Group #22606404 — subgroups

Subgroups — C₇₃×#2251 #22606404

Order 9344.

178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 72 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	2	yes	maximal, complemented
2	C₂×SD₃₂×C₇₃	C₂	1	2	yes	maximal
3	#495×C₇₃	C₂	1	1	yes	maximal, characteristic
4	#494×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆×C₇₃	—	2	1	no	—
7	C₂×Q₁₆×C₇₃	C₂²	1	2	yes	—
8	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
9	C₂×SD₁₆×C₇₃	—	2	1	no	—
10	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₄⋊Q₈×C₇₃	—	2	1	no	—
12	#124×C₇₃	—	2	1	no	—
13	Q₃₂×C₇₃	—	2	4	no	—
14	SD₃₂×C₇₃	—	2	4	no	—
15	C₂×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₇₃	—	2	1	no	—
18	C₂×Q₈×C₇₃	—	2	2	no	—
19	C₂×D₄×C₇₃	—	2	1	no	—
20	Q₁₆×C₇₃	—	2	2	no	—
21	Q₁₆×C₇₃	—	2	2	no	—
22	Q₁₆×C₇₃	—	2	2	no	—
23	Q₁₆×C₇₃	—	4	2	no	—
24	SD₁₆×C₇₃	—	4	2	no	—
25	D₈×C₇₃	—	2	2	no	—
26	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₇₃	—	4	2	no	—
29	C₂²⋊C₄×C₇₃	—	4	2	no	—
30	C₄²×C₇₃	D₄	1	1	yes	characteristic
31	C₁₆×C₇₃	D₄	1	4	yes	—
32	C₂³×C₇₃	—	4	1	no	—
33	Q₈×C₇₃	—	4	2	no	—
34	Q₈×C₇₃	—	4	1	no	—
35	Q₈×C₇₃	—	4	1	no	—
36	Q₈×C₇₃	—	4	2	no	—
37	D₄×C₇₃	—	4	1	no	—
38	D₄×C₇₃	—	4	1	no	—
39	C₂×C₄×C₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	—	4	2	no	—
41	C₂×C₄×C₇₃	D₈	1	2	yes	—
42	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₈×C₇₃	D₈	1	2	yes	—
44	C₈×C₇₃	C₂×D₄	1	2	yes	—
45	C₂²×C₇₃	—	8	1	no	—
46	C₂²×C₇₃	—	8	2	no	—
47	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
48	C₄×C₇₃	—	2	2	no	—
49	C₄×C₇₃	—	8	1	no	—
50	C₄×C₇₃	—	8	2	no	—
51	C₄×C₇₃	#127	1	1	yes	characteristic
52	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₇₃	—	16	1	no	—
54	C₂×C₇₃	#493	1	1	yes	characteristic
55	C₂×C₇₃	#510	1	2	yes	—
56	#2251	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₇₃	#2251	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×Q₃₂	C₂×C₇₃	1	2	yes	complemented
59	C₂×SD₃₂	C₂×C₇₃	1	2	yes	—
60	#495	C₂×C₇₃	1	1	yes	characteristic
61	#494	C₂×C₇₃	1	1	yes	characteristic, complemented
62	#346	C₂×C₇₃	1	1	yes	characteristic, complemented
63	C₂×Q₁₆	—	2	1	no	—
64	C₂×Q₁₆	C₂²×C₇₃	1	2	yes	—
65	C₂×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
66	C₂×SD₁₆	—	2	1	no	—
67	C₂×D₈	C₂²×C₇₃	1	1	yes	characteristic
68	C₄⋊Q₈	—	2	1	no	—
69	#124	—	2	1	no	—
70	Q₃₂	—	2	4	no	—
71	SD₃₂	—	2	4	no	—
72	C₂×C₁₆	C₂²×C₇₃	1	2	yes	—
73	C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
74	C₂×Q₈	—	2	1	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×D₄	—	2	1	no	—
77	Q₁₆	—	2	2	no	—
78	Q₁₆	—	2	2	no	—
79	Q₁₆	—	2	2	no	—
80	Q₁₆	—	4	2	no	—
81	SD₁₆	—	4	2	no	—
82	D₈	—	2	2	no	—
83	C₂×C₈	D₄×C₇₃	1	1	yes	characteristic
84	C₂×C₈	C₂³×C₇₃	1	1	yes	characteristic
85	C₄⋊C₄	—	4	2	no	—
86	C₂²⋊C₄	—	4	2	no	—
87	C₄²	D₄×C₇₃	1	1	yes	characteristic
88	C₁₆	D₄×C₇₃	1	4	yes	—
89	C₂³	—	4	1	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	4	1	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	2	no	—
94	D₄	—	4	1	no	—
95	D₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	D₈×C₇₃	1	2	yes	—
99	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
100	C₈	D₈×C₇₃	1	2	yes	—
101	C₈	C₂×D₄×C₇₃	1	2	yes	—
102	C₂²	—	8	1	no	—
103	C₂²	—	8	2	no	—
104	C₂²	C₂×D₈×C₇₃	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	8	1	no	—
107	C₄	—	8	2	no	—
108	C₄	#127×C₇₃	1	1	yes	characteristic
109	C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
110	C₂	—	16	1	no	—
111	C₂	#493×C₇₃	1	1	yes	characteristic
112	C₂	#510×C₇₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₃₂×C₇₃

C₂

yes

maximal, complemented

C₂×SD₃₂×C₇₃

C₂

yes

maximal

#495×C₇₃

C₂

yes

maximal, characteristic

#494×C₇₃

C₂

yes

maximal, characteristic, complemented

#346×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

C₂²

yes

—

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

—

#124×C₇₃

—

Q₃₂×C₇₃

—

SD₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₁₆×C₇₃

D₄

yes

—

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

D₈

yes

—

C₈×C₇₃

C₂×D₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#127

yes

characteristic

C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

#493

yes

characteristic

C₂×C₇₃

#510

yes

—

#2251

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₇₃

#2251

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂×C₇₃

yes

complemented

C₂×SD₃₂

C₂×C₇₃

yes

—

#495

C₂×C₇₃

yes

characteristic

#494

C₂×C₇₃

yes

characteristic, complemented

#346

C₂×C₇₃

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×Q₁₆

C₂²×C₇₃

yes

—

C₂×Q₁₆

C₂²×C₇₃

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂²×C₇₃

yes

characteristic

C₄⋊Q₈

—

#124

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

C₂²×C₇₃

yes

—

C₄×C₈

C₂²×C₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

D₄×C₇₃

yes

characteristic

C₂×C₈

C₂³×C₇₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₇₃

yes

characteristic

C₁₆

D₄×C₇₃

yes

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×C₇₃

yes

—

C₂×C₄

C₂×D₄×C₇₃

yes

characteristic

100

C₈

D₈×C₇₃

yes

—

101

C₈

C₂×D₄×C₇₃

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×D₈×C₇₃

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#127×C₇₃

yes

characteristic

109

C₄

C₂×D₈×C₇₃

yes

characteristic

110

C₂

—

111

C₂

#493×C₇₃

yes

characteristic

112

C₂

#510×C₇₃

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₇₃
prime radical	C₂×D₈×C₇₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₇₃

prime radical

C₂×D₈×C₇₃

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. C₂×Q₃₂×C₇₃	C₂	semidirect
4. #494×C₇₃	C₂	semidirect
5. #346×C₇₃	C₂	semidirect
56. #2251	C₇₃	direct (N × H)
57. C₇₃	#2251	direct (N × H)
58. C₂×Q₃₂	C₂×C₇₃	semidirect
61. #494	C₂×C₇₃	semidirect
62. #346	C₂×C₇₃	semidirect

H ≅ G/N

type

1. C₂×Q₃₂×C₇₃

C₂

semidirect

4. #494×C₇₃

C₂

semidirect

5. #346×C₇₃

C₂

semidirect

56. #2251

C₇₃

direct (N × H)

57. C₇₃

#2251

direct (N × H)

58. C₂×Q₃₂

C₂×C₇₃

semidirect

61. #494

C₂×C₇₃

semidirect

62. #346

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×SD₃₂×C₇₃	2
#495×C₇₃	2
#494×C₇₃	2
#346×C₇₃	2
#2251	73

subgroup

index

C₂×Q₃₂×C₇₃

C₂×SD₃₂×C₇₃

#495×C₇₃

#494×C₇₃

#346×C₇₃

#2251

Subgroups — C₇₃×#2251 #22606404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#2251 #22606404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#2251 #22606404