Group #49095184 — subgroups

Subgroups — #49095184

Order 35968.

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

normal subgroups only

This group has 232 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	C₂×#22582815	C₂	1	1	yes	maximal, characteristic
2	#31823419	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	C₄	1	4	yes	complemented
10	#22582790	—	2	2	no	—
11	C₄×#14555886	—	2	2	no	—
12	C₈×Dic₂₈₁	C₄	1	4	yes	complemented
13	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
14	#117×C₂₈₁	C₂²	1	2	yes	—
15	#107×C₂₈₁	C₄	1	2	yes	complemented
16	#106×C₂₈₁	C₄	1	2	yes	complemented
17	C₂²×Dic₂₈₁	—	2	1	no	—
18	#17985122	—	2	2	no	—
19	#17985122	C₂×C₄	1	4	yes	—
20	C₂×#14555886	—	2	2	no	—
21	C₂×#14555886	C₂×C₄	1	2	yes	—
22	C₂×#14555886	D₄	1	1	yes	characteristic
23	C₂×#14555886	D₄	1	1	yes	characteristic
24	C₂×#14555886	Q₈	1	2	yes	—
25	C₄×Dic₂₈₁	—	2	1	no	—
26	C₄×Dic₂₈₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₈₁	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₂₈₁	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₂₈₁	—	2	2	no	—
34	C₄²×C₂₈₁	—	2	2	no	—
35	C₂×Dic₂₈₁	—	2	4	no	—
36	C₂×Dic₂₈₁	—	2	2	no	—
37	#14555886	—	2	4	no	—
38	#14555886	C₄⋊C₄	1	2	yes	—
39	#14555886	C₄⋊C₄	1	2	yes	—
40	C₂³×C₂₈₁	#41	1	1	yes	characteristic
41	C₂×C₄×C₂₈₁	—	2	4	no	—
42	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
48	C₈×C₂₈₁	C₄²	1	2	yes	—
49	C₈×C₂₈₁	C₄²	1	2	yes	—
50	C₂²×C₂₈₁	—	2	2	no	—
51	C₂²×C₂₈₁	#117	1	1	yes	characteristic
52	C₂²×C₂₈₁	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₈₁	—	4	4	no	—
55	C₄×C₂₈₁	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₂₈₁	C₄×D₄	1	1	yes	characteristic
58	C₄×C₂₈₁	C₄×Q₈	1	1	yes	characteristic
59	Dic₂₈₁	—	2	4	no	—
60	C₂×C₂₈₁	—	2	1	no	—
61	C₂×C₂₈₁	—	2	1	no	—
62	C₂×C₂₈₁	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₂₈₁	#443	1	2	yes	—
64	C₂₈₁	#1772	1	1	yes	characteristic, complemented, Sylow
65	#1772	—	281	1	no	maximal, Sylow
66	C₂×#108	—	281	1	no	—
67	#427	D₂₈₁	1	1	yes	characteristic
68	#405	—	281	2	no	—
69	C₂×C₄×C₈	—	281	1	no	—
70	#339	—	281	2	no	—
71	C₂×M₁₆	—	281	2	no	—
72	C₂²×C₈	—	281	1	no	—
73	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
74	#117	C₂×D₂₈₁	1	2	yes	—
75	C₂×C₄²	—	281	1	no	—
76	#108	—	281	4	no	—
77	#107	Dic₂₈₁	1	2	yes	complemented
78	#106	Dic₂₈₁	1	2	yes	complemented
79	#97	—	562	2	no	—
80	C₄×C₈	—	281	4	no	—
81	C₄×C₈	—	562	2	no	—
82	C₂²×C₄	—	562	1	no	—
83	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
84	M₁₆	—	281	4	no	—
85	M₁₆	—	562	2	no	—
86	C₂×C₈	—	281	2	no	—
87	C₂×C₈	—	281	1	no	—
88	C₂×C₈	—	281	1	no	—
89	C₂×C₈	—	281	2	no	—
90	C₂×C₈	—	562	2	no	—
91	C₂×C₈	C₄×D₂₈₁	1	2	yes	—
92	C₂×C₈	C₄×D₂₈₁	1	2	yes	—
93	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₂₈₁	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₂₈₁	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	281	2	no	—
99	C₄²	—	562	1	no	—
100	C₂³	#17985137	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	562	4	no	—
103	C₂×C₄	—	562	2	no	—
104	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₂₈₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
107	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₂₈₁	1	1	yes	characteristic
109	C₂×C₄	#17985139	1	1	yes	characteristic
110	C₈	—	281	2	no	—
111	C₈	—	281	2	no	—
112	C₈	—	562	4	no	—
113	C₈	C₄×Dic₂₈₁	1	2	yes	—
114	C₈	C₄×Dic₂₈₁	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#22582837	1	1	yes	characteristic
117	C₂²	D₄×Dic₂₈₁	1	1	yes	characteristic
118	C₂²	#22582844	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	562	4	no	—
121	C₄	C₂×C₄×Dic₂₈₁	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₂₈₁	1	1	yes	characteristic
123	C₄	#22582841	1	1	yes	characteristic
124	C₄	Q₈×Dic₂₈₁	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₂₈₁	1	1	yes	characteristic
128	C₂	#31823580	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582815

C₂

yes

maximal, characteristic

#31823419

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

yes

complemented

#22582790

—

C₄×#14555886

—

C₈×Dic₂₈₁

C₄

yes

complemented

C₂²×C₈×C₂₈₁

C₂²

yes

characteristic

#117×C₂₈₁

C₂²

yes

—

#107×C₂₈₁

C₄

yes

complemented

#106×C₂₈₁

C₄

yes

complemented

C₂²×Dic₂₈₁

—

#17985122

—

#17985122

C₂×C₄

yes

—

C₂×#14555886

—

C₂×#14555886

C₂×C₄

yes

—

C₂×#14555886

D₄

yes

characteristic

C₂×#14555886

D₄

yes

characteristic

C₂×#14555886

Q₈

yes

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

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

C₂×C₄

yes

—

C₂²⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

#14555886

C₄⋊C₄

yes

—

#14555886

C₄⋊C₄

yes

—

C₂³×C₂₈₁

#41

yes

characteristic

C₂×C₄×C₂₈₁

—

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

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

—

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

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₈₁

#443

yes

—

C₂₈₁

#1772

yes

characteristic, complemented, Sylow

#1772

—

281

maximal, Sylow

C₂×#108

—

281

—

#427

D₂₈₁

yes

characteristic

#405

—

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

—

281

—

#107

Dic₂₈₁

yes

complemented

#106

Dic₂₈₁

yes

complemented

#97

—

562

—

C₄×C₈

—

281

—

C₄×C₈

—

562

—

C₂²×C₄

—

562

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

281

—

M₁₆

—

562

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

—

281

—

C₂×C₈

—

562

—

C₂×C₈

C₄×D₂₈₁

yes

—

C₂×C₈

C₄×D₂₈₁

yes

—

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₂×C₈

C₂×Dic₂₈₁

yes

characteristic

C₄⋊C₄

C₂×Dic₂₈₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

281

—

C₄²

—

562

—

100

C₂³

#17985137

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

562

—

103

C₂×C₄

—

562

—

104

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

105

C₂×C₄

C₂×C₄×D₂₈₁

yes

characteristic

106

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

107

C₂×C₄

D₄×D₂₈₁

yes

characteristic

108

C₂×C₄

Q₈×D₂₈₁

yes

characteristic

109

C₂×C₄

#17985139

yes

characteristic

110

C₈

—

281

—

111

C₈

—

281

—

112

C₈

—

562

—

113

C₈

C₄×Dic₂₈₁

yes

—

114

C₈

C₄×Dic₂₈₁

yes

—

115

C₂²

—

116

C₂²

#22582837

yes

characteristic

117

C₂²

D₄×Dic₂₈₁

yes

characteristic

118

C₂²

#22582844

yes

characteristic

119

C₄

—

120

C₄

—

562

—

121

C₄

C₂×C₄×Dic₂₈₁

yes

characteristic

122

C₄

C₄⋊C₄×D₂₈₁

yes

characteristic

123

C₄

#22582841

yes

characteristic

124

C₄

Q₈×Dic₂₈₁

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₂₈₁

yes

characteristic

128

C₂

#31823580

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₈₁
commutator	C₄×C₂₈₁	C₂×C₄²
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	#1772
LCS	G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₈₁

commutator

C₄×C₂₈₁

C₂×C₄²

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 ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #22582815	C₄	semidirect
12. C₈×Dic₂₈₁	C₄	semidirect
15. #107×C₂₈₁	C₄	semidirect
16. #106×C₂₈₁	C₄	semidirect
64. C₂₈₁	#1772	semidirect
77. #107	Dic₂₈₁	semidirect
78. #106	Dic₂₈₁	semidirect

H ≅ G/N

type

9. #22582815

C₄

semidirect

12. C₈×Dic₂₈₁

C₄

semidirect

15. #107×C₂₈₁

C₄

semidirect

16. #106×C₂₈₁

C₄

semidirect

64. C₂₈₁

#1772

semidirect

77. #107

Dic₂₈₁

semidirect

78. #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
#31823419	2
C₂×C₈×Dic₂₈₁	2
#31823222	2
#427×C₂₈₁	2
#1772	281

subgroup

index

C₂×C₈×Dic₂₈₁

#427×C₂₈₁

281

Subgroups — #49095184

Conjugacy classes of subgroups

Complements

Maximal subgroups