Finite Group Databasebeta

← C5×#71884 #14287903

Subgroups — C5×#71884 #14287903

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#538C211yesmaximal, characteristic, complemented
2C5×#5046C211yesmaximal, characteristic, complemented
3C5×#5036C211yesmaximal, characteristic, complemented
4C5×#4960C211yesmaximal, characteristic, complemented
5C5×#4538C211yesmaximal, characteristic, complemented
6C5×#4537C211yesmaximal, characteristic, complemented
7C5×#4525C211yesmaximal, characteristic, complemented
8C5×#1600—31nomaximal, Hall
9C22×C5×#67—21no—
10C22×C5×D12C2211yescharacteristic
11C3×C5×#126—21no—
12C3×C5×#124—21no—
13C3×C5×#123—21no—
14C3×C5×C22⋊Q8C2211yescharacteristic
15C3×C5×#121—21no—
16C3×C5×#121C2211yescharacteristic
17C3×C5×#120—21no—
18C3×C4×C5×D4—21no—
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×C5×#215—21no—
21C2×C5×#213—21no—
22C2×C5×(C2×D3)⋊C4—21no—
23C5×C3⋊#95C2211yescharacteristic
24C5×#830C2211yescharacteristic
25C5×#830C2211yescharacteristic
26C5×#809—21no—
27C5×#809—21no—
28#71884C511yesmaximal, characteristic, direct product factor, Hall
29C5×#538D311yescharacteristic, complemented
30C5×#450—31no—
31C5×#447—31no—
32C5×#392—31no—
33C5×#328—31no—
34C5×#327—31no—
35C5×#323—31no—
36C24×C3×C5D411yescharacteristic
37C23×C5×D3—21no—
38C2×C3×C5×Q8D411yescharacteristic, complemented
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4D411yescharacteristic, complemented
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5D411yescharacteristic
45C22×C3×C4×C5C2311yescharacteristic
46C2×C5×#67—42no—
47C2×C5×#67—42no—
48C2×C5×#67—42no—
49C22×C5×Dic3—21no—
50C2×C5×D12—21no—
51C2×C5×D12—21no—
52C2×C5×D12—42no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4D411yescharacteristic, complemented
56C3×C5×C4⋊C4D411yescharacteristic, complemented
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—21no—
63C3×C5×C22⋊C4—21no—
64C3×C42×C5—21no—
65C5×#215—42no—
66C5×#213—42no—
67C5×(C2×D3)⋊C4—42no—
68C2×C5×C3⋊C8—21no—
69C2×C5×C3⋊C8—21no—
70C3×#538C2×C511yescharacteristic, complemented
71#5046C2×C511yescharacteristic, complemented
72#5036C2×C511yescharacteristic, complemented
73#4960C2×C511yescharacteristic, complemented
74#4538C2×C511yescharacteristic, complemented
75#4537C2×C511yescharacteristic, complemented
76#4525C2×C511yescharacteristic, complemented
77C22×C5×D4—31no—
78C22×C5×D4—61no—
79C2×C5×SD16—61no—
80C2×C5×D8—61no—
81C5×#126—21no—
82C5×#124—21no—
83C5×#123—21no—
84C5×C22⋊Q8C2×D311yescharacteristic
85C5×#121—21no—
86C5×#121C2×D311yescharacteristic
87C5×#120—21no—
88C4×C5×D4—21no—
89C2×C5×C22⋊C4—61no—
90C2×C5×C22⋊C4C2×D311yescharacteristic
91C5×#102—61no—
92C5×#102—61no—
93C5×C22⋊C8—31no—
94C5×C22⋊C8—31no—
95C5×#95—31no—
96#1600—31noSylow
97C23×C3×C5—21no—
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5—21no—
101C23×C3×C5—41no—
102C23×C3×C5—41no—
103C23×C3×C5C2×D411yescharacteristic
104C22×C5×D3—22no—
105C22×C5×D3—41no—
106C22×C5×D3—41no—
107C22×C5×D3—41no—
108C22×C5×D3—41no—
109C22×C5×D3—41no—
110C22×C5×D3—41no—
111C3×C5×Q8—41no—
112C3×C5×D4—41no—
113C3×C5×D4—41no—
114C3×C5×D4—41no—
115C3×C5×D4—81no—
116C3×C5×D4—81no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5—41no—
123C2×C3×C4×C5—81no—
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126C5×#67—48no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3—41no—
129C2×C5×Dic3—41no—
130C5×D12—22no—
131C5×D12—22no—
132C5×D12—41no—
133C5×D12—41no—
134C5×C3⋊C8—41no—
135C5×C3⋊C8—41no—
136C22×#67—21no—
137C22×D12C22×C511yescharacteristic
138C3×#126—21no—
139C3×#124—21no—
140C3×#123—21no—
141C3×C22⋊Q8C22×C511yescharacteristic
142C3×#121—21no—
143C3×#121C22×C511yescharacteristic
144C3×#120—21no—
145C3×C4×D4—21no—
146C2×C3×C22⋊C4C22×C511yescharacteristic
147C2×#215—21no—
148C2×#213—21no—
149C2×(C2×D3)⋊C4—21no—
150C3⋊#95C22×C511yescharacteristic
151#830C22×C511yescharacteristic
152#830C22×C511yescharacteristic
153#809—21no—
154#809—21no—
155C24×C5—61no—
156C24×C5#6711yescharacteristic
157C2×C5×Q8#6711yescharacteristic, complemented
158C2×C5×D4—21no—
159C2×C5×D4—21no—
160C2×C5×D4—21no—
161C2×C5×D4—61no—
162C2×C5×D4—61no—
163C2×C5×D4—122no—
164C2×C5×D4—122no—
165C2×C5×D4—122no—
166C2×C5×D4—122no—
167C2×C5×D4#6711yescharacteristic, complemented
168C22×C4×C5—21no—
169C22×C4×C5—61no—
170C22×C4×C5#6711yescharacteristic
171C22×C4×C5C22×D311yescharacteristic
172C5×SD16—122no—
173C5×D8—122no—
174C2×C5×C8—61no—
175C2×C5×C8—61no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4#6711yescharacteristic, complemented
179C5×C4⋊C4#6711yescharacteristic, complemented
180C5×C22⋊C4—21no—
181C5×C22⋊C4—21no—
182C5×C22⋊C4—21no—
183C5×C22⋊C4—21no—
184C5×C22⋊C4—21no—
185C5×C22⋊C4—21no—
186C5×C22⋊C4—21no—
187C5×C22⋊C4—122no—
188C42×C5—21no—
189#538C5×D311yescharacteristic, complemented
190#450—31no—
191#447—31no—
192#392—31no—
193#328—31no—
194#327—31no—
195#323—31no—
196C22×C3×C5—21no—
197C22×C3×C5—21no—
198C22×C3×C5—42no—
199C22×C3×C5—41no—
200C22×C3×C5—41no—
201C22×C3×C5—41no—
202C22×C3×C5—41no—
203C22×C3×C5—41no—
204C22×C3×C5—41no—
205C22×C3×C5—81no—
206C22×C3×C5#12011yescharacteristic
207C22×C3×C5#13511yescharacteristic
208C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
209C2×C5×D3—42no—
210C2×C5×D3—42no—
211C2×C5×D3—42no—
212C2×C5×D3—81no—
213C2×C5×D3—81no—
214C2×C5×D3—81no—
215C2×C5×D3—81no—
216C3×C4×C5—21no—
217C3×C4×C5—21no—
218C3×C4×C5—41no—
219C3×C4×C5—41no—
220C3×C4×C5—41no—
221C3×C4×C5—41no—
222C5×Dic3—42no—
223C24×C3C5×D411yescharacteristic
224C23×D3—21no—
225C2×C3×Q8C5×D411yescharacteristic, complemented
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4C5×D411yescharacteristic, complemented
230C22×C3×C4—21no—
231C22×C3×C4C5×D411yescharacteristic
232C22×C3×C4C23×C511yescharacteristic
233C2×#67—42no—
234C2×#67—42no—
235C2×#67—42no—
236C22×Dic3—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—42no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C5×D411yescharacteristic, complemented
243C3×C4⋊C4C5×D411yescharacteristic, complemented
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C42—21no—
252#215—42no—
253#213—42no—
254(C2×D3)⋊C4—42no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5—41no—
262C23×C5—41no—
263C23×C5—62no—
264C23×C5—121no—
265C23×C5—121no—
266C23×C5—121no—
267C23×C5—121no—
268C23×C5—121no—
269C23×C5—121no—
270C23×C5C2×#6711yescharacteristic
271C5×Q8—41no—
272C5×D4—41no—
273C5×D4—41no—
274C5×D4—41no—
275C5×D4—62no—
276C5×D4—62no—
277C5×D4—81no—
278C5×D4—81no—
279C5×D4—121no—
280C5×D4—121no—
281C5×D4—128no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—62no—
289C2×C4×C5—81no—
290C2×C4×C5—121no—
291C2×C4×C5—121no—
292C2×C4×C5C2×#6711yescharacteristic
293C2×C4×C5C2×#6711yescharacteristic
294C5×C8—121no—
295C5×C8—121no—
296C22×D4—31no—
297C22×D4—61no—
298C2×SD16—61no—
299C2×D8—61no—
300#126—21no—
301#124—21no—
302#123—21no—
303C22⋊Q8C2×C5×D311yescharacteristic
304#121—21no—
305#121C2×C5×D311yescharacteristic
306#120—21no—
307C4×D4—21no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4C2×C5×D311yescharacteristic
310#102—61no—
311#102—61no—
312C22⋊C8—31no—
313C22⋊C8—31no—
314#95—31no—
315C2×C3×C5—21no—
316C2×C3×C5—21no—
317C2×C3×C5—81no—
318C2×C3×C5—81no—
319C2×C3×C5#44911yescharacteristic
320C2×C3×C5#45311yescharacteristic
321C2×C3×C5B(4,𝔽2)11yescharacteristic
322C5×D3—82no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C2×C5×D411yescharacteristic
330C22×D3—22no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C3×Q8—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—81no—
342C3×D4—81no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—81no—
350C2×C3×C4C2×C5×D411yescharacteristic
351C2×C3×C4C2×C5×D411yescharacteristic
352#67—48no—
353C2×Dic3—22no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356D12—22no—
357D12—22no—
358D12—41no—
359D12—41no—
360C3⋊C8—41no—
361C3⋊C8—41no—
362C22×C5—21no—
363C22×C5—21no—
364C22×C5—42no—
365C22×C5—41no—
366C22×C5—41no—
367C22×C5—41no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—41no—
371C22×C5—81no—
372C22×C5—122no—
373C22×C5—122no—
374C22×C5—122no—
375C22×C5—241no—
376C22×C5—241no—
377C22×C5—241no—
378C22×C5—241no—
379C22×C5#94911yescharacteristic
380C22×C5#95011yescharacteristic
381C22×C5#95311yescharacteristic
382C4×C5—21no—
383C4×C5—21no—
384C4×C5—41no—
385C4×C5—41no—
386C4×C5—41no—
387C4×C5—41no—
388C4×C5—122no—
389C24—61no—
390C24C5×#6711yescharacteristic
391C2×Q8C5×#6711yescharacteristic, complemented
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—122no—
398C2×D4—122no—
399C2×D4—122no—
400C2×D4—122no—
401C2×D4C5×#6711yescharacteristic, complemented
402C22×C4—21no—
403C22×C4—61no—
404C22×C4C5×#6711yescharacteristic
405C22×C4C22×C5×D311yescharacteristic
406SD16—122no—
407D8—122no—
408C2×C8—61no—
409C2×C8—61no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4C5×#6711yescharacteristic, complemented
413C4⋊C4C5×#6711yescharacteristic, complemented
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—122no—
422C42—21no—
423C3×C5#160011yescharacteristic, complemented, Hall
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—42no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—81no—
434C22×C3C5×#12011yescharacteristic
435C22×C3C5×#13511yescharacteristic
436C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—42no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450Dic3—42no—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—81no—
454C2×C5—81no—
455C2×C5—242no—
456C2×C5#523811yescharacteristic
457C2×C5#524111yescharacteristic
458C2×C5#524411yescharacteristic
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—41no—
464C23—41no—
465C23—62no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23—121no—
470C23—121no—
471C23—121no—
472C23C2×C5×#6711yescharacteristic
473Q8—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—62no—
478D4—62no—
479D4—81no—
480D4—81no—
481D4—121no—
482D4—121no—
483D4—128no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—62no—
491C2×C4—81no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4C2×C5×#6711yescharacteristic
495C2×C4C2×C5×#6711yescharacteristic
496C8—121no—
497C8—121no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—81no—
501C2×C3—81no—
502C2×C3C5×#44911yescharacteristic
503C2×C3C5×#45311yescharacteristic
504C2×C3C5×B(4,𝔽2)11yescharacteristic
505D3—82no—
506C5#7188411yescharacteristic, direct product factor, Sylow
507C22—21no—
508C22—21no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—81no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22—241no—
521C22—241no—
522C22—241no—
523C22—241no—
524C22C5×#94911yescharacteristic
525C22C5×#95011yescharacteristic
526C22C5×#95311yescharacteristic
527C4—21no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—122no—
534C3C5×#160011yescharacteristic, complemented, Sylow
535C2—21no—
536C2—21no—
537C2—81no—
538C2—81no—
539C2—242no—
540C2C5×#523811yescharacteristic
541C2C5×#524111yescharacteristic
542C2C5×#524411yescharacteristic
subgroupG/H
centerC22×C5#953
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#538C2
socleC22×C3×C5#120
hypercenterC5×#538D3
hypocenterC3C5×#1600
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#538

Complements

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

NH ≅ G/Ntype
1. C3×C5×#538C2semidirect
2. C5×#5046C2semidirect
3. C5×#5036C2semidirect
4. C5×#4960C2semidirect
5. C5×#4538C2semidirect
6. C5×#4537C2semidirect
7. C5×#4525C2semidirect
28. #71884C5direct (N × H)
29. C5×#538D3semidirect
38. C2×C3×C5×Q8D4semidirect
42. C2×C3×C5×D4D4semidirect
55. C3×C5×C4⋊C4D4semidirect
56. C3×C5×C4⋊C4D4semidirect
70. C3×#538C2×C5semidirect
71. #5046C2×C5semidirect
72. #5036C2×C5semidirect
73. #4960C2×C5semidirect
74. #4538C2×C5semidirect
75. #4537C2×C5semidirect
76. #4525C2×C5semidirect
157. C2×C5×Q8#67semidirect
167. C2×C5×D4#67semidirect
178. C5×C4⋊C4#67semidirect
179. C5×C4⋊C4#67semidirect
189. #538C5×D3semidirect
225. C2×C3×Q8C5×D4semidirect
229. C2×C3×D4C5×D4semidirect
242. C3×C4⋊C4C5×D4semidirect
243. C3×C4⋊C4C5×D4semidirect
391. C2×Q8C5×#67semidirect
401. C2×D4C5×#67semidirect
412. C4⋊C4C5×#67semidirect
413. C4⋊C4C5×#67semidirect
423. C3×C5#1600semidirect
506. C5#71884direct (N × H)
534. C3C5×#1600semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5382
C5×#50462
C5×#50362
C5×#49602
C5×#45382
C5×#45372
C5×#45252
C5×#16003
#718845