Group #49096052 — subgroups

Subgroups — #49096052

Order 35968.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22582815	C₂	1	1	yes	maximal, characteristic
2	#31823490	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₈₁	C₂	1	1	yes	maximal, characteristic
4	#31823222	C₂	1	2	yes	maximal
5	#427×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#17985122	C₂²	1	2	yes	—
7	C₂²×#14555886	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₈₁	C₂²	1	1	yes	characteristic
9	#22582815	—	2	2	no	—
10	C₄⋊#14555886	—	2	4	no	—
11	C₈×Dic₂₈₁	—	2	2	no	—
12	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
13	#117×C₂₈₁	C₂²	1	2	yes	—
14	#107×C₂₈₁	C₄	1	2	yes	complemented
15	#106×C₂₈₁	C₄	1	2	yes	complemented
16	C₂²×Dic₂₈₁	—	2	1	no	—
17	#17985122	—	2	2	no	—
18	#17985122	—	2	2	no	—
19	C₂×#14555886	—	2	1	no	—
20	C₂×#14555886	—	2	2	no	—
21	C₂×#14555886	—	2	1	no	—
22	C₂×#14555886	D₄	1	2	yes	—
23	C₄×Dic₂₈₁	—	2	1	no	—
24	C₄×Dic₂₈₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
29	C₂×C₈×C₂₈₁	Q₈	1	2	yes	—
30	C₄⋊C₄×C₂₈₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₂₈₁	—	2	2	no	—
32	C₄²×C₂₈₁	—	2	2	no	—
33	C₂×Dic₂₈₁	—	2	4	no	—
34	C₂×Dic₂₈₁	—	2	2	no	—
35	#14555886	—	2	2	no	—
36	#14555886	—	4	2	no	—
37	C₂³×C₂₈₁	#41	1	1	yes	characteristic
38	C₂×C₄×C₂₈₁	—	2	4	no	—
39	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
45	C₈×C₂₈₁	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₈₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₈₁	—	2	2	no	—
48	C₂²×C₂₈₁	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₈₁	#125	1	1	yes	characteristic
51	C₄×C₂₈₁	—	4	4	no	—
52	C₄×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₈₁	#121	1	1	yes	characteristic
55	C₄×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₂₈₁	—	2	4	no	—
57	C₂×C₂₈₁	—	2	1	no	—
58	C₂×C₂₈₁	—	2	1	no	—
59	C₂×C₂₈₁	#389	1	1	yes	characteristic
60	C₂×C₂₈₁	#443	1	2	yes	—
61	C₂₈₁	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	281	1	no	maximal, Sylow
63	C₂×#108	—	281	1	no	—
64	#427	D₂₈₁	1	1	yes	characteristic
65	#424	—	281	2	no	—
66	C₂×C₄×C₈	—	281	1	no	—
67	#339	—	281	2	no	—
68	C₂×M₁₆	—	281	2	no	—
69	C₂²×C₈	—	281	1	no	—
70	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
71	#117	C₂×D₂₈₁	1	2	yes	—
72	C₂×C₄²	—	281	1	no	—
73	#108	—	562	2	no	—
74	#107	Dic₂₈₁	1	2	yes	complemented
75	#106	Dic₂₈₁	1	2	yes	complemented
76	C₄⋊C₈	—	562	4	no	—
77	C₄×C₈	—	562	2	no	—
78	C₂²×C₄	—	562	1	no	—
79	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
80	M₁₆	—	562	2	no	—
81	M₁₆	—	562	2	no	—
82	C₂×C₈	—	281	2	no	—
83	C₂×C₈	—	562	1	no	—
84	C₂×C₈	—	562	1	no	—
85	C₂×C₈	—	562	2	no	—
86	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
88	C₂×C₈	D₁₁₂₄	1	2	yes	—
89	C₂×C₈	Dic₅₆₂	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₂₈₁	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	281	2	no	—
94	C₄²	—	562	1	no	—
95	C₂³	#17985137	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	562	4	no	—
98	C₂×C₄	—	562	2	no	—
99	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₂₈₁	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₅₆₂	1	1	yes	characteristic
104	C₂×C₄	#17985139	1	1	yes	characteristic
105	C₈	—	562	2	no	—
106	C₈	—	1124	2	no	—
107	C₈	C₄⋊Dic₂₈₁	1	2	yes	—
108	C₈	C₄⋊Dic₂₈₁	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₂₈₁	1	1	yes	characteristic
111	C₂²	#22582859	1	1	yes	characteristic
112	C₂²	#22582870	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	562	4	no	—
115	C₄	C₂×C₄⋊Dic₂₈₁	1	1	yes	characteristic
116	C₄	Q₈×Dic₂₈₁	1	1	yes	characteristic
117	C₄	#22582854	1	1	yes	characteristic
118	C₄	#22582878	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#31823362	1	1	yes	characteristic
122	C₂	#31823580	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582815

C₂

yes

maximal, characteristic

#31823490

C₂

yes

maximal

C₂×C₈×Dic₂₈₁

C₂

yes

maximal, characteristic

#31823222

C₂

yes

maximal

#427×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985122

C₂²

yes

—

C₂²×#14555886

C₂²

yes

characteristic

C₂×C₄×Dic₂₈₁

C₂²

yes

characteristic

#22582815

—

C₄⋊#14555886

—

C₈×Dic₂₈₁

—

C₂²×C₈×C₂₈₁

C₂²

yes

characteristic

#117×C₂₈₁

C₂²

yes

—

#107×C₂₈₁

C₄

yes

complemented

#106×C₂₈₁

C₄

yes

complemented

C₂²×Dic₂₈₁

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

Q₈

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

D₄

yes

—

C₂×C₈×C₂₈₁

Q₈

yes

—

C₄⋊C₄×C₂₈₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

#14555886

—

C₂³×C₂₈₁

#41

yes

characteristic

C₂×C₄×C₂₈₁

—

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

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

characteristic

C₈×C₂₈₁

C₄⋊C₄

yes

—

C₈×C₂₈₁

C₄⋊C₄

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₄×D₄

yes

characteristic

C₂²×C₂₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₈₁

#125

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₈₁

C₄×Q₈

yes

characteristic

C₄×C₂₈₁

#121

yes

characteristic

C₄×C₂₈₁

C₄⋊Q₈

yes

characteristic

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#389

yes

characteristic

C₂×C₂₈₁

#443

yes

—

C₂₈₁

#1945

yes

characteristic, complemented, Sylow

#1945

—

281

maximal, Sylow

C₂×#108

—

281

—

#427

D₂₈₁

yes

characteristic

#424

—

281

—

C₂×C₄×C₈

—

281

—

#339

—

281

—

C₂×M₁₆

—

281

—

C₂²×C₈

—

281

—

C₂²×C₈

C₂×D₂₈₁

yes

characteristic

#117

C₂×D₂₈₁

yes

—

C₂×C₄²

—

281

—

#108

—

562

—

#107

Dic₂₈₁

yes

complemented

#106

Dic₂₈₁

yes

complemented

C₄⋊C₈

—

562

—

C₄×C₈

—

562

—

C₂²×C₄

—

562

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

562

—

M₁₆

—

562

—

C₂×C₈

—

281

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₂×C₈

D₁₁₂₄

yes

—

C₂×C₈

Dic₅₆₂

yes

—

C₄⋊C₄

C₂×Dic₂₈₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

281

—

C₄²

—

562

—

C₂³

#17985137

yes

characteristic

C₂×C₄

—

C₂×C₄

—

562

—

C₂×C₄

—

562

—

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

100

C₂×C₄

D₄×D₂₈₁

yes

characteristic

101

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

102

C₂×C₄

Q₈×D₂₈₁

yes

characteristic

103

C₂×C₄

C₂×Dic₅₆₂

yes

characteristic

104

C₂×C₄

#17985139

yes

characteristic

105

C₈

—

562

—

106

C₈

—

1124

—

107

C₈

C₄⋊Dic₂₈₁

yes

—

108

C₈

C₄⋊Dic₂₈₁

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₂₈₁

yes

characteristic

111

C₂²

#22582859

yes

characteristic

112

C₂²

#22582870

yes

characteristic

113

C₄

—

114

C₄

—

562

—

115

C₄

C₂×C₄⋊Dic₂₈₁

yes

characteristic

116

C₄

Q₈×Dic₂₈₁

yes

characteristic

117

C₄

#22582854

yes

characteristic

118

C₄

#22582878

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#31823362

yes

characteristic

122

C₂

#31823580

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₈₁
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	#427×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₄×D₄
prime radical	C₂³×C₂₈₁	#41
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	#427	D₂₈₁
hypocenter	C₂₈₁	#1945
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₈₁

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

#427×C₂₈₁

C₂

socle

C₂²×C₂₈₁

C₄×D₄

prime radical

C₂³×C₂₈₁

#41

cosocle

C₂²×C₄×C₂₈₁

C₂³

D₂₈₁

C₂₈₁

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₂₈₁	C₄	semidirect
15. #106×C₂₈₁	C₄	semidirect
61. C₂₈₁	#1945	semidirect
74. #107	Dic₂₈₁	semidirect
75. #106	Dic₂₈₁	semidirect

H ≅ G/N

type

14. #107×C₂₈₁

C₄

semidirect

15. #106×C₂₈₁

C₄

semidirect

61. C₂₈₁

#1945

semidirect

74. #107

Dic₂₈₁

semidirect

75. #106

Dic₂₈₁

semidirect

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

subgroup	index
C₂×#22582815	2
#31823490	2
C₂×C₈×Dic₂₈₁	2
#31823222	2
#427×C₂₈₁	2
#1945	281

subgroup

index

C₂×C₈×Dic₂₈₁

#427×C₂₈₁

281

Subgroups — #49096052

Conjugacy classes of subgroups

Complements

Maximal subgroups