Group #33244086 — subgroups

Subgroups — C₂²×C₃₂×D₇₃ #33244086

Order 18688.

542 conjugacy classes of subgroups; only the 331 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 331 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	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556497	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×D₇₃	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
6	C₂²×#11924003	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₇₃	C₄	1	6	yes	—
8	C₂×C₁₆×D₇₃	C₂²	1	6	yes	—
9	C₂×#14556497	C₂²	1	6	yes	complemented
10	C₃₂×D₇₃	C₂²	1	16	yes	direct product factor
11	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₇₃	C₂²	1	6	yes	complemented
13	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
14	C₂²×#10596831	C₈	1	1	yes	characteristic
15	C₂×C₈×D₇₃	C₈	1	6	yes	—
16	C₂×C₈×D₇₃	C₂×C₄	1	6	yes	—
17	C₂×#11924003	C₂×C₄	1	6	yes	—
18	C₁₆×D₇₃	C₂×C₄	1	12	yes	—
19	C₁₆×D₇₃	C₂³	1	4	yes	—
20	#14556497	C₂³	1	4	yes	complemented
21	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₃	C₂×C₄	1	3	yes	—
23	C₂×C₁₆×C₇₃	C₂³	1	3	yes	—
24	C₃₂×C₇₃	C₂³	1	4	yes	complemented
25	C₂³×D₇₃	C₁₆	1	1	yes	characteristic
26	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
27	C₂×C₄×D₇₃	C₁₆	1	6	yes	—
28	C₂×C₄×D₇₃	C₂×C₈	1	6	yes	—
29	C₂×#10596831	C₂×C₈	1	6	yes	—
30	C₈×D₇₃	C₂×C₈	1	12	yes	—
31	C₈×D₇₃	C₂²×C₄	1	4	yes	—
32	#11924003	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂×C₈	1	3	yes	—
35	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
36	C₁₆×C₇₃	C₂²×C₄	1	3	yes	—
37	C₁₆×C₇₃	C₂⁴	1	1	yes	characteristic
38	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂²×D₇₃	C₃₂	1	8	yes	direct product factor
40	C₂²×D₇₃	C₂×C₁₆	1	6	yes	—
41	C₂×C₄×C₇₃	C₂×C₁₆	1	3	yes	—
42	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
43	C₂×Dic₇₃	C₂×C₁₆	1	6	yes	—
44	C₄×D₇₃	C₂×C₁₆	1	12	yes	—
45	C₄×D₇₃	C₂²×C₈	1	4	yes	—
46	C₈×C₇₃	C₂²×C₈	1	3	yes	—
47	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
48	#10596831	C₂²×C₈	1	4	yes	—
49	C₂²×C₇₃	C₂×C₃₂	1	4	yes	complemented
50	C₂²×C₇₃	C₂²×C₁₆	1	3	yes	—
51	C₂×D₇₃	C₂×C₃₂	1	24	yes	direct product factor
52	C₂×D₇₃	C₂²×C₁₆	1	4	yes	—
53	C₄×C₇₃	C₂²×C₁₆	1	3	yes	—
54	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
55	Dic₇₃	C₂²×C₁₆	1	4	yes	—
56	C₂³×C₃₂	—	73	1	no	maximal, Sylow
57	C₂×C₇₃	C₂²×C₃₂	1	6	yes	complemented
58	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
59	D₇₃	C₂²×C₃₂	1	8	yes	direct product factor
60	C₂³×C₁₆	—	73	1	no	—
61	C₂²×C₃₂	—	73	1	no	—
62	C₂²×C₃₂	—	73	12	no	—
63	C₂²×C₃₂	D₇₃	1	1	yes	characteristic, direct product factor
64	C₇₃	C₂³×C₃₂	1	1	yes	characteristic, complemented, Sylow
65	C₂³×C₈	—	73	1	no	—
66	C₂²×C₁₆	—	73	1	no	—
67	C₂²×C₁₆	—	73	6	no	—
68	C₂²×C₁₆	—	73	6	no	—
69	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
70	C₂×C₃₂	—	73	6	no	—
71	C₂×C₃₂	—	73	16	no	—
72	C₂×C₃₂	C₂×D₇₃	1	6	yes	direct product factor
73	C₂³×C₄	—	73	1	no	—
74	C₂²×C₈	—	73	1	no	—
75	C₂²×C₈	—	73	6	no	—
76	C₂²×C₈	—	73	6	no	—
77	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
78	C₂×C₁₆	—	73	6	no	—
79	C₂×C₁₆	—	73	12	no	—
80	C₂×C₁₆	—	73	4	no	—
81	C₂×C₁₆	C₄×D₇₃	1	3	yes	—
82	C₂×C₁₆	C₂²×D₇₃	1	3	yes	—
83	C₃₂	—	73	4	no	—
84	C₃₂	C₂²×D₇₃	1	4	yes	direct product factor
85	C₂⁴	—	73	1	no	—
86	C₂²×C₄	—	73	1	no	—
87	C₂²×C₄	—	73	6	no	—
88	C₂²×C₄	—	73	6	no	—
89	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
90	C₂×C₈	—	73	6	no	—
91	C₂×C₈	—	73	12	no	—
92	C₂×C₈	—	73	4	no	—
93	C₂×C₈	C₈×D₇₃	1	3	yes	—
94	C₂×C₈	C₂×C₄×D₇₃	1	3	yes	—
95	C₁₆	—	73	4	no	—
96	C₁₆	C₂×C₄×D₇₃	1	3	yes	—
97	C₁₆	C₂³×D₇₃	1	1	yes	characteristic
98	C₂³	—	73	8	no	—
99	C₂³	—	73	6	no	—
100	C₂³	C₁₆×D₇₃	1	1	yes	characteristic
101	C₂×C₄	—	73	6	no	—
102	C₂×C₄	—	73	12	no	—
103	C₂×C₄	—	73	4	no	—
104	C₂×C₄	C₁₆×D₇₃	1	3	yes	—
105	C₂×C₄	C₂×C₈×D₇₃	1	3	yes	—
106	C₈	—	73	4	no	—
107	C₈	C₂×C₈×D₇₃	1	3	yes	—
108	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
109	C₂²	—	73	4	no	—
110	C₂²	—	73	24	no	—
111	C₂²	C₃₂×D₇₃	1	4	yes	direct product factor
112	C₂²	C₂×C₁₆×D₇₃	1	3	yes	—
113	C₄	—	73	4	no	—
114	C₄	C₂×C₁₆×D₇₃	1	3	yes	—
115	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
116	C₂	—	73	8	no	—
117	C₂	C₂×C₃₂×D₇₃	1	6	yes	direct product factor
118	C₂	C₂²×C₁₆×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₇₃

C₂

yes

maximal, characteristic

C₂²×#14556497

C₂

yes

maximal, characteristic, complemented

C₂×C₃₂×D₇₃

C₂

yes

maximal, direct product factor

C₂²×C₃₂×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₃

C₄

yes

characteristic

C₂²×#11924003

C₄

yes

characteristic

C₂×C₁₆×D₇₃

C₄

yes

—

C₂×C₁₆×D₇₃

C₂²

yes

—

C₂×#14556497

C₂²

yes

complemented

C₃₂×D₇₃

C₂²

yes

direct product factor

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

C₂²

yes

complemented

C₂²×C₄×D₇₃

C₈

yes

characteristic

C₂²×#10596831

C₈

yes

characteristic

C₂×C₈×D₇₃

C₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×#11924003

C₂×C₄

yes

—

C₁₆×D₇₃

C₂×C₄

yes

—

C₁₆×D₇₃

C₂³

yes

—

#14556497

C₂³

yes

complemented

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

—

C₃₂×C₇₃

C₂³

yes

complemented

C₂³×D₇₃

C₁₆

yes

characteristic

C₂²×Dic₇₃

C₁₆

yes

characteristic

C₂×C₄×D₇₃

C₁₆

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

C₂²×C₄

yes

—

#11924003

C₂²×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

—

C₁₆×C₇₃

C₂²×C₄

yes

—

C₁₆×C₇₃

C₂⁴

yes

characteristic

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂²×D₇₃

C₃₂

yes

direct product factor

C₂²×D₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

C₂²×C₈

yes

—

C₈×C₇₃

C₂²×C₈

yes

—

C₈×C₇₃

C₂³×C₄

yes

characteristic

#10596831

C₂²×C₈

yes

—

C₂²×C₇₃

C₂×C₃₂

yes

complemented

C₂²×C₇₃

C₂²×C₁₆

yes

—

C₂×D₇₃

C₂×C₃₂

yes

direct product factor

C₂×D₇₃

C₂²×C₁₆

yes

—

C₄×C₇₃

C₂²×C₁₆

yes

—

C₄×C₇₃

C₂³×C₈

yes

characteristic

Dic₇₃

C₂²×C₁₆

yes

—

C₂³×C₃₂

—

maximal, Sylow

C₂×C₇₃

C₂²×C₃₂

yes

complemented

C₂×C₇₃

C₂³×C₁₆

yes

characteristic

D₇₃

C₂²×C₃₂

yes

direct product factor

C₂³×C₁₆

—

C₂²×C₃₂

—

C₂²×C₃₂

—

C₂²×C₃₂

D₇₃

yes

characteristic, direct product factor

C₇₃

C₂³×C₃₂

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

direct product factor

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₃

yes

—

C₂×C₁₆

C₂²×D₇₃

yes

—

C₃₂

—

C₃₂

C₂²×D₇₃

yes

direct product factor

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₈×D₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₇₃

yes

—

C₂×C₈

C₂×C₄×D₇₃

yes

—

C₁₆

—

C₁₆

C₂×C₄×D₇₃

yes

—

C₁₆

C₂³×D₇₃

yes

characteristic

C₂³

—

C₂³

—

100

C₂³

C₁₆×D₇₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₁₆×D₇₃

yes

—

105

C₂×C₄

C₂×C₈×D₇₃

yes

—

106

C₈

—

107

C₈

C₂×C₈×D₇₃

yes

—

108

C₈

C₂²×C₄×D₇₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

C₃₂×D₇₃

yes

direct product factor

112

C₂²

C₂×C₁₆×D₇₃

yes

—

113

C₄

—

114

C₄

C₂×C₁₆×D₇₃

yes

—

115

C₄

C₂²×C₈×D₇₃

yes

characteristic

116

C₂

—

117

C₂

C₂×C₃₂×D₇₃

yes

direct product factor

118

C₂

C₂²×C₁₆×D₇₃

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃₂	D₇₃
commutator, hypocenter	C₇₃	C₂³×C₃₂
Frattini	C₁₆	C₂³×D₇₃
Fitting	C₂²×C₃₂×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₁₆
prime radical	C₂³×D₇₃	C₁₆
cosocle	C₁₆×C₇₃	C₂⁴
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂²×C₃₂

subgroup

G/H

center, hypercenter

C₂²×C₃₂

D₇₃

commutator, hypocenter

C₇₃

C₂³×C₃₂

Frattini

C₁₆

C₂³×D₇₃

Fitting

C₂²×C₃₂×C₇₃

C₂

socle

C₂³×C₇₃

C₂×C₁₆

prime radical

C₂³×D₇₃

C₁₆

cosocle

C₁₆×C₇₃

C₂⁴

LCS

G ▷ C₇₃

UCS

C₁ ◁ C₂²×C₃₂

N	H ≅ G/N	type
2. C₂²×#14556497	C₂	semidirect
3. C₂×C₃₂×D₇₃	C₂	direct (N × H)
4. C₂²×C₃₂×C₇₃	C₂	semidirect
9. C₂×#14556497	C₂²	semidirect
10. C₃₂×D₇₃	C₂²	direct (N × H)
12. C₂×C₃₂×C₇₃	C₂²	semidirect
20. #14556497	C₂³	semidirect
24. C₃₂×C₇₃	C₂³	semidirect
39. C₂²×D₇₃	C₃₂	direct (N × H)
49. C₂²×C₇₃	C₂×C₃₂	semidirect
51. C₂×D₇₃	C₂×C₃₂	direct (N × H)
57. C₂×C₇₃	C₂²×C₃₂	semidirect
59. D₇₃	C₂²×C₃₂	direct (N × H)
63. C₂²×C₃₂	D₇₃	direct (N × H)
64. C₇₃	C₂³×C₃₂	semidirect
72. C₂×C₃₂	C₂×D₇₃	direct (N × H)
84. C₃₂	C₂²×D₇₃	direct (N × H)
111. C₂²	C₃₂×D₇₃	direct (N × H)
117. C₂	C₂×C₃₂×D₇₃	direct (N × H)

H ≅ G/N

type

2. C₂²×#14556497

C₂

semidirect

3. C₂×C₃₂×D₇₃

C₂

direct (N × H)

4. C₂²×C₃₂×C₇₃

C₂

semidirect

9. C₂×#14556497

C₂²

semidirect

10. C₃₂×D₇₃

C₂²

direct (N × H)

12. C₂×C₃₂×C₇₃

C₂²

semidirect

20. #14556497

C₂³

semidirect

24. C₃₂×C₇₃

C₂³

semidirect

39. C₂²×D₇₃

C₃₂

direct (N × H)

49. C₂²×C₇₃

C₂×C₃₂

semidirect

51. C₂×D₇₃

C₂×C₃₂

direct (N × H)

57. C₂×C₇₃

C₂²×C₃₂

semidirect

59. D₇₃

C₂²×C₃₂

direct (N × H)

63. C₂²×C₃₂

D₇₃

direct (N × H)

64. C₇₃

C₂³×C₃₂

semidirect

72. C₂×C₃₂

C₂×D₇₃

direct (N × H)

84. C₃₂

C₂²×D₇₃

direct (N × H)

111. C₂²

C₃₂×D₇₃

direct (N × H)

117. C₂

C₂×C₃₂×D₇₃

direct (N × H)

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

subgroup	index
C₂²×C₁₆×D₇₃	2
C₂²×#14556497	2
C₂×C₃₂×D₇₃	2
C₂²×C₃₂×C₇₃	2
C₂³×C₃₂	73

subgroup

index

C₂²×C₁₆×D₇₃

C₂²×#14556497

C₂×C₃₂×D₇₃

C₂²×C₃₂×C₇₃

C₂³×C₃₂

Subgroups — C₂²×C₃₂×D₇₃ #33244086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C32×D73 #33244086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₂×D₇₃ #33244086