Group #49133065 — subgroups

Subgroups — #49133065

Order 36224.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#31827278	C₂	1	4	yes	maximal, complemented
2	#31826737	C₂	1	4	yes	maximal, complemented
3	#31826647	C₂	1	4	yes	maximal, complemented
4	#31826583	C₂	1	2	yes	maximal, complemented
5	#494×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#22583658	C₂²	1	4	yes	—
7	#22583642	—	2	8	no	—
8	#22583641	C₂²	1	8	yes	—
9	#22583633	—	2	4	no	—
10	#22583632	C₂²	1	2	yes	—
11	Q₁₆×D₂₈₃	—	2	4	no	—
12	#22583616	C₂²	1	4	yes	—
13	#22583546	C₂²	1	2	yes	—
14	#22583538	C₂²	1	4	yes	—
15	#22583529	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₂₈₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₂₈₃	C₂²	1	2	yes	—
18	C₄×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
19	#17985393	—	2	8	no	—
20	#17985392	—	4	4	no	—
21	#17985392	C₂³	1	4	yes	—
22	Q₈×D₂₈₃	—	2	8	no	—
23	Dic₁₁₃₂	D₄	1	2	yes	—
24	#17985382	—	2	8	no	—
25	#17985381	D₄	1	4	yes	—
26	#17985379	—	2	8	no	—
27	D₂₂₆₄	D₄	1	2	yes	—
28	#17985376	C₂³	1	4	yes	—
29	#17985375	—	2	4	no	—
30	C₈×D₂₈₃	—	2	4	no	—
31	C₂×Q₈×C₂₈₃	C₂³	1	4	yes	—
32	Q₁₆×C₂₈₃	—	2	4	no	—
33	Q₁₆×C₂₈₃	—	2	4	no	—
34	C₂×C₈×C₂₈₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₂₈₃	—	4	2	no	—
36	C₄²×C₂₈₃	C₂³	1	1	yes	characteristic
37	Dic₅₆₆	—	4	4	no	—
38	Dic₅₆₆	C₂×D₄	1	4	yes	—
39	D₁₁₃₂	—	4	4	no	—
40	D₁₁₃₂	C₂×D₄	1	4	yes	—
41	C₂²⋊D₂₈₃	—	2	4	no	—
42	C₄×D₂₈₃	—	2	4	no	—
43	C₄×D₂₈₃	—	4	8	no	—
44	C₂₈₃⋊C₈	—	2	4	no	—
45	Q₈×C₂₈₃	—	2	8	no	—
46	C₂×C₄×C₂₈₃	—	2	1	no	—
47	C₂×C₄×C₂₈₃	—	2	4	no	—
48	C₂×C₄×C₂₈₃	C₂⁴	1	1	yes	characteristic
49	C₈×C₂₈₃	C₂×D₄	1	2	yes	—
50	C₈×C₂₈₃	C₂×D₄	1	2	yes	—
51	C₂²×C₂₈₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₂₈₃	—	2	4	no	—
53	C₄×C₂₈₃	—	2	2	no	—
54	C₄×C₂₈₃	—	4	4	no	—
55	C₄×C₂₈₃	C₂²×D₄	1	2	yes	—
56	Dic₂₈₃	—	2	4	no	—
57	C₂×C₂₈₃	—	2	1	no	—
58	C₂×C₂₈₃	D₄²	1	1	yes	characteristic
59	D₂₈₃	—	4	4	no	—
60	C₂₈₃	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	283	1	no	maximal, Sylow
62	#578	—	283	4	no	—
63	#494	D₂₈₃	1	1	yes	characteristic, complemented
64	#473	—	283	4	no	—
65	#456	—	283	4	no	—
66	#443	—	283	2	no	—
67	2₋¹⁺⁴	—	283	4	no	—
68	#137	—	283	8	no	—
69	#137	—	566	8	no	—
70	#135	—	283	2	no	—
71	#135	—	566	4	no	—
72	C₂×Q₁₆	—	566	4	no	—
73	C₂×Q₁₆	C₂×D₂₈₃	1	4	yes	—
74	#131	—	283	4	no	—
75	C₄⋊Q₈	C₂×D₂₈₃	1	2	yes	—
76	#108	—	283	2	no	—
77	#104	—	283	4	no	—
78	#101	—	283	4	no	—
79	C₄×C₈	C₂×D₂₈₃	1	1	yes	characteristic
80	#41	—	283	4	no	—
81	#41	—	566	8	no	—
82	#41	—	1132	4	no	—
83	C₂×Q₈	—	566	8	no	—
84	C₂×Q₈	C₂²×D₂₈₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	283	2	no	—
88	Q₁₆	—	566	8	no	—
89	SD₁₆	—	283	4	no	—
90	SD₁₆	—	566	8	no	—
91	D₈	—	283	2	no	—
92	M₁₆	—	283	4	no	—
93	M₁₆	—	566	4	no	—
94	C₂×C₈	—	566	4	no	—
95	C₂×C₈	C₂²×D₂₈₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₂₈₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	283	4	no	—
100	Q₈	—	1132	4	no	—
101	D₄	—	283	4	no	—
102	D₄	—	566	4	no	—
103	D₄	—	1132	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	566	4	no	—
107	C₂×C₄	—	1132	8	no	—
108	C₂×C₄	C₂³×D₂₈₃	1	1	yes	characteristic
109	C₈	—	566	4	no	—
110	C₈	D₄×D₂₈₃	1	2	yes	—
111	C₈	D₄×D₂₈₃	1	2	yes	—
112	C₂²	—	566	4	no	—
113	C₂²	#22583655	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	566	4	no	—
117	C₄	C₂×D₄×D₂₈₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	1132	4	no	—
120	C₂	#31827100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31827278

C₂

yes

maximal, complemented

#31826737

C₂

yes

maximal, complemented

#31826647

C₂

yes

maximal, complemented

#31826583

C₂

yes

maximal, complemented

#494×C₂₈₃

C₂

yes

maximal, characteristic, complemented

#22583658

C₂²

yes

—

#22583642

—

#22583641

C₂²

yes

—

#22583633

—

#22583632

C₂²

yes

—

Q₁₆×D₂₈₃

—

#22583616

C₂²

yes

—

#22583546

C₂²

yes

—

#22583538

C₂²

yes

—

#22583529

C₂²

yes

—

C₂×Q₁₆×C₂₈₃

C₂²

yes

—

C₄⋊Q₈×C₂₈₃

C₂²

yes

—

C₄×C₈×C₂₈₃

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₂₈₃

—

Dic₁₁₃₂

D₄

yes

—

#17985382

—

#17985381

D₄

yes

—

#17985379

—

D₂₂₆₄

D₄

yes

—

#17985376

C₂³

yes

—

#17985375

—

C₈×D₂₈₃

—

C₂×Q₈×C₂₈₃

C₂³

yes

—

Q₁₆×C₂₈₃

—

Q₁₆×C₂₈₃

—

C₂×C₈×C₂₈₃

C₂³

yes

—

C₄⋊C₄×C₂₈₃

—

C₄²×C₂₈₃

C₂³

yes

characteristic

Dic₅₆₆

—

Dic₅₆₆

C₂×D₄

yes

—

D₁₁₃₂

—

D₁₁₃₂

C₂×D₄

yes

—

C₂²⋊D₂₈₃

—

C₄×D₂₈₃

—

C₄×D₂₈₃

—

C₂₈₃⋊C₈

—

Q₈×C₂₈₃

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂⁴

yes

characteristic

C₈×C₂₈₃

C₂×D₄

yes

—

C₈×C₂₈₃

C₂×D₄

yes

—

C₂²×C₂₈₃

H(5,𝔽₂)

yes

characteristic

C₂×D₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₂²×D₄

yes

—

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

D₄²

yes

characteristic

D₂₈₃

—

C₂₈₃

#3293

yes

characteristic, complemented, Sylow

#3293

—

283

maximal, Sylow

#578

—

283

—

#494

D₂₈₃

yes

characteristic, complemented

#473

—

283

—

#456

—

283

—

#443

—

283

—

2₋¹⁺⁴

—

283

—

#137

—

283

—

#137

—

566

—

#135

—

283

—

#135

—

566

—

C₂×Q₁₆

—

566

—

C₂×Q₁₆

C₂×D₂₈₃

yes

—

#131

—

283

—

C₄⋊Q₈

C₂×D₂₈₃

yes

—

#108

—

283

—

#104

—

283

—

#101

—

283

—

C₄×C₈

C₂×D₂₈₃

yes

characteristic

#41

—

283

—

#41

—

566

—

#41

—

1132

—

C₂×Q₈

—

566

—

C₂×Q₈

C₂²×D₂₈₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

283

—

Q₁₆

—

566

—

SD₁₆

—

283

—

SD₁₆

—

566

—

D₈

—

283

—

M₁₆

—

283

—

M₁₆

—

566

—

C₂×C₈

—

566

—

C₂×C₈

C₂²×D₂₈₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₈₃

yes

characteristic

Q₈

—

Q₈

—

283

—

100

Q₈

—

1132

—

101

D₄

—

283

—

102

D₄

—

566

—

103

D₄

—

1132

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

566

—

107

C₂×C₄

—

1132

—

108

C₂×C₄

C₂³×D₂₈₃

yes

characteristic

109

C₈

—

566

—

110

C₈

D₄×D₂₈₃

yes

—

111

C₈

D₄×D₂₈₃

yes

—

112

C₂²

—

566

—

113

C₂²

#22583655

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

566

—

117

C₄

C₂×D₄×D₂₈₃

yes

—

118

C₂

—

119

C₂

—

1132

—

120

C₂

#31827100

yes

characteristic

	subgroup	G/H
center	C₂	#31827100
commutator, cosocle	C₂×C₄×C₂₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₈₃
Fitting	#494×C₂₈₃	C₂
socle	C₂×C₂₈₃	D₄²
hypercenter	#494	D₂₈₃
hypocenter	C₂₈₃	#3293
LCS	G ▷ C₂×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#31827100

commutator, cosocle

C₂×C₄×C₂₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₈₃

Fitting

#494×C₂₈₃

C₂

socle

C₂×C₂₈₃

D₄²

D₂₈₃

C₂₈₃

G ▷ C₂×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #31827278	C₂	semidirect
2. #31826737	C₂	semidirect
3. #31826647	C₂	semidirect
4. #31826583	C₂	semidirect
5. #494×C₂₈₃	C₂	semidirect
60. C₂₈₃	#3293	semidirect
63. #494	D₂₈₃	semidirect

H ≅ G/N

type

1. #31827278

C₂

semidirect

2. #31826737

C₂

semidirect

3. #31826647

C₂

semidirect

4. #31826583

C₂

semidirect

5. #494×C₂₈₃

C₂

semidirect

60. C₂₈₃

#3293

semidirect

63. #494

D₂₈₃

semidirect

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

subgroup	index
#31827278	2
#31826737	2
#31826647	2
#31826583	2
#494×C₂₈₃	2
#3293	283

subgroup

index

#494×C₂₈₃

283

Subgroups — #49133065

Conjugacy classes of subgroups

Complements

Maximal subgroups