Group #49095190 — subgroups

Subgroups — #49095190

Order 35968.

224 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 117 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	#31823501	C₂	1	1	yes	maximal, characteristic
2	#31823421	C₂	1	2	yes	maximal
3	C₂×#22582789	C₂	1	1	yes	maximal, characteristic
4	#31823224	C₂	1	2	yes	maximal
5	C₂×#108×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555886	C₂²	1	1	yes	characteristic
7	#22582838	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₈₁	C₂²	1	1	yes	characteristic
9	#22582813	C₄	1	2	yes	complemented
10	#22582811	C₄	1	2	yes	complemented
11	#22582789	—	2	2	no	—
12	#22582789	C₄	1	4	yes	complemented
13	C₈×Dic₂₈₁	—	2	2	no	—
14	C₂×M₁₆×C₂₈₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
16	#108×C₂₈₁	C₄	1	4	yes	complemented
17	C₂²×Dic₂₈₁	—	2	1	no	—
18	C₂×#14555886	C₂×C₄	1	2	yes	—
19	C₂×#14555886	D₄	1	2	yes	—
20	C₂×#14555886	Q₈	1	2	yes	—
21	C₄⋊Dic₂₈₁	C₂×C₄	1	4	yes	—
22	C₂²⋊Dic₂₈₁	—	2	2	no	—
23	C₄×Dic₂₈₁	—	2	2	no	—
24	C₄×Dic₂₈₁	—	2	1	no	—
25	C₄×Dic₂₈₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₈₁	—	2	2	no	—
28	M₁₆×C₂₈₁	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₂₈₁	—	2	2	no	—
30	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₈₁	—	2	2	no	—
35	C₂×Dic₂₈₁	—	2	4	no	—
36	C₂×Dic₂₈₁	—	4	2	no	—
37	#14555886	C₄⋊C₄	1	4	yes	—
38	C₂³×C₂₈₁	#41	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₈₁	C₂×D₄	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	—	2	4	no	—
46	C₈×C₂₈₁	C₄²	1	2	yes	—
47	C₈×C₂₈₁	C₄²	1	2	yes	—
48	C₂²×C₂₈₁	—	2	2	no	—
49	C₂²×C₂₈₁	#117	1	1	yes	characteristic
50	C₂²×C₂₈₁	C₄×D₄	1	1	yes	characteristic
51	C₂²×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
52	C₄×C₂₈₁	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₂₈₁	C₄×D₄	1	1	yes	characteristic
55	C₄×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
56	Dic₂₈₁	—	4	2	no	—
57	Dic₂₈₁	—	4	4	no	—
58	C₂×C₂₈₁	—	2	1	no	—
59	C₂×C₂₈₁	—	2	1	no	—
60	C₂×C₂₈₁	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₂₈₁	#444	1	2	yes	—
62	C₂₈₁	#1773	1	1	yes	characteristic, complemented, Sylow
63	#1773	—	281	1	no	maximal, Sylow
64	C₂×#108	D₂₈₁	1	1	yes	characteristic
65	#427	—	281	1	no	—
66	#405	—	281	2	no	—
67	C₂×#97	—	281	1	no	—
68	#339	—	281	2	no	—
69	C₂×M₁₆	C₂×D₂₈₁	1	2	yes	—
70	C₂²×C₈	—	281	1	no	—
71	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
72	#117	—	281	2	no	—
73	C₂×C₄²	—	281	1	no	—
74	#108	Dic₂₈₁	1	4	yes	complemented
75	#107	—	281	2	no	—
76	#106	—	281	2	no	—
77	#97	—	281	4	no	—
78	#97	—	562	2	no	—
79	C₄×C₈	—	562	2	no	—
80	C₂²×C₄	—	562	1	no	—
81	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	C₂×Dic₂₈₁	1	4	yes	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	281	2	no	—
86	C₂×C₈	—	281	2	no	—
87	C₂×C₈	—	281	2	no	—
88	C₂×C₈	C₄×D₂₈₁	1	2	yes	—
89	C₂×C₈	C₄×D₂₈₁	1	1	yes	characteristic
90	C₂×C₈	C₄×D₂₈₁	1	1	yes	characteristic
91	C₂×C₈	C₂×Dic₂₈₁	1	2	yes	—
92	C₄⋊C₄	—	281	4	no	—
93	C₂²⋊C₄	—	562	2	no	—
94	C₄²	—	281	2	no	—
95	C₄²	—	562	2	no	—
96	C₄²	—	562	1	no	—
97	C₂³	#17985139	1	1	yes	characteristic
98	C₂×C₄	—	562	2	no	—
99	C₂×C₄	—	562	4	no	—
100	C₂×C₄	—	1124	2	no	—
101	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
103	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
104	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₂₈₁	1	1	yes	characteristic
106	C₂×C₄	#17985137	1	1	yes	characteristic
107	C₈	—	2	4	no	—
108	C₈	—	281	4	no	—
109	C₈	C₄×Dic₂₈₁	1	2	yes	—
110	C₈	C₄×Dic₂₈₁	1	2	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	#22582837	1	1	yes	characteristic
113	C₂²	#22582841	1	1	yes	characteristic
114	C₂²	Q₈×Dic₂₈₁	1	1	yes	characteristic
115	C₄	—	1124	2	no	—
116	C₄	—	1124	4	no	—
117	C₄	C₂×C₄×Dic₂₈₁	1	1	yes	characteristic
118	C₄	C₄⋊C₄×D₂₈₁	1	1	yes	characteristic
119	C₄	D₄×Dic₂₈₁	1	1	yes	characteristic
120	C₄	#22582844	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	C₄⋊C₄×Dic₂₈₁	1	1	yes	characteristic
124	C₂	#31823584	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31823501

C₂

yes

maximal, characteristic

#31823421

C₂

yes

maximal

C₂×#22582789

C₂

yes

maximal, characteristic

#31823224

C₂

yes

maximal

C₂×#108×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×#14555886

C₂²

yes

characteristic

#22582838

C₂²

yes

—

C₂×C₄×Dic₂₈₁

C₂²

yes

characteristic

#22582813

C₄

yes

complemented

#22582811

C₄

yes

complemented

#22582789

—

#22582789

C₄

yes

complemented

C₈×Dic₂₈₁

—

C₂×M₁₆×C₂₈₁

C₂²

yes

—

C₂²×C₈×C₂₈₁

C₂²

yes

characteristic

#108×C₂₈₁

C₄

yes

complemented

C₂²×Dic₂₈₁

—

C₂×#14555886

C₂×C₄

yes

—

C₂×#14555886

D₄

yes

—

C₂×#14555886

Q₈

yes

—

C₄⋊Dic₂₈₁

C₂×C₄

yes

—

C₂²⋊Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

M₁₆×C₂₈₁

—

M₁₆×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

C₄⋊C₄

yes

—

C₂³×C₂₈₁

#41

yes

characteristic

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

characteristic

C₈×C₂₈₁

—

C₈×C₂₈₁

C₄²

yes

—

C₈×C₂₈₁

C₄²

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#117

yes

characteristic

C₂²×C₂₈₁

C₄×D₄

yes

characteristic

C₂²×C₂₈₁

C₄×Q₈

yes

characteristic

C₄×C₂₈₁

C₂×C₄²

yes

characteristic

C₄×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₈₁

C₄×D₄

yes

characteristic

C₄×C₂₈₁

C₄×Q₈

yes

characteristic

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₈₁

#444

yes

—

C₂₈₁

#1773

yes

characteristic, complemented, Sylow

#1773

—

281

maximal, Sylow

C₂×#108

D₂₈₁

yes

characteristic

#427

—

281

—

#405

—

281

—

C₂×#97

—

281

—

#339

—

281

—

C₂×M₁₆

C₂×D₂₈₁

yes

—

C₂²×C₈

—

281

—

C₂²×C₈

C₂×D₂₈₁

yes

characteristic

#117

—

281

—

C₂×C₄²

—

281

—

#108

Dic₂₈₁

yes

complemented

#107

—

281

—

#106

—

281

—

#97

—

281

—

#97

—

562

—

C₄×C₈

—

562

—

C₂²×C₄

—

562

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

M₁₆

C₂×Dic₂₈₁

yes

—

C₂×C₈

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

C₄×D₂₈₁

yes

—

C₂×C₈

C₄×D₂₈₁

yes

characteristic

C₂×C₈

C₄×D₂₈₁

yes

characteristic

C₂×C₈

C₂×Dic₂₈₁

yes

—

C₄⋊C₄

—

281

—

C₂²⋊C₄

—

562

—

C₄²

—

281

—

C₄²

—

562

—

C₄²

—

562

—

C₂³

#17985139

yes

characteristic

C₂×C₄

—

562

—

C₂×C₄

—

562

—

100

C₂×C₄

—

1124

—

101

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

102

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

103

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

104

C₂×C₄

D₄×D₂₈₁

yes

characteristic

105

C₂×C₄

Q₈×D₂₈₁

yes

characteristic

106

C₂×C₄

#17985137

yes

characteristic

107

C₈

—

108

C₈

—

281

—

109

C₈

C₄×Dic₂₈₁

yes

—

110

C₈

C₄×Dic₂₈₁

yes

—

111

C₂²

—

112

C₂²

#22582837

yes

characteristic

113

C₂²

#22582841

yes

characteristic

114

C₂²

Q₈×Dic₂₈₁

yes

characteristic

115

C₄

—

1124

—

116

C₄

—

1124

—

117

C₄

C₂×C₄×Dic₂₈₁

yes

characteristic

118

C₄

C₄⋊C₄×D₂₈₁

yes

characteristic

119

C₄

D₄×Dic₂₈₁

yes

characteristic

120

C₄

#22582844

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₄⋊C₄×Dic₂₈₁

yes

characteristic

124

C₂

#31823584

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₈₁
commutator	C₄×C₂₈₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂×#108×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₄×D₄
prime radical	C₂³×C₂₈₁	#41
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×#108	D₂₈₁
hypocenter	C₂₈₁	#1773
LCS	G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₈₁

commutator

C₄×C₂₈₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂×#108×C₂₈₁

C₂

socle

C₂²×C₂₈₁

C₄×D₄

prime radical

C₂³×C₂₈₁

#41

cosocle

C₂²×C₄×C₂₈₁

C₂³

D₂₈₁

C₂₈₁

G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #22582813	C₄	semidirect
10. #22582811	C₄	semidirect
12. #22582789	C₄	semidirect
16. #108×C₂₈₁	C₄	semidirect
62. C₂₈₁	#1773	semidirect
74. #108	Dic₂₈₁	semidirect

H ≅ G/N

type

9. #22582813

C₄

semidirect

10. #22582811

C₄

semidirect

12. #22582789

C₄

semidirect

16. #108×C₂₈₁

C₄

semidirect

62. C₂₈₁

#1773

semidirect

74. #108

Dic₂₈₁

semidirect

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

subgroup	index
#31823501	2
#31823421	2
C₂×#22582789	2
#31823224	2
C₂×#108×C₂₈₁	2
#1773	281

subgroup

index

C₂×#108×C₂₈₁

281

Subgroups — #49095190

Conjugacy classes of subgroups

Complements

Maximal subgroups