Finite Group Databasebeta

← C5×#71708 #14282967

Subgroups — C5×#71708 #14282967

Order 1920.

668 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#525C211yesmaximal, characteristic
2C2×C5×C3⋊#95C211yesmaximal, characteristic, complemented
3C5×#5207C211yesmaximal, characteristic, complemented
4C5×#4538C212yesmaximal, complemented
5C5×#4537C212yesmaximal, complemented
6C5×#1504—31nomaximal, Hall
7C2×C3×C5×#41C411yescharacteristic, complemented
8C23×C3×C4×C5C2211yescharacteristic
9C23×C5×Dic3—21no—
10C3×C5×#123—21no—
11C3×C5×C22⋊Q8C411yescharacteristic, complemented
12C3×C5×C22⋊Q8C2211yescharacteristic
13C3×C5×#121C411yescharacteristic, complemented
14C3×C5×#121C2211yescharacteristic
15C3×C5×#120—21no—
16C3×C4×C5×D4—21no—
17C3×C4×C5×D4—21no—
18C3×C5×#117C411yescharacteristic, complemented
19C2×C5×#208—21no—
20C5×C3⋊#95—21no—
21C5×C3⋊#95C2212yes—
22C5×#830—21no—
23C5×#830C2212yes—
24#71708C511yesmaximal, characteristic, direct product factor, Hall
25C5×#525D311yescharacteristic
26C5×#407—31no—
27C2×C5×#95—31no—
28C5×#328—32no—
29C5×#327—32no—
30C24×C3×C5D411yescharacteristic
31C3×C5×#41—42no—
32C2×C3×C5×Q8C2×C411yescharacteristic, complemented
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4C2×C411yescharacteristic, complemented
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5—42no—
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×C4×C5C2×C411yescharacteristic
41C22×C3×C4×C5D411yescharacteristic
42C22×C3×C4×C5D411yescharacteristic
43C22×C3×C4×C5D411yescharacteristic
44C22×C3×C4×C5C2311yescharacteristic
45C22×C5×Dic3—24no—
46C22×C5×Dic3—22no—
47C22×C5×Dic3—42no—
48C22×C5×Dic3—42no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4C2×C411yescharacteristic, complemented
52C3×C5×C4⋊C4C2×C411yescharacteristic, complemented
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C42×C5—21no—
59C5×#208—42no—
60C2×C5×C3⋊C8—22no—
61C3×#525C2×C511yescharacteristic
62C2×C3⋊#95C2×C511yescharacteristic, complemented
63#5207C2×C511yescharacteristic, complemented
64#4538C2×C512yescomplemented
65#4537C2×C512yescomplemented
66C2×C5×#41Dic311yescharacteristic, complemented
67C23×C4×C5—61no—
68C23×C4×C5C2×D311yescharacteristic
69C2×C5×M16—61no—
70C5×#123—21no—
71C5×C22⋊Q8Dic311yescharacteristic, complemented
72C5×C22⋊Q8C2×D311yescharacteristic
73C5×#121Dic311yescharacteristic, complemented
74C5×#121C2×D311yescharacteristic
75C5×#120—21no—
76C4×C5×D4—21no—
77C4×C5×D4—21no—
78C5×#117Dic311yescharacteristic, complemented
79C5×C22⋊C8—32no—
80C5×C22⋊C8—61no—
81C5×#95—32no—
82C5×#95—61no—
83#1504—31noSylow
84C23×C3×C5—22no—
85C23×C3×C5—22no—
86C23×C3×C5—21no—
87C23×C3×C5—21no—
88C23×C3×C5—21no—
89C23×C3×C5C22⋊C411yescharacteristic
90C23×C3×C5C22⋊C411yescharacteristic
91C23×C3×C5C2×D411yescharacteristic
92C3×C5×Q8—41no—
93C3×C5×D4—41no—
94C3×C5×D4—42no—
95C3×C5×D4—42no—
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C2×C3×C4×C5—22no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—41no—
107C2×C3×C4×C5—42no—
108C2×C3×C4×C5C22⋊C411yescharacteristic
109C2×C3×C4×C5C22⋊C411yescharacteristic
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C5×Dic3—24no—
113C2×C5×Dic3—48no—
114C2×C5×Dic3—42no—
115C2×C5×Dic3—42no—
116C5×C3⋊C8—42no—
117C2×C3×#41C4×C511yescharacteristic, complemented
118C23×C3×C4C22×C511yescharacteristic
119C23×Dic3—21no—
120C3×#123—21no—
121C3×C22⋊Q8C4×C511yescharacteristic, complemented
122C3×C22⋊Q8C22×C511yescharacteristic
123C3×#121C4×C511yescharacteristic, complemented
124C3×#121C22×C511yescharacteristic
125C3×#120—21no—
126C3×C4×D4—21no—
127C3×C4×D4—21no—
128C3×#117C4×C511yescharacteristic, complemented
129C2×#208—21no—
130C3⋊#95—21no—
131C3⋊#95C22×C512yes—
132#830—21no—
133#830C22×C512yes—
134C24×C5#6711yescharacteristic
135C5×#41—42no—
136C2×C5×Q8C2×Dic311yescharacteristic, complemented
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4C2×Dic311yescharacteristic, complemented
141C22×C4×C5—21no—
142C22×C4×C5—42no—
143C22×C4×C5—64no—
144C22×C4×C5—62no—
145C22×C4×C5—122no—
146C22×C4×C5—122no—
147C22×C4×C5C2×Dic311yescharacteristic
148C22×C4×C5C2×Dic311yescharacteristic
149C22×C4×C5#6711yescharacteristic
150C22×C4×C5#6711yescharacteristic
151C22×C4×C5#6711yescharacteristic
152C22×C4×C5C22×D311yescharacteristic
153C5×M16—122no—
154C2×C5×C8—62no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4—21no—
157C5×C4⋊C4C2×Dic311yescharacteristic, complemented
158C5×C4⋊C4C2×Dic311yescharacteristic, complemented
159C5×C22⋊C4—21no—
160C5×C22⋊C4—21no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4—21no—
164C42×C5—21no—
165#525C5×D311yescharacteristic
166#407—31no—
167C2×#95—31no—
168#328—32no—
169#327—32no—
170C22×C3×C5—24no—
171C22×C3×C5—24no—
172C22×C3×C5—21no—
173C22×C3×C5—21no—
174C22×C3×C5—21no—
175C22×C3×C5—21no—
176C22×C3×C5—21no—
177C22×C3×C5—21no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5#9912yes—
182C22×C3×C5#10412yes—
183C22×C3×C5C2×C22⋊C411yescharacteristic
184C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
185C22×C3×C5#13711yescharacteristic
186C3×C4×C5—21no—
187C3×C4×C5—21no—
188C3×C4×C5—21no—
189C3×C4×C5—21no—
190C3×C4×C5—41no—
191C3×C4×C5—41no—
192C3×C4×C5—41no—
193C5×Dic3—44no—
194C24×C3C5×D411yescharacteristic
195C3×#41—42no—
196C2×C3×Q8C2×C4×C511yescharacteristic, complemented
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4C2×C4×C511yescharacteristic, complemented
201C22×C3×C4—21no—
202C22×C3×C4—42no—
203C22×C3×C4C2×C4×C511yescharacteristic
204C22×C3×C4C2×C4×C511yescharacteristic
205C22×C3×C4C5×D411yescharacteristic
206C22×C3×C4C5×D411yescharacteristic
207C22×C3×C4C5×D411yescharacteristic
208C22×C3×C4C23×C511yescharacteristic
209C22×Dic3—24no—
210C22×Dic3—22no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4C2×C4×C511yescharacteristic, complemented
216C3×C4⋊C4C2×C4×C511yescharacteristic, complemented
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—21no—
222C3×C42—21no—
223#208—42no—
224C2×C3⋊C8—22no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—21no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5#21711yescharacteristic
231C23×C5#21711yescharacteristic
232C23×C5C2×#6711yescharacteristic
233C5×Q8—41no—
234C5×D4—41no—
235C5×D4—42no—
236C5×D4—42no—
237C5×D4—41no—
238C5×D4—41no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—42no—
249C2×C4×C5—64no—
250C2×C4×C5—128no—
251C2×C4×C5—122no—
252C2×C4×C5—122no—
253C2×C4×C5#21711yescharacteristic
254C2×C4×C5#21711yescharacteristic
255C2×C4×C5C22×Dic311yescharacteristic
256C2×C4×C5C2×#6711yescharacteristic
257C5×C8—122no—
258C2×#41C5×Dic311yescharacteristic, complemented
259C23×C4—61no—
260C23×C4C2×C5×D311yescharacteristic
261C2×M16—61no—
262#123—21no—
263C22⋊Q8C5×Dic311yescharacteristic, complemented
264C22⋊Q8C2×C5×D311yescharacteristic
265#121C5×Dic311yescharacteristic, complemented
266#121C2×C5×D311yescharacteristic
267#120—21no—
268C4×D4—21no—
269C4×D4—21no—
270#117C5×Dic311yescharacteristic, complemented
271C22⋊C8—32no—
272C22⋊C8—61no—
273#95—32no—
274#95—61no—
275C2×C3×C5—21no—
276C2×C3×C5—22no—
277C2×C3×C5—22no—
278C2×C3×C5—21no—
279C2×C3×C5—81no—
280C2×C3×C5C2×#9911yescharacteristic
281C2×C3×C5#41711yescharacteristic
282C2×C3×C5C2×#10411yescharacteristic
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—21no—
288C23×C3C5×C22⋊C411yescharacteristic
289C23×C3C5×C22⋊C411yescharacteristic
290C23×C3C2×C5×D411yescharacteristic
291C3×Q8—41no—
292C3×D4—41no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—42no—
296C3×D4—42no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4C5×C22⋊C411yescharacteristic
308C2×C3×C4C5×C22⋊C411yescharacteristic
309C2×C3×C4C22×C4×C511yescharacteristic
310C2×C3×C4C2×C5×D411yescharacteristic
311C2×Dic3—24no—
312C2×Dic3—48no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C3⋊C8—42no—
316C22×C5—24no—
317C22×C5—24no—
318C22×C5—21no—
319C22×C5—21no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—41no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5#83412yes—
328C22×C5#83712yes—
329C22×C5#94911yescharacteristic
330C22×C5#95111yescharacteristic
331C22×C5C2×#21711yescharacteristic
332C4×C5—21no—
333C4×C5—21no—
334C4×C5—21no—
335C4×C5—21no—
336C4×C5—41no—
337C4×C5—41no—
338C4×C5—41no—
339C4×C5—124no—
340C24C5×#6711yescharacteristic
341#41—42no—
342C2×Q8C2×C5×Dic311yescharacteristic, complemented
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×C5×Dic311yescharacteristic, complemented
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—64no—
350C22×C4—62no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4C2×C5×Dic311yescharacteristic
354C22×C4C2×C5×Dic311yescharacteristic
355C22×C4C5×#6711yescharacteristic
356C22×C4C5×#6711yescharacteristic
357C22×C4C5×#6711yescharacteristic
358C22×C4C22×C5×D311yescharacteristic
359M16—122no—
360C2×C8—62no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4C2×C5×Dic311yescharacteristic, complemented
364C4⋊C4C2×C5×Dic311yescharacteristic, complemented
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C42—21no—
371C3×C5#150411yescharacteristic, complemented, Hall
372C22×C3—24no—
373C22×C3—24no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3C5×#9912yes—
384C22×C3C5×#10412yes—
385C22×C3C2×C5×C22⋊C411yescharacteristic
386C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
387C22×C3C5×#13711yescharacteristic
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—41no—
395Dic3—44no—
396C2×C5—21no—
397C2×C5—22no—
398C2×C5—22no—
399C2×C5—21no—
400C2×C5—81no—
401C2×C5C2×#83411yescharacteristic
402C2×C5#523211yescharacteristic
403C2×C5C2×#83711yescharacteristic
404C23—21no—
405C23—21no—
406C23—22no—
407C23—22no—
408C23—21no—
409C23C5×#21711yescharacteristic
410C23C5×#21711yescharacteristic
411C23C2×C5×#6711yescharacteristic
412Q8—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416D4—42no—
417D4—42no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—64no—
429C2×C4—128no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4C5×#21711yescharacteristic
433C2×C4C5×#21711yescharacteristic
434C2×C4C22×C5×Dic311yescharacteristic
435C2×C4C2×C5×#6711yescharacteristic
436C8—122no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—22no—
440C2×C3—22no—
441C2×C3—81no—
442C2×C3C2×C5×#9911yescharacteristic
443C2×C3C5×#41711yescharacteristic
444C2×C3C2×C5×#10411yescharacteristic
445C5#7170811yescharacteristic, direct product factor, Sylow
446C22—24no—
447C22—24no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22C5×#83412yes—
458C22C5×#83712yes—
459C22C5×#94911yescharacteristic
460C22C5×#95111yescharacteristic
461C22C2×C5×#21711yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—124no—
470C3C5×#150411yescharacteristic, complemented, Sylow
471C2—21no—
472C2—21no—
473C2—22no—
474C2—22no—
475C2—81no—
476C2C2×C5×#83411yescharacteristic
477C2C5×#523211yescharacteristic
478C2C2×C5×#83711yescharacteristic
subgroupG/H
centerC22×C5C2×#217
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#525C2
socleC22×C3×C5C2×C22⋊C4
cosocleC22×C3×C4C23×C5
hypercenterC5×#525D3
hypocenterC3C5×#1504
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C5×#525

Complements

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

NH ≅ G/Ntype
2. C2×C5×C3⋊#95C2semidirect
3. C5×#5207C2semidirect
4. C5×#4538C2semidirect
5. C5×#4537C2semidirect
7. C2×C3×C5×#41C4semidirect
11. C3×C5×C22⋊Q8C4semidirect
13. C3×C5×#121C4semidirect
18. C3×C5×#117C4semidirect
24. #71708C5direct (N × H)
32. C2×C3×C5×Q8C2×C4semidirect
36. C2×C3×C5×D4C2×C4semidirect
51. C3×C5×C4⋊C4C2×C4semidirect
52. C3×C5×C4⋊C4C2×C4semidirect
62. C2×C3⋊#95C2×C5semidirect
63. #5207C2×C5semidirect
64. #4538C2×C5semidirect
65. #4537C2×C5semidirect
66. C2×C5×#41Dic3semidirect
71. C5×C22⋊Q8Dic3semidirect
73. C5×#121Dic3semidirect
78. C5×#117Dic3semidirect
117. C2×C3×#41C4×C5semidirect
121. C3×C22⋊Q8C4×C5semidirect
123. C3×#121C4×C5semidirect
128. C3×#117C4×C5semidirect
136. C2×C5×Q8C2×Dic3semidirect
140. C2×C5×D4C2×Dic3semidirect
157. C5×C4⋊C4C2×Dic3semidirect
158. C5×C4⋊C4C2×Dic3semidirect
196. C2×C3×Q8C2×C4×C5semidirect
200. C2×C3×D4C2×C4×C5semidirect
215. C3×C4⋊C4C2×C4×C5semidirect
216. C3×C4⋊C4C2×C4×C5semidirect
258. C2×#41C5×Dic3semidirect
263. C22⋊Q8C5×Dic3semidirect
265. #121C5×Dic3semidirect
270. #117C5×Dic3semidirect
342. C2×Q8C2×C5×Dic3semidirect
346. C2×D4C2×C5×Dic3semidirect
363. C4⋊C4C2×C5×Dic3semidirect
364. C4⋊C4C2×C5×Dic3semidirect
371. C3×C5#1504semidirect
445. C5#71708direct (N × H)
470. C3C5×#1504semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5252
C2×C5×C3⋊#952
C5×#52072
C5×#45382
C5×#45372
C5×#15043
#717085