Finite Group Databasebeta

← C73×#10213 #32919617

Subgroups — C73×#10213 #32919617

Order 18688.

498 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3253C211yesmaximal, characteristic
2C73×#3216C212yesmaximal, complemented
3C73×#1995C211yesmaximal, characteristic, complemented
4C73×#1505C212yesmaximal, complemented
5C73×#1307C211yesmaximal, characteristic, complemented
6#533×C73C2211yescharacteristic
7#532×C73C2212yes—
8#497×C73—22no—
9#489×C73—22no—
10#476×C73—22no—
11#469×C73—22no—
12#462×C73—22no—
13#460×C73—22no—
14#459×C73—22no—
15#452×C73—24no—
16#432×C73C2212yes—
17#423×C73C2211yescharacteristic
18#376×C73C2211yescharacteristic
19#339×C73—22no—
20#327×C73—24no—
21C2×#41×C73—22no—
22C2×Q16×C73—24no—
23C2×SD16×C73—24no—
24C2×D8×C73—22no—
25C2×M16×C73D412yes—
26C4⋊Q8×C73D412yes—
27C4⋊Q8×C73D412yes—
28#127×C73D412yes—
29#125×C73—21no—
30#124×C73—22no—
31C22⋊Q8×C73—22no—
32#121×C73—22no—
33#121×C73—22no—
34C4×Q8×C73—22no—
35C4×D4×C73—22no—
36C4×D4×C73—22no—
37#117×C73—21no—
38C2×C42×C73—21no—
39C2×C42×C73C2311yescharacteristic
40#106×C73—22no—
41C4⋊C8×C73—22no—
42C4⋊C8×C73—22no—
43#103×C73—24no—
44#102×C73—24no—
45#102×C73—24no—
46C22⋊C8×C73D414yes—
47#97×C73—22no—
48#95×C73—22no—
49#41×C73—84no—
50C2×Q8×C73—22no—
51C2×Q8×C73—22no—
52C2×D4×C73—22no—
53C2×D4×C73—22no—
54C2×D4×C73—22no—
55C2×D4×C73—42no—
56C22×C4×C73—41no—
57C22×C4×C73—42no—
58C22×C4×C73C2×D411yescharacteristic
59C22×C4×C73C2×D412yes—
60Q16×C73—84no—
61SD16×C73—84no—
62D8×C73—82no—
63M16×C73—42no—
64C2×C8×C73—24no—
65C2×C8×C73—22no—
66C4⋊C4×C73—22no—
67C4⋊C4×C73—22no—
68C4⋊C4×C73—22no—
69C4⋊C4×C73—22no—
70C4⋊C4×C73—42no—
71C22⋊C4×C73—41no—
72C22⋊C4×C73—42no—
73C22⋊C4×C73—42no—
74C42×C73—22no—
75C42×C73—41no—
76C42×C73—41no—
77C42×C73C2×D411yescharacteristic
78C42×C73C2×D412yes—
79C42×C73#4111yescharacteristic
80C23×C73—42no—
81C23×C73#12711yescharacteristic
82Q8×C73—42no—
83Q8×C73—42no—
84Q8×C73—44no—
85D4×C73—44no—
86D4×C73—42no—
87D4×C73—42no—
88D4×C73—82no—
89D4×C73—84no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—42no—
93C2×C4×C73—42no—
94C2×C4×C73—42no—
95C2×C4×C73—42no—
96C2×C4×C73—84no—
97C2×C4×C73—82no—
98C2×C4×C73#12012yes—
99C2×C4×C73#12011yescharacteristic
100C2×C4×C73#12111yescharacteristic
101C2×C4×C73#12112yes—
102C8×C73—44no—
103C8×C73—42no—
104C22×C73—21no—
105C22×C73—22no—
106C22×C73—82no—
107C22×C73—82no—
108C22×C73—82no—
109C22×C73#39211yescharacteristic
110C4×C73—21no—
111C4×C73—22no—
112C4×C73—22no—
113C4×C73—82no—
114C4×C73—82no—
115C4×C73—82no—
116C4×C73#45511yescharacteristic
117C4×C73#45611yescharacteristic
118#10213C7311yesmaximal, characteristic, direct product factor, Sylow
119C2×C73—41no—
120C2×C73—162no—
121C2×C73#200711yescharacteristic
122C2×C73#200812yes—
123#3253C2×C7311yescharacteristic
124#3216C2×C7312yescomplemented
125#1995C2×C7311yescharacteristic, complemented
126#1505C2×C7312yescomplemented
127#1307C2×C7311yescharacteristic, complemented
128C73#1021311yescharacteristic, direct product factor, Sylow
129#533C22×C7311yescharacteristic
130#532C22×C7312yes—
131#497—22no—
132#489—22no—
133#476—22no—
134#469—22no—
135#462—22no—
136#460—22no—
137#459—22no—
138#452—24no—
139#432C22×C7312yes—
140#423C22×C7311yescharacteristic
141#376C22×C7311yescharacteristic
142#339—22no—
143#327—24no—
144C2×#41—22no—
145C2×Q16—24no—
146C2×SD16—24no—
147C2×D8—22no—
148C2×M16D4×C7312yes—
149C4⋊Q8D4×C7312yes—
150C4⋊Q8D4×C7312yes—
151#127D4×C7312yes—
152#125—21no—
153#124—22no—
154C22⋊Q8—22no—
155#121—22no—
156#121—22no—
157C4×Q8—22no—
158C4×D4—22no—
159C4×D4—22no—
160#117—21no—
161C2×C42—21no—
162C2×C42C23×C7311yescharacteristic
163#106—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166#103—24no—
167#102—24no—
168#102—24no—
169C22⋊C8D4×C7314yes—
170#97—22no—
171#95—22no—
172#41—84no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4C2×D4×C7311yescharacteristic
182C22×C4C2×D4×C7312yes—
183Q16—84no—
184SD16—84no—
185D8—82no—
186M16—42no—
187C2×C8—24no—
188C2×C8—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—41no—
199C42—41no—
200C42C2×D4×C7311yescharacteristic
201C42C2×D4×C7312yes—
202C42#41×C7311yescharacteristic
203C23—42no—
204C23#127×C7311yescharacteristic
205Q8—42no—
206Q8—42no—
207Q8—44no—
208D4—44no—
209D4—42no—
210D4—42no—
211D4—82no—
212D4—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4#120×C7312yes—
222C2×C4#120×C7311yescharacteristic
223C2×C4#121×C7311yescharacteristic
224C2×C4#121×C7312yes—
225C8—44no—
226C8—42no—
227C22—21no—
228C22—22no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22#392×C7311yescharacteristic
233C4—21no—
234C4—22no—
235C4—22no—
236C4—82no—
237C4—82no—
238C4—82no—
239C4#455×C7311yescharacteristic
240C4#456×C7311yescharacteristic
241C2—41no—
242C2—162no—
243C2C73×#200711yescharacteristic
244C2C73×#200812yes—
subgroupG/H
center, socleC22×C73#392
commutator, Frattini, cosocleC2×C42C23×C73
prime radicalC73×#3253C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C73×#3216C2semidirect
3. C73×#1995C2semidirect
4. C73×#1505C2semidirect
5. C73×#1307C2semidirect
118. #10213C73direct (N × H)
124. #3216C2×C73semidirect
125. #1995C2×C73semidirect
126. #1505C2×C73semidirect
127. #1307C2×C73semidirect
128. C73#10213direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#32532
C73×#32162
C73×#19952
C73×#15052
C73×#13072
#1021373