Group #32973100 — subgroups

Subgroups — #32973100

Order 18688.

244 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 71 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	#22608275	C₂	1	2	yes	maximal
2	C₇₃×#1984	C₂	1	1	yes	maximal, characteristic
3	#17988644	—	2	2	no	—
4	#17988313	—	2	2	no	—
5	#17988313	C₄	1	4	yes	—
6	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
9	#336×C₇₃	—	2	2	no	—
10	C₄⋊#10596831	—	4	2	no	—
11	C₂²⋊#10596831	—	2	4	no	—
12	#14556502	—	4	2	no	—
13	C₂²×Q₈×C₇₃	Q₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	—	2	1	no	—
15	C₂²×C₈×C₇₃	—	2	1	no	—
16	C₄×Q₈×C₇₃	—	4	1	no	—
17	C₄×Q₈×C₇₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₃	D₄	1	2	yes	—
22	C₄⋊C₈×C₇₃	—	4	1	no	—
23	C₄×C₈×C₇₃	—	4	1	no	—
24	C₂×#10596831	—	4	4	no	—
25	C₂×Q₈×C₇₃	—	2	2	no	—
26	C₂×Q₈×C₇₃	—	4	1	no	—
27	C₂×Q₈×C₇₃	—	4	1	no	—
28	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	4	1	no	—
36	C₂×C₈×C₇₃	—	4	2	no	—
37	C₂×C₈×C₇₃	—	4	1	no	—
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄⋊C₄×C₇₃	—	4	1	no	—
40	C₄²×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	—	2	1	no	—
42	C₄²×C₇₃	—	2	2	no	—
43	C₄²×C₇₃	—	2	2	no	—
44	C₂³×C₇₃	#95	1	1	yes	characteristic
45	Q₈×C₇₃	—	4	4	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	—	4	2	no	—
57	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#108	1	2	yes	—
60	C₈×C₇₃	—	4	2	no	—
61	C₈×C₇₃	—	8	1	no	—
62	#10596831	—	8	4	no	—
63	C₂²×C₇₃	—	2	1	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	#338	1	1	yes	characteristic
66	C₂²×C₇₃	#341	1	1	yes	characteristic
67	C₂²×C₇₃	#342	1	1	yes	characteristic
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	4	2	no	—
71	C₄×C₇₃	—	4	2	no	—
72	C₄×C₇₃	—	4	1	no	—
73	C₄×C₇₃	#355	1	1	yes	characteristic
74	C₄×C₇₃	#356	1	1	yes	characteristic
75	#7248	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	2	1	no	—
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	#1293	1	1	yes	characteristic
79	C₂×C₇₃	#1391	1	1	yes	characteristic
80	C₂×C₇₃	#1394	1	1	yes	characteristic
81	#1984	D₇₃	1	1	yes	characteristic
82	#1467	—	73	2	no	—
83	C₇₃	#7248	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
85	#432	—	146	2	no	—
86	C₂×C₄⋊C₈	Dic₇₃	1	1	yes	characteristic
87	C₂×C₄×C₈	Dic₇₃	1	1	yes	characteristic
88	#336	—	2	2	no	—
89	#324	—	73	4	no	—
90	#324	—	146	2	no	—
91	C₂²×Q₈	Dic₁₄₆	1	1	yes	characteristic
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₄×Q₈	—	4	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	C₂×C₄⋊C₄	C₄×D₇₃	1	2	yes	—
97	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
98	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
100	C₄⋊C₈	—	4	1	no	—
101	C₄⋊C₈	—	292	2	no	—
102	C₂²⋊C₈	—	146	4	no	—
103	#97	—	292	2	no	—
104	C₄×C₈	—	4	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
109	C₂²×C₄	#11924005	1	2	yes	—
110	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
111	C₂²×C₄	#11924007	1	2	yes	—
112	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	292	4	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	2	no	—
125	C₂³	#14556498	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#14556506	1	1	yes	characteristic
139	C₂×C₄	#14556506	1	1	yes	characteristic
140	C₂×C₄	#14556527	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₈	—	8	1	no	—
143	C₈	—	584	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#17988345	1	1	yes	characteristic
147	C₂²	#17988352	1	1	yes	characteristic
148	C₂²	#17988354	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	1	no	—
153	C₄	—	4	2	no	—
154	C₄	#17988384	1	1	yes	characteristic
155	C₄	#17988386	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	#22607808	1	1	yes	characteristic
159	C₂	#22608052	1	1	yes	characteristic
160	C₂	#22608058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22608275

C₂

yes

maximal

C₇₃×#1984

C₂

yes

maximal, characteristic

#17988644

—

#17988313

—

#17988313

C₄

yes

—

C₂×C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

#336×C₇₃

—

C₄⋊#10596831

—

C₂²⋊#10596831

—

#14556502

—

C₂²×Q₈×C₇₃

Q₈

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

C₂×#10596831

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#95

yes

characteristic

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

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

#1467

—

C₇₃

#7248

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

C₂×D₇₃

yes

characteristic

#432

—

146

—

C₂×C₄⋊C₈

Dic₇₃

yes

characteristic

C₂×C₄×C₈

Dic₇₃

yes

—

—

—

146

—

C₂²×Q₈

Dic₁₄₆

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×D₇₃

yes

—

C₂×C₄⋊C₄

D₂₉₂

yes

characteristic

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

C₂×C₄²

C₂²⋊D₇₃

yes

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

292

—

102

C₂²⋊C₈

—

146

—

103

#97

—

292

—

104

C₄×C₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

C₄×Dic₇₃

yes

characteristic

109

C₂²×C₄

#11924005

yes

—

110

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

111

C₂²×C₄

#11924007

yes

—

112

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

292

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₂³

#14556498

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14556506

yes

characteristic

139

C₂×C₄

#14556506

yes

characteristic

140

C₂×C₄

#14556527

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

584

—

144

C₂²

—

145

C₂²

—

146

C₂²

#17988345

yes

characteristic

147

C₂²

#17988352

yes

characteristic

148

C₂²

#17988354

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#17988384

yes

characteristic

155

C₄

#17988386

yes

characteristic

156

C₂

—

157

C₂

—

158

C₂

#22607808

yes

characteristic

159

C₂

#22608052

yes

characteristic

160

C₂

#22608058

yes

characteristic

	subgroup	G/H
center	C₂²	#17988354
commutator	C₂²×C₄×C₇₃	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₇₃
Fitting	C₇₃×#1984	C₂
socle	C₂²×C₇₃	#342
prime radical	C₂³×C₇₃	#95
cosocle	C₂×C₄×Q₈×C₇₃	C₂²
hypercenter	#1984	D₇₃
hypocenter	C₇₃	#7248
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1984

subgroup

G/H

center

C₂²

#17988354

commutator

C₂²×C₄×C₇₃

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₇₃

Fitting

C₇₃×#1984

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×Q₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1984

N	H ≅ G/N	type
83. C₇₃	#7248	semidirect

H ≅ G/N

type

83. C₇₃

#7248

semidirect

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

subgroup	index
#22608275	2
C₇₃×#1984	2
#7248	73

subgroup

index

#22608275

C₇₃×#1984

#7248

Subgroups — #32973100

Conjugacy classes of subgroups

Complements

Maximal subgroups