Group #33000722 — subgroups

Subgroups — #33000722

Order 18688.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22612211	C₂	1	2	yes	maximal
2	#22612204	C₂	1	2	yes	maximal
3	#22609541	C₂	1	1	yes	maximal, characteristic
4	C₈×#11924003	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1710	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988938	—	2	2	no	—
8	#17988414	—	2	4	no	—
9	#17988412	—	2	4	no	—
10	C₄×#11924003	C₂²	1	1	yes	characteristic
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	#500×C₇₃	C₂²	1	2	yes	—
13	#487×C₇₃	—	2	2	no	—
14	C₈²×C₇₃	C₂²	1	1	yes	characteristic
15	#14556591	—	2	2	no	—
16	C₂×#11924003	D₄	1	2	yes	—
17	C₂×#11924003	D₄	1	2	yes	—
18	#14556525	—	2	2	no	—
19	#14556525	—	2	2	no	—
20	#14556516	—	4	4	no	—
21	C₄⋊Q₈×C₇₃	—	2	2	no	—
22	#107×C₇₃	—	2	2	no	—
23	#107×C₇₃	—	2	2	no	—
24	#103×C₇₃	—	4	4	no	—
25	C₄×C₈×C₇₃	D₄	1	2	yes	—
26	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₄₆	—	4	2	no	—
28	C₄⋊Dic₇₃	—	4	2	no	—
29	C₄⋊Dic₇₃	—	4	2	no	—
30	#11924003	Q₁₆	1	4	yes	—
31	#11924003	Q₁₆	1	4	yes	—
32	C₂×Q₈×C₇₃	—	4	2	no	—
33	C₂×C₈×C₇₃	D₈	1	4	yes	—
34	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	4	2	no	—
37	C₄⋊C₄×C₇₃	—	4	2	no	—
38	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₇₃	—	8	4	no	—
40	C₂×C₄×C₇₃	—	8	2	no	—
41	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
43	C₂×Dic₇₃	—	8	2	no	—
44	Dic₁₄₆	—	8	4	no	—
45	C₈×C₇₃	—	2	4	no	—
46	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
47	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
48	C₂²×C₇₃	#493	1	1	yes	characteristic
49	C₄×C₇₃	—	16	2	no	—
50	C₄×C₇₃	#494	1	1	yes	characteristic
51	C₄×C₇₃	#494	1	1	yes	characteristic
52	C₄×C₇₃	#508	1	4	yes	—
53	Dic₇₃	—	16	2	no	—
54	#12522	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	#1713	1	1	yes	characteristic
56	C₂×C₇₃	#2249	1	2	yes	—
57	#2254	—	73	2	no	—
58	#2252	—	73	2	no	—
59	#1710	—	73	1	no	—
60	#1710	D₇₃	1	1	yes	characteristic
61	C₈×C₁₆	—	73	1	no	—
62	C₇₃	#12522	1	1	yes	characteristic, complemented, Sylow
63	#500	—	73	2	no	—
64	#500	C₂×D₇₃	1	2	yes	—
65	#487	—	2	2	no	—
66	#487	—	146	2	no	—
67	#367	—	146	4	no	—
68	#366	—	146	4	no	—
69	C₄×C₁₆	—	73	1	no	—
70	C₄×C₁₆	—	73	1	no	—
71	C₈²	C₂×D₇₃	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	—	146	2	no	—
74	C₂×C₁₆	—	73	2	no	—
75	C₂×C₁₆	—	73	2	no	—
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	146	2	no	—
79	#107	—	146	2	no	—
80	#103	—	4	4	no	—
81	#103	—	292	4	no	—
82	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
83	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	292	2	no	—
86	C₂×C₈	#11924013	1	4	yes	—
87	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
88	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	292	2	no	—
92	C₄⋊C₄	—	292	2	no	—
93	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
94	C₁₆	—	73	4	no	—
95	C₁₆	—	73	4	no	—
96	Q₈	—	8	4	no	—
97	Q₈	—	584	4	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	584	2	no	—
100	C₂×C₄	#14556587	1	1	yes	characteristic
101	C₂×C₄	C₂×#11924013	1	2	yes	—
102	C₈	—	2	4	no	—
103	C₈	Q₁₆×D₇₃	1	2	yes	—
104	C₈	Q₁₆×D₇₃	1	2	yes	—
105	C₂²	#17988963	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	1168	2	no	—
108	C₄	#17988965	1	1	yes	characteristic
109	C₄	#17988965	1	1	yes	characteristic
110	C₄	#17989017	1	4	yes	—
111	C₂	#22609547	1	1	yes	characteristic
112	C₂	#22612190	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612211

C₂

yes

maximal

#22612204

C₂

yes

maximal

#22609541

C₂

yes

maximal, characteristic

C₈×#11924003

C₂

yes

maximal, characteristic

C₇₃×#1710

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#11924003

C₂²

yes

characteristic

#500×C₇₃

C₂²

yes

—

#487×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

#14556591

—

C₂×#11924003

D₄

yes

—

C₂×#11924003

D₄

yes

—

#14556525

—

#14556525

—

#14556516

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

Q₁₆

yes

—

#11924003

Q₁₆

yes

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₁₆

yes

—

C₈×C₇₃

C₂×Q₁₆

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

D₇₃

yes

characteristic

C₈×C₁₆

—

C₇₃

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₇₃

yes

—

#487

—

#487

—

146

—

#367

—

146

—

#366

—

146

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

146

—

#107

—

146

—

#103

—

#103

—

292

—

C₄×C₈

C₂²⋊D₇₃

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

C₂×C₈

#11924013

yes

—

C₂×C₈

D₄×D₇₃

yes

characteristic

C₂×C₈

D₄×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

584

—

C₂×C₄

—

C₂×C₄

—

584

—

100

C₂×C₄

#14556587

yes

characteristic

101

C₂×C₄

C₂×#11924013

yes

—

102

C₈

—

103

C₈

Q₁₆×D₇₃

yes

—

104

C₈

Q₁₆×D₇₃

yes

—

105

C₂²

#17988963

yes

characteristic

106

C₄

—

107

C₄

—

1168

—

108

C₄

#17988965

yes

characteristic

109

C₄

#17988965

yes

characteristic

110

C₄

#17989017

yes

—

111

C₂

#22609547

yes

characteristic

112

C₂

#22612190

yes

—

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1710	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1710	D₇₃
hypocenter	C₇₃	#12522
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

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
62. C₇₃	#12522	semidirect

H ≅ G/N

type

62. C₇₃

#12522

semidirect

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

subgroup	index
#22612211	2
#22612204	2
#22609541	2
C₈×#11924003	2
C₇₃×#1710	2
#12522	73

subgroup

index

C₇₃×#1710

Subgroups — #33000722

Conjugacy classes of subgroups

Complements

Maximal subgroups