Group #49122241 — subgroups

Subgroups — #49122241

Order 36224.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#31827022	C₂	1	1	yes	maximal, characteristic
2	#31826311	C₂	1	1	yes	maximal, characteristic
3	#31826226	C₂	1	4	yes	maximal
4	#423×C₂₈₃	C₂	1	1	yes	maximal, characteristic
5	#22583610	C₄	1	2	yes	complemented
6	#22583602	C₄	1	2	yes	complemented
7	#22583589	—	2	2	no	—
8	C₄×Dic₅₆₆	—	2	2	no	—
9	#22583569	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₈₃	C₂²	1	2	yes	—
11	C₂₈₃⋊#95	—	2	2	no	—
12	C₂×M₁₆×C₂₈₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₈₃	—	2	2	no	—
15	C₂×Dic₅₆₆	—	2	2	no	—
16	C₂²×Dic₂₈₃	—	2	2	no	—
17	C₄⋊Dic₂₈₃	—	2	2	no	—
18	C₄⋊Dic₂₈₃	C₂×C₄	1	4	yes	—
19	Dic₂₈₃⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₂₈₃	—	2	2	no	—
21	C₄×Dic₂₈₃	—	2	2	no	—
22	C₄×Dic₂₈₃	—	2	2	no	—
23	C₄×Dic₂₈₃	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₂₈₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₈₃	—	2	4	no	—
27	C₂×C₈×C₂₈₃	—	2	2	no	—
28	C₄²×C₂₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₈₃	D₄	1	2	yes	—
31	Dic₅₆₆	—	4	4	no	—
32	C₂×Dic₂₈₃	—	2	4	no	—
33	C₂×Dic₂₈₃	—	2	4	no	—
34	C₂×Dic₂₈₃	—	4	4	no	—
35	C₂³×C₂₈₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₈₃	—	4	2	no	—
37	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₈₃	C₂²×C₄	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	C₈×C₂₈₃	—	4	2	no	—
44	C₂²×C₂₈₃	—	2	2	no	—
45	C₂²×C₂₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₈₃	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₂₈₃	—	2	2	no	—
48	C₄×C₂₈₃	—	2	2	no	—
49	C₄×C₂₈₃	C₄×Q₈	1	2	yes	—
50	C₄×C₂₈₃	#124	1	1	yes	characteristic
51	C₄×C₂₈₃	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₂₈₃	—	4	4	no	—
53	Dic₂₈₃	—	4	4	no	—
54	C₂×C₂₈₃	—	2	2	no	—
55	C₂×C₂₈₃	#391	1	1	yes	characteristic
56	C₂×C₂₈₃	#421	1	2	yes	—
57	C₂₈₃	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	283	1	no	maximal, Sylow
59	#533	—	283	1	no	—
60	#423	D₂₈₃	1	1	yes	characteristic
61	#376	—	283	1	no	—
62	#339	—	283	4	no	—
63	C₂×M₁₆	C₂×D₂₈₃	1	2	yes	—
64	C₄⋊Q₈	—	283	2	no	—
65	#125	—	283	2	no	—
66	C₂²⋊Q₈	—	566	2	no	—
67	C₄×Q₈	—	566	2	no	—
68	#117	—	283	2	no	—
69	C₂×C₄²	—	283	2	no	—
70	C₂×C₄²	C₂×D₂₈₃	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	566	2	no	—
73	C₂×Q₈	—	566	2	no	—
74	C₂²×C₄	—	566	2	no	—
75	C₂²×C₄	D₁₁₃₂	1	2	yes	—
76	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	283	4	no	—
80	C₄⋊C₄	—	566	4	no	—
81	C₄⋊C₄	—	566	2	no	—
82	C₂²⋊C₄	—	566	2	no	—
83	C₄²	—	283	4	no	—
84	C₄²	—	566	2	no	—
85	C₄²	—	566	2	no	—
86	C₄²	C₄×D₂₈₃	1	1	yes	characteristic
87	C₄²	C₄×D₂₈₃	1	1	yes	characteristic
88	C₄²	C₂²⋊D₂₈₃	1	2	yes	—
89	C₂³	C₂×D₁₁₃₂	1	1	yes	characteristic
90	Q₈	—	1132	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	566	4	no	—
93	C₂×C₄	—	566	4	no	—
94	C₂×C₄	—	1132	4	no	—
95	C₂×C₄	(C₂×D₂₈₃)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₂₈₃)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₂₈₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₂₈₃	1	2	yes	—
100	C₂×C₄	#17985391	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×(C₂×D₂₈₃)⋊C₄	1	1	yes	characteristic
104	C₂²	#22583590	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	1132	4	no	—
108	C₄	—	1132	4	no	—
109	C₄	#22583575	1	2	yes	—
110	C₄	#22583597	1	1	yes	characteristic
111	C₄	#22583608	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#31826370	1	1	yes	characteristic
114	C₂	#31826481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31827022

C₂

yes

maximal, characteristic

#31826311

C₂

yes

maximal, characteristic

#31826226

C₂

yes

maximal

#423×C₂₈₃

C₂

yes

maximal, characteristic

#22583610

C₄

yes

complemented

#22583602

C₄

yes

complemented

#22583589

—

C₄×Dic₅₆₆

—

#22583569

C₂²

yes

—

C₂×C₄×Dic₂₈₃

C₂²

yes

—

C₂₈₃⋊#95

—

C₂×M₁₆×C₂₈₃

C₂²

yes

—

C₂×C₄²×C₂₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₈₃

—

C₂×Dic₅₆₆

—

C₂²×Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

C₂×C₄

yes

—

Dic₂₈₃⋊C₄

—

C₂²⋊Dic₂₈₃

—

C₄×Dic₂₈₃

—

C₄×Dic₂₈₃

—

C₄×Dic₂₈₃

Q₈

yes

complemented

C₂²×C₄×C₂₈₃

D₄

yes

—

C₂²×C₄×C₂₈₃

C₂³

yes

characteristic

M₁₆×C₂₈₃

—

C₂×C₈×C₂₈₃

—

C₄²×C₂₈₃

C₂×C₄

yes

characteristic

C₄²×C₂₈₃

C₂×C₄

yes

characteristic

C₄²×C₂₈₃

D₄

yes

—

Dic₅₆₆

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂³×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₃

C₂²×C₄

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

C₂²⋊Q₈

yes

—

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₄×Q₈

yes

—

C₄×C₂₈₃

#124

yes

characteristic

C₄×C₂₈₃

C₄⋊Q₈

yes

characteristic

Dic₂₈₃

—

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

#391

yes

characteristic

C₂×C₂₈₃

#421

yes

—

C₂₈₃

#1995

yes

characteristic, complemented, Sylow

#1995

—

283

maximal, Sylow

#533

—

283

—

#423

D₂₈₃

yes

characteristic

#376

—

283

—

#339

—

283

—

C₂×M₁₆

C₂×D₂₈₃

yes

—

C₄⋊Q₈

—

283

—

#125

—

283

—

C₂²⋊Q₈

—

566

—

C₄×Q₈

—

566

—

#117

—

283

—

C₂×C₄²

—

283

—

C₂×C₄²

C₂×D₂₈₃

yes

characteristic

C₄⋊C₈

—

#95

—

566

—

C₂×Q₈

—

566

—

C₂²×C₄

—

566

—

C₂²×C₄

D₁₁₃₂

yes

—

C₂²×C₄

C₂²×D₂₈₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

283

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₂²⋊C₄

—

566

—

C₄²

—

283

—

C₄²

—

566

—

C₄²

—

566

—

C₄²

C₄×D₂₈₃

yes

characteristic

C₄²

C₄×D₂₈₃

yes

characteristic

C₄²

C₂²⋊D₂₈₃

yes

—

C₂³

C₂×D₁₁₃₂

yes

characteristic

Q₈

—

1132

—

C₂×C₄

—

C₂×C₄

—

566

—

C₂×C₄

—

566

—

C₂×C₄

—

1132

—

C₂×C₄

(C₂×D₂₈₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₈₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₈₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

C₂×C₄

Q₈×D₂₈₃

yes

—

100

C₂×C₄

#17985391

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×(C₂×D₂₈₃)⋊C₄

yes

characteristic

104

C₂²

#22583590

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

1132

—

108

C₄

—

1132

—

109

C₄

#22583575

yes

—

110

C₄

#22583597

yes

characteristic

111

C₄

#22583608

yes

characteristic

112

C₂

—

113

C₂

#31826370

yes

characteristic

114

C₂

#31826481

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₈₃
commutator	C₂×C₄×C₂₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₃
Fitting	#423×C₂₈₃	C₂
socle	C₂²×C₂₈₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₈₃	C₂×D₄
cosocle	C₂²×C₄×C₂₈₃	C₂³
hypercenter	#423	D₂₈₃
hypocenter	C₂₈₃	#1995
LCS	G ▷ C₂×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₈₃

commutator

C₂×C₄×C₂₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₃

Fitting

#423×C₂₈₃

C₂

socle

C₂²×C₂₈₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₈₃

C₂×D₄

cosocle

C₂²×C₄×C₂₈₃

C₂³

D₂₈₃

C₂₈₃

G ▷ C₂×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #22583610	C₄	semidirect
6. #22583602	C₄	semidirect
23. C₄×Dic₂₈₃	Q₈	semidirect
57. C₂₈₃	#1995	semidirect

H ≅ G/N

type

5. #22583610

C₄

semidirect

6. #22583602

C₄

semidirect

23. C₄×Dic₂₈₃

Q₈

semidirect

57. C₂₈₃

#1995

semidirect

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

subgroup	index
#31827022	2
#31826311	2
#31826226	2
#423×C₂₈₃	2
#1995	283

subgroup

index

#423×C₂₈₃

283

Subgroups — #49122241

Conjugacy classes of subgroups

Complements

Maximal subgroups