Group #32942280 — subgroups

Subgroups — C₇₃×#32876 #32942280

Order 18688.

584 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 186 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×#2091	C₂	1	4	yes	maximal
2	C₇₃×#2070	C₂	1	8	yes	maximal, complemented
3	C₇₃×#1784	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
5	#430×C₇₃	C₂²	1	12	yes	—
6	C₂×#101×C₇₃	C₂²	1	12	yes	—
7	#407×C₇₃	C₂²	1	6	yes	—
8	#338×C₇₃	C₂²	1	4	yes	—
9	C₂²×Q₈×C₇₃	—	4	6	no	—
10	C₂²×Q₈×C₇₃	Q₈	1	2	yes	—
11	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
12	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
13	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
14	C₂×M₁₆×C₇₃	C₂³	1	12	yes	—
15	#108×C₇₃	—	4	12	no	—
16	#101×C₇₃	—	4	12	no	—
17	C₂²⋊C₈×C₇₃	—	4	6	no	—
18	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
19	C₂×Q₈×C₇₃	—	2	12	no	—
20	C₂×Q₈×C₇₃	—	4	6	no	—
21	C₂×Q₈×C₇₃	—	4	16	no	—
22	C₂²×C₄×C₇₃	—	2	6	no	—
23	C₂²×C₄×C₇₃	—	4	6	no	—
24	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
25	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
27	M₁₆×C₇₃	—	2	24	no	—
28	C₂×C₈×C₇₃	—	2	12	no	—
29	C₂³×C₇₃	—	2	3	no	—
30	C₂³×C₇₃	—	4	2	no	—
31	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
32	Q₈×C₇₃	—	4	8	no	—
33	Q₈×C₇₃	—	4	8	no	—
34	C₂×C₄×C₇₃	—	2	3	no	—
35	C₂×C₄×C₇₃	—	2	12	no	—
36	C₂×C₄×C₇₃	—	4	8	no	—
37	C₂×C₄×C₇₃	—	4	4	no	—
38	C₂×C₄×C₇₃	H(5,𝔽₂)	1	6	yes	—
39	C₈×C₇₃	—	4	12	no	—
40	C₂²×C₇₃	—	2	6	no	—
41	C₂²×C₇₃	—	4	1	no	—
42	C₂²×C₇₃	—	4	3	no	—
43	C₂²×C₇₃	—	4	1	no	—
44	C₂²×C₇₃	#543	1	3	yes	—
45	C₄×C₇₃	—	4	1	no	—
46	C₄×C₇₃	—	4	4	no	—
47	C₄×C₇₃	#560	1	4	yes	—
48	#32876	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×C₇₃	—	2	3	no	—
50	C₂×C₇₃	—	4	2	no	—
51	C₂×C₇₃	#2842	1	1	yes	characteristic
52	#2091	C₂×C₇₃	1	4	yes	—
53	#2070	C₂×C₇₃	1	8	yes	complemented
54	#1784	C₂×C₇₃	1	3	yes	—
55	C₇₃	#32876	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×Q₈	C₂²×C₇₃	1	1	yes	characteristic
57	#430	C₂²×C₇₃	1	12	yes	—
58	C₂×#101	C₂²×C₇₃	1	12	yes	—
59	#407	C₂²×C₇₃	1	6	yes	—
60	#338	C₂²×C₇₃	1	4	yes	—
61	C₂²×Q₈	—	4	6	no	—
62	C₂²×Q₈	Q₈×C₇₃	1	2	yes	—
63	C₂²×Q₈	C₂³×C₇₃	1	2	yes	—
64	C₂³×C₄	Q₈×C₇₃	1	2	yes	—
65	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
66	C₂×M₁₆	C₂³×C₇₃	1	12	yes	—
67	#108	—	4	12	no	—
68	#101	—	4	12	no	—
69	C₂²⋊C₈	—	4	6	no	—
70	C₂⁴	C₂×Q₈×C₇₃	1	1	yes	characteristic
71	C₂×Q₈	—	2	12	no	—
72	C₂×Q₈	—	4	6	no	—
73	C₂×Q₈	—	4	16	no	—
74	C₂²×C₄	—	2	6	no	—
75	C₂²×C₄	—	4	6	no	—
76	C₂²×C₄	C₂×Q₈×C₇₃	1	4	yes	—
77	C₂²×C₄	C₂×Q₈×C₇₃	1	1	yes	characteristic
78	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
79	M₁₆	—	2	24	no	—
80	C₂×C₈	—	2	12	no	—
81	C₂³	—	2	3	no	—
82	C₂³	—	4	2	no	—
83	C₂³	C₂²×Q₈×C₇₃	1	1	yes	characteristic
84	Q₈	—	4	8	no	—
85	Q₈	—	4	8	no	—
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	12	no	—
88	C₂×C₄	—	4	8	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	H(5,𝔽₂)×C₇₃	1	6	yes	—
91	C₈	—	4	12	no	—
92	C₂²	—	2	6	no	—
93	C₂²	—	4	1	no	—
94	C₂²	—	4	3	no	—
95	C₂²	—	4	1	no	—
96	C₂²	#543×C₇₃	1	3	yes	—
97	C₄	—	4	1	no	—
98	C₄	—	4	4	no	—
99	C₄	#560×C₇₃	1	4	yes	—
100	C₂	—	2	3	no	—
101	C₂	—	4	2	no	—
102	C₂	C₇₃×#2842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2091

C₂

yes

maximal

C₇₃×#2070

C₂

yes

maximal, complemented

C₇₃×#1784

C₂

yes

maximal

C₂³×Q₈×C₇₃

C₂²

yes

characteristic

#430×C₇₃

C₂²

yes

—

C₂×#101×C₇₃

C₂²

yes

—

#407×C₇₃

C₂²

yes

—

#338×C₇₃

C₂²

yes

—

C₂²×Q₈×C₇₃

—

C₂²×Q₈×C₇₃

Q₈

yes

—

C₂²×Q₈×C₇₃

C₂³

yes

—

C₂³×C₄×C₇₃

Q₈

yes

—

C₂³×C₄×C₇₃

C₂³

yes

characteristic

C₂×M₁₆×C₇₃

C₂³

yes

—

#108×C₇₃

—

#101×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₂⁴×C₇₃

C₂×Q₈

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×Q₈

yes

—

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂⁴

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂²×Q₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

H(5,𝔽₂)

yes

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#543

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#560

yes

—

#32876

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2842

yes

characteristic

#2091

C₂×C₇₃

yes

—

#2070

C₂×C₇₃

yes

complemented

#1784

C₂×C₇₃

yes

—

C₇₃

#32876

yes

characteristic, direct product factor, Sylow

C₂³×Q₈

C₂²×C₇₃

yes

characteristic

#430

C₂²×C₇₃

yes

—

C₂×#101

C₂²×C₇₃

yes

—

#407

C₂²×C₇₃

yes

—

#338

C₂²×C₇₃

yes

—

C₂²×Q₈

—

C₂²×Q₈

Q₈×C₇₃

yes

—

C₂²×Q₈

C₂³×C₇₃

yes

—

C₂³×C₄

Q₈×C₇₃

yes

—

C₂³×C₄

C₂³×C₇₃

yes

characteristic

C₂×M₁₆

C₂³×C₇₃

yes

—

#108

—

#101

—

C₂²⋊C₈

—

C₂⁴

C₂×Q₈×C₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Q₈×C₇₃

yes

—

C₂²×C₄

C₂×Q₈×C₇₃

yes

characteristic

C₂²×C₄

C₂⁴×C₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

C₂²×Q₈×C₇₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

H(5,𝔽₂)×C₇₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#543×C₇₃

yes

—

C₄

—

C₄

—

C₄

#560×C₇₃

yes

—

100

C₂

—

101

C₂

—

102

C₂

C₇₃×#2842

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇₃	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂⁴×C₇₃	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂×C₇₃

#2842

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₇₃

prime radical

C₂⁴×C₇₃

C₂×Q₈

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ G

N	H ≅ G/N	type
2. C₇₃×#2070	C₂	semidirect
48. #32876	C₇₃	direct (N × H)
53. #2070	C₂×C₇₃	semidirect
55. C₇₃	#32876	direct (N × H)

H ≅ G/N

type

2. C₇₃×#2070

C₂

semidirect

48. #32876

C₇₃

direct (N × H)

53. #2070

C₂×C₇₃

semidirect

55. C₇₃

#32876

direct (N × H)

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

subgroup	index
C₇₃×#2091	2
C₇₃×#2070	2
C₇₃×#1784	2
#32876	73

subgroup

index

C₇₃×#2091

C₇₃×#2070

C₇₃×#1784

#32876

Subgroups — C₇₃×#32876 #32942280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#32876 #32942280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#32876 #32942280