Finite Group Databasebeta

← #44487685

Subgroups — #44487685

Order 30592.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#21439928C211yesmaximal, characteristic
2#29504672C212yesmaximal, complemented
3#29504663C212yesmaximal, complemented
4#29504662C212yesmaximal, complemented
5#29504660C212yesmaximal, complemented
6#29504652C211yesmaximal, characteristic, complemented
7#29504649C212yesmaximal, complemented
8#29504627C212yesmaximal, complemented
9#390×C239C211yesmaximal, characteristic, complemented
10C24×D239C2211yescharacteristic
11C22×C22⋊D239C2212yes—
12C23×Dic239C2211yescharacteristic, complemented
13#21439928C418yescomplemented
14C2×Dic239⋊C4C2212yescomplemented
15C2×C22⋊Dic239C2212yes—
16C2×C22⋊Dic239C2211yescharacteristic
17C2×C22⋊Dic239C2211yescharacteristic, complemented
18C2×(C2×D239)⋊C4C2212yes—
19C2×(C2×D239)⋊C4C2212yes—
20C2×(C2×D239)⋊C4C2212yes—
21C2×(C2×D239)⋊C4C2212yes—
22C2×(C2×D239)⋊C4C2212yes—
23C2×C4×Dic239C2212yescomplemented
24C239⋊#95C2212yescomplemented
25C239⋊#95C2212yescomplemented
26C239⋊#95C2212yescomplemented
27C22×D4×C239C2211yescharacteristic
28C2×C4⋊C4×C239C2211yescharacteristic, complemented
29C2×C22⋊C4×C239C2212yes—
30C2×C22⋊C4×C239C2212yescomplemented
31C2×C42×C239C2211yescharacteristic, complemented
32C23×D239—42no—
33C23×D239—41no—
34C23×D239—41no—
35C23×D239—42no—
36C23×D239C2×C414yescomplemented
37C23×D239C2312yes—
38C2×C22⋊D239—42no—
39C2×C22⋊D239—42no—
40C2×C22⋊D239C2×C418yes—
41C22×Dic239—41no—
42C22×Dic239—42no—
43C22×Dic239C2312yes—
44C22×Dic239C2312yes—
45C22×Dic239C2312yes—
46Dic239⋊C4—42no—
47C22⋊Dic239—42no—
48C22⋊Dic239—41no—
49C22⋊Dic239C2×C414yescomplemented
50(C2×D239)⋊C4—42no—
51(C2×D239)⋊C4—42no—
52(C2×D239)⋊C4—42no—
53(C2×D239)⋊C4—42no—
54(C2×D239)⋊C4C2×C418yescomplemented
55C4×Dic239—42no—
56C24×C239C2311yescharacteristic
57C24×C239C2311yescharacteristic
58C2×D4×C239—41no—
59C2×D4×C239—41no—
60C2×D4×C239C2×C414yes—
61C22×C4×C239C2312yes—
62C22×C4×C239C2312yes—
63C22×C4×C239C2311yescharacteristic
64C4⋊C4×C239—41no—
65C22⋊C4×C239—42no—
66C22⋊C4×C239—42no—
67C42×C239—41no—
68C22×D239—22no—
69C22×D239—22no—
70C22×D239—22no—
71C22×D239—22no—
72C22×D239—22no—
73C22×D239—22no—
74C22×D239—48no—
75C22×D239—44no—
76C22×D239—44no—
77C22×D239—48no—
78C22×D239—44no—
79C22×D239C22×C414yes—
80C22⋊D239—48no—
81C2×Dic239—22no—
82C2×Dic239—22no—
83C2×Dic239—22no—
84C2×Dic239—22no—
85C2×Dic239—22no—
86C2×Dic239—22no—
87C2×Dic239—22no—
88C2×Dic239—22no—
89C2×Dic239—44no—
90C2×Dic239C22×C414yes—
91C23×C239—22no—
92C23×C239—22no—
93C23×C239—21no—
94C23×C239—21no—
95C23×C239—21no—
96C23×C239—21no—
97C23×C239—22no—
98C23×C239—22no—
99C23×C239C22×C412yes—
100C23×C239C22×C412yes—
101C23×C239C2411yescharacteristic
102D4×C239—44no—
103C2×C4×C239—21no—
104C2×C4×C239—21no—
105C2×C4×C239—22no—
106C2×C4×C239—22no—
107C2×C4×C239—22no—
108C2×C4×C239—22no—
109C2×C4×C239—22no—
110C2×C4×C239—22no—
111C2×C4×C239C22×C412yes—
112C22×C239—22no—
113C22×C239—22no—
114C22×C239—24no—
115C22×C239—24no—
116C22×C239—41no—
117C22×C239—42no—
118C22×C239—41no—
119C22×C239—42no—
120C22×C239—41no—
121C22×C239—41no—
122C22×C239C23×C411yescharacteristic
123C22×C239H(5,𝔽2)12yes—
124C22×C239H(5,𝔽2)11yescharacteristic
125C22×C239H(5,𝔽2)11yescharacteristic
126C22×C239H(5,𝔽2)12yes—
127C2×D239—24no—
128C2×D239—24no—
129C2×D239—24no—
130C2×D239—416no—
131C2×D239—42no—
132C2×D239—42no—
133C2×D239—42no—
134C2×D239—42no—
135C4×C239—22no—
136C4×C239—42no—
137C4×C239—42no—
138Dic239—24no—
139Dic239—42no—
140Dic239—42no—
141C2×C239—42no—
142C2×C239—42no—
143C2×C239#51812yes—
144C2×C239#51811yescharacteristic
145C2×C239#56011yescharacteristic
146C2×C239#56012yes—
147C2×C239H(3,𝔽4)11yescharacteristic
148D239—44no—
149C239#237511yescharacteristic, complemented, Sylow
150#2375—2391nomaximal, Sylow
151C2×#120—2391no—
152#390—2392no—
153#390D23911yescharacteristic, complemented
154#388—2392no—
155#388—2392no—
156#388—2392no—
157#386—2391no—
158#386—2392no—
159#379—2392no—
160C25—2391no—
161C22×D4—2392no—
162C22×D4C2×D23911yescharacteristic
163C23×C4—2391no—
164#120—2398no—
165C2×C4⋊C4—2392no—
166C2×C4⋊C4C2×D23911yescharacteristic, complemented
167C2×C22⋊C4—2392no—
168C2×C22⋊C4—2391no—
169C2×C22⋊C4—2392no—
170C2×C22⋊C4—2392no—
171C2×C22⋊C4—2392no—
172C2×C22⋊C4—2392no—
173C2×C22⋊C4—2391no—
174C2×C22⋊C4—2392no—
175C2×C22⋊C4C2×D23912yes—
176C2×C22⋊C4C2×D23912yescomplemented
177C2×C42—2392no—
178C2×C42C2×D23911yescharacteristic, complemented
179#95—2392no—
180#95—2392no—
181#95—2392no—
182C24—9562no—
183C24—9561no—
184C24—9561no—
185C24—9562no—
186C24—2394no—
187C24—2392no—
188C24C22×D23911yescharacteristic
189C24C22×D23911yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—9562no—
193C2×D4—9562no—
194C2×D4—2398no—
195C2×D4C4×D23914yes—
196C22×C4—9561no—
197C22×C4—9562no—
198C22×C4—2392no—
199C22×C4—2392no—
200C22×C4—2392no—
201C22×C4C22×D23912yes—
202C22×C4C22×D23912yes—
203C22×C4C22×D23911yescharacteristic
204C4⋊C4—41no—
205C4⋊C4—9562no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—9562no—
209C22⋊C4—9562no—
210C22⋊C4—9562no—
211C22⋊C4—9562no—
212C22⋊C4—9561no—
213C22⋊C4—9562no—
214C22⋊C4—2394no—
215C22⋊C4—2398no—
216C42—41no—
217C42—9562no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—9568no—
227C23—9564no—
228C23—9564no—
229C23—9568no—
230C23—9564no—
231C23—4782no—
232C23—4782no—
233C23—4782no—
234C23—4782no—
235C23—4782no—
236C23—4782no—
237C23—2394no—
238C23C2×C4×D23912yes—
239C23C2×C4×D23912yes—
240C23C23×D23911yescharacteristic
241D4—44no—
242D4—9568no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—9564no—
252C2×C4—4782no—
253C2×C4—4782no—
254C2×C4—4782no—
255C2×C4—4782no—
256C2×C4—4782no—
257C2×C4—4782no—
258C2×C4—4782no—
259C2×C4—4782no—
260C2×C4—2394no—
261C2×C4C2×C4×D23912yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—95616no—
273C22—9562no—
274C22—9562no—
275C22—9562no—
276C22—9562no—
277C22—4784no—
278C22—4784no—
279C22—4784no—
280C22C22×C4×D23911yescharacteristic
281C22#2144000512yes—
282C22#2144000511yescharacteristic
283C22#2144000511yescharacteristic
284C22#2144000612yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—9562no—
289C4—9562no—
290C4—4784no—
291C2—42no—
292C2—42no—
293C2—9564no—
294C2#2950524412yes—
295C2#2950524511yescharacteristic
296C2#2950550911yescharacteristic
297C2#2950551012yes—
298C2#2950551411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
Fitting#390×C239C2
socle, cosocleC23×C239C24
prime radicalC2×#21439928C2
hypercenter#390D239
hypocenterC239#2375
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #29504672C2semidirect
3. #29504663C2semidirect
4. #29504662C2semidirect
5. #29504660C2semidirect
6. #29504652C2semidirect
7. #29504649C2semidirect
8. #29504627C2semidirect
9. #390×C239C2semidirect
12. C23×Dic239C22semidirect
13. #21439928C4semidirect
14. C2×Dic239⋊C4C22semidirect
17. C2×C22⋊Dic239C22semidirect
23. C2×C4×Dic239C22semidirect
24. C239⋊#95C22semidirect
25. C239⋊#95C22semidirect
26. C239⋊#95C22semidirect
28. C2×C4⋊C4×C239C22semidirect
30. C2×C22⋊C4×C239C22semidirect
31. C2×C42×C239C22semidirect
36. C23×D239C2×C4semidirect
49. C22⋊Dic239C2×C4semidirect
54. (C2×D239)⋊C4C2×C4semidirect
149. C239#2375semidirect
153. #390D239semidirect
166. C2×C4⋊C4C2×D239semidirect
176. C2×C22⋊C4C2×D239semidirect
178. C2×C42C2×D239semidirect

Maximal subgroups

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

subgroupindex
C2×#214399282
#295046722
#295046632
#295046622
#295046602
#295046522
#295046492
#295046272
#390×C2392
#2375239