Group #49094929 — subgroups

Subgroups — #49094929

Order 35968.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#31823861	C₂	1	1	yes	maximal, characteristic, complemented
2	#31823835	C₂	1	2	yes	maximal, complemented
3	#31823809	C₂	1	2	yes	maximal, complemented
4	#31823184	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#17985126	—	2	4	no	—
7	#22582879	C₂²	1	2	yes	—
8	#22582811	—	2	2	no	—
9	#22582800	—	2	4	no	—
10	C₄×#14555886	C₂²	1	2	yes	—
11	C₂×D₈×C₂₈₁	—	2	2	no	—
12	#127×C₂₈₁	C₂²	1	1	yes	characteristic
13	#127×C₂₈₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₆₂	—	2	2	no	—
16	#17985126	—	4	8	no	—
17	C₂×#14555886	D₄	1	4	yes	—
18	C₄⋊Dic₂₈₁	—	2	2	no	—
19	C₄⋊Dic₂₈₁	—	2	2	no	—
20	C₂×D₄×C₂₈₁	—	2	2	no	—
21	C₂×D₄×C₂₈₁	—	2	2	no	—
22	C₂×D₄×C₂₈₁	—	2	1	no	—
23	C₂×D₄×C₂₈₁	—	2	1	no	—
24	D₈×C₂₈₁	—	4	4	no	—
25	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
26	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
27	Dic₅₆₂	—	4	4	no	—
28	C₂×Dic₂₈₁	—	4	2	no	—
29	#14555886	—	2	4	no	—
30	C₂³×C₂₈₁	—	4	1	no	—
31	C₂³×C₂₈₁	—	4	1	no	—
32	D₄×C₂₈₁	—	4	2	no	—
33	D₄×C₂₈₁	—	4	2	no	—
34	D₄×C₂₈₁	—	4	4	no	—
35	D₄×C₂₈₁	—	4	4	no	—
36	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
38	C₈×C₂₈₁	SD₁₆	1	4	yes	complemented
39	C₂²×C₂₈₁	—	8	1	no	—
40	C₂²×C₂₈₁	—	8	1	no	—
41	C₂²×C₂₈₁	—	8	2	no	—
42	C₂²×C₂₈₁	—	8	2	no	—
43	C₂²×C₂₈₁	#127	1	1	yes	characteristic
44	C₄×C₂₈₁	C₂×SD₁₆	1	2	yes	—
45	C₄×C₂₈₁	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₂₈₁	—	8	2	no	—
47	C₂×C₂₈₁	—	16	1	no	—
48	C₂×C₂₈₁	—	16	1	no	—
49	C₂×C₂₈₁	#492	1	1	yes	characteristic
50	C₂×C₂₈₁	#496	1	2	yes	—
51	C₂₈₁	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	281	1	no	maximal, Sylow
53	#498	—	281	1	no	—
54	#493	D₂₈₁	1	1	yes	characteristic
55	#492	—	281	2	no	—
56	#486	—	281	2	no	—
57	#322	—	281	1	no	—
58	C₂×SD₁₆	—	562	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	281	2	no	—
61	#127	C₂×D₂₈₁	1	1	yes	characteristic
62	#127	C₂×D₂₈₁	1	1	yes	characteristic
63	#106	—	562	2	no	—
64	#102	—	562	4	no	—
65	C₄×C₈	—	281	2	no	—
66	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
67	C₂×Q₈	—	562	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	1124	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	281	4	no	—
75	C₂×C₈	C₂²⋊D₂₈₁	1	2	yes	—
76	C₄⋊C₄	—	562	2	no	—
77	C₄⋊C₄	—	562	2	no	—
78	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	1124	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	1124	2	no	—
87	C₂×C₄	D₄×D₂₈₁	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
89	C₈	—	562	4	no	—
90	C₈	#17985126	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#22582875	1	1	yes	characteristic
96	C₄	—	2248	2	no	—
97	C₄	C₂×#17985126	1	2	yes	—
98	C₄	#22582903	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#31823835	1	1	yes	characteristic
102	C₂	#31823851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31823861

C₂

yes

maximal, characteristic, complemented

#31823835

C₂

yes

maximal, complemented

#31823809

C₂

yes

maximal, complemented

#31823184

C₂

yes

maximal, characteristic, complemented

#493×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985126

—

#22582879

C₂²

yes

—

#22582811

—

#22582800

—

C₄×#14555886

C₂²

yes

—

C₂×D₈×C₂₈₁

—

#127×C₂₈₁

C₂²

yes

characteristic

#127×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₂×Dic₅₆₂

—

#17985126

—

C₂×#14555886

D₄

yes

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

—

D₈×C₂₈₁

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₄²×C₂₈₁

C₂³

yes

characteristic

Dic₅₆₂

—

C₂×Dic₂₈₁

—

#14555886

—

C₂³×C₂₈₁

—

C₂³×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

SD₁₆

yes

complemented

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#127

yes

characteristic

C₄×C₂₈₁

C₂×SD₁₆

yes

—

C₄×C₂₈₁

AGL(1,ℤ/8)

yes

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#492

yes

characteristic

C₂×C₂₈₁

#496

yes

—

C₂₈₁

#1715

yes

characteristic, complemented, Sylow

#1715

—

281

maximal, Sylow

#498

—

281

—

#493

D₂₈₁

yes

characteristic

#492

—

281

—

#486

—

281

—

#322

—

281

—

C₂×SD₁₆

—

562

—

C₂×D₈

—

C₄⋊Q₈

—

281

—

#127

C₂×D₂₈₁

yes

characteristic

#127

C₂×D₂₈₁

yes

characteristic

#106

—

562

—

#102

—

562

—

C₄×C₈

—

281

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₂×Q₈

—

562

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

1124

—

D₈

—

C₂×C₈

—

281

—

C₂×C₈

C₂²⋊D₂₈₁

yes

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

562

—

C₄²

C₂²×D₂₈₁

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

1124

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

1124

—

C₂×C₄

D₄×D₂₈₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

C₈

—

562

—

C₈

#17985126

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#22582875

yes

characteristic

C₄

—

2248

—

C₄

C₂×#17985126

yes

—

C₄

#22582903

yes

—

C₂

—

100

C₂

—

101

C₂

#31823835

yes

characteristic

102

C₂

#31823851

yes

—

	subgroup	G/H
center	C₂²	#22582875
commutator, cosocle	C₄²×C₂₈₁	C₂³
Frattini	C₄²	C₂²×D₂₈₁
Fitting, prime radical	#493×C₂₈₁	C₂
socle	C₂²×C₂₈₁	#127
hypercenter	#493	D₂₈₁
hypocenter	C₂₈₁	#1715
LCS	G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#22582875

commutator, cosocle

C₄²×C₂₈₁

C₂³

Frattini

C₄²

C₂²×D₂₈₁

Fitting, prime radical

#493×C₂₈₁

C₂

C₂²×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #31823861	C₂	semidirect
2. #31823835	C₂	semidirect
3. #31823809	C₂	semidirect
4. #31823184	C₂	semidirect
38. C₈×C₂₈₁	SD₁₆	semidirect
51. C₂₈₁	#1715	semidirect
90. C₈	#17985126	semidirect

H ≅ G/N

type

1. #31823861

C₂

semidirect

2. #31823835

C₂

semidirect

3. #31823809

C₂

semidirect

4. #31823184

C₂

semidirect

38. C₈×C₂₈₁

SD₁₆

semidirect

51. C₂₈₁

#1715

semidirect

90. C₈

#17985126

semidirect

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

subgroup	index
#31823861	2
#31823835	2
#31823809	2
#31823184	2
#493×C₂₈₁	2
#1715	281

subgroup

index

#493×C₂₈₁

281

Subgroups — #49094929

Conjugacy classes of subgroups

Complements

Maximal subgroups