Group #44551209 — subgroups

Subgroups — #44551209

Order 32128.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22548715	C₂	1	1	yes	maximal, characteristic
2	#30631711	C₂	1	1	yes	maximal, characteristic
3	C₂×#22548633	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₀₂	C₂²	1	2	yes	—
6	C₂×#16860794	C₂²	1	2	yes	—
7	#22548715	—	2	1	no	—
8	#22548715	—	2	1	no	—
9	#22548715	—	2	2	no	—
10	C₂×C₄⋊Dic₂₅₁	C₄	1	4	yes	—
11	C₄⋊(C₂₅₁⋊C₈)	—	2	2	no	—
12	#22548633	—	2	8	no	—
13	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
15	#97×C₂₅₁	—	2	1	no	—
16	C₄×C₈×C₂₅₁	—	2	1	no	—
17	C₂×Dic₅₀₂	—	2	4	no	—
18	C₂×Dic₅₀₂	—	2	2	no	—
19	C₂×Dic₅₀₂	—	2	2	no	—
20	C₂×Dic₅₀₂	—	2	4	no	—
21	C₂²×Dic₂₅₁	C₂×C₄	1	4	yes	—
22	#16860794	—	2	4	no	—
23	C₂×C₂₅₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₂₅₁	—	2	4	no	—
25	C₄⋊Dic₂₅₁	—	2	4	no	—
26	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₂₅₁	—	2	2	no	—
29	M₁₆×C₂₅₁	D₄	1	4	yes	—
30	C₂×C₈×C₂₅₁	—	2	2	no	—
31	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
32	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
33	C₄²×C₂₅₁	D₄	1	2	yes	—
34	Dic₅₀₂	—	4	4	no	—
35	Dic₅₀₂	—	4	2	no	—
36	Dic₅₀₂	—	4	2	no	—
37	C₂×Dic₂₅₁	—	2	4	no	—
38	C₂×Dic₂₅₁	—	2	8	no	—
39	C₂₅₁⋊C₈	—	4	2	no	—
40	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₅₁	—	2	4	no	—
42	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₅₁	#41	1	2	yes	—
48	C₈×C₂₅₁	—	2	4	no	—
49	C₂²×C₂₅₁	—	2	2	no	—
50	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
52	C₄×C₂₅₁	—	2	2	no	—
53	C₄×C₂₅₁	#101	1	4	yes	complemented
54	C₄×C₂₅₁	#121	1	2	yes	—
55	C₄×C₂₅₁	#124	1	1	yes	characteristic
56	C₄×C₂₅₁	#127	1	1	yes	characteristic
57	Dic₂₅₁	—	4	4	no	—
58	C₂×C₂₅₁	—	2	2	no	—
59	C₂×C₂₅₁	#390	1	1	yes	characteristic
60	C₂×C₂₅₁	C₂×#101	1	2	yes	—
61	C₂₅₁	#1966	1	1	yes	characteristic, complemented, Sylow
62	#1966	—	251	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	251	1	no	—
64	#423	—	251	1	no	—
65	C₂×#101	—	251	4	no	—
66	C₄×M₁₆	D₂₅₁	1	1	yes	characteristic
67	C₂²×Q₈	—	251	2	no	—
68	C₂×M₁₆	—	251	2	no	—
69	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
70	C₄⋊Q₈	—	502	1	no	—
71	C₄⋊Q₈	—	502	1	no	—
72	C₄⋊Q₈	—	502	2	no	—
73	C₂×C₄⋊C₄	—	251	4	no	—
74	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
75	C₄⋊C₈	—	502	2	no	—
76	#101	—	502	8	no	—
77	#97	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₂×Q₈	—	502	2	no	—
80	C₂×Q₈	—	502	4	no	—
81	C₂×Q₈	—	502	2	no	—
82	C₂×Q₈	—	502	4	no	—
83	C₂²×C₄	—	251	4	no	—
84	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	502	4	no	—
88	M₁₆	D₁₀₀₄	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	502	2	no	—
91	C₄⋊C₄	—	502	4	no	—
92	C₄⋊C₄	—	502	4	no	—
93	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
94	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
95	C₄²	D₁₀₀₄	1	2	yes	—
96	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
97	Q₈	—	1004	4	no	—
98	Q₈	—	1004	2	no	—
99	Q₈	—	1004	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	502	4	no	—
102	C₂×C₄	—	502	8	no	—
103	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
104	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
106	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
107	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
108	C₂×C₄	#16860810	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	1004	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
113	C₂²	C₄×D₁₀₀₄	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	1004	4	no	—
116	C₄	#22548633	1	4	yes	complemented
117	C₄	#22548689	1	2	yes	—
118	C₄	#22548703	1	1	yes	characteristic
119	C₄	#22548712	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#30631589	1	1	yes	characteristic
122	C₂	C₂×#22548633	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548715

C₂

yes

maximal, characteristic

#30631711

C₂

yes

maximal, characteristic

C₂×#22548633

C₂

yes

maximal

C₄×M₁₆×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

—

C₂×#16860794

C₂²

yes

—

#22548715

—

#22548715

—

#22548715

—

C₂×C₄⋊Dic₂₅₁

C₄

yes

—

C₄⋊(C₂₅₁⋊C₈)

—

#22548633

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

#97×C₂₅₁

—

C₄×C₈×C₂₅₁

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

#16860794

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

D₄

yes

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#101

yes

complemented

C₄×C₂₅₁

#121

yes

—

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#390

yes

characteristic

C₂×C₂₅₁

C₂×#101

yes

—

C₂₅₁

#1966

yes

characteristic, complemented, Sylow

#1966

—

251

maximal, Sylow

C₂×C₄⋊Q₈

—

251

—

#423

—

251

—

C₂×#101

—

251

—

C₄×M₁₆

D₂₅₁

yes

characteristic

C₂²×Q₈

—

251

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

502

—

#101

—

502

—

#97

—

C₄×C₈

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂²×C₄

—

251

—

C₂²×C₄

C₄×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

M₁₆

—

502

—

M₁₆

D₁₀₀₄

yes

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄²

C₂²⋊D₂₅₁

yes

characteristic

C₄²

C₂²⋊D₂₅₁

yes

characteristic

C₄²

D₁₀₀₄

yes

—

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

Q₈

—

1004

—

Q₈

—

1004

—

Q₈

—

1004

—

100

C₂×C₄

—

101

C₂×C₄

—

502

—

102

C₂×C₄

—

502

—

103

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

104

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

105

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

106

C₂×C₄

C₂×D₁₀₀₄

yes

—

107

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

108

C₂×C₄

#16860810

yes

—

109

C₈

—

110

C₈

—

1004

—

111

C₂²

—

112

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

113

C₂²

C₄×D₁₀₀₄

yes

—

114

C₄

—

115

C₄

—

1004

—

116

C₄

#22548633

yes

complemented

117

C₄

#22548689

yes

—

118

C₄

#22548703

yes

characteristic

119

C₄

#22548712

yes

characteristic

120

C₂

—

121

C₂

#30631589

yes

characteristic

122

C₂

C₂×#22548633

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₂₅₁)⋊C₄
commutator, prime radical	C₂³×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₄×M₁₆×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₄×M₁₆	D₂₅₁
hypocenter	C₂₅₁	#1966
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

commutator, prime radical

C₂³×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₄×M₁₆×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₄×M₁₆

D₂₅₁

hypocenter

C₂₅₁

#1966

LCS

G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
53. C₄×C₂₅₁	#101	semidirect
61. C₂₅₁	#1966	semidirect
116. C₄	#22548633	semidirect

H ≅ G/N

type

53. C₄×C₂₅₁

#101

semidirect

61. C₂₅₁

#1966

semidirect

116. C₄

#22548633

semidirect

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

subgroup	index
C₂×#22548715	2
#30631711	2
C₂×#22548633	2
C₄×M₁₆×C₂₅₁	2
#1966	251

subgroup

index

C₄×M₁₆×C₂₅₁

251

Subgroups — #44551209

Conjugacy classes of subgroups

Complements

Maximal subgroups