Group #44550266 — subgroups

Subgroups — #44550266

Order 32128.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×#22548650	C₂	1	2	yes	maximal
2	M₁₆×Dic₂₅₁	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
4	#30631452	C₂	1	1	yes	maximal, characteristic
5	#341×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16860794	C₂²	1	2	yes	—
7	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
9	#22548650	C₄	1	8	yes	complemented
10	#22548625	—	2	2	no	—
11	C₈×Dic₂₅₁	—	2	2	no	—
12	C₈×Dic₂₅₁	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₅₁	Q₈	1	2	yes	—
16	#16860794	—	2	2	no	—
17	#16860794	C₂×C₄	1	4	yes	—
18	C₂×C₂₅₁⋊C₈	—	2	2	no	—
19	C₂×C₂₅₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₂₅₁⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
22	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
23	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₅₁	—	2	2	no	—
26	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₂₅₁	—	2	2	no	—
28	C₂×C₈×C₂₅₁	—	2	2	no	—
29	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₅₁	—	2	4	no	—
31	C₂×Dic₂₅₁	C₄⋊C₄	1	2	yes	—
32	C₂×Dic₂₅₁	C₄⋊C₄	1	2	yes	—
33	C₂₅₁⋊C₈	—	2	4	no	—
34	C₂₅₁⋊C₈	C₄²	1	4	yes	—
35	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₅₁	#41	1	2	yes	—
40	C₈×C₂₅₁	—	2	2	no	—
41	C₈×C₂₅₁	—	2	4	no	—
42	C₂²×C₂₅₁	—	2	2	no	—
43	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₂₅₁	C₄×Q₈	1	2	yes	—
45	C₄×C₂₅₁	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₂₅₁	#117	1	1	yes	characteristic
47	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
48	Dic₂₅₁	—	2	2	no	—
49	Dic₂₅₁	#108	1	4	yes	complemented
50	C₂×C₂₅₁	—	2	2	no	—
51	C₂×C₂₅₁	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₂₅₁	C₂×#108	1	2	yes	—
53	C₂₅₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
54	C₄×#108	—	251	1	no	maximal, Sylow
55	C₂×#108	—	251	2	no	—
56	C₄×M₁₆	—	251	2	no	—
57	C₂×C₄×C₈	—	251	1	no	—
58	#341	—	251	1	no	—
59	#341	D₂₅₁	1	1	yes	characteristic
60	C₂×M₁₆	—	251	2	no	—
61	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
62	C₂²×C₈	—	251	1	no	—
63	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
64	C₂×C₄²	—	251	1	no	—
65	#108	—	251	8	no	—
66	#97	—	502	2	no	—
67	C₄×C₈	—	502	2	no	—
68	C₄×C₈	—	251	4	no	—
69	C₂²×C₄	—	251	2	no	—
70	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	502	2	no	—
73	M₁₆	—	251	4	no	—
74	M₁₆	C₄×D₂₅₁	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	502	2	no	—
78	C₂×C₈	—	251	2	no	—
79	C₂×C₈	—	251	4	no	—
80	C₂×C₈	C₄×D₂₅₁	1	2	yes	—
81	C₄²	—	251	2	no	—
82	C₄²	—	251	1	no	—
83	C₄²	—	251	1	no	—
84	C₂³	Q₈×D₂₅₁	1	1	yes	characteristic
85	C₂×C₄	—	502	4	no	—
86	C₂×C₄	—	251	2	no	—
87	C₂×C₄	—	251	2	no	—
88	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
90	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
91	C₂×C₄	#16860809	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	502	4	no	—
95	C₈	—	251	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
98	C₂²	#22548680	1	2	yes	—
99	C₄	—	502	2	no	—
100	C₄	—	251	4	no	—
101	C₄	C₄²×D₂₅₁	1	1	yes	characteristic
102	C₄	#22548671	1	1	yes	characteristic
103	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#30631538	1	1	yes	characteristic
106	C₂	#108×D₂₅₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548650

C₂

yes

maximal

M₁₆×Dic₂₅₁

C₂

yes

maximal

C₂×C₈×Dic₂₅₁

C₂

yes

maximal, characteristic

#30631452

C₂

yes

maximal, characteristic

#341×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

#22548650

C₄

yes

complemented

#22548625

—

C₈×Dic₂₅₁

—

C₈×Dic₂₅₁

C₄

yes

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

C₂²×Dic₂₅₁

Q₈

yes

—

#16860794

—

#16860794

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₅₁

C₂×C₄

yes

—

C₄×Dic₂₅₁

D₄

yes

characteristic

C₄×Dic₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₄⋊C₄

yes

—

C₂×Dic₂₅₁

C₄⋊C₄

yes

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

C₄²

yes

—

C₂³×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₄×Q₈

yes

—

C₄×C₂₅₁

C₂×C₄²

yes

characteristic

C₄×C₂₅₁

#117

yes

characteristic

C₄×C₂₅₁

C₄×D₄

yes

—

Dic₂₅₁

—

Dic₂₅₁

#108

yes

complemented

C₂×C₂₅₁

—

C₂×C₂₅₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₅₁

C₂×#108

yes

—

C₂₅₁

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

251

maximal, Sylow

C₂×#108

—

251

—

C₄×M₁₆

—

251

—

C₂×C₄×C₈

—

251

—

#341

—

251

—

#341

D₂₅₁

yes

characteristic

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₂²×C₈

—

251

—

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

C₂×C₄²

—

251

—

#108

—

251

—

#97

—

502

—

C₄×C₈

—

502

—

C₄×C₈

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

M₁₆

—

502

—

M₁₆

—

251

—

M₁₆

C₄×D₂₅₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

502

—

C₂×C₈

—

251

—

C₂×C₈

—

251

—

C₂×C₈

C₄×D₂₅₁

yes

—

C₄²

—

251

—

C₄²

—

251

—

C₄²

—

251

—

C₂³

Q₈×D₂₅₁

yes

characteristic

C₂×C₄

—

502

—

C₂×C₄

—

251

—

C₂×C₄

—

251

—

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

C₂×C₄

C₂×C₄×D₂₅₁

yes

—

C₂×C₄

D₄×D₂₅₁

yes

characteristic

C₂×C₄

#16860809

yes

—

C₈

—

C₈

—

C₈

—

502

—

C₈

—

251

—

C₂²

—

C₂²

C₄⋊C₄×D₂₅₁

yes

characteristic

C₂²

#22548680

yes

—

C₄

—

502

—

100

C₄

—

251

—

101

C₄

C₄²×D₂₅₁

yes

characteristic

102

C₄

#22548671

yes

characteristic

103

C₄

(C₂²⋊D₂₅₁)⋊C₄

yes

—

104

C₂

—

105

C₂

#30631538

yes

characteristic

106

C₂

#108×D₂₅₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#341×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#341	D₂₅₁
hypocenter	C₂₅₁	C₄×#108
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₂₅₁

commutator

C₄×C₂₅₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#341×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₄⋊C₄

prime radical

C₂³×C₂₅₁

C₂×Q₈

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #22548650	C₄	semidirect
49. Dic₂₅₁	#108	semidirect
53. C₂₅₁	C₄×#108	semidirect

H ≅ G/N

type

9. #22548650

C₄

semidirect

49. Dic₂₅₁

#108

semidirect

53. C₂₅₁

C₄×#108

semidirect

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

subgroup	index
C₂×#22548650	2
M₁₆×Dic₂₅₁	2
C₂×C₈×Dic₂₅₁	2
#30631452	2
#341×C₂₅₁	2
C₄×#108	251

subgroup

index

C₂×#22548650

M₁₆×Dic₂₅₁

C₂×C₈×Dic₂₅₁

#30631452

#341×C₂₅₁

C₄×#108

251

Subgroups — #44550266

Conjugacy classes of subgroups

Complements

Maximal subgroups