Group #32923058 — subgroups

Subgroups — C₇₃×#13654 #32923058

Order 18688.

212 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 90 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×D₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₆×C₇₃	C₂	1	2	yes	maximal, complemented
3	C₇₃×#1424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1415	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆×C₇₃	C₂²	1	2	yes	—
7	C₄×D₈×C₇₃	—	2	2	no	—
8	D₃₂×C₇₃	C₄	1	4	yes	complemented
9	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
11	#366×C₇₃	C₂²	1	2	yes	—
12	#357×C₇₃	—	2	2	no	—
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×D₈×C₇₃	—	2	2	no	—
15	C₄×D₄×C₇₃	—	4	2	no	—
16	D₁₆×C₇₃	—	2	2	no	—
17	D₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
18	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
20	#107×C₇₃	—	2	2	no	—
21	#102×C₇₃	—	4	2	no	—
22	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₃₂×C₇₃	—	2	1	no	—
24	C₃₂×C₇₃	C₂×C₄	1	2	yes	complemented
25	C₂×D₄×C₇₃	—	4	2	no	—
26	C₂²×C₄×C₇₃	—	8	2	no	—
27	D₈×C₇₃	—	2	4	no	—
28	D₈×C₇₃	—	4	2	no	—
29	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
30	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₇₃	—	4	2	no	—
32	C₂²⋊C₄×C₇₃	—	8	2	no	—
33	C₄²×C₇₃	D₈	1	1	yes	characteristic
34	C₁₆×C₇₃	—	2	1	no	—
35	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₇₃	#41	1	1	yes	characteristic
37	C₂³×C₇₃	—	8	2	no	—
38	D₄×C₇₃	—	4	4	no	—
39	D₄×C₇₃	—	8	2	no	—
40	C₂×C₄×C₇₃	—	8	2	no	—
41	C₂×C₄×C₇₃	—	16	4	no	—
42	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
44	C₈×C₇₃	—	2	1	no	—
45	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
46	C₈×C₇₃	#135	1	1	yes	characteristic
47	C₂²×C₇₃	—	8	4	no	—
48	C₂²×C₇₃	—	16	2	no	—
49	C₂²×C₇₃	—	16	2	no	—
50	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₇₃	—	2	2	no	—
52	C₄×C₇₃	—	16	2	no	—
53	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
54	C₄×C₇₃	#508	1	1	yes	characteristic
55	#13654	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₇₃	—	16	4	no	—
57	C₂×C₇₃	C₄×D₁₆	1	1	yes	characteristic
58	C₂×C₇₃	#2263	1	2	yes	—
59	C₂×D₃₂	C₂×C₇₃	1	1	yes	characteristic
60	C₄×D₁₆	C₂×C₇₃	1	2	yes	complemented
61	#1424	C₂×C₇₃	1	1	yes	characteristic, complemented
62	#1415	C₂×C₇₃	1	2	yes	complemented
63	#1397	C₂×C₇₃	1	1	yes	characteristic, complemented
64	C₇₃	#13654	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×D₁₆	C₂²×C₇₃	1	2	yes	—
66	C₄×D₈	—	2	2	no	—
67	D₃₂	C₄×C₇₃	1	4	yes	complemented
68	C₂×C₃₂	C₂²×C₇₃	1	1	yes	characteristic
69	C₂×C₃₂	C₂²×C₇₃	1	1	yes	characteristic
70	#366	C₂²×C₇₃	1	2	yes	—
71	#357	—	2	2	no	—
72	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
73	C₂×D₈	—	2	2	no	—
74	C₄×D₄	—	4	2	no	—
75	D₁₆	—	2	2	no	—
76	D₁₆	C₂×C₄×C₇₃	1	4	yes	complemented
77	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
78	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
79	#107	—	2	2	no	—
80	#102	—	4	2	no	—
81	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
82	C₃₂	—	2	1	no	—
83	C₃₂	C₂×C₄×C₇₃	1	2	yes	complemented
84	C₂×D₄	—	4	2	no	—
85	C₂²×C₄	—	8	2	no	—
86	D₈	—	2	4	no	—
87	D₈	—	4	2	no	—
88	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
89	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
90	C₄⋊C₄	—	4	2	no	—
91	C₂²⋊C₄	—	8	2	no	—
92	C₄²	D₈×C₇₃	1	1	yes	characteristic
93	C₁₆	—	2	1	no	—
94	C₁₆	C₂²×C₄×C₇₃	1	1	yes	characteristic
95	C₁₆	#41×C₇₃	1	1	yes	characteristic
96	C₂³	—	8	2	no	—
97	D₄	—	4	4	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	16	4	no	—
101	C₂×C₄	D₁₆×C₇₃	1	2	yes	—
102	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	C₄×D₄×C₇₃	1	1	yes	characteristic
105	C₈	#135×C₇₃	1	1	yes	characteristic
106	C₂²	—	8	4	no	—
107	C₂²	—	16	2	no	—
108	C₂²	—	16	2	no	—
109	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
110	C₄	—	2	2	no	—
111	C₄	—	16	2	no	—
112	C₄	C₄×D₈×C₇₃	1	1	yes	characteristic
113	C₄	#508×C₇₃	1	1	yes	characteristic
114	C₂	—	16	4	no	—
115	C₂	C₄×D₁₆×C₇₃	1	1	yes	characteristic
116	C₂	C₇₃×#2263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₃₂×C₇₃

C₂

yes

maximal, characteristic

C₄×D₁₆×C₇₃

C₂

yes

maximal, complemented

C₇₃×#1424

C₂

yes

maximal, characteristic, complemented

C₇₃×#1415

C₂

yes

maximal, complemented

C₇₃×#1397

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₇₃

C₂²

yes

—

C₄×D₈×C₇₃

—

D₃₂×C₇₃

C₄

yes

complemented

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

#366×C₇₃

C₂²

yes

—

#357×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

—

C₄×D₄×C₇₃

—

D₁₆×C₇₃

—

D₁₆×C₇₃

C₂×C₄

yes

complemented

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#102×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₃₂×C₇₃

C₂×C₄

yes

complemented

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₄×D₄

yes

characteristic

C₈×C₇₃

#135

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₈

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₄×D₁₆

yes

characteristic

C₂×C₇₃

#2263

yes

—

C₂×D₃₂

C₂×C₇₃

yes

characteristic

C₄×D₁₆

C₂×C₇₃

yes

complemented

#1424

C₂×C₇₃

yes

characteristic, complemented

#1415

C₂×C₇₃

yes

complemented

#1397

C₂×C₇₃

yes

characteristic, complemented

C₇₃

#13654

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

C₂²×C₇₃

yes

—

C₄×D₈

—

D₃₂

C₄×C₇₃

yes

complemented

C₂×C₃₂

C₂²×C₇₃

yes

characteristic

C₂×C₃₂

C₂²×C₇₃

yes

characteristic

#366

C₂²×C₇₃

yes

—

#357

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×D₈

—

C₄×D₄

—

D₁₆

—

D₁₆

C₂×C₄×C₇₃

yes

complemented

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

#107

—

#102

—

C₄×C₈

D₄×C₇₃

yes

characteristic

C₃₂

—

C₃₂

C₂×C₄×C₇₃

yes

complemented

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

C₂×C₈

D₈×C₇₃

yes

characteristic

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₈×C₇₃

yes

characteristic

C₁₆

—

C₁₆

C₂²×C₄×C₇₃

yes

characteristic

C₁₆

#41×C₇₃

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₁₆×C₇₃

yes

—

102

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

103

C₈

—

104

C₈

C₄×D₄×C₇₃

yes

characteristic

105

C₈

#135×C₇₃

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

C₄×D₈×C₇₃

yes

characteristic

113

C₄

#508×C₇₃

yes

characteristic

114

C₂

—

115

C₂

C₄×D₁₆×C₇₃

yes

characteristic

116

C₂

C₇₃×#2263

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₂×D₃₂×C₇₃	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂×D₁₆

commutator

C₁₆

C₂²×C₄×C₇₃

Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

prime radical

C₂×D₃₂×C₇₃

C₂

LCS

G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
2. C₄×D₁₆×C₇₃	C₂	semidirect
3. C₇₃×#1424	C₂	semidirect
4. C₇₃×#1415	C₂	semidirect
5. C₇₃×#1397	C₂	semidirect
8. D₃₂×C₇₃	C₄	semidirect
17. D₁₆×C₇₃	C₂×C₄	semidirect
24. C₃₂×C₇₃	C₂×C₄	semidirect
55. #13654	C₇₃	direct (N × H)
60. C₄×D₁₆	C₂×C₇₃	semidirect
61. #1424	C₂×C₇₃	semidirect
62. #1415	C₂×C₇₃	semidirect
63. #1397	C₂×C₇₃	semidirect
64. C₇₃	#13654	direct (N × H)
67. D₃₂	C₄×C₇₃	semidirect
76. D₁₆	C₂×C₄×C₇₃	semidirect
83. C₃₂	C₂×C₄×C₇₃	semidirect

H ≅ G/N

type

2. C₄×D₁₆×C₇₃

C₂

semidirect

3. C₇₃×#1424

C₂

semidirect

4. C₇₃×#1415

C₂

semidirect

5. C₇₃×#1397

C₂

semidirect

8. D₃₂×C₇₃

C₄

semidirect

17. D₁₆×C₇₃

C₂×C₄

semidirect

24. C₃₂×C₇₃

C₂×C₄

semidirect

55. #13654

C₇₃

direct (N × H)

60. C₄×D₁₆

C₂×C₇₃

semidirect

61. #1424

C₂×C₇₃

semidirect

62. #1415

C₂×C₇₃

semidirect

63. #1397

C₂×C₇₃

semidirect

64. C₇₃

#13654

direct (N × H)

67. D₃₂

C₄×C₇₃

semidirect

76. D₁₆

C₂×C₄×C₇₃

semidirect

83. C₃₂

C₂×C₄×C₇₃

semidirect

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

subgroup	index
C₂×D₃₂×C₇₃	2
C₄×D₁₆×C₇₃	2
C₇₃×#1424	2
C₇₃×#1415	2
C₇₃×#1397	2
#13654	73

subgroup

index

C₂×D₃₂×C₇₃

C₄×D₁₆×C₇₃

C₇₃×#1424

C₇₃×#1415

C₇₃×#1397

#13654

Subgroups — C₇₃×#13654 #32923058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13654 #32923058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13654 #32923058