Group #32918220 — subgroups

Subgroups — C₇₃×#8816 #32918220

Order 18688.

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

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 238 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×#1845	C₂	1	2	yes	maximal
2	C₇₃×#1834	C₂	1	4	yes	maximal
3	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#107×C₇₃	—	2	4	no	—
5	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
7	#384×C₇₃	C₂²	1	4	yes	—
8	#340×C₇₃	—	2	8	no	—
9	C₈²×C₇₃	C₄	1	4	yes	complemented
10	C₂²×C₈×C₇₃	D₄	1	4	yes	—
11	C₂²×C₈×C₇₃	D₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
13	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
14	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
15	#107×C₇₃	—	4	8	no	—
16	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₇₃	Q₈	1	4	yes	—
19	C₄×C₈×C₇₃	Q₈	1	2	yes	—
20	C₂²×C₄×C₇₃	—	4	4	no	—
21	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
23	C₂×C₈×C₇₃	—	2	8	no	—
24	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₇₃	D₈	1	4	yes	—
27	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
28	C₄⋊C₄×C₇₃	—	4	8	no	—
29	C₄⋊C₄×C₇₃	—	4	16	no	—
30	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
33	C₂³×C₇₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₇₃	—	8	4	no	—
35	C₂×C₄×C₇₃	—	8	8	no	—
36	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
39	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
41	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
42	C₂×C₄×C₇₃	#135	1	8	yes	—
43	C₈×C₇₃	—	2	8	no	—
44	C₈×C₇₃	#107	1	8	yes	—
45	C₂²×C₇₃	#384	1	1	yes	characteristic
46	C₂²×C₇₃	#493	1	1	yes	characteristic
47	C₂²×C₇₃	#494	1	1	yes	characteristic
48	C₂²×C₇₃	#495	1	4	yes	—
49	C₄×C₇₃	—	16	4	no	—
50	C₄×C₇₃	C₂×#107	1	2	yes	—
51	C₄×C₇₃	#427	1	4	yes	—
52	C₄×C₇₃	#499	1	4	yes	—
53	C₄×C₇₃	#500	1	2	yes	—
54	#8816	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₇₃	#1709	1	2	yes	—
56	C₂×C₇₃	#1710	1	2	yes	—
57	C₂×C₇₃	#1845	1	1	yes	characteristic
58	C₂×C₇₃	#1846	1	2	yes	—
59	#1845	C₂×C₇₃	1	2	yes	—
60	#1834	C₂×C₇₃	1	4	yes	—
61	C₂×C₈²	C₂×C₇₃	1	1	yes	characteristic
62	C₇₃	#8816	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×#107	—	2	4	no	—
64	C₂×C₄×C₈	C₂²×C₇₃	1	2	yes	—
65	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
66	#384	C₂²×C₇₃	1	4	yes	—
67	#340	—	2	8	no	—
68	C₈²	C₄×C₇₃	1	4	yes	complemented
69	C₂²×C₈	D₄×C₇₃	1	4	yes	—
70	C₂²×C₈	D₄×C₇₃	1	2	yes	—
71	C₂×C₄⋊C₄	—	2	8	no	—
72	C₂×C₄⋊C₄	—	2	4	no	—
73	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
74	#107	—	4	8	no	—
75	C₄×C₈	C₂×C₄×C₇₃	1	4	yes	—
76	C₄×C₈	C₂×C₄×C₇₃	1	2	yes	—
77	C₄×C₈	Q₈×C₇₃	1	4	yes	—
78	C₄×C₈	Q₈×C₇₃	1	2	yes	—
79	C₂²×C₄	—	4	4	no	—
80	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₄⋊C₄×C₇₃	1	8	yes	—
84	C₂×C₈	C₄⋊C₄×C₇₃	1	4	yes	—
85	C₂×C₈	D₈×C₇₃	1	4	yes	—
86	C₂×C₈	Q₁₆×C₇₃	1	4	yes	—
87	C₄⋊C₄	—	4	8	no	—
88	C₄⋊C₄	—	4	16	no	—
89	C₄²	C₂²×C₄×C₇₃	1	1	yes	characteristic
90	C₄²	C₂×Q₈×C₇₃	1	1	yes	characteristic
91	C₄²	C₂×Q₈×C₇₃	1	2	yes	—
92	C₂³	#127×C₇₃	1	1	yes	characteristic
93	C₂×C₄	—	8	4	no	—
94	C₂×C₄	—	8	8	no	—
95	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
97	C₂×C₄	C₄⋊Q₈×C₇₃	1	2	yes	—
98	C₂×C₄	C₄⋊Q₈×C₇₃	1	1	yes	characteristic
99	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
100	C₂×C₄	C₂×Q₁₆×C₇₃	1	2	yes	—
101	C₂×C₄	#135×C₇₃	1	8	yes	—
102	C₈	—	2	8	no	—
103	C₈	#107×C₇₃	1	8	yes	—
104	C₂²	#384×C₇₃	1	1	yes	characteristic
105	C₂²	#493×C₇₃	1	1	yes	characteristic
106	C₂²	#494×C₇₃	1	1	yes	characteristic
107	C₂²	#495×C₇₃	1	4	yes	—
108	C₄	—	16	4	no	—
109	C₄	C₂×#107×C₇₃	1	2	yes	—
110	C₄	#427×C₇₃	1	4	yes	—
111	C₄	#499×C₇₃	1	4	yes	—
112	C₄	#500×C₇₃	1	2	yes	—
113	C₂	C₇₃×#1709	1	2	yes	—
114	C₂	C₇₃×#1710	1	2	yes	—
115	C₂	C₇₃×#1845	1	1	yes	characteristic
116	C₂	C₇₃×#1846	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#1845

C₂

yes

maximal

C₇₃×#1834

C₂

yes

maximal

C₂×C₈²×C₇₃

C₂

yes

maximal, characteristic

C₂×#107×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#384×C₇₃

C₂²

yes

—

#340×C₇₃

—

C₈²×C₇₃

C₄

yes

complemented

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

Q₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

—

C₂³×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

—

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₈×C₇₃

—

C₈×C₇₃

#107

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#107

yes

—

C₄×C₇₃

#427

yes

—

C₄×C₇₃

#499

yes

—

C₄×C₇₃

#500

yes

—

#8816

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

#1709

yes

—

C₂×C₇₃

#1710

yes

—

C₂×C₇₃

#1845

yes

characteristic

C₂×C₇₃

#1846

yes

—

#1845

C₂×C₇₃

yes

—

#1834

C₂×C₇₃

yes

—

C₂×C₈²

C₂×C₇₃

yes

characteristic

C₇₃

#8816

yes

characteristic, direct product factor, Sylow

C₂×#107

—

C₂×C₄×C₈

C₂²×C₇₃

yes

—

C₂×C₄×C₈

C₂²×C₇₃

yes

characteristic

#384

C₂²×C₇₃

yes

—

#340

—

C₈²

C₄×C₇₃

yes

complemented

C₂²×C₈

D₄×C₇₃

yes

—

C₂²×C₈

D₄×C₇₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₇₃

yes

characteristic

#107

—

C₄×C₈

C₂×C₄×C₇₃

yes

—

C₄×C₈

C₂×C₄×C₇₃

yes

—

C₄×C₈

Q₈×C₇₃

yes

—

C₄×C₈

Q₈×C₇₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₇₃

yes

characteristic

C₂²×C₄

C₂×D₄×C₇₃

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₇₃

yes

—

C₂×C₈

C₄⋊C₄×C₇₃

yes

—

C₂×C₈

D₈×C₇₃

yes

—

C₂×C₈

Q₁₆×C₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₇₃

yes

characteristic

C₄²

C₂×Q₈×C₇₃

yes

characteristic

C₄²

C₂×Q₈×C₇₃

yes

—

C₂³

#127×C₇₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄×C₇₃

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄×C₇₃

yes

—

C₂×C₄

C₄⋊Q₈×C₇₃

yes

—

C₂×C₄

C₄⋊Q₈×C₇₃

yes

characteristic

C₂×C₄

C₂×D₈×C₇₃

yes

—

100

C₂×C₄

C₂×Q₁₆×C₇₃

yes

—

101

C₂×C₄

#135×C₇₃

yes

—

102

C₈

—

103

C₈

#107×C₇₃

yes

—

104

C₂²

#384×C₇₃

yes

characteristic

105

C₂²

#493×C₇₃

yes

characteristic

106

C₂²

#494×C₇₃

yes

characteristic

107

C₂²

#495×C₇₃

yes

—

108

C₄

—

109

C₄

C₂×#107×C₇₃

yes

—

110

C₄

#427×C₇₃

yes

—

111

C₄

#499×C₇₃

yes

—

112

C₄

#500×C₇₃

yes

—

113

C₂

C₇₃×#1709

yes

—

114

C₂

C₇₃×#1710

yes

—

115

C₂

C₇₃×#1845

yes

characteristic

116

C₂

C₇₃×#1846

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₃	#127
commutator	C₄²	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇₃

#127

commutator

C₄²

C₂²×C₄×C₇₃

Frattini, cosocle

C₂×C₄²

C₂³×C₇₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

N	H ≅ G/N	type
9. C₈²×C₇₃	C₄	semidirect
54. #8816	C₇₃	direct (N × H)
62. C₇₃	#8816	direct (N × H)
68. C₈²	C₄×C₇₃	semidirect

H ≅ G/N

type

9. C₈²×C₇₃

C₄

semidirect

54. #8816

C₇₃

direct (N × H)

62. C₇₃

#8816

direct (N × H)

68. C₈²

C₄×C₇₃

semidirect

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

subgroup	index
C₇₃×#1845	2
C₇₃×#1834	2
C₂×C₈²×C₇₃	2
#8816	73

subgroup

index

C₇₃×#1845

C₇₃×#1834

C₂×C₈²×C₇₃

#8816

Subgroups — C₇₃×#8816 #32918220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#8816 #32918220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#8816 #32918220