Group #32973223 — subgroups

Subgroups — #32973223

Order 18688.

180 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22608694	C₂	1	1	yes	maximal, characteristic
2	#22608679	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2163	C₂	1	1	yes	maximal, characteristic
4	#17988644	—	2	1	no	—
5	#17988640	—	2	1	no	—
6	#17988340	—	2	1	no	—
7	#17988340	—	2	1	no	—
8	#17988338	C₄	1	2	yes	—
9	#17988338	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₇₃	—	2	1	no	—
12	C₂²⋊C₁₆×C₇₃	C₄	1	2	yes	—
13	#346×C₇₃	—	2	1	no	—
14	C₄⋊#10596831	—	2	2	no	—
15	C₄⋊#10596831	—	2	2	no	—
16	C₂²⋊#10596831	—	4	1	no	—
17	C₂²⋊#10596831	—	4	1	no	—
18	#14556502	—	4	1	no	—
19	C₄×#10596831	—	4	1	no	—
20	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₃	—	2	2	no	—
24	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂×#10596831	—	4	2	no	—
29	C₂×#10596831	—	4	2	no	—
30	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
42	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
43	C₁₆×C₇₃	—	4	2	no	—
44	C₂³×C₇₃	#107	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	—	2	1	no	—
58	C₈×C₇₃	#108	1	2	yes	—
59	#10596831	—	8	2	no	—
60	#10596831	—	8	2	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	#340	1	1	yes	characteristic
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	#341	1	1	yes	characteristic
71	C₄×C₇₃	#344	1	1	yes	characteristic
72	#7293	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	4	1	no	—
74	C₂×C₇₃	#1301	1	1	yes	characteristic
75	C₂×C₇₃	#1384	1	1	yes	characteristic
76	C₂×C₇₃	#1388	1	1	yes	characteristic
77	#2163	D₇₃	1	1	yes	characteristic
78	#1568	—	73	1	no	—
79	#1564	—	73	1	no	—
80	C₇₃	#7293	1	1	yes	characteristic, complemented, Sylow
81	#432	—	146	1	no	—
82	#431	—	146	1	no	—
83	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
84	C₄⋊C₁₆	—	2	1	no	—
85	C₂²⋊C₁₆	Dic₇₃	1	2	yes	—
86	#346	—	2	1	no	—
87	#335	—	146	1	no	—
88	#335	—	146	1	no	—
89	#334	—	73	2	no	—
90	#334	—	73	2	no	—
91	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
92	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
93	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
94	C₂×C₁₆	—	2	2	no	—
95	C₄⋊C₈	—	146	2	no	—
96	C₄⋊C₈	—	146	2	no	—
97	C₂²⋊C₈	—	292	1	no	—
98	C₂²⋊C₈	—	292	1	no	—
99	#97	—	292	1	no	—
100	C₄×C₈	—	292	1	no	—
101	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
102	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
103	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
104	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
106	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
107	C₂²×C₄	Dic₂₉₂	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	292	2	no	—
111	C₂×C₈	—	292	2	no	—
112	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
114	C₂×C₈	#11924007	1	1	yes	characteristic
115	C₂×C₈	#11924007	1	1	yes	characteristic
116	C₄²	#11924005	1	1	yes	characteristic
117	C₄²	#11924005	1	1	yes	characteristic
118	C₄²	#11924017	1	1	yes	characteristic
119	C₄²	#11924017	1	1	yes	characteristic
120	C₁₆	—	4	2	no	—
121	C₂³	#14556525	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	#14556498	1	1	yes	characteristic
127	C₂×C₄	#14556513	1	1	yes	characteristic
128	C₂×C₄	#14556513	1	1	yes	characteristic
129	C₂×C₄	#14556516	1	1	yes	characteristic
130	C₂×C₄	#14556516	1	1	yes	characteristic
131	C₂×C₄	#14556523	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	584	2	no	—
136	C₈	—	584	2	no	—
137	C₈	#14556526	1	2	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#17988350	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	#17988351	1	1	yes	characteristic
148	C₄	#17988357	1	1	yes	characteristic
149	C₂	—	4	1	no	—
150	C₂	#22607827	1	1	yes	characteristic
151	C₂	#22608033	1	1	yes	characteristic
152	C₂	#22608043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22608694

C₂

yes

maximal, characteristic

#22608679

C₂

yes

maximal, characteristic

C₇₃×#2163

C₂

yes

maximal, characteristic

—

—

—

—

C₄

yes

—

#17988338

C₄

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₁₆×C₇₃

—

C₂²⋊C₁₆×C₇₃

C₄

yes

—

#346×C₇₃

—

—

—

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

Q₈

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂×#10596831

—

C₂×#10596831

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

Q₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#107

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#108

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

#1568

—

#1564

—

C₇₃

#7293

yes

characteristic, complemented, Sylow

#432

—

146

—

#431

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

Dic₇₃

yes

—

#346

—

#335

—

146

—

#335

—

146

—

#334

—

#334

—

C₂²×C₈

Dic₁₄₆

yes

characteristic

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂×C₄²

D₂₉₂

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

C₂²⋊C₈

—

292

—

#97

—

292

—

100

C₄×C₈

—

292

—

101

C₄×C₈

C₄×D₇₃

yes

characteristic

102

C₄×C₈

C₄×D₇₃

yes

characteristic

103

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

104

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

105

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

106

C₂²×C₄

D₅₈₄

yes

characteristic

107

C₂²×C₄

Dic₂₉₂

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

292

—

111

C₂×C₈

—

292

—

112

C₂×C₈

C₄×Dic₇₃

yes

characteristic

113

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

114

C₂×C₈

#11924007

yes

characteristic

115

C₂×C₈

#11924007

yes

characteristic

116

C₄²

#11924005

yes

characteristic

117

C₄²

#11924005

yes

characteristic

118

C₄²

#11924017

yes

characteristic

119

C₄²

#11924017

yes

characteristic

120

C₁₆

—

121

C₂³

#14556525

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#14556498

yes

characteristic

127

C₂×C₄

#14556513

yes

characteristic

128

C₂×C₄

#14556513

yes

characteristic

129

C₂×C₄

#14556516

yes

characteristic

130

C₂×C₄

#14556516

yes

characteristic

131

C₂×C₄

#14556523

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

584

—

136

C₈

—

584

—

137

C₈

#14556526

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#17988350

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#17988351

yes

characteristic

148

C₄

#17988357

yes

characteristic

149

C₂

—

150

C₂

#22607827

yes

characteristic

151

C₂

#22608033

yes

characteristic

152

C₂

#22608043

yes

characteristic

	subgroup	G/H
center	C₂²	#17988350
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₇₃×#2163	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#107
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	#2163	D₇₃
hypocenter	C₇₃	#7293
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

#17988350

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₇₃×#2163

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
80. C₇₃	#7293	semidirect

H ≅ G/N

type

80. C₇₃

#7293

semidirect

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

subgroup	index
#22608694	2
#22608679	2
C₇₃×#2163	2
#7293	73

subgroup

index

#22608694

#22608679

C₇₃×#2163

#7293

Subgroups — #32973223

Conjugacy classes of subgroups

Complements

Maximal subgroups