Finite Group Databasebeta

← #13089408

Subgroups — #13089408

Order 1440.

696 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3828C211yesmaximal, characteristic, complemented
2C3×#6817C211yesmaximal, characteristic, complemented
3C2×C3×C4×D15C211yesmaximal, characteristic, complemented
4#10624235C212yesmaximal, complemented
5#10624230C212yesmaximal, complemented
6#10624226C211yesmaximal, characteristic, complemented
7C4×(C3⋊D3)⋊D5C212yesmaximal, complemented
8#10624224C211yesmaximal, characteristic, complemented
9#10624223C212yesmaximal, complemented
10#10624221C211yesmaximal, characteristic, complemented
11#10624219C211yesmaximal, characteristic, complemented
12#92016—31nomaximal
13D5×#235—31nomaximal
14C22×C3×D15C2211yescharacteristic
15C2×(C3⋊D3)⋊D5C2212yes—
16C2×C32×C4×C5C2211yescharacteristic, complemented
17C5×#638C2212yescomplemented
18C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C4×C5×C3⋊D3C2212yescomplemented
20C5×#634C2211yescharacteristic, complemented
21C3×#1220C2212yescomplemented
22C2×C3×Dic15C2211yescharacteristic, complemented
23C3×D60C2211yescharacteristic, complemented
24C3×C4×D15C2211yescharacteristic
25C3×C4×D15C2211yescharacteristic
26C3×C4×D15C2211yescharacteristic, complemented
27C3×C4×D15C2211yescharacteristic, complemented
28C3×C4×D15C2212yescomplemented
29C3×Dic30C2211yescharacteristic, complemented
30#67158C2212yescomplemented
31#67157C2212yescomplemented
32#67156C2212yescomplemented
33#67156C2212yescomplemented
34#67155C2212yescomplemented
35#67154C2212yes—
36#67154C2212yescomplemented
37D3×#235—51nomaximal, Hall
38#6817D311yescharacteristic, complemented
39C2×C4×D15D311yescharacteristic, complemented
40C5×#235—31no—
41C5×#235—31no—
42C5×#235—61no—
43C3×#709—31no—
44C2×C3×C4×D5—31no—
45#6790—32no—
46D5×#67—32no—
47#6782—32no—
48#6780—32no—
49#6777—31no—
50D5×D12—31no—
51C4×D3×D5—32no—
52C4×D3×D5—32no—
53#6773—31no—
54#6772—31no—
55#6771—32no—
56#6771—32no—
57D15⋊Q8—31no—
58#6768—31no—
59#6766—31no—
60D5×Dic6—31no—
61C22×C32×C5C2311yescharacteristic
62C2×C5×C3⋊D3C2312yes—
63C2×C3×D15—22no—
64C2×C3×D15C2312yes—
65C2×C3×D15C2311yescharacteristic
66C2×C3×D15C2311yescharacteristic
67(C3⋊D3)⋊D5—24no—
68C32×C4×C5C2311yescharacteristic
69C32×C4×C5C2311yescharacteristic
70C5×C3⋊Dic3C2312yes—
71C3×Dic15C2312yes—
72C3×Dic15C2311yescharacteristic
73C3×Dic15C2311yescharacteristic
74D5×#41—91noHall
75#3828D511yescharacteristic, complemented
76C3×#235—51no—
77C2×C3×C4×D3—51no—
78D3×#67—52no—
79#3804—52no—
80D3×D12—51no—
81C4×D32—52no—
82#3799—51no—
83#3798—52no—
84#3795—51no—
85D3×Dic6—51no—
86C22×D15C2×D311yescharacteristic
87C22×C3×D5—31no—
88C2×D3×D5—32no—
89C2×D3×D5—32no—
90C2×C3×C4×C5—21no—
91C2×C3×C4×C5C2×D311yescharacteristic, complemented
92C2×C3×C4×C5C2×D311yescharacteristic, complemented
93#1220C2×D312yescomplemented
94C2×Dic15C2×D311yescharacteristic, complemented
95D60C2×D311yescharacteristic, complemented
96C4×D15C2×D311yescharacteristic
97C4×D15C2×D311yescharacteristic
98C4×D15C2×D311yescharacteristic, complemented
99C4×D15C2×D311yescharacteristic, complemented
100C4×D15C2×D312yescomplemented
101Dic30C2×D311yescharacteristic, complemented
102C5×#67—32no—
103C5×#67—32no—
104C5×#67—62no—
105C5×D12—31no—
106C5×D12—31no—
107C5×D12—61no—
108C4×C5×D3—32no—
109C4×C5×D3—32no—
110C4×C5×D3—62no—
111C5×Dic6—31no—
112C5×Dic6—31no—
113C5×Dic6—61no—
114C3×C22⋊D5—32no—
115C2×C3×Dic5—31no—
116C3×D20—31no—
117C3×C4×D5—31no—
118C3×C4×D5—31no—
119C3×C4×D5—31no—
120C3×C4×D5—32no—
121C3×C4×D5—31no—
122C3×Dic10—31no—
123#1204—32no—
124#1204—32no—
125#1203—32no—
126#1203—32no—
127#1202—32no—
128#1202—32no—
129#1201—32no—
130#1201—32no—
131#1200—32no—
132#1200—32no—
133D3×Dic5—32no—
134D3×Dic5—32no—
135D5×Dic3—32no—
136D5×Dic3—32no—
137D3×#41—151no—
138C2×#235—151no—
139C2×C32×C5—21no—
140C2×C32×C5C2411yescharacteristic
141C5×C3⋊D3—22no—
142C3×D15—22no—
143C3×D15—21no—
144C3×D15#4112yescomplemented
145C5×#41—91no—
146#713—91no—
147Q8×D5—91no—
148#711—91no—
149#711—92no—
150D4×D5—91no—
151D4×D5—92no—
152#709—31no—
153#709—92no—
154C2×C4×D5—31no—
155C2×C4×D5—92no—
156C22×C3×D3—51no—
157C2×D32—52no—
158C2×C32×C4C2×D511yescharacteristic, complemented
159#638C2×D512yescomplemented
160C4⋊(C3⋊D3)C2×D511yescharacteristic, complemented
161C4×C3⋊D3C2×D512yescomplemented
162#634C2×D511yescharacteristic, complemented
163C3×#67—52no—
164C2×C3×Dic3—51no—
165C3×D12—51no—
166C3×C4×D3—51no—
167C3×C4×D3—51no—
168C3×C4×D3—51no—
169C3×C4×D3—51no—
170C3×C4×D3—52no—
171C3×Dic6—51no—
172#627—52no—
173#626—52no—
174#626—52no—
175#625—52no—
176#624—52no—
177D3×Dic3—52no—
178D3×Dic3—52no—
179C22×C3×C5—21no—
180C22×C3×C5C22×D311yescharacteristic
181C22×C3×C5C22×D311yescharacteristic
182C2×D15—22no—
183C2×D15C22×D312yes—
184C2×D15C22×D311yescharacteristic
185C2×D15C22×D311yescharacteristic
186C2×C5×D3—32no—
187C2×C5×D3—32no—
188C2×C5×D3—62no—
189C2×C3×D5—31no—
190C2×C3×D5—32no—
191C2×C3×D5—31no—
192C2×C3×D5—62no—
193D3×D5—64no—
194D3×D5—64no—
195C3×C4×C5—21no—
196C3×C4×C5—21no—
197C3×C4×C5C22×D311yescharacteristic
198C3×C4×C5C22×D311yescharacteristic
199C3×C4×C5C22×D311yescharacteristic
200C3×C4×C5C22×D311yescharacteristic
201Dic15C22×D311yescharacteristic
202Dic15C22×D312yes—
203Dic15C22×D311yescharacteristic
204C3×Dic5—31no—
205C3×Dic5—32no—
206C3×Dic5—31no—
207C5×Dic3—32no—
208C5×Dic3—32no—
209C5×Dic3—62no—
210C3×#41—151no—
211C22×C3×C4—151no—
212C2×#67—152no—
213#239—151no—
214D3×Q8—151no—
215#237—151no—
216#237—152no—
217D3×D4—151no—
218D3×D4—152no—
219#235—31no—
220#235—31no—
221#235—51no—
222#235—61no—
223#235—151no—
224#235—151no—
225#235—152no—
226#235—152no—
227#235—152no—
228C2×D12—151no—
229C2×C4×D3—51no—
230C2×C4×D3—152no—
231C2×C4×D3—152no—
232C2×Dic6—151no—
233C32×C5C2×#4111yescharacteristic, complemented, Hall
234C22×D5—31no—
235C22×D5—92no—
236C5×Q8—91no—
237C5×D4—92no—
238C5×D4—91no—
239C2×C4×C5—92no—
240C2×C4×C5D3211yescharacteristic, complemented
241C22⋊D5—32no—
242C22⋊D5—92no—
243C22⋊D5—92no—
244C2×Dic5—31no—
245C2×Dic5—92no—
246D20—31no—
247D20—92no—
248C4×D5—31no—
249C4×D5—31no—
250C4×D5—31no—
251C4×D5—31no—
252C4×D5—32no—
253C4×D5—92no—
254C4×D5—92no—
255Dic10—31no—
256Dic10—92no—
257C22×C32C22×D511yescharacteristic
258C2×C3⋊D3C22×D512yes—
259C2×C3×D3—51no—
260C2×C3×D3—52no—
261C2×C3×D3—51no—
262C2×C3×D3—102no—
263D32—104no—
264C32×C4C22×D511yescharacteristic
265C32×C4C22×D511yescharacteristic
266C3⋊Dic3C22×D512yes—
267C3×Dic3—51no—
268C3×Dic3—52no—
269C3×Dic3—51no—
270C2×#41—451noSylow
271C2×C3×C5—21no—
272C2×C3×C5—21no—
273C2×C3×C5—21no—
274C2×C3×C5—41no—
275C2×C3×C5C23×D311yescharacteristic
276C2×C3×C5C23×D311yescharacteristic
277D15—21no—
278D15—22no—
279D15#23512yescomplemented
280C3×D5—32no—
281C3×D5—62no—
282C3×D5—61no—
283C5×D3—62no—
284C5×D3—62no—
285C5×D3—122no—
286C23×C3—151no—
287C22×D3—51no—
288C22×D3—152no—
289C22×D3—152no—
290C3×Q8—151no—
291C3×D4—152no—
292C3×D4—151no—
293C2×C3×C4—21no—
294C2×C3×C4—151no—
295C2×C3×C4—151no—
296C2×C3×C4—151no—
297C2×C3×C4—152no—
298C2×C3×C4—151no—
299C2×C3×C4—151no—
300C2×C3×C4D3×D511yescharacteristic, complemented
301C2×C3×C4D3×D511yescharacteristic, complemented
302#67—32no—
303#67—32no—
304#67—52no—
305#67—62no—
306#67—152no—
307#67—152no—
308#67—152no—
309#67—152no—
310C2×Dic3—51no—
311C2×Dic3—152no—
312C2×Dic3—152no—
313D12—31no—
314D12—31no—
315D12—51no—
316D12—61no—
317D12—152no—
318D12—152no—
319C4×D3—32no—
320C4×D3—32no—
321C4×D3—51no—
322C4×D3—51no—
323C4×D3—51no—
324C4×D3—51no—
325C4×D3—52no—
326C4×D3—62no—
327C4×D3—152no—
328C4×D3—152no—
329C4×D3—152no—
330C4×D3—152no—
331Dic6—31no—
332Dic6—31no—
333Dic6—51no—
334Dic6—61no—
335Dic6—152no—
336Dic6—152no—
337C22×C5—92no—
338C22×C5C2×D3211yescharacteristic
339C2×D5—31no—
340C2×D5—31no—
341C2×D5—32no—
342C2×D5—62no—
343C2×D5—184no—
344C4×C5—92no—
345C4×C5C2×D3211yescharacteristic
346C4×C5C2×D3211yescharacteristic
347Dic5—32no—
348Dic5—31no—
349Dic5—31no—
350C2×C32—21no—
351C2×C32C23×D511yescharacteristic
352C3⋊D3—22no—
353C3×D3—52no—
354C3×D3—102no—
355C3×D3—101no—
356#41—91no—
357#41—151no—
358#41—452no—
359#41—452no—
360#41—451no—
361#41—451no—
362C2×Q8—451no—
363C2×D4—451no—
364C2×D4—452no—
365C22×C4—151no—
366C22×C4—452no—
367C3×C5—21no—
368C3×C5C2×#23511yescharacteristic, complemented
369C3×C5D3×#4111yescharacteristic, complemented
370C22×C3—21no—
371C22×C3—151no—
372C22×C3—152no—
373C22×C3—151no—
374C22×C3—302no—
375C22×C3C2×D3×D511yescharacteristic
376C22×C3C2×D3×D511yescharacteristic
377C2×D3—32no—
378C2×D3—32no—
379C2×D3—51no—
380C2×D3—51no—
381C2×D3—52no—
382C2×D3—62no—
383C2×D3—102no—
384C2×D3—304no—
385C2×D3—304no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—151no—
389C3×C4—152no—
390C3×C4—151no—
391C3×C4C2×D3×D511yescharacteristic
392C3×C4C2×D3×D511yescharacteristic
393C3×C4C2×D3×D511yescharacteristic
394C3×C4C2×D3×D511yescharacteristic
395Dic3—32no—
396Dic3—32no—
397Dic3—52no—
398Dic3—51no—
399Dic3—51no—
400Dic3—62no—
401C2×C5—21no—
402C2×C5—182no—
403C2×C5C22×D3211yescharacteristic
404D5—32no—
405D5—61no—
406D5—62no—
407C32D5×#4111yescharacteristic, complemented, Sylow
408C23—151no—
409C23—452no—
410Q8—91no—
411Q8—151no—
412Q8—452no—
413D4—92no—
414D4—91no—
415D4—152no—
416D4—151no—
417D4—452no—
418D4—452no—
419D4—452no—
420C2×C4—92no—
421C2×C4—151no—
422C2×C4—151no—
423C2×C4—151no—
424C2×C4—151no—
425C2×C4—151no—
426C2×C4—152no—
427C2×C4—452no—
428C2×C4—452no—
429C2×C4—452no—
430C2×C4(C3⋊D3)⋊D511yescharacteristic, complemented
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—41no—
435C2×C3—152no—
436C2×C3—302no—
437C2×C3—301no—
438C2×C3C22×D3×D511yescharacteristic
439C2×C3C22×D3×D511yescharacteristic
440D3—52no—
441D3—62no—
442D3—62no—
443D3—101no—
444D3—102no—
445D3—122no—
446C5D3×#23511yescharacteristic, complemented, Sylow
447C22—92no—
448C22—151no—
449C22—151no—
450C22—152no—
451C22—302no—
452C22—904no—
453C22C2×(C3⋊D3)⋊D511yescharacteristic
454C4—92no—
455C4—151no—
456C4—151no—
457C4—152no—
458C4C2×(C3⋊D3)⋊D511yescharacteristic
459C4C2×(C3⋊D3)⋊D511yescharacteristic
460C3—21no—
461C3D5×#23511yescharacteristic, complemented
462C3#9201611yescharacteristic, complemented
463C2—21no—
464C2—152no—
465C2—182no—
466C2—301no—
467C2—302no—
468C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
centerC4C2×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FrattiniC2C22×(C3⋊D3)⋊D5
FittingC2×C32×C4×C5C22
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C5×#3828C2semidirect
2. C3×#6817C2semidirect
3. C2×C3×C4×D15C2semidirect
4. #10624235C2semidirect
5. #10624230C2semidirect
6. #10624226C2semidirect
7. C4×(C3⋊D3)⋊D5C2semidirect
8. #10624224C2semidirect
9. #10624223C2semidirect
10. #10624221C2semidirect
11. #10624219C2semidirect
16. C2×C32×C4×C5C22semidirect
17. C5×#638C22semidirect
18. C5×C4⋊(C3⋊D3)C22semidirect
19. C4×C5×C3⋊D3C22semidirect
20. C5×#634C22semidirect
21. C3×#1220C22semidirect
22. C2×C3×Dic15C22semidirect
23. C3×D60C22semidirect
26. C3×C4×D15C22semidirect
27. C3×C4×D15C22semidirect
28. C3×C4×D15C22semidirect
29. C3×Dic30C22semidirect
30. #67158C22semidirect
31. #67157C22semidirect
32. #67156C22semidirect
33. #67156C22semidirect
34. #67155C22semidirect
36. #67154C22semidirect
38. #6817D3semidirect
39. C2×C4×D15D3semidirect
75. #3828D5semidirect
91. C2×C3×C4×C5C2×D3semidirect
92. C2×C3×C4×C5C2×D3semidirect
93. #1220C2×D3semidirect
94. C2×Dic15C2×D3semidirect
95. D60C2×D3semidirect
98. C4×D15C2×D3semidirect
99. C4×D15C2×D3semidirect
100. C4×D15C2×D3semidirect
101. Dic30C2×D3semidirect
144. C3×D15#41semidirect
158. C2×C32×C4C2×D5semidirect
159. #638C2×D5semidirect
160. C4⋊(C3⋊D3)C2×D5semidirect
161. C4×C3⋊D3C2×D5semidirect
162. #634C2×D5semidirect
233. C32×C5C2×#41semidirect
240. C2×C4×C5D32semidirect
279. D15#235semidirect
300. C2×C3×C4D3×D5semidirect
301. C2×C3×C4D3×D5semidirect
368. C3×C5C2×#235semidirect
369. C3×C5D3×#41semidirect
407. C32D5×#41semidirect
430. C2×C4(C3⋊D3)⋊D5semidirect
446. C5D3×#235semidirect
461. C3D5×#235semidirect
462. C3#92016semidirect

Maximal subgroups

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

subgroupindex
C5×#38282
C3×#68172
C2×C3×C4×D152
#106242352
#106242302
#106242262
C4×(C3⋊D3)⋊D52
#106242242
#106242232
#106242212
#106242192
#920163
D5×#2353
D3×#2355