Group #44549503 — subgroups

Subgroups — #44549503

Order 32128.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#30632417	C₂	1	1	yes	maximal, characteristic
2	#30631979	C₂	1	2	yes	maximal
3	#30631436	C₂	1	1	yes	maximal, characteristic
4	#30631419	C₂	1	2	yes	maximal
5	#494×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	#22548714	—	2	2	no	—
7	#22548713	—	2	2	no	—
8	C₄×Dic₅₀₂	C₂²	1	2	yes	—
9	#22548680	—	2	2	no	—
10	#22548648	—	2	2	no	—
11	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
12	#22548638	—	2	2	no	—
13	C₂×Q₁₆×C₂₅₁	—	2	2	no	—
14	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₀₂	—	2	2	no	—
18	C₂×Dic₅₀₂	D₄	1	2	yes	—
19	C₂×C₂₅₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
21	Dic₂₅₁⋊C₄	—	2	4	no	—
22	Dic₂₅₁⋊C₄	—	2	2	no	—
23	C₄×Dic₂₅₁	—	2	2	no	—
24	C₄×Dic₂₅₁	—	2	2	no	—
25	C₂×Q₈×C₂₅₁	—	2	1	no	—
26	C₂×Q₈×C₂₅₁	D₄	1	2	yes	—
27	Q₁₆×C₂₅₁	—	4	4	no	—
28	C₂×C₈×C₂₅₁	—	2	1	no	—
29	C₄⋊C₄×C₂₅₁	—	2	2	no	—
30	C₄⋊C₄×C₂₅₁	—	2	2	no	—
31	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
32	Dic₅₀₂	—	2	2	no	—
33	Dic₅₀₂	—	8	2	no	—
34	Dic₅₀₂	Q₁₆	1	4	yes	complemented
35	C₂×Dic₂₅₁	—	2	2	no	—
36	C₂×Dic₂₅₁	—	2	2	no	—
37	C₂₅₁⋊C₈	—	4	2	no	—
38	Q₈×C₂₅₁	—	4	2	no	—
39	Q₈×C₂₅₁	—	4	2	no	—
40	C₂×C₄×C₂₅₁	—	2	2	no	—
41	C₂×C₄×C₂₅₁	—	4	1	no	—
42	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₅₁	—	2	2	no	—
45	C₂²×C₂₅₁	#120	1	1	yes	characteristic
46	C₄×C₂₅₁	—	2	1	no	—
47	C₄×C₂₅₁	—	4	2	no	—
48	C₄×C₂₅₁	—	8	1	no	—
49	C₄×C₂₅₁	C₂×Q₁₆	1	2	yes	—
50	C₄×C₂₅₁	#137	1	2	yes	—
51	Dic₂₅₁	—	2	4	no	—
52	Dic₂₅₁	—	4	2	no	—
53	C₂×C₂₅₁	#451	1	2	yes	—
54	C₂×C₂₅₁	#453	1	1	yes	characteristic
55	C₂₅₁	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	251	1	no	maximal, Sylow
57	Q₈²	—	251	1	no	—
58	#494	D₂₅₁	1	1	yes	characteristic
59	#477	—	251	2	no	—
60	#333	—	251	1	no	—
61	Q₈⋊C₈	—	251	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	502	2	no	—
64	C₄⋊Q₈	—	502	2	no	—
65	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
67	C₄×Q₈	—	502	2	no	—
68	C₄×Q₈	—	251	2	no	—
69	#107	—	502	2	no	—
70	C₄⋊C₈	—	251	2	no	—
71	#103	—	502	2	no	—
72	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	502	2	no	—
75	C₂×Q₈	—	251	2	no	—
76	C₂×Q₈	C₂²⋊D₂₅₁	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	502	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	502	4	no	—
83	C₄⋊C₄	—	502	2	no	—
84	C₄⋊C₄	—	251	2	no	—
85	C₄²	—	502	2	no	—
86	C₄²	—	502	2	no	—
87	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	2008	2	no	—
91	Q₈	—	502	2	no	—
92	Q₈	—	251	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	502	2	no	—
96	C₂×C₄	—	502	2	no	—
97	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	1004	2	no	—
101	C₂²	#22548683	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	1004	2	no	—
106	C₄	—	502	4	no	—
107	C₄	Q₁₆×D₂₅₁	1	2	yes	—
108	C₄	#22548747	1	2	yes	—
109	C₂	#30631841	1	2	yes	—
110	C₂	#30631854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632417

C₂

yes

maximal, characteristic

#30631979

C₂

yes

maximal

#30631436

C₂

yes

maximal, characteristic

#30631419

C₂

yes

maximal

#494×C₂₅₁

C₂

yes

maximal, characteristic

#22548714

—

#22548713

—

C₄×Dic₅₀₂

C₂²

yes

—

#22548680

—

#22548648

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

#22548638

—

C₂×Q₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

D₄

yes

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

D₄

yes

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

D₄

yes

—

Q₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

Q₁₆

yes

complemented

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

#120

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×Q₁₆

yes

—

C₄×C₂₅₁

#137

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#451

yes

—

C₂×C₂₅₁

#453

yes

characteristic

C₂₅₁

#1633

yes

characteristic, complemented, Sylow

#1633

—

251

maximal, Sylow

Q₈²

—

251

—

#494

D₂₅₁

yes

characteristic

#477

—

251

—

#333

—

251

—

Q₈⋊C₈

—

251

—

C₂×Q₁₆

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄×Q₈

—

502

—

C₄×Q₈

—

251

—

#107

—

502

—

C₄⋊C₈

—

251

—

#103

—

502

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

C₂×Q₈

C₂²⋊D₂₅₁

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

251

—

C₄²

—

502

—

C₄²

—

502

—

C₄²

C₂²×D₂₅₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

2008

—

Q₈

—

502

—

Q₈

—

251

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

D₄×D₂₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

C₈

—

100

C₈

—

1004

—

101

C₂²

#22548683

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

1004

—

106

C₄

—

502

—

107

C₄

Q₁₆×D₂₅₁

yes

—

108

C₄

#22548747

yes

—

109

C₂

#30631841

yes

—

110

C₂

#30631854

yes

characteristic

	subgroup	G/H
center	C₂²	#22548683
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#494×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#120
hypercenter	#494	D₂₅₁
hypocenter	C₂₅₁	#1633
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#22548683

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#494×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₅₀₂	Q₁₆	semidirect
55. C₂₅₁	#1633	semidirect

H ≅ G/N

type

34. Dic₅₀₂

Q₁₆

semidirect

55. C₂₅₁

#1633

semidirect

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

subgroup	index
#30632417	2
#30631979	2
#30631436	2
#30631419	2
#494×C₂₅₁	2
#1633	251

subgroup

index

#494×C₂₅₁

251

Subgroups — #44549503

Conjugacy classes of subgroups

Complements

Maximal subgroups