Group #44551345 — subgroups

Subgroups — #44551345

Order 32128.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#30632244	C₂	1	1	yes	maximal, characteristic
2	#30631533	C₂	1	1	yes	maximal, characteristic
3	#30631448	C₂	1	4	yes	maximal
4	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	#22548715	C₄	1	2	yes	complemented
6	#22548707	C₄	1	2	yes	complemented
7	#22548694	—	2	2	no	—
8	C₄×Dic₅₀₂	—	2	2	no	—
9	#22548674	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
11	C₂₅₁⋊#95	—	2	2	no	—
12	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₅₁	—	2	2	no	—
15	C₂×Dic₅₀₂	—	2	2	no	—
16	C₂²×Dic₂₅₁	—	2	2	no	—
17	C₄⋊Dic₂₅₁	—	2	2	no	—
18	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
19	Dic₂₅₁⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₂₅₁	—	2	2	no	—
21	C₄×Dic₂₅₁	—	2	2	no	—
22	C₄×Dic₂₅₁	—	2	2	no	—
23	C₄×Dic₂₅₁	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
25	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₅₁	—	2	4	no	—
27	C₂×C₈×C₂₅₁	—	2	2	no	—
28	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₅₁	D₄	1	2	yes	—
31	Dic₅₀₂	—	4	4	no	—
32	C₂×Dic₂₅₁	—	2	4	no	—
33	C₂×Dic₂₅₁	—	2	4	no	—
34	C₂×Dic₂₅₁	—	4	4	no	—
35	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₅₁	—	4	2	no	—
37	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₂₅₁	#41	1	2	yes	—
43	C₈×C₂₅₁	—	4	2	no	—
44	C₂²×C₂₅₁	—	2	2	no	—
45	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₂₅₁	—	2	2	no	—
48	C₄×C₂₅₁	—	2	2	no	—
49	C₄×C₂₅₁	C₄×Q₈	1	2	yes	—
50	C₄×C₂₅₁	#124	1	1	yes	characteristic
51	C₄×C₂₅₁	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₂₅₁	—	4	4	no	—
53	Dic₂₅₁	—	4	4	no	—
54	C₂×C₂₅₁	—	2	2	no	—
55	C₂×C₂₅₁	#391	1	1	yes	characteristic
56	C₂×C₂₅₁	#421	1	2	yes	—
57	C₂₅₁	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	251	1	no	maximal, Sylow
59	#533	—	251	1	no	—
60	#423	D₂₅₁	1	1	yes	characteristic
61	#376	—	251	1	no	—
62	#339	—	251	4	no	—
63	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
64	C₄⋊Q₈	—	251	2	no	—
65	#125	—	251	2	no	—
66	C₂²⋊Q₈	—	502	2	no	—
67	C₄×Q₈	—	502	2	no	—
68	#117	—	251	2	no	—
69	C₂×C₄²	—	251	2	no	—
70	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	502	2	no	—
73	C₂×Q₈	—	502	2	no	—
74	C₂²×C₄	—	502	2	no	—
75	C₂²×C₄	D₁₀₀₄	1	2	yes	—
76	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	502	4	no	—
80	C₄⋊C₄	—	502	2	no	—
81	C₄⋊C₄	—	251	4	no	—
82	C₂²⋊C₄	—	502	2	no	—
83	C₄²	—	502	2	no	—
84	C₄²	—	502	2	no	—
85	C₄²	—	251	4	no	—
86	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
87	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
88	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
89	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
90	Q₈	—	1004	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	1004	4	no	—
93	C₂×C₄	—	502	4	no	—
94	C₂×C₄	—	502	4	no	—
95	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
100	C₂×C₄	#16860809	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
104	C₂²	#22548695	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	1004	4	no	—
108	C₄	—	1004	4	no	—
109	C₄	#22548680	1	2	yes	—
110	C₄	#22548702	1	1	yes	characteristic
111	C₄	#22548713	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#30631592	1	1	yes	characteristic
114	C₂	#30631703	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632244

C₂

yes

maximal, characteristic

#30631533

C₂

yes

maximal, characteristic

#30631448

C₂

yes

maximal

#423×C₂₅₁

C₂

yes

maximal, characteristic

#22548715

C₄

yes

complemented

#22548707

C₄

yes

complemented

#22548694

—

C₄×Dic₅₀₂

—

#22548674

C₂²

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

—

C₂₅₁⋊#95

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

—

C₂×Dic₅₀₂

—

C₂²×Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

Q₈

yes

complemented

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₂²⋊Q₈

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×Q₈

yes

—

C₄×C₂₅₁

#124

yes

characteristic

C₄×C₂₅₁

C₄⋊Q₈

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#391

yes

characteristic

C₂×C₂₅₁

#421

yes

—

C₂₅₁

#1995

yes

characteristic, complemented, Sylow

#1995

—

251

maximal, Sylow

#533

—

251

—

#423

D₂₅₁

yes

characteristic

#376

—

251

—

#339

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₄⋊Q₈

—

251

—

#125

—

251

—

C₂²⋊Q₈

—

502

—

C₄×Q₈

—

502

—

#117

—

251

—

C₂×C₄²

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

#95

—

502

—

C₂×Q₈

—

502

—

C₂²×C₄

—

502

—

C₂²×C₄

D₁₀₀₄

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

251

—

C₂²⋊C₄

—

502

—

C₄²

—

502

—

C₄²

—

502

—

C₄²

—

251

—

C₄²

C₄×D₂₅₁

yes

characteristic

C₄²

C₄×D₂₅₁

yes

characteristic

C₄²

C₂²⋊D₂₅₁

yes

—

C₂³

C₂×D₁₀₀₄

yes

characteristic

Q₈

—

1004

—

C₂×C₄

—

C₂×C₄

—

1004

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

C₂×C₄

Q₈×D₂₅₁

yes

—

100

C₂×C₄

#16860809

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

104

C₂²

#22548695

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1004

—

108

C₄

—

1004

—

109

C₄

#22548680

yes

—

110

C₄

#22548702

yes

characteristic

111

C₄

#22548713

yes

characteristic

112

C₂

—

113

C₂

#30631592

yes

characteristic

114

C₂

#30631703

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#423×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₅₁	C₂×D₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#1995
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#423×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₅₁

C₂×D₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #22548715	C₄	semidirect
6. #22548707	C₄	semidirect
23. C₄×Dic₂₅₁	Q₈	semidirect
57. C₂₅₁	#1995	semidirect

H ≅ G/N

type

5. #22548715

C₄

semidirect

6. #22548707

C₄

semidirect

23. C₄×Dic₂₅₁

Q₈

semidirect

57. C₂₅₁

#1995

semidirect

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

subgroup	index
#30632244	2
#30631533	2
#30631448	2
#423×C₂₅₁	2
#1995	251

subgroup

index

#423×C₂₅₁

251

Subgroups — #44551345

Conjugacy classes of subgroups

Complements

Maximal subgroups