Group #32916521 — subgroups

Subgroups — C₇₃×#7117 #32916521

Order 18688.

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

normal subgroups only

This group has 230 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×#1960	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1375	C₂	1	2	yes	maximal
3	C₂×C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
4	C₂×M₃₂×C₇₃	—	2	2	no	—
5	C₂×#103×C₇₃	—	2	2	no	—
6	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
7	#382×C₇₃	C₄	1	1	yes	characteristic
8	C₂²×Q₈×C₇₃	—	2	1	no	—
9	C₂²×C₈×C₇₃	D₄	1	2	yes	—
10	C₄⋊Q₈×C₇₃	—	2	1	no	—
11	C₄⋊Q₈×C₇₃	—	4	1	no	—
12	C₄⋊Q₈×C₇₃	C₈	1	2	yes	—
13	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
14	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
15	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
16	M₃₂×C₇₃	—	4	2	no	—
17	M₃₂×C₇₃	—	4	2	no	—
18	C₂×C₁₆×C₇₃	—	2	4	no	—
19	#103×C₇₃	—	4	2	no	—
20	#103×C₇₃	—	4	2	no	—
21	#97×C₇₃	—	4	1	no	—
22	#95×C₇₃	—	4	1	no	—
23	C₂×Q₈×C₇₃	—	2	2	no	—
24	C₂×Q₈×C₇₃	—	4	1	no	—
25	C₂×Q₈×C₇₃	—	4	2	no	—
26	C₂×Q₈×C₇₃	—	4	2	no	—
27	C₂²×C₄×C₇₃	—	2	1	no	—
28	C₂²×C₄×C₇₃	—	4	1	no	—
29	C₂²×C₄×C₇₃	—	4	1	no	—
30	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₃	—	2	2	no	—
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₂×C₈×C₇₃	D₈	1	2	yes	—
34	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
35	C₄⋊C₄×C₇₃	—	4	2	no	—
36	C₄⋊C₄×C₇₃	—	4	2	no	—
37	C₄⋊C₄×C₇₃	—	4	2	no	—
38	C₄²×C₇₃	—	2	1	no	—
39	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
41	C₁₆×C₇₃	—	4	4	no	—
42	C₂³×C₇₃	#100	1	1	yes	characteristic
43	Q₈×C₇₃	—	4	4	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	—	4	2	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	8	1	no	—
50	C₂×C₄×C₇₃	—	8	1	no	—
51	C₂×C₄×C₇₃	—	8	2	no	—
52	C₂×C₄×C₇₃	—	8	1	no	—
53	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#102	1	2	yes	—
56	C₂×C₄×C₇₃	#104	1	2	yes	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	—	2	2	no	—
59	C₂²×C₇₃	—	2	2	no	—
60	C₂²×C₇₃	#323	1	1	yes	characteristic
61	C₂²×C₇₃	#329	1	1	yes	characteristic
62	C₂²×C₇₃	#331	1	1	yes	characteristic
63	C₄×C₇₃	—	4	1	no	—
64	C₄×C₇₃	—	4	1	no	—
65	C₄×C₇₃	—	8	2	no	—
66	C₄×C₇₃	—	16	1	no	—
67	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
68	C₄×C₇₃	#327	1	2	yes	—
69	#7117	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×C₇₃	—	2	1	no	—
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	#1270	1	1	yes	characteristic
73	C₂×C₇₃	#1356	1	2	yes	—
74	#1960	C₂×C₇₃	1	1	yes	characteristic
75	#1375	C₂×C₇₃	1	2	yes	—
76	C₇₃	#7117	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×C₄⋊Q₈	C₄×C₇₃	1	1	yes	characteristic
78	C₂×M₃₂	—	2	2	no	—
79	C₂×#103	—	2	2	no	—
80	C₂×#97	C₂²×C₇₃	1	1	yes	characteristic
81	#382	C₄×C₇₃	1	1	yes	characteristic
82	C₂²×Q₈	—	2	1	no	—
83	C₂²×C₈	D₄×C₇₃	1	2	yes	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	4	1	no	—
86	C₄⋊Q₈	C₈×C₇₃	1	2	yes	—
87	C₂×C₄⋊C₄	—	2	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
90	M₃₂	—	4	2	no	—
91	M₃₂	—	4	2	no	—
92	C₂×C₁₆	—	2	4	no	—
93	#103	—	4	2	no	—
94	#103	—	4	2	no	—
95	#97	—	4	1	no	—
96	#95	—	4	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	4	1	no	—
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	D₈×C₇₃	1	2	yes	—
108	C₂×C₈	SD₁₆×C₇₃	1	2	yes	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	C₂×C₈×C₇₃	1	1	yes	characteristic
114	C₄²	M₁₆×C₇₃	1	1	yes	characteristic
115	C₁₆	—	4	4	no	—
116	C₂³	#100×C₇₃	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	8	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	C₂²⋊C₈×C₇₃	1	1	yes	characteristic
128	C₂×C₄	#99×C₇₃	1	1	yes	characteristic
129	C₂×C₄	#102×C₇₃	1	2	yes	—
130	C₂×C₄	#104×C₇₃	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	#323×C₇₃	1	1	yes	characteristic
135	C₂²	#329×C₇₃	1	1	yes	characteristic
136	C₂²	#331×C₇₃	1	1	yes	characteristic
137	C₄	—	4	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	8	2	no	—
140	C₄	—	16	1	no	—
141	C₄	D₄⋊C₈×C₇₃	1	2	yes	—
142	C₄	#327×C₇₃	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	C₇₃×#1270	1	1	yes	characteristic
146	C₂	C₇₃×#1356	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#1960

C₂

yes

maximal, characteristic

C₇₃×#1375

C₂

yes

maximal

C₂×C₄⋊Q₈×C₇₃

C₄

yes

characteristic

C₂×M₃₂×C₇₃

—

C₂×#103×C₇₃

—

C₂×#97×C₇₃

C₂²

yes

characteristic

#382×C₇₃

C₄

yes

characteristic

C₂²×Q₈×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

C₈

yes

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

#103×C₇₃

—

#103×C₇₃

—

#97×C₇₃

—

#95×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#100

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#104

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

D₄⋊C₈

yes

—

C₄×C₇₃

#327

yes

—

#7117

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1270

yes

characteristic

C₂×C₇₃

#1356

yes

—

#1960

C₂×C₇₃

yes

characteristic

#1375

C₂×C₇₃

yes

—

C₇₃

#7117

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₄×C₇₃

yes

characteristic

C₂×M₃₂

—

C₂×#103

—

C₂×#97

C₂²×C₇₃

yes

characteristic

#382

C₄×C₇₃

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

D₄×C₇₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₈×C₇₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₇₃

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊C₄×C₇₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

D₈×C₇₃

yes

—

108

C₂×C₈

SD₁₆×C₇₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂×C₈×C₇₃

yes

characteristic

114

C₄²

M₁₆×C₇₃

yes

characteristic

115

C₁₆

—

116

C₂³

#100×C₇₃

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂²⋊C₈×C₇₃

yes

characteristic

128

C₂×C₄

#99×C₇₃

yes

characteristic

129

C₂×C₄

#102×C₇₃

yes

—

130

C₂×C₄

#104×C₇₃

yes

—

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

#323×C₇₃

yes

characteristic

135

C₂²

#329×C₇₃

yes

characteristic

136

C₂²

#331×C₇₃

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

D₄⋊C₈×C₇₃

yes

—

142

C₄

#327×C₇₃

yes

—

143

C₂

—

144

C₂

—

145

C₂

C₇₃×#1270

yes

characteristic

146

C₂

C₇₃×#1356

yes

—

	subgroup	G/H
center, socle	C₂²×C₇₃	#329
commutator	C₄²	C₂×C₈×C₇₃
Frattini, cosocle	C₂×#97	C₂²×C₇₃
prime radical	C₂³×C₇₃	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#97×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

#329

commutator

C₄²

C₂×C₈×C₇₃

Frattini, cosocle

C₂×#97

C₂²×C₇₃

prime radical

C₂³×C₇₃

#100

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂²×C₄×C₇₃ ◁ C₂×#97×C₇₃ ◁ G

N	H ≅ G/N	type
69. #7117	C₇₃	direct (N × H)
76. C₇₃	#7117	direct (N × H)

H ≅ G/N

type

69. #7117

C₇₃

direct (N × H)

76. C₇₃

#7117

direct (N × H)

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

subgroup	index
C₇₃×#1960	2
C₇₃×#1375	2
#7117	73

subgroup

index

C₇₃×#1960

C₇₃×#1375

#7117

Subgroups — C₇₃×#7117 #32916521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#7117 #32916521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7117 #32916521