Finite Group Databasebeta

← C73×#32881 #32942285

Subgroups — C73×#32881 #32942285

Order 18688.

5942 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 5942 conjugacy classes of subgroups — too many to draw. Only its 438 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
1C24×D4×C73C211yesmaximal, characteristic
2C2×#450×C73C216yesmaximal, complemented
3C73×#2848C211yesmaximal, characteristic, complemented
4C73×#1786C216yesmaximal, complemented
5C73×#1779C211yesmaximal, characteristic, complemented
6C26×C73—21no—
7C23×D4×C73—224no—
8C23×D4×C73—23no—
9C23×D4×C73C2216yes—
10C24×C4×C73C2211yescharacteristic
11C22×SD16×C73C2214yescomplemented
12C2×C22⋊Q8×C73C2213yescomplemented
13#450×C73—224no—
14C2×#102×C73C22112yescomplemented
15C2×C22⋊C8×C73C2216yescomplemented
16#385×C73—23no—
17#380×C73C2213yescomplemented
18#379×C73—21no—
19C25×C73—224no—
20C25×C73—26no—
21C25×C73—412no—
22C25×C73—44no—
23C25×C73D411yescharacteristic
24C22×Q8×C73C2311yescharacteristic, complemented
25C22×D4×C73—2128no—
26C22×D4×C73—224no—
27C22×D4×C73—248no—
28C22×D4×C73—248no—
29C22×D4×C73—26no—
30C22×D4×C73—212no—
31C22×D4×C73D4124yes—
32C22×D4×C73C2314yes—
33C23×C4×C73—212no—
34C23×C4×C73D413yes—
35C23×C4×C73C2313yes—
36C2×SD16×C73—224no—
37C22×C8×C73C2314yescomplemented
38C22⋊Q8×C73—212no—
39C2×C4⋊C4×C73—23no—
40C2×C4⋊C4×C73C2313yescomplemented
41C2×C22⋊C4×C73—23no—
42C2×C22⋊C4×C73—23no—
43#102×C73—224no—
44C22⋊C8×C73—212no—
45#95×C73—23no—
46C24×C73—26no—
47C24×C73—26no—
48C24×C73—264no—
49C24×C73—248no—
50C24×C73—224no—
51C24×C73—24no—
52C24×C73—496no—
53C24×C73—424no—
54C24×C73—496no—
55C24×C73—412no—
56C24×C73—412no—
57C24×C73SD1614yescomplemented
58C24×C73C2×D413yes—
59C2×Q8×C73—26no—
60C2×D4×C73—2192no—
61C2×D4×C73—248no—
62C2×D4×C73—296no—
63C2×D4×C73—296no—
64C2×D4×C73—224no—
65C2×D4×C73—212no—
66C2×D4×C73C2×D4124yes—
67C22×C4×C73—232no—
68C22×C4×C73—26no—
69C22×C4×C73—212no—
70C22×C4×C73—212no—
71C22×C4×C73—23no—
72C22×C4×C73—21no—
73C22×C4×C73C2×D413yes—
74C22×C4×C73C2×D4112yes—
75C22×C4×C73C2411yescharacteristic
76SD16×C73—416no—
77C2×C8×C73—212no—
78C4⋊C4×C73—26no—
79C4⋊C4×C73—46no—
80C22⋊C4×C73—46no—
81C22⋊C4×C73—46no—
82C23×C73—28no—
83C23×C73—224no—
84C23×C73—26no—
85C23×C73—26no—
86C23×C73—224no—
87C23×C73—296no—
88C23×C73—224no—
89C23×C73—4256no—
90C23×C73—496no—
91C23×C73—448no—
92C23×C73—496no—
93C23×C73—44no—
94C23×C73—412no—
95C23×C73—448no—
96C23×C73#12016yes—
97C23×C73C2×SD1616yes—
98C23×C73AGL(1,ℤ/8)16yes—
99C23×C73C22×D411yescharacteristic
100Q8×C73—44no—
101D4×C73—296no—
102D4×C73—224no—
103D4×C73#120116yes—
104C2×C4×C73—224no—
105C2×C4×C73—26no—
106C2×C4×C73—224no—
107C2×C4×C73—43no—
108C2×C4×C73—43no—
109C2×C4×C73—46no—
110C2×C4×C73#12016yes—
111C2×C4×C73C22×D416yes—
112C8×C73—44no—
113C22×C73—232no—
114C22×C73—26no—
115C22×C73—212no—
116C22×C73—26no—
117C22×C73—212no—
118C22×C73—216no—
119C22×C73—4192no—
120C22×C73—424no—
121C22×C73—424no—
122C22×C73#450112yescomplemented
123C22×C73C2×#12013yes—
124C22×C73C22×SD1611yescharacteristic
125C22×C73C2×AGL(1,ℤ/8)13yes—
126C4×C73—26no—
127C4×C73—81no—
128C4×C73—83no—
129C4×C73C2×#12014yes—
130#32881C7311yesmaximal, characteristic, direct product factor, Sylow
131C2×C73—212no—
132C2×C73—416no—
133C2×C73#284611yescharacteristic
134C2×C73C2×#45013yes—
135C2×C73#300313yes—
136C24×D4C2×C7311yescharacteristic
137C2×#450C2×C7316yescomplemented
138#2848C2×C7311yescharacteristic, complemented
139#1786C2×C7316yescomplemented
140#1779C2×C7311yescharacteristic, complemented
141C73#3288111yescharacteristic, direct product factor, Sylow
142C26—21no—
143C23×D4—224no—
144C23×D4—23no—
145C23×D4C22×C7316yes—
146C24×C4C22×C7311yescharacteristic
147C22×SD16C22×C7314yescomplemented
148C2×C22⋊Q8C22×C7313yescomplemented
149#450—224no—
150C2×#102C22×C73112yescomplemented
151C2×C22⋊C8C22×C7316yescomplemented
152#385—23no—
153#380C22×C7313yescomplemented
154#379—21no—
155C25—224no—
156C25—26no—
157C25—412no—
158C25—44no—
159C25D4×C7311yescharacteristic
160C22×Q8C23×C7311yescharacteristic, complemented
161C22×D4—2128no—
162C22×D4—224no—
163C22×D4—248no—
164C22×D4—248no—
165C22×D4—26no—
166C22×D4—212no—
167C22×D4D4×C73124yes—
168C22×D4C23×C7314yes—
169C23×C4—212no—
170C23×C4D4×C7313yes—
171C23×C4C23×C7313yes—
172C2×SD16—224no—
173C22×C8C23×C7314yescomplemented
174C22⋊Q8—212no—
175C2×C4⋊C4—23no—
176C2×C4⋊C4C23×C7313yescomplemented
177C2×C22⋊C4—23no—
178C2×C22⋊C4—23no—
179#102—224no—
180C22⋊C8—212no—
181#95—23no—
182C24—26no—
183C24—26no—
184C24—264no—
185C24—248no—
186C24—224no—
187C24—24no—
188C24—496no—
189C24—496no—
190C24—424no—
191C24—412no—
192C24—412no—
193C24SD16×C7314yescomplemented
194C24C2×D4×C7313yes—
195C2×Q8—26no—
196C2×D4—2192no—
197C2×D4—248no—
198C2×D4—296no—
199C2×D4—296no—
200C2×D4—224no—
201C2×D4—212no—
202C2×D4C2×D4×C73124yes—
203C22×C4—232no—
204C22×C4—26no—
205C22×C4—212no—
206C22×C4—212no—
207C22×C4—23no—
208C22×C4—21no—
209C22×C4C2×D4×C7313yes—
210C22×C4C2×D4×C73112yes—
211C22×C4C24×C7311yescharacteristic
212SD16—416no—
213C2×C8—212no—
214C4⋊C4—26no—
215C4⋊C4—46no—
216C22⋊C4—46no—
217C22⋊C4—46no—
218C23—28no—
219C23—224no—
220C23—224no—
221C23—26no—
222C23—26no—
223C23—296no—
224C23—224no—
225C23—4256no—
226C23—496no—
227C23—448no—
228C23—496no—
229C23—448no—
230C23—44no—
231C23—412no—
232C23#120×C7316yes—
233C23C2×SD16×C7316yes—
234C23AGL(1,ℤ/8)×C7316yes—
235C23C22×D4×C7311yescharacteristic
236Q8—44no—
237D4—296no—
238D4—224no—
239D4#120×C73116yes—
240C2×C4—224no—
241C2×C4—26no—
242C2×C4—224no—
243C2×C4—43no—
244C2×C4—43no—
245C2×C4—46no—
246C2×C4#120×C7316yes—
247C2×C4C22×D4×C7316yes—
248C8—44no—
249C22—232no—
250C22—212no—
251C22—26no—
252C22—212no—
253C22—26no—
254C22—216no—
255C22—4192no—
256C22—424no—
257C22—424no—
258C22#450×C73112yescomplemented
259C22C2×#120×C7313yes—
260C22C22×SD16×C7311yescharacteristic
261C22C2×AGL(1,ℤ/8)×C7313yes—
262C4—26no—
263C4—83no—
264C4—81no—
265C4C2×#120×C7314yes—
266C2—212no—
267C2—416no—
268C2C73×#284611yescharacteristic
269C2C2×#450×C7313yes—
270C2C73×#300313yes—
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC24×D4×C73C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C24×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#450×C73C2semidirect
3. C73×#2848C2semidirect
4. C73×#1786C2semidirect
5. C73×#1779C2semidirect
11. C22×SD16×C73C22semidirect
12. C2×C22⋊Q8×C73C22semidirect
14. C2×#102×C73C22semidirect
15. C2×C22⋊C8×C73C22semidirect
17. #380×C73C22semidirect
24. C22×Q8×C73C23semidirect
37. C22×C8×C73C23semidirect
40. C2×C4⋊C4×C73C23semidirect
57. C24×C73SD16semidirect
122. C22×C73#450semidirect
130. #32881C73direct (N × H)
137. C2×#450C2×C73semidirect
138. #2848C2×C73semidirect
139. #1786C2×C73semidirect
140. #1779C2×C73semidirect
141. C73#32881direct (N × H)
147. C22×SD16C22×C73semidirect
148. C2×C22⋊Q8C22×C73semidirect
150. C2×#102C22×C73semidirect
151. C2×C22⋊C8C22×C73semidirect
153. #380C22×C73semidirect
160. C22×Q8C23×C73semidirect
173. C22×C8C23×C73semidirect
176. C2×C4⋊C4C23×C73semidirect
193. C24SD16×C73semidirect
258. C22#450×C73semidirect

Maximal subgroups

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

subgroupindex
C24×D4×C732
C2×#450×C732
C73×#28482
C73×#17862
C73×#17792
#3288173