Group #49097320 — subgroups

Subgroups — #49097320

Order 35968.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#31823635	C₂	1	3	yes	maximal, complemented
2	#31823260	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22582927	C₂²	1	1	yes	characteristic, complemented
5	#22582905	—	2	3	no	—
6	#22582807	—	2	3	no	—
7	#22582798	C₂²	1	3	yes	complemented
8	#124×C₂₈₁	—	2	3	no	—
9	#124×C₂₈₁	C₂²	1	3	yes	—
10	#120×C₂₈₁	—	2	3	no	—
11	#17985139	—	4	1	no	—
12	#17985138	—	2	3	no	—
13	Q₈×D₂₈₁	—	4	1	no	—
14	#17985126	—	4	3	no	—
15	#17985123	—	4	3	no	—
16	#17985122	—	2	3	no	—
17	C₂⁴×C₂₈₁	—	2	1	no	—
18	C₂×Q₈×C₂₈₁	—	2	1	no	—
19	C₂×Q₈×C₂₈₁	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₂₈₁	—	2	3	no	—
21	C₂×D₄×C₂₈₁	D₄	1	3	yes	—
22	C₂²⋊C₄×C₂₈₁	—	2	3	no	—
23	C₂²⋊C₄×C₂₈₁	—	2	6	no	—
24	C₄²×C₂₈₁	D₄	1	3	yes	—
25	Dic₅₆₂	—	2	3	no	—
26	D₁₁₂₄	—	2	3	no	—
27	C₂²⋊D₂₈₁	—	4	1	no	—
28	C₄×D₂₈₁	—	4	3	no	—
29	#14555886	—	4	3	no	—
30	C₂³×C₂₈₁	—	2	3	no	—
31	C₂³×C₂₈₁	—	8	2	no	—
32	C₂³×C₂₈₁	—	8	1	no	—
33	Q₈×C₂₈₁	—	4	1	no	—
34	Q₈×C₂₈₁	—	8	1	no	—
35	D₄×C₂₈₁	—	4	3	no	—
36	D₄×C₂₈₁	—	8	3	no	—
37	C₂×C₄×C₂₈₁	—	2	3	no	—
38	C₂×C₄×C₂₈₁	C₂×D₄	1	3	yes	—
39	C₂²×C₂₈₁	—	4	6	no	—
40	C₂²×C₂₈₁	—	4	3	no	—
41	C₂²×C₂₈₁	—	8	4	no	—
42	C₂²×C₂₈₁	#120	1	1	yes	characteristic
43	C₂×D₂₈₁	—	4	1	no	—
44	C₄×C₂₈₁	—	2	3	no	—
45	C₄×C₂₈₁	—	4	3	no	—
46	Dic₂₈₁	—	4	1	no	—
47	C₂×C₂₈₁	—	2	1	no	—
48	C₂×C₂₈₁	—	8	3	no	—
49	C₂×C₂₈₁	B(4,𝔽₂)	1	1	yes	characteristic
50	D₂₈₁	—	8	1	no	—
51	C₂₈₁	#2202	1	1	yes	characteristic, complemented, Sylow
52	#2202	—	281	1	no	maximal, Sylow
53	H(3,𝔽₄)	D₂₈₁	1	1	yes	characteristic, complemented
54	#454	—	281	3	no	—
55	#355	—	281	3	no	—
56	2₋¹⁺⁴	—	281	1	no	—
57	AGL(1,ℤ/8)	—	562	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₂₈₁	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	562	3	no	—
62	#101	—	281	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	562	3	no	—
65	#41	—	1124	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	1124	1	no	—
68	C₂×Q₈	C₂²×D₂₈₁	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₂₈₁	1	3	yes	—
71	SD₁₆	—	1124	3	no	—
72	D₈	—	1124	3	no	—
73	M₁₆	—	562	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₂₈₁	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	562	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	562	3	no	—
86	D₄	—	1124	1	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	1124	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	3	yes	—
90	C₈	—	1124	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	1124	1	no	—
95	C₂²	#22582848	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	1124	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	2248	1	no	—
102	C₂	#31823650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823635

C₂

yes

maximal, complemented

#31823260

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₂₈₁

C₂

yes

maximal, characteristic, complemented

#22582927

C₂²

yes

characteristic, complemented

#22582905

—

#22582807

—

#22582798

C₂²

yes

complemented

#124×C₂₈₁

—

#124×C₂₈₁

C₂²

yes

—

#120×C₂₈₁

—

#17985139

—

#17985138

—

Q₈×D₂₈₁

—

#17985126

—

#17985123

—

#17985122

—

C₂⁴×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

C₂³

yes

characteristic

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

D₄

yes

—

C₂²⋊C₄×C₂₈₁

—

C₂²⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

D₄

yes

—

Dic₅₆₂

—

D₁₁₂₄

—

C₂²⋊D₂₈₁

—

C₄×D₂₈₁

—

#14555886

—

C₂³×C₂₈₁

—

C₂³×C₂₈₁

—

C₂³×C₂₈₁

—

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#120

yes

characteristic

C₂×D₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

B(4,𝔽₂)

yes

characteristic

D₂₈₁

—

C₂₈₁

#2202

yes

characteristic, complemented, Sylow

#2202

—

281

maximal, Sylow

H(3,𝔽₄)

D₂₈₁

yes

characteristic, complemented

#454

—

281

—

#355

—

281

—

2₋¹⁺⁴

—

281

—

AGL(1,ℤ/8)

—

562

—

#124

—

#124

C₂×D₂₈₁

yes

—

#120

—

#104

—

562

—

#101

—

281

—

C₂⁴

—

#41

—

562

—

#41

—

1124

—

C₂×Q₈

—

C₂×Q₈

—

1124

—

C₂×Q₈

C₂²×D₂₈₁

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₈₁

yes

—

SD₁₆

—

1124

—

D₈

—

1124

—

M₁₆

—

562

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₂₈₁

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

562

—

D₄

—

D₄

—

D₄

—

562

—

D₄

—

1124

—

C₂×C₄

—

C₂×C₄

—

1124

—

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

—

C₈

—

1124

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

1124

—

C₂²

#22582848

yes

characteristic

C₄

—

C₄

—

C₄

—

1124

—

C₂

—

100

C₂

—

101

C₂

—

2248

—

102

C₂

#31823650

yes

characteristic

	subgroup	G/H
center	C₂	#31823650
commutator, cosocle	C₂×Q₈×C₂₈₁	C₂³
Frattini	C₂×Q₈	C₂²×D₂₈₁
Fitting	H(3,𝔽₄)×C₂₈₁	C₂
socle	C₂×C₂₈₁	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₂₈₁
hypocenter	C₂₈₁	#2202
LCS	G ▷ C₂×Q₈×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#31823650

commutator, cosocle

C₂×Q₈×C₂₈₁

C₂³

Frattini

C₂×Q₈

C₂²×D₂₈₁

Fitting

H(3,𝔽₄)×C₂₈₁

C₂

C₂×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₂×Q₈×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #31823635	C₂	semidirect
2. #31823260	C₂	semidirect
3. H(3,𝔽₄)×C₂₈₁	C₂	semidirect
4. #22582927	C₂²	semidirect
7. #22582798	C₂²	semidirect
51. C₂₈₁	#2202	semidirect
53. H(3,𝔽₄)	D₂₈₁	semidirect

H ≅ G/N

type

1. #31823635

C₂

semidirect

2. #31823260

C₂

semidirect

3. H(3,𝔽₄)×C₂₈₁

C₂

semidirect

4. #22582927

C₂²

semidirect

7. #22582798

C₂²

semidirect

51. C₂₈₁

#2202

semidirect

53. H(3,𝔽₄)

D₂₈₁

semidirect

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

subgroup	index
#31823635	2
#31823260	2
H(3,𝔽₄)×C₂₈₁	2
#2202	281

subgroup

index

#31823635

#31823260

H(3,𝔽₄)×C₂₈₁

#2202

281

Subgroups — #49097320

Conjugacy classes of subgroups

Complements

Maximal subgroups