Finite Group Databasebeta

← C5×D3×#235 #13089318

Subgroups — C5×D3×#235 #13089318

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×C5×#235C211yesmaximal, characteristic, complemented
3C2×C3×C4×C5×D3C211yesmaximal, characteristic, complemented
4C5×D3×#67C212yesmaximal, complemented
5C5×#3804C212yesmaximal, complemented
6C5×D3×D12C211yesmaximal, characteristic, complemented
7C4×C5×D32C212yesmaximal, complemented
8C5×#3799C211yesmaximal, characteristic, complemented
9C5×#3798C212yesmaximal, complemented
10C5×#3795C211yesmaximal, characteristic, complemented
11C5×D3×Dic6C211yesmaximal, characteristic, complemented
12C5×D3×#41—31nomaximal
13C2×C5×#235—31nomaximal
14C22×C3×C5×D3C2211yescharacteristic
15C2×C5×D32C2212yes—
16C2×C32×C4×C5C2211yescharacteristic, complemented
17C5×#638C2212yescomplemented
18C5×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19C4×C5×C3⋊D3C2212yescomplemented
20C5×#634C2211yescharacteristic, complemented
21C3×C5×#67C2212yescomplemented
22C2×C3×C5×Dic3C2211yescharacteristic, complemented
23C3×C5×D12C2211yescharacteristic, complemented
24C3×C4×C5×D3C2211yescharacteristic
25C3×C4×C5×D3C2211yescharacteristic
26C3×C4×C5×D3C2211yescharacteristic, complemented
27C3×C4×C5×D3C2211yescharacteristic, complemented
28C3×C4×C5×D3C2212yescomplemented
29C3×C5×Dic6C2211yescharacteristic, complemented
30C5×#627C2212yescomplemented
31C5×#626C2212yescomplemented
32C5×#626C2212yescomplemented
33C5×#625C2212yescomplemented
34C5×#624C2212yescomplemented
35C5×D3×Dic3C2212yes—
36C5×D3×Dic3C2212yescomplemented
37D3×#235C511yesmaximal, characteristic, direct product factor, Hall
38C3×C5×#41—31no—
39C22×C3×C4×C5—31no—
40C2×C5×#67—32no—
41C5×#239—31no—
42C5×D3×Q8—31no—
43C5×#237—32no—
44C5×#237—31no—
45C5×D3×D4—32no—
46C5×D3×D4—31no—
47C5×#235—31no—
48C5×#235—31no—
49C5×#235—32no—
50C5×#235—31no—
51C5×#235—32no—
52C5×#235—32no—
53C5×#235—31no—
54C5×#235—61no—
55C5×#235D311yescharacteristic, direct product factor
56C2×C5×D12—31no—
57C2×C4×C5×D3—32no—
58C2×C4×C5×D3—32no—
59C2×C4×C5×D3D311yescharacteristic, complemented
60C2×C5×Dic6—31no—
61C22×C32×C5C2311yescharacteristic
62C2×C5×C3⋊D3C2312yes—
63C2×C3×C5×D3—22no—
64C2×C3×C5×D3C2312yes—
65C2×C3×C5×D3C2311yescharacteristic
66C2×C3×C5×D3C2311yescharacteristic
67C5×D32—24no—
68C32×C4×C5C2311yescharacteristic
69C32×C4×C5C2311yescharacteristic
70C5×C3⋊Dic3C2312yes—
71C3×C5×Dic3C2312yes—
72C3×C5×Dic3C2311yescharacteristic
73C3×C5×Dic3C2311yescharacteristic
74C2×C5×#41—91noHall
75#3828C2×C511yescharacteristic, complemented
76C3×#235C2×C511yescharacteristic, complemented
77C2×C3×C4×D3C2×C511yescharacteristic, complemented
78D3×#67C2×C512yescomplemented
79#3804C2×C512yescomplemented
80D3×D12C2×C511yescharacteristic, complemented
81C4×D32C2×C512yescomplemented
82#3799C2×C511yescharacteristic, complemented
83#3798C2×C512yescomplemented
84#3795C2×C511yescharacteristic, complemented
85D3×Dic6C2×C511yescharacteristic, complemented
86C23×C3×C5—31no—
87C22×C5×D3—32no—
88C22×C5×D3—32no—
89C22×C5×D3C2×D311yescharacteristic
90C3×C5×Q8—31no—
91C3×C5×D4—32no—
92C3×C5×D4—31no—
93C2×C3×C4×C5—21no—
94C2×C3×C4×C5—31no—
95C2×C3×C4×C5—31no—
96C2×C3×C4×C5—31no—
97C2×C3×C4×C5—32no—
98C2×C3×C4×C5—31no—
99C2×C3×C4×C5—31no—
100C2×C3×C4×C5C2×D311yescharacteristic, complemented
101C2×C3×C4×C5C2×D311yescharacteristic, complemented
102C5×#67—32no—
103C5×#67—32no—
104C5×#67—32no—
105C5×#67—32no—
106C5×#67—32no—
107C5×#67—32no—
108C5×#67—62no—
109C5×#67C2×D312yescomplemented
110C2×C5×Dic3—32no—
111C2×C5×Dic3—32no—
112C2×C5×Dic3C2×D311yescharacteristic, complemented
113C5×D12—32no—
114C5×D12—32no—
115C5×D12—31no—
116C5×D12—31no—
117C5×D12—61no—
118C5×D12C2×D311yescharacteristic, complemented
119C4×C5×D3—32no—
120C4×C5×D3—32no—
121C4×C5×D3—32no—
122C4×C5×D3—32no—
123C4×C5×D3—32no—
124C4×C5×D3—32no—
125C4×C5×D3—62no—
126C4×C5×D3C2×D311yescharacteristic
127C4×C5×D3C2×D311yescharacteristic
128C4×C5×D3C2×D312yescomplemented
129C4×C5×D3C2×D311yescharacteristic, complemented
130C4×C5×D3C2×D311yescharacteristic, complemented
131C5×Dic6—31no—
132C5×Dic6—31no—
133C5×Dic6—32no—
134C5×Dic6—32no—
135C5×Dic6—61no—
136C5×Dic6C2×D311yescharacteristic, complemented
137D3×#41—31no—
138C2×#235—31no—
139C2×C32×C5—21no—
140C2×C32×C5C2411yescharacteristic
141C5×C3⋊D3—22no—
142C3×C5×D3—22no—
143C3×C5×D3—21no—
144C3×C5×D3#4112yescomplemented
145C5×#41—31no—
146C5×#41—92no—
147C5×#41—92no—
148C5×#41—91no—
149C5×#41—91no—
150C5×#41—91no—
151C2×C5×Q8—91no—
152C2×C5×D4—91no—
153C2×C5×D4—92no—
154C22×C4×C5—31no—
155C22×C4×C5—92no—
156C22×C3×D3C22×C511yescharacteristic
157C2×D32C22×C512yes—
158C2×C32×C4C22×C511yescharacteristic, complemented
159#638C22×C512yescomplemented
160C4⋊(C3⋊D3)C22×C511yescharacteristic, complemented
161C4×C3⋊D3C22×C512yescomplemented
162#634C22×C511yescharacteristic, complemented
163C3×#67C22×C512yescomplemented
164C2×C3×Dic3C22×C511yescharacteristic, complemented
165C3×D12C22×C511yescharacteristic, complemented
166C3×C4×D3C22×C511yescharacteristic
167C3×C4×D3C22×C511yescharacteristic
168C3×C4×D3C22×C511yescharacteristic, complemented
169C3×C4×D3C22×C511yescharacteristic, complemented
170C3×C4×D3C22×C512yescomplemented
171C3×Dic6C22×C511yescharacteristic, complemented
172#627C22×C512yescomplemented
173#626C22×C512yescomplemented
174#626C22×C512yescomplemented
175#625C22×C512yescomplemented
176#624C22×C512yescomplemented
177D3×Dic3C22×C512yes—
178D3×Dic3C22×C512yescomplemented
179C22×C3×C5—21no—
180C22×C3×C5—31no—
181C22×C3×C5—32no—
182C22×C3×C5—31no—
183C22×C3×C5—62no—
184C22×C3×C5C22×D311yescharacteristic
185C22×C3×C5C22×D311yescharacteristic
186C2×C5×D3—22no—
187C2×C5×D3—32no—
188C2×C5×D3—32no—
189C2×C5×D3—62no—
190C2×C5×D3—64no—
191C2×C5×D3—64no—
192C2×C5×D3C22×D312yes—
193C2×C5×D3C22×D311yescharacteristic
194C2×C5×D3C22×D311yescharacteristic
195C3×C4×C5—21no—
196C3×C4×C5—21no—
197C3×C4×C5—31no—
198C3×C4×C5—32no—
199C3×C4×C5—31no—
200C3×C4×C5C22×D311yescharacteristic
201C3×C4×C5C22×D311yescharacteristic
202C3×C4×C5C22×D311yescharacteristic
203C3×C4×C5C22×D311yescharacteristic
204C5×Dic3—32no—
205C5×Dic3—32no—
206C5×Dic3—62no—
207C5×Dic3C22×D311yescharacteristic
208C5×Dic3C22×D312yes—
209C5×Dic3C22×D311yescharacteristic
210C3×#41—31no—
211C22×C3×C4—31no—
212C2×#67—32no—
213#239—31no—
214D3×Q8—31no—
215#237—32no—
216#237—31no—
217D3×D4—32no—
218D3×D4—31no—
219#235—31no—
220#235—31no—
221#235—32no—
222#235—31no—
223#235—32no—
224#235—32no—
225#235—31no—
226#235—61no—
227#235C5×D311yescharacteristic, direct product factor
228C2×D12—31no—
229C2×C4×D3—32no—
230C2×C4×D3—32no—
231C2×C4×D3C5×D311yescharacteristic, complemented
232C2×Dic6—31no—
233C32×C5C2×#4111yescharacteristic, complemented, Hall
234C23×C5—31no—
235C23×C5—92no—
236C5×Q8—31no—
237C5×Q8—92no—
238C5×Q8—91no—
239C5×D4—32no—
240C5×D4—31no—
241C5×D4—92no—
242C5×D4—92no—
243C5×D4—92no—
244C5×D4—92no—
245C5×D4—91no—
246C2×C4×C5—31no—
247C2×C4×C5—31no—
248C2×C4×C5—31no—
249C2×C4×C5—31no—
250C2×C4×C5—31no—
251C2×C4×C5—32no—
252C2×C4×C5—92no—
253C2×C4×C5—92no—
254C2×C4×C5—92no—
255C2×C4×C5—92no—
256C2×C4×C5D3211yescharacteristic, complemented
257C22×C32C23×C511yescharacteristic
258C2×C3⋊D3C23×C512yes—
259C2×C3×D3—22no—
260C2×C3×D3C23×C512yes—
261C2×C3×D3C23×C511yescharacteristic
262C2×C3×D3C23×C511yescharacteristic
263D32—24no—
264C32×C4C23×C511yescharacteristic
265C32×C4C23×C511yescharacteristic
266C3⋊Dic3C23×C512yes—
267C3×Dic3C23×C512yes—
268C3×Dic3C23×C511yescharacteristic
269C3×Dic3C23×C511yescharacteristic
270C2×#41—91noSylow
271C2×C3×C5—21no—
272C2×C3×C5—21no—
273C2×C3×C5—21no—
274C2×C3×C5—32no—
275C2×C3×C5—41no—
276C2×C3×C5—62no—
277C2×C3×C5—61no—
278C2×C3×C5C23×D311yescharacteristic
279C2×C3×C5C23×D311yescharacteristic
280C5×D3—21no—
281C5×D3—22no—
282C5×D3—62no—
283C5×D3—62no—
284C5×D3—122no—
285C5×D3#23512yesdirect product factor
286C23×C3—31no—
287C22×D3—32no—
288C22×D3—32no—
289C22×D3C2×C5×D311yescharacteristic
290C3×Q8—31no—
291C3×D4—32no—
292C3×D4—31no—
293C2×C3×C4—21no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—32no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4C2×C5×D311yescharacteristic, complemented
301C2×C3×C4C2×C5×D311yescharacteristic, complemented
302#67—32no—
303#67—32no—
304#67—32no—
305#67—32no—
306#67—32no—
307#67—32no—
308#67—62no—
309#67C2×C5×D312yescomplemented
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3C2×C5×D311yescharacteristic, complemented
313D12—32no—
314D12—32no—
315D12—31no—
316D12—31no—
317D12—61no—
318D12C2×C5×D311yescharacteristic, complemented
319C4×D3—32no—
320C4×D3—32no—
321C4×D3—32no—
322C4×D3—32no—
323C4×D3—32no—
324C4×D3—32no—
325C4×D3—62no—
326C4×D3C2×C5×D311yescharacteristic
327C4×D3C2×C5×D311yescharacteristic
328C4×D3C2×C5×D312yescomplemented
329C4×D3C2×C5×D311yescharacteristic, complemented
330C4×D3C2×C5×D311yescharacteristic, complemented
331Dic6—31no—
332Dic6—31no—
333Dic6—32no—
334Dic6—32no—
335Dic6—61no—
336Dic6C2×C5×D311yescharacteristic, complemented
337C22×C5—31no—
338C22×C5—31no—
339C22×C5—32no—
340C22×C5—62no—
341C22×C5—92no—
342C22×C5—184no—
343C22×C5C2×D3211yescharacteristic
344C4×C5—31no—
345C4×C5—31no—
346C4×C5—32no—
347C4×C5—92no—
348C4×C5C2×D3211yescharacteristic
349C4×C5C2×D3211yescharacteristic
350C2×C32—21no—
351C2×C32C24×C511yescharacteristic
352C3⋊D3—22no—
353C3×D3—22no—
354C3×D3—21no—
355C3×D3C5×#4112yescomplemented
356#41—31no—
357#41—92no—
358#41—92no—
359#41—91no—
360#41—91no—
361#41—91no—
362C2×Q8—91no—
363C2×D4—91no—
364C2×D4—92no—
365C22×C4—31no—
366C22×C4—92no—
367C3×C5—21no—
368C3×C5C2×#23511yescharacteristic, complemented
369C3×C5D3×#4111yescharacteristic, complemented
370C22×C3—21no—
371C22×C3—31no—
372C22×C3—32no—
373C22×C3—31no—
374C22×C3—62no—
375C22×C3C22×C5×D311yescharacteristic
376C22×C3C22×C5×D311yescharacteristic
377C2×D3—22no—
378C2×D3—32no—
379C2×D3—32no—
380C2×D3—62no—
381C2×D3—64no—
382C2×D3—64no—
383C2×D3C22×C5×D312yes—
384C2×D3C22×C5×D311yescharacteristic
385C2×D3C22×C5×D311yescharacteristic
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—31no—
389C3×C4—32no—
390C3×C4—31no—
391C3×C4C22×C5×D311yescharacteristic
392C3×C4C22×C5×D311yescharacteristic
393C3×C4C22×C5×D311yescharacteristic
394C3×C4C22×C5×D311yescharacteristic
395Dic3—32no—
396Dic3—32no—
397Dic3—62no—
398Dic3C22×C5×D311yescharacteristic
399Dic3C22×C5×D312yes—
400Dic3C22×C5×D311yescharacteristic
401C2×C5—21no—
402C2×C5—32no—
403C2×C5—61no—
404C2×C5—62no—
405C2×C5—182no—
406C2×C5C22×D3211yescharacteristic
407C32C2×C5×#4111yescharacteristic, complemented, Sylow
408C23—31no—
409C23—92no—
410Q8—31no—
411Q8—92no—
412Q8—91no—
413D4—32no—
414D4—31no—
415D4—92no—
416D4—92no—
417D4—92no—
418D4—92no—
419D4—91no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—32no—
426C2×C4—92no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4C5×D3211yescharacteristic, complemented
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—32no—
435C2×C3—41no—
436C2×C3—62no—
437C2×C3—61no—
438C2×C3C23×C5×D311yescharacteristic
439C2×C3C23×C5×D311yescharacteristic
440D3—21no—
441D3—22no—
442D3—62no—
443D3—62no—
444D3—122no—
445D3C5×#23512yesdirect product factor
446C5D3×#23511yescharacteristic, direct product factor, Sylow
447C22—31no—
448C22—31no—
449C22—32no—
450C22—62no—
451C22—92no—
452C22—184no—
453C22C2×C5×D3211yescharacteristic
454C4—31no—
455C4—31no—
456C4—32no—
457C4—92no—
458C4C2×C5×D3211yescharacteristic
459C4C2×C5×D3211yescharacteristic
460C3—21no—
461C3C2×C5×#23511yescharacteristic, complemented
462C3C5×D3×#4111yescharacteristic, complemented
463C2—21no—
464C2—32no—
465C2—61no—
466C2—62no—
467C2—182no—
468C2C22×C5×D3211yescharacteristic
subgroupG/H
centerC4×C5C2×D32
commutator, cosocleC2×C32C24×C5
FrattiniC2C22×C5×D32
FittingC2×C32×C4×C5C22
socleC2×C32×C5C24
hypercenterC2×C4×C5D32
hypocenterC32C2×C5×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C4×C5 ◁ C2×C4×C5

Complements

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

NH ≅ G/Ntype
1. C5×#3828C2semidirect
2. C3×C5×#235C2semidirect
3. C2×C3×C4×C5×D3C2semidirect
4. C5×D3×#67C2semidirect
5. C5×#3804C2semidirect
6. C5×D3×D12C2semidirect
7. C4×C5×D32C2semidirect
8. C5×#3799C2semidirect
9. C5×#3798C2semidirect
10. C5×#3795C2semidirect
11. C5×D3×Dic6C2semidirect
16. C2×C32×C4×C5C22semidirect
17. C5×#638C22semidirect
18. C5×C4⋊(C3⋊D3)C22semidirect
19. C4×C5×C3⋊D3C22semidirect
20. C5×#634C22semidirect
21. C3×C5×#67C22semidirect
22. C2×C3×C5×Dic3C22semidirect
23. C3×C5×D12C22semidirect
26. C3×C4×C5×D3C22semidirect
27. C3×C4×C5×D3C22semidirect
28. C3×C4×C5×D3C22semidirect
29. C3×C5×Dic6C22semidirect
30. C5×#627C22semidirect
31. C5×#626C22semidirect
32. C5×#626C22semidirect
33. C5×#625C22semidirect
34. C5×#624C22semidirect
36. C5×D3×Dic3C22semidirect
37. D3×#235C5direct (N × H)
55. C5×#235D3direct (N × H)
59. C2×C4×C5×D3D3semidirect
75. #3828C2×C5semidirect
76. C3×#235C2×C5semidirect
77. C2×C3×C4×D3C2×C5semidirect
78. D3×#67C2×C5semidirect
79. #3804C2×C5semidirect
80. D3×D12C2×C5semidirect
81. C4×D32C2×C5semidirect
82. #3799C2×C5semidirect
83. #3798C2×C5semidirect
84. #3795C2×C5semidirect
85. D3×Dic6C2×C5semidirect
100. C2×C3×C4×C5C2×D3semidirect
101. C2×C3×C4×C5C2×D3semidirect
109. C5×#67C2×D3semidirect
112. C2×C5×Dic3C2×D3semidirect
118. C5×D12C2×D3semidirect
128. C4×C5×D3C2×D3semidirect
129. C4×C5×D3C2×D3semidirect
130. C4×C5×D3C2×D3semidirect
136. C5×Dic6C2×D3semidirect
144. C3×C5×D3#41semidirect
158. C2×C32×C4C22×C5semidirect
159. #638C22×C5semidirect
160. C4⋊(C3⋊D3)C22×C5semidirect
161. C4×C3⋊D3C22×C5semidirect
162. #634C22×C5semidirect
163. C3×#67C22×C5semidirect
164. C2×C3×Dic3C22×C5semidirect
165. C3×D12C22×C5semidirect
168. C3×C4×D3C22×C5semidirect
169. C3×C4×D3C22×C5semidirect
170. C3×C4×D3C22×C5semidirect
171. C3×Dic6C22×C5semidirect
172. #627C22×C5semidirect
173. #626C22×C5semidirect
174. #626C22×C5semidirect
175. #625C22×C5semidirect
176. #624C22×C5semidirect
178. D3×Dic3C22×C5semidirect
227. #235C5×D3direct (N × H)
231. C2×C4×D3C5×D3semidirect
233. C32×C5C2×#41semidirect
256. C2×C4×C5D32semidirect
285. C5×D3#235direct (N × H)
300. C2×C3×C4C2×C5×D3semidirect
301. C2×C3×C4C2×C5×D3semidirect
309. #67C2×C5×D3semidirect
312. C2×Dic3C2×C5×D3semidirect
318. D12C2×C5×D3semidirect
328. C4×D3C2×C5×D3semidirect
329. C4×D3C2×C5×D3semidirect
330. C4×D3C2×C5×D3semidirect
336. Dic6C2×C5×D3semidirect
355. C3×D3C5×#41semidirect
368. C3×C5C2×#235semidirect
369. C3×C5D3×#41semidirect
407. C32C2×C5×#41semidirect
430. C2×C4C5×D32semidirect
445. D3C5×#235direct (N × H)
446. C5D3×#235direct (N × H)
461. C3C2×C5×#235semidirect
462. C3C5×D3×#41semidirect

Maximal subgroups

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

subgroupindex
C5×#38282
C3×C5×#2352
C2×C3×C4×C5×D32
C5×D3×#672
C5×#38042
C5×D3×D122
C4×C5×D322
C5×#37992
C5×#37982
C5×#37952
C5×D3×Dic62
C5×D3×#413
C2×C5×#2353
D3×#2355