Group #49095020 — subgroups

Subgroups — #49095020

Order 35968.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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×#22582838	C₂	1	2	yes	maximal
2	#31823226	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
4	#22582838	—	2	4	no	—
5	#22582838	C₄	1	8	yes	complemented
6	C₂×C₄⋊Dic₂₈₁	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₂₈₁	—	2	2	no	—
8	C₂×C₄×Dic₂₈₁	—	2	2	no	—
9	C₂³×C₄×C₂₈₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₈₁	—	2	4	no	—
11	C₂²×C₈×C₂₈₁	C₄	1	4	yes	complemented
12	C₂²×C₈×C₂₈₁	C₂²	1	2	yes	—
13	C₂²×Dic₂₈₁	—	2	4	no	—
14	C₄⋊Dic₂₈₁	—	2	4	no	—
15	C₄⋊Dic₂₈₁	C₂×C₄	1	8	yes	—
16	C₂²⋊Dic₂₈₁	—	2	8	no	—
17	C₄×Dic₂₈₁	—	2	8	no	—
18	C₂⁴×C₂₈₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₈₁	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₂₈₁	D₄	1	4	yes	—
22	C₂²×C₄×C₂₈₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₈₁	Q₈	1	2	yes	—
24	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₈₁	—	2	8	no	—
26	C₂×C₈×C₂₈₁	—	2	4	no	—
27	C₂×C₈×C₂₈₁	C₂×C₄	1	4	yes	—
28	C₂×Dic₂₈₁	—	2	8	no	—
29	C₂×Dic₂₈₁	—	4	8	no	—
30	C₂³×C₂₈₁	—	2	4	no	—
31	C₂³×C₂₈₁	C₂²⋊C₄	1	4	yes	—
32	C₂³×C₂₈₁	C₄⋊C₄	1	2	yes	—
33	C₂³×C₂₈₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₈₁	C₄²	1	4	yes	—
35	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	4	yes	—
36	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₂₈₁	C₄⋊C₄	1	8	yes	—
38	C₂×C₄×C₂₈₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₈₁	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₂₈₁	—	2	4	no	—
44	C₂²×C₂₈₁	—	2	8	no	—
45	C₂²×C₂₈₁	—	2	4	no	—
46	C₂²×C₂₈₁	#95	1	4	yes	—
47	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	2	yes	—
48	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₈₁	#135	1	4	yes	—
50	C₄×C₂₈₁	#95	1	4	yes	—
51	C₄×C₂₈₁	C₂×C₄²	1	1	yes	characteristic
52	C₄×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₂₈₁	C₂×C₄⋊C₄	1	2	yes	—
54	Dic₂₈₁	—	4	8	no	—
55	C₂×C₂₈₁	—	2	4	no	—
56	C₂×C₂₈₁	C₂×#95	1	1	yes	characteristic
57	C₂×C₂₈₁	#416	1	4	yes	—
58	C₂×C₂₈₁	#427	1	2	yes	—
59	C₂₈₁	#1735	1	1	yes	characteristic, complemented, Sylow
60	#1735	—	281	1	no	maximal, Sylow
61	C₂³×C₈	D₂₈₁	1	1	yes	characteristic
62	C₂×#117	—	281	2	no	—
63	#340	—	281	4	no	—
64	C₂³×C₄	C₂×D₂₈₁	1	1	yes	characteristic
65	C₂²×C₈	—	2	4	no	—
66	C₂²×C₈	Dic₂₈₁	1	4	yes	complemented
67	C₂²×C₈	C₂×D₂₈₁	1	2	yes	—
68	#117	—	281	8	no	—
69	#117	—	562	4	no	—
70	C₂×C₄⋊C₄	—	281	4	no	—
71	C₂×C₂²⋊C₄	—	562	2	no	—
72	C₂×C₄²	—	562	2	no	—
73	C₂⁴	D₁₁₂₄	1	1	yes	characteristic
74	C₂²×C₄	—	562	4	no	—
75	C₂²×C₄	C₄×D₂₈₁	1	4	yes	—
76	C₂²×C₄	C₂×Dic₂₈₁	1	2	yes	—
77	C₂²×C₄	C₂²⋊D₂₈₁	1	4	yes	—
78	C₂²×C₄	D₁₁₂₄	1	1	yes	characteristic
79	C₂²×C₄	Dic₅₆₂	1	2	yes	—
80	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₂₈₁	1	4	yes	—
84	C₄⋊C₄	—	281	8	no	—
85	C₄⋊C₄	—	562	4	no	—
86	C₂²⋊C₄	—	562	8	no	—
87	C₄²	—	562	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	#17985115	1	4	yes	—
90	C₂³	C₄⋊Dic₂₈₁	1	2	yes	—
91	C₂³	C₂×D₁₁₂₄	1	1	yes	characteristic
92	C₂×C₄	—	562	8	no	—
93	C₂×C₄	—	1124	8	no	—
94	C₂×C₄	C₄×Dic₂₈₁	1	4	yes	—
95	C₂×C₄	#17985115	1	4	yes	—
96	C₂×C₄	C₂²⋊Dic₂₈₁	1	4	yes	—
97	C₂×C₄	#17985117	1	8	yes	—
98	C₂×C₄	C₄⋊Dic₂₈₁	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₂₈₁	1	2	yes	—
100	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	2	yes	—
102	C₂×C₄	C₂×Dic₅₆₂	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	#22582786	1	4	yes	—
107	C₂²	C₂×#17985115	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₂₈₁	1	1	yes	characteristic
109	C₂²	#22582901	1	4	yes	—
110	C₄	—	1124	8	no	—
111	C₄	#22582786	1	4	yes	—
112	C₄	C₂×C₄×Dic₂₈₁	1	1	yes	characteristic
113	C₄	C₂×C₂²⋊Dic₂₈₁	1	1	yes	characteristic
114	C₄	C₂×#17985117	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	C₂×#22582786	1	1	yes	characteristic
117	C₂	#31823460	1	4	yes	—
118	C₂	#31823501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582838

C₂

yes

maximal

#31823226

C₂

yes

maximal, complemented

C₂³×C₈×C₂₈₁

C₂

yes

maximal, characteristic

#22582838

—

#22582838

C₄

yes

complemented

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

C₂×C₂²⋊Dic₂₈₁

—

C₂×C₄×Dic₂₈₁

—

C₂³×C₄×C₂₈₁

C₂²

yes

characteristic

C₂²×C₈×C₂₈₁

—

C₂²×C₈×C₂₈₁

C₄

yes

complemented

C₂²×C₈×C₂₈₁

C₂²

yes

—

C₂²×Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

C₂×C₄

yes

—

C₂²⋊Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₂⁴×C₂₈₁

D₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

D₄

yes

—

C₂²×C₄×C₂₈₁

D₄

yes

characteristic

C₂²×C₄×C₂₈₁

Q₈

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂³×C₂₈₁

—

C₂³×C₂₈₁

C₂²⋊C₄

yes

—

C₂³×C₂₈₁

C₄⋊C₄

yes

—

C₂³×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₄²

yes

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂×C₄×C₂₈₁

C₂×Q₈

yes

characteristic

C₈×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#95

yes

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₈₁

#135

yes

—

C₄×C₂₈₁

#95

yes

—

C₄×C₂₈₁

C₂×C₄²

yes

characteristic

C₄×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₈₁

C₂×C₄⋊C₄

yes

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

C₂×#95

yes

characteristic

C₂×C₂₈₁

#416

yes

—

C₂×C₂₈₁

#427

yes

—

C₂₈₁

#1735

yes

characteristic, complemented, Sylow

#1735

—

281

maximal, Sylow

C₂³×C₈

D₂₈₁

yes

characteristic

C₂×#117

—

281

—

#340

—

281

—

C₂³×C₄

C₂×D₂₈₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₂₈₁

yes

complemented

C₂²×C₈

C₂×D₂₈₁

yes

—

#117

—

281

—

#117

—

562

—

C₂×C₄⋊C₄

—

281

—

C₂×C₂²⋊C₄

—

562

—

C₂×C₄²

—

562

—

C₂⁴

D₁₁₂₄

yes

characteristic

C₂²×C₄

—

562

—

C₂²×C₄

C₄×D₂₈₁

yes

—

C₂²×C₄

C₂×Dic₂₈₁

yes

—

C₂²×C₄

C₂²⋊D₂₈₁

yes

—

C₂²×C₄

D₁₁₂₄

yes

characteristic

C₂²×C₄

Dic₅₆₂

yes

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₂₈₁

yes

—

C₄⋊C₄

—

281

—

C₄⋊C₄

—

562

—

C₂²⋊C₄

—

562

—

C₄²

—

562

—

C₂³

—

C₂³

#17985115

yes

—

C₂³

C₄⋊Dic₂₈₁

yes

—

C₂³

C₂×D₁₁₂₄

yes

characteristic

C₂×C₄

—

562

—

C₂×C₄

—

1124

—

C₂×C₄

C₄×Dic₂₈₁

yes

—

C₂×C₄

#17985115

yes

—

C₂×C₄

C₂²⋊Dic₂₈₁

yes

—

C₂×C₄

#17985117

yes

—

C₂×C₄

C₄⋊Dic₂₈₁

yes

—

C₂×C₄

C₂×C₄×D₂₈₁

yes

—

100

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

—

102

C₂×C₄

C₂×Dic₅₆₂

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

#22582786

yes

—

107

C₂²

C₂×#17985115

yes

—

108

C₂²

C₂×C₄⋊Dic₂₈₁

yes

characteristic

109

C₂²

#22582901

yes

—

110

C₄

—

1124

—

111

C₄

#22582786

yes

—

112

C₄

C₂×C₄×Dic₂₈₁

yes

characteristic

113

C₄

C₂×C₂²⋊Dic₂₈₁

yes

characteristic

114

C₄

C₂×#17985117

yes

—

115

C₂

—

116

C₂

C₂×#22582786

yes

characteristic

117

C₂

#31823460

yes

—

118

C₂

#31823501

yes

—

	subgroup	G/H
center	C₂²×C₄	D₁₁₂₄
commutator	C₄×C₂₈₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂³×C₈×C₂₈₁	C₂
socle	C₂³×C₂₈₁	C₂×D₄
prime radical	C₂⁴×C₂₈₁	D₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂³×C₈	D₂₈₁
hypocenter	C₂₈₁	#1735
LCS	G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₁₁₂₄

commutator

C₄×C₂₈₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂³×C₈×C₂₈₁

C₂

socle

C₂³×C₂₈₁

C₂×D₄

prime radical

C₂⁴×C₂₈₁

D₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂³×C₈

D₂₈₁

hypocenter

C₂₈₁

#1735

LCS

G ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #31823226	C₂	semidirect
5. #22582838	C₄	semidirect
11. C₂²×C₈×C₂₈₁	C₄	semidirect
59. C₂₈₁	#1735	semidirect
66. C₂²×C₈	Dic₂₈₁	semidirect

H ≅ G/N

type

2. #31823226

C₂

semidirect

5. #22582838

C₄

semidirect

11. C₂²×C₈×C₂₈₁

C₄

semidirect

59. C₂₈₁

#1735

semidirect

66. C₂²×C₈

Dic₂₈₁

semidirect

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

subgroup	index
C₂×#22582838	2
#31823226	2
C₂³×C₈×C₂₈₁	2
#1735	281

subgroup

index

C₂×#22582838

#31823226

C₂³×C₈×C₂₈₁

#1735

281

Subgroups — #49095020

Conjugacy classes of subgroups

Complements

Maximal subgroups