Group #49096857 — subgroups

Subgroups — #49096857

Order 35968.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#31823512	C₂	1	1	yes	maximal, characteristic, complemented
2	#31823490	C₂	1	2	yes	maximal, complemented
3	#31823481	C₂	1	2	yes	maximal
4	#31823431	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985138	C₂²	1	1	yes	characteristic
7	C₂×#17985122	C₂²	1	2	yes	—
8	C₂²×#14555886	C₂²	1	1	yes	characteristic
9	#22582815	—	2	2	no	—
10	C₄⋊#14555886	—	2	4	no	—
11	#22582807	—	2	4	no	—
12	#22582791	—	2	2	no	—
13	C₂×M₁₆×C₂₈₁	C₂²	1	1	yes	characteristic
14	#117×C₂₈₁	C₂²	1	2	yes	—
15	#17985138	—	2	2	no	—
16	#17985138	—	2	2	no	—
17	C₂×Dic₅₆₂	Q₈	1	1	yes	characteristic
18	C₂×D₁₁₂₄	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₈₁	—	2	1	no	—
20	C₂×C₄×D₂₈₁	—	2	1	no	—
21	#17985122	—	2	2	no	—
22	#17985122	—	2	2	no	—
23	C₂×#14555886	—	2	1	no	—
24	C₂×#14555886	—	2	2	no	—
25	C₂×#14555886	—	2	2	no	—
26	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₈₁	—	2	2	no	—
28	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₈₁	—	2	2	no	—
30	C₂²⋊C₄×C₂₈₁	—	2	2	no	—
31	C₄²×C₂₈₁	D₄	1	4	yes	—
32	C₂²×D₂₈₁	—	2	1	no	—
33	Dic₅₆₂	—	2	2	no	—
34	D₁₁₂₄	—	2	2	no	—
35	C₂²⋊D₂₈₁	—	4	2	no	—
36	C₂×Dic₂₈₁	—	2	1	no	—
37	C₄×D₂₈₁	—	4	1	no	—
38	C₄×D₂₈₁	—	4	1	no	—
39	#14555886	—	4	1	no	—
40	#14555886	—	4	2	no	—
41	C₂³×C₂₈₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₈₁	—	2	4	no	—
43	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₈₁	#41	1	2	yes	—
47	C₈×C₂₈₁	—	2	2	no	—
48	C₂²×C₂₈₁	—	4	1	no	—
49	C₂²×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₈₁	#123	1	2	yes	—
51	C₂×D₂₈₁	—	4	2	no	—
52	C₂×D₂₈₁	—	4	1	no	—
53	C₄×C₂₈₁	—	4	4	no	—
54	C₄×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₂₈₁	#123	1	1	yes	characteristic
56	C₄×C₂₈₁	#127	1	1	yes	characteristic
57	Dic₂₈₁	—	4	1	no	—
58	C₂×C₂₈₁	—	2	2	no	—
59	C₂×C₂₈₁	—	2	1	no	—
60	C₂×C₂₈₁	#399	1	1	yes	characteristic
61	D₂₈₁	—	8	1	no	—
62	C₂₈₁	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	281	1	no	maximal, Sylow
64	#430	—	281	1	no	—
65	#424	—	281	2	no	—
66	#421	—	281	2	no	—
67	#408	—	281	1	no	—
68	H(3,ℤ/4)	D₂₈₁	1	1	yes	characteristic, complemented
69	C₂×#41	—	281	1	no	—
70	C₂×M₁₆	—	281	2	no	—
71	C₂×M₁₆	C₂×D₂₈₁	1	1	yes	characteristic
72	C₂²×C₈	—	281	1	no	—
73	#117	C₂×D₂₈₁	1	2	yes	—
74	#108	—	562	2	no	—
75	C₄⋊C₈	—	562	4	no	—
76	#104	—	562	4	no	—
77	C₂²⋊C₈	—	562	2	no	—
78	#41	—	562	2	no	—
79	#41	—	562	2	no	—
80	C₂×Q₈	—	281	1	no	—
81	C₂×D₄	—	281	1	no	—
82	C₂×D₄	—	562	1	no	—
83	C₂²×C₄	—	562	1	no	—
84	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	562	2	no	—
87	M₁₆	—	562	2	no	—
88	C₂×C₈	—	562	1	no	—
89	C₂×C₈	—	562	2	no	—
90	C₂×C₈	—	562	2	no	—
91	C₂×C₈	D₁₁₂₄	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₁₁₂₄	1	4	yes	—
95	C₂³	—	562	1	no	—
96	C₂³	#17985137	1	1	yes	characteristic
97	Q₈	—	562	2	no	—
98	D₄	—	562	2	no	—
99	D₄	—	1124	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	562	1	no	—
102	C₂×C₄	—	1124	1	no	—
103	C₂×C₄	—	1124	1	no	—
104	C₂×C₄	C₂×D₁₁₂₄	1	2	yes	—
105	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₈₁	1	1	yes	characteristic
107	C₂×C₄	#17985137	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	1124	1	no	—
110	C₈	—	1124	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	1124	2	no	—
113	C₂²	—	1124	1	no	—
114	C₂²	#22582859	1	1	yes	characteristic
115	C₂²	#22582863	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	1124	1	no	—
118	C₄	#22582859	1	2	yes	—
119	C₄	#22582863	1	1	yes	characteristic
120	C₄	#22582876	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	2248	1	no	—
124	C₂	#31823400	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823512

C₂

yes

maximal, characteristic, complemented

#31823490

C₂

yes

maximal, complemented

#31823481

C₂

yes

maximal

#31823431

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₂₈₁

C₂

yes

maximal, characteristic, complemented

C₂×#17985138

C₂²

yes

characteristic

C₂×#17985122

C₂²

yes

—

C₂²×#14555886

C₂²

yes

—

—

—

—

C₂×M₁₆×C₂₈₁

C₂²

yes

characteristic

#117×C₂₈₁

C₂²

yes

—

#17985138

—

#17985138

—

C₂×Dic₅₆₂

Q₈

yes

characteristic

C₂×D₁₁₂₄

Q₈

yes

characteristic

C₂×C₂²⋊D₂₈₁

—

C₂×C₄×D₂₈₁

—

—

—

—

—

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

M₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₄⋊C₄×C₂₈₁

—

C₂²⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

D₄

yes

—

C₂²×D₂₈₁

—

Dic₅₆₂

—

D₁₁₂₄

—

C₂²⋊D₂₈₁

—

C₂×Dic₂₈₁

—

C₄×D₂₈₁

—

C₄×D₂₈₁

—

#14555886

—

#14555886

—

C₂³×C₂₈₁

#41

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

—

C₈×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₈₁

#123

yes

—

C₂×D₂₈₁

—

C₂×D₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

—

C₄×C₂₈₁

yes

C₄×C₂₈₁

yes

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#399

yes

characteristic

D₂₈₁

—

C₂₈₁

#2098

yes

characteristic, complemented, Sylow

#2098

—

281

maximal, Sylow

#430

—

281

—

#424

—

281

—

#421

—

281

—

#408

—

281

—

H(3,ℤ/4)

D₂₈₁

yes

characteristic, complemented

C₂×#41

—

281

—

C₂×M₁₆

—

281

—

C₂×M₁₆

C₂×D₂₈₁

yes

characteristic

C₂²×C₈

—

281

—

#117

C₂×D₂₈₁

yes

—

#108

—

562

—

C₄⋊C₈

—

562

—

#104

—

562

—

C₂²⋊C₈

—

562

—

#41

—

562

—

#41

—

562

—

C₂×Q₈

—

281

—

C₂×D₄

—

281

—

C₂×D₄

—

562

—

C₂²×C₄

—

562

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

M₁₆

—

562

—

M₁₆

—

562

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

D₁₁₂₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₁₁₂₄

yes

—

C₂³

—

562

—

C₂³

#17985137

yes

characteristic

Q₈

—

562

—

D₄

—

562

—

D₄

—

1124

—

100

C₂×C₄

—

101

C₂×C₄

—

562

—

102

C₂×C₄

—

1124

—

103

C₂×C₄

—

1124

—

104

C₂×C₄

C₂×D₁₁₂₄

yes

—

105

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

106

C₂×C₄

Q₈×D₂₈₁

yes

characteristic

107

C₂×C₄

#17985137

yes

—

108

C₈

—

109

C₈

—

1124

—

110

C₈

—

1124

—

111

C₂²

—

112

C₂²

—

1124

—

113

C₂²

—

1124

—

114

C₂²

#22582859

yes

characteristic

115

C₂²

#22582863

yes

—

116

C₄

—

117

C₄

—

1124

—

118

C₄

#22582859

yes

—

119

C₄

#22582863

yes

characteristic

120

C₄

#22582876

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

2248

—

124

C₂

#31823400

yes

characteristic

	subgroup	G/H
center	C₄	#22582876
commutator, cosocle	C₂²×C₄×C₂₈₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	H(3,ℤ/4)×C₂₈₁	C₂
socle	C₂×C₂₈₁	#399
prime radical	C₂×#17985138	C₂²
hypercenter	H(3,ℤ/4)	D₂₈₁
hypocenter	C₂₈₁	#2098
LCS	G ▷ C₂²×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#22582876

commutator, cosocle

C₂²×C₄×C₂₈₁

C₂³

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

H(3,ℤ/4)×C₂₈₁

C₂

C₂×C₂₈₁

C₂²

D₂₈₁

C₂₈₁

G ▷ C₂²×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #31823512	C₂	semidirect
2. #31823490	C₂	semidirect
5. H(3,ℤ/4)×C₂₈₁	C₂	semidirect
62. C₂₈₁	#2098	semidirect
68. H(3,ℤ/4)	D₂₈₁	semidirect

H ≅ G/N

type

1. #31823512

C₂

semidirect

2. #31823490

C₂

semidirect

5. H(3,ℤ/4)×C₂₈₁

C₂

semidirect

62. C₂₈₁

#2098

semidirect

68. H(3,ℤ/4)

D₂₈₁

semidirect

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

subgroup	index
#31823512	2
#31823490	2
#31823481	2
#31823431	2
H(3,ℤ/4)×C₂₈₁	2
#2098	281

subgroup

index

H(3,ℤ/4)×C₂₈₁

281

Subgroups — #49096857

Conjugacy classes of subgroups

Complements

Maximal subgroups