Group #32923123 — subgroups

Subgroups — C₇₃×#13719 #32923123

Order 18688.

202 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#2246	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1415	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₇₃	—	2	1	no	—
6	C₂×D₁₆×C₇₃	C₂²	1	2	yes	—
7	#500×C₇₃	—	2	1	no	—
8	#493×C₇₃	—	2	1	no	—
9	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
10	#366×C₇₃	—	2	1	no	—
11	#366×C₇₃	C₂²	1	2	yes	—
12	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₇₃	—	2	2	no	—
14	C₂×D₈×C₇₃	—	4	1	no	—
15	C₄⋊Q₈×C₇₃	—	4	1	no	—
16	#127×C₇₃	—	4	1	no	—
17	D₁₆×C₇₃	—	2	4	no	—
18	D₁₆×C₇₃	—	4	2	no	—
19	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
21	#107×C₇₃	—	2	2	no	—
22	#107×C₇₃	—	4	1	no	—
23	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₃₂×C₇₃	—	2	2	no	—
25	C₂×Q₈×C₇₃	—	8	1	no	—
26	C₂×D₄×C₇₃	—	4	2	no	—
27	C₂×D₄×C₇₃	—	8	1	no	—
28	C₂×D₄×C₇₃	—	8	1	no	—
29	D₈×C₇₃	—	4	2	no	—
30	D₈×C₇₃	—	4	2	no	—
31	D₈×C₇₃	—	8	2	no	—
32	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₇₃	—	4	2	no	—
35	C₄⋊C₄×C₇₃	—	8	1	no	—
36	C₄²×C₇₃	D₈	1	1	yes	characteristic
37	C₁₆×C₇₃	SD₁₆	1	2	yes	—
38	C₁₆×C₇₃	#41	1	2	yes	—
39	C₂³×C₇₃	—	8	2	no	—
40	Q₈×C₇₃	—	16	2	no	—
41	D₄×C₇₃	—	8	2	no	—
42	D₄×C₇₃	—	8	2	no	—
43	D₄×C₇₃	—	16	2	no	—
44	D₄×C₇₃	—	16	2	no	—
45	C₂×C₄×C₇₃	—	8	2	no	—
46	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
49	C₈×C₇₃	SD₃₂	1	2	yes	—
50	C₈×C₇₃	#124	1	1	yes	characteristic
51	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
52	C₂²×C₇₃	—	16	2	no	—
53	C₂²×C₇₃	—	16	2	no	—
54	C₂²×C₇₃	—	16	2	no	—
55	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₇₃	—	16	2	no	—
57	C₄×C₇₃	D₃₂	1	2	yes	—
58	C₄×C₇₃	SD₆₄	1	2	yes	—
59	C₄×C₇₃	#486	1	1	yes	characteristic
60	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
61	#13719	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₇₃	—	32	2	no	—
63	C₂×C₇₃	#2240	1	1	yes	characteristic
64	C₂×C₇₃	C₂×D₃₂	1	1	yes	characteristic
65	C₂×C₇₃	C₂×SD₆₄	1	1	yes	characteristic
66	#2252	C₂×C₇₃	1	1	yes	characteristic, complemented
67	#2246	C₂×C₇₃	1	1	yes	characteristic
68	#1415	C₂×C₇₃	1	4	yes	complemented
69	C₄×C₃₂	C₂×C₇₃	1	1	yes	characteristic, complemented
70	C₇₃	#13719	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×D₁₆	—	2	1	no	—
72	C₂×D₁₆	C₂²×C₇₃	1	2	yes	—
73	#500	—	2	1	no	—
74	#493	—	2	1	no	—
75	C₂×C₃₂	C₂²×C₇₃	1	2	yes	—
76	#366	—	2	1	no	—
77	#366	C₂²×C₇₃	1	2	yes	—
78	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
79	C₂×D₈	—	2	2	no	—
80	C₂×D₈	—	4	1	no	—
81	C₄⋊Q₈	—	4	1	no	—
82	#127	—	4	1	no	—
83	D₁₆	—	2	4	no	—
84	D₁₆	—	4	2	no	—
85	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
87	#107	—	2	2	no	—
88	#107	—	4	1	no	—
89	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
90	C₃₂	—	2	2	no	—
91	C₂×Q₈	—	8	1	no	—
92	C₂×D₄	—	4	2	no	—
93	C₂×D₄	—	8	1	no	—
94	C₂×D₄	—	8	1	no	—
95	D₈	—	4	2	no	—
96	D₈	—	4	2	no	—
97	D₈	—	8	2	no	—
98	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
99	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₄²	D₈×C₇₃	1	1	yes	characteristic
103	C₁₆	SD₁₆×C₇₃	1	2	yes	—
104	C₁₆	#41×C₇₃	1	2	yes	—
105	C₂³	—	8	2	no	—
106	Q₈	—	16	2	no	—
107	D₄	—	8	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	16	2	no	—
110	D₄	—	16	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
113	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
114	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
115	C₈	SD₃₂×C₇₃	1	2	yes	—
116	C₈	#124×C₇₃	1	1	yes	characteristic
117	C₈	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
118	C₂²	—	16	2	no	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
122	C₄	—	16	2	no	—
123	C₄	D₃₂×C₇₃	1	2	yes	—
124	C₄	SD₆₄×C₇₃	1	2	yes	—
125	C₄	#486×C₇₃	1	1	yes	characteristic
126	C₄	C₂×SD₃₂×C₇₃	1	1	yes	characteristic
127	C₂	—	32	2	no	—
128	C₂	C₇₃×#2240	1	1	yes	characteristic
129	C₂	C₂×D₃₂×C₇₃	1	1	yes	characteristic
130	C₂	C₂×SD₆₄×C₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2252

C₂

yes

maximal, characteristic, complemented

C₇₃×#2246

C₂

yes

maximal, characteristic

C₇₃×#1415

C₂

yes

maximal, complemented

C₄×C₃₂×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₇₃

—

C₂×D₁₆×C₇₃

C₂²

yes

—

#500×C₇₃

—

#493×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

—

#366×C₇₃

—

#366×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#127×C₇₃

—

D₁₆×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

SD₁₆

yes

—

C₁₆×C₇₃

#41

yes

—

C₂³×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

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

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

D₃₂

yes

—

C₄×C₇₃

SD₆₄

yes

—

C₄×C₇₃

#486

yes

characteristic

C₄×C₇₃

C₂×SD₃₂

yes

characteristic

#13719

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2240

yes

characteristic

C₂×C₇₃

C₂×D₃₂

yes

characteristic

C₂×C₇₃

C₂×SD₆₄

yes

characteristic

#2252

C₂×C₇₃

yes

characteristic, complemented

#2246

C₂×C₇₃

yes

characteristic

#1415

C₂×C₇₃

yes

complemented

C₄×C₃₂

C₂×C₇₃

yes

characteristic, complemented

C₇₃

#13719

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

—

C₂×D₁₆

C₂²×C₇₃

yes

—

#500

—

#493

—

C₂×C₃₂

C₂²×C₇₃

yes

—

#366

—

#366

C₂²×C₇₃

yes

—

C₄×C₁₆

C₂²×C₇₃

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

D₁₆

—

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

#107

—

#107

—

C₄×C₈

D₄×C₇₃

yes

characteristic

C₃₂

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

D₈×C₇₃

yes

characteristic

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

D₈×C₇₃

yes

characteristic

103

C₁₆

SD₁₆×C₇₃

yes

—

104

C₁₆

#41×C₇₃

yes

—

105

C₂³

—

106

Q₈

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

D₁₆×C₇₃

yes

characteristic

113

C₂×C₄

D₁₆×C₇₃

yes

characteristic

114

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

115

C₈

SD₃₂×C₇₃

yes

—

116

C₈

#124×C₇₃

yes

characteristic

117

C₈

C₂×SD₁₆×C₇₃

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

122

C₄

—

123

C₄

D₃₂×C₇₃

yes

—

124

C₄

SD₆₄×C₇₃

yes

—

125

C₄

#486×C₇₃

yes

characteristic

126

C₄

C₂×SD₃₂×C₇₃

yes

characteristic

127

C₂

—

128

C₂

C₇₃×#2240

yes

characteristic

129

C₂

C₂×D₃₂×C₇₃

yes

characteristic

130

C₂

C₂×SD₆₄×C₇₃

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₇₃×#2246	C₂
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

C₇₃×#2246

C₂

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×#2252	C₂	semidirect
3. C₇₃×#1415	C₂	semidirect
4. C₄×C₃₂×C₇₃	C₂	semidirect
61. #13719	C₇₃	direct (N × H)
66. #2252	C₂×C₇₃	semidirect
68. #1415	C₂×C₇₃	semidirect
69. C₄×C₃₂	C₂×C₇₃	semidirect
70. C₇₃	#13719	direct (N × H)

H ≅ G/N

type

1. C₇₃×#2252

C₂

semidirect

3. C₇₃×#1415

C₂

semidirect

4. C₄×C₃₂×C₇₃

C₂

semidirect

61. #13719

C₇₃

direct (N × H)

66. #2252

C₂×C₇₃

semidirect

68. #1415

C₂×C₇₃

semidirect

69. C₄×C₃₂

C₂×C₇₃

semidirect

70. C₇₃

#13719

direct (N × H)

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

subgroup	index
C₇₃×#2252	2
C₇₃×#2246	2
C₇₃×#1415	2
C₄×C₃₂×C₇₃	2
#13719	73

subgroup

index

C₇₃×#2252

C₇₃×#2246

C₇₃×#1415

C₄×C₃₂×C₇₃

#13719

Subgroups — C₇₃×#13719 #32923123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13719 #32923123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13719 #32923123