Group #44551186 — subgroups

Subgroups — #44551186

Order 32128.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#22548639	C₂	1	4	yes	maximal
2	C₂×#22548626	C₂	1	1	yes	maximal, characteristic
3	#30631553	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
7	#22548639	—	2	8	no	—
8	#22548626	—	2	2	no	—
9	C₂₅₁⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₂₅₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₅₁	—	2	2	no	—
12	C₄⋊Q₈×C₂₅₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₅₁	—	2	2	no	—
14	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₅₁	—	2	2	no	—
16	C₂×C₂₅₁⋊C₈	—	2	2	no	—
17	C₂×C₂₅₁⋊C₈	—	2	2	no	—
18	C₂×C₂₅₁⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₂₅₁	—	2	4	no	—
20	C₂×Q₈×C₂₅₁	—	2	2	no	—
21	C₂×Q₈×C₂₅₁	—	2	4	no	—
22	C₂×Q₈×C₂₅₁	—	2	4	no	—
23	C₂×Q₈×C₂₅₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₅₁	—	2	2	no	—
25	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
26	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₅₁	—	2	8	no	—
28	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
29	C₄²×C₂₅₁	D₄	1	2	yes	—
30	C₂×Dic₂₅₁	—	4	2	no	—
31	C₂×Dic₂₅₁	—	4	4	no	—
32	C₂₅₁⋊C₈	—	2	4	no	—
33	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₂₅₁	—	2	8	no	—
35	Q₈×C₂₅₁	—	4	4	no	—
36	C₂×C₄×C₂₅₁	—	2	4	no	—
37	C₂×C₄×C₂₅₁	—	2	4	no	—
38	C₂×C₄×C₂₅₁	—	4	2	no	—
39	C₂×C₄×C₂₅₁	—	4	2	no	—
40	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₅₁	#41	1	2	yes	—
45	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₅₁	#124	1	1	yes	characteristic
47	C₂²×C₂₅₁	#127	1	1	yes	characteristic
48	C₂²×C₂₅₁	#137	1	4	yes	—
49	C₄×C₂₅₁	—	2	4	no	—
50	C₄×C₂₅₁	—	4	4	no	—
51	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
52	C₄×C₂₅₁	#121	1	2	yes	—
53	Dic₂₅₁	—	8	2	no	—
54	C₂×C₂₅₁	#390	1	1	yes	characteristic
55	C₂×C₂₅₁	#419	1	2	yes	—
56	C₂×C₂₅₁	#491	1	2	yes	—
57	C₂×C₂₅₁	#497	1	2	yes	—
58	C₂₅₁	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	251	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₂₅₁	1	1	yes	characteristic
61	C₂×#103	—	251	4	no	—
62	C₂×#97	—	251	1	no	—
63	#382	—	251	1	no	—
64	C₂²×Q₈	C₂×D₂₅₁	1	2	yes	—
65	C₂²×C₈	—	251	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₂₅₁	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	251	2	no	—
70	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
71	#103	—	502	8	no	—
72	#97	—	502	2	no	—
73	#95	—	502	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₂₅₁	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	502	2	no	—
80	C₂²×C₄	C₂²⋊D₂₅₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
82	C₂×C₈	—	502	2	no	—
83	C₂×C₈	—	502	2	no	—
84	C₂×C₈	—	251	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	502	4	no	—
87	C₄²	C₂×Dic₂₅₁	1	2	yes	—
88	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	1004	4	no	—
97	C₂×C₄	—	1004	2	no	—
98	C₂×C₄	C₂²⋊Dic₂₅₁	1	4	yes	—
99	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
102	C₂×C₄	#16860809	1	2	yes	—
103	C₈	—	502	4	no	—
104	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
105	C₂²	#22548702	1	1	yes	characteristic
106	C₂²	#22548711	1	1	yes	characteristic
107	C₂²	#22548746	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	2008	2	no	—
111	C₄	D₄×Dic₂₅₁	1	2	yes	—
112	C₄	#22548686	1	2	yes	—
113	C₂	#30631590	1	1	yes	characteristic
114	C₂	#30631696	1	2	yes	—
115	C₂	#30632055	1	2	yes	—
116	C₂	#30632079	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548639

C₂

yes

maximal

C₂×#22548626

C₂

yes

maximal, characteristic

#30631553

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

#22548639

—

#22548626

—

C₂₅₁⋊#95

—

C₂²×Q₈×C₂₅₁

C₂²

yes

—

C₄⋊Q₈×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂²×Dic₂₅₁

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

—

C₄²×C₂₅₁

D₄

yes

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊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

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

#121

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

#390

yes

characteristic

C₂×C₂₅₁

#419

yes

—

C₂×C₂₅₁

#491

yes

—

C₂×C₂₅₁

#497

yes

—

C₂₅₁

#1960

yes

characteristic, complemented, Sylow

#1960

—

251

maximal, Sylow

C₂×C₄⋊Q₈

D₂₅₁

yes

characteristic

C₂×#103

—

251

—

C₂×#97

—

251

—

#382

—

251

—

C₂²×Q₈

C₂×D₂₅₁

yes

—

C₂²×C₈

—

251

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₅₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#103

—

502

—

#97

—

502

—

#95

—

502

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₅₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

502

—

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂×C₈

—

502

—

C₂×C₈

—

502

—

C₂×C₈

—

251

—

C₄⋊C₄

—

C₄⋊C₄

—

502

—

C₄²

C₂×Dic₂₅₁

yes

—

C₄²

C₂²⋊D₂₅₁

yes

—

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1004

—

C₂×C₄

—

1004

—

C₂×C₄

C₂²⋊Dic₂₅₁

yes

—

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

100

C₂×C₄

D₄×D₂₅₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

102

C₂×C₄

#16860809

yes

—

103

C₈

—

502

—

104

C₂²

C₂×C₂²⋊Dic₂₅₁

yes

characteristic

105

C₂²

#22548702

yes

characteristic

106

C₂²

#22548711

yes

characteristic

107

C₂²

#22548746

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

2008

—

111

C₄

D₄×Dic₂₅₁

yes

—

112

C₄

#22548686

yes

—

113

C₂

#30631590

yes

characteristic

114

C₂

#30631696

yes

—

115

C₂

#30632055

yes

—

116

C₂

#30632079

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×C₄⋊Q₈×C₂₅₁	C₂
socle, prime radical	C₂³×C₂₅₁	C₂×D₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₅₁
hypocenter	C₂₅₁	#1960
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₂×C₄⋊Q₈×C₂₅₁

C₂

socle, prime radical

C₂³×C₂₅₁

C₂×D₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₅₁

hypocenter

C₂₅₁

#1960

LCS

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₂₅₁	C₄	semidirect
58. C₂₅₁	#1960	semidirect
67. C₄⋊Q₈	Dic₂₅₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₅₁

C₄

semidirect

58. C₂₅₁

#1960

semidirect

67. C₄⋊Q₈

Dic₂₅₁

semidirect

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

subgroup	index
C₂×#22548639	2
C₂×#22548626	2
#30631553	2
C₂×C₄⋊Q₈×C₂₅₁	2
#1960	251

subgroup

index

C₂×C₄⋊Q₈×C₂₅₁

251

Subgroups — #44551186

Conjugacy classes of subgroups

Complements

Maximal subgroups