Group #32923102 — subgroups

Subgroups — C₇₃×#13698 #32923102

Order 18688.

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

normal subgroups only

This group has 190 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	C₂×Q₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2216	C₂	1	2	yes	maximal
4	C₇₃×#1424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1416	C₂	1	2	yes	maximal, complemented
6	C₂×Q₃₂×C₇₃	C₂²	1	2	yes	—
7	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
8	#467×C₇₃	—	2	2	no	—
9	Q₆₄×C₇₃	—	2	2	no	—
10	M₆₄×C₇₃	—	2	2	no	—
11	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
13	#366×C₇₃	C₂²	1	2	yes	—
14	#358×C₇₃	—	2	2	no	—
15	C₂×Q₁₆×C₇₃	—	2	2	no	—
16	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
17	C₂²⋊Q₈×C₇₃	—	4	2	no	—
18	Q₃₂×C₇₃	—	2	2	no	—
19	Q₃₂×C₇₃	—	2	2	no	—
20	C₂×C₁₆×C₇₃	—	2	1	no	—
21	C₂×C₁₆×C₇₃	—	2	1	no	—
22	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
24	#107×C₇₃	—	2	2	no	—
25	#103×C₇₃	—	4	2	no	—
26	C₃₂×C₇₃	—	2	1	no	—
27	C₃₂×C₇₃	D₄	1	2	yes	—
28	C₂×Q₈×C₇₃	—	4	2	no	—
29	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
30	Q₁₆×C₇₃	—	4	2	no	—
31	Q₁₆×C₇₃	—	4	2	no	—
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	4	2	no	—
37	C₄⋊C₄×C₇₃	—	8	2	no	—
38	C₂²⋊C₄×C₇₃	—	8	2	no	—
39	C₁₆×C₇₃	—	2	1	no	—
40	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	#41	1	1	yes	characteristic
42	C₂³×C₇₃	D₁₆	1	1	yes	characteristic
43	Q₈×C₇₃	—	8	2	no	—
44	Q₈×C₇₃	—	8	2	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	8	2	no	—
48	C₂×C₄×C₇₃	—	8	2	no	—
49	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
51	C₈×C₇₃	—	2	1	no	—
52	C₈×C₇₃	#121	1	1	yes	characteristic
53	C₈×C₇₃	#135	1	1	yes	characteristic
54	C₂²×C₇₃	—	2	1	no	—
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	16	2	no	—
59	C₄×C₇₃	—	16	2	no	—
60	C₄×C₇₃	#466	1	1	yes	characteristic
61	C₄×C₇₃	#508	1	1	yes	characteristic
62	#13698	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₇₃	—	4	1	no	—
64	C₂×C₇₃	#2215	1	1	yes	characteristic
65	C₂×C₇₃	#2264	1	2	yes	—
66	C₂×Q₆₄	C₂×C₇₃	1	1	yes	characteristic, complemented
67	C₂×M₆₄	C₂×C₇₃	1	1	yes	characteristic
68	#2216	C₂×C₇₃	1	2	yes	—
69	#1424	C₂×C₇₃	1	1	yes	characteristic, complemented
70	#1416	C₂×C₇₃	1	2	yes	complemented
71	C₇₃	#13698	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×Q₃₂	C₂²×C₇₃	1	2	yes	—
73	C₂²×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
74	#467	—	2	2	no	—
75	Q₆₄	—	2	2	no	—
76	M₆₄	—	2	2	no	—
77	C₂×C₃₂	C₂²×C₇₃	1	1	yes	characteristic
78	C₂×C₃₂	C₂²×C₇₃	1	1	yes	characteristic
79	#366	C₂²×C₇₃	1	2	yes	—
80	#358	—	2	2	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
83	C₂²⋊Q₈	—	4	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	C₂×C₁₆	—	2	1	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
90	#107	—	2	2	no	—
91	#103	—	4	2	no	—
92	C₃₂	—	2	1	no	—
93	C₃₂	D₄×C₇₃	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
101	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₂²⋊C₄	—	8	2	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	C₂×D₄×C₇₃	1	1	yes	characteristic
107	C₁₆	#41×C₇₃	1	1	yes	characteristic
108	C₂³	D₁₆×C₇₃	1	1	yes	characteristic
109	Q₈	—	8	2	no	—
110	Q₈	—	8	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	8	2	no	—
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	D₁₆×C₇₃	1	1	yes	characteristic
116	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	#121×C₇₃	1	1	yes	characteristic
119	C₈	#135×C₇₃	1	1	yes	characteristic
120	C₂²	—	2	1	no	—
121	C₂²	—	2	2	no	—
122	C₂²	C₂×D₁₆×C₇₃	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	16	2	no	—
125	C₄	—	16	2	no	—
126	C₄	#466×C₇₃	1	1	yes	characteristic
127	C₄	#508×C₇₃	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	C₇₃×#2215	1	1	yes	characteristic
130	C₂	C₇₃×#2264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₆₄×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×M₆₄×C₇₃

C₂

yes

maximal, characteristic

C₇₃×#2216

C₂

yes

maximal

C₇₃×#1424

C₂

yes

maximal, characteristic, complemented

C₇₃×#1416

C₂

yes

maximal, complemented

C₂×Q₃₂×C₇₃

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

#467×C₇₃

—

Q₆₄×C₇₃

—

M₆₄×C₇₃

—

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

C₂²

yes

characteristic

#366×C₇₃

C₂²

yes

—

#358×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²⋊Q₈×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#103×C₇₃

—

C₃₂×C₇₃

—

C₃₂×C₇₃

D₄

yes

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₈

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂³×C₇₃

D₁₆

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₁₆

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

#2215

yes

characteristic

C₂×C₇₃

#2264

yes

—

C₂×Q₆₄

C₂×C₇₃

yes

characteristic, complemented

C₂×M₆₄

C₂×C₇₃

yes

characteristic

#2216

C₂×C₇₃

yes

—

#1424

C₂×C₇₃

yes

characteristic, complemented

#1416

C₂×C₇₃

yes

complemented

C₇₃

#13698

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂²×C₇₃

yes

—

C₂²×C₁₆

C₂²×C₇₃

yes

characteristic

#467

—

Q₆₄

—

M₆₄

—

C₂×C₃₂

C₂²×C₇₃

yes

characteristic

C₂×C₃₂

C₂²×C₇₃

yes

characteristic

#366

C₂²×C₇₃

yes

—

#358

—

C₂×Q₁₆

—

C₂²×C₈

D₄×C₇₃

yes

characteristic

C₂²⋊Q₈

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

characteristic

#107

—

#103

—

C₃₂

—

C₃₂

D₄×C₇₃

yes

—

C₂×Q₈

—

C₂²×C₄

D₈×C₇₃

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

D₈×C₇₃

yes

characteristic

101

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₁₆

—

106

C₁₆

C₂×D₄×C₇₃

yes

characteristic

107

C₁₆

#41×C₇₃

yes

characteristic

108

C₂³

D₁₆×C₇₃

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

D₁₆×C₇₃

yes

characteristic

116

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

117

C₈

—

118

C₈

#121×C₇₃

yes

characteristic

119

C₈

#135×C₇₃

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×D₁₆×C₇₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#466×C₇₃

yes

characteristic

127

C₄

#508×C₇₃

yes

characteristic

128

C₂

—

129

C₂

C₇₃×#2215

yes

characteristic

130

C₂

C₇₃×#2264

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	C₂³×C₇₃	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂²×C₈×C₇₃ ◁ C₂²×C₁₆×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

prime radical

C₂³×C₇₃

D₁₆

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. C₂×Q₆₄×C₇₃	C₂	semidirect
4. C₇₃×#1424	C₂	semidirect
5. C₇₃×#1416	C₂	semidirect
62. #13698	C₇₃	direct (N × H)
66. C₂×Q₆₄	C₂×C₇₃	semidirect
69. #1424	C₂×C₇₃	semidirect
70. #1416	C₂×C₇₃	semidirect
71. C₇₃	#13698	direct (N × H)

H ≅ G/N

type

1. C₂×Q₆₄×C₇₃

C₂

semidirect

4. C₇₃×#1424

C₂

semidirect

5. C₇₃×#1416

C₂

semidirect

62. #13698

C₇₃

direct (N × H)

66. C₂×Q₆₄

C₂×C₇₃

semidirect

69. #1424

C₂×C₇₃

semidirect

70. #1416

C₂×C₇₃

semidirect

71. C₇₃

#13698

direct (N × H)

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

subgroup	index
C₂×Q₆₄×C₇₃	2
C₂×M₆₄×C₇₃	2
C₇₃×#2216	2
C₇₃×#1424	2
C₇₃×#1416	2
#13698	73

subgroup

index

C₂×Q₆₄×C₇₃

C₂×M₆₄×C₇₃

C₇₃×#2216

C₇₃×#1424

C₇₃×#1416

#13698

Subgroups — C₇₃×#13698 #32923102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13698 #32923102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13698 #32923102