Finite Group Databasebeta

← D15×#237 #13089369

Subgroups — D15×#237 #13089369

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
1#10624285C211yesmaximal, characteristic, complemented
2C3×D4×D15C211yesmaximal, characteristic, complemented
3C3×C5×#237C211yesmaximal, characteristic, complemented
4#67×D15C212yesmaximal, complemented
5#10624208C212yesmaximal, complemented
6#10624207C212yesmaximal, complemented
7C2×Dic3×D15C212yesmaximal, complemented
8C4×D3×D15C211yesmaximal, characteristic, complemented
9#10624194C211yesmaximal, characteristic, complemented
10#10624193C211yesmaximal, characteristic, complemented
11Dic6×D15C211yesmaximal, characteristic, complemented
12#41×D15—31nomaximal
13D5×#237—31nomaximal
14C22×C3×D15C2212yes—
15C2×D3×D15C2211yescharacteristic
16C32×C5×D4C2211yescharacteristic, complemented
17#67184C2212yescomplemented
18C2×C3⋊Dic15C2212yescomplemented
19C4×C3⋊D15C2211yescharacteristic, complemented
20#67180C2211yescharacteristic, complemented
21C3×#1220C2212yescomplemented
22C3×D60C2211yescharacteristic, complemented
23C3×C4×D15C2211yescharacteristic
24C3×C5×#67C2212yescomplemented
25C2×C3×C5×Dic3C2212yescomplemented
26C3×C4×C5×D3C2211yescharacteristic, complemented
27C3×C5×Dic6C2211yescharacteristic, complemented
28#67153C2212yescomplemented
29#67151C2212yescomplemented
30#67150C2212yescomplemented
31#67149C2211yescharacteristic, complemented
32D3×Dic15C2211yescharacteristic, complemented
33Dic3×D15C2212yes—
34Dic3×D15C2211yescharacteristic
35Dic3×D15C2212yescomplemented
36Dic3×D15C2212yescomplemented
37D3×#237—51nomaximal, Hall
38C3×C5×#41—31no—
39#6821—31no—
40Q8×D15—31no—
41#6819—31no—
42#6819—31no—
43#6819—32no—
44#6819—61no—
45D4×D15—32no—
46D4×D15D311yescharacteristic, complemented
47#6817—32no—
48#6817—31no—
49C2×C4×D15—32no—
50C2×C4×D15—31no—
51C5×#237D311yescharacteristic, complemented
52C3×D4×D5—31no—
53D5×#67—32no—
54#6780—32no—
55#6779—32no—
56C2×D5×Dic3—32no—
57C4×D3×D5—31no—
58#6766—31no—
59#6765—31no—
60D5×Dic6—31no—
61C22×C32×C5C2312yes—
62C2×C3⋊D15C2311yescharacteristic
63C2×C3×D15—24no—
64C2×C3×D15C2312yes—
65C2×C3×D15C2311yescharacteristic
66C2×C3×C5×D3C2311yescharacteristic
67D3×D15—22no—
68C32×C4×C5C2311yescharacteristic
69C3⋊Dic15C2311yescharacteristic
70C3⋊Dic15C2312yes—
71C3×Dic15C2311yescharacteristic
72C3×C5×Dic3C2311yescharacteristic
73C3×C5×Dic3C2312yes—
74D5×#41—91noHall
75#3830—51no—
76C3×#237D511yescharacteristic, complemented
77C3×D3×D4—51no—
78D3×#67—52no—
79#3805—52no—
80#3804—52no—
81C2×D3×Dic3—52no—
82C4×D32—51no—
83#3796—51no—
84#3795—51no—
85D3×Dic6—51no—
86C22×D15—31no—
87C22×D15C2×D312yes—
88C22×C3×D5—32no—
89C2×D3×D5—31no—
90C3×C5×Q8—31no—
91C3×C5×D4—21no—
92C3×C5×D4—32no—
93C3×C5×D4C2×D311yescharacteristic, complemented
94C3×C5×D4C2×D311yescharacteristic, complemented
95C2×C3×C4×C5—32no—
96C2×C3×C4×C5—31no—
97#1220—32no—
98#1220—32no—
99#1220—32no—
100#1220—62no—
101#1220C2×D312yescomplemented
102C2×Dic15—32no—
103C2×Dic15—32no—
104C2×Dic15—31no—
105C2×Dic15—62no—
106D60—32no—
107D60C2×D311yescharacteristic, complemented
108C4×D15—31no—
109C4×D15—32no—
110C4×D15—32no—
111C4×D15—32no—
112C4×D15—31no—
113C4×D15—31no—
114C4×D15—31no—
115C4×D15—61no—
116C4×D15C2×D311yescharacteristic
117Dic30—31no—
118Dic30—31no—
119Dic30—32no—
120Dic30—61no—
121C5×#67C2×D312yescomplemented
122C2×C5×Dic3C2×D312yescomplemented
123C4×C5×D3C2×D311yescharacteristic, complemented
124C5×Dic6C2×D311yescharacteristic, complemented
125C3×C22⋊D5—32no—
126C3×D20—31no—
127C3×C4×D5—31no—
128#1204—32no—
129#1202—32no—
130#1201—32no—
131#1200—31no—
132D3×Dic5—31no—
133D5×Dic3—32no—
134D5×Dic3—31no—
135D5×Dic3—32no—
136D5×Dic3—32no—
137D3×#41—151no—
138C2×#237—151no—
139C2×C32×C5—22no—
140C2×C32×C5C2411yescharacteristic
141C3⋊D15—21no—
142C3×D15—22no—
143C3×D15#4112yescomplemented
144C3×C5×D3—21no—
145C5×#41—31no—
146#713—91no—
147Q8×D5—91no—
148#711—92no—
149#711—91no—
150D4×D5—31no—
151D4×D5—92no—
152#709—91no—
153#709—92no—
154C2×C4×D5—91no—
155C2×C4×D5—92no—
156C22×C3×D3—52no—
157C2×D32—51no—
158C32×D4C2×D511yescharacteristic, complemented
159#638—52no—
160C2×C3⋊Dic3—52no—
161C4×C3⋊D3—51no—
162#634—51no—
163C3×#67—52no—
164C3×#67C2×D512yescomplemented
165C2×C3×Dic3C2×D512yescomplemented
166C3×D12—51no—
167C3×C4×D3—51no—
168C3×C4×D3C2×D511yescharacteristic, complemented
169C3×Dic6C2×D511yescharacteristic, complemented
170#627—52no—
171#626—52no—
172#625—52no—
173#624—51no—
174D3×Dic3—51no—
175D3×Dic3—52no—
176D3×Dic3—51no—
177D3×Dic3—52no—
178D3×Dic3—52no—
179C22×C3×C5—22no—
180C22×C3×C5—31no—
181C22×C3×C5C22×D312yes—
182C22×C3×C5C22×D312yes—
183C2×D15—24no—
184C2×D15—31no—
185C2×D15—31no—
186C2×D15—61no—
187C2×D15—62no—
188C2×D15C22×D312yes—
189C2×D15C22×D311yescharacteristic
190C2×C5×D3C22×D311yescharacteristic
191C2×C3×D5—31no—
192C2×C3×D5—32no—
193C2×C3×D5—64no—
194D3×D5—62no—
195C3×C4×C5—21no—
196C3×C4×C5—32no—
197C3×C4×C5—31no—
198C3×C4×C5C22×D311yescharacteristic
199C3×C4×C5C22×D311yescharacteristic
200Dic15—31no—
201Dic15—32no—
202Dic15—32no—
203Dic15—31no—
204Dic15—62no—
205Dic15—61no—
206Dic15C22×D311yescharacteristic
207C3×Dic5—31no—
208C5×Dic3C22×D311yescharacteristic
209C5×Dic3C22×D312yes—
210C3×#41—31no—
211C2×C3×D4—151no—
212C2×#67—152no—
213C22×Dic3—152no—
214#239—151no—
215D3×Q8—151no—
216#237—152no—
217#237—151no—
218#237—151no—
219#237—152no—
220#237—152no—
221#237—151no—
222#237—151no—
223#237—301no—
224#237D1511yescharacteristic, direct product factor
225D3×D4—51no—
226D3×D4—152no—
227#235—151no—
228#235—152no—
229C2×C4×D3—151no—
230C2×C4×D3—152no—
231C2×C4×D3—151no—
232C2×Dic6—151no—
233C32×C5C2×#4111yescharacteristic, complemented, Hall
234C22×D5—32no—
235C22×D5—91no—
236C5×Q8—31no—
237C5×D4—32no—
238C5×D4D3211yescharacteristic, complemented
239C2×C4×C5—32no—
240C2×C4×C5—31no—
241C22⋊D5—32no—
242C22⋊D5—92no—
243C22⋊D5—92no—
244C2×Dic5—91no—
245C2×Dic5—92no—
246D20—31no—
247D20—92no—
248C4×D5—31no—
249C4×D5—91no—
250C4×D5—91no—
251C4×D5—92no—
252C4×D5—92no—
253C4×D5—92no—
254C4×D5—91no—
255Dic10—92no—
256Dic10—91no—
257C22×C32C22×D512yes—
258C2×C3⋊D3—51no—
259C2×C3×D3—51no—
260C2×C3×D3—52no—
261C2×C3×D3—104no—
262C2×C3×D3C22×D511yescharacteristic
263D32—102no—
264C32×C4C22×D511yescharacteristic
265C3⋊Dic3—51no—
266C3⋊Dic3—52no—
267C3×Dic3—51no—
268C3×Dic3C22×D511yescharacteristic
269C3×Dic3C22×D512yes—
270C2×#41—451noSylow
271C2×C3×C5—22no—
272C2×C3×C5—22no—
273C2×C3×C5—21no—
274C2×C3×C5—42no—
275C2×C3×C5—61no—
276C2×C3×C5C23×D311yescharacteristic
277C2×C3×C5C23×D311yescharacteristic
278D15—22no—
279D15—61no—
280D15—61no—
281D15—121no—
282D15#23712yesdirect product factor
283C3×D5—32no—
284C3×D5—62no—
285C5×D3—21no—
286C23×C3—152no—
287C22×D3—52no—
288C22×D3—151no—
289C22×D3—151no—
290C3×Q8—31no—
291C3×D4—21no—
292C3×D4—32no—
293C3×D4—152no—
294C3×D4—151no—
295C3×D4D3×D511yescharacteristic, complemented
296C3×D4C2×D1511yescharacteristic, complemented
297C2×C3×C4—32no—
298C2×C3×C4—31no—
299C2×C3×C4—151no—
300#67—52no—
301#67—152no—
302#67—152no—
303#67—152no—
304#67—152no—
305#67—152no—
306#67—302no—
307#67C2×D1512yescomplemented
308C2×Dic3—151no—
309C2×Dic3—152no—
310C2×Dic3—152no—
311C2×Dic3—152no—
312C2×Dic3—151no—
313C2×Dic3—152no—
314C2×Dic3—152no—
315C2×Dic3—302no—
316C2×Dic3C2×D1512yescomplemented
317D12—51no—
318D12—152no—
319C4×D3—51no—
320C4×D3—151no—
321C4×D3—151no—
322C4×D3—152no—
323C4×D3—152no—
324C4×D3—152no—
325C4×D3—151no—
326C4×D3—151no—
327C4×D3—151no—
328C4×D3—151no—
329C4×D3—301no—
330C4×D3C2×D1511yescharacteristic, complemented
331Dic6—152no—
332Dic6—151no—
333Dic6—151no—
334Dic6—152no—
335Dic6—301no—
336Dic6C2×D1511yescharacteristic, complemented
337C22×C5—31no—
338C22×C5C2×D3212yes—
339C2×D5—31no—
340C2×D5—32no—
341C2×D5—64no—
342C2×D5—91no—
343C2×D5—182no—
344C4×C5—32no—
345C4×C5—31no—
346C4×C5C2×D3211yescharacteristic
347Dic5—31no—
348Dic5—92no—
349Dic5—91no—
350C2×C32—22no—
351C2×C32C23×D511yescharacteristic
352C3⋊D3—101no—
353C3×D3—21no—
354C3×D3—52no—
355C3×D3—102no—
356#41—31no—
357#41—452no—
358#41—451no—
359#41—452no—
360#41—451no—
361#41—451no—
362C2×Q8—451no—
363C2×D4—151no—
364C2×D4—452no—
365C22×C4—451no—
366C22×C4—452no—
367C3×C5—21no—
368C3×C5C2×#23711yescharacteristic, complemented
369C3×C5D3×#4111yescharacteristic, complemented
370C22×C3—22no—
371C22×C3—31no—
372C22×C3—151no—
373C22×C3—152no—
374C22×C3—304no—
375C22×C3C2×D3×D512yes—
376C22×C3C22×D1512yes—
377C2×D3—51no—
378C2×D3—52no—
379C2×D3—104no—
380C2×D3—151no—
381C2×D3—151no—
382C2×D3—301no—
383C2×D3—302no—
384C2×D3—302no—
385C2×D3C22×D1511yescharacteristic
386C3×C4—21no—
387C3×C4—32no—
388C3×C4—31no—
389C3×C4—151no—
390C3×C4C2×D3×D511yescharacteristic
391C3×C4C22×D1511yescharacteristic
392Dic3—51no—
393Dic3—152no—
394Dic3—151no—
395Dic3—152no—
396Dic3—151no—
397Dic3—302no—
398Dic3—301no—
399Dic3C22×D1512yes—
400Dic3C22×D1511yescharacteristic
401C2×C5—22no—
402C2×C5—61no—
403C2×C5C22×D3211yescharacteristic
404D5—32no—
405D5—62no—
406D5—181no—
407C32D5×#4111yescharacteristic, complemented, Sylow
408C23—152no—
409C23—451no—
410Q8—31no—
411Q8—452no—
412Q8—451no—
413D4—32no—
414D4—152no—
415D4—151no—
416D4—452no—
417D4—452no—
418D4—452no—
419D4D3×D1511yescharacteristic, complemented
420C2×C4—32no—
421C2×C4—31no—
422C2×C4—151no—
423C2×C4—451no—
424C2×C4—451no—
425C2×C4—452no—
426C2×C4—452no—
427C2×C4—452no—
428C2×C4—451no—
429C2×C4—452no—
430C2×C4—451no—
431C2×C3—22no—
432C2×C3—22no—
433C2×C3—21no—
434C2×C3—42no—
435C2×C3—61no—
436C2×C3—152no—
437C2×C3—302no—
438C2×C3C22×D3×D511yescharacteristic
439C2×C3C23×D1511yescharacteristic
440D3—21no—
441D3—52no—
442D3—102no—
443D3—301no—
444D3—301no—
445D3—601no—
446C5D3×#23711yescharacteristic, complemented, Sylow
447C22—31no—
448C22—151no—
449C22—152no—
450C22—304no—
451C22—451no—
452C22—902no—
453C22C2×D3×D1512yes—
454C4—31no—
455C4—32no—
456C4—151no—
457C4—451no—
458C4—452no—
459C4C2×D3×D1511yescharacteristic
460C3—21no—
461C3D5×#23711yescharacteristic, complemented
462C3#41×D1511yescharacteristic, complemented
463C2—22no—
464C2—61no—
465C2—152no—
466C2—302no—
467C2—901no—
468C2C22×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D15
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4D3×D15
hypocenterC32×C5C2×#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. #10624285C2semidirect
2. C3×D4×D15C2semidirect
3. C3×C5×#237C2semidirect
4. #67×D15C2semidirect
5. #10624208C2semidirect
6. #10624207C2semidirect
7. C2×Dic3×D15C2semidirect
8. C4×D3×D15C2semidirect
9. #10624194C2semidirect
10. #10624193C2semidirect
11. Dic6×D15C2semidirect
16. C32×C5×D4C22semidirect
17. #67184C22semidirect
18. C2×C3⋊Dic15C22semidirect
19. C4×C3⋊D15C22semidirect
20. #67180C22semidirect
21. C3×#1220C22semidirect
22. C3×D60C22semidirect
24. C3×C5×#67C22semidirect
25. C2×C3×C5×Dic3C22semidirect
26. C3×C4×C5×D3C22semidirect
27. C3×C5×Dic6C22semidirect
28. #67153C22semidirect
29. #67151C22semidirect
30. #67150C22semidirect
31. #67149C22semidirect
32. D3×Dic15C22semidirect
35. Dic3×D15C22semidirect
36. Dic3×D15C22semidirect
46. D4×D15D3semidirect
51. C5×#237D3semidirect
76. C3×#237D5semidirect
93. C3×C5×D4C2×D3semidirect
94. C3×C5×D4C2×D3semidirect
101. #1220C2×D3semidirect
107. D60C2×D3semidirect
121. C5×#67C2×D3semidirect
122. C2×C5×Dic3C2×D3semidirect
123. C4×C5×D3C2×D3semidirect
124. C5×Dic6C2×D3semidirect
143. C3×D15#41semidirect
158. C32×D4C2×D5semidirect
164. C3×#67C2×D5semidirect
165. C2×C3×Dic3C2×D5semidirect
168. C3×C4×D3C2×D5semidirect
169. C3×Dic6C2×D5semidirect
224. #237D15direct (N × H)
233. C32×C5C2×#41semidirect
238. C5×D4D32semidirect
282. D15#237direct (N × H)
295. C3×D4D3×D5semidirect
296. C3×D4C2×D15semidirect
307. #67C2×D15semidirect
316. C2×Dic3C2×D15semidirect
330. C4×D3C2×D15semidirect
336. Dic6C2×D15semidirect
368. C3×C5C2×#237semidirect
369. C3×C5D3×#41semidirect
407. C32D5×#41semidirect
419. D4D3×D15semidirect
446. C5D3×#237semidirect
461. C3D5×#237semidirect
462. C3#41×D15semidirect

Maximal subgroups

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

subgroupindex
#106242852
C3×D4×D152
C3×C5×#2372
#67×D152
#106242082
#106242072
C2×Dic3×D152
C4×D3×D152
#106241942
#106241932
Dic6×D152
#41×D153
D5×#2373
D3×#2375