Group #33000673 — subgroups

Subgroups — #33000673

Order 18688.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22612158	C₂	1	4	yes	maximal
2	#22609549	C₂	1	1	yes	maximal, characteristic
3	C₈×#11924003	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic
5	#17988992	—	2	1	no	—
6	#17988992	C₂²	1	2	yes	—
7	#17988967	—	2	1	no	—
8	#17988391	—	2	8	no	—
9	C₄×#11924003	C₂²	1	2	yes	—
10	#500×C₇₃	—	2	1	no	—
11	#500×C₇₃	—	2	1	no	—
12	#494×C₇₃	C₂²	1	2	yes	—
13	C₈²×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₉₂	—	4	2	no	—
15	#14556591	—	2	2	no	—
16	C₂×#11924003	D₄	1	4	yes	—
17	#14556525	—	2	4	no	—
18	#14556525	—	4	2	no	—
19	C₂×Q₁₆×C₇₃	—	2	4	no	—
20	C₄⋊Q₈×C₇₃	—	2	2	no	—
21	#107×C₇₃	—	4	2	no	—
22	#107×C₇₃	—	4	2	no	—
23	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₄₆	—	4	2	no	—
27	Dic₂₉₂	—	8	4	no	—
28	C₄⋊Dic₇₃	—	4	2	no	—
29	C₄⋊Dic₇₃	—	4	2	no	—
30	#11924003	—	2	4	no	—
31	C₂×Q₈×C₇₃	—	4	2	no	—
32	Q₁₆×C₇₃	—	4	8	no	—
33	C₂×C₈×C₇₃	D₈	1	2	yes	—
34	C₂×C₈×C₇₃	D₈	1	2	yes	—
35	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₇₃	—	8	2	no	—
45	Dic₁₄₆	—	8	4	no	—
46	C₈×C₇₃	SD₃₂	1	4	yes	—
47	C₈×C₇₃	Q₃₂	1	4	yes	—
48	C₈×C₇₃	#135	1	4	yes	—
49	C₂²×C₇₃	#493	1	1	yes	characteristic
50	C₄×C₇₃	—	16	2	no	—
51	C₄×C₇₃	#495	1	2	yes	—
52	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
53	C₄×C₇₃	C₂×Q₃₂	1	2	yes	—
54	Dic₇₃	—	16	2	no	—
55	#12510	—	73	1	no	maximal, Sylow
56	C₂×C₇₃	#1709	1	1	yes	characteristic
57	C₂×C₇₃	#2247	1	1	yes	characteristic
58	C₂×C₇₃	#2248	1	1	yes	characteristic
59	#2241	—	73	4	no	—
60	#1713	—	73	1	no	—
61	#1713	D₇₃	1	1	yes	characteristic
62	C₈×C₁₆	—	73	1	no	—
63	C₇₃	#12510	1	1	yes	characteristic, complemented, Sylow
64	#500	—	2	1	no	—
65	#500	—	2	1	no	—
66	#500	—	73	2	no	—
67	#500	—	146	1	no	—
68	#494	—	146	1	no	—
69	#494	C₂×D₇₃	1	2	yes	—
70	#358	—	146	8	no	—
71	C₄×C₁₆	—	73	2	no	—
72	C₈²	C₂×D₇₃	1	1	yes	characteristic
73	C₂×Q₁₆	—	2	4	no	—
74	C₂×Q₁₆	—	292	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	146	2	no	—
77	C₂×C₁₆	—	73	4	no	—
78	#107	—	4	2	no	—
79	#107	—	4	2	no	—
80	#107	—	292	2	no	—
81	#107	—	146	4	no	—
82	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	292	2	no	—
87	Q₁₆	—	4	8	no	—
88	Q₁₆	—	584	4	no	—
89	C₂×C₈	#11924013	1	2	yes	—
90	C₂×C₈	#11924013	1	2	yes	—
91	C₂×C₈	D₄×D₇₃	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	292	2	no	—
95	C₄⋊C₄	—	292	2	no	—
96	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
97	C₁₆	—	146	4	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	584	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	584	2	no	—
102	C₂×C₄	#14556587	1	1	yes	characteristic
103	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
104	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
105	C₈	#14556534	1	4	yes	—
106	C₈	#14556537	1	4	yes	—
107	C₈	#14556610	1	4	yes	—
108	C₂²	#17988963	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	1168	2	no	—
111	C₄	#17988970	1	2	yes	—
112	C₄	C₂×#14556534	1	2	yes	—
113	C₄	C₂×#14556537	1	2	yes	—
114	C₂	#22609536	1	1	yes	characteristic
115	C₂	#22612183	1	1	yes	characteristic
116	C₂	#22612188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612158

C₂

yes

maximal

#22609549

C₂

yes

maximal, characteristic

C₈×#11924003

C₂

yes

maximal, characteristic

C₇₃×#1713

C₂

yes

maximal, characteristic

#17988992

—

#17988992

C₂²

yes

—

#17988967

—

#17988391

—

C₄×#11924003

C₂²

yes

—

#500×C₇₃

—

#500×C₇₃

—

#494×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

#14556591

—

C₂×#11924003

D₄

yes

—

#14556525

—

#14556525

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

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

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

Q₃₂

yes

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#495

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

C₄×C₇₃

C₂×Q₃₂

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

C₈×C₁₆

—

C₇₃

#12510

yes

characteristic, complemented, Sylow

—

—

—

—

146

—

#494

—

146

—

#494

C₂×D₇₃

yes

—

#358

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

292

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

292

—

#107

—

146

—

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

Q₁₆

—

Q₁₆

—

584

—

C₂×C₈

#11924013

yes

—

C₂×C₈

#11924013

yes

—

C₂×C₈

D₄×D₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

Q₈

—

Q₈

—

584

—

100

C₂×C₄

—

101

C₂×C₄

—

584

—

102

C₂×C₄

#14556587

yes

characteristic

103

C₂×C₄

C₂×#11924013

yes

characteristic

104

C₂×C₄

C₂×#11924013

yes

characteristic

105

C₈

#14556534

yes

—

106

C₈

#14556537

yes

—

107

C₈

#14556610

yes

—

108

C₂²

#17988963

yes

characteristic

109

C₄

—

110

C₄

—

1168

—

111

C₄

#17988970

yes

—

112

C₄

C₂×#14556534

yes

—

113

C₄

C₂×#14556537

yes

—

114

C₂

#22609536

yes

characteristic

115

C₂

#22612183

yes

characteristic

116

C₂

#22612188

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1713	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1713	D₇₃
hypocenter	C₇₃	#12510
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1713

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
63. C₇₃	#12510	semidirect

H ≅ G/N

type

63. C₇₃

#12510

semidirect

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

subgroup	index
#22612158	2
#22609549	2
C₈×#11924003	2
C₇₃×#1713	2
#12510	73

subgroup

index

C₇₃×#1713

Subgroups — #33000673

Conjugacy classes of subgroups

Complements

Maximal subgroups