Group #33000689 — subgroups

Subgroups — #33000689

Order 18688.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22612156	C₂	1	4	yes	maximal, complemented
2	#22609546	C₂	1	1	yes	maximal, characteristic
3	#22607848	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988964	—	2	2	no	—
6	#17988964	C₂²	1	2	yes	—
7	#17988387	—	2	8	no	—
8	C₄×#11924003	C₂²	1	2	yes	—
9	#500×C₇₃	C₂²	1	2	yes	—
10	#487×C₇₃	—	2	2	no	—
11	C₈²×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×D₅₈₄	—	2	4	no	—
13	C₂×D₅₈₄	—	4	4	no	—
14	#14556588	—	2	2	no	—
15	C₂×#11924003	D₄	1	4	yes	—
16	C₄⋊Q₈×C₇₃	—	2	2	no	—
17	#107×C₇₃	—	2	4	no	—
18	#103×C₇₃	—	4	4	no	—
19	C₄×C₈×C₇₃	D₄	1	2	yes	—
20	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
21	C₂×D₂₉₂	—	4	2	no	—
22	C₂×D₂₉₂	—	4	4	no	—
23	D₅₈₄	—	4	8	no	—
24	D₅₈₄	—	8	8	no	—
25	#11924003	—	2	4	no	—
26	C₂×Q₈×C₇₃	—	4	2	no	—
27	C₂×C₈×C₇₃	D₈	1	4	yes	—
28	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
29	C₄⋊C₄×C₇₃	—	4	2	no	—
30	C₄⋊C₄×C₇₃	—	4	2	no	—
31	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₇₃	—	8	2	no	—
33	Q₈×C₇₃	—	8	4	no	—
34	C₂×C₄×C₇₃	—	8	2	no	—
35	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
37	D₂₉₂	—	8	4	no	—
38	D₂₉₂	—	8	8	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	—	16	2	no	—
45	C₄×C₇₃	#495	1	2	yes	—
46	C₄×C₇₃	#509	1	4	yes	—
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	D₇₃	1	1	yes	characteristic, complemented
54	#1311	—	73	1	no	—
55	C₇₃	#12513	1	1	yes	characteristic, complemented, Sylow
56	#500	C₂×D₇₃	1	2	yes	—
57	#493	—	73	2	no	—
58	#493	—	146	2	no	—
59	#487	—	2	2	no	—
60	#357	—	146	8	no	—
61	C₄×C₁₆	—	73	2	no	—
62	C₈²	C₂×D₇₃	1	1	yes	characteristic
63	C₂×D₈	—	292	4	no	—
64	C₂×D₈	—	146	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	#127	—	146	2	no	—
67	C₂×C₁₆	—	73	4	no	—
68	#107	—	2	4	no	—
69	#103	—	4	4	no	—
70	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
71	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
72	C₂×Q₈	—	4	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	#11924013	1	4	yes	—
78	C₂×C₈	D₄×D₇₃	1	2	yes	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
82	C₁₆	—	146	4	no	—
83	C₂³	—	584	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	584	4	no	—
86	D₄	—	584	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	#14556587	1	1	yes	characteristic
89	C₂×C₄	C₂×#11924013	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#14556614	1	4	yes	—
92	C₂²	—	1168	2	no	—
93	C₂²	—	1168	4	no	—
94	C₂²	#17988963	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#17988972	1	2	yes	—
97	C₄	#17989022	1	4	yes	—
98	C₂	—	2336	2	no	—
99	C₂	#22609538	1	1	yes	characteristic
100	C₂	#22612198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612156

C₂

yes

maximal, complemented

#22609546

C₂

yes

maximal, characteristic

#22607848

C₂

yes

maximal, characteristic, complemented

C₇₃×#1710

C₂

yes

maximal, characteristic, complemented

#17988964

—

#17988964

C₂²

yes

—

#17988387

—

C₄×#11924003

C₂²

yes

—

#500×C₇₃

C₂²

yes

—

#487×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

—

C₂×#11924003

D₄

yes

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

D₅₈₄

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#495

yes

—

C₄×C₇₃

#509

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2240

—

#1712

—

#1710

D₇₃

yes

characteristic, complemented

#1311

—

C₇₃

#12513

yes

characteristic, complemented, Sylow

#500

C₂×D₇₃

yes

—

#493

—

#493

—

146

—

#487

—

#357

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×D₈

—

292

—

C₂×D₈

—

146

—

C₄⋊Q₈

—

#127

—

146

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₇₃

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

D₈

—

292

—

D₈

—

584

—

C₂×C₈

#11924013

yes

—

C₂×C₈

D₄×D₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

C₂³

—

584

—

Q₈

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

—

C₂×C₄

#14556587

yes

characteristic

C₂×C₄

C₂×#11924013

yes

—

C₈

—

C₈

#14556614

yes

—

C₂²

—

1168

—

C₂²

—

1168

—

C₂²

#17988963

yes

characteristic

C₄

—

C₄

#17988972

yes

—

C₄

#17989022

yes

—

C₂

—

2336

—

C₂

#22609538

yes

characteristic

100

C₂

#22612198

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1710	C₂
socle	C₂²×C₇₃	#493
prime radical	#22609546	C₂
hypercenter	#1710	D₇₃
hypocenter	C₇₃	#12513
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1710

C₂

C₂²×C₇₃

C₂

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #22612156	C₂	semidirect
3. #22607848	C₂	semidirect
4. C₇₃×#1710	C₂	semidirect
53. #1710	D₇₃	semidirect
55. C₇₃	#12513	semidirect

H ≅ G/N

type

1. #22612156

C₂

semidirect

3. #22607848

C₂

semidirect

4. C₇₃×#1710

C₂

semidirect

53. #1710

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
#22612156	2
#22609546	2
#22607848	2
C₇₃×#1710	2
#12513	73

subgroup

index

C₇₃×#1710

Subgroups — #33000689

Conjugacy classes of subgroups

Complements

Maximal subgroups