Group #32923125 — subgroups

Subgroups — C₇₃×#13721 #32923125

Order 18688.

180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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×#2249	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1416	C₂	1	2	yes	maximal, complemented
4	C₇₃×#1415	C₂	1	2	yes	maximal
5	C₄×C₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×SD₃₂×C₇₃	—	2	1	no	—
8	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
9	#499×C₇₃	—	2	1	no	—
10	#495×C₇₃	—	2	1	no	—
11	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
12	#367×C₇₃	—	2	1	no	—
13	#366×C₇₃	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₇₃	—	2	1	no	—
16	C₂×SD₁₆×C₇₃	—	4	1	no	—
17	C₂×D₈×C₇₃	—	2	1	no	—
18	#125×C₇₃	—	4	1	no	—
19	#124×C₇₃	—	4	1	no	—
20	Q₃₂×C₇₃	—	2	2	no	—
21	SD₃₂×C₇₃	—	4	2	no	—
22	D₁₆×C₇₃	—	2	2	no	—
23	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
25	#107×C₇₃	—	2	2	no	—
26	#106×C₇₃	—	4	1	no	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₃₂×C₇₃	—	2	2	no	—
29	C₂×Q₈×C₇₃	—	4	1	no	—
30	C₂×D₄×C₇₃	—	4	1	no	—
31	Q₁₆×C₇₃	—	4	1	no	—
32	Q₁₆×C₇₃	—	4	1	no	—
33	SD₁₆×C₇₃	—	8	2	no	—
34	D₈×C₇₃	—	4	1	no	—
35	D₈×C₇₃	—	4	1	no	—
36	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄⋊C₄×C₇₃	—	8	2	no	—
40	C₂²⋊C₄×C₇₃	—	8	2	no	—
41	C₄²×C₇₃	D₈	1	1	yes	characteristic
42	C₁₆×C₇₃	SD₁₆	1	2	yes	—
43	C₁₆×C₇₃	#41	1	2	yes	—
44	C₂³×C₇₃	—	8	1	no	—
45	Q₈×C₇₃	—	8	1	no	—
46	Q₈×C₇₃	—	8	1	no	—
47	D₄×C₇₃	—	8	1	no	—
48	D₄×C₇₃	—	8	1	no	—
49	C₂×C₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	—	8	2	no	—
51	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₈×C₇₃	SD₃₂	1	2	yes	—
54	C₈×C₇₃	#124	1	1	yes	characteristic
55	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₇₃	—	16	1	no	—
57	C₂²×C₇₃	—	16	2	no	—
58	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
59	C₄×C₇₃	—	2	2	no	—
60	C₄×C₇₃	—	16	1	no	—
61	C₄×C₇₃	—	16	2	no	—
62	C₄×C₇₃	#486	1	1	yes	characteristic
63	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
64	#13721	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₇₃	—	32	1	no	—
66	C₂×C₇₃	#2240	1	1	yes	characteristic
67	C₂×C₇₃	#2262	1	2	yes	—
68	#2253	C₂×C₇₃	1	1	yes	characteristic, complemented
69	#2249	C₂×C₇₃	1	1	yes	characteristic
70	#1416	C₂×C₇₃	1	2	yes	complemented
71	#1415	C₂×C₇₃	1	2	yes	—
72	C₄×C₃₂	C₂×C₇₃	1	1	yes	characteristic, complemented
73	C₇₃	#13721	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×Q₃₂	C₂²×C₇₃	1	1	yes	characteristic
75	C₂×SD₃₂	—	2	1	no	—
76	C₂×D₁₆	C₂²×C₇₃	1	1	yes	characteristic
77	#499	—	2	1	no	—
78	#495	—	2	1	no	—
79	C₂×C₃₂	C₂²×C₇₃	1	2	yes	—
80	#367	—	2	1	no	—
81	#366	C₂²×C₇₃	1	2	yes	—
82	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
83	C₂×Q₁₆	—	2	1	no	—
84	C₂×SD₁₆	—	4	1	no	—
85	C₂×D₈	—	2	1	no	—
86	#125	—	4	1	no	—
87	#124	—	4	1	no	—
88	Q₃₂	—	2	2	no	—
89	SD₃₂	—	4	2	no	—
90	D₁₆	—	2	2	no	—
91	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
92	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
93	#107	—	2	2	no	—
94	#106	—	4	1	no	—
95	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
96	C₃₂	—	2	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂×D₄	—	4	1	no	—
99	Q₁₆	—	4	1	no	—
100	Q₁₆	—	4	1	no	—
101	SD₁₆	—	8	2	no	—
102	D₈	—	4	1	no	—
103	D₈	—	4	1	no	—
104	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	8	2	no	—
108	C₂²⋊C₄	—	8	2	no	—
109	C₄²	D₈×C₇₃	1	1	yes	characteristic
110	C₁₆	SD₁₆×C₇₃	1	2	yes	—
111	C₁₆	#41×C₇₃	1	2	yes	—
112	C₂³	—	8	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	8	1	no	—
115	D₄	—	8	1	no	—
116	D₄	—	8	1	no	—
117	C₂×C₄	—	8	1	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	D₁₆×C₇₃	1	2	yes	—
120	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
121	C₈	SD₃₂×C₇₃	1	2	yes	—
122	C₈	#124×C₇₃	1	1	yes	characteristic
123	C₈	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
124	C₂²	—	16	1	no	—
125	C₂²	—	16	2	no	—
126	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	16	1	no	—
129	C₄	—	16	2	no	—
130	C₄	#486×C₇₃	1	1	yes	characteristic
131	C₄	C₂×SD₃₂×C₇₃	1	1	yes	characteristic
132	C₂	—	32	1	no	—
133	C₂	C₇₃×#2240	1	1	yes	characteristic
134	C₂	C₇₃×#2262	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2253

C₂

yes

maximal, characteristic, complemented

C₇₃×#2249

C₂

yes

maximal, characteristic

C₇₃×#1416

C₂

yes

maximal, complemented

C₇₃×#1415

C₂

yes

maximal

C₄×C₃₂×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

C₂×SD₃₂×C₇₃

—

C₂×D₁₆×C₇₃

C₂²

yes

characteristic

#499×C₇₃

—

#495×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

—

#367×C₇₃

—

#366×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Q₁₆×C₇₃

—

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

—

#125×C₇₃

—

#124×C₇₃

—

Q₃₂×C₇₃

—

SD₃₂×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#106×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

#41

yes

—

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×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

SD₃₂

yes

—

C₈×C₇₃

#124

yes

characteristic

C₈×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#486

yes

characteristic

C₄×C₇₃

C₂×SD₃₂

yes

characteristic

#13721

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2240

yes

characteristic

C₂×C₇₃

#2262

yes

—

#2253

C₂×C₇₃

yes

characteristic, complemented

#2249

C₂×C₇₃

yes

characteristic

#1416

C₂×C₇₃

yes

complemented

#1415

C₂×C₇₃

yes

—

C₄×C₃₂

C₂×C₇₃

yes

characteristic, complemented

C₇₃

#13721

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂²×C₇₃

yes

characteristic

C₂×SD₃₂

—

C₂×D₁₆

C₂²×C₇₃

yes

characteristic

#499

—

#495

—

C₂×C₃₂

C₂²×C₇₃

yes

—

#367

—

#366

C₂²×C₇₃

yes

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

#125

—

#124

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

#107

—

#106

—

C₄×C₈

D₄×C₇₃

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

101

SD₁₆

—

102

D₈

—

103

D₈

—

104

C₂×C₈

D₈×C₇₃

yes

characteristic

105

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

D₈×C₇₃

yes

characteristic

110

C₁₆

SD₁₆×C₇₃

yes

—

111

C₁₆

#41×C₇₃

yes

—

112

C₂³

—

113

Q₈

—

114

Q₈

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₁₆×C₇₃

yes

—

120

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

121

C₈

SD₃₂×C₇₃

yes

—

122

C₈

#124×C₇₃

yes

characteristic

123

C₈

C₂×SD₁₆×C₇₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#486×C₇₃

yes

characteristic

131

C₄

C₂×SD₃₂×C₇₃

yes

characteristic

132

C₂

—

133

C₂

C₇₃×#2240

yes

characteristic

134

C₂

C₇₃×#2262

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₂×D₁₆×C₇₃	C₂²
LCS	G ▷ C₂×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, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

prime radical

C₂×D₁₆×C₇₃

C₂²

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. C₇₃×#2253	C₂	semidirect
3. C₇₃×#1416	C₂	semidirect
5. C₄×C₃₂×C₇₃	C₂	semidirect
64. #13721	C₇₃	direct (N × H)
68. #2253	C₂×C₇₃	semidirect
70. #1416	C₂×C₇₃	semidirect
72. C₄×C₃₂	C₂×C₇₃	semidirect
73. C₇₃	#13721	direct (N × H)

H ≅ G/N

type

1. C₇₃×#2253

C₂

semidirect

3. C₇₃×#1416

C₂

semidirect

5. C₄×C₃₂×C₇₃

C₂

semidirect

64. #13721

C₇₃

direct (N × H)

68. #2253

C₂×C₇₃

semidirect

70. #1416

C₂×C₇₃

semidirect

72. C₄×C₃₂

C₂×C₇₃

semidirect

73. C₇₃

#13721

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×#2249	2
C₇₃×#1416	2
C₇₃×#1415	2
C₄×C₃₂×C₇₃	2
#13721	73

subgroup

index

C₇₃×#2253

C₇₃×#2249

C₇₃×#1416

C₇₃×#1415

C₄×C₃₂×C₇₃

#13721

Subgroups — C₇₃×#13721 #32923125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13721 #32923125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13721 #32923125