Group #33000688 — subgroups

Subgroups — #33000688

Order 18688.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22612155	C₂	1	4	yes	maximal, complemented
2	#22609546	C₂	1	1	yes	maximal, characteristic
3	#22609541	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988992	C₂²	1	2	yes	—
6	#17988964	—	2	2	no	—
7	#17988964	C₂²	1	2	yes	—
8	#17988938	—	2	2	no	—
9	#17988389	—	2	8	no	—
10	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
11	C₈²×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×D₅₈₄	—	2	4	no	—
13	C₂×D₅₈₄	—	4	4	no	—
14	#14556591	—	2	2	no	—
15	#14556588	—	2	2	no	—
16	#14556525	—	2	4	no	—
17	#14556516	—	4	4	no	—
18	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
19	C₄×C₈×C₇₃	D₄	1	2	yes	—
20	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₄₆	—	4	2	no	—
22	C₂×D₂₉₂	—	4	2	no	—
23	C₂×D₂₉₂	—	4	4	no	—
24	D₅₈₄	—	4	8	no	—
25	D₅₈₄	—	8	8	no	—
26	C₄⋊Dic₇₃	—	4	2	no	—
27	C₄⋊Dic₇₃	—	4	2	no	—
28	C₂×C₈×C₇₃	D₈	1	4	yes	—
29	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
30	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
31	C₁₆×C₇₃	—	2	4	no	—
32	C₂²×D₇₃	—	8	2	no	—
33	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
35	C₂×Dic₇₃	—	8	2	no	—
36	D₂₉₂	—	8	4	no	—
37	D₂₉₂	—	8	8	no	—
38	Dic₁₄₆	—	8	4	no	—
39	C₈×C₇₃	—	2	4	no	—
40	C₈×C₇₃	#135	1	4	yes	—
41	C₂²×C₇₃	#493	1	1	yes	characteristic
42	C₂×D₇₃	—	16	2	no	—
43	C₂×D₇₃	—	16	4	no	—
44	C₄×C₇₃	#495	1	2	yes	—
45	C₄×C₇₃	#509	1	4	yes	—
46	Dic₇₃	—	16	2	no	—
47	#12513	—	73	1	no	maximal, Sylow
48	C₂×C₇₃	#1709	1	1	yes	characteristic
49	C₂×C₇₃	#2250	1	2	yes	—
50	D₇₃	—	32	2	no	—
51	#2240	—	73	4	no	—
52	#1712	—	73	1	no	—
53	#1710	—	73	1	no	—
54	#1311	D₇₃	1	1	yes	characteristic, complemented
55	C₇₃	#12513	1	1	yes	characteristic, complemented, Sylow
56	#500	—	73	2	no	—
57	#493	—	73	2	no	—
58	#493	—	146	2	no	—
59	#487	—	146	2	no	—
60	#357	—	146	8	no	—
61	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
62	C₈²	C₂×D₇₃	1	1	yes	characteristic
63	C₂×D₈	—	292	4	no	—
64	C₂×D₈	—	146	4	no	—
65	C₄⋊Q₈	—	146	2	no	—
66	#127	—	146	2	no	—
67	C₂×C₁₆	D₂₉₂	1	4	yes	—
68	#107	—	146	4	no	—
69	#103	—	292	4	no	—
70	C₄×C₈	D₂₉₂	1	2	yes	—
71	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
72	C₂×Q₈	—	292	2	no	—
73	C₂×D₄	—	292	2	no	—
74	C₂×D₄	—	292	4	no	—
75	D₈	—	292	8	no	—
76	D₈	—	584	8	no	—
77	C₂×C₈	D₅₈₄	1	4	yes	—
78	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
79	C₄⋊C₄	—	292	2	no	—
80	C₄⋊C₄	—	292	2	no	—
81	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	584	2	no	—
84	Q₈	—	584	4	no	—
85	D₄	—	584	4	no	—
86	D₄	—	584	8	no	—
87	C₂×C₄	—	584	2	no	—
88	C₂×C₄	#14556588	1	1	yes	characteristic
89	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14556613	1	4	yes	—
92	C₂²	—	1168	2	no	—
93	C₂²	—	1168	4	no	—
94	C₂²	#17988964	1	1	yes	characteristic
95	C₄	—	1168	2	no	—
96	C₄	#17988971	1	2	yes	—
97	C₄	#17989025	1	4	yes	—
98	C₂	—	2336	2	no	—
99	C₂	#22609537	1	1	yes	characteristic
100	C₂	#22612197	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612155

C₂

yes

maximal, complemented

#22609546

C₂

yes

maximal, characteristic

#22609541

C₂

yes

maximal, characteristic, complemented

C₇₃×#1311

C₂

yes

maximal, characteristic, complemented

#17988992

C₂²

yes

—

#17988964

—

#17988964

C₂²

yes

—

#17988938

—

#17988389

—

C₄×C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

—

—

—

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

#495

yes

—

C₄×C₇₃

#509

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#12513

yes

characteristic, complemented, Sylow

#500

—

#493

—

#493

—

146

—

#487

—

146

—

#357

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×D₈

—

292

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

146

—

#127

—

146

—

C₂×C₁₆

D₂₉₂

yes

—

#107

—

146

—

#103

—

292

—

C₄×C₈

D₂₉₂

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

D₈

—

292

—

D₈

—

584

—

C₂×C₈

D₅₈₄

yes

—

C₂×C₈

C₂×D₂₉₂

yes

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄²

C₂×D₂₉₂

yes

characteristic

C₁₆

—

C₂³

—

584

—

Q₈

—

584

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

—

584

—

C₂×C₄

#14556588

yes

characteristic

C₂×C₄

C₂×D₅₈₄

yes

—

C₈

—

C₈

#14556613

yes

—

C₂²

—

1168

—

C₂²

—

1168

—

C₂²

#17988964

yes

characteristic

C₄

—

1168

—

C₄

#17988971

yes

—

C₄

#17989025

yes

—

C₂

—

2336

—

C₂

#22609537

yes

characteristic

100

C₂

#22612197

yes

—

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1311	C₂
socle	C₂²×C₇₃	#493
prime radical	#22609546	C₂
hypercenter	#1311	D₇₃
hypocenter	C₇₃	#12513
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1311

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #22612155	C₂	semidirect
3. #22609541	C₂	semidirect
4. C₇₃×#1311	C₂	semidirect
54. #1311	D₇₃	semidirect
55. C₇₃	#12513	semidirect

H ≅ G/N

type

1. #22612155

C₂

semidirect

3. #22609541

C₂

semidirect

4. C₇₃×#1311

C₂

semidirect

54. #1311

D₇₃

semidirect

55. C₇₃

#12513

semidirect

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

subgroup	index
#22612155	2
#22609546	2
#22609541	2
C₇₃×#1311	2
#12513	73

subgroup

index

C₇₃×#1311

Subgroups — #33000688

Conjugacy classes of subgroups

Complements

Maximal subgroups