Finite Group Databasebeta

← C5×#18695 #11941415

Subgroups — C5×#18695 #11941415

Order 1280.

866 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3552C211yesmaximal, characteristic
2C5×#3238C211yesmaximal, characteristic, complemented
3C5×#3238C211yesmaximal, characteristic, complemented
4C2×C5×#487C211yesmaximal, characteristic, complemented
5C5×#1638C212yesmaximal, complemented
6C5×#1637C212yesmaximal, complemented
7C5×#1636C212yesmaximal, complemented
8C5×#1635C212yesmaximal, complemented
9C5×#1556C211yesmaximal, characteristic, complemented
10C5×#1494C211yesmaximal, characteristic, complemented
11C5×#1494C211yesmaximal, characteristic, complemented
12C5×#563—22no—
13C5×#557C2211yescharacteristic
14C5×#556C2211yescharacteristic
15C5×#554—21no—
16C5×#553—21no—
17C5×D4×Q8—21no—
18C5×D4×Q8C2211yescharacteristic
19C5×#548—21no—
20C5×#547C2211yescharacteristic
21C5×#544—21no—
22C5×#542—21no—
23C5×#541—21no—
24C5×#539—22no—
25C5×#536—21no—
26C2×C5×C4⋊Q8C2211yescharacteristic
27C2×C5×C4⋊Q8C2211yescharacteristic
28C5×#529C2211yescharacteristic
29C5×#529C2211yescharacteristic
30C5×#501—21no—
31C5×#501—21no—
32C5×#489—21no—
33C5×#489—21no—
34C5×#487—22no—
35C5×#487C2212yes—
36C5×#487C2212yes—
37C5×#483—21no—
38C5×#483—21no—
39C5×#480—21no—
40C5×#480—21no—
41C5×#477C2212yes—
42C5×#477C2212yes—
43C5×#476C2212yes—
44C5×#476C2212yes—
45C5×#432C2211yescharacteristic
46C5×#432C2211yescharacteristic
47C2×C5×#103—22no—
48C2×C4×C5×C8C2211yescharacteristic
49C5×#332C2212yes—
50C5×#332C2212yes—
51C5×Q8⋊C8C2212yes—
52C5×Q8⋊C8C2212yes—
53C5×D4⋊C8C2212yes—
54C5×D4⋊C8C2212yes—
55#18695C511yesmaximal, characteristic, direct product factor, Sylow
56C2×C5×#41—21no—
57C22×C5×Q8—21no—
58C22×C5×Q8—21no—
59C22×C5×Q8—21no—
60C22×C5×C8—21no—
61C5×C4⋊Q8—22no—
62C5×C4⋊Q8—21no—
63C5×C4⋊Q8—21no—
64C5×C4⋊Q8—21no—
65C5×C4⋊Q8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8D412yes—
70C5×C4⋊Q8C2312yes—
71C5×C4⋊Q8C2311yescharacteristic
72C5×C4⋊Q8C2311yescharacteristic
73C5×#126—22no—
74C5×#126—22no—
75C5×#126—22no—
76C5×#126—21no—
77C5×#126—21no—
78C5×#125—22no—
79C5×#125—22no—
80C5×#125—22no—
81C5×#125—21no—
82C5×#125D411yescharacteristic
83C5×#125D411yescharacteristic
84C5×#124—22no—
85C5×#124—21no—
86C5×#124D411yescharacteristic
87C5×#124D411yescharacteristic
88C5×#123—22no—
89C5×#123—22no—
90C5×#123—21no—
91C5×#123—21no—
92C5×#123—22no—
93C5×C22⋊Q8—22no—
94C5×C22⋊Q8—22no—
95C5×C22⋊Q8—22no—
96C5×C22⋊Q8—22no—
97C5×C22⋊Q8—22no—
98C5×C22⋊Q8—21no—
99C5×C22⋊Q8—22no—
100C5×C22⋊Q8—21no—
101C5×C22⋊Q8—21no—
102C5×C22⋊Q8D411yescharacteristic
103C5×C22⋊Q8D411yescharacteristic
104C5×#121—22no—
105C5×#121—21no—
106C5×#121D411yescharacteristic
107C5×#121D411yescharacteristic
108C4×C5×Q8—21no—
109C4×C5×Q8—21no—
110C4×C5×Q8—21no—
111C4×C5×Q8C2311yescharacteristic
112C4×C5×Q8C2311yescharacteristic
113C4×C5×D4—22no—
114C4×C5×D4—22no—
115C4×C5×D4—21no—
116C4×C5×D4—21no—
117C4×C5×D4—21no—
118C4×C5×D4C2311yescharacteristic
119C4×C5×D4C2311yescharacteristic
120C5×#117—21no—
121C5×#117—21no—
122C5×#117—21no—
123C2×C5×C4⋊C4—21no—
124C2×C5×C4⋊C4—21no—
125C2×C5×C4⋊C4—21no—
126C2×C5×C4⋊C4—21no—
127C2×C5×C4⋊C4D412yes—
128C2×C42×C5C2311yescharacteristic
129C5×#107—22no—
130C5×#107—22no—
131C5×#106—22no—
132C5×#106—22no—
133C5×C4⋊C8C2312yes—
134C5×C4⋊C8C2312yes—
135C5×#103—22no—
136C5×#103—22no—
137C5×#103—24no—
138C5×#103—24no—
139C5×#102—22no—
140C5×#102—22no—
141C5×C22⋊C8—21no—
142C5×C22⋊C8—21no—
143C5×#97—21no—
144C5×#97—21no—
145C4×C5×C8—21no—
146C4×C5×C8C2312yes—
147#3552C2×C511yescharacteristic
148#3238C2×C511yescharacteristic, complemented
149#3238C2×C511yescharacteristic, complemented
150C2×#487C2×C511yescharacteristic, complemented
151#1638C2×C512yescomplemented
152#1637C2×C512yescomplemented
153#1636C2×C512yescomplemented
154#1635C2×C512yescomplemented
155#1556C2×C511yescharacteristic, complemented
156#1494C2×C511yescharacteristic, complemented
157#1494C2×C511yescharacteristic, complemented
158C5×#41—82no—
159C2×C5×Q8—22no—
160C2×C5×Q8—22no—
161C2×C5×Q8—21no—
162C2×C5×Q8—21no—
163C2×C5×Q8—22no—
164C2×C5×Q8—44no—
165C2×C5×Q8—41no—
166C2×C5×Q8—82no—
167C2×C5×Q8—82no—
168C2×C5×Q8C2×D411yescharacteristic
169C2×C5×Q8C2×D411yescharacteristic
170C2×C5×D4—21no—
171C2×C5×D4—21no—
172C2×C5×D4—22no—
173C2×C5×D4C2×D411yescharacteristic
174C2×C5×D4C2×D411yescharacteristic
175C22×C4×C5—22no—
176C22×C4×C5—22no—
177C22×C4×C5—22no—
178C22×C4×C5—21no—
179C22×C4×C5—21no—
180C22×C4×C5—41no—
181C22×C4×C5C2×D411yescharacteristic
182C22×C4×C5C2×D411yescharacteristic
183C22×C4×C5C2×D411yescharacteristic
184C2×C5×C8—22no—
185C2×C5×C8—22no—
186C2×C5×C8—22no—
187C2×C5×C8—42no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—22no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—22no—
194C5×C4⋊C4—22no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—21no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—21no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4—22no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—41no—
209C5×C4⋊C4—41no—
210C5×C4⋊C4C2×D411yescharacteristic
211C5×C4⋊C4C2×D412yes—
212C5×C4⋊C4C2×D412yes—
213C5×C4⋊C4C2×D411yescharacteristic
214C5×C4⋊C4C2×D411yescharacteristic
215C5×C4⋊C4C2×D411yescharacteristic
216C5×C22⋊C4—22no—
217C5×C22⋊C4—22no—
218C5×C22⋊C4—22no—
219C5×C22⋊C4—22no—
220C5×C22⋊C4—21no—
221C5×C22⋊C4—21no—
222C5×C22⋊C4—21no—
223C5×C22⋊C4—21no—
224C5×C22⋊C4—21no—
225C5×C22⋊C4—21no—
226C5×C22⋊C4—21no—
227C5×C22⋊C4—21no—
228C5×C22⋊C4—21no—
229C5×C22⋊C4—21no—
230C42×C5—22no—
231C42×C5—22no—
232C42×C5—21no—
233C42×C5—21no—
234C42×C5C2×D411yescharacteristic
235C42×C5C2×D411yescharacteristic
236C42×C5C2×D411yescharacteristic
237C42×C5C2411yescharacteristic
238#563—22no—
239#557C22×C511yescharacteristic
240#556C22×C511yescharacteristic
241#554—21no—
242#553—21no—
243D4×Q8—21no—
244D4×Q8C22×C511yescharacteristic
245#548—21no—
246#547C22×C511yescharacteristic
247#544—21no—
248#542—21no—
249#541—21no—
250#539—22no—
251#536—21no—
252C2×C4⋊Q8C22×C511yescharacteristic
253C2×C4⋊Q8C22×C511yescharacteristic
254#529C22×C511yescharacteristic
255#529C22×C511yescharacteristic
256#501—21no—
257#501—21no—
258#489—21no—
259#489—21no—
260#487—22no—
261#487C22×C512yes—
262#487C22×C512yes—
263#483—21no—
264#483—21no—
265#480—21no—
266#480—21no—
267#477C22×C512yes—
268#477C22×C512yes—
269#476C22×C512yes—
270#476C22×C512yes—
271#432C22×C511yescharacteristic
272#432C22×C511yescharacteristic
273C2×#103—22no—
274C2×C4×C8C22×C511yescharacteristic
275#332C22×C512yes—
276#332C22×C512yes—
277Q8⋊C8C22×C512yes—
278Q8⋊C8C22×C512yes—
279D4⋊C8C22×C512yes—
280D4⋊C8C22×C512yes—
281C23×C5—21no—
282C23×C5—21no—
283C23×C5#12011yescharacteristic
284C5×Q8—22no—
285C5×Q8—22no—
286C5×Q8—44no—
287C5×Q8—42no—
288C5×Q8—42no—
289C5×Q8—82no—
290C5×Q8—82no—
291C5×Q8—82no—
292C5×D4—22no—
293C5×D4—22no—
294C5×D4—42no—
295C5×D4—42no—
296C5×D4—82no—
297C2×C4×C5—22no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—21no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—21no—
308C2×C4×C5—42no—
309C2×C4×C5—42no—
310C2×C4×C5—42no—
311C2×C4×C5—82no—
312C2×C4×C5—82no—
313C2×C4×C5—82no—
314C2×C4×C5—82no—
315C2×C4×C5#12011yescharacteristic
316C2×C4×C5#12011yescharacteristic
317C2×C4×C5#12011yescharacteristic
318C2×C4×C5C22×D411yescharacteristic
319C2×C4×C5C22×D411yescharacteristic
320C2×C4×C5C22×D411yescharacteristic
321C5×C8—42no—
322C5×C8—42no—
323C5×C8—42no—
324C2×#41—21no—
325C22×Q8—21no—
326C22×Q8—21no—
327C22×Q8—21no—
328C22×C8—21no—
329C4⋊Q8—22no—
330C4⋊Q8—21no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8C5×D412yes—
338C4⋊Q8C23×C512yes—
339C4⋊Q8C23×C511yescharacteristic
340C4⋊Q8C23×C511yescharacteristic
341#126—22no—
342#126—22no—
343#126—22no—
344#126—21no—
345#126—21no—
346#125—22no—
347#125—22no—
348#125—22no—
349#125—21no—
350#125C5×D411yescharacteristic
351#125C5×D411yescharacteristic
352#124—22no—
353#124—21no—
354#124C5×D411yescharacteristic
355#124C5×D411yescharacteristic
356#123—22no—
357#123—22no—
358#123—21no—
359#123—21no—
360#123—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—21no—
367C22⋊Q8—22no—
368C22⋊Q8—21no—
369C22⋊Q8—21no—
370C22⋊Q8C5×D411yescharacteristic
371C22⋊Q8C5×D411yescharacteristic
372#121—22no—
373#121—21no—
374#121C5×D411yescharacteristic
375#121C5×D411yescharacteristic
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C23×C511yescharacteristic
380C4×Q8C23×C511yescharacteristic
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4C23×C511yescharacteristic
387C4×D4C23×C511yescharacteristic
388#117—21no—
389#117—21no—
390#117—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4C5×D412yes—
396C2×C42C23×C511yescharacteristic
397#107—22no—
398#107—22no—
399#106—22no—
400#106—22no—
401C4⋊C8C23×C512yes—
402C4⋊C8C23×C512yes—
403#103—22no—
404#103—22no—
405#103—24no—
406#103—24no—
407#102—22no—
408#102—22no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411#97—21no—
412#97—21no—
413C4×C8—21no—
414C4×C8C23×C512yes—
415C22×C5—21no—
416C22×C5—21no—
417C22×C5—41no—
418C22×C5—41no—
419C22×C5—41no—
420C22×C5—41no—
421C22×C5—41no—
422C22×C5—41no—
423C22×C5#45412yes—
424C22×C5C2×#12011yescharacteristic
425C4×C5—21no—
426C4×C5—21no—
427C4×C5—21no—
428C4×C5—21no—
429C4×C5—41no—
430C4×C5—42no—
431C4×C5—41no—
432C4×C5—42no—
433C4×C5—41no—
434C4×C5—41no—
435C4×C5—41no—
436C4×C5—41no—
437C4×C5—82no—
438C4×C5#57712yes—
439C4×C5#57812yes—
440#41—82no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—44no—
447C2×Q8—41no—
448C2×Q8—82no—
449C2×Q8—82no—
450C2×Q8C2×C5×D411yescharacteristic
451C2×Q8C2×C5×D411yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4C2×C5×D411yescharacteristic
456C2×D4C2×C5×D411yescharacteristic
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4C2×C5×D411yescharacteristic
464C22×C4C2×C5×D411yescharacteristic
465C22×C4C2×C5×D411yescharacteristic
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—42no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4C2×C5×D412yes—
493C4⋊C4C2×C5×D412yes—
494C4⋊C4C2×C5×D411yescharacteristic
495C4⋊C4C2×C5×D411yescharacteristic
496C4⋊C4C2×C5×D411yescharacteristic
497C4⋊C4C2×C5×D411yescharacteristic
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—21no—
516C42C2×C5×D411yescharacteristic
517C42C2×C5×D411yescharacteristic
518C42C2×C5×D411yescharacteristic
519C42C24×C511yescharacteristic
520C2×C5—22no—
521C2×C5—81no—
522C2×C5—81no—
523C2×C5#300911yescharacteristic
524C2×C5#301111yescharacteristic
525C2×C5C2×#45411yescharacteristic
526C23—21no—
527C23—21no—
528C23C5×#12011yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—44no—
532Q8—42no—
533Q8—42no—
534Q8—82no—
535Q8—82no—
536Q8—82no—
537D4—22no—
538D4—22no—
539D4—42no—
540D4—42no—
541D4—82no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4C5×#12011yescharacteristic
561C2×C4C5×#12011yescharacteristic
562C2×C4C5×#12011yescharacteristic
563C2×C4C22×C5×D411yescharacteristic
564C2×C4C22×C5×D411yescharacteristic
565C2×C4C22×C5×D411yescharacteristic
566C8—42no—
567C8—42no—
568C8—42no—
569C5#1869511yescharacteristic, direct product factor, Sylow
570C22—21no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22C5×#45412yes—
579C22C2×C5×#12011yescharacteristic
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—82no—
593C4C5×#57712yes—
594C4C5×#57812yes—
595C2—22no—
596C2—81no—
597C2—81no—
598C2C5×#300911yescharacteristic
599C2C5×#301111yescharacteristic
600C2C2×C5×#45411yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3552C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3238C2semidirect
3. C5×#3238C2semidirect
4. C2×C5×#487C2semidirect
5. C5×#1638C2semidirect
6. C5×#1637C2semidirect
7. C5×#1636C2semidirect
8. C5×#1635C2semidirect
9. C5×#1556C2semidirect
10. C5×#1494C2semidirect
11. C5×#1494C2semidirect
55. #18695C5direct (N × H)
148. #3238C2×C5semidirect
149. #3238C2×C5semidirect
150. C2×#487C2×C5semidirect
151. #1638C2×C5semidirect
152. #1637C2×C5semidirect
153. #1636C2×C5semidirect
154. #1635C2×C5semidirect
155. #1556C2×C5semidirect
156. #1494C2×C5semidirect
157. #1494C2×C5semidirect
569. C5#18695direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35522
C5×#32382
C2×C5×#4872
C5×#16382
C5×#16372
C5×#16362
C5×#16352
C5×#15562
C5×#14942
#186955