Group #49122255 — subgroups

Subgroups — #49122255

Order 36224.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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	#31826489	C₂	1	1	yes	maximal, characteristic
2	C₄×#17985376	C₂	1	1	yes	maximal, characteristic
3	#31826234	C₂	1	4	yes	maximal
4	C₂×#125×C₂₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985376	C₂²	1	2	yes	—
6	C₂×#17985376	C₂²	1	2	yes	—
7	C₄⋊(C₂₈₃⋊C₈)	—	2	2	no	—
8	#22583521	—	2	1	no	—
9	C₄×C₂₈₃⋊C₈	—	2	1	no	—
10	#125×C₂₈₃	—	2	1	no	—
11	#125×C₂₈₃	—	2	1	no	—
12	#125×C₂₈₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₈₃	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₈₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
16	#17985376	—	2	2	no	—
17	#17985376	—	2	4	no	—
18	#17985376	Q₈	1	4	yes	—
19	C₂×C₂₈₃⋊C₈	—	2	2	no	—
20	C₂×C₂₈₃⋊C₈	—	2	2	no	—
21	C₂²×C₄×C₂₈₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₈₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₂₈₃	—	2	4	no	—
25	C₄⋊C₄×C₂₈₃	—	2	4	no	—
26	C₄⋊C₄×C₂₈₃	—	2	2	no	—
27	C₄⋊C₄×C₂₈₃	—	2	2	no	—
28	C₄²×C₂₈₃	D₄	1	1	yes	characteristic
29	C₄²×C₂₈₃	D₄	1	1	yes	characteristic
30	C₄²×C₂₈₃	D₄	1	2	yes	—
31	C₂₈₃⋊C₈	—	2	4	no	—
32	C₂₈₃⋊C₈	—	4	2	no	—
33	C₂³×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₈₃	—	2	4	no	—
35	C₂×C₄×C₂₈₃	—	4	2	no	—
36	C₂×C₄×C₂₈₃	—	4	4	no	—
37	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₈₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₂₈₃	#41	1	2	yes	—
43	C₂²×C₂₈₃	—	2	2	no	—
44	C₂²×C₂₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₂₈₃	C₄×Q₈	1	2	yes	—
46	C₄×C₂₈₃	—	2	2	no	—
47	C₄×C₂₈₃	—	2	2	no	—
48	C₄×C₂₈₃	—	4	4	no	—
49	C₄×C₂₈₃	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₂₈₃	#124	1	1	yes	characteristic
51	C₄×C₂₈₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₂₈₃	—	2	2	no	—
53	C₂×C₂₈₃	#391	1	1	yes	characteristic
54	C₂×C₂₈₃	#413	1	2	yes	—
55	C₂₈₃	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	283	1	no	maximal, Sylow
57	C₂×#125	D₂₈₃	1	1	yes	characteristic
58	#423	—	283	1	no	—
59	C₄×M₁₆	—	283	1	no	—
60	#343	—	283	4	no	—
61	C₂×M₁₆	—	283	2	no	—
62	C₂×M₁₆	—	283	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₂₈₃	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₂₈₃	1	2	yes	—
68	C₂×C₄²	C₂×D₂₈₃	1	1	yes	characteristic
69	C₄⋊C₈	—	566	2	no	—
70	#97	—	566	1	no	—
71	C₄×C₈	—	566	1	no	—
72	C₂²×C₄	C₂×Dic₂₈₃	1	4	yes	—
73	C₂²×C₄	C₂×Dic₂₈₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
75	M₁₆	—	283	4	no	—
76	M₁₆	—	566	2	no	—
77	M₁₆	—	566	4	no	—
78	C₂×C₈	—	566	2	no	—
79	C₂×C₈	—	566	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	C₂²⋊D₂₈₃	1	1	yes	characteristic
85	C₄²	C₂²⋊D₂₈₃	1	1	yes	characteristic
86	C₄²	C₂²⋊D₂₈₃	1	2	yes	—
87	C₂³	C₂²×Dic₂₈₃	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₂₈₃	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₂₈₃	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₂₈₃	1	2	yes	—
96	C₂×C₄	#17985393	1	2	yes	—
97	C₈	—	566	4	no	—
98	C₈	—	1132	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₂₈₃	1	1	yes	characteristic
101	C₂²	Q₈×Dic₂₈₃	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#22583586	1	2	yes	—
106	C₄	#22583599	1	1	yes	characteristic
107	C₄	#22583608	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#31826372	1	1	yes	characteristic
110	C₂	#31826449	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31826489

C₂

yes

maximal, characteristic

C₄×#17985376

C₂

yes

maximal, characteristic

#31826234

C₂

yes

maximal

C₂×#125×C₂₈₃

C₂

yes

maximal, characteristic

C₂×#17985376

C₂²

yes

—

C₂×#17985376

C₂²

yes

—

C₄⋊(C₂₈₃⋊C₈)

—

#22583521

—

C₄×C₂₈₃⋊C₈

—

#125×C₂₈₃

—

#125×C₂₈₃

—

#125×C₂₈₃

—

C₂×C₄⋊C₄×C₂₈₃

C₄

yes

—

C₂×C₄⋊C₄×C₂₈₃

C₂²

yes

—

C₂×C₄²×C₂₈₃

C₂²

yes

—

—

Q₈

yes

—

C₂×C₂₈₃⋊C₈

—

C₂×C₂₈₃⋊C₈

—

C₂²×C₄×C₂₈₃

C₂×C₄

yes

—

C₂²×C₄×C₂₈₃

C₂×C₄

yes

—

C₂²×C₄×C₂₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₈₃

—

C₄⋊C₄×C₂₈₃

—

C₄⋊C₄×C₂₈₃

—

C₄⋊C₄×C₂₈₃

—

C₄²×C₂₈₃

D₄

yes

characteristic

C₄²×C₂₈₃

D₄

yes

characteristic

C₄²×C₂₈₃

D₄

yes

—

C₂₈₃⋊C₈

—

C₂₈₃⋊C₈

—

C₂³×C₂₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×Q₈

yes

—

C₂×C₄×C₂₈₃

#41

yes

—

C₂²×C₂₈₃

—

C₂²×C₂₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₃

C₄×Q₈

yes

—

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₂²⋊Q₈

yes

—

C₄×C₂₈₃

#124

yes

characteristic

C₄×C₂₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₂₈₃

—

C₂×C₂₈₃

#391

yes

characteristic

C₂×C₂₈₃

#413

yes

—

C₂₈₃

#1998

yes

characteristic, complemented, Sylow

#1998

—

283

maximal, Sylow

C₂×#125

D₂₈₃

yes

characteristic

#423

—

283

—

C₄×M₁₆

—

283

—

#343

—

283

—

C₂×M₁₆

—

283

—

C₂×M₁₆

—

283

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₂₈₃

yes

—

C₂×C₄⋊C₄

C₂×D₂₈₃

yes

—

C₂×C₄²

C₂×D₂₈₃

yes

characteristic

C₄⋊C₈

—

566

—

#97

—

566

—

C₄×C₈

—

566

—

C₂²×C₄

C₂×Dic₂₈₃

yes

—

C₂²×C₄

C₂×Dic₂₈₃

yes

—

C₂²×C₄

C₂²×D₂₈₃

yes

characteristic

M₁₆

—

283

—

M₁₆

—

566

—

M₁₆

—

566

—

C₂×C₈

—

566

—

C₂×C₈

—

566

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₂₈₃

yes

characteristic

C₄²

C₂²⋊D₂₈₃

yes

characteristic

C₄²

C₂²⋊D₂₈₃

yes

—

C₂³

C₂²×Dic₂₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₈₃

yes

—

C₂×C₄

C₂²⋊Dic₂₈₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

C₂×C₄

Q₈×D₂₈₃

yes

—

C₂×C₄

#17985393

yes

—

C₈

—

566

—

C₈

—

1132

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₂₈₃

yes

characteristic

101

C₂²

Q₈×Dic₂₈₃

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#22583586

yes

—

106

C₄

#22583599

yes

characteristic

107

C₄

#22583608

yes

characteristic

108

C₂

—

109

C₂

#31826372

yes

characteristic

110

C₂

#31826449

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₈₃
commutator, prime radical	C₂³×C₂₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₃
Fitting	C₂×#125×C₂₈₃	C₂
socle	C₂²×C₂₈₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₈₃	C₂³
hypercenter	C₂×#125	D₂₈₃
hypocenter	C₂₈₃	#1998
LCS	G ▷ C₂³×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₈₃

commutator, prime radical

C₂³×C₂₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₃

Fitting

C₂×#125×C₂₈₃

C₂

socle

C₂²×C₂₈₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₈₃

C₂³

D₂₈₃

C₂₈₃

G ▷ C₂³×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
55. C₂₈₃	#1998	semidirect

H ≅ G/N

type

55. C₂₈₃

#1998

semidirect

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

subgroup	index
#31826489	2
C₄×#17985376	2
#31826234	2
C₂×#125×C₂₈₃	2
#1998	283

subgroup

index

C₂×#125×C₂₈₃

283

Subgroups — #49122255

Conjugacy classes of subgroups

Complements

Maximal subgroups