Group #44558844 — subgroups

Subgroups — #44558844

Order 32128.

756 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 289 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	#30632454	C₂	1	2	yes	maximal, complemented
2	#30631671	C₂	1	6	yes	maximal, complemented
3	C₂×#22548632	C₂	1	6	yes	maximal, complemented
4	C₂×H(5,𝔽₂)×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#22548722	—	2	8	no	—
6	C₂×#16860794	C₂²	1	12	yes	complemented
7	#22548634	C₂²	1	4	yes	complemented
8	#22548632	C₂²	1	12	yes	complemented
9	H(5,𝔽₂)×C₂₅₁	—	2	8	no	—
10	C₂×#41×C₂₅₁	C₄	1	2	yes	—
11	C₂×#41×C₂₅₁	C₂²	1	3	yes	—
12	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₅₁	C₄	1	6	yes	—
14	C₂²×D₄×C₂₅₁	C₂²	1	3	yes	—
15	#16860794	—	2	12	no	—
16	#16860794	C₂³	1	8	yes	—
17	C₂×C₂₅₁⋊C₈	—	2	12	no	—
18	C₂⁴×C₂₅₁	C₂×C₄	1	6	yes	—
19	#41×C₂₅₁	—	2	8	no	—
20	#41×C₂₅₁	—	2	12	no	—
21	#41×C₂₅₁	D₄	1	8	yes	—
22	C₂×Q₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic
23	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
24	C₂×D₄×C₂₅₁	—	2	24	no	—
25	C₂×D₄×C₂₅₁	—	2	12	no	—
26	C₂×D₄×C₂₅₁	C₂×C₄	1	6	yes	—
27	C₂×D₄×C₂₅₁	C₂×C₄	1	6	yes	—
28	C₂×D₄×C₂₅₁	C₂×C₄	1	3	yes	—
29	C₂×D₄×C₂₅₁	C₂³	1	3	yes	—
30	C₂²×C₄×C₂₅₁	C₂×C₄	1	6	yes	—
31	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
32	C₂₅₁⋊C₈	—	2	8	no	—
33	C₂³×C₂₅₁	—	2	12	no	—
34	C₂³×C₂₅₁	—	8	6	no	—
35	C₂³×C₂₅₁	C₂²×C₄	1	6	yes	—
36	C₂³×C₂₅₁	C₂²×C₄	1	3	yes	—
37	Q₈×C₂₅₁	—	2	2	no	—
38	Q₈×C₂₅₁	C₂²⋊C₄	1	4	yes	—
39	D₄×C₂₅₁	—	2	12	no	—
40	D₄×C₂₅₁	—	2	12	no	—
41	D₄×C₂₅₁	—	2	6	no	—
42	D₄×C₂₅₁	C₂²⋊C₄	1	12	yes	—
43	C₂×C₄×C₂₅₁	—	2	12	no	—
44	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₂₅₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₂₅₁	C₂×D₄	1	12	yes	—
47	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
48	C₂²×C₂₅₁	—	2	6	no	—
49	C₂²×C₂₅₁	—	4	3	no	—
50	C₂²×C₂₅₁	—	4	6	no	—
51	C₂²×C₂₅₁	—	8	12	no	—
52	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	6	yes	—
53	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₂₅₁	—	2	2	no	—
55	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	6	yes	—
56	C₄×C₂₅₁	C₂²×D₄	1	2	yes	—
57	C₂×C₂₅₁	—	2	1	no	—
58	C₂×C₂₅₁	—	2	6	no	—
59	C₂×C₂₅₁	—	4	6	no	—
60	C₂×C₂₅₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
61	C₂₅₁	#2888	1	1	yes	characteristic, complemented, Sylow
62	#2888	—	251	1	no	maximal, Sylow
63	C₂×H(5,𝔽₂)	D₂₅₁	1	1	yes	characteristic
64	#568	—	251	2	no	—
65	#413	—	251	6	no	—
66	C₂×#100	—	251	6	no	—
67	H(5,𝔽₂)	—	2	8	no	—
68	C₂×#41	Dic₂₅₁	1	2	yes	—
69	C₂×#41	C₂×D₂₅₁	1	3	yes	—
70	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
71	C₂²×D₄	Dic₂₅₁	1	6	yes	—
72	C₂²×D₄	C₂×D₂₅₁	1	3	yes	—
73	#131	—	502	8	no	—
74	C₂×M₁₆	—	251	12	no	—
75	#101	—	251	4	no	—
76	#100	—	251	12	no	—
77	C₂⁴	C₂×Dic₂₅₁	1	6	yes	—
78	#41	—	2	8	no	—
79	#41	—	2	12	no	—
80	#41	C₂²⋊D₂₅₁	1	8	yes	—
81	C₂×Q₈	C₂×Dic₂₅₁	1	1	yes	characteristic
82	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
83	C₂×D₄	—	2	24	no	—
84	C₂×D₄	—	2	12	no	—
85	C₂×D₄	C₂×Dic₂₅₁	1	6	yes	—
86	C₂×D₄	C₂×Dic₂₅₁	1	6	yes	—
87	C₂×D₄	C₂×Dic₂₅₁	1	3	yes	—
88	C₂×D₄	C₂²×D₂₅₁	1	3	yes	—
89	C₂²×C₄	C₂×Dic₂₅₁	1	6	yes	—
90	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
91	M₁₆	—	502	12	no	—
92	M₁₆	—	251	8	no	—
93	C₂×C₈	—	502	12	no	—
94	C₂³	—	2	12	no	—
95	C₂³	—	8	6	no	—
96	C₂³	C₂²×Dic₂₅₁	1	6	yes	—
97	C₂³	C₂²×Dic₂₅₁	1	3	yes	—
98	Q₈	—	2	2	no	—
99	Q₈	C₂²⋊Dic₂₅₁	1	4	yes	—
100	D₄	—	2	12	no	—
101	D₄	—	2	6	no	—
102	D₄	—	2	12	no	—
103	D₄	C₂²⋊Dic₂₅₁	1	12	yes	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
106	C₂×C₄	C₂²×Dic₂₅₁	1	3	yes	—
107	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	12	yes	—
108	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
109	C₈	—	502	8	no	—
110	C₂²	—	2	6	no	—
111	C₂²	—	4	3	no	—
112	C₂²	—	4	6	no	—
113	C₂²	—	8	12	no	—
114	C₂²	C₂×C₂²⋊Dic₂₅₁	1	6	yes	—
115	C₂²	C₂³×Dic₂₅₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	C₂×C₂²⋊Dic₂₅₁	1	6	yes	—
118	C₄	C₂²×C₂²⋊D₂₅₁	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	6	no	—
121	C₂	—	4	6	no	—
122	C₂	C₂²×C₂²⋊Dic₂₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632454

C₂

yes

maximal, complemented

#30631671

C₂

yes

maximal, complemented

C₂×#22548632

C₂

yes

maximal, complemented

C₂×H(5,𝔽₂)×C₂₅₁

C₂

yes

maximal, characteristic

#22548722

—

C₂×#16860794

C₂²

yes

complemented

#22548634

C₂²

yes

complemented

#22548632

C₂²

yes

complemented

H(5,𝔽₂)×C₂₅₁

—

C₂×#41×C₂₅₁

C₄

yes

—

C₂×#41×C₂₅₁

C₂²

yes

—

C₂×#41×C₂₅₁

C₂²

yes

characteristic

C₂²×D₄×C₂₅₁

C₄

yes

—

C₂²×D₄×C₂₅₁

C₂²

yes

—

#16860794

—

#16860794

C₂³

yes

—

C₂×C₂₅₁⋊C₈

—

C₂⁴×C₂₅₁

C₂×C₄

yes

—

#41×C₂₅₁

—

#41×C₂₅₁

—

#41×C₂₅₁

D₄

yes

—

C₂×Q₈×C₂₅₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₂₅₁

C₂³

yes

characteristic

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂²×C₄

yes

—

C₂³×C₂₅₁

C₂²×C₄

yes

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

C₂²⋊C₄

yes

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₅₁

C₂³×C₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×C₂²⋊C₄

yes

—

C₄×C₂₅₁

C₂²×D₄

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂₅₁

#2888

yes

characteristic, complemented, Sylow

#2888

—

251

maximal, Sylow

C₂×H(5,𝔽₂)

D₂₅₁

yes

characteristic

#568

—

251

—

#413

—

251

—

C₂×#100

—

251

—

H(5,𝔽₂)

—

C₂×#41

Dic₂₅₁

yes

—

C₂×#41

C₂×D₂₅₁

yes

—

C₂×#41

C₂×D₂₅₁

yes

characteristic

C₂²×D₄

Dic₂₅₁

yes

—

C₂²×D₄

C₂×D₂₅₁

yes

—

#131

—

502

—

C₂×M₁₆

—

251

—

#101

—

251

—

#100

—

251

—

C₂⁴

C₂×Dic₂₅₁

yes

—

#41

—

#41

—

#41

C₂²⋊D₂₅₁

yes

—

C₂×Q₈

C₂×Dic₂₅₁

yes

characteristic

C₂×Q₈

C₂²×D₂₅₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₂₅₁

yes

—

C₂×D₄

C₂×Dic₂₅₁

yes

—

C₂×D₄

C₂×Dic₂₅₁

yes

—

C₂×D₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂×Dic₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

M₁₆

—

502

—

M₁₆

—

251

—

C₂×C₈

—

502

—

C₂³

—

C₂³

—

C₂³

C₂²×Dic₂₅₁

yes

—

C₂³

C₂²×Dic₂₅₁

yes

—

Q₈

—

Q₈

C₂²⋊Dic₂₅₁

yes

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

C₂²⋊Dic₂₅₁

yes

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₂₅₁

yes

—

106

C₂×C₄

C₂²×Dic₂₅₁

yes

—

107

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

108

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

109

C₈

—

502

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₂²⋊Dic₂₅₁

yes

—

115

C₂²

C₂³×Dic₂₅₁

yes

characteristic

116

C₄

—

117

C₄

C₂×C₂²⋊Dic₂₅₁

yes

—

118

C₄

C₂²×C₂²⋊D₂₅₁

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

C₂²×C₂²⋊Dic₂₅₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting, prime radical	C₂×H(5,𝔽₂)×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂²×C₂²⋊C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×H(5,𝔽₂)	D₂₅₁
hypocenter	C₂₅₁	#2888
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊Dic₂₅₁

commutator

C₂²×C₂₅₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₅₁

Fitting, prime radical

C₂×H(5,𝔽₂)×C₂₅₁

C₂

socle

C₂×C₂₅₁

C₂²×C₂²⋊C₄

cosocle

C₂×C₄×C₂₅₁

C₂⁴

hypercenter

C₂×H(5,𝔽₂)

D₂₅₁

hypocenter

C₂₅₁

#2888

LCS

G ▷ C₂²×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

N	H ≅ G/N	type
1. #30632454	C₂	semidirect
2. #30631671	C₂	semidirect
3. C₂×#22548632	C₂	semidirect
6. C₂×#16860794	C₂²	semidirect
7. #22548634	C₂²	semidirect
8. #22548632	C₂²	semidirect
61. C₂₅₁	#2888	semidirect

H ≅ G/N

type

1. #30632454

C₂

semidirect

2. #30631671

C₂

semidirect

3. C₂×#22548632

C₂

semidirect

6. C₂×#16860794

C₂²

semidirect

7. #22548634

C₂²

semidirect

8. #22548632

C₂²

semidirect

61. C₂₅₁

#2888

semidirect

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

subgroup	index
#30632454	2
#30631671	2
C₂×#22548632	2
C₂×H(5,𝔽₂)×C₂₅₁	2
#2888	251

subgroup

index

C₂×H(5,𝔽₂)×C₂₅₁

251

Subgroups — #44558844

Conjugacy classes of subgroups

Complements

Maximal subgroups