Finite Group Databasebeta

← C5×#18257 #11940977

Subgroups — C5×#18257 #11940977

Order 1280.

1192 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3536C211yesmaximal, characteristic
2C5×#3004C211yesmaximal, characteristic, complemented
3C2×C5×#451C211yesmaximal, characteristic, complemented
4C5×#2436C211yesmaximal, characteristic, complemented
5C5×#1606C214yesmaximal, complemented
6C5×#1602C214yesmaximal, complemented
7C5×#1508C211yesmaximal, characteristic, complemented
8C5×#1501C211yesmaximal, characteristic, complemented
9C5×#1471C211yesmaximal, characteristic, complemented
10C23×C5×Q8C2211yescharacteristic
11C2×C5×#137—21no—
12C22×C5×Q16—21no—
13C5×#562—22no—
14C5×#554—21no—
15C5×#548—21no—
16C5×#547—21no—
17C5×#546—22no—
18C5×#546—21no—
19C5×#543C2212yes—
20C5×#542—21no—
21C5×#539—21no—
22C5×#538C2212yes—
23C5×#537—21no—
24C5×#537—21no—
25C2×C5×#124—21no—
26C5×#525C2211yescharacteristic
27C2×C5×C22⋊Q8C2211yescharacteristic
28C2×C5×#121C2211yescharacteristic
29C5×#451—22no—
30C5×#451—21no—
31C5×#451C2214yes—
32C5×#451C2212yes—
33C5×#448—21no—
34C5×#448C2212yes—
35C5×#419—21no—
36C2×C5×#103—21no—
37C5×#407C2211yescharacteristic
38C2×C5×C22⋊C8C2211yescharacteristic
39C5×#393C2214yes—
40C5×#391—21no—
41C5×#386—22no—
42C4×C5×C22⋊C4C2211yescharacteristic
43C5×#328C2212yes—
44C5×#328C2214yes—
45C5×#327C2212yes—
46C5×#323C2214yes—
47#18257C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41—22no—
49C2×C5×#41—21no—
50C2×C5×#41D411yescharacteristic
51C22×C5×Q8—21no—
52C22×C5×Q8—21no—
53C22×C5×Q8—21no—
54C22×C5×Q8—21no—
55C22×C5×Q8—21no—
56C22×C5×Q8—44no—
57C22×C5×Q8D411yescharacteristic
58C22×C5×Q8C2312yes—
59C22×C5×D4D412yes—
60C22×C5×D4D411yescharacteristic
61C23×C4×C5—21no—
62C23×C4×C5C2311yescharacteristic
63C5×#137—44no—
64C2×C5×Q16—24no—
65C2×C5×Q16—22no—
66C2×C5×Q16—44no—
67C2×C5×SD16—22no—
68C2×C5×M16—21no—
69C22×C5×C8—21no—
70C5×C4⋊Q8—21no—
71C5×#126—22no—
72C5×#125—22no—
73C5×#124—21no—
74C5×#124—22no—
75C5×#124—22no—
76C5×#124—44no—
77C5×#123—22no—
78C5×#123—22no—
79C5×#123—22no—
80C5×#123—21no—
81C5×#123—22no—
82C5×C22⋊Q8—21no—
83C5×C22⋊Q8—22no—
84C5×C22⋊Q8—22no—
85C5×C22⋊Q8—22no—
86C5×C22⋊Q8—21no—
87C5×C22⋊Q8—21no—
88C5×C22⋊Q8—22no—
89C5×C22⋊Q8—22no—
90C5×C22⋊Q8—22no—
91C5×C22⋊Q8D412yes—
92C5×C22⋊Q8D411yescharacteristic
93C5×C22⋊Q8C2312yes—
94C5×C22⋊Q8C2311yescharacteristic
95C5×#121—21no—
96C5×#121—22no—
97C5×#121—22no—
98C5×#121—22no—
99C5×#121—22no—
100C5×#121—22no—
101C5×#121—21no—
102C5×#121—21no—
103C5×#121—21no—
104C5×#121—21no—
105C5×#121D411yescharacteristic
106C5×#121C2311yescharacteristic
107C5×#120—21no—
108C5×#120—22no—
109C4×C5×Q8—21no—
110C4×C5×D4—22no—
111C4×C5×D4—21no—
112C4×C5×D4—21no—
113C4×C5×D4—21no—
114C4×C5×D4—22no—
115C5×#117D411yescharacteristic
116C2×C5×C4⋊C4—21no—
117C2×C5×C4⋊C4—21no—
118C2×C5×C4⋊C4D411yescharacteristic
119C2×C5×C4⋊C4D411yescharacteristic
120C2×C5×C22⋊C4—21no—
121C2×C5×C22⋊C4—22no—
122C2×C5×C22⋊C4—21no—
123C2×C5×C22⋊C4—24no—
124C2×C5×C22⋊C4C2312yes—
125C2×C42×C5—21no—
126C5×#103—24no—
127C5×#103—22no—
128C5×#103—22no—
129C5×C22⋊C8—21no—
130C5×C22⋊C8—21no—
131C5×C22⋊C8C2312yes—
132C5×C22⋊C8C2312yes—
133C5×#95—24no—
134C5×#95C2312yes—
135#3536C2×C511yescharacteristic
136#3004C2×C511yescharacteristic, complemented
137C2×#451C2×C511yescharacteristic, complemented
138#2436C2×C511yescharacteristic, complemented
139#1606C2×C514yescomplemented
140#1602C2×C514yescomplemented
141#1508C2×C511yescharacteristic, complemented
142#1501C2×C511yescharacteristic, complemented
143#1471C2×C511yescharacteristic, complemented
144C24×C5C2×D411yescharacteristic
145C24×C5C2×D412yes—
146C5×#41—42no—
147C5×#41—84no—
148C5×#41—82no—
149C2×C5×Q8—24no—
150C2×C5×Q8—22no—
151C2×C5×Q8—21no—
152C2×C5×Q8—24no—
153C2×C5×Q8—24no—
154C2×C5×Q8—22no—
155C2×C5×Q8—22no—
156C2×C5×Q8—44no—
157C2×C5×Q8—42no—
158C2×C5×Q8—42no—
159C2×C5×Q8—42no—
160C2×C5×Q8—42no—
161C2×C5×Q8—44no—
162C2×C5×Q8—44no—
163C2×C5×Q8—41no—
164C2×C5×Q8—41no—
165C2×C5×Q8C2×D411yescharacteristic
166C2×C5×Q8C2×D412yes—
167C2×C5×D4—21no—
168C2×C5×D4—22no—
169C2×C5×D4—21no—
170C2×C5×D4—22no—
171C2×C5×D4—22no—
172C2×C5×D4—21no—
173C2×C5×D4—22no—
174C2×C5×D4—21no—
175C2×C5×D4—21no—
176C2×C5×D4—42no—
177C2×C5×D4—44no—
178C2×C5×D4C2×D411yescharacteristic
179C22×C4×C5—21no—
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5—22no—
183C22×C4×C5—22no—
184C22×C4×C5—21no—
185C22×C4×C5—21no—
186C22×C4×C5—22no—
187C22×C4×C5—42no—
188C22×C4×C5—44no—
189C22×C4×C5—41no—
190C22×C4×C5—41no—
191C22×C4×C5C2×D411yescharacteristic
192C22×C4×C5C2×D411yescharacteristic
193C22×C4×C5C2×D411yescharacteristic
194C22×C4×C5C2×D412yes—
195C22×C4×C5C2×D411yescharacteristic
196C22×C4×C5C2×D411yescharacteristic
197C22×C4×C5C2411yescharacteristic
198C5×Q16—48no—
199C5×Q16—44no—
200C5×SD16—44no—
201C5×M16—42no—
202C2×C5×C8—22no—
203C2×C5×C8—22no—
204C2×C5×C8—42no—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—22no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—22no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—21no—
211C5×C4⋊C4—22no—
212C5×C4⋊C4—21no—
213C5×C4⋊C4—21no—
214C5×C4⋊C4—22no—
215C5×C4⋊C4—21no—
216C5×C4⋊C4C2×D412yes—
217C5×C4⋊C4C2×D411yescharacteristic
218C5×C4⋊C4C2×D411yescharacteristic
219C5×C22⋊C4—22no—
220C5×C22⋊C4—22no—
221C5×C22⋊C4—22no—
222C5×C22⋊C4—22no—
223C5×C22⋊C4—22no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4—21no—
226C5×C22⋊C4—21no—
227C5×C22⋊C4—22no—
228C5×C22⋊C4—21no—
229C5×C22⋊C4—21no—
230C5×C22⋊C4—22no—
231C5×C22⋊C4—22no—
232C5×C22⋊C4—21no—
233C5×C22⋊C4—48no—
234C5×C22⋊C4—42no—
235C42×C5—22no—
236C42×C5—21no—
237C42×C5—42no—
238C23×Q8C22×C511yescharacteristic
239C2×#137—21no—
240C22×Q16—21no—
241#562—22no—
242#554—21no—
243#548—21no—
244#547—21no—
245#546—22no—
246#546—21no—
247#543C22×C512yes—
248#542—21no—
249#539—21no—
250#538C22×C512yes—
251#537—21no—
252#537—21no—
253C2×#124—21no—
254#525C22×C511yescharacteristic
255C2×C22⋊Q8C22×C511yescharacteristic
256C2×#121C22×C511yescharacteristic
257#451—22no—
258#451—21no—
259#451C22×C512yes—
260#451C22×C514yes—
261#448—21no—
262#448C22×C512yes—
263#419—21no—
264C2×#103—21no—
265#407C22×C511yescharacteristic
266C2×C22⋊C8C22×C511yescharacteristic
267#393C22×C514yes—
268#391—21no—
269#386—22no—
270C4×C22⋊C4C22×C511yescharacteristic
271#328C22×C512yes—
272#328C22×C514yes—
273#327C22×C512yes—
274#323C22×C514yes—
275C23×C5—22no—
276C23×C5—22no—
277C23×C5—22no—
278C23×C5—21no—
279C23×C5—24no—
280C23×C5—41no—
281C23×C5—42no—
282C23×C5—42no—
283C23×C5#12011yescharacteristic
284C23×C5#12011yescharacteristic
285C23×C5C22×D411yescharacteristic
286C5×Q8—24no—
287C5×Q8—24no—
288C5×Q8—42no—
289C5×Q8—41no—
290C5×Q8—42no—
291C5×Q8—42no—
292C5×Q8—44no—
293C5×Q8—44no—
294C5×Q8—42no—
295C5×Q8—82no—
296C5×D4—41no—
297C5×D4—42no—
298C5×D4—44no—
299C5×D4—42no—
300C5×D4—42no—
301C5×D4—81no—
302C5×D4—82no—
303C5×D4—81no—
304C5×D4—82no—
305C5×D4—82no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—24no—
310C2×C4×C5—21no—
311C2×C4×C5—22no—
312C2×C4×C5—21no—
313C2×C4×C5—22no—
314C2×C4×C5—21no—
315C2×C4×C5—22no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—42no—
320C2×C4×C5—44no—
321C2×C4×C5—44no—
322C2×C4×C5—42no—
323C2×C4×C5—42no—
324C2×C4×C5—42no—
325C2×C4×C5—42no—
326C2×C4×C5—42no—
327C2×C4×C5—42no—
328C2×C4×C5—42no—
329C2×C4×C5—81no—
330C2×C4×C5—82no—
331C2×C4×C5—82no—
332C2×C4×C5—82no—
333C2×C4×C5#12011yescharacteristic
334C2×C4×C5#12011yescharacteristic
335C2×C4×C5C22×D411yescharacteristic
336C2×C4×C5C22×D411yescharacteristic
337C5×C8—42no—
338C5×C8—42no—
339C2×#41—22no—
340C2×#41—21no—
341C2×#41C5×D411yescharacteristic
342C22×Q8—21no—
343C22×Q8—21no—
344C22×Q8—21no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×Q8—44no—
348C22×Q8C5×D411yescharacteristic
349C22×Q8C23×C512yes—
350C22×D4C5×D411yescharacteristic
351C22×D4C5×D412yes—
352C23×C4—21no—
353C23×C4C23×C511yescharacteristic
354#137—44no—
355C2×Q16—24no—
356C2×Q16—22no—
357C2×Q16—44no—
358C2×SD16—22no—
359C2×M16—21no—
360C22×C8—21no—
361C4⋊Q8—21no—
362#126—22no—
363#125—22no—
364#124—21no—
365#124—22no—
366#124—22no—
367#124—44no—
368#123—22no—
369#123—22no—
370#123—22no—
371#123—21no—
372#123—22no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8C5×D412yes—
383C22⋊Q8C5×D411yescharacteristic
384C22⋊Q8C23×C512yes—
385C22⋊Q8C23×C511yescharacteristic
386#121—21no—
387#121—22no—
388#121—22no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—21no—
393#121—21no—
394#121—21no—
395#121—21no—
396#121C5×D411yescharacteristic
397#121C23×C511yescharacteristic
398#120—21no—
399#120—22no—
400C4×Q8—21no—
401C4×D4—22no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4—22no—
406#117C5×D411yescharacteristic
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4C5×D411yescharacteristic
410C2×C4⋊C4C5×D411yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—22no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4C23×C512yes—
416C2×C42—21no—
417#103—24no—
418#103—22no—
419#103—22no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C22⋊C8C23×C512yes—
423C22⋊C8C23×C512yes—
424#95—24no—
425#95C23×C512yes—
426C22×C5—21no—
427C22×C5—21no—
428C22×C5—21no—
429C22×C5—21no—
430C22×C5—21no—
431C22×C5—21no—
432C22×C5—21no—
433C22×C5—44no—
434C22×C5—41no—
435C22×C5—42no—
436C22×C5—41no—
437C22×C5—42no—
438C22×C5—41no—
439C22×C5—42no—
440C22×C5—84no—
441C22×C5#45512yes—
442C22×C5C2×#12011yescharacteristic
443C22×C5#57812yes—
444C4×C5—21no—
445C4×C5—21no—
446C4×C5—21no—
447C4×C5—41no—
448C4×C5—42no—
449C4×C5—42no—
450C4×C5—41no—
451C4×C5—44no—
452C4×C5—41no—
453C4×C5—42no—
454C4×C5—82no—
455C4×C5B(4,𝔽2)12yes—
456C24C2×C5×D411yescharacteristic
457C24C2×C5×D412yes—
458#41—42no—
459#41—84no—
460#41—82no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—21no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8—41no—
476C2×Q8—41no—
477C2×Q8C2×C5×D411yescharacteristic
478C2×Q8C2×C5×D412yes—
479C2×D4—21no—
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—22no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—42no—
489C2×D4—44no—
490C2×D4C2×C5×D411yescharacteristic
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4C2×C5×D411yescharacteristic
504C22×C4C2×C5×D411yescharacteristic
505C22×C4C2×C5×D411yescharacteristic
506C22×C4C2×C5×D412yes—
507C22×C4C2×C5×D411yescharacteristic
508C22×C4C2×C5×D411yescharacteristic
509C22×C4C24×C511yescharacteristic
510Q16—48no—
511Q16—44no—
512SD16—44no—
513M16—42no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—42no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4C2×C5×D412yes—
529C4⋊C4C2×C5×D411yescharacteristic
530C4⋊C4C2×C5×D411yescharacteristic
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—21no—
545C22⋊C4—48no—
546C22⋊C4—42no—
547C42—22no—
548C42—21no—
549C42—42no—
550C2×C5—22no—
551C2×C5—22no—
552C2×C5—41no—
553C2×C5—81no—
554C2×C5—82no—
555C2×C5#301111yescharacteristic
556C2×C5C2×#45511yescharacteristic
557C2×C5C2×B(4,𝔽2)11yescharacteristic
558C23—22no—
559C23—22no—
560C23—22no—
561C23—21no—
562C23—24no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23C5×#12011yescharacteristic
567C23C5×#12011yescharacteristic
568C23C22×C5×D411yescharacteristic
569Q8—24no—
570Q8—24no—
571Q8—42no—
572Q8—41no—
573Q8—42no—
574Q8—42no—
575Q8—44no—
576Q8—44no—
577Q8—42no—
578Q8—82no—
579D4—41no—
580D4—44no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—81no—
585D4—82no—
586D4—81no—
587D4—82no—
588D4—82no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—81no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4C5×#12011yescharacteristic
617C2×C4C5×#12011yescharacteristic
618C2×C4C22×C5×D411yescharacteristic
619C2×C4C22×C5×D411yescharacteristic
620C8—42no—
621C8—42no—
622C5#1825711yescharacteristic, direct product factor, Sylow
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—44no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—84no—
638C22C5×#45512yes—
639C22C2×C5×#12011yescharacteristic
640C22C5×#57812yes—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—41no—
645C4—42no—
646C4—42no—
647C4—41no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—82no—
652C4C5×B(4,𝔽2)12yes—
653C2—22no—
654C2—22no—
655C2—41no—
656C2—82no—
657C2—81no—
658C2C5×#301111yescharacteristic
659C2C2×C5×#45511yescharacteristic
660C2C2×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3004C2semidirect
3. C2×C5×#451C2semidirect
4. C5×#2436C2semidirect
5. C5×#1606C2semidirect
6. C5×#1602C2semidirect
7. C5×#1508C2semidirect
8. C5×#1501C2semidirect
9. C5×#1471C2semidirect
47. #18257C5direct (N × H)
136. #3004C2×C5semidirect
137. C2×#451C2×C5semidirect
138. #2436C2×C5semidirect
139. #1606C2×C5semidirect
140. #1602C2×C5semidirect
141. #1508C2×C5semidirect
142. #1501C2×C5semidirect
143. #1471C2×C5semidirect
622. C5#18257direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35362
C5×#30042
C2×C5×#4512
C5×#24362
C5×#16062
C5×#16022
C5×#15082
C5×#15012
C5×#14712
#182575