Group #49095058 — subgroups

Subgroups — #49095058

Order 35968.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#31823418	C₂	1	2	yes	maximal
2	#31823221	C₂	1	1	yes	maximal, characteristic
3	#31823220	C₂	1	1	yes	maximal, characteristic
4	#31823219	C₂	1	2	yes	maximal
5	#405×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#17985122	C₂²	1	2	yes	—
7	#22582838	C₂²	1	2	yes	—
8	#22582789	C₄	1	4	yes	complemented
9	C₈×Dic₂₈₁	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₂₈₁	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
12	#117×C₂₈₁	C₂²	1	1	yes	characteristic
13	#97×C₂₈₁	C₄	1	2	yes	complemented
14	C₄×C₈×C₂₈₁	C₄	1	2	yes	complemented
15	#17985122	—	2	2	no	—
16	#17985122	—	2	2	no	—
17	C₂×#14555886	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₂₈₁	—	2	2	no	—
19	C₂²⋊Dic₂₈₁	—	2	2	no	—
20	C₄×Dic₂₈₁	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₂₈₁	—	2	2	no	—
23	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₂₈₁	—	2	1	no	—
29	C₂²⋊C₄×C₂₈₁	—	2	1	no	—
30	C₄²×C₂₈₁	C₂×C₄	1	2	yes	—
31	C₂×Dic₂₈₁	—	2	4	no	—
32	#14555886	—	2	4	no	—
33	C₂³×C₂₈₁	#41	1	1	yes	characteristic
34	C₂×C₄×C₂₈₁	—	2	2	no	—
35	C₂×C₄×C₂₈₁	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
38	C₂×C₄×C₂₈₁	#41	1	2	yes	—
39	C₈×C₂₈₁	—	2	2	no	—
40	C₈×C₂₈₁	C₄²	1	2	yes	—
41	C₈×C₂₈₁	C₄²	1	1	yes	characteristic
42	C₈×C₂₈₁	C₄²	1	1	yes	characteristic
43	C₂²×C₂₈₁	—	4	1	no	—
44	C₂²×C₂₈₁	#117	1	2	yes	—
45	C₂²×C₂₈₁	#117	1	1	yes	characteristic
46	C₄×C₂₈₁	—	4	2	no	—
47	C₄×C₂₈₁	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₂₈₁	#117	1	2	yes	—
49	C₄×C₂₈₁	#117	1	1	yes	characteristic
50	Dic₂₈₁	—	4	4	no	—
51	C₂×C₂₈₁	—	2	2	no	—
52	C₂×C₂₈₁	—	2	1	no	—
53	C₂×C₂₈₁	#376	1	1	yes	characteristic
54	C₂₈₁	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	281	1	no	maximal, Sylow
56	#405	—	281	2	no	—
57	#405	D₂₈₁	1	1	yes	characteristic
58	#338	—	281	1	no	—
59	H(3,ℤ/4)	—	281	1	no	—
60	H(3,ℤ/4)	—	281	2	no	—
61	C₂×M₁₆	—	281	2	no	—
62	C₂×M₁₆	C₂×D₂₈₁	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
64	#117	—	281	2	no	—
65	#117	C₂×D₂₈₁	1	1	yes	characteristic
66	#97	—	281	4	no	—
67	#97	Dic₂₈₁	1	2	yes	complemented
68	C₄×C₈	—	281	4	no	—
69	C₄×C₈	Dic₂₈₁	1	2	yes	complemented
70	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	562	2	no	—
73	M₁₆	—	562	2	no	—
74	C₂×C₈	—	281	4	no	—
75	C₂×C₈	C₄×D₂₈₁	1	2	yes	—
76	C₂×C₈	C₄×D₂₈₁	1	2	yes	—
77	C₂×C₈	C₂×Dic₂₈₁	1	2	yes	—
78	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	562	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	562	2	no	—
84	C₄²	—	281	4	no	—
85	C₄²	C₂×Dic₂₈₁	1	2	yes	—
86	C₂³	#17985138	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	562	4	no	—
89	C₂×C₄	C₂×C₄×D₂₈₁	1	2	yes	—
90	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
91	C₂×C₄	#17985138	1	1	yes	characteristic
92	C₂×C₄	#17985138	1	2	yes	—
93	C₈	—	2	2	no	—
94	C₈	—	562	4	no	—
95	C₈	C₄×Dic₂₈₁	1	1	yes	characteristic
96	C₈	C₄×Dic₂₈₁	1	2	yes	—
97	C₈	C₄×Dic₂₈₁	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#22582836	1	2	yes	—
100	C₂²	#22582838	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	1124	4	no	—
103	C₄	C₂×C₄×Dic₂₈₁	1	1	yes	characteristic
104	C₄	#22582836	1	2	yes	—
105	C₄	#22582838	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#31823309	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823418

C₂

yes

maximal

#31823221

C₂

yes

maximal, characteristic

#31823220

C₂

yes

maximal, characteristic

#31823219

C₂

yes

maximal

#405×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985122

C₂²

yes

—

#22582838

C₂²

yes

—

#22582789

C₄

yes

complemented

C₈×Dic₂₈₁

C₄

yes

complemented

C₂×M₁₆×C₂₈₁

C₂²

yes

characteristic

C₂²×C₈×C₂₈₁

C₂²

yes

characteristic

#117×C₂₈₁

C₂²

yes

characteristic

#97×C₂₈₁

C₄

yes

complemented

C₄×C₈×C₂₈₁

C₄

yes

—

—

C₂×C₄

yes

—

C₄⋊Dic₂₈₁

—

C₂²⋊Dic₂₈₁

—

C₄×Dic₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

M₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₈₁

—

C₂²⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

C₂×C₄

yes

—

C₂×Dic₂₈₁

—

#14555886

—

C₂³×C₂₈₁

#41

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

C₂×C₄×C₂₈₁

yes

C₂×C₄×C₂₈₁

yes

—

C₈×C₂₈₁

—

C₈×C₂₈₁

C₄²

yes

—

C₈×C₂₈₁

C₄²

yes

characteristic

C₈×C₂₈₁

C₄²

yes

characteristic

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#117

yes

—

C₂²×C₂₈₁

#117

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×C₄²

yes

characteristic

C₄×C₂₈₁

#117

yes

—

C₄×C₂₈₁

#117

yes

characteristic

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#376

yes

characteristic

C₂₈₁

#1747

yes

characteristic, complemented, Sylow

#1747

—

281

maximal, Sylow

#405

—

281

—

#405

D₂₈₁

yes

characteristic

#338

—

281

—

H(3,ℤ/4)

—

281

—

H(3,ℤ/4)

—

281

—

C₂×M₁₆

—

281

—

C₂×M₁₆

C₂×D₂₈₁

yes

characteristic

C₂²×C₈

C₂×D₂₈₁

yes

characteristic

#117

—

281

—

#117

C₂×D₂₈₁

yes

characteristic

#97

—

281

—

#97

Dic₂₈₁

yes

complemented

C₄×C₈

—

281

—

C₄×C₈

Dic₂₈₁

yes

complemented

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

M₁₆

—

562

—

M₁₆

—

562

—

C₂×C₈

—

281

—

C₂×C₈

C₄×D₂₈₁

yes

—

C₂×C₈

C₄×D₂₈₁

yes

—

C₂×C₈

C₂×Dic₂₈₁

yes

—

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

562

—

C₂²⋊C₄

—

C₂²⋊C₄

—

562

—

C₄²

—

281

—

C₄²

C₂×Dic₂₈₁

yes

—

C₂³

#17985138

yes

characteristic

C₂×C₄

—

C₂×C₄

—

562

—

C₂×C₄

C₂×C₄×D₂₈₁

yes

—

C₂×C₄

C₂²×Dic₂₈₁

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

562

—

C₈

C₄×Dic₂₈₁

yes

characteristic

C₈

C₄×Dic₂₈₁

yes

—

C₈

C₄×Dic₂₈₁

yes

characteristic

C₂²

—

C₂²

#22582836

yes

—

100

C₂²

#22582838

yes

characteristic

101

C₄

—

102

C₄

—

1124

—

103

C₄

C₂×C₄×Dic₂₈₁

yes

characteristic

104

C₄

#22582836

yes

—

105

C₄

#22582838

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#31823309

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₂₈₁
commutator	C₄×C₂₈₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	#405×C₂₈₁	C₂
socle	C₂×C₂₈₁	#376
prime radical	C₂³×C₂₈₁	#41
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	#405	D₂₈₁
hypocenter	C₂₈₁	#1747
LCS	G ▷ C₄×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₂₈₁

commutator

C₄×C₂₈₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

#405×C₂₈₁

C₂

C₂×C₂₈₁

C₂³×C₂₈₁

C₂²×C₄×C₂₈₁

C₂³

D₂₈₁

C₂₈₁

G ▷ C₄×C₂₈₁ ▷ C₂₈₁

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #22582789	C₄	semidirect
9. C₈×Dic₂₈₁	C₄	semidirect
13. #97×C₂₈₁	C₄	semidirect
14. C₄×C₈×C₂₈₁	C₄	semidirect
54. C₂₈₁	#1747	semidirect
67. #97	Dic₂₈₁	semidirect
69. C₄×C₈	Dic₂₈₁	semidirect

H ≅ G/N

type

8. #22582789

C₄

semidirect

9. C₈×Dic₂₈₁

C₄

semidirect

13. #97×C₂₈₁

C₄

semidirect

14. C₄×C₈×C₂₈₁

C₄

semidirect

54. C₂₈₁

#1747

semidirect

67. #97

Dic₂₈₁

semidirect

69. C₄×C₈

Dic₂₈₁

semidirect

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

subgroup	index
#31823418	2
#31823221	2
#31823220	2
#31823219	2
#405×C₂₈₁	2
#1747	281

subgroup

index

#405×C₂₈₁

281

Subgroups — #49095058

Conjugacy classes of subgroups

Complements

Maximal subgroups