Group #33006582 — subgroups

Subgroups — #33006582

Order 18688.

188 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 73 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	Q₃₂×Dic₇₃	C₂	1	2	yes	maximal
2	#22608133	C₂	1	1	yes	maximal, characteristic
3	#22608112	C₂	1	2	yes	maximal
4	#22608064	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×Dic₇₃	—	2	2	no	—
7	C₂×#14556497	C₂²	1	1	yes	characteristic
8	#17988413	C₂²	1	2	yes	—
9	#17988391	—	2	2	no	—
10	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
12	Q₆₄×C₇₃	C₄	1	4	yes	complemented
13	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
14	Q₈×Dic₇₃	—	4	2	no	—
15	C₂×#11924003	D₄	1	1	yes	characteristic
16	#14556525	—	2	2	no	—
17	#14556515	—	4	2	no	—
18	C₈×Dic₇₃	D₄	1	1	yes	characteristic
19	#14556497	—	2	1	no	—
20	C₂×Q₁₆×C₇₃	—	2	2	no	—
21	Q₃₂×C₇₃	—	2	2	no	—
22	Q₃₂×C₇₃	C₂×C₄	1	4	yes	—
23	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
24	C₃₂×C₇₃	C₂×C₄	1	2	yes	—
25	C₂×#10596831	D₈	1	1	yes	characteristic
26	C₄⋊Dic₇₃	—	4	2	no	—
27	C₄⋊Dic₇₃	—	8	2	no	—
28	C₄×Dic₇₃	—	8	2	no	—
29	C₄×Dic₇₃	D₈	1	1	yes	characteristic
30	#11924003	—	2	1	no	—
31	C₂×Q₈×C₇₃	—	4	2	no	—
32	Q₁₆×C₇₃	—	2	4	no	—
33	Q₁₆×C₇₃	—	4	2	no	—
34	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₇₃	#41	1	1	yes	characteristic
37	Q₈×C₇₃	—	4	4	no	—
38	Q₈×C₇₃	—	8	2	no	—
39	C₂×C₄×C₇₃	—	8	2	no	—
40	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
41	C₂×Dic₇₃	—	8	2	no	—
42	C₂×Dic₇₃	D₁₆	1	2	yes	—
43	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
44	C₈×C₇₃	#135	1	1	yes	characteristic
45	#10596831	—	2	1	no	—
46	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
47	C₄×C₇₃	—	8	4	no	—
48	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
49	C₄×C₇₃	#508	1	1	yes	characteristic
50	Dic₇₃	—	2	2	no	—
51	Dic₇₃	—	16	2	no	—
52	#13653	—	73	1	no	maximal, Sylow
53	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
54	C₂×C₇₃	#2264	1	2	yes	—
55	C₂×Q₆₄	D₇₃	1	1	yes	characteristic
56	C₄×Q₃₂	—	73	2	no	—
57	#1424	—	73	1	no	—
58	#1416	—	73	2	no	—
59	#1397	—	73	1	no	—
60	C₇₃	#13653	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₃₂	C₂×D₇₃	1	2	yes	—
62	C₄×Q₁₆	—	146	2	no	—
63	Q₆₄	Dic₇₃	1	4	yes	complemented
64	C₂×C₃₂	—	73	1	no	—
65	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
66	#366	—	73	2	no	—
67	#358	—	146	2	no	—
68	C₄×C₁₆	—	73	1	no	—
69	C₂×Q₁₆	—	2	2	no	—
70	C₄×Q₈	—	292	2	no	—
71	Q₃₂	—	2	2	no	—
72	Q₃₂	C₂×Dic₇₃	1	4	yes	—
73	C₂×C₁₆	—	73	1	no	—
74	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
75	#107	—	146	2	no	—
76	#103	—	292	2	no	—
77	C₄×C₈	—	73	1	no	—
78	C₃₂	—	146	1	no	—
79	C₃₂	C₂×Dic₇₃	1	2	yes	—
80	C₂×Q₈	—	4	2	no	—
81	Q₁₆	—	2	4	no	—
82	Q₁₆	—	4	2	no	—
83	C₂×C₈	—	73	1	no	—
84	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
85	C₄⋊C₄	—	292	2	no	—
86	C₄⋊C₄	—	584	2	no	—
87	C₄²	—	584	2	no	—
88	C₄²	—	73	1	no	—
89	C₁₆	—	146	1	no	—
90	C₁₆	C₂²×Dic₇₃	1	1	yes	characteristic
91	C₁₆	#11924027	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	Q₈	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	584	2	no	—
96	C₂×C₄	—	73	2	no	—
97	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
98	C₈	—	146	1	no	—
99	C₈	D₄×Dic₇₃	1	1	yes	characteristic
100	C₈	#14556610	1	1	yes	characteristic
101	C₂²	D₁₆×D₇₃	1	1	yes	characteristic
102	C₄	—	8	4	no	—
103	C₄	—	1168	2	no	—
104	C₄	—	146	2	no	—
105	C₄	D₈×Dic₇₃	1	1	yes	characteristic
106	C₄	#17989016	1	1	yes	characteristic
107	C₂	D₁₆×Dic₇₃	1	1	yes	characteristic
108	C₂	#22612250	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×Dic₇₃

C₂

yes

maximal

#22608133

C₂

yes

maximal, characteristic

#22608112

C₂

yes

maximal

#22608064

C₂

yes

maximal, characteristic

C₂×Q₆₄×C₇₃

C₂

yes

maximal, characteristic

Q₁₆×Dic₇₃

—

C₂×#14556497

C₂²

yes

characteristic

#17988413

C₂²

yes

—

#17988391

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×Q₃₂×C₇₃

C₂²

yes

—

Q₆₄×C₇₃

C₄

yes

complemented

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

Q₈×Dic₇₃

—

C₂×#11924003

D₄

yes

characteristic

#14556525

—

#14556515

—

C₈×Dic₇₃

D₄

yes

characteristic

#14556497

—

C₂×Q₁₆×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₃₂×C₇₃

C₂×C₄

yes

—

C₂×#10596831

D₈

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₈

yes

characteristic

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

D₁₆

yes

—

C₈×C₇₃

C₄×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

#508

yes

characteristic

Dic₇₃

—

Dic₇₃

—

#13653

—

maximal, Sylow

C₂×C₇₃

C₄×D₁₆

yes

characteristic

C₂×C₇₃

#2264

yes

—

C₂×Q₆₄

D₇₃

yes

characteristic

C₄×Q₃₂

—

#1424

—

#1416

—

#1397

—

C₇₃

#13653

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₃

yes

—

C₄×Q₁₆

—

146

—

Q₆₄

Dic₇₃

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

characteristic

#366

—

#358

—

146

—

C₄×C₁₆

—

C₂×Q₁₆

—

C₄×Q₈

—

292

—

Q₃₂

—

Q₃₂

C₂×Dic₇₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#107

—

146

—

#103

—

292

—

C₄×C₈

—

C₃₂

—

146

—

C₃₂

C₂×Dic₇₃

yes

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₇₃

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

584

—

C₄²

—

584

—

C₄²

—

C₁₆

—

146

—

C₁₆

C₂²×Dic₇₃

yes

characteristic

C₁₆

#11924027

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

584

—

C₂×C₄

—

C₂×C₄

D₈×D₇₃

yes

characteristic

C₈

—

146

—

C₈

D₄×Dic₇₃

yes

characteristic

100

C₈

#14556610

yes

characteristic

101

C₂²

D₁₆×D₇₃

yes

characteristic

102

C₄

—

103

C₄

—

1168

—

104

C₄

—

146

—

105

C₄

D₈×Dic₇₃

yes

characteristic

106

C₄

#17989016

yes

characteristic

107

C₂

D₁₆×Dic₇₃

yes

characteristic

108

C₂

#22612250

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₇₃
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₂×Q₆₄×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	C₂×Q₆₄	D₇₃
hypocenter	C₇₃	#13653
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₇₃

commutator

C₁₆×C₇₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₂×Q₆₄×C₇₃

C₂

socle, prime radical

C₂²×C₇₃

C₂×D₁₆

cosocle

C₂×C₁₆×C₇₃

C₂³

hypercenter

C₂×Q₆₄

D₇₃

hypocenter

C₇₃

#13653

LCS

G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
12. Q₆₄×C₇₃	C₄	semidirect
60. C₇₃	#13653	semidirect
63. Q₆₄	Dic₇₃	semidirect

H ≅ G/N

type

12. Q₆₄×C₇₃

C₄

semidirect

60. C₇₃

#13653

semidirect

63. Q₆₄

Dic₇₃

semidirect

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

subgroup	index
Q₃₂×Dic₇₃	2
#22608133	2
#22608112	2
#22608064	2
C₂×Q₆₄×C₇₃	2
#13653	73

subgroup

index

Q₃₂×Dic₇₃

C₂×Q₆₄×C₇₃

Subgroups — #33006582

Conjugacy classes of subgroups

Complements

Maximal subgroups