Group #44509462 — subgroups

Subgroups — #44509462

Order 30848.

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	#29507889	C₂	1	3	yes	maximal, complemented
2	#29507514	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#21440690	C₂²	1	1	yes	characteristic, complemented
5	#21440668	—	2	3	no	—
6	#21440570	—	2	3	no	—
7	#21440561	C₂²	1	3	yes	complemented
8	#124×C₂₄₁	—	2	3	no	—
9	#124×C₂₄₁	C₂²	1	3	yes	—
10	#120×C₂₄₁	—	2	3	no	—
11	#16838771	—	4	1	no	—
12	#16838770	—	2	3	no	—
13	Q₈×D₂₄₁	—	4	1	no	—
14	#16838758	—	4	3	no	—
15	#16838755	—	4	3	no	—
16	#16838754	—	2	3	no	—
17	C₂⁴×C₂₄₁	—	2	1	no	—
18	C₂×Q₈×C₂₄₁	—	2	1	no	—
19	C₂×Q₈×C₂₄₁	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₂₄₁	—	2	3	no	—
21	C₂×D₄×C₂₄₁	D₄	1	3	yes	—
22	C₂²⋊C₄×C₂₄₁	—	2	3	no	—
23	C₂²⋊C₄×C₂₄₁	—	2	6	no	—
24	C₄²×C₂₄₁	D₄	1	3	yes	—
25	Dic₄₈₂	—	2	3	no	—
26	D₉₆₄	—	2	3	no	—
27	C₂²⋊D₂₄₁	—	4	1	no	—
28	C₄×D₂₄₁	—	4	3	no	—
29	#14515349	—	4	3	no	—
30	C₂³×C₂₄₁	—	2	3	no	—
31	C₂³×C₂₄₁	—	8	2	no	—
32	C₂³×C₂₄₁	—	8	1	no	—
33	Q₈×C₂₄₁	—	4	1	no	—
34	Q₈×C₂₄₁	—	8	1	no	—
35	D₄×C₂₄₁	—	4	3	no	—
36	D₄×C₂₄₁	—	8	3	no	—
37	C₂×C₄×C₂₄₁	—	2	3	no	—
38	C₂×C₄×C₂₄₁	C₂×D₄	1	3	yes	—
39	C₂²×C₂₄₁	—	4	6	no	—
40	C₂²×C₂₄₁	—	4	3	no	—
41	C₂²×C₂₄₁	—	8	4	no	—
42	C₂²×C₂₄₁	#120	1	1	yes	characteristic
43	C₂×D₂₄₁	—	4	1	no	—
44	C₄×C₂₄₁	—	2	3	no	—
45	C₄×C₂₄₁	—	4	3	no	—
46	Dic₂₄₁	—	4	1	no	—
47	C₂×C₂₄₁	—	2	1	no	—
48	C₂×C₂₄₁	—	8	3	no	—
49	C₂×C₂₄₁	B(4,𝔽₂)	1	1	yes	characteristic
50	D₂₄₁	—	8	1	no	—
51	C₂₄₁	#2202	1	1	yes	characteristic, complemented, Sylow
52	#2202	—	241	1	no	maximal, Sylow
53	H(3,𝔽₄)	D₂₄₁	1	1	yes	characteristic, complemented
54	#454	—	241	3	no	—
55	#355	—	241	3	no	—
56	2₋¹⁺⁴	—	241	1	no	—
57	AGL(1,ℤ/8)	—	482	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₂₄₁	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	482	3	no	—
62	#101	—	241	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	964	1	no	—
65	#41	—	482	3	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	964	1	no	—
68	C₂×Q₈	C₂²×D₂₄₁	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₂₄₁	1	3	yes	—
71	SD₁₆	—	964	3	no	—
72	D₈	—	964	3	no	—
73	M₁₆	—	482	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₂₄₁	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	—	482	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	964	1	no	—
86	D₄	—	482	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	964	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	3	yes	—
90	C₈	—	964	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	964	1	no	—
95	C₂²	#21440611	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	964	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	1928	1	no	—
102	C₂	#29507904	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29507889

C₂

yes

maximal, complemented

#29507514

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₂₄₁

C₂

yes

maximal, characteristic, complemented

#21440690

C₂²

yes

characteristic, complemented

#21440668

—

#21440570

—

#21440561

C₂²

yes

complemented

#124×C₂₄₁

—

#124×C₂₄₁

C₂²

yes

—

#120×C₂₄₁

—

#16838771

—

#16838770

—

Q₈×D₂₄₁

—

#16838758

—

#16838755

—

#16838754

—

C₂⁴×C₂₄₁

—

C₂×Q₈×C₂₄₁

—

C₂×Q₈×C₂₄₁

C₂³

yes

characteristic

C₂×D₄×C₂₄₁

—

C₂×D₄×C₂₄₁

D₄

yes

—

C₂²⋊C₄×C₂₄₁

—

C₂²⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

D₄

yes

—

Dic₄₈₂

—

D₉₆₄

—

C₂²⋊D₂₄₁

—

C₄×D₂₄₁

—

#14515349

—

C₂³×C₂₄₁

—

C₂³×C₂₄₁

—

C₂³×C₂₄₁

—

Q₈×C₂₄₁

—

Q₈×C₂₄₁

—

D₄×C₂₄₁

—

D₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂×D₄

yes

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

#120

yes

characteristic

C₂×D₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

B(4,𝔽₂)

yes

characteristic

D₂₄₁

—

C₂₄₁

#2202

yes

characteristic, complemented, Sylow

#2202

—

241

maximal, Sylow

H(3,𝔽₄)

D₂₄₁

yes

characteristic, complemented

#454

—

241

—

#355

—

241

—

2₋¹⁺⁴

—

241

—

AGL(1,ℤ/8)

—

482

—

#124

—

#124

C₂×D₂₄₁

yes

—

#120

—

#104

—

482

—

#101

—

241

—

C₂⁴

—

#41

—

964

—

#41

—

482

—

C₂×Q₈

—

C₂×Q₈

—

964

—

C₂×Q₈

C₂²×D₂₄₁

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₄₁

yes

—

SD₁₆

—

964

—

D₈

—

964

—

M₁₆

—

482

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₂₄₁

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

482

—

D₄

—

D₄

—

D₄

—

964

—

D₄

—

482

—

C₂×C₄

—

C₂×C₄

—

964

—

C₂×C₄

C₂×C₂²⋊D₂₄₁

yes

—

C₈

—

964

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

964

—

C₂²

#21440611

yes

characteristic

C₄

—

C₄

—

C₄

—

964

—

C₂

—

100

C₂

—

101

C₂

—

1928

—

102

C₂

#29507904

yes

characteristic

	subgroup	G/H
center	C₂	#29507904
commutator, cosocle	C₂×Q₈×C₂₄₁	C₂³
Frattini	C₂×Q₈	C₂²×D₂₄₁
Fitting	H(3,𝔽₄)×C₂₄₁	C₂
socle	C₂×C₂₄₁	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₂₄₁
hypocenter	C₂₄₁	#2202
LCS	G ▷ C₂×Q₈×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#29507904

commutator, cosocle

C₂×Q₈×C₂₄₁

C₂³

Frattini

C₂×Q₈

C₂²×D₂₄₁

Fitting

H(3,𝔽₄)×C₂₄₁

C₂

C₂×C₂₄₁

D₂₄₁

C₂₄₁

G ▷ C₂×Q₈×C₂₄₁ ▷ C₂²×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #29507889	C₂	semidirect
2. #29507514	C₂	semidirect
3. H(3,𝔽₄)×C₂₄₁	C₂	semidirect
4. #21440690	C₂²	semidirect
7. #21440561	C₂²	semidirect
51. C₂₄₁	#2202	semidirect
53. H(3,𝔽₄)	D₂₄₁	semidirect

H ≅ G/N

type

1. #29507889

C₂

semidirect

2. #29507514

C₂

semidirect

3. H(3,𝔽₄)×C₂₄₁

C₂

semidirect

4. #21440690

C₂²

semidirect

7. #21440561

C₂²

semidirect

51. C₂₄₁

#2202

semidirect

53. H(3,𝔽₄)

D₂₄₁

semidirect

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

subgroup	index
#29507889	2
#29507514	2
H(3,𝔽₄)×C₂₄₁	2
#2202	241

subgroup

index

#29507889

#29507514

H(3,𝔽₄)×C₂₄₁

#2202

241

Subgroups — #44509462

Conjugacy classes of subgroups

Complements

Maximal subgroups