Group #49096091 — subgroups

Subgroups — #49096091

Order 35968.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#31823511	C₂	1	2	yes	maximal
2	C₂×#22582815	C₂	1	1	yes	maximal, characteristic
3	#31823418	C₂	1	1	yes	maximal, characteristic
4	#31823234	C₂	1	2	yes	maximal
5	C₂×#108×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#17985122	C₂²	1	2	yes	—
7	C₂×#17985122	C₂²	1	1	yes	characteristic
8	#22582838	C₂²	1	1	yes	characteristic
9	#22582815	C₄	1	4	yes	complemented
10	#22582814	—	2	4	no	—
11	#22582789	—	2	1	no	—
12	C₈×Dic₂₈₁	—	2	1	no	—
13	C₂×M₁₆×C₂₈₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
15	#108×C₂₈₁	—	2	1	no	—
16	#108×C₂₈₁	—	2	1	no	—
17	#17985122	—	2	1	no	—
18	#17985122	—	2	2	no	—
19	#17985122	C₂×C₄	1	4	yes	—
20	#17985122	D₄	1	2	yes	—
21	C₂×#14555886	—	2	1	no	—
22	C₂×#14555886	—	2	2	no	—
23	C₄⋊Dic₂₈₁	—	2	1	no	—
24	C₂²⋊Dic₂₈₁	Q₈	1	2	yes	—
25	C₄×Dic₂₈₁	—	2	1	no	—
26	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₈₁	—	2	2	no	—
28	M₁₆×C₂₈₁	—	2	2	no	—
29	C₂×C₈×C₂₈₁	—	2	2	no	—
30	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₂₈₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₈₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₈₁	Q₈	1	2	yes	—
34	C₂×Dic₂₈₁	—	2	2	no	—
35	#14555886	—	2	4	no	—
36	#14555886	—	2	2	no	—
37	C₂³×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
44	C₈×C₂₈₁	—	4	2	no	—
45	C₈×C₂₈₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₈₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₈₁	—	4	1	no	—
48	C₂²×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₂₈₁	C₄×D₄	1	1	yes	characteristic
53	C₄×C₂₈₁	#121	1	1	yes	characteristic
54	C₄×C₂₈₁	#125	1	1	yes	characteristic
55	Dic₂₈₁	—	4	2	no	—
56	C₂×C₂₈₁	—	2	1	no	—
57	C₂×C₂₈₁	—	2	1	no	—
58	C₂×C₂₈₁	—	2	1	no	—
59	C₂×C₂₈₁	#389	1	1	yes	characteristic
60	C₂₈₁	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	281	1	no	maximal, Sylow
62	#430	—	281	2	no	—
63	C₂×#108	—	281	1	no	—
64	C₂×#108	D₂₈₁	1	1	yes	characteristic
65	#405	—	281	1	no	—
66	#344	—	281	2	no	—
67	C₂×M₁₆	—	281	2	no	—
68	C₂×M₁₆	—	281	1	no	—
69	C₂×M₁₆	C₂×D₂₈₁	1	2	yes	—
70	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
71	#117	—	281	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	281	4	no	—
75	#108	—	562	4	no	—
76	#97	—	562	1	no	—
77	C₄×C₈	—	562	1	no	—
78	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	281	4	no	—
82	M₁₆	—	281	2	no	—
83	M₁₆	—	562	1	no	—
84	M₁₆	—	562	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	562	2	no	—
87	C₂×C₈	—	562	1	no	—
88	C₂×C₈	C₄×D₂₈₁	1	2	yes	—
89	C₂×C₈	C₂²⋊D₂₈₁	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₂₈₁	1	1	yes	characteristic
91	C₂×C₈	Dic₅₆₂	1	2	yes	—
92	C₄⋊C₄	—	562	1	no	—
93	C₂²⋊C₄	—	281	2	no	—
94	C₄²	—	562	1	no	—
95	C₂³	Q₈×D₂₈₁	1	1	yes	characteristic
96	C₂×C₄	—	562	2	no	—
97	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
98	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₅₆₂	1	1	yes	characteristic
101	C₂×C₄	#17985137	1	1	yes	characteristic
102	C₂×C₄	#17985139	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	562	4	no	—
105	C₈	—	562	2	no	—
106	C₈	#17985117	1	2	yes	—
107	C₈	#17985117	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#22582844	1	1	yes	characteristic
110	C₂²	#22582855	1	1	yes	characteristic
111	C₂²	#22582878	1	1	yes	characteristic
112	C₄	—	1124	2	no	—
113	C₄	C₂×#17985117	1	1	yes	characteristic
114	C₄	#22582841	1	1	yes	characteristic
115	C₄	#22582850	1	1	yes	characteristic
116	C₄	#22582870	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#31823359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823511

C₂

yes

maximal

C₂×#22582815

C₂

yes

maximal, characteristic

#31823418

C₂

yes

maximal, characteristic

#31823234

C₂

yes

maximal

C₂×#108×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985122

C₂²

yes

—

C₂×#17985122

C₂²

yes

characteristic

#22582838

C₂²

yes

characteristic

#22582815

C₄

yes

complemented

#22582814

—

#22582789

—

C₈×Dic₂₈₁

—

C₂×M₁₆×C₂₈₁

C₂²

yes

—

C₂²×C₈×C₂₈₁

C₂²

yes

characteristic

#108×C₂₈₁

—

#108×C₂₈₁

—

#17985122

—

#17985122

—

#17985122

C₂×C₄

yes

—

#17985122

D₄

yes

—

C₂×#14555886

—

C₂×#14555886

—

C₄⋊Dic₂₈₁

—

C₂²⋊Dic₂₈₁

Q₈

yes

—

C₄×Dic₂₈₁

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

M₁₆×C₂₈₁

—

M₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

D₄

yes

characteristic

C₂×C₈×C₂₈₁

D₄

yes

characteristic

C₂×C₈×C₂₈₁

Q₈

yes

—

C₂×Dic₂₈₁

—

#14555886

—

#14555886

—

C₂³×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

C₂×C₄×C₂₈₁

yes

C₂×C₄×C₂₈₁

yes

C₈×C₂₈₁

—

C₈×C₂₈₁

C₄⋊C₄

yes

—

C₈×C₂₈₁

C₄⋊C₄

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₄×Q₈

yes

characteristic

C₂²×C₂₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₈₁

C₄⋊Q₈

yes

characteristic

C₄×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₈₁

C₄×D₄

yes

C₄×C₂₈₁

yes

C₄×C₂₈₁

yes

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#389

yes

characteristic

C₂₈₁

#1953

yes

characteristic, complemented, Sylow

#1953

—

281

maximal, Sylow

#430

—

281

—

C₂×#108

—

281

—

C₂×#108

D₂₈₁

yes

characteristic

#405

—

281

—

#344

—

281

—

C₂×M₁₆

—

281

—

C₂×M₁₆

—

281

—

C₂×M₁₆

C₂×D₂₈₁

yes

—

C₂²×C₈

C₂×D₂₈₁

yes

characteristic

#117

—

281

—

#108

—

#108

—

#108

—

281

—

#108

—

562

—

#97

—

562

—

C₄×C₈

—

562

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

281

—

M₁₆

—

281

—

M₁₆

—

562

—

M₁₆

—

562

—

C₂×C₈

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

C₄×D₂₈₁

yes

—

C₂×C₈

C₂²⋊D₂₈₁

yes

characteristic

C₂×C₈

C₂²⋊D₂₈₁

yes

characteristic

C₂×C₈

Dic₅₆₂

yes

—

C₄⋊C₄

—

562

—

C₂²⋊C₄

—

281

—

C₄²

—

562

—

C₂³

Q₈×D₂₈₁

yes

characteristic

C₂×C₄

—

562

—

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

C₂×C₄

D₄×D₂₈₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

100

C₂×C₄

C₂×Dic₅₆₂

yes

characteristic

101

C₂×C₄

#17985137

yes

characteristic

102

C₂×C₄

#17985139

yes

characteristic

103

C₈

—

104

C₈

—

562

—

105

C₈

—

562

—

106

C₈

#17985117

yes

—

107

C₈

#17985117

yes

—

108

C₂²

—

109

C₂²

#22582844

yes

characteristic

110

C₂²

#22582855

yes

characteristic

111

C₂²

#22582878

yes

characteristic

112

C₄

—

1124

—

113

C₄

C₂×#17985117

yes

characteristic

114

C₄

#22582841

yes

characteristic

115

C₄

#22582850

yes

characteristic

116

C₄

#22582870

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#31823359

yes

characteristic

	subgroup	G/H
center	C₄	#22582850
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂×#108×C₂₈₁	C₂
socle	C₂×C₂₈₁	#389
prime radical	C₂³×C₂₈₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×#108	D₂₈₁
hypocenter	C₂₈₁	#1953
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#22582850

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂×#108×C₂₈₁

C₂

socle

C₂×C₂₈₁

#389

prime radical

C₂³×C₂₈₁

C₂×Q₈

cosocle

C₂²×C₄×C₂₈₁

C₂³

D₂₈₁

C₂₈₁

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #22582815	C₄	semidirect
60. C₂₈₁	#1953	semidirect

H ≅ G/N

type

9. #22582815

C₄

semidirect

60. C₂₈₁

#1953

semidirect

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

subgroup	index
#31823511	2
C₂×#22582815	2
#31823418	2
#31823234	2
C₂×#108×C₂₈₁	2
#1953	281

subgroup

index

C₂×#108×C₂₈₁

281

Subgroups — #49096091

Conjugacy classes of subgroups

Complements

Maximal subgroups