Finite Group Databasebeta

← C5×#86286 #14495600

Subgroups — C5×#86286 #14495600

Order 1920.

1860 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#533C211yesmaximal, characteristic, complemented
2C5×#5735C214yesmaximal, complemented
3C5×#5700C211yesmaximal, characteristic, complemented
4C5×#5699C212yesmaximal, complemented
5C5×C3⋊#564C211yesmaximal, characteristic, complemented
6C5×#5647C214yesmaximal, complemented
7C5×#5646C214yesmaximal, complemented
8C5×#5580C214yesmaximal, complemented
9C5×#5578C214yesmaximal, complemented
10C5×#5528C214yesmaximal, complemented
11C5×#5479C212yesmaximal, complemented
12C5×#3531—31nomaximal, Hall
13C3×C5×C4⋊Q8C2212yes—
14C3×C5×#125C2211yescharacteristic
15C3×C5×#125C2211yescharacteristic
16C3×C5×C22⋊Q8C2214yes—
17C3×C4×C5×Q8C2214yes—
18C3×C5×#117C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20C5×#947C2214yes—
21C5×#946C2214yes—
22C5×#946C2214yes—
23C5×#944C2214yes—
24C5×#930C2212yes—
25C5×#929C2214yes—
26C4×C5×#67C2214yes—
27C5×#924C2212yes—
28C5×#898C2214yes—
29C5×#898C2214yes—
30C5×#898C2214yes—
31C5×#897C2214yes—
32C5×#896C2214yes—
33C5×#896C2214yes—
34C5×#896C2214yes—
35C5×#895C2214yes—
36C5×#894C2214yes—
37C5×#894C2214yes—
38C5×#894C2214yes—
39C5×#890C2214yes—
40C5×#889C2214yes—
41C5×#889C2214yes—
42C5×#889C2214yes—
43C5×#888C2214yes—
44C5×#885C2214yes—
45C5×#884C2214yes—
46C5×#883C2214yes—
47C5×#879C2214yes—
48C5×#876C2214yes—
49C5×#875C2212yes—
50C4×C5×D12C2212yes—
51C4×C5×D12C2214yes—
52C5×#872C2214yes—
53C5×#872C2214yes—
54C42×C5×D3C2214yes—
55C5×#870C2212yes—
56C4×C5×Dic6C2212yes—
57C4×C5×Dic6C2214yes—
58#86286C511yesmaximal, characteristic, direct product factor, Hall
59C5×#564—31no—
60C5×#563—34no—
61C5×#563—32no—
62C5×#563—34no—
63C5×#562—31no—
64C5×#562—34no—
65C5×#533—34no—
66C5×#533D311yescharacteristic, complemented
67C5×#529—34no—
68C5×#529—34no—
69C5×#529—32no—
70C2×C3×C5×Q8C2314yes—
71C22×C3×C4×C5C2311yescharacteristic
72C22×C3×C4×C5C2312yes—
73C2×C5×#67C2314yes—
74C2×C5×D12C2312yes—
75C2×C5×D12C2314yes—
76C2×C4×C5×D3C2314yes—
77C2×C4×C5×D3C2314yes—
78C2×C4×C5×D3C2314yes—
79C2×C5×Dic6C2312yes—
80C2×C5×Dic6C2314yes—
81C3×C5×C4⋊C4C2314yes—
82C3×C5×C4⋊C4C2312yes—
83C3×C5×C4⋊C4C2314yes—
84C3×C5×C4⋊C4C2314yes—
85C3×C5×C4⋊C4C2312yes—
86C3×C5×C22⋊C4C2314yes—
87C3×C42×C5C2314yes—
88C3×C42×C5C2312yes—
89C3×C42×C5C2311yescharacteristic
90C3×C42×C5C2311yescharacteristic
91C5×#217C2314yes—
92C5×(C2×D3)⋊C4C2314yes—
93C5×(C2×D3)⋊C4C2314yes—
94C5×(C2×D3)⋊C4C2314yes—
95C5×(C2×D3)⋊C4C2314yes—
96C5×(C2×D3)⋊C4C2314yes—
97C5×(C2×D3)⋊C4C2314yes—
98C5×(C2×D3)⋊C4C2314yes—
99C5×(C2×D3)⋊C4C2314yes—
100C5×C4⋊Dic3C2312yes—
101C5×C4⋊Dic3C2314yes—
102C5×C4⋊Dic3C2312yes—
103C5×C4⋊Dic3C2314yes—
104C5×Dic3⋊C4C2314yes—
105C5×Dic3⋊C4C2314yes—
106C5×Dic3⋊C4C2314yes—
107C5×Dic3⋊C4C2314yes—
108C5×Dic3⋊C4C2314yes—
109C5×Dic3⋊C4C2314yes—
110C5×Dic3⋊C4C2314yes—
111C5×Dic3⋊C4C2314yes—
112C4×C5×Dic3C2314yes—
113C4×C5×Dic3C2314yes—
114C4×C5×Dic3C2314yes—
115C3×#533C2×C511yescharacteristic, complemented
116#5735C2×C514yescomplemented
117#5700C2×C511yescharacteristic, complemented
118#5699C2×C512yescomplemented
119C3⋊#564C2×C511yescharacteristic, complemented
120#5647C2×C514yescomplemented
121#5646C2×C514yescomplemented
122#5580C2×C514yescomplemented
123#5578C2×C514yescomplemented
124#5528C2×C514yescomplemented
125#5479C2×C512yescomplemented
126C5×C4⋊Q8—34no—
127C5×C4⋊Q8—34no—
128C5×C4⋊Q8C2×D312yes—
129C5×#126—34no—
130C5×#126—34no—
131C5×#126—34no—
132C5×#126—34no—
133C5×#126—34no—
134C5×#125—34no—
135C5×#125—34no—
136C5×#125—34no—
137C5×#125—34no—
138C5×#125—32no—
139C5×#125C2×D311yescharacteristic
140C5×#125C2×D311yescharacteristic
141C5×#124—34no—
142C5×#124—34no—
143C5×#124—32no—
144C5×#123—34no—
145C5×#123—34no—
146C5×#123—34no—
147C5×#123—34no—
148C5×#123—34no—
149C5×C22⋊Q8—34no—
150C5×C22⋊Q8—34no—
151C5×C22⋊Q8—32no—
152C5×C22⋊Q8—34no—
153C5×C22⋊Q8—34no—
154C5×C22⋊Q8—34no—
155C5×C22⋊Q8—34no—
156C5×C22⋊Q8C2×D314yes—
157C5×#121—32no—
158C5×#121—34no—
159C5×#121—34no—
160C4×C5×Q8—34no—
161C4×C5×Q8—32no—
162C4×C5×Q8C2×D314yes—
163C4×C5×D4—34no—
164C4×C5×D4—34no—
165C4×C5×D4—32no—
166C5×#117—34no—
167C5×#117—34no—
168C5×#117C2×D312yes—
169C2×C42×C5—34no—
170C2×C42×C5C2×D311yescharacteristic
171#3531—31noSylow
172C23×C3×C5C2411yescharacteristic
173C22×C5×D3C2414yes—
174C3×C5×Q8—44no—
175C2×C3×C4×C5—22no—
176C2×C3×C4×C5—42no—
177C2×C3×C4×C5C2414yes—
178C2×C3×C4×C5C2414yes—
179C2×C3×C4×C5C2411yescharacteristic
180C2×C3×C4×C5C2412yes—
181C2×C3×C4×C5C2412yes—
182C2×C3×C4×C5C2411yescharacteristic
183C5×#67—44no—
184C2×C5×Dic3C2414yes—
185C2×C5×Dic3C2414yes—
186C2×C5×Dic3C2414yes—
187C5×D12—42no—
188C5×D12—44no—
189C4×C5×D3—28no—
190C4×C5×D3—44no—
191C4×C5×D3—44no—
192C5×Dic6—42no—
193C5×Dic6—44no—
194C3×C4⋊Q8C22×C512yes—
195C3×#125C22×C511yescharacteristic
196C3×#125C22×C511yescharacteristic
197C3×C22⋊Q8C22×C514yes—
198C3×C4×Q8C22×C514yes—
199C3×#117C22×C512yes—
200C2×C3×C42C22×C511yescharacteristic
201#947C22×C514yes—
202#946C22×C514yes—
203#946C22×C514yes—
204#944C22×C514yes—
205#930C22×C512yes—
206#929C22×C514yes—
207C4×#67C22×C514yes—
208#924C22×C512yes—
209#898C22×C514yes—
210#898C22×C514yes—
211#898C22×C514yes—
212#897C22×C514yes—
213#896C22×C514yes—
214#896C22×C514yes—
215#896C22×C514yes—
216#895C22×C514yes—
217#894C22×C514yes—
218#894C22×C514yes—
219#894C22×C514yes—
220#890C22×C514yes—
221#889C22×C514yes—
222#889C22×C514yes—
223#889C22×C514yes—
224#888C22×C514yes—
225#885C22×C514yes—
226#884C22×C514yes—
227#883C22×C514yes—
228#879C22×C514yes—
229#876C22×C514yes—
230#875C22×C512yes—
231C4×D12C22×C512yes—
232C4×D12C22×C514yes—
233#872C22×C514yes—
234#872C22×C514yes—
235C42×D3C22×C514yes—
236#870C22×C512yes—
237C4×Dic6C22×C512yes—
238C4×Dic6C22×C514yes—
239C2×C5×Q8—32no—
240C2×C5×Q8—34no—
241C2×C5×Q8C22×D314yes—
242C2×C5×D4—34no—
243C2×C5×D4—34no—
244C2×C5×D4—32no—
245C22×C4×C5—34no—
246C22×C4×C5—34no—
247C22×C4×C5—34no—
248C22×C4×C5C22×D311yescharacteristic
249C22×C4×C5C22×D312yes—
250C5×C4⋊C4—34no—
251C5×C4⋊C4—34no—
252C5×C4⋊C4—34no—
253C5×C4⋊C4—34no—
254C5×C4⋊C4—34no—
255C5×C4⋊C4—34no—
256C5×C4⋊C4—32no—
257C5×C4⋊C4—34no—
258C5×C4⋊C4—32no—
259C5×C4⋊C4—34no—
260C5×C4⋊C4—34no—
261C5×C4⋊C4—34no—
262C5×C4⋊C4C22×D312yes—
263C5×C4⋊C4C22×D312yes—
264C5×C4⋊C4C22×D314yes—
265C5×C4⋊C4C22×D314yes—
266C5×C4⋊C4C22×D314yes—
267C5×C22⋊C4—34no—
268C5×C22⋊C4—34no—
269C5×C22⋊C4—34no—
270C5×C22⋊C4—34no—
271C5×C22⋊C4—34no—
272C5×C22⋊C4—34no—
273C5×C22⋊C4—34no—
274C5×C22⋊C4—34no—
275C5×C22⋊C4—34no—
276C5×C22⋊C4C22×D314yes—
277C42×C5—34no—
278C42×C5—34no—
279C42×C5—34no—
280C42×C5C22×D311yescharacteristic
281C42×C5C22×D311yescharacteristic
282C42×C5C22×D314yes—
283C42×C5C22×D312yes—
284#564—31no—
285#563—34no—
286#563—32no—
287#563—34no—
288#562—31no—
289#562—34no—
290#533—34no—
291#533C5×D311yescharacteristic, complemented
292#529—34no—
293#529—34no—
294#529—32no—
295C22×C3×C5—21no—
296C22×C3×C5—22no—
297C22×C3×C5C2511yescharacteristic
298C2×C5×D3—24no—
299C2×C5×D3—24no—
300C2×C5×D3—24no—
301C3×C4×C5—21no—
302C3×C4×C5—24no—
303C3×C4×C5—22no—
304C3×C4×C5—24no—
305C3×C4×C5—22no—
306C3×C4×C52−1+412yes—
307C5×Dic3—24no—
308C5×Dic3—24no—
309C5×Dic3—24no—
310C2×C3×Q8C23×C514yes—
311C22×C3×C4C23×C511yescharacteristic
312C22×C3×C4C23×C512yes—
313C2×#67C23×C514yes—
314C2×D12C23×C512yes—
315C2×D12C23×C514yes—
316C2×C4×D3C23×C514yes—
317C2×C4×D3C23×C514yes—
318C2×C4×D3C23×C514yes—
319C2×Dic6C23×C512yes—
320C2×Dic6C23×C514yes—
321C3×C4⋊C4C23×C514yes—
322C3×C4⋊C4C23×C512yes—
323C3×C4⋊C4C23×C514yes—
324C3×C4⋊C4C23×C514yes—
325C3×C4⋊C4C23×C512yes—
326C3×C22⋊C4C23×C514yes—
327C3×C42C23×C514yes—
328C3×C42C23×C512yes—
329C3×C42C23×C511yescharacteristic
330C3×C42C23×C511yescharacteristic
331#217C23×C514yes—
332(C2×D3)⋊C4C23×C514yes—
333(C2×D3)⋊C4C23×C514yes—
334(C2×D3)⋊C4C23×C514yes—
335(C2×D3)⋊C4C23×C514yes—
336(C2×D3)⋊C4C23×C514yes—
337(C2×D3)⋊C4C23×C514yes—
338(C2×D3)⋊C4C23×C514yes—
339(C2×D3)⋊C4C23×C514yes—
340C4⋊Dic3C23×C512yes—
341C4⋊Dic3C23×C514yes—
342C4⋊Dic3C23×C512yes—
343C4⋊Dic3C23×C514yes—
344Dic3⋊C4C23×C514yes—
345Dic3⋊C4C23×C514yes—
346Dic3⋊C4C23×C514yes—
347Dic3⋊C4C23×C514yes—
348Dic3⋊C4C23×C514yes—
349Dic3⋊C4C23×C514yes—
350Dic3⋊C4C23×C514yes—
351Dic3⋊C4C23×C514yes—
352C4×Dic3C23×C514yes—
353C4×Dic3C23×C514yes—
354C4×Dic3C23×C514yes—
355C23×C5—34no—
356C23×C5C23×D311yescharacteristic
357C5×Q8—44no—
358C5×Q8—122no—
359C5×Q8—124no—
360C5×D4—122no—
361C5×D4—124no—
362C5×D4—124no—
363C2×C4×C5—22no—
364C2×C4×C5—34no—
365C2×C4×C5—34no—
366C2×C4×C5—34no—
367C2×C4×C5—42no—
368C2×C4×C5—68no—
369C2×C4×C5—124no—
370C2×C4×C5—124no—
371C2×C4×C5C23×D314yes—
372C2×C4×C5C23×D314yes—
373C2×C4×C5C23×D311yescharacteristic
374C2×C4×C5C23×D312yes—
375C2×C4×C5C23×D312yes—
376C2×C4×C5C23×D311yescharacteristic
377C4⋊Q8—34no—
378C4⋊Q8—34no—
379C4⋊Q8C2×C5×D312yes—
380#126—34no—
381#126—34no—
382#126—34no—
383#126—34no—
384#126—34no—
385#125—34no—
386#125—34no—
387#125—34no—
388#125—34no—
389#125—32no—
390#125C2×C5×D311yescharacteristic
391#125C2×C5×D311yescharacteristic
392#124—34no—
393#124—34no—
394#124—32no—
395#123—34no—
396#123—34no—
397#123—34no—
398#123—34no—
399#123—34no—
400C22⋊Q8—34no—
401C22⋊Q8—34no—
402C22⋊Q8—32no—
403C22⋊Q8—34no—
404C22⋊Q8—34no—
405C22⋊Q8—34no—
406C22⋊Q8—34no—
407C22⋊Q8C2×C5×D314yes—
408#121—32no—
409#121—34no—
410#121—34no—
411C4×Q8—34no—
412C4×Q8—32no—
413C4×Q8C2×C5×D314yes—
414C4×D4—34no—
415C4×D4—34no—
416C4×D4—32no—
417#117—34no—
418#117—34no—
419#117C2×C5×D312yes—
420C2×C42—34no—
421C2×C42C2×C5×D311yescharacteristic
422C2×C3×C5—41no—
423C2×C3×C5C2×2−1+411yescharacteristic
424C2×C3×C5#58512yes—
425C5×D3—44no—
426C23×C3C24×C511yescharacteristic
427C22×D3C24×C514yes—
428C3×Q8—44no—
429C2×C3×C4—22no—
430C2×C3×C4—42no—
431C2×C3×C4C24×C514yes—
432C2×C3×C4C24×C514yes—
433C2×C3×C4C24×C511yescharacteristic
434C2×C3×C4C24×C512yes—
435C2×C3×C4C24×C512yes—
436C2×C3×C4C24×C511yescharacteristic
437#67—44no—
438C2×Dic3C24×C514yes—
439C2×Dic3C24×C514yes—
440C2×Dic3C24×C514yes—
441D12—42no—
442D12—44no—
443C4×D3—28no—
444C4×D3—44no—
445C4×D3—44no—
446Dic6—42no—
447Dic6—44no—
448C22×C5—21no—
449C22×C5—22no—
450C22×C5—64no—
451C22×C5—64no—
452C22×C5—64no—
453C22×C5C24×D311yescharacteristic
454C4×C5—21no—
455C4×C5—22no—
456C4×C5—22no—
457C4×C5—24no—
458C4×C5—24no—
459C4×C5—64no—
460C4×C5—64no—
461C4×C5—64no—
462C4×C5#100712yes—
463C2×Q8—32no—
464C2×Q8—34no—
465C2×Q8C22×C5×D314yes—
466C2×D4—34no—
467C2×D4—34no—
468C2×D4—32no—
469C22×C4—34no—
470C22×C4—34no—
471C22×C4—34no—
472C22×C4C22×C5×D311yescharacteristic
473C22×C4C22×C5×D312yes—
474C4⋊C4—34no—
475C4⋊C4—34no—
476C4⋊C4—34no—
477C4⋊C4—34no—
478C4⋊C4—34no—
479C4⋊C4—34no—
480C4⋊C4—32no—
481C4⋊C4—34no—
482C4⋊C4—32no—
483C4⋊C4—34no—
484C4⋊C4—34no—
485C4⋊C4—34no—
486C4⋊C4C22×C5×D312yes—
487C4⋊C4C22×C5×D312yes—
488C4⋊C4C22×C5×D314yes—
489C4⋊C4C22×C5×D314yes—
490C4⋊C4C22×C5×D314yes—
491C22⋊C4—34no—
492C22⋊C4—34no—
493C22⋊C4—34no—
494C22⋊C4—34no—
495C22⋊C4—34no—
496C22⋊C4—34no—
497C22⋊C4—34no—
498C22⋊C4—34no—
499C22⋊C4—34no—
500C22⋊C4C22×C5×D314yes—
501C42—34no—
502C42—34no—
503C42—34no—
504C42C22×C5×D314yes—
505C42C22×C5×D312yes—
506C42C22×C5×D311yescharacteristic
507C42C22×C5×D311yescharacteristic
508C3×C5#353111yescharacteristic, complemented, Hall
509C22×C3—21no—
510C22×C3—22no—
511C22×C3C25×C511yescharacteristic
512C2×D3—24no—
513C2×D3—24no—
514C2×D3—24no—
515C3×C4—21no—
516C3×C4—24no—
517C3×C4—22no—
518C3×C4—24no—
519C3×C4—22no—
520C3×C4C5×2−1+412yes—
521Dic3—24no—
522Dic3—24no—
523Dic3—24no—
524C2×C5—41no—
525C2×C5—124no—
526C2×C5C2×#100711yescharacteristic
527C2×C5#596412yes—
528C23—34no—
529C23C23×C5×D311yescharacteristic
530Q8—44no—
531Q8—122no—
532Q8—124no—
533D4—122no—
534D4—124no—
535D4—124no—
536C2×C4—22no—
537C2×C4—34no—
538C2×C4—34no—
539C2×C4—34no—
540C2×C4—42no—
541C2×C4—68no—
542C2×C4—124no—
543C2×C4—124no—
544C2×C4C23×C5×D314yes—
545C2×C4C23×C5×D314yes—
546C2×C4C23×C5×D311yescharacteristic
547C2×C4C23×C5×D312yes—
548C2×C4C23×C5×D312yes—
549C2×C4C23×C5×D311yescharacteristic
550C2×C3—41no—
551C2×C3C2×C5×2−1+411yescharacteristic
552C2×C3C5×#58512yes—
553D3—44no—
554C5#8628611yescharacteristic, direct product factor, Sylow
555C22—21no—
556C22—22no—
557C22—64no—
558C22—64no—
559C22—64no—
560C22C24×C5×D311yescharacteristic
561C4—21no—
562C4—22no—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—64no—
567C4—64no—
568C4—64no—
569C4C5×#100712yes—
570C3C5×#353111yescharacteristic, complemented, Sylow
571C2—41no—
572C2—124no—
573C2C2×C5×#100711yescharacteristic
574C2C5×#596412yes—
subgroupG/H
centerC2×C4×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC3×C5×#533C2
socleC22×C3×C5C25
hypercenterC5×#533D3
hypocenterC3C5×#3531
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C5×#533

Complements

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

NH ≅ G/Ntype
1. C3×C5×#533C2semidirect
2. C5×#5735C2semidirect
3. C5×#5700C2semidirect
4. C5×#5699C2semidirect
5. C5×C3⋊#564C2semidirect
6. C5×#5647C2semidirect
7. C5×#5646C2semidirect
8. C5×#5580C2semidirect
9. C5×#5578C2semidirect
10. C5×#5528C2semidirect
11. C5×#5479C2semidirect
58. #86286C5direct (N × H)
66. C5×#533D3semidirect
115. C3×#533C2×C5semidirect
116. #5735C2×C5semidirect
117. #5700C2×C5semidirect
118. #5699C2×C5semidirect
119. C3⋊#564C2×C5semidirect
120. #5647C2×C5semidirect
121. #5646C2×C5semidirect
122. #5580C2×C5semidirect
123. #5578C2×C5semidirect
124. #5528C2×C5semidirect
125. #5479C2×C5semidirect
291. #533C5×D3semidirect
508. C3×C5#3531semidirect
554. C5#86286direct (N × H)
570. C3C5×#3531semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5332
C5×#57352
C5×#57002
C5×#56992
C5×C3⋊#5642
C5×#56472
C5×#56462
C5×#55802
C5×#55782
C5×#55282
C5×#54792
C5×#35313
#862865