Group #44549508 — subgroups

Subgroups — #44549508

Order 32128.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#30632403	C₂	1	1	yes	maximal, characteristic
2	#30631974	C₂	1	2	yes	maximal, complemented
3	#30631436	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631416	C₂	1	2	yes	maximal, complemented
5	#494×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22548704	—	2	2	no	—
7	#22548690	—	2	2	no	—
8	C₄×D₁₀₀₄	C₂²	1	2	yes	—
9	#22548673	—	2	2	no	—
10	#22548646	—	2	2	no	—
11	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
12	#22548635	—	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×D₁₀₀₄	—	2	2	no	—
18	C₂×D₁₀₀₄	D₄	1	2	yes	—
19	C₂×C₄×D₂₅₁	—	2	2	no	—
20	C₂×C₂₅₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
22	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
23	(C₂×D₂₅₁)⋊C₄	—	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	C₂²×D₂₅₁	—	2	2	no	—
33	D₁₀₀₄	—	2	2	no	—
34	D₁₀₀₄	—	2	2	no	—
35	D₁₀₀₄	—	8	2	no	—
36	C₂×Dic₂₅₁	—	2	2	no	—
37	C₄×D₂₅₁	—	4	2	no	—
38	C₄×D₂₅₁	—	4	2	no	—
39	C₂₅₁⋊C₈	—	4	2	no	—
40	Q₈×C₂₅₁	—	4	2	no	—
41	Q₈×C₂₅₁	—	4	2	no	—
42	C₂×C₄×C₂₅₁	—	2	2	no	—
43	C₂×C₄×C₂₅₁	—	4	1	no	—
44	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
46	C₈×C₂₅₁	—	2	2	no	—
47	C₂²×C₂₅₁	#120	1	1	yes	characteristic
48	C₂×D₂₅₁	—	4	2	no	—
49	C₂×D₂₅₁	—	4	2	no	—
50	C₂×D₂₅₁	—	4	2	no	—
51	C₄×C₂₅₁	—	2	1	no	—
52	C₄×C₂₅₁	—	4	2	no	—
53	C₄×C₂₅₁	—	8	1	no	—
54	C₄×C₂₅₁	#135	1	2	yes	—
55	C₄×C₂₅₁	#137	1	2	yes	—
56	Dic₂₅₁	—	4	2	no	—
57	C₂×C₂₅₁	#452	1	2	yes	—
58	C₂×C₂₅₁	#453	1	1	yes	characteristic
59	D₂₅₁	—	8	2	no	—
60	C₂₅₁	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	251	1	no	maximal, Sylow
62	#555	—	251	1	no	—
63	#494	D₂₅₁	1	1	yes	characteristic, complemented
64	#476	—	251	2	no	—
65	#333	—	251	1	no	—
66	D₄⋊C₈	—	251	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
70	#124	—	502	2	no	—
71	#121	—	502	2	no	—
72	C₄×D₄	—	251	2	no	—
73	#117	—	502	2	no	—
74	#106	—	502	2	no	—
75	C₄⋊C₈	—	251	2	no	—
76	#102	—	502	2	no	—
77	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₂₅₁	1	2	yes	—
80	C₂×D₄	—	502	2	no	—
81	C₂×D₄	—	251	2	no	—
82	C₂²×C₄	—	502	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	502	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	251	2	no	—
89	C₂²⋊C₄	—	502	4	no	—
90	C₂²⋊C₄	—	502	2	no	—
91	C₄²	—	502	2	no	—
92	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
93	C₂³	—	502	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	2008	2	no	—
97	D₄	—	502	2	no	—
98	D₄	—	502	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	1004	2	no	—
102	C₂×C₄	—	1004	2	no	—
103	C₂×C₄	—	502	2	no	—
104	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	1004	2	no	—
108	C₂²	—	1004	2	no	—
109	C₂²	—	1004	2	no	—
110	C₂²	—	1004	2	no	—
111	C₂²	#22548683	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	1004	2	no	—
116	C₄	#22548738	1	2	yes	—
117	C₄	#22548747	1	2	yes	—
118	C₂	—	2008	2	no	—
119	C₂	#30631848	1	2	yes	—
120	C₂	#30631854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632403

C₂

yes

maximal, characteristic

#30631974

C₂

yes

maximal, complemented

#30631436

C₂

yes

maximal, characteristic, complemented

#30631416

C₂

yes

maximal, complemented

#494×C₂₅₁

C₂

yes

maximal, characteristic, complemented

#22548704

—

#22548690

—

C₄×D₁₀₀₄

C₂²

yes

—

#22548673

—

#22548646

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

#22548635

—

C₂×Q₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄×C₈×C₂₅₁

C₂²

yes

characteristic

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

D₄

yes

—

C₂×C₄×D₂₅₁

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

D₄

yes

—

(C₂×D₂₅₁)⋊C₄

—

(C₂×D₂₅₁)⋊C₄

—

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

C₂²×D₂₅₁

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂×Dic₂₅₁

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

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

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#135

yes

—

C₄×C₂₅₁

#137

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

#452

yes

—

C₂×C₂₅₁

#453

yes

characteristic

D₂₅₁

—

C₂₅₁

#1634

yes

characteristic, complemented, Sylow

#1634

—

251

maximal, Sylow

#555

—

251

—

#494

D₂₅₁

yes

characteristic, complemented

#476

—

251

—

#333

—

251

—

D₄⋊C₈

—

251

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

#124

—

502

—

#121

—

502

—

C₄×D₄

—

251

—

#117

—

502

—

#106

—

502

—

C₄⋊C₈

—

251

—

#102

—

502

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₅₁

yes

—

C₂×D₄

—

502

—

C₂×D₄

—

251

—

C₂²×C₄

—

502

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

251

—

C₂²⋊C₄

—

502

—

C₂²⋊C₄

—

502

—

C₄²

—

502

—

C₄²

C₂²×D₂₅₁

yes

characteristic

C₂³

—

502

—

Q₈

—

Q₈

—

D₄

—

2008

—

D₄

—

502

—

D₄

—

502

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

1004

—

102

C₂×C₄

—

1004

—

103

C₂×C₄

—

502

—

104

C₂×C₄

D₄×D₂₅₁

yes

—

105

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

106

C₈

—

107

C₈

—

1004

—

108

C₂²

—

1004

—

109

C₂²

—

1004

—

110

C₂²

—

1004

—

111

C₂²

#22548683

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

1004

—

116

C₄

#22548738

yes

—

117

C₄

#22548747

yes

—

118

C₂

—

2008

—

119

C₂

#30631848

yes

—

120

C₂

#30631854

yes

characteristic

	subgroup	G/H
center	C₂²	#22548683
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#494×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#120
prime radical	#30632403	C₂
hypercenter	#494	D₂₅₁
hypocenter	C₂₅₁	#1634
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

C₂

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #30631974	C₂	semidirect
3. #30631436	C₂	semidirect
4. #30631416	C₂	semidirect
5. #494×C₂₅₁	C₂	semidirect
60. C₂₅₁	#1634	semidirect
63. #494	D₂₅₁	semidirect

H ≅ G/N

type

2. #30631974

C₂

semidirect

3. #30631436

C₂

semidirect

4. #30631416

C₂

semidirect

5. #494×C₂₅₁

C₂

semidirect

60. C₂₅₁

#1634

semidirect

63. #494

D₂₅₁

semidirect

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

subgroup	index
#30632403	2
#30631974	2
#30631436	2
#30631416	2
#494×C₂₅₁	2
#1634	251

subgroup

index

#494×C₂₅₁

251

Subgroups — #44549508

Conjugacy classes of subgroups

Complements

Maximal subgroups