Group #33007019 — subgroups

Subgroups — M₁₂₈×D₇₃ #33007019

Order 18688.

134 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
2	#22608142	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608141	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₇₃	C₂	1	2	yes	maximal, complemented
5	C₇₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
7	C₂×#14556497	C₄	1	1	yes	characteristic
8	C₃₂×D₇₃	C₄	1	1	yes	characteristic
9	C₃₂×D₇₃	C₄	1	1	yes	characteristic
10	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
11	C₃₂×D₇₃	C₂²	1	1	yes	characteristic
12	#17988306	C₂²	1	2	yes	complemented
13	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
14	C₆₄×C₇₃	C₂²	1	2	yes	complemented
15	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic
16	C₂×#11924003	C₈	1	1	yes	characteristic
17	C₁₆×D₇₃	C₈	1	1	yes	characteristic
18	C₁₆×D₇₃	C₈	1	1	yes	characteristic
19	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
21	#14556497	C₂×C₄	1	1	yes	characteristic
22	#14556497	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₃₂×C₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₇₃	C₁₆	1	1	yes	characteristic
27	C₂×#10596831	C₁₆	1	1	yes	characteristic
28	C₈×D₇₃	C₁₆	1	1	yes	characteristic
29	C₈×D₇₃	C₁₆	1	1	yes	characteristic
30	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
31	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
32	#11924003	C₂×C₈	1	1	yes	characteristic
33	#11924003	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
35	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×D₇₃	C₃₂	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×Dic₇₃	C₃₂	1	1	yes	characteristic
40	C₄×D₇₃	C₃₂	1	1	yes	characteristic
41	C₄×D₇₃	C₃₂	1	1	yes	characteristic
42	C₄×D₇₃	C₂×C₁₆	1	1	yes	characteristic
43	C₄×D₇₃	C₂×C₁₆	1	1	yes	characteristic
44	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	#10596831	C₂×C₁₆	1	1	yes	characteristic
47	#10596831	C₂×C₁₆	1	1	yes	characteristic
48	C₂²×C₇₃	C₂×C₃₂	1	1	yes	characteristic
49	C₂×D₇₃	—	2	2	no	—
50	C₂×D₇₃	C₂×C₃₂	1	1	yes	characteristic
51	C₂×D₇₃	C₂×C₃₂	1	1	yes	characteristic
52	C₄×C₇₃	C₂×C₃₂	1	1	yes	characteristic
53	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
54	Dic₇₃	C₂×C₃₂	1	1	yes	characteristic
55	Dic₇₃	C₂×C₃₂	1	1	yes	characteristic
56	C₂×M₁₂₈	—	73	1	no	maximal, Sylow
57	C₂×C₇₃	—	2	1	no	—
58	C₂×C₇₃	C₂²×C₃₂	1	1	yes	characteristic
59	D₇₃	—	2	1	no	—
60	D₇₃	M₁₂₈	1	2	yes	direct product factor
61	C₂²×C₃₂	—	73	1	no	—
62	M₁₂₈	—	73	1	no	—
63	M₁₂₈	—	73	2	no	—
64	M₁₂₈	D₇₃	1	1	yes	characteristic, direct product factor
65	C₂×C₆₄	—	73	2	no	—
66	C₇₃	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₁₆	—	73	1	no	—
68	C₂×C₃₂	—	73	1	no	—
69	C₂×C₃₂	—	73	1	no	—
70	C₂×C₃₂	—	73	1	no	—
71	C₂×C₃₂	—	73	1	no	—
72	C₂×C₃₂	—	73	1	no	—
73	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
74	C₆₄	—	73	2	no	—
75	C₆₄	C₂×D₇₃	1	2	yes	complemented
76	C₂²×C₈	—	73	1	no	—
77	C₂×C₁₆	—	73	1	no	—
78	C₂×C₁₆	—	73	1	no	—
79	C₂×C₁₆	—	73	1	no	—
80	C₂×C₁₆	—	73	1	no	—
81	C₂×C₁₆	—	73	1	no	—
82	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
83	C₃₂	—	73	1	no	—
84	C₃₂	—	73	1	no	—
85	C₃₂	C₄×D₇₃	1	1	yes	characteristic
86	C₃₂	C₂²×D₇₃	1	1	yes	characteristic
87	C₂²×C₄	—	73	1	no	—
88	C₂×C₈	—	73	1	no	—
89	C₂×C₈	—	73	1	no	—
90	C₂×C₈	—	73	1	no	—
91	C₂×C₈	—	73	1	no	—
92	C₂×C₈	—	73	1	no	—
93	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
94	C₁₆	—	73	1	no	—
95	C₁₆	—	73	1	no	—
96	C₁₆	C₈×D₇₃	1	1	yes	characteristic
97	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
98	C₂³	—	73	1	no	—
99	C₂×C₄	—	73	1	no	—
100	C₂×C₄	—	73	1	no	—
101	C₂×C₄	—	73	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
105	C₈	—	73	1	no	—
106	C₈	—	73	1	no	—
107	C₈	C₁₆×D₇₃	1	1	yes	characteristic
108	C₈	C₂×C₈×D₇₃	1	1	yes	characteristic
109	C₂²	—	73	1	no	—
110	C₂²	—	73	1	no	—
111	C₂²	—	146	2	no	—
112	C₂²	C₃₂×D₇₃	1	1	yes	characteristic
113	C₄	—	73	1	no	—
114	C₄	—	73	1	no	—
115	C₄	C₃₂×D₇₃	1	1	yes	characteristic
116	C₄	C₂×C₁₆×D₇₃	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	73	2	no	—
119	C₂	—	146	1	no	—
120	C₂	C₂×C₃₂×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₂×D₇₃

C₂

yes

maximal, characteristic

#22608142

C₂

yes

maximal, characteristic, complemented

#22608141

C₂

yes

maximal, complemented

C₆₄×D₇₃

C₂

yes

maximal, complemented

C₇₃×M₁₂₈

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₃

C₄

yes

characteristic

C₂×#14556497

C₄

yes

characteristic

C₃₂×D₇₃

C₄

yes

characteristic

C₃₂×D₇₃

C₄

yes

characteristic

C₃₂×D₇₃

C₂²

yes

characteristic

C₃₂×D₇₃

C₂²

yes

characteristic

#17988306

C₂²

yes

complemented

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

C₆₄×C₇₃

C₂²

yes

complemented

C₂×C₈×D₇₃

C₈

yes

characteristic

C₂×#11924003

C₈

yes

characteristic

C₁₆×D₇₃

C₈

yes

characteristic

C₁₆×D₇₃

C₈

yes

characteristic

C₁₆×D₇₃

C₂×C₄

yes

characteristic

C₁₆×D₇₃

C₂×C₄

yes

characteristic

#14556497

C₂×C₄

yes

characteristic

#14556497

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

C₁₆

yes

characteristic

C₂×#10596831

C₁₆

yes

characteristic

C₈×D₇₃

C₁₆

yes

characteristic

C₈×D₇₃

C₁₆

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

#11924003

C₂×C₈

yes

characteristic

#11924003

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₃₂

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×Dic₇₃

C₃₂

yes

characteristic

C₄×D₇₃

C₃₂

yes

characteristic

C₄×D₇₃

C₃₂

yes

characteristic

C₄×D₇₃

C₂×C₁₆

yes

characteristic

C₄×D₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₂²×C₈

yes

characteristic

#10596831

C₂×C₁₆

yes

characteristic

#10596831

C₂×C₁₆

yes

characteristic

C₂²×C₇₃

C₂×C₃₂

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₂×C₃₂

yes

characteristic

C₂×D₇₃

C₂×C₃₂

yes

characteristic

C₄×C₇₃

C₂×C₃₂

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

Dic₇₃

C₂×C₃₂

yes

characteristic

Dic₇₃

C₂×C₃₂

yes

characteristic

C₂×M₁₂₈

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂²×C₃₂

yes

characteristic

D₇₃

—

D₇₃

M₁₂₈

yes

direct product factor

C₂²×C₃₂

—

M₁₂₈

—

M₁₂₈

—

M₁₂₈

D₇₃

yes

characteristic, direct product factor

C₂×C₆₄

—

C₇₃

C₂×M₁₂₈

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

characteristic

C₆₄

—

C₆₄

C₂×D₇₃

yes

complemented

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

C₄×D₇₃

yes

characteristic

C₃₂

C₂²×D₇₃

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₇₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₈×D₇₃

yes

characteristic

C₁₆

C₂×C₄×D₇₃

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₁₆×D₇₃

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

C₁₆×D₇₃

yes

characteristic

108

C₈

C₂×C₈×D₇₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

146

—

112

C₂²

C₃₂×D₇₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

C₃₂×D₇₃

yes

characteristic

116

C₄

C₂×C₁₆×D₇₃

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

146

—

120

C₂

C₂×C₃₂×D₇₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₇₃
commutator, socle	C₂×C₇₃	C₂²×C₃₂
Fitting	C₇₃×M₁₂₈	C₂
prime radical	C₂²×D₇₃	C₃₂
cosocle	C₃₂×C₇₃	C₂³
hypercenter	M₁₂₈	D₇₃
hypocenter	C₇₃	C₂×M₁₂₈
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₃₂ ◁ M₁₂₈

subgroup

G/H

center, Frattini

C₃₂

C₂²×D₇₃

commutator, socle

C₂×C₇₃

C₂²×C₃₂

Fitting

C₇₃×M₁₂₈

C₂

prime radical

C₂²×D₇₃

C₃₂

cosocle

C₃₂×C₇₃

C₂³

hypercenter

M₁₂₈

D₇₃

hypocenter

C₇₃

C₂×M₁₂₈

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₃₂ ◁ M₁₂₈

N	H ≅ G/N	type
2. #22608142	C₂	semidirect
3. #22608141	C₂	semidirect
4. C₆₄×D₇₃	C₂	semidirect
5. C₇₃×M₁₂₈	C₂	semidirect
12. #17988306	C₂²	semidirect
14. C₆₄×C₇₃	C₂²	semidirect
60. D₇₃	M₁₂₈	direct (N × H)
64. M₁₂₈	D₇₃	direct (N × H)
66. C₇₃	C₂×M₁₂₈	semidirect
75. C₆₄	C₂×D₇₃	semidirect

H ≅ G/N

type

2. #22608142

C₂

semidirect

3. #22608141

C₂

semidirect

4. C₆₄×D₇₃

C₂

semidirect

5. C₇₃×M₁₂₈

C₂

semidirect

12. #17988306

C₂²

semidirect

14. C₆₄×C₇₃

C₂²

semidirect

60. D₇₃

M₁₂₈

direct (N × H)

64. M₁₂₈

D₇₃

direct (N × H)

66. C₇₃

C₂×M₁₂₈

semidirect

75. C₆₄

C₂×D₇₃

semidirect

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

subgroup	index
C₂×C₃₂×D₇₃	2
#22608142	2
#22608141	2
C₆₄×D₇₃	2
C₇₃×M₁₂₈	2
C₂×M₁₂₈	73

subgroup

index

C₂×C₃₂×D₇₃

#22608142

#22608141

C₆₄×D₇₃

C₇₃×M₁₂₈

C₂×M₁₂₈

Subgroups — M₁₂₈×D₇₃ #33007019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M128×D73 #33007019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₂₈×D₇₃ #33007019