860811

yes

characteristic

102

C₂²

C₂×D₂₀₀₈

yes

characteristic

103

C₄

—

104

C₄

—

2008

—

105

C₄

—

2008

—

106

C₄

—

1004

—

107

C₄

Dic₂₀₀₈

yes

complemented

108

C₄

#22548690

yes

characteristic

109

C₄

#22548743

yes

characteristic

110

C₂

#30631883

yes

characteristic

111

C₂

C₂×Dic₂₀₀₈

yes

characteristic

112

C₂

#30632130

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₀₀₈
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	C₄⋊C₁₆×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₂₅₁
hypocenter	C₂₅₁	#2209
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₂₀₀₈

commutator, cosocle

C₂×C₈×C₂₅₁

C₂³

Frattini

C₂×C₈

C₂²×D₂₅₁

Fitting

C₄⋊C₁₆×C₂₅₁

C₂

socle, prime radical

C₂²×C₂₅₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₂₅₁

hypocenter

C₂₅₁

#2209

LCS

G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₂₅₁	Q₃₂	semidirect
56. C₂₅₁	#2209	semidirect
107. C₄	Dic₂₀₀₈	semidirect

H ≅ G/N

type

47. C₄×C₂₅₁

Q₃₂

semidirect

56. C₂₅₁

#2209

semidirect

107. C₄

Dic₂₀₀₈

semidirect

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

subgroup	index
C₂×Dic₂₀₀₈	2
#30632067	2
C₄×Dic₁₀₀₄	2
#30631492	2
C₄⋊C₁₆×C₂₅₁	2
#2209	251

subgroup

index

C₂×Dic₂₀₀₈

#30632067

C₄×Dic₁₀₀₄

#30631492

C₄⋊C₁₆×C₂₅₁

#2209

251

Subgroups — #44552416

Conjugacy classes of subgroups

Complements

Maximal subgroups