Group #22606372 — subgroups

Subgroups — C₇₃×#2219 #22606372

Order 9344.

150 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₂×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
3	#467×C₇₃	C₂	1	2	yes	maximal
4	#367×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#358×C₇₃	C₂	1	2	yes	maximal, complemented
6	C₂×Q₁₆×C₇₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
8	C₂²⋊Q₈×C₇₃	—	2	2	no	—
9	Q₃₂×C₇₃	—	2	2	no	—
10	M₃₂×C₇₃	—	2	2	no	—
11	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	#107×C₇₃	C₂²	1	2	yes	—
14	#103×C₇₃	—	2	2	no	—
15	C₂×Q₈×C₇₃	—	2	2	no	—
16	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
17	Q₁₆×C₇₃	—	2	2	no	—
18	Q₁₆×C₇₃	—	2	2	no	—
19	C₂×C₈×C₇₃	—	2	1	no	—
20	C₂×C₈×C₇₃	—	2	1	no	—
21	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₇₃	—	2	2	no	—
24	C₄⋊C₄×C₇₃	—	4	2	no	—
25	C₂²⋊C₄×C₇₃	—	4	2	no	—
26	C₁₆×C₇₃	—	2	1	no	—
27	C₁₆×C₇₃	D₄	1	2	yes	—
28	C₂³×C₇₃	D₈	1	1	yes	characteristic
29	Q₈×C₇₃	—	4	2	no	—
30	Q₈×C₇₃	—	4	2	no	—
31	C₂×C₄×C₇₃	—	2	1	no	—
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	4	2	no	—
34	C₂×C₄×C₇₃	—	4	2	no	—
35	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
37	C₈×C₇₃	—	2	1	no	—
38	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₇₃	#41	1	1	yes	characteristic
40	C₂²×C₇₃	—	2	1	no	—
41	C₂²×C₇₃	—	2	2	no	—
42	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
43	C₄×C₇₃	—	2	1	no	—
44	C₄×C₇₃	—	8	2	no	—
45	C₄×C₇₃	—	8	2	no	—
46	C₄×C₇₃	#121	1	1	yes	characteristic
47	C₄×C₇₃	#135	1	1	yes	characteristic
48	C₂×C₇₃	—	4	1	no	—
49	C₂×C₇₃	#466	1	1	yes	characteristic
50	C₂×C₇₃	#510	1	2	yes	—
51	#2219	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₇₃	#2219	1	1	yes	characteristic, direct product factor, Sylow
53	C₂×Q₃₂	C₂×C₇₃	1	1	yes	characteristic, complemented
54	C₂×M₃₂	C₂×C₇₃	1	1	yes	characteristic
55	#467	C₂×C₇₃	1	2	yes	—
56	#367	C₂×C₇₃	1	1	yes	characteristic, complemented
57	#358	C₂×C₇₃	1	2	yes	complemented
58	C₂×Q₁₆	C₂²×C₇₃	1	2	yes	—
59	C₂²×C₈	C₂²×C₇₃	1	1	yes	characteristic
60	C₂²⋊Q₈	—	2	2	no	—
61	Q₃₂	—	2	2	no	—
62	M₃₂	—	2	2	no	—
63	C₂×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
65	#107	C₂²×C₇₃	1	2	yes	—
66	#103	—	2	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
69	Q₁₆	—	2	2	no	—
70	Q₁₆	—	2	2	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	D₄×C₇₃	1	1	yes	characteristic
74	C₂×C₈	C₂³×C₇₃	1	1	yes	characteristic
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₂²⋊C₄	—	4	2	no	—
78	C₁₆	—	2	1	no	—
79	C₁₆	D₄×C₇₃	1	2	yes	—
80	C₂³	D₈×C₇₃	1	1	yes	characteristic
81	Q₈	—	4	2	no	—
82	Q₈	—	4	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₈×C₇₃	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
91	C₈	#41×C₇₃	1	1	yes	characteristic
92	C₂²	—	2	1	no	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂×D₈×C₇₃	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	8	2	no	—
97	C₄	—	8	2	no	—
98	C₄	#121×C₇₃	1	1	yes	characteristic
99	C₄	#135×C₇₃	1	1	yes	characteristic
100	C₂	—	4	1	no	—
101	C₂	#466×C₇₃	1	1	yes	characteristic
102	C₂	#510×C₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₃₂×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×M₃₂×C₇₃

C₂

yes

maximal, characteristic

#467×C₇₃

C₂

yes

maximal

#367×C₇₃

C₂

yes

maximal, characteristic, complemented

#358×C₇₃

C₂

yes

maximal, complemented

C₂×Q₁₆×C₇₃

C₂²

yes

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

—

#103×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

D₄

yes

—

C₂³×C₇₃

D₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

#510

yes

—

#2219

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₇₃

#2219

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂×C₇₃

yes

characteristic, complemented

C₂×M₃₂

C₂×C₇₃

yes

characteristic

#467

C₂×C₇₃

yes

—

#367

C₂×C₇₃

yes

characteristic, complemented

#358

C₂×C₇₃

yes

complemented

C₂×Q₁₆

C₂²×C₇₃

yes

—

C₂²×C₈

C₂²×C₇₃

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

M₃₂

—

C₂×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×C₁₆

C₂²×C₇₃

yes

characteristic

#107

C₂²×C₇₃

yes

—

#103

—

C₂×Q₈

—

C₂²×C₄

D₄×C₇₃

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₇₃

yes

characteristic

C₂×C₈

C₂³×C₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

C₁₆

D₄×C₇₃

yes

—

C₂³

D₈×C₇₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×C₇₃

yes

characteristic

C₂×C₄

C₂×D₄×C₇₃

yes

characteristic

C₈

—

C₈

C₂×D₄×C₇₃

yes

characteristic

C₈

#41×C₇₃

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×D₈×C₇₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#121×C₇₃

yes

characteristic

C₄

#135×C₇₃

yes

characteristic

100

C₂

—

101

C₂

#466×C₇₃

yes

characteristic

102

C₂

#510×C₇₃

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇₃
prime radical	C₂³×C₇₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂²×C₈×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₇₃

prime radical

C₂³×C₇₃

D₈

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. C₂×Q₃₂×C₇₃	C₂	semidirect
4. #367×C₇₃	C₂	semidirect
5. #358×C₇₃	C₂	semidirect
51. #2219	C₇₃	direct (N × H)
52. C₇₃	#2219	direct (N × H)
53. C₂×Q₃₂	C₂×C₇₃	semidirect
56. #367	C₂×C₇₃	semidirect
57. #358	C₂×C₇₃	semidirect

H ≅ G/N

type

1. C₂×Q₃₂×C₇₃

C₂

semidirect

4. #367×C₇₃

C₂

semidirect

5. #358×C₇₃

C₂

semidirect

51. #2219

C₇₃

direct (N × H)

52. C₇₃

#2219

direct (N × H)

53. C₂×Q₃₂

C₂×C₇₃

semidirect

56. #367

C₂×C₇₃

semidirect

57. #358

C₂×C₇₃

semidirect

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

subgroup	index
C₂×Q₃₂×C₇₃	2
C₂×M₃₂×C₇₃	2
#467×C₇₃	2
#367×C₇₃	2
#358×C₇₃	2
#2219	73

subgroup

index

C₂×Q₃₂×C₇₃

C₂×M₃₂×C₇₃

#467×C₇₃

#367×C₇₃

#358×C₇₃

#2219

Subgroups — C₇₃×#2219 #22606372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#2219 #22606372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#2219 #22606372