Group #44550403 — subgroups

Subgroups — #44550403

Order 32128.

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

normal subgroups only

This group has 292 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	C₂×#22548651	C₂	1	2	yes	maximal, complemented
2	#30631651	C₂	1	2	yes	maximal
3	#30631452	C₂	1	2	yes	maximal, complemented
4	C₂³×C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16860794	C₂²	1	4	yes	—
6	#22548651	—	2	4	no	—
7	C₂²⋊(C₂₅₁⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₅₁	—	2	4	no	—
10	C₂²×C₈×C₂₅₁	C₄	1	4	yes	—
11	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
13	#16860794	—	2	4	no	—
14	#16860794	—	2	4	no	—
15	C₂×C₂₅₁⋊C₈	—	2	4	no	—
16	C₂⁴×C₂₅₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₅₁	—	2	2	no	—
18	C₂²×C₄×C₂₅₁	—	2	2	no	—
19	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₅₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₅₁	—	2	2	no	—
24	C₂×C₈×C₂₅₁	—	2	8	no	—
25	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₅₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₅₁	D₄	1	4	yes	—
28	C₂₅₁⋊C₈	—	4	4	no	—
29	C₂³×C₂₅₁	—	2	4	no	—
30	C₂³×C₂₅₁	—	2	2	no	—
31	C₂³×C₂₅₁	C₄⋊C₄	1	2	yes	—
32	C₂³×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₅₁	—	2	2	no	—
34	C₂×C₄×C₂₅₁	—	2	2	no	—
35	C₂×C₄×C₂₅₁	—	2	2	no	—
36	C₂×C₄×C₂₅₁	—	2	4	no	—
37	C₂×C₄×C₂₅₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₅₁	#41	1	2	yes	—
42	C₈×C₂₅₁	—	2	2	no	—
43	C₈×C₂₅₁	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₂₅₁	—	2	8	no	—
45	C₂²×C₂₅₁	—	2	2	no	—
46	C₂²×C₂₅₁	—	2	2	no	—
47	C₂²×C₂₅₁	—	2	2	no	—
48	C₂²×C₂₅₁	#108	1	4	yes	complemented
49	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₂₅₁	—	2	2	no	—
52	C₄×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₅₁	#117	1	1	yes	characteristic
54	C₄×C₂₅₁	#121	1	2	yes	—
55	C₂×C₂₅₁	—	2	4	no	—
56	C₂×C₂₅₁	—	2	2	no	—
57	C₂×C₂₅₁	#380	1	1	yes	characteristic
58	C₂×C₂₅₁	C₂×#108	1	2	yes	—
59	C₂₅₁	#1810	1	1	yes	characteristic, complemented, Sylow
60	#1810	—	251	1	no	maximal, Sylow
61	C₂³×C₈	D₂₅₁	1	1	yes	characteristic
62	C₂×#108	—	251	2	no	—
63	#407	—	251	2	no	—
64	#341	—	251	2	no	—
65	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
66	C₂×M₁₆	—	251	4	no	—
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	Dic₂₅₁	1	4	yes	—
69	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
71	#108	—	502	4	no	—
72	C₂²⋊C₈	—	502	4	no	—
73	C₂⁴	Dic₅₀₂	1	1	yes	characteristic
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
77	C₂²×C₄	D₁₀₀₄	1	2	yes	—
78	C₂²×C₄	Dic₅₀₂	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
80	M₁₆	—	502	4	no	—
81	M₁₆	—	502	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	8	no	—
84	C₂×C₈	—	502	4	no	—
85	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
86	C₂×C₈	C₂×Dic₂₅₁	1	2	yes	—
87	C₂×C₈	C₂²⋊D₂₅₁	1	4	yes	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	C₄⋊Dic₂₅₁	1	2	yes	—
91	C₂³	C₂×Dic₅₀₂	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	C₄⋊Dic₂₅₁	1	2	yes	—
97	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
99	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
100	C₂×C₄	#16860810	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	1004	4	no	—
103	C₈	C₂²⋊Dic₂₅₁	1	4	yes	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	#22548651	1	4	yes	complemented
109	C₂²	C₂×C₄⋊Dic₂₅₁	1	1	yes	characteristic
110	C₂²	#22548694	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
113	C₄	#22548674	1	1	yes	characteristic
114	C₄	#22548689	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	—	2	2	no	—
117	C₂	#30631547	1	1	yes	characteristic
118	C₂	C₂×#22548651	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548651

C₂

yes

maximal, complemented

#30631651

C₂

yes

maximal

#30631452

C₂

yes

maximal, complemented

C₂³×C₈×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

—

#22548651

—

C₂²⋊(C₂₅₁⋊C₈)

—

C₂³×C₄×C₂₅₁

C₂²

yes

characteristic

C₂²×C₈×C₂₅₁

—

C₂²×C₈×C₂₅₁

C₄

yes

—

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

C₂²×C₈×C₂₅₁

C₂²

yes

characteristic

#16860794

—

#16860794

—

C₂×C₂₅₁⋊C₈

—

C₂⁴×C₂₅₁

Q₈

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

Q₈

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

D₄

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₄⋊C₄

yes

—

C₂³×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₈×C₂₅₁

C₂²⋊C₄

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#108

yes

complemented

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₂²⋊Q₈

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×C₂²⋊C₄

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#380

yes

characteristic

C₂×C₂₅₁

C₂×#108

yes

—

C₂₅₁

#1810

yes

characteristic, complemented, Sylow

#1810

—

251

maximal, Sylow

C₂³×C₈

D₂₅₁

yes

characteristic

C₂×#108

—

251

—

#407

—

251

—

#341

—

251

—

C₂³×C₄

C₂×D₂₅₁

yes

characteristic

C₂×M₁₆

—

251

—

C₂²×C₈

—

C₂²×C₈

Dic₂₅₁

yes

—

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

C₂²×C₈

C₂×D₂₅₁

yes

characteristic

#108

—

502

—

C₂²⋊C₈

—

502

—

C₂⁴

Dic₅₀₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₅₁

yes

—

C₂²×C₄

D₁₀₀₄

yes

—

C₂²×C₄

Dic₅₀₂

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

502

—

M₁₆

—

502

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

502

—

C₂×C₈

C₂×Dic₂₅₁

yes

—

C₂×C₈

C₂×Dic₂₅₁

yes

—

C₂×C₈

C₂²⋊D₂₅₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₂₅₁

yes

—

C₂³

C₂×Dic₅₀₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₅₁

yes

—

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

100

C₂×C₄

#16860810

yes

—

101

C₈

—

102

C₈

—

1004

—

103

C₈

C₂²⋊Dic₂₅₁

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#22548651

yes

complemented

109

C₂²

C₂×C₄⋊Dic₂₅₁

yes

characteristic

110

C₂²

#22548694

yes

—

111

C₄

—

112

C₄

C₂×C₂²⋊Dic₂₅₁

yes

characteristic

113

C₄

#22548674

yes

characteristic

114

C₄

#22548689

yes

—

115

C₂

—

116

C₂

—

117

C₂

#30631547

yes

characteristic

118

C₂

C₂×#22548651

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₀₄
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂³×C₈×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₅₁	Q₈
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂³×C₈	D₂₅₁
hypocenter	C₂₅₁	#1810
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₀₀₄

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₂³×C₈×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₅₁

Q₈

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₂³×C₈

D₂₅₁

hypocenter

C₂₅₁

#1810

LCS

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#22548651	C₂	semidirect
3. #30631452	C₂	semidirect
48. C₂²×C₂₅₁	#108	semidirect
59. C₂₅₁	#1810	semidirect
108. C₂²	#22548651	semidirect

H ≅ G/N

type

1. C₂×#22548651

C₂

semidirect

3. #30631452

C₂

semidirect

48. C₂²×C₂₅₁

#108

semidirect

59. C₂₅₁

#1810

semidirect

108. C₂²

#22548651

semidirect

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

subgroup	index
C₂×#22548651	2
#30631651	2
#30631452	2
C₂³×C₈×C₂₅₁	2
#1810	251

subgroup

index

C₂³×C₈×C₂₅₁

251

Subgroups — #44550403

Conjugacy classes of subgroups

Complements

Maximal subgroups