Group #44562169 — subgroups

Subgroups — #44562169

Order 32128.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#30632500	C₂	1	4	yes	maximal, complemented
2	#30631959	C₂	1	4	yes	maximal, complemented
3	#30631869	C₂	1	4	yes	maximal, complemented
4	#30631805	C₂	1	2	yes	maximal, complemented
5	#494×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22548763	C₂²	1	4	yes	—
7	#22548747	—	2	8	no	—
8	#22548746	C₂²	1	8	yes	—
9	#22548738	—	2	4	no	—
10	#22548737	C₂²	1	2	yes	—
11	Q₁₆×D₂₅₁	—	2	4	no	—
12	#22548721	C₂²	1	4	yes	—
13	#22548651	C₂²	1	2	yes	—
14	#22548643	C₂²	1	4	yes	—
15	#22548634	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₂₅₁	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₂₅₁	C₂²	1	2	yes	—
18	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
19	#16860811	—	2	8	no	—
20	#16860810	—	4	4	no	—
21	#16860810	C₂³	1	4	yes	—
22	Q₈×D₂₅₁	—	2	8	no	—
23	Dic₁₀₀₄	D₄	1	2	yes	—
24	#16860800	—	2	8	no	—
25	#16860799	D₄	1	4	yes	—
26	#16860797	—	2	8	no	—
27	D₂₀₀₈	D₄	1	2	yes	—
28	#16860794	C₂³	1	4	yes	—
29	#16860793	—	2	4	no	—
30	C₈×D₂₅₁	—	2	4	no	—
31	C₂×Q₈×C₂₅₁	C₂³	1	4	yes	—
32	Q₁₆×C₂₅₁	—	2	4	no	—
33	Q₁₆×C₂₅₁	—	2	4	no	—
34	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
35	C₄⋊C₄×C₂₅₁	—	4	2	no	—
36	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
37	Dic₅₀₂	—	4	4	no	—
38	Dic₅₀₂	C₂×D₄	1	4	yes	—
39	D₁₀₀₄	—	4	4	no	—
40	D₁₀₀₄	C₂×D₄	1	4	yes	—
41	C₂²⋊D₂₅₁	—	2	4	no	—
42	C₄×D₂₅₁	—	2	4	no	—
43	C₄×D₂₅₁	—	4	8	no	—
44	C₂₅₁⋊C₈	—	2	4	no	—
45	Q₈×C₂₅₁	—	2	8	no	—
46	C₂×C₄×C₂₅₁	—	2	1	no	—
47	C₂×C₄×C₂₅₁	—	2	4	no	—
48	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
49	C₈×C₂₅₁	C₂×D₄	1	2	yes	—
50	C₈×C₂₅₁	C₂×D₄	1	2	yes	—
51	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₂₅₁	—	2	4	no	—
53	C₄×C₂₅₁	—	2	2	no	—
54	C₄×C₂₅₁	—	4	4	no	—
55	C₄×C₂₅₁	C₂²×D₄	1	2	yes	—
56	Dic₂₅₁	—	2	4	no	—
57	C₂×C₂₅₁	—	2	1	no	—
58	C₂×C₂₅₁	D₄²	1	1	yes	characteristic
59	D₂₅₁	—	4	4	no	—
60	C₂₅₁	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	251	1	no	maximal, Sylow
62	#578	—	251	4	no	—
63	#494	D₂₅₁	1	1	yes	characteristic, complemented
64	#473	—	251	4	no	—
65	#456	—	251	4	no	—
66	#443	—	251	2	no	—
67	2₋¹⁺⁴	—	251	4	no	—
68	#137	—	502	8	no	—
69	#137	—	251	8	no	—
70	#135	—	502	4	no	—
71	#135	—	251	2	no	—
72	C₂×Q₁₆	—	502	4	no	—
73	C₂×Q₁₆	C₂×D₂₅₁	1	4	yes	—
74	#131	—	251	4	no	—
75	C₄⋊Q₈	C₂×D₂₅₁	1	2	yes	—
76	#108	—	251	2	no	—
77	#104	—	251	4	no	—
78	#101	—	251	4	no	—
79	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
80	#41	—	1004	4	no	—
81	#41	—	502	8	no	—
82	#41	—	251	4	no	—
83	C₂×Q₈	—	502	8	no	—
84	C₂×Q₈	C₂²×D₂₅₁	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	502	8	no	—
88	Q₁₆	—	251	2	no	—
89	SD₁₆	—	502	8	no	—
90	SD₁₆	—	251	4	no	—
91	D₈	—	251	2	no	—
92	M₁₆	—	502	4	no	—
93	M₁₆	—	251	4	no	—
94	C₂×C₈	—	502	4	no	—
95	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	1004	4	no	—
100	Q₈	—	251	4	no	—
101	D₄	—	1004	4	no	—
102	D₄	—	502	4	no	—
103	D₄	—	251	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	1004	8	no	—
107	C₂×C₄	—	502	4	no	—
108	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
109	C₈	—	502	4	no	—
110	C₈	D₄×D₂₅₁	1	2	yes	—
111	C₈	D₄×D₂₅₁	1	2	yes	—
112	C₂²	—	502	4	no	—
113	C₂²	#22548760	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	502	4	no	—
117	C₄	C₂×D₄×D₂₅₁	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	1004	4	no	—
120	C₂	#30632322	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632500

C₂

yes

maximal, complemented

#30631959

C₂

yes

maximal, complemented

#30631869

C₂

yes

maximal, complemented

#30631805

C₂

yes

maximal, complemented

#494×C₂₅₁

C₂

yes

maximal, characteristic, complemented

#22548763

C₂²

yes

—

#22548747

—

#22548746

C₂²

yes

—

#22548738

—

#22548737

C₂²

yes

—

Q₁₆×D₂₅₁

—

#22548721

C₂²

yes

—

#22548651

C₂²

yes

—

#22548643

C₂²

yes

—

#22548634

C₂²

yes

—

C₂×Q₁₆×C₂₅₁

C₂²

yes

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

—

C₄×C₈×C₂₅₁

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₂₅₁

—

Dic₁₀₀₄

D₄

yes

—

#16860800

—

#16860799

D₄

yes

—

#16860797

—

D₂₀₀₈

D₄

yes

—

#16860794

C₂³

yes

—

#16860793

—

C₈×D₂₅₁

—

C₂×Q₈×C₂₅₁

C₂³

yes

—

Q₁₆×C₂₅₁

—

Q₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

Dic₅₀₂

C₂×D₄

yes

—

D₁₀₀₄

—

D₁₀₀₄

C₂×D₄

yes

—

C₂²⋊D₂₅₁

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

C₂×D₄

yes

—

C₈×C₂₅₁

C₂×D₄

yes

—

C₂²×C₂₅₁

H(5,𝔽₂)

yes

characteristic

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂²×D₄

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

D₄²

yes

characteristic

D₂₅₁

—

C₂₅₁

#3293

yes

characteristic, complemented, Sylow

#3293

—

251

maximal, Sylow

#578

—

251

—

#494

D₂₅₁

yes

characteristic, complemented

#473

—

251

—

#456

—

251

—

#443

—

251

—

2₋¹⁺⁴

—

251

—

#137

—

502

—

#137

—

251

—

#135

—

502

—

#135

—

251

—

C₂×Q₁₆

—

502

—

C₂×Q₁₆

C₂×D₂₅₁

yes

—

#131

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

—

#108

—

251

—

#104

—

251

—

#101

—

251

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic

#41

—

1004

—

#41

—

502

—

#41

—

251

—

C₂×Q₈

—

502

—

C₂×Q₈

C₂²×D₂₅₁

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

502

—

Q₁₆

—

251

—

SD₁₆

—

502

—

SD₁₆

—

251

—

D₈

—

251

—

M₁₆

—

502

—

M₁₆

—

251

—

C₂×C₈

—

502

—

C₂×C₈

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₅₁

yes

characteristic

Q₈

—

Q₈

—

1004

—

100

Q₈

—

251

—

101

D₄

—

1004

—

102

D₄

—

502

—

103

D₄

—

251

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

1004

—

107

C₂×C₄

—

502

—

108

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

109

C₈

—

502

—

110

C₈

D₄×D₂₅₁

yes

—

111

C₈

D₄×D₂₅₁

yes

—

112

C₂²

—

502

—

113

C₂²

#22548760

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

502

—

117

C₄

C₂×D₄×D₂₅₁

yes

—

118

C₂

—

119

C₂

—

1004

—

120

C₂

#30632322

yes

characteristic

	subgroup	G/H
center	C₂	#30632322
commutator, cosocle	C₂×C₄×C₂₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#494×C₂₅₁	C₂
socle	C₂×C₂₅₁	D₄²
hypercenter	#494	D₂₅₁
hypocenter	C₂₅₁	#3293
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#30632322

commutator, cosocle

C₂×C₄×C₂₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting

#494×C₂₅₁

C₂

socle

C₂×C₂₅₁

D₄²

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #30632500	C₂	semidirect
2. #30631959	C₂	semidirect
3. #30631869	C₂	semidirect
4. #30631805	C₂	semidirect
5. #494×C₂₅₁	C₂	semidirect
60. C₂₅₁	#3293	semidirect
63. #494	D₂₅₁	semidirect

H ≅ G/N

type

1. #30632500

C₂

semidirect

2. #30631959

C₂

semidirect

3. #30631869

C₂

semidirect

4. #30631805

C₂

semidirect

5. #494×C₂₅₁

C₂

semidirect

60. C₂₅₁

#3293

semidirect

63. #494

D₂₅₁

semidirect

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

subgroup	index
#30632500	2
#30631959	2
#30631869	2
#30631805	2
#494×C₂₅₁	2
#3293	251

subgroup

index

#494×C₂₅₁

251

Subgroups — #44562169

Conjugacy classes of subgroups

Complements

Maximal subgroups