Group #49120401 — subgroups

Subgroups — #49120401

Order 36224.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#31826844	C₂	1	1	yes	maximal, characteristic
2	#31826760	C₂	1	2	yes	maximal
3	#31826214	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₂₈₃⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₂₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#17985382	—	2	2	no	—
7	#22583610	C₂²	1	1	yes	characteristic
8	#22583543	—	2	2	no	—
9	C₄⋊(C₂₈₃⋊C₈)	C₂²	1	2	yes	—
10	#22583534	—	2	2	no	—
11	C₄×C₂₈₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₂₈₃	—	2	2	no	—
13	C₄⋊Q₈×C₂₈₃	—	2	2	no	—
14	C₄⋊Q₈×C₂₈₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₈₃	—	2	2	no	—
16	C₄×Q₈×C₂₈₃	C₂²	1	2	yes	—
17	C₂×Dic₅₆₆	—	2	1	no	—
18	#17985382	—	4	4	no	—
19	C₂×C₂₈₃⋊C₈	—	2	1	no	—
20	C₂×C₂₈₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₈₃	—	2	2	no	—
22	C₂×Q₈×C₂₈₃	—	2	2	no	—
23	C₂×Q₈×C₂₈₃	D₄	1	2	yes	—
24	C₂×Q₈×C₂₈₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₂₈₃	—	2	4	no	—
26	C₄⋊C₄×C₂₈₃	—	2	2	no	—
27	C₄⋊C₄×C₂₈₃	—	2	2	no	—
28	C₄⋊C₄×C₂₈₃	D₄	1	2	yes	—
29	C₄²×C₂₈₃	—	2	2	no	—
30	C₄²×C₂₈₃	—	2	2	no	—
31	C₄²×C₂₈₃	C₂³	1	1	yes	characteristic
32	Dic₅₆₆	—	4	2	no	—
33	C₂×Dic₂₈₃	—	4	1	no	—
34	C₂₈₃⋊C₈	—	2	2	no	—
35	C₂₈₃⋊C₈	—	4	2	no	—
36	Q₈×C₂₈₃	—	2	2	no	—
37	Q₈×C₂₈₃	—	4	2	no	—
38	Q₈×C₂₈₃	—	8	2	no	—
39	Q₈×C₂₈₃	Q₁₆	1	4	yes	complemented
40	C₂×C₄×C₂₈₃	—	2	2	no	—
41	C₂×C₄×C₂₈₃	—	2	2	no	—
42	C₂×C₄×C₂₈₃	—	2	2	no	—
43	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₈₃	C₂×D₄	1	2	yes	—
45	C₂²×C₂₈₃	#120	1	1	yes	characteristic
46	C₄×C₂₈₃	—	2	4	no	—
47	C₄×C₂₈₃	—	2	1	no	—
48	C₄×C₂₈₃	—	4	2	no	—
49	C₄×C₂₈₃	—	4	2	no	—
50	C₄×C₂₈₃	C₂×Q₁₆	1	2	yes	—
51	C₄×C₂₈₃	#137	1	2	yes	—
52	Dic₂₈₃	—	8	1	no	—
53	C₂×C₂₈₃	#451	1	2	yes	—
54	C₂×C₂₈₃	#453	1	1	yes	characteristic
55	C₂₈₃	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	283	1	no	maximal, Sylow
57	Q₈²	D₂₈₃	1	1	yes	characteristic
58	#494	—	283	1	no	—
59	#477	—	283	2	no	—
60	#333	—	283	1	no	—
61	Q₈⋊C₈	—	283	2	no	—
62	C₂×Q₁₆	—	566	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	283	1	no	—
66	C₄⋊Q₈	C₂×D₂₈₃	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₂₈₃	1	2	yes	—
69	#107	—	566	2	no	—
70	C₄⋊C₈	—	283	2	no	—
71	#103	—	566	2	no	—
72	C₄×C₈	—	283	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	566	1	no	—
75	C₂×Q₈	C₂²⋊D₂₈₃	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₂₈₃	1	2	yes	—
77	Q₁₆	—	1132	4	no	—
78	C₂×C₈	—	566	1	no	—
79	C₂×C₈	—	566	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	566	2	no	—
84	C₄⋊C₄	C₂²⋊D₂₈₃	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₂₈₃	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	1132	2	no	—
92	Q₈	#17985382	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	1132	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	2	yes	—
99	C₈	—	566	2	no	—
100	C₈	—	1132	2	no	—
101	C₂²	#22583579	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	2264	1	no	—
107	C₄	C₂×#17985382	1	2	yes	—
108	C₄	#22583641	1	2	yes	—
109	C₂	#31826621	1	2	yes	—
110	C₂	#31826633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31826844

C₂

yes

maximal, characteristic

#31826760

C₂

yes

maximal

#31826214

C₂

yes

maximal, characteristic

Q₈⋊(C₂₈₃⋊C₈)

C₂

yes

maximal

Q₈²×C₂₈₃

C₂

yes

maximal, characteristic

C₂×#17985382

—

#22583610

C₂²

yes

characteristic

#22583543

—

C₄⋊(C₂₈₃⋊C₈)

C₂²

yes

—

#22583534

—

C₄×C₂₈₃⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₈₃

—

C₄⋊Q₈×C₂₈₃

—

C₄⋊Q₈×C₂₈₃

C₂²

yes

characteristic

C₄×Q₈×C₂₈₃

—

C₄×Q₈×C₂₈₃

C₂²

yes

—

C₂×Dic₅₆₆

—

#17985382

—

C₂×C₂₈₃⋊C₈

—

C₂×C₂₈₃⋊C₈

—

C₄⋊Dic₂₈₃

—

C₂×Q₈×C₂₈₃

—

C₂×Q₈×C₂₈₃

D₄

yes

—

C₂×Q₈×C₂₈₃

D₄

yes

—

C₄⋊C₄×C₂₈₃

—

C₄⋊C₄×C₂₈₃

—

C₄⋊C₄×C₂₈₃

—

C₄⋊C₄×C₂₈₃

D₄

yes

—

C₄²×C₂₈₃

—

C₄²×C₂₈₃

—

C₄²×C₂₈₃

C₂³

yes

characteristic

Dic₅₆₆

—

C₂×Dic₂₈₃

—

C₂₈₃⋊C₈

—

C₂₈₃⋊C₈

—

Q₈×C₂₈₃

—

Q₈×C₂₈₃

—

Q₈×C₂₈₃

—

Q₈×C₂₈₃

Q₁₆

yes

complemented

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×D₄

yes

—

C₂²×C₂₈₃

#120

yes

characteristic

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₂×Q₁₆

yes

—

C₄×C₂₈₃

#137

yes

—

Dic₂₈₃

—

C₂×C₂₈₃

#451

yes

—

C₂×C₂₈₃

#453

yes

characteristic

C₂₈₃

#1633

yes

characteristic, complemented, Sylow

#1633

—

283

maximal, Sylow

Q₈²

D₂₈₃

yes

characteristic

#494

—

283

—

#477

—

283

—

#333

—

283

—

Q₈⋊C₈

—

283

—

C₂×Q₁₆

—

566

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

283

—

C₄⋊Q₈

C₂×D₂₈₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₂₈₃

yes

—

#107

—

566

—

C₄⋊C₈

—

283

—

#103

—

566

—

C₄×C₈

—

283

—

C₂×Q₈

—

C₂×Q₈

—

566

—

C₂×Q₈

C₂²⋊D₂₈₃

yes

—

C₂×Q₈

C₂²⋊D₂₈₃

yes

—

Q₁₆

—

1132

—

C₂×C₈

—

566

—

C₂×C₈

—

566

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

566

—

C₄⋊C₄

C₂²⋊D₂₈₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₈₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

1132

—

Q₈

#17985382

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1132

—

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

—

C₈

—

566

—

100

C₈

—

1132

—

101

C₂²

#22583579

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

2264

—

107

C₄

C₂×#17985382

yes

—

108

C₄

#22583641

yes

—

109

C₂

#31826621

yes

—

110

C₂

#31826633

yes

characteristic

	subgroup	G/H
center	C₂²	#22583579
commutator, cosocle	C₄²×C₂₈₃	C₂³
Frattini	C₄²	C₂²×D₂₈₃
Fitting	Q₈²×C₂₈₃	C₂
socle, prime radical	C₂²×C₂₈₃	#120
hypercenter	Q₈²	D₂₈₃
hypocenter	C₂₈₃	#1633
LCS	G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#22583579

commutator, cosocle

C₄²×C₂₈₃

C₂³

Frattini

C₄²

C₂²×D₂₈₃

Fitting

Q₈²×C₂₈₃

C₂

socle, prime radical

C₂²×C₂₈₃

#120

hypercenter

Q₈²

D₂₈₃

hypocenter

C₂₈₃

#1633

LCS

G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
39. Q₈×C₂₈₃	Q₁₆	semidirect
55. C₂₈₃	#1633	semidirect
92. Q₈	#17985382	semidirect

H ≅ G/N

type

39. Q₈×C₂₈₃

Q₁₆

semidirect

55. C₂₈₃

#1633

semidirect

92. Q₈

#17985382

semidirect

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

subgroup	index
#31826844	2
#31826760	2
#31826214	2
Q₈⋊(C₂₈₃⋊C₈)	2
Q₈²×C₂₈₃	2
#1633	283

subgroup

index

#31826844

#31826760

#31826214

Q₈⋊(C₂₈₃⋊C₈)

Q₈²×C₂₈₃

#1633

283

Subgroups — #49120401

Conjugacy classes of subgroups

Complements

Maximal subgroups