Group #32923124 — subgroups

Subgroups — C₇₃×#13720 #32923124

Order 18688.

182 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₇₃×#2252	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#2247	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1416	C₂	1	4	yes	maximal
4	C₄×C₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₇₃	—	2	1	no	—
6	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
7	#500×C₇₃	—	2	1	no	—
8	#494×C₇₃	—	2	1	no	—
9	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
10	#366×C₇₃	—	2	1	no	—
11	#366×C₇₃	C₂²	1	2	yes	—
12	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₇₃	—	2	2	no	—
14	C₂×Q₁₆×C₇₃	—	4	1	no	—
15	C₄⋊Q₈×C₇₃	—	4	1	no	—
16	C₄⋊Q₈×C₇₃	—	4	1	no	—
17	Q₃₂×C₇₃	—	2	4	no	—
18	Q₃₂×C₇₃	—	4	2	no	—
19	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
21	#107×C₇₃	—	2	2	no	—
22	#107×C₇₃	—	4	1	no	—
23	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₃₂×C₇₃	—	2	2	no	—
25	C₂×Q₈×C₇₃	—	4	2	no	—
26	C₂×Q₈×C₇₃	—	8	1	no	—
27	Q₁₆×C₇₃	—	4	2	no	—
28	Q₁₆×C₇₃	—	4	2	no	—
29	Q₁₆×C₇₃	—	8	2	no	—
30	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
31	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₇₃	—	4	2	no	—
33	C₄⋊C₄×C₇₃	—	8	1	no	—
34	C₄⋊C₄×C₇₃	—	8	1	no	—
35	C₄⋊C₄×C₇₃	—	8	1	no	—
36	C₄²×C₇₃	D₈	1	1	yes	characteristic
37	C₁₆×C₇₃	SD₁₆	1	2	yes	—
38	C₁₆×C₇₃	#41	1	2	yes	—
39	Q₈×C₇₃	—	8	2	no	—
40	Q₈×C₇₃	—	8	2	no	—
41	Q₈×C₇₃	—	16	2	no	—
42	C₂×C₄×C₇₃	—	8	2	no	—
43	C₂×C₄×C₇₃	—	8	2	no	—
44	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
47	C₈×C₇₃	SD₃₂	1	2	yes	—
48	C₈×C₇₃	#124	1	1	yes	characteristic
49	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
50	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₇₃	—	16	2	no	—
52	C₄×C₇₃	—	16	2	no	—
53	C₄×C₇₃	SD₆₄	1	2	yes	—
54	C₄×C₇₃	Q₆₄	1	2	yes	—
55	C₄×C₇₃	#486	1	1	yes	characteristic
56	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
57	#13720	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×C₇₃	#2240	1	1	yes	characteristic
59	C₂×C₇₃	C₂×SD₆₄	1	1	yes	characteristic
60	C₂×C₇₃	C₂×Q₆₄	1	1	yes	characteristic
61	#2252	C₂×C₇₃	1	1	yes	characteristic
62	#2247	C₂×C₇₃	1	1	yes	characteristic
63	#1416	C₂×C₇₃	1	4	yes	—
64	C₄×C₃₂	C₂×C₇₃	1	1	yes	characteristic
65	C₇₃	#13720	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×Q₃₂	—	2	1	no	—
67	C₂×Q₃₂	C₂²×C₇₃	1	2	yes	—
68	#500	—	2	1	no	—
69	#494	—	2	1	no	—
70	C₂×C₃₂	C₂²×C₇₃	1	2	yes	—
71	#366	—	2	1	no	—
72	#366	C₂²×C₇₃	1	2	yes	—
73	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
74	C₂×Q₁₆	—	2	2	no	—
75	C₂×Q₁₆	—	4	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	4	1	no	—
78	Q₃₂	—	2	4	no	—
79	Q₃₂	—	4	2	no	—
80	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
81	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
82	#107	—	2	2	no	—
83	#107	—	4	1	no	—
84	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
85	C₃₂	—	2	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	8	1	no	—
88	Q₁₆	—	4	2	no	—
89	Q₁₆	—	4	2	no	—
90	Q₁₆	—	8	2	no	—
91	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
92	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	8	1	no	—
96	C₄⋊C₄	—	8	1	no	—
97	C₄²	D₈×C₇₃	1	1	yes	characteristic
98	C₁₆	SD₁₆×C₇₃	1	2	yes	—
99	C₁₆	#41×C₇₃	1	2	yes	—
100	Q₈	—	8	2	no	—
101	Q₈	—	8	2	no	—
102	Q₈	—	16	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	8	2	no	—
105	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
106	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
107	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
108	C₈	SD₃₂×C₇₃	1	2	yes	—
109	C₈	#124×C₇₃	1	1	yes	characteristic
110	C₈	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
111	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	16	2	no	—
114	C₄	SD₆₄×C₇₃	1	2	yes	—
115	C₄	Q₆₄×C₇₃	1	2	yes	—
116	C₄	#486×C₇₃	1	1	yes	characteristic
117	C₄	C₂×SD₃₂×C₇₃	1	1	yes	characteristic
118	C₂	C₇₃×#2240	1	1	yes	characteristic
119	C₂	C₂×SD₆₄×C₇₃	1	1	yes	characteristic
120	C₂	C₂×Q₆₄×C₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2252

C₂

yes

maximal, characteristic

C₇₃×#2247

C₂

yes

maximal, characteristic

C₇₃×#1416

C₂

yes

maximal

C₄×C₃₂×C₇₃

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₇₃

—

C₂×Q₃₂×C₇₃

C₂²

yes

—

#500×C₇₃

—

#494×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

—

#366×C₇₃

—

#366×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

#41

yes

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

SD₃₂

yes

—

C₈×C₇₃

#124

yes

characteristic

C₈×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

SD₆₄

yes

—

C₄×C₇₃

Q₆₄

yes

—

C₄×C₇₃

#486

yes

characteristic

C₄×C₇₃

C₂×SD₃₂

yes

characteristic

#13720

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

#2240

yes

characteristic

C₂×C₇₃

C₂×SD₆₄

yes

characteristic

C₂×C₇₃

C₂×Q₆₄

yes

characteristic

#2252

C₂×C₇₃

yes

characteristic

#2247

C₂×C₇₃

yes

characteristic

#1416

C₂×C₇₃

yes

—

C₄×C₃₂

C₂×C₇₃

yes

characteristic

C₇₃

#13720

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₇₃

yes

—

#500

—

#494

—

C₂×C₃₂

C₂²×C₇₃

yes

—

#366

—

#366

C₂²×C₇₃

yes

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

#107

—

#107

—

C₄×C₈

D₄×C₇₃

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₈×C₇₃

yes

characteristic

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₈×C₇₃

yes

characteristic

C₁₆

SD₁₆×C₇₃

yes

—

C₁₆

#41×C₇₃

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

D₁₆×C₇₃

yes

characteristic

106

C₂×C₄

D₁₆×C₇₃

yes

characteristic

107

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

108

C₈

SD₃₂×C₇₃

yes

—

109

C₈

#124×C₇₃

yes

characteristic

110

C₈

C₂×SD₁₆×C₇₃

yes

characteristic

111

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

SD₆₄×C₇₃

yes

—

115

C₄

Q₆₄×C₇₃

yes

—

116

C₄

#486×C₇₃

yes

characteristic

117

C₄

C₂×SD₃₂×C₇₃

yes

characteristic

118

C₂

C₇₃×#2240

yes

characteristic

119

C₂

C₂×SD₆₄×C₇₃

yes

characteristic

120

C₂

C₂×Q₆₄×C₇₃

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₇₃

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

N	H ≅ G/N	type
57. #13720	C₇₃	direct (N × H)
65. C₇₃	#13720	direct (N × H)

H ≅ G/N

type

57. #13720

C₇₃

direct (N × H)

65. C₇₃

#13720

direct (N × H)

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

subgroup	index
C₇₃×#2252	2
C₇₃×#2247	2
C₇₃×#1416	2
C₄×C₃₂×C₇₃	2
#13720	73

subgroup

index

C₇₃×#2252

C₇₃×#2247

C₇₃×#1416

C₄×C₃₂×C₇₃

#13720

Subgroups — C₇₃×#13720 #32923124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13720 #32923124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13720 #32923124