Group #44549089 — subgroups

Subgroups — #44549089

Order 32128.

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄⋊Q₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#30631743	C₂	1	2	yes	maximal
3	#30631436	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631435	C₂	1	2	yes	maximal, complemented
5	#333×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₂₅₁	C₄	1	2	yes	—
7	#22548715	C₂²	1	1	yes	characteristic
8	#22548714	—	2	2	no	—
9	#22548713	C₄	1	2	yes	—
10	C₄⋊C₄×D₂₅₁	—	2	2	no	—
11	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
12	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
13	(C₂×D₂₅₁)⋊C₈	—	2	2	no	—
14	#22548625	—	2	2	no	—
15	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
17	C₂×Dic₅₀₂	—	2	2	no	—
18	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
19	Q₈×D₂₅₁	—	4	4	no	—
20	C₂×C₄×D₂₅₁	—	2	2	no	—
21	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₂₅₁	D₄	1	2	yes	—
23	C₂×C₂₅₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₂₅₁	—	2	2	no	—
25	Dic₂₅₁⋊C₄	—	2	4	no	—
26	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₂₅₁	D₄	1	2	yes	—
28	C₂×Q₈×C₂₅₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₅₁	—	2	2	no	—
30	C₄⋊C₄×C₂₅₁	—	2	2	no	—
31	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
32	C₂²×D₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₅₀₂	—	4	4	no	—
34	Dic₅₀₂	—	4	2	no	—
35	C₂×Dic₂₅₁	—	2	2	no	—
36	C₂×Dic₂₅₁	C₂²⋊C₄	1	2	yes	—
37	C₂×Dic₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₂₅₁	—	2	2	no	—
39	C₄×D₂₅₁	—	2	2	no	—
40	C₄×D₂₅₁	—	2	2	no	—
41	C₄×D₂₅₁	—	4	4	no	—
42	C₂₅₁⋊C₈	—	4	2	no	—
43	Q₈×C₂₅₁	—	4	2	no	—
44	C₂×C₄×C₂₅₁	—	2	2	no	—
45	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
47	C₈×C₂₅₁	—	4	2	no	—
48	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₂₅₁	—	2	2	no	—
50	C₂×D₂₅₁	#100	1	2	yes	—
51	C₄×C₂₅₁	—	2	1	no	—
52	C₄×C₂₅₁	—	4	2	no	—
53	C₄×C₂₅₁	#137	1	2	yes	—
54	C₄×C₂₅₁	#137	1	2	yes	—
55	Dic₂₅₁	—	2	1	no	—
56	Dic₂₅₁	—	2	2	no	—
57	Dic₂₅₁	—	4	2	no	—
58	C₂×C₂₅₁	C₂×#100	1	1	yes	characteristic
59	C₂×C₂₅₁	#419	1	2	yes	—
60	D₂₅₁	—	2	2	no	—
61	C₂₅₁	#1558	1	1	yes	characteristic, complemented, Sylow
62	#1558	—	251	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	—	251	1	no	—
64	#432	—	251	2	no	—
65	#333	—	251	1	no	—
66	#333	—	251	2	no	—
67	#333	D₂₅₁	1	1	yes	characteristic, complemented
68	C₂²×Q₈	—	251	2	no	—
69	C₄⋊Q₈	—	502	2	no	—
70	C₄⋊Q₈	—	251	2	no	—
71	C₄⋊Q₈	—	251	1	no	—
72	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	502	2	no	—
74	C₂×C₄²	—	251	1	no	—
75	C₄⋊C₈	—	251	2	no	—
76	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
77	C₂²⋊C₈	—	502	2	no	—
78	#97	—	502	2	no	—
79	C₂×Q₈	—	1004	4	no	—
80	C₂×Q₈	—	502	2	no	—
81	C₂×Q₈	—	251	2	no	—
82	C₂×Q₈	C₄×D₂₅₁	1	2	yes	—
83	C₂²×C₄	—	502	2	no	—
84	C₂²×C₄	—	251	1	no	—
85	C₂²×C₄	—	251	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	502	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	502	4	no	—
90	C₄⋊C₄	—	502	2	no	—
91	C₄²	—	251	1	no	—
92	C₄²	—	251	2	no	—
93	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
94	C₂³	—	251	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	1004	2	no	—
97	Q₈	—	1004	4	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	1004	4	no	—
100	C₂×C₄	—	502	2	no	—
101	C₂×C₄	—	502	2	no	—
102	C₂×C₄	—	502	2	no	—
103	C₂×C₄	—	502	2	no	—
104	C₂×C₄	—	251	2	no	—
105	C₂×C₄	—	251	1	no	—
106	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
108	C₈	—	4	2	no	—
109	C₈	—	1004	2	no	—
110	C₂²	—	502	2	no	—
111	C₂²	—	251	2	no	—
112	C₂²	C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	4	2	no	—
115	C₄	—	1004	2	no	—
116	C₄	—	502	2	no	—
117	C₄	—	502	1	no	—
118	C₄	#22548744	1	2	yes	—
119	C₄	#22548747	1	2	yes	—
120	C₂	—	502	2	no	—
121	C₂	#100×D₂₅₁	1	1	yes	characteristic
122	C₂	#30631694	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₂₅₁

C₂

yes

maximal, characteristic

#30631743

C₂

yes

maximal

#30631436

C₂

yes

maximal, characteristic, complemented

#30631435

C₂

yes

maximal, complemented

#333×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₂₅₁

C₄

yes

—

#22548715

C₂²

yes

characteristic

#22548714

—

#22548713

C₄

yes

—

C₄⋊C₄×D₂₅₁

—

C₄²×D₂₅₁

C₂²

yes

characteristic

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

(C₂×D₂₅₁)⋊C₈

—

#22548625

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

Q₈×D₂₅₁

—

C₂×C₄×D₂₅₁

—

C₂×C₄×D₂₅₁

C₂×C₄

yes

characteristic

C₂×C₄×D₂₅₁

D₄

yes

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

C₂×C₄

yes

characteristic

C₄×Dic₂₅₁

D₄

yes

—

C₂×Q₈×C₂₅₁

C₂×C₄

yes

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

C₂²⋊C₄

yes

characteristic

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂²⋊C₄

yes

—

C₂×Dic₂₅₁

C₂²⋊C₄

yes

characteristic

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

#100

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#137

yes

—

C₄×C₂₅₁

#137

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

C₂×#100

yes

characteristic

C₂×C₂₅₁

#419

yes

—

D₂₅₁

—

C₂₅₁

#1558

yes

characteristic, complemented, Sylow

#1558

—

251

maximal, Sylow

C₂×C₄⋊Q₈

—

251

—

#432

—

251

—

#333

—

251

—

#333

—

251

—

#333

D₂₅₁

yes

characteristic, complemented

C₂²×Q₈

—

251

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₂×C₄⋊C₄

—

502

—

C₂×C₄²

—

251

—

C₄⋊C₈

—

251

—

C₄⋊C₈

C₂×D₂₅₁

yes

—

C₂²⋊C₈

—

502

—

#97

—

502

—

C₂×Q₈

—

1004

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

C₂×Q₈

C₄×D₂₅₁

yes

—

C₂²×C₄

—

502

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄²

—

251

—

C₄²

—

251

—

C₄²

C₂²×D₂₅₁

yes

characteristic

C₂³

—

251

—

Q₈

—

Q₈

—

1004

—

Q₈

—

1004

—

C₂×C₄

—

C₂×C₄

—

1004

—

100

C₂×C₄

—

502

—

101

C₂×C₄

—

502

—

102

C₂×C₄

—

502

—

103

C₂×C₄

—

502

—

104

C₂×C₄

—

251

—

105

C₂×C₄

—

251

—

106

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

107

C₂×C₄

D₄×D₂₅₁

yes

—

108

C₈

—

109

C₈

—

1004

—

110

C₂²

—

502

—

111

C₂²

—

251

—

112

C₂²

C₂²⋊C₄×D₂₅₁

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

1004

—

116

C₄

—

502

—

117

C₄

—

502

—

118

C₄

#22548744

yes

—

119

C₄

#22548747

yes

—

120

C₂

—

502

—

121

C₂

#100×D₂₅₁

yes

characteristic

122

C₂

#30631694

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#333×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₅₁	C₂²⋊C₄
cosocle	C₄²×C₂₅₁	C₂³
hypercenter	#333	D₂₅₁
hypocenter	C₂₅₁	#1558
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#333×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₅₁

C₂²⋊C₄

cosocle

C₄²×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
3. #30631436	C₂	semidirect
4. #30631435	C₂	semidirect
5. #333×C₂₅₁	C₂	semidirect
61. C₂₅₁	#1558	semidirect
67. #333	D₂₅₁	semidirect

H ≅ G/N

type

3. #30631436

C₂

semidirect

4. #30631435

C₂

semidirect

5. #333×C₂₅₁

C₂

semidirect

61. C₂₅₁

#1558

semidirect

67. #333

D₂₅₁

semidirect

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

subgroup	index
C₄⋊Q₈×D₂₅₁	2
#30631743	2
#30631436	2
#30631435	2
#333×C₂₅₁	2
#1558	251

subgroup

index

C₄⋊Q₈×D₂₅₁

#333×C₂₅₁

251

Subgroups — #44549089

Conjugacy classes of subgroups

Complements

Maximal subgroups