Group #49095341 — subgroups

Subgroups — #49095341

Order 35968.

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

normal subgroups only

This group has 292 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	2	yes	maximal, complemented
2	#31823427	C₂	1	2	yes	maximal
3	#31823228	C₂	1	2	yes	maximal, complemented
4	C₂³×C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985122	C₂²	1	4	yes	—
6	#22582815	—	2	4	no	—
7	C₂²⋊#14555886	—	2	4	no	—
8	C₂³×C₄×C₂₈₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₈₁	—	2	4	no	—
10	C₂²×C₈×C₂₈₁	C₄	1	4	yes	—
11	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
13	#17985122	—	2	4	no	—
14	#17985122	—	2	4	no	—
15	C₂×#14555886	—	2	4	no	—
16	C₂⁴×C₂₈₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₂₈₁	—	2	2	no	—
18	C₂²×C₄×C₂₈₁	—	2	2	no	—
19	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₈₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₈₁	—	2	2	no	—
24	C₂×C₈×C₂₈₁	—	2	8	no	—
25	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₂₈₁	D₄	1	4	yes	—
28	#14555886	—	4	4	no	—
29	C₂³×C₂₈₁	—	2	4	no	—
30	C₂³×C₂₈₁	—	2	2	no	—
31	C₂³×C₂₈₁	C₄⋊C₄	1	2	yes	—
32	C₂³×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₈₁	—	2	2	no	—
34	C₂×C₄×C₂₈₁	—	2	2	no	—
35	C₂×C₄×C₂₈₁	—	2	2	no	—
36	C₂×C₄×C₂₈₁	—	2	4	no	—
37	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₁	#41	1	2	yes	—
42	C₈×C₂₈₁	—	2	2	no	—
43	C₈×C₂₈₁	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₂₈₁	—	2	8	no	—
45	C₂²×C₂₈₁	—	2	2	no	—
46	C₂²×C₂₈₁	—	2	2	no	—
47	C₂²×C₂₈₁	—	2	2	no	—
48	C₂²×C₂₈₁	#108	1	4	yes	complemented
49	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₂₈₁	—	2	2	no	—
52	C₄×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₈₁	#117	1	1	yes	characteristic
54	C₄×C₂₈₁	#121	1	2	yes	—
55	C₂×C₂₈₁	—	2	4	no	—
56	C₂×C₂₈₁	—	2	2	no	—
57	C₂×C₂₈₁	#380	1	1	yes	characteristic
58	C₂×C₂₈₁	C₂×#108	1	2	yes	—
59	C₂₈₁	#1810	1	1	yes	characteristic, complemented, Sylow
60	#1810	—	281	1	no	maximal, Sylow
61	C₂³×C₈	D₂₈₁	1	1	yes	characteristic
62	C₂×#108	—	281	2	no	—
63	#407	—	281	2	no	—
64	#341	—	281	2	no	—
65	C₂³×C₄	C₂×D₂₈₁	1	1	yes	characteristic
66	C₂×M₁₆	—	281	4	no	—
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	Dic₂₈₁	1	4	yes	—
69	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
71	#108	—	562	4	no	—
72	C₂²⋊C₈	—	562	4	no	—
73	C₂⁴	Dic₅₆₂	1	1	yes	characteristic
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂×Dic₂₈₁	1	2	yes	—
77	C₂²×C₄	D₁₁₂₄	1	2	yes	—
78	C₂²×C₄	Dic₅₆₂	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
80	M₁₆	—	562	4	no	—
81	M₁₆	—	562	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	8	no	—
84	C₂×C₈	—	562	4	no	—
85	C₂×C₈	C₂×Dic₂₈₁	1	2	yes	—
86	C₂×C₈	C₂×Dic₂₈₁	1	2	yes	—
87	C₂×C₈	C₂²⋊D₂₈₁	1	4	yes	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	C₄⋊Dic₂₈₁	1	2	yes	—
91	C₂³	C₂×Dic₅₆₂	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	C₄⋊Dic₂₈₁	1	2	yes	—
97	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	2	yes	—
99	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
100	C₂×C₄	#17985138	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	1124	4	no	—
103	C₈	C₂²⋊Dic₂₈₁	1	4	yes	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	#22582815	1	4	yes	complemented
109	C₂²	C₂×C₄⋊Dic₂₈₁	1	1	yes	characteristic
110	C₂²	#22582858	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	C₂×C₂²⋊Dic₂₈₁	1	1	yes	characteristic
113	C₄	#22582838	1	1	yes	characteristic
114	C₄	#22582853	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	—	2	2	no	—
117	C₂	#31823323	1	1	yes	characteristic
118	C₂	C₂×#22582815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582815

C₂

yes

maximal, complemented

#31823427

C₂

yes

maximal

#31823228

C₂

yes

maximal, complemented

C₂³×C₈×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985122

C₂²

yes

—

#22582815

—

C₂²⋊#14555886

—

C₂³×C₄×C₂₈₁

C₂²

yes

characteristic

C₂²×C₈×C₂₈₁

—

C₂²×C₈×C₂₈₁

C₄

yes

—

C₂²×C₈×C₂₈₁

C₂²

yes

characteristic

C₂²×C₈×C₂₈₁

C₂²

yes

—

—

—

C₂⁴×C₂₈₁

Q₈

yes

characteristic

C₂²×C₄×C₂₈₁

—

C₂²×C₄×C₂₈₁

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

D₄

yes

—

C₂²×C₄×C₂₈₁

Q₈

yes

characteristic

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

D₄

yes

—

#14555886

—

C₂³×C₂₈₁

—

C₂³×C₂₈₁

—

C₂³×C₂₈₁

C₄⋊C₄

yes

—

C₂³×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

—

C₈×C₂₈₁

—

C₈×C₂₈₁

C₂²⋊C₄

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#108

yes

complemented

C₂²×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₈₁

C₂²⋊Q₈

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×C₂²⋊C₄

yes

C₄×C₂₈₁

yes

C₄×C₂₈₁

yes

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#380

yes

characteristic

C₂×C₂₈₁

C₂×#108

yes

—

C₂₈₁

#1810

yes

characteristic, complemented, Sylow

#1810

—

281

maximal, Sylow

C₂³×C₈

D₂₈₁

yes

characteristic

C₂×#108

—

281

—

#407

—

281

—

#341

—

281

—

C₂³×C₄

C₂×D₂₈₁

yes

characteristic

C₂×M₁₆

—

281

—

C₂²×C₈

—

C₂²×C₈

Dic₂₈₁

yes

—

C₂²×C₈

C₂×D₂₈₁

yes

characteristic

C₂²×C₈

C₂×D₂₈₁

yes

characteristic

#108

—

562

—

C₂²⋊C₈

—

562

—

C₂⁴

Dic₅₆₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₂₈₁

yes

—

C₂²×C₄

D₁₁₂₄

yes

—

C₂²×C₄

Dic₅₆₂

yes

characteristic

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

562

—

M₁₆

—

562

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

562

—

C₂×C₈

C₂×Dic₂₈₁

yes

—

C₂×C₈

C₂×Dic₂₈₁

yes

—

C₂×C₈

C₂²⋊D₂₈₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₂₈₁

yes

—

C₂³

C₂×Dic₅₆₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₈₁

yes

—

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

—

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

100

C₂×C₄

#17985138

yes

—

101

C₈

—

102

C₈

—

1124

—

103

C₈

C₂²⋊Dic₂₈₁

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#22582815

yes

complemented

109

C₂²

C₂×C₄⋊Dic₂₈₁

yes

characteristic

110

C₂²

#22582858

yes

—

111

C₄

—

112

C₄

C₂×C₂²⋊Dic₂₈₁

yes

characteristic

113

C₄

#22582838

yes

characteristic

114

C₄

#22582853

yes

—

115

C₂

—

116

C₂

—

117

C₂

#31823323

yes

characteristic

118

C₂

C₂×#22582815

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₂₄
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂³×C₈×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₂₈₁	Q₈
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂³×C₈	D₂₈₁
hypocenter	C₂₈₁	#1810
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₁₂₄

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂³×C₈×C₂₈₁

C₂

socle

C₂²×C₂₈₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₂₈₁

Q₈

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂³×C₈

D₂₈₁

hypocenter

C₂₈₁

#1810

LCS

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#22582815	C₂	semidirect
3. #31823228	C₂	semidirect
48. C₂²×C₂₈₁	#108	semidirect
59. C₂₈₁	#1810	semidirect
108. C₂²	#22582815	semidirect

H ≅ G/N

type

1. C₂×#22582815

C₂

semidirect

3. #31823228

C₂

semidirect

48. C₂²×C₂₈₁

#108

semidirect

59. C₂₈₁

#1810

semidirect

108. C₂²

#22582815

semidirect

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

subgroup	index
C₂×#22582815	2
#31823427	2
#31823228	2
C₂³×C₈×C₂₈₁	2
#1810	281

subgroup

index

C₂³×C₈×C₂₈₁

281

Subgroups — #49095341

Conjugacy classes of subgroups

Complements

Maximal subgroups