Group #44506264 — subgroups

Subgroups — #44506264

Order 30848.

160 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 87 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	Q₈×#14515349	C₂	1	2	yes	maximal
2	#29507468	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#14515349	C₂	1	2	yes	maximal
4	#29507438	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₂₄₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#14515349	—	2	2	no	—
7	C₄⋊#14515349	C₂²	1	2	yes	—
8	C₄×#14515349	—	2	2	no	—
9	C₄×#14515349	C₂²	1	1	yes	characteristic
10	C₄×#14515349	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₂₄₁	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₂₄₁	C₂²	1	2	yes	—
13	#107×C₂₄₁	C₄	1	1	yes	characteristic
14	#103×C₂₄₁	C₄	1	2	yes	—
15	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
16	C₂×#14515349	—	2	2	no	—
17	C₂×#14515349	—	2	1	no	—
18	C₂×#14515349	D₄	1	2	yes	—
19	C₂×Q₈×C₂₄₁	C₂×C₄	1	2	yes	—
20	Q₁₆×C₂₄₁	C₈	1	4	yes	complemented
21	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₂₄₁	—	2	2	no	—
24	C₄⋊C₄×C₂₄₁	C₂×C₄	1	2	yes	—
25	C₄²×C₂₄₁	—	2	2	no	—
26	C₄²×C₂₄₁	C₂³	1	1	yes	characteristic
27	#14515349	—	2	2	no	—
28	#14515349	—	4	1	no	—
29	#14515349	—	4	2	no	—
30	Q₈×C₂₄₁	—	2	2	no	—
31	Q₈×C₂₄₁	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₂₄₁	—	2	2	no	—
33	C₂×C₄×C₂₄₁	—	2	2	no	—
34	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
37	C₈×C₂₄₁	—	2	1	no	—
38	C₈×C₂₄₁	C₂×C₈	1	2	yes	—
39	C₂²×C₂₄₁	C₄×D₄	1	1	yes	characteristic
40	C₄×C₂₄₁	—	2	4	no	—
41	C₄×C₂₄₁	—	2	1	no	—
42	C₄×C₂₄₁	—	4	2	no	—
43	C₄×C₂₄₁	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₂₄₁	#131	1	1	yes	characteristic
45	C₄×C₂₄₁	#137	1	2	yes	—
46	C₂×C₂₄₁	C₈×D₄	1	1	yes	characteristic
47	C₂×C₂₄₁	#441	1	1	yes	characteristic
48	C₂×C₂₄₁	#444	1	1	yes	characteristic
49	C₂₄₁	#1579	1	1	yes	characteristic, complemented, Sylow
50	#1579	—	241	1	no	maximal, Sylow
51	C₈×Q₈	—	241	2	no	—
52	C₄×Q₁₆	D₂₄₁	1	1	yes	characteristic
53	#334	—	241	1	no	—
54	Q₈⋊C₈	—	241	2	no	—
55	#322	—	241	1	no	—
56	C₂×Q₁₆	Dic₂₄₁	1	1	yes	characteristic
57	C₄×Q₈	C₂×D₂₄₁	1	2	yes	—
58	#107	Dic₂₄₁	1	1	yes	characteristic
59	C₄⋊C₈	—	482	2	no	—
60	C₄⋊C₈	—	241	2	no	—
61	#103	Dic₂₄₁	1	2	yes	—
62	C₄×C₈	—	482	2	no	—
63	C₄×C₈	—	241	1	no	—
64	C₄×C₈	—	241	1	no	—
65	C₄×C₈	C₂×D₂₄₁	1	1	yes	characteristic
66	C₂×Q₈	C₂×Dic₂₄₁	1	2	yes	—
67	Q₁₆	#14515349	1	4	yes	complemented
68	C₂×C₈	—	482	2	no	—
69	C₂×C₈	—	482	1	no	—
70	C₂×C₈	—	241	2	no	—
71	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
72	C₂×C₈	C₂×Dic₂₄₁	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×Dic₂₄₁	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂²×D₂₄₁	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	C₂×#14515349	1	4	yes	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×Dic₂₄₁	1	1	yes	characteristic
82	C₂×C₄	D₄×D₂₄₁	1	1	yes	characteristic
83	C₂×C₄	#16838769	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	964	1	no	—
86	C₈	—	964	2	no	—
87	C₈	—	482	2	no	—
88	C₈	C₂×#14515349	1	2	yes	—
89	C₂²	D₄×Dic₂₄₁	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	2	1	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂²×#14515349	1	1	yes	characteristic
94	C₄	#21440649	1	1	yes	characteristic
95	C₄	#21440674	1	2	yes	—
96	C₂	D₄×#14515349	1	1	yes	characteristic
97	C₂	#29507824	1	1	yes	characteristic
98	C₂	#29507839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#14515349

C₂

yes

maximal

#29507468

C₂

yes

maximal, characteristic

Q₈⋊#14515349

C₂

yes

maximal

#29507438

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₂₄₁

C₂

yes

maximal, characteristic

C₄⋊#14515349

—

C₄⋊#14515349

C₂²

yes

—

C₄×#14515349

—

C₄×#14515349

C₂²

yes

characteristic

C₄×#14515349

C₂²

yes

characteristic

C₂×Q₁₆×C₂₄₁

C₄

yes

characteristic

C₄×Q₈×C₂₄₁

C₂²

yes

—

#107×C₂₄₁

C₄

yes

characteristic

#103×C₂₄₁

C₄

yes

—

C₄×C₈×C₂₄₁

C₂²

yes

—

—

D₄

yes

—

C₂×Q₈×C₂₄₁

C₂×C₄

yes

—

Q₁₆×C₂₄₁

C₈

yes

complemented

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

C₂×C₄

yes

—

C₄²×C₂₄₁

—

C₄²×C₂₄₁

C₂³

yes

—

—

—

Q₈×C₂₄₁

—

Q₈×C₂₄₁

C₂×C₈

yes

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

characteristic

C₈×C₂₄₁

—

C₈×C₂₄₁

C₂×C₈

yes

—

C₂²×C₂₄₁

C₄×D₄

yes

characteristic

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

C₂²×C₈

yes

C₄×C₂₄₁

yes

C₄×C₂₄₁

yes

—

C₂×C₂₄₁

C₈×D₄

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂₄₁

yes

characteristic, complemented, Sylow

#1579

—

241

maximal, Sylow

C₈×Q₈

—

241

—

C₄×Q₁₆

D₂₄₁

yes

characteristic

#334

—

241

—

Q₈⋊C₈

—

241

—

#322

—

241

—

C₂×Q₁₆

Dic₂₄₁

yes

characteristic

C₄×Q₈

C₂×D₂₄₁

yes

—

#107

Dic₂₄₁

yes

characteristic

C₄⋊C₈

—

482

—

C₄⋊C₈

—

241

—

#103

Dic₂₄₁

yes

—

C₄×C₈

—

482

—

C₄×C₈

—

241

—

C₄×C₈

—

241

—

C₄×C₈

C₂×D₂₄₁

yes

characteristic

C₂×Q₈

C₂×Dic₂₄₁

yes

—

Q₁₆

#14515349

yes

complemented

C₂×C₈

—

482

—

C₂×C₈

—

482

—

C₂×C₈

—

241

—

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₂×C₈

C₂×Dic₂₄₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₂₄₁

yes

—

C₄²

—

C₄²

C₂²×D₂₄₁

yes

characteristic

Q₈

—

Q₈

C₂×#14515349

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₂₄₁

yes

characteristic

C₂×C₄

D₄×D₂₄₁

yes

characteristic

C₂×C₄

#16838769

yes

characteristic

C₈

—

C₈

—

964

—

C₈

—

964

—

C₈

—

482

—

C₈

C₂×#14515349

yes

—

C₂²

D₄×Dic₂₄₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₂₄₁
commutator	C₄×C₂₄₁	C₂²×C₈
Frattini	C₄²	C₂²×D₂₄₁
Fitting	C₄×Q₁₆×C₂₄₁	C₂
socle, prime radical	C₂²×C₂₄₁	C₄×D₄
cosocle	C₄²×C₂₄₁	C₂³
hypercenter	C₄×Q₁₆	D₂₄₁
hypocenter	C₂₄₁	#1579
LCS	G ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₂₄₁

commutator

C₄×C₂₄₁

C₂²×C₈

Frattini

C₄²

C₂²×D₂₄₁

Fitting

C₄×Q₁₆×C₂₄₁

C₂

socle, prime radical

C₂²×C₂₄₁

C₄×D₄

cosocle

C₄²×C₂₄₁

C₂³

hypercenter

C₄×Q₁₆

D₂₄₁

hypocenter

C₂₄₁

#1579

LCS

G ▷ C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
20. Q₁₆×C₂₄₁	C₈	semidirect
49. C₂₄₁	#1579	semidirect
67. Q₁₆	#14515349	semidirect

H ≅ G/N

type

20. Q₁₆×C₂₄₁

C₈

semidirect

49. C₂₄₁

#1579

semidirect

67. Q₁₆

#14515349

semidirect

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

subgroup	index
Q₈×#14515349	2
#29507468	2
Q₈⋊#14515349	2
#29507438	2
C₄×Q₁₆×C₂₄₁	2
#1579	241

subgroup

index

C₄×Q₁₆×C₂₄₁

241

Subgroups — #44506264

Conjugacy classes of subgroups

Complements

Maximal subgroups