Group #49095210 — subgroups

Subgroups — #49095210

Order 35968.

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

normal subgroups only

This group has 220 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×#22582814	C₂	1	2	yes	maximal
2	C₄×#17985122	C₂	1	1	yes	maximal, characteristic
3	C₂×#22582790	C₂	1	1	yes	maximal, characteristic
4	#31823227	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#17985122	C₂²	1	2	yes	—
7	C₂²×#14555886	C₂²	1	1	yes	characteristic
8	C₂²×#14555886	C₂²	1	1	yes	characteristic
9	#22582814	C₄	1	8	yes	complemented
10	#22582790	—	2	1	no	—
11	#22582790	C₄	1	4	yes	—
12	C₄×#14555886	—	2	1	no	—
13	C₂×M₁₆×C₂₈₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
15	#97×C₂₈₁	—	2	1	no	—
16	C₄×C₈×C₂₈₁	—	2	1	no	—
17	#17985122	—	2	2	no	—
18	#17985122	C₂×C₄	1	4	yes	—
19	C₂×#14555886	—	2	2	no	—
20	C₂×#14555886	—	2	2	no	—
21	C₂×#14555886	C₂×C₄	1	2	yes	—
22	C₂×#14555886	C₂×C₄	1	2	yes	—
23	C₂×#14555886	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₈₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₈₁	—	2	2	no	—
27	M₁₆×C₂₈₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₈₁	—	2	2	no	—
29	C₄²×C₂₈₁	C₂×C₄	1	2	yes	—
30	C₄²×C₂₈₁	D₄	1	1	yes	characteristic
31	C₄²×C₂₈₁	D₄	1	1	yes	characteristic
32	#14555886	—	2	2	no	—
33	#14555886	—	2	4	no	—
34	#14555886	C₄²	1	4	yes	—
35	C₂³×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₈₁	—	4	2	no	—
37	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₈₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₈₁	#41	1	2	yes	—
43	C₈×C₂₈₁	—	2	4	no	—
44	C₂²×C₂₈₁	—	2	2	no	—
45	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₈₁	C₄×Q₈	1	2	yes	—
47	C₄×C₂₈₁	—	2	2	no	—
48	C₄×C₂₈₁	—	2	2	no	—
49	C₄×C₂₈₁	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₂₈₁	#117	1	1	yes	characteristic
51	C₄×C₂₈₁	C₄×D₄	1	2	yes	—
52	C₂×C₂₈₁	—	2	2	no	—
53	C₂×C₂₈₁	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₂₈₁	#430	1	2	yes	—
55	C₂₈₁	#1778	1	1	yes	characteristic, complemented, Sylow
56	#1778	—	281	1	no	maximal, Sylow
57	C₂×#108	—	281	2	no	—
58	C₄×M₁₆	—	281	1	no	—
59	C₄×M₁₆	D₂₈₁	1	1	yes	characteristic
60	C₂×#97	—	281	1	no	—
61	#341	—	281	2	no	—
62	C₂×M₁₆	—	281	2	no	—
63	C₂×M₁₆	C₂×D₂₈₁	1	2	yes	—
64	C₂²×C₈	—	281	1	no	—
65	C₂²×C₈	—	281	1	no	—
66	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
67	#108	—	281	8	no	—
68	#97	—	2	1	no	—
69	#97	—	281	4	no	—
70	#97	—	562	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	562	1	no	—
73	C₂²×C₄	Dic₅₆₂	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	281	4	no	—
77	M₁₆	—	562	2	no	—
78	M₁₆	C₄×D₂₈₁	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	281	2	no	—
81	C₂×C₈	—	281	4	no	—
82	C₂×C₈	—	281	2	no	—
83	C₂×C₈	—	562	2	no	—
84	C₂×C₈	—	562	2	no	—
85	C₄²	C₄×D₂₈₁	1	2	yes	—
86	C₄²	C₂²⋊D₂₈₁	1	1	yes	characteristic
87	C₄²	C₂²⋊D₂₈₁	1	1	yes	characteristic
88	C₂³	C₂×Dic₅₆₂	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#17985117	1	2	yes	—
91	C₂×C₄	#17985117	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₂₈₁	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
95	C₂×C₄	#17985138	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	281	4	no	—
98	C₈	—	562	2	no	—
99	C₈	—	562	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#17985117	1	1	yes	characteristic
102	C₂²	C₄×Dic₅₆₂	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₂₈₁	1	1	yes	characteristic
106	C₄	#22582836	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₂₈₁	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#17985117	1	1	yes	characteristic
110	C₂	#31823511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582814

C₂

yes

maximal

C₄×#17985122

C₂

yes

maximal, characteristic

C₂×#22582790

C₂

yes

maximal, characteristic

#31823227

C₂

yes

maximal

C₄×M₁₆×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985122

C₂²

yes

—

C₂²×#14555886

C₂²

yes

characteristic

C₂²×#14555886

C₂²

yes

characteristic

#22582814

C₄

yes

complemented

#22582790

—

#22582790

C₄

yes

—

C₄×#14555886

—

C₂×M₁₆×C₂₈₁

C₂²

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

#97×C₂₈₁

—

C₄×C₈×C₂₈₁

—

#17985122

—

#17985122

C₂×C₄

yes

—

C₂×#14555886

—

C₂×#14555886

—

C₂×#14555886

C₂×C₄

yes

—

C₂×#14555886

C₂×C₄

yes

—

C₂×#14555886

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

Q₈

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

M₁₆×C₂₈₁

—

M₁₆×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

—

C₄²×C₂₈₁

C₂×C₄

yes

—

C₄²×C₂₈₁

D₄

yes

characteristic

C₄²×C₂₈₁

D₄

yes

—

—

C₄²

yes

—

C₂³×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂²×C₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

—

C₈×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₈₁

C₄×Q₈

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×C₄²

yes

characteristic

C₄×C₂₈₁

#117

yes

characteristic

C₄×C₂₈₁

C₄×D₄

yes

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₈₁

#430

yes

—

C₂₈₁

#1778

yes

characteristic, complemented, Sylow

#1778

—

281

maximal, Sylow

C₂×#108

—

281

—

C₄×M₁₆

—

281

—

C₄×M₁₆

D₂₈₁

yes

characteristic

C₂×#97

—

281

—

#341

—

281

—

C₂×M₁₆

—

281

—

C₂×M₁₆

C₂×D₂₈₁

yes

—

C₂²×C₈

—

281

—

C₂²×C₈

—

281

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

#108

—

281

—

#97

—

#97

—

281

—

#97

—

562

—

C₄×C₈

—

C₄×C₈

—

562

—

C₂²×C₄

Dic₅₆₂

yes

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

M₁₆

—

281

—

M₁₆

—

562

—

M₁₆

C₄×D₂₈₁

yes

—

C₂×C₈

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₄²

C₄×D₂₈₁

yes

—

C₄²

C₂²⋊D₂₈₁

yes

characteristic

C₄²

C₂²⋊D₂₈₁

yes

characteristic

C₂³

C₂×Dic₅₆₂

yes

characteristic

C₂×C₄

—

C₂×C₄

#17985117

yes

—

C₂×C₄

#17985117

yes

—

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

C₂×C₄

C₂×C₄×D₂₈₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

C₂×C₄

#17985138

yes

—

C₈

—

C₈

—

281

—

C₈

—

562

—

C₈

—

562

—

100

C₂²

—

101

C₂²

C₂×#17985117

yes

characteristic

102

C₂²

C₄×Dic₅₆₂

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₂₈₁

yes

characteristic

106

C₄

#22582836

yes

characteristic

107

C₄

C₄×C₂²⋊D₂₈₁

yes

—

108

C₂

—

109

C₂

C₄×#17985117

yes

characteristic

110

C₂

#31823511

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₈₁
commutator	C₄×C₂₈₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₄×M₁₆×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₈₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₄×M₁₆	D₂₈₁
hypocenter	C₂₈₁	#1778
LCS	G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₈₁

commutator

C₄×C₂₈₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₄×M₁₆×C₂₈₁

C₂

socle

C₂²×C₂₈₁

C₂×C₄⋊C₄

prime radical

C₂³×C₂₈₁

C₂×Q₈

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₄×M₁₆

D₂₈₁

hypocenter

C₂₈₁

#1778

LCS

G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #22582814	C₄	semidirect
55. C₂₈₁	#1778	semidirect

H ≅ G/N

type

9. #22582814

C₄

semidirect

55. C₂₈₁

#1778

semidirect

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

subgroup	index
C₂×#22582814	2
C₄×#17985122	2
C₂×#22582790	2
#31823227	2
C₄×M₁₆×C₂₈₁	2
#1778	281

subgroup

index

C₄×M₁₆×C₂₈₁

281

Subgroups — #49095210

Conjugacy classes of subgroups

Complements

Maximal subgroups