Group #49122254 — subgroups

Subgroups — #49122254

Order 36224.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#22583602	C₂	1	1	yes	maximal, characteristic
2	#31826489	C₂	1	1	yes	maximal, characteristic
3	#31826233	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₂₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985376	C₂²	1	2	yes	—
6	#22583602	—	2	1	no	—
7	#22583602	—	2	1	no	—
8	#22583602	—	2	2	no	—
9	C₂×C₄⋊Dic₂₈₃	C₂²	1	2	yes	—
10	C₂×Dic₂₈₃⋊C₄	C₄	1	4	yes	—
11	C₄⋊(C₂₈₃⋊C₈)	—	2	2	no	—
12	C₂×M₁₆×C₂₈₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
14	#97×C₂₈₃	—	2	1	no	—
15	C₄×C₈×C₂₈₃	—	2	1	no	—
16	C₂²×Dic₂₈₃	C₂×C₄	1	4	yes	—
17	#17985376	—	2	4	no	—
18	C₂×C₂₈₃⋊C₈	—	2	2	no	—
19	C₄⋊Dic₂₈₃	—	2	2	no	—
20	C₄⋊Dic₂₈₃	—	2	2	no	—
21	Dic₂₈₃⋊C₄	—	2	4	no	—
22	Dic₂₈₃⋊C₄	—	2	4	no	—
23	C₂²×C₄×C₂₈₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₈₃	—	2	2	no	—
26	M₁₆×C₂₈₃	Q₈	1	4	yes	—
27	C₂×C₈×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	C₂×Dic₂₈₃	—	2	4	no	—
32	C₂×Dic₂₈₃	—	4	4	no	—
33	C₂₈₃⋊C₈	—	4	2	no	—
34	C₂³×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₈₃	—	4	2	no	—
36	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₈₃	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₂₈₃	#41	1	2	yes	—
42	C₈×C₂₈₃	—	2	4	no	—
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	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₂₈₃	#124	1	1	yes	characteristic
50	C₄×C₂₈₃	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₂₈₃	—	4	4	no	—
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	—	283	1	no	—
58	#423	—	283	1	no	—
59	C₄×M₁₆	D₂₈₃	1	1	yes	characteristic
60	#343	—	283	4	no	—
61	C₂×M₁₆	—	283	2	no	—
62	C₂×M₁₆	C₂×D₂₈₃	1	2	yes	—
63	#125	—	566	1	no	—
64	#125	—	566	1	no	—
65	#125	—	566	2	no	—
66	C₂×C₄⋊C₄	—	283	4	no	—
67	C₂×C₄⋊C₄	—	283	2	no	—
68	C₂×C₄²	C₂×D₂₈₃	1	1	yes	characteristic
69	C₄⋊C₈	—	566	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	283	4	no	—
73	C₂²×C₄	C₄×D₂₈₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	566	4	no	—
77	M₁₆	Dic₅₆₆	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	566	2	no	—
80	C₄⋊C₄	—	566	4	no	—
81	C₄⋊C₄	—	566	4	no	—
82	C₄⋊C₄	—	566	2	no	—
83	C₄⋊C₄	—	566	2	no	—
84	C₄²	C₂²⋊D₂₈₃	1	1	yes	characteristic
85	C₄²	C₂²⋊D₂₈₃	1	1	yes	characteristic
86	C₄²	D₁₁₃₂	1	2	yes	—
87	C₂³	C₂×C₄×D₂₈₃	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	566	4	no	—
90	C₂×C₄	—	1132	4	no	—
91	C₂×C₄	(C₂×D₂₈₃)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₂₈₃)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
94	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₅₆₆	1	2	yes	—
96	C₂×C₄	#17985392	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	1132	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₂₈₃)⋊C₄	1	1	yes	characteristic
101	C₂²	C₄×Dic₅₆₆	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	1132	4	no	—
105	C₄	#22583589	1	2	yes	—
106	C₄	#22583598	1	1	yes	characteristic
107	C₄	#22583610	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#31826371	1	1	yes	characteristic
110	C₂	#31826447	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22583602

C₂

yes

maximal, characteristic

#31826489

C₂

yes

maximal, characteristic

#31826233

C₂

yes

maximal

C₄×M₁₆×C₂₈₃

C₂

yes

maximal, characteristic

C₂×#17985376

C₂²

yes

—

#22583602

—

#22583602

—

#22583602

—

C₂×C₄⋊Dic₂₈₃

C₂²

yes

—

C₂×Dic₂₈₃⋊C₄

C₄

yes

—

C₄⋊(C₂₈₃⋊C₈)

—

C₂×M₁₆×C₂₈₃

C₂²

yes

—

C₂×C₄²×C₂₈₃

C₂²

yes

characteristic

#97×C₂₈₃

—

C₄×C₈×C₂₈₃

—

C₂²×Dic₂₈₃

C₂×C₄

yes

—

#17985376

—

C₂×C₂₈₃⋊C₈

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

Dic₂₈₃⋊C₄

—

Dic₂₈₃⋊C₄

—

C₂²×C₄×C₂₈₃

C₂×C₄

yes

—

C₂²×C₄×C₂₈₃

C₂³

yes

characteristic

M₁₆×C₂₈₃

—

M₁₆×C₂₈₃

Q₈

yes

—

C₂×C₈×C₂₈₃

—

C₄²×C₂₈₃

D₄

yes

characteristic

C₄²×C₂₈₃

D₄

yes

characteristic

C₄²×C₂₈₃

D₄

yes

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂₈₃⋊C₈

—

C₂³×C₂₈₃

C₂²×C₄

yes

characteristic

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

—

C₈×C₂₈₃

—

C₂²×C₂₈₃

—

C₂²×C₂₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₃

C₄×Q₈

yes

—

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₂²⋊Q₈

yes

—

C₄×C₂₈₃

#124

yes

characteristic

C₄×C₂₈₃

C₄⋊Q₈

yes

characteristic

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

#391

yes

characteristic

C₂×C₂₈₃

#413

yes

—

C₂₈₃

#1998

yes

characteristic, complemented, Sylow

#1998

—

283

maximal, Sylow

C₂×#125

—

283

—

#423

—

283

—

C₄×M₁₆

D₂₈₃

yes

characteristic

#343

—

283

—

C₂×M₁₆

—

283

—

C₂×M₁₆

C₂×D₂₈₃

yes

—

#125

—

566

—

#125

—

566

—

#125

—

566

—

C₂×C₄⋊C₄

—

283

—

C₂×C₄⋊C₄

—

283

—

C₂×C₄²

C₂×D₂₈₃

yes

characteristic

C₄⋊C₈

—

566

—

#97

—

C₄×C₈

—

C₂²×C₄

—

283

—

C₂²×C₄

C₄×D₂₈₃

yes

—

C₂²×C₄

C₂²×D₂₈₃

yes

characteristic

M₁₆

—

M₁₆

—

566

—

M₁₆

Dic₅₆₆

yes

—

C₂×C₈

—

C₂×C₈

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄²

C₂²⋊D₂₈₃

yes

characteristic

C₄²

C₂²⋊D₂₈₃

yes

characteristic

C₄²

D₁₁₃₂

yes

—

C₂³

C₂×C₄×D₂₈₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

566

—

C₂×C₄

—

1132

—

C₂×C₄

(C₂×D₂₈₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₈₃)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

C₂×C₄

C₂×Dic₅₆₆

yes

—

C₂×C₄

#17985392

yes

—

C₈

—

C₈

—

1132

—

C₂²

—

100

C₂²

C₂×(C₂×D₂₈₃)⋊C₄

yes

characteristic

101

C₂²

C₄×Dic₅₆₆

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

1132

—

105

C₄

#22583589

yes

—

106

C₄

#22583598

yes

characteristic

107

C₄

#22583610

yes

characteristic

108

C₂

—

109

C₂

#31826371

yes

characteristic

110

C₂

#31826447

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₂₈₃)⋊C₄
commutator, prime radical	C₂³×C₂₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₃
Fitting	C₄×M₁₆×C₂₈₃	C₂
socle	C₂²×C₂₈₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₈₃	C₂³
hypercenter	C₄×M₁₆	D₂₈₃
hypocenter	C₂₈₃	#1998
LCS	G ▷ C₂³×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₂₈₃)⋊C₄

commutator, prime radical

C₂³×C₂₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₃

Fitting

C₄×M₁₆×C₂₈₃

C₂

socle

C₂²×C₂₈₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₈₃

C₂³

hypercenter

C₄×M₁₆

D₂₈₃

hypocenter

C₂₈₃

#1998

LCS

G ▷ C₂³×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

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
C₂×#22583602	2
#31826489	2
#31826233	2
C₄×M₁₆×C₂₈₃	2
#1998	283

subgroup

index

C₄×M₁₆×C₂₈₃

283

Subgroups — #49122254

Conjugacy classes of subgroups

Complements

Maximal subgroups