Group #44558073 — subgroups

Subgroups — #44558073

Order 32128.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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	#30631604	C₂	1	3	yes	maximal
2	#30631598	C₂	1	2	yes	maximal, complemented
3	#30631579	C₂	1	6	yes	maximal, complemented
4	#30631552	C₂	1	3	yes	maximal, complemented
5	#395×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
8	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	6	yes	—
9	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	6	yes	—
10	C₂×C₄×Dic₂₅₁	C₂²	1	3	yes	—
11	C₂₅₁⋊#95	C₂²	1	3	yes	—
12	C₂₅₁⋊#95	C₂²	1	6	yes	—
13	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
15	#95×C₂₅₁	C₂²	1	3	yes	—
16	C₂³×D₂₅₁	C₂³	1	2	yes	—
17	C₂×D₁₀₀₄	—	4	3	no	—
18	C₂²×Dic₂₅₁	C₂³	1	6	yes	—
19	C₄⋊Dic₂₅₁	—	4	3	no	—
20	(C₂×D₂₅₁)⋊C₄	—	2	12	no	—
21	(C₂×D₂₅₁)⋊C₄	—	4	6	no	—
22	C₄×Dic₂₅₁	—	2	6	no	—
23	C₂×Q₈×C₂₅₁	—	4	3	no	—
24	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
26	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
27	C₄⋊C₄×C₂₅₁	—	4	3	no	—
28	C₂²×D₂₅₁	—	2	6	no	—
29	C₂²×D₂₅₁	—	2	6	no	—
30	C₂²×D₂₅₁	—	2	2	no	—
31	D₁₀₀₄	—	4	4	no	—
32	C₂×Dic₂₅₁	—	2	6	no	—
33	C₂×Dic₂₅₁	—	2	6	no	—
34	C₂×Dic₂₅₁	—	2	6	no	—
35	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
36	Q₈×C₂₅₁	—	4	4	no	—
37	C₂×C₄×C₂₅₁	—	2	3	no	—
38	C₂×C₄×C₂₅₁	—	2	3	no	—
39	C₂×C₄×C₂₅₁	—	2	3	no	—
40	C₂×C₄×C₂₅₁	—	2	3	no	—
41	C₂×C₄×C₂₅₁	—	2	6	no	—
42	C₂×C₄×C₂₅₁	#41	1	6	yes	—
43	C₂²×C₂₅₁	C₂×#41	1	3	yes	—
44	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
46	C₂×D₂₅₁	—	4	6	no	—
47	C₂×D₂₅₁	—	4	6	no	—
48	C₂×D₂₅₁	—	4	2	no	—
49	C₄×C₂₅₁	—	2	6	no	—
50	C₄×C₂₅₁	—	4	3	no	—
51	C₄×C₂₅₁	—	4	1	no	—
52	Dic₂₅₁	—	4	6	no	—
53	C₂×C₂₅₁	#539	1	3	yes	—
54	C₂×C₂₅₁	#555	1	3	yes	—
55	C₂×C₂₅₁	H(3,𝔽₄)	1	1	yes	characteristic
56	D₂₅₁	—	8	2	no	—
57	C₂₅₁	#2785	1	1	yes	characteristic, complemented, Sylow
58	#2785	—	251	1	no	maximal, Sylow
59	#395	D₂₅₁	1	1	yes	characteristic, complemented
60	#394	—	251	3	no	—
61	#393	—	251	2	no	—
62	#388	—	251	6	no	—
63	#382	—	251	3	no	—
64	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
65	C₂²×D₄	—	251	1	no	—
66	C₂×C₄⋊C₄	—	251	3	no	—
67	C₂×C₄⋊C₄	C₂×D₂₅₁	1	3	yes	—
68	C₂×C₂²⋊C₄	—	251	6	no	—
69	C₂×C₂²⋊C₄	—	251	6	no	—
70	C₂×C₄²	—	251	3	no	—
71	#95	—	251	3	no	—
72	#95	—	251	6	no	—
73	#95	C₂×D₂₅₁	1	3	yes	—
74	C₂⁴	—	251	2	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	1004	3	no	—
77	C₂²×C₄	—	251	6	no	—
78	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
80	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	1004	3	no	—
83	C₂²⋊C₄	—	1004	6	no	—
84	C₂²⋊C₄	—	502	12	no	—
85	C₄²	—	502	6	no	—
86	C₂³	—	502	6	no	—
87	C₂³	—	502	6	no	—
88	C₂³	—	502	2	no	—
89	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	1004	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	502	6	no	—
98	C₂×C₄	—	502	6	no	—
99	C₂×C₄	—	502	6	no	—
100	C₂×C₄	#16860811	1	6	yes	—
101	C₂²	—	1004	6	no	—
102	C₂²	—	1004	6	no	—
103	C₂²	—	1004	2	no	—
104	C₂²	C₂×#16860811	1	3	yes	—
105	C₂²	#22548760	1	3	yes	—
106	C₂²	#22548762	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	1004	6	no	—
111	C₂	—	2008	2	no	—
112	C₂	#30632286	1	3	yes	—
113	C₂	#30632403	1	3	yes	—
114	C₂	#30632430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30631604

C₂

yes

maximal

#30631598

C₂

yes

maximal, complemented

#30631579

C₂

yes

maximal, complemented

#30631552

C₂

yes

maximal, complemented

#395×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₀₀₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂²×Q₈×C₂₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

#95×C₂₅₁

C₂²

yes

—

C₂³×D₂₅₁

C₂³

yes

—

C₂×D₁₀₀₄

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₄⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

(C₂×D₂₅₁)⋊C₄

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

D₁₀₀₄

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

C₂⁴

yes

characteristic

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

#41

yes

—

C₂²×C₂₅₁

C₂×#41

yes

—

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#539

yes

—

C₂×C₂₅₁

#555

yes

—

C₂×C₂₅₁

H(3,𝔽₄)

yes

characteristic

D₂₅₁

—

C₂₅₁

#2785

yes

characteristic, complemented, Sylow

#2785

—

251

maximal, Sylow

#395

D₂₅₁

yes

characteristic, complemented

#394

—

251

—

#393

—

251

—

#388

—

251

—

#382

—

251

—

C₂²×Q₈

C₂×D₂₅₁

yes

characteristic

C₂²×D₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₄²

—

251

—

#95

—

251

—

#95

—

251

—

#95

C₂×D₂₅₁

yes

—

C₂⁴

—

251

—

C₂×Q₈

—

C₂×D₄

—

1004

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

1004

—

C₂²⋊C₄

—

1004

—

C₂²⋊C₄

—

502

—

C₄²

—

502

—

C₂³

—

502

—

C₂³

—

502

—

C₂³

—

502

—

C₂³

C₂³×D₂₅₁

yes

characteristic

Q₈

—

D₄

—

1004

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

100

C₂×C₄

#16860811

yes

—

101

C₂²

—

1004

—

102

C₂²

—

1004

—

103

C₂²

—

1004

—

104

C₂²

C₂×#16860811

yes

—

105

C₂²

#22548760

yes

—

106

C₂²

#22548762

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

1004

—

111

C₂

—

2008

—

112

C₂

#30632286

yes

—

113

C₂

#30632403

yes

—

114

C₂

#30632430

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#395×C₂₅₁	C₂
prime radical	C₂²×D₁₀₀₄	C₂²
hypercenter	#395	D₂₅₁
hypocenter	C₂₅₁	#2785
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#395×C₂₅₁

C₂

prime radical

C₂²×D₁₀₀₄

C₂²

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
2. #30631598	C₂	semidirect
3. #30631579	C₂	semidirect
4. #30631552	C₂	semidirect
5. #395×C₂₅₁	C₂	semidirect
57. C₂₅₁	#2785	semidirect
59. #395	D₂₅₁	semidirect

H ≅ G/N

type

2. #30631598

C₂

semidirect

3. #30631579

C₂

semidirect

4. #30631552

C₂

semidirect

5. #395×C₂₅₁

C₂

semidirect

57. C₂₅₁

#2785

semidirect

59. #395

D₂₅₁

semidirect

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

subgroup	index
#30631604	2
#30631598	2
#30631579	2
#30631552	2
#395×C₂₅₁	2
#2785	251

subgroup

index

#395×C₂₅₁

251

Subgroups — #44558073

Conjugacy classes of subgroups

Complements

Maximal subgroups