Group #49094967 — subgroups

Subgroups — #49094967

Order 35968.

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

normal subgroups only

This group has 242 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	#31823840	C₂	1	1	yes	maximal, characteristic, complemented
2	#31823839	C₂	1	2	yes	maximal, complemented
3	#31823811	C₂	1	2	yes	maximal, complemented
4	#31823184	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₂₄₈	—	2	2	no	—
7	C₂×#17985123	—	2	4	no	—
8	#22582876	C₂²	1	2	yes	—
9	#22582799	—	2	4	no	—
10	C₄×#14555886	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₂₈₁	—	2	2	no	—
12	C₄⋊Q₈×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×D₁₁₂₄	—	2	2	no	—
16	C₂×D₁₁₂₄	—	2	2	no	—
17	C₂×D₁₁₂₄	—	2	2	no	—
18	D₂₂₄₈	—	4	4	no	—
19	#17985123	—	4	8	no	—
20	C₂×#14555886	D₄	1	4	yes	—
21	C₂×Q₈×C₂₈₁	—	2	1	no	—
22	C₂×D₄×C₂₈₁	—	2	2	no	—
23	C₂×D₄×C₂₈₁	—	2	1	no	—
24	SD₁₆×C₂₈₁	—	4	4	no	—
25	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
26	C₄⋊C₄×C₂₈₁	—	2	2	no	—
27	C₄²×C₂₈₁	C₂³	1	1	yes	characteristic
28	C₂²×D₂₈₁	—	4	2	no	—
29	D₁₁₂₄	—	4	4	no	—
30	D₁₁₂₄	—	4	4	no	—
31	D₁₁₂₄	—	4	4	no	—
32	#14555886	—	2	4	no	—
33	C₂³×C₂₈₁	—	4	1	no	—
34	Q₈×C₂₈₁	—	4	2	no	—
35	D₄×C₂₈₁	—	4	2	no	—
36	D₄×C₂₈₁	—	4	4	no	—
37	C₂×C₄×C₂₈₁	—	4	1	no	—
38	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₈₁	D₈	1	4	yes	complemented
41	C₂²×C₂₈₁	—	8	1	no	—
42	C₂²×C₂₈₁	—	8	2	no	—
43	C₂²×C₂₈₁	#127	1	1	yes	characteristic
44	C₂×D₂₈₁	—	8	2	no	—
45	C₂×D₂₈₁	—	8	2	no	—
46	C₂×D₂₈₁	—	8	2	no	—
47	C₄×C₂₈₁	—	8	1	no	—
48	C₄×C₂₈₁	C₂×D₈	1	2	yes	—
49	C₄×C₂₈₁	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₂₈₁	—	16	1	no	—
51	C₂×C₂₈₁	#493	1	1	yes	characteristic
52	C₂×C₂₈₁	#496	1	2	yes	—
53	D₂₈₁	—	16	2	no	—
54	C₂₈₁	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	281	1	no	maximal, Sylow
56	#493	—	281	2	no	—
57	#493	—	281	1	no	—
58	#492	D₂₈₁	1	1	yes	characteristic, complemented
59	#486	—	281	2	no	—
60	#322	—	281	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	562	4	no	—
63	C₂×D₈	—	562	2	no	—
64	C₄⋊Q₈	C₂×D₂₈₁	1	1	yes	characteristic
65	#127	—	281	2	no	—
66	#127	C₂×D₂₈₁	1	1	yes	characteristic
67	#102	—	562	4	no	—
68	C₄×C₈	—	281	2	no	—
69	C₄×C₈	C₂×D₂₈₁	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	562	2	no	—
74	C₂×D₄	—	562	2	no	—
75	C₂×D₄	—	562	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	1124	8	no	—
78	D₈	—	1124	4	no	—
79	C₂×C₈	—	281	4	no	—
80	C₂×C₈	C₂²⋊D₂₈₁	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₂₈₁	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	1124	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	1124	4	no	—
89	D₄	—	1124	4	no	—
90	D₄	—	1124	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₂₈₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
94	C₈	—	562	4	no	—
95	C₈	#17985123	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	2248	2	no	—
99	C₂²	—	2248	2	no	—
100	C₂²	—	2248	2	no	—
101	C₂²	#22582875	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#17985123	1	2	yes	—
104	C₄	#22582906	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	4496	2	no	—
107	C₂	#31823839	1	1	yes	characteristic
108	C₂	#31823849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31823840

C₂

yes

maximal, characteristic, complemented

#31823839

C₂

yes

maximal, complemented

#31823811

C₂

yes

maximal, complemented

#31823184

C₂

yes

maximal, characteristic, complemented

#492×C₂₈₁

C₂

yes

maximal, characteristic, complemented

C₂×D₂₂₄₈

—

C₂×#17985123

—

#22582876

C₂²

yes

—

#22582799

—

C₄×#14555886

C₂²

yes

—

C₂×SD₁₆×C₂₈₁

—

C₄⋊Q₈×C₂₈₁

C₂²

yes

characteristic

#127×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

C₂²

yes

characteristic

C₂×D₁₁₂₄

—

C₂×D₁₁₂₄

—

C₂×D₁₁₂₄

—

D₂₂₄₈

—

#17985123

—

C₂×#14555886

D₄

yes

—

C₂×Q₈×C₂₈₁

—

C₂×D₄×C₂₈₁

—

C₂×D₄×C₂₈₁

—

SD₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₄⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

C₂³

yes

characteristic

C₂²×D₂₈₁

—

D₁₁₂₄

—

D₁₁₂₄

—

D₁₁₂₄

—

#14555886

—

C₂³×C₂₈₁

—

Q₈×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

D₈

yes

complemented

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

#127

yes

characteristic

C₂×D₂₈₁

—

C₂×D₂₈₁

—

C₂×D₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×D₈

yes

—

C₄×C₂₈₁

AGL(1,ℤ/8)

yes

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#493

yes

characteristic

C₂×C₂₈₁

#496

yes

—

D₂₈₁

—

C₂₈₁

#1721

yes

characteristic, complemented, Sylow

#1721

—

281

maximal, Sylow

#493

—

281

—

#493

—

281

—

#492

D₂₈₁

yes

characteristic, complemented

#486

—

281

—

#322

—

281

—

C₂×SD₁₆

—

C₂×D₈

—

562

—

C₂×D₈

—

562

—

C₄⋊Q₈

C₂×D₂₈₁

yes

characteristic

#127

—

281

—

#127

C₂×D₂₈₁

yes

characteristic

#102

—

562

—

C₄×C₈

—

281

—

C₄×C₈

C₂×D₂₈₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

562

—

C₂×D₄

—

562

—

C₂×D₄

—

562

—

SD₁₆

—

D₈

—

1124

—

D₈

—

1124

—

C₂×C₈

—

281

—

C₂×C₈

C₂²⋊D₂₈₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₈₁

yes

characteristic

C₂³

—

C₂³

—

1124

—

Q₈

—

D₄

—

D₄

—

D₄

—

1124

—

D₄

—

1124

—

D₄

—

1124

—

C₂×C₄

—

C₂×C₄

D₄×D₂₈₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

C₈

—

562

—

C₈

#17985123

yes

complemented

C₂²

—

C₂²

—

C₂²

—

2248

—

C₂²

—

2248

—

100

C₂²

—

2248

—

101

C₂²

#22582875

yes

characteristic

102

C₄

—

103

C₄

C₂×#17985123

yes

—

104

C₄

#22582906

yes

—

105

C₂

—

106

C₂

—

4496

—

107

C₂

#31823839

yes

characteristic

108

C₂

#31823849

yes

—

	subgroup	G/H
center	C₂²	#22582875
commutator, cosocle	C₄²×C₂₈₁	C₂³
Frattini	C₄²	C₂²×D₂₈₁
Fitting	#492×C₂₈₁	C₂
socle	C₂²×C₂₈₁	#127
hypercenter	#492	D₂₈₁
hypocenter	C₂₈₁	#1721
LCS	G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22582875

commutator, cosocle

C₄²×C₂₈₁

C₂³

Frattini

C₄²

C₂²×D₂₈₁

Fitting

#492×C₂₈₁

C₂

C₂²×C₂₈₁

D₂₈₁

C₂₈₁

G ▷ C₄²×C₂₈₁ ▷ C₂²×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #31823840	C₂	semidirect
2. #31823839	C₂	semidirect
3. #31823811	C₂	semidirect
4. #31823184	C₂	semidirect
5. #492×C₂₈₁	C₂	semidirect
40. C₈×C₂₈₁	D₈	semidirect
54. C₂₈₁	#1721	semidirect
58. #492	D₂₈₁	semidirect
95. C₈	#17985123	semidirect

H ≅ G/N

type

1. #31823840

C₂

semidirect

2. #31823839

C₂

semidirect

3. #31823811

C₂

semidirect

4. #31823184

C₂

semidirect

5. #492×C₂₈₁

C₂

semidirect

40. C₈×C₂₈₁

D₈

semidirect

54. C₂₈₁

#1721

semidirect

58. #492

D₂₈₁

semidirect

95. C₈

#17985123

semidirect

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

subgroup	index
#31823840	2
#31823839	2
#31823811	2
#31823184	2
#492×C₂₈₁	2
#1721	281

subgroup

index

#492×C₂₈₁

281

Subgroups — #49094967

Conjugacy classes of subgroups

Complements

Maximal subgroups