Group #49123338 — subgroups

Subgroups — #49123338

Order 36224.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#31826903	C₂	1	2	yes	maximal, complemented
2	#31826842	C₂	1	1	yes	maximal, characteristic, complemented
3	#31826583	C₂	1	1	yes	maximal, characteristic, complemented
4	#31826279	C₂	1	2	yes	maximal, complemented
5	#364×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#22583632	C₂²	1	1	yes	characteristic
7	C₂×D₂₂₆₄	—	2	1	no	—
8	C₂×D₂₂₆₄	C₂²	1	2	yes	—
9	#22583616	—	2	1	no	—
10	#22583607	—	2	1	no	—
11	#22583555	—	2	2	no	—
12	D₄₅₂₈	—	2	2	no	—
13	#22583546	C₂²	1	1	yes	characteristic
14	#22583538	—	2	1	no	—
15	M₃₂×C₂₈₃	C₂²	1	2	yes	—
16	C₄×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
17	#17985392	—	2	1	no	—
18	C₂×D₁₁₃₂	—	2	2	no	—
19	C₂×D₁₁₃₂	—	4	2	no	—
20	Dic₁₁₃₂	D₄	1	1	yes	characteristic, complemented
21	#17985381	—	2	1	no	—
22	D₂₂₆₄	—	2	2	no	—
23	D₂₂₆₄	—	2	2	no	—
24	D₂₂₆₄	—	4	1	no	—
25	D₂₂₆₄	—	4	1	no	—
26	D₂₂₆₄	D₄	1	1	yes	characteristic, complemented
27	#17985376	—	2	1	no	—
28	#17985375	—	4	1	no	—
29	C₈×D₂₈₃	—	4	1	no	—
30	C₂×C₈×C₂₈₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₈₃	C₂³	1	1	yes	characteristic
32	C₄²×C₂₈₃	D₄	1	1	yes	characteristic
33	C₁₆×C₂₈₃	—	2	2	no	—
34	C₂²×D₂₈₃	—	4	2	no	—
35	Dic₅₆₆	—	2	1	no	—
36	D₁₁₃₂	—	2	1	no	—
37	D₁₁₃₂	—	4	2	no	—
38	D₁₁₃₂	—	4	2	no	—
39	D₁₁₃₂	—	8	4	no	—
40	C₂²⋊D₂₈₃	—	4	1	no	—
41	C₄×D₂₈₃	—	4	1	no	—
42	C₂₈₃⋊C₈	—	4	1	no	—
43	C₂×C₄×C₂₈₃	—	2	1	no	—
44	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
45	C₈×C₂₈₃	D₈	1	1	yes	characteristic
46	C₈×C₂₈₃	D₈	1	1	yes	characteristic
47	C₈×C₂₈₃	C₂×D₄	1	1	yes	characteristic
48	C₈×C₂₈₃	#41	1	1	yes	characteristic
49	C₂²×C₂₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₈₃	—	4	1	no	—
51	C₂×D₂₈₃	—	8	4	no	—
52	C₂×D₂₈₃	—	8	2	no	—
53	C₄×C₂₈₃	—	2	2	no	—
54	C₄×C₂₈₃	#121	1	1	yes	characteristic
55	C₄×C₂₈₃	C₂×D₈	1	1	yes	characteristic
56	Dic₂₈₃	—	4	1	no	—
57	C₂×C₂₈₃	—	2	1	no	—
58	C₂×C₂₈₃	#459	1	1	yes	characteristic
59	D₂₈₃	—	8	1	no	—
60	D₂₈₃	—	16	2	no	—
61	C₂₈₃	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	283	1	no	maximal, Sylow
63	#509	—	283	2	no	—
64	#493	—	283	1	no	—
65	#443	—	283	1	no	—
66	#364	D₂₈₃	1	1	yes	characteristic, complemented
67	#361	—	283	2	no	—
68	#135	—	283	1	no	—
69	C₂×D₈	—	283	2	no	—
70	C₂×D₈	—	566	1	no	—
71	#131	—	566	1	no	—
72	#127	—	566	1	no	—
73	SD₃₂	—	566	2	no	—
74	D₁₆	—	566	2	no	—
75	M₃₂	C₂×D₂₈₃	1	2	yes	—
76	#108	—	283	1	no	—
77	#104	—	566	1	no	—
78	C₄×C₈	C₂×D₂₈₃	1	1	yes	characteristic
79	#41	—	566	1	no	—
80	C₂×D₄	—	566	2	no	—
81	C₂×D₄	—	1132	2	no	—
82	Q₁₆	—	283	1	no	—
83	SD₁₆	—	566	1	no	—
84	D₈	—	283	1	no	—
85	D₈	—	566	2	no	—
86	D₈	—	566	2	no	—
87	D₈	—	1132	1	no	—
88	D₈	—	1132	1	no	—
89	M₁₆	—	566	1	no	—
90	M₁₆	—	1132	1	no	—
91	C₂×C₈	—	1132	1	no	—
92	C₂×C₈	D₁₁₃₂	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₂₈₃	1	1	yes	characteristic
94	C₄²	D₁₁₃₂	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	1132	2	no	—
97	Q₈	—	566	1	no	—
98	D₄	—	566	1	no	—
99	D₄	—	1132	1	no	—
100	D₄	—	1132	2	no	—
101	D₄	—	1132	2	no	—
102	D₄	—	2264	4	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	1132	1	no	—
105	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
106	C₈	—	1132	1	no	—
107	C₈	D₂₂₆₄	1	1	yes	characteristic
108	C₈	D₂₂₆₄	1	1	yes	characteristic
109	C₈	D₄×D₂₈₃	1	1	yes	characteristic
110	C₈	#17985393	1	1	yes	characteristic
111	C₂²	—	1132	1	no	—
112	C₂²	—	2264	4	no	—
113	C₂²	—	2264	2	no	—
114	C₂²	#22583638	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	1132	1	no	—
117	C₄	#22583585	1	1	yes	characteristic
118	C₄	C₂×D₂₂₆₄	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	4528	2	no	—
121	C₂	—	2264	1	no	—
122	C₂	#31826661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31826903

C₂

yes

maximal, complemented

#31826842

C₂

yes

maximal, characteristic, complemented

#31826583

C₂

yes

maximal, characteristic, complemented

#31826279

C₂

yes

maximal, complemented

#364×C₂₈₃

C₂

yes

maximal, characteristic, complemented

#22583632

C₂²

yes

characteristic

C₂×D₂₂₆₄

—

C₂×D₂₂₆₄

C₂²

yes

—

#22583616

—

#22583607

—

#22583555

—

D₄₅₂₈

—

#22583546

C₂²

yes

characteristic

#22583538

—

M₃₂×C₂₈₃

C₂²

yes

—

C₄×C₈×C₂₈₃

C₂²

yes

characteristic

#17985392

—

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

Dic₁₁₃₂

D₄

yes

characteristic, complemented

#17985381

—

D₂₂₆₄

—

D₂₂₆₄

—

D₂₂₆₄

—

D₂₂₆₄

—

D₂₂₆₄

D₄

yes

characteristic, complemented

#17985376

—

#17985375

—

C₈×D₂₈₃

—

C₂×C₈×C₂₈₃

D₄

yes

characteristic

C₂×C₈×C₂₈₃

C₂³

yes

characteristic

C₄²×C₂₈₃

D₄

yes

characteristic

C₁₆×C₂₈₃

—

C₂²×D₂₈₃

—

Dic₅₆₆

—

D₁₁₃₂

—

D₁₁₃₂

—

D₁₁₃₂

—

D₁₁₃₂

—

C₂²⋊D₂₈₃

—

C₄×D₂₈₃

—

C₂₈₃⋊C₈

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₈×C₂₈₃

D₈

yes

characteristic

C₈×C₂₈₃

D₈

yes

characteristic

C₈×C₂₈₃

C₂×D₄

yes

C₈×C₂₈₃

yes

C₂²×C₂₈₃

yes

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

#121

yes

characteristic

C₄×C₂₈₃

C₂×D₈

yes

characteristic

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

#459

yes

characteristic

D₂₈₃

—

D₂₈₃

—

C₂₈₃

#2213

yes

characteristic, complemented, Sylow

#2213

—

283

maximal, Sylow

#509

—

283

—

#493

—

283

—

#443

—

283

—

#364

D₂₈₃

yes

characteristic, complemented

#361

—

283

—

#135

—

283

—

C₂×D₈

—

283

—

C₂×D₈

—

566

—

#131

—

566

—

#127

—

566

—

SD₃₂

—

566

—

D₁₆

—

566

—

M₃₂

C₂×D₂₈₃

yes

—

#108

—

283

—

#104

—

566

—

C₄×C₈

C₂×D₂₈₃

yes

characteristic

#41

—

566

—

C₂×D₄

—

566

—

C₂×D₄

—

1132

—

Q₁₆

—

283

—

SD₁₆

—

566

—

D₈

—

283

—

D₈

—

566

—

D₈

—

566

—

D₈

—

1132

—

D₈

—

1132

—

M₁₆

—

566

—

M₁₆

—

1132

—

C₂×C₈

—

1132

—

C₂×C₈

D₁₁₃₂

yes

characteristic

C₂×C₈

C₂²×D₂₈₃

yes

characteristic

C₄²

D₁₁₃₂

yes

characteristic

C₁₆

—

C₂³

—

1132

—

Q₈

—

566

—

D₄

—

566

—

D₄

—

1132

—

100

D₄

—

1132

—

101

D₄

—

1132

—

102

D₄

—

2264

—

103

C₂×C₄

—

104

C₂×C₄

—

1132

—

105

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

106

C₈

—

1132

—

107

C₈

D₂₂₆₄

yes

characteristic

108

C₈

D₂₂₆₄

yes

characteristic

109

C₈

D₄×D₂₈₃

yes

characteristic

110

C₈

#17985393

yes

characteristic

111

C₂²

—

1132

—

112

C₂²

—

2264

—

113

C₂²

—

2264

—

114

C₂²

#22583638

yes

characteristic

115

C₄

—

116

C₄

—

1132

—

117

C₄

#22583585

yes

characteristic

118

C₄

C₂×D₂₂₆₄

yes

characteristic

119

C₂

—

120

C₂

—

4528

—

121

C₂

—

2264

—

122

C₂

#31826661

yes

characteristic

	subgroup	G/H
center	C₂	#31826661
commutator, cosocle	C₂×C₈×C₂₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₈₃
Fitting	#364×C₂₈₃	C₂
socle	C₂×C₂₈₃	#459
hypercenter	#364	D₂₈₃
hypocenter	C₂₈₃	#2213
LCS	G ▷ C₂×C₈×C₂₈₃ ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#31826661

commutator, cosocle

C₂×C₈×C₂₈₃

C₂³

Frattini

C₂×C₈

C₂²×D₂₈₃

Fitting

#364×C₂₈₃

C₂

C₂×C₂₈₃

D₂₈₃

C₂₈₃

G ▷ C₂×C₈×C₂₈₃ ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #31826903	C₂	semidirect
2. #31826842	C₂	semidirect
3. #31826583	C₂	semidirect
4. #31826279	C₂	semidirect
5. #364×C₂₈₃	C₂	semidirect
20. Dic₁₁₃₂	D₄	semidirect
26. D₂₂₆₄	D₄	semidirect
61. C₂₈₃	#2213	semidirect
66. #364	D₂₈₃	semidirect

H ≅ G/N

type

1. #31826903

C₂

semidirect

2. #31826842

C₂

semidirect

3. #31826583

C₂

semidirect

4. #31826279

C₂

semidirect

5. #364×C₂₈₃

C₂

semidirect

20. Dic₁₁₃₂

D₄

semidirect

26. D₂₂₆₄

D₄

semidirect

61. C₂₈₃

#2213

semidirect

66. #364

D₂₈₃

semidirect

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

subgroup	index
#31826903	2
#31826842	2
#31826583	2
#31826279	2
#364×C₂₈₃	2
#2213	283

subgroup

index

#364×C₂₈₃

283

Subgroups — #49123338

Conjugacy classes of subgroups

Complements

Maximal subgroups