Group #32921868 — subgroups

Subgroups — C₇₃×#12464 #32921868

Order 18688.

172 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₇₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#2231	C₂	1	2	yes	maximal
3	C₇₃×#2170	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1373	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1366	C₂	1	2	yes	maximal, complemented
6	#504×C₇₃	—	2	1	no	—
7	#499×C₇₃	C₂²	1	2	yes	—
8	#443×C₇₃	C₂²	1	1	yes	characteristic
9	#367×C₇₃	—	2	1	no	—
10	#366×C₇₃	—	2	1	no	—
11	#365×C₇₃	—	2	2	no	—
12	#364×C₇₃	C₂²	1	2	yes	—
13	#364×C₇₃	C₂²	1	1	yes	characteristic
14	#360×C₇₃	—	2	2	no	—
15	#350×C₇₃	—	2	1	no	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	#135×C₇₃	D₄	1	1	yes	characteristic
18	#131×C₇₃	—	2	1	no	—
19	#125×C₇₃	—	2	2	no	—
20	M₃₂×C₇₃	—	2	1	no	—
21	M₃₂×C₇₃	—	2	2	no	—
22	M₃₂×C₇₃	—	4	1	no	—
23	C₂×C₁₆×C₇₃	—	4	1	no	—
24	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
26	#108×C₇₃	D₄	1	1	yes	characteristic
27	#107×C₇₃	—	2	2	no	—
28	#106×C₇₃	—	2	2	no	—
29	#104×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	#41×C₇₃	—	2	1	no	—
32	Q₁₆×C₇₃	Q₁₆	1	1	yes	characteristic
33	SD₁₆×C₇₃	—	2	1	no	—
34	D₈×C₇₃	Q₁₆	1	1	yes	characteristic
35	M₁₆×C₇₃	—	2	1	no	—
36	M₁₆×C₇₃	—	4	1	no	—
37	C₂×C₈×C₇₃	—	4	1	no	—
38	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₇₃	—	4	2	no	—
41	C₄⋊C₄×C₇₃	—	8	2	no	—
42	C₄²×C₇₃	#41	1	1	yes	characteristic
43	C₁₆×C₇₃	—	2	1	no	—
44	C₁₆×C₇₃	—	4	1	no	—
45	C₁₆×C₇₃	—	4	2	no	—
46	C₁₆×C₇₃	SD₁₆	1	1	yes	characteristic
47	C₁₆×C₇₃	SD₁₆	1	1	yes	characteristic
48	Q₈×C₇₃	—	2	1	no	—
49	D₄×C₇₃	—	2	1	no	—
50	D₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	8	2	no	—
54	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
55	C₈×C₇₃	—	4	1	no	—
56	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
58	C₈×C₇₃	#135	1	1	yes	characteristic
59	C₈×C₇₃	#137	1	1	yes	characteristic
60	C₂²×C₇₃	—	4	1	no	—
61	C₂²×C₇₃	#472	1	1	yes	characteristic
62	C₄×C₇₃	—	4	1	no	—
63	C₄×C₇₃	—	4	1	no	—
64	C₄×C₇₃	—	16	2	no	—
65	C₄×C₇₃	#461	1	1	yes	characteristic
66	C₄×C₇₃	#467	1	1	yes	characteristic
67	#12464	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₇₃	—	2	1	no	—
69	C₂×C₇₃	—	8	1	no	—
70	C₂×C₇₃	#1675	1	1	yes	characteristic
71	#2253	C₂×C₇₃	1	1	yes	characteristic, complemented
72	#2231	C₂×C₇₃	1	2	yes	—
73	#2170	C₂×C₇₃	1	1	yes	characteristic
74	#1373	C₂×C₇₃	1	1	yes	characteristic, complemented
75	#1366	C₂×C₇₃	1	2	yes	complemented
76	C₇₃	#12464	1	1	yes	characteristic, direct product factor, Sylow
77	#504	—	2	1	no	—
78	#499	C₂²×C₇₃	1	2	yes	—
79	#443	C₂²×C₇₃	1	1	yes	characteristic
80	#367	—	2	1	no	—
81	#366	—	2	1	no	—
82	#365	—	2	2	no	—
83	#364	C₂²×C₇₃	1	1	yes	characteristic
84	#364	C₂²×C₇₃	1	2	yes	—
85	#360	—	2	2	no	—
86	#350	—	2	1	no	—
87	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
88	#135	D₄×C₇₃	1	1	yes	characteristic
89	#131	—	2	1	no	—
90	#125	—	2	2	no	—
91	M₃₂	—	2	1	no	—
92	M₃₂	—	2	2	no	—
93	M₃₂	—	4	1	no	—
94	C₂×C₁₆	—	4	1	no	—
95	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
96	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
97	#108	D₄×C₇₃	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#106	—	2	2	no	—
100	#104	—	2	1	no	—
101	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
102	#41	—	2	1	no	—
103	Q₁₆	Q₁₆×C₇₃	1	1	yes	characteristic
104	SD₁₆	—	2	1	no	—
105	D₈	Q₁₆×C₇₃	1	1	yes	characteristic
106	M₁₆	—	2	1	no	—
107	M₁₆	—	4	1	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
110	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄²	#41×C₇₃	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	SD₁₆×C₇₃	1	1	yes	characteristic
118	C₁₆	SD₁₆×C₇₃	1	1	yes	characteristic
119	Q₈	—	2	1	no	—
120	D₄	—	2	1	no	—
121	D₄	—	4	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#121×C₇₃	1	1	yes	characteristic
126	C₈	—	4	1	no	—
127	C₈	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
128	C₈	C₂×Q₁₆×C₇₃	1	1	yes	characteristic
129	C₈	#135×C₇₃	1	1	yes	characteristic
130	C₈	#137×C₇₃	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	#472×C₇₃	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	4	1	no	—
135	C₄	—	16	2	no	—
136	C₄	#461×C₇₃	1	1	yes	characteristic
137	C₄	#467×C₇₃	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	8	1	no	—
140	C₂	C₇₃×#1675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2253

C₂

yes

maximal, characteristic, complemented

C₇₃×#2231

C₂

yes

maximal

C₇₃×#2170

C₂

yes

maximal, characteristic

C₇₃×#1373

C₂

yes

maximal, characteristic, complemented

C₇₃×#1366

C₂

yes

maximal, complemented

#504×C₇₃

—

#499×C₇₃

C₂²

yes

—

#443×C₇₃

C₂²

yes

characteristic

#367×C₇₃

—

#366×C₇₃

—

#365×C₇₃

—

#364×C₇₃

C₂²

yes

—

#364×C₇₃

C₂²

yes

characteristic

#360×C₇₃

—

#350×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#135×C₇₃

D₄

yes

characteristic

#131×C₇₃

—

#125×C₇₃

—

M₃₂×C₇₃

—

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

#108×C₇₃

D₄

yes

characteristic

#107×C₇₃

—

#106×C₇₃

—

#104×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

#41×C₇₃

—

Q₁₆×C₇₃

Q₁₆

yes

characteristic

SD₁₆×C₇₃

—

D₈×C₇₃

Q₁₆

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

#41

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

SD₁₆

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂×SD₁₆

yes

characteristic

C₈×C₇₃

C₂×Q₁₆

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

#472

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1675

yes

characteristic

#2253

C₂×C₇₃

yes

characteristic, complemented

#2231

C₂×C₇₃

yes

—

#2170

C₂×C₇₃

yes

characteristic

#1373

C₂×C₇₃

yes

characteristic, complemented

#1366

C₂×C₇₃

yes

complemented

C₇₃

#12464

yes

characteristic, direct product factor, Sylow

#504

—

#499

C₂²×C₇₃

yes

—

#443

C₂²×C₇₃

yes

—

—

—

C₂²×C₇₃

yes

characteristic

#364

C₂²×C₇₃

yes

—

#360

—

#350

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

#135

D₄×C₇₃

yes

characteristic

#131

—

#125

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

D₄×C₇₃

yes

characteristic

#108

D₄×C₇₃

yes

characteristic

#107

—

#106

—

100

#104

—

101

C₄×C₈

C₂³×C₇₃

yes

characteristic

102

#41

—

103

Q₁₆

Q₁₆×C₇₃

yes

characteristic

104

SD₁₆

—

105

D₈

Q₁₆×C₇₃

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

110

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

#41×C₇₃

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

SD₁₆×C₇₃

yes

characteristic

118

C₁₆

SD₁₆×C₇₃

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#121×C₇₃

yes

characteristic

126

C₈

—

127

C₈

C₂×SD₁₆×C₇₃

yes

characteristic

128

C₈

C₂×Q₁₆×C₇₃

yes

characteristic

129

C₈

#135×C₇₃

yes

characteristic

130

C₈

#137×C₇₃

yes

characteristic

131

C₂²

—

132

C₂²

#472×C₇₃

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#461×C₇₃

yes

characteristic

137

C₄

#467×C₇₃

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

C₇₃×#1675

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇₃	#1675
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇₃
prime radical	#135×C₇₃	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄×C₈×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂×C₇₃

#1675

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₇₃

prime radical

#135×C₇₃

D₄

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄×C₈×C₇₃ ◁ G

N	H ≅ G/N	type
1. C₇₃×#2253	C₂	semidirect
4. C₇₃×#1373	C₂	semidirect
5. C₇₃×#1366	C₂	semidirect
67. #12464	C₇₃	direct (N × H)
71. #2253	C₂×C₇₃	semidirect
74. #1373	C₂×C₇₃	semidirect
75. #1366	C₂×C₇₃	semidirect
76. C₇₃	#12464	direct (N × H)

H ≅ G/N

type

1. C₇₃×#2253

C₂

semidirect

4. C₇₃×#1373

C₂

semidirect

5. C₇₃×#1366

C₂

semidirect

67. #12464

C₇₃

direct (N × H)

71. #2253

C₂×C₇₃

semidirect

74. #1373

C₂×C₇₃

semidirect

75. #1366

C₂×C₇₃

semidirect

76. C₇₃

#12464

direct (N × H)

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

subgroup	index
C₇₃×#2253	2
C₇₃×#2231	2
C₇₃×#2170	2
C₇₃×#1373	2
C₇₃×#1366	2
#12464	73

subgroup

index

C₇₃×#2253

C₇₃×#2231

C₇₃×#2170

C₇₃×#1373

C₇₃×#1366

#12464

Subgroups — C₇₃×#12464 #32921868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#12464 #32921868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#12464 #32921868