Finite Group Databasebeta

← C5×#25866 #11948586

Subgroups — C5×#25866 #11948586

Order 1280.

1238 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#536C211yesmaximal, characteristic
2C2×C5×#489C211yesmaximal, characteristic, complemented
3C2×C5×#487C211yesmaximal, characteristic, complemented
4C5×#2628C211yesmaximal, characteristic, complemented
5C5×#2043C212yesmaximal, complemented
6C5×#2042C212yesmaximal, complemented
7C5×#1960C211yesmaximal, characteristic, complemented
8C5×#1958C211yesmaximal, characteristic, complemented
9C5×#1788C212yesmaximal
10C5×#1787C212yesmaximal, complemented
11C5×#1750C211yesmaximal, characteristic, complemented
12C22×C5×#41C2211yescharacteristic
13C23×C5×Q8C2211yescharacteristic
14C5×#536—24no—
15C5×#536—24no—
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×C4⋊Q8C2211yescharacteristic
19C2×C5×#124C2211yescharacteristic
20C2×C5×#124C2211yescharacteristic
21C2×C5×#123—21no—
22C2×C5×#123—21no—
23C2×C5×C22⋊Q8—21no—
24C2×C5×C22⋊Q8—21no—
25C2×C5×#117C2211yescharacteristic
26C5×#489—24no—
27C5×#487—24no—
28C2×C5×#103C2212yes—
29C2×C5×#103C2212yes—
30C2×C5×#103C2212yes—
31C2×C5×#103C2212yes—
32C2×C5×#103C2212yes—
33C2×C5×#103C2212yes—
34C2×C5×#102C2212yes—
35C2×C5×#102C2212yes—
36C2×C5×C22⋊C8C2212yes—
37C2×C5×#97C2211yescharacteristic
38C2×C4×C5×C8C2211yescharacteristic
39C5×#400—21no—
40C5×#399—21no—
41C5×#395—21no—
42C5×#393—21no—
43C5×#389C2212yes—
44C5×#382C2211yescharacteristic
45C5×#380C2212yes—
46C5×#336C2212yes—
47#25866C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41—22no—
49C2×C5×#41—22no—
50C2×C5×#41—24no—
51C2×C5×#41—42no—
52C22×C5×Q8—22no—
53C22×C5×Q8—22no—
54C22×C5×Q8—21no—
55C22×C5×Q8—21no—
56C22×C5×Q8—24no—
57C22×C5×Q8—42no—
58C22×C5×Q8C2311yescharacteristic
59C22×C5×Q8C2311yescharacteristic
60C22×C5×Q8C2311yescharacteristic
61C22×C5×D4—21no—
62C22×C5×D4C2311yescharacteristic
63C23×C4×C5—21no—
64C23×C4×C5—21no—
65C23×C4×C5C2311yescharacteristic
66C22×C5×C8C2312yes—
67C22×C5×C8C2312yes—
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8—22no—
70C5×C4⋊Q8—22no—
71C5×C4⋊Q8—22no—
72C5×C4⋊Q8—22no—
73C5×C4⋊Q8D414yes—
74C5×#124—22no—
75C5×#124—22no—
76C5×#124—22no—
77C5×#124D414yes—
78C5×#123—28no—
79C5×#123—28no—
80C5×C22⋊Q8—28no—
81C5×C22⋊Q8—28no—
82C5×#117—22no—
83C5×#117—22no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—22no—
90C2×C5×C4⋊C4D411yescharacteristic
91C2×C5×C4⋊C4D411yescharacteristic
92C2×C5×C4⋊C4C2312yes—
93C2×C5×C4⋊C4C2312yes—
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—22no—
99C2×C5×C22⋊C4D411yescharacteristic
100C2×C5×C22⋊C4D411yescharacteristic
101C2×C42×C5C2311yescharacteristic
102C2×C42×C5C2311yescharacteristic
103C5×#103—24no—
104C5×#103—24no—
105C5×#103—24no—
106C5×#103—42no—
107C5×#103—42no—
108C5×#103—42no—
109C5×#102—24no—
110C5×#102—42no—
111C5×C22⋊C8—24no—
112C5×#97—22no—
113C4×C5×C8—22no—
114C5×#95—22no—
115C5×#95—21no—
116C5×#95—21no—
117C5×#95—22no—
118C2×#536C2×C511yescharacteristic
119C2×#489C2×C511yescharacteristic, complemented
120C2×#487C2×C511yescharacteristic, complemented
121#2628C2×C511yescharacteristic, complemented
122#2043C2×C512yescomplemented
123#2042C2×C512yescomplemented
124#1960C2×C511yescharacteristic, complemented
125#1958C2×C511yescharacteristic, complemented
126#1788C2×C512yes—
127#1787C2×C512yescomplemented
128#1750C2×C511yescharacteristic, complemented
129C24×C5—21no—
130C24×C5C2×D411yescharacteristic
131C5×#41—48no—
132C5×#41—48no—
133C2×C5×Q8—24no—
134C2×C5×Q8—21no—
135C2×C5×Q8—21no—
136C2×C5×Q8—21no—
137C2×C5×Q8—22no—
138C2×C5×Q8—22no—
139C2×C5×Q8—22no—
140C2×C5×Q8—22no—
141C2×C5×Q8—22no—
142C2×C5×Q8—22no—
143C2×C5×Q8—48no—
144C2×C5×Q8—48no—
145C2×C5×Q8—42no—
146C2×C5×Q8—42no—
147C2×C5×Q8—42no—
148C2×C5×Q8—82no—
149C2×C5×Q8C2×D412yes—
150C2×C5×Q8C2×D412yes—
151C2×C5×Q8C2×D412yes—
152C2×C5×D4—24no—
153C2×C5×D4—21no—
154C2×C5×D4—22no—
155C2×C5×D4—22no—
156C2×C5×D4—42no—
157C2×C5×D4—42no—
158C2×C5×D4C2×D412yes—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5—22no—
162C22×C4×C5—24no—
163C22×C4×C5—24no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—21no—
167C22×C4×C5—22no—
168C22×C4×C5—22no—
169C22×C4×C5—42no—
170C22×C4×C5—42no—
171C22×C4×C5—42no—
172C22×C4×C5—42no—
173C22×C4×C5C2×D411yescharacteristic
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×D411yescharacteristic
177C22×C4×C5C2×D411yescharacteristic
178C22×C4×C5#4114yes—
179C22×C4×C5C2411yescharacteristic
180C2×C5×C8—22no—
181C2×C5×C8—22no—
182C2×C5×C8—22no—
183C2×C5×C8—22no—
184C2×C5×C8—22no—
185C2×C5×C8—22no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—24no—
188C5×C4⋊C4—24no—
189C5×C4⋊C4—24no—
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—42no—
197C5×C4⋊C4—82no—
198C5×C22⋊C4—24no—
199C5×C22⋊C4—24no—
200C5×C22⋊C4—24no—
201C5×C22⋊C4—24no—
202C5×C22⋊C4—22no—
203C5×C22⋊C4—22no—
204C5×C22⋊C4—82no—
205C42×C5—21no—
206C42×C5—21no—
207C42×C5C2×D412yes—
208C42×C5C2×D412yes—
209C22×#41C22×C511yescharacteristic
210C23×Q8C22×C511yescharacteristic
211#536—24no—
212#536—24no—
213C2×C4⋊Q8C22×C511yescharacteristic
214C2×C4⋊Q8C22×C511yescharacteristic
215C2×C4⋊Q8C22×C511yescharacteristic
216C2×#124C22×C511yescharacteristic
217C2×#124C22×C511yescharacteristic
218C2×#123—21no—
219C2×#123—21no—
220C2×C22⋊Q8—21no—
221C2×C22⋊Q8—21no—
222C2×#117C22×C511yescharacteristic
223#489—24no—
224#487—24no—
225C2×#103C22×C512yes—
226C2×#103C22×C512yes—
227C2×#103C22×C512yes—
228C2×#103C22×C512yes—
229C2×#103C22×C512yes—
230C2×#103C22×C512yes—
231C2×#102C22×C512yes—
232C2×#102C22×C512yes—
233C2×C22⋊C8C22×C512yes—
234C2×#97C22×C511yescharacteristic
235C2×C4×C8C22×C511yescharacteristic
236#400—21no—
237#399—21no—
238#395—21no—
239#393—21no—
240#389C22×C512yes—
241#382C22×C511yescharacteristic
242#380C22×C512yes—
243#336C22×C512yes—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—41no—
250C23×C5—41no—
251C23×C5—42no—
252C23×C5—42no—
253C23×C5#12412yes—
254C23×C5C22×D411yescharacteristic
255C5×Q8—24no—
256C5×Q8—24no—
257C5×Q8—24no—
258C5×Q8—48no—
259C5×Q8—42no—
260C5×Q8—42no—
261C5×Q8—42no—
262C5×Q8—84no—
263C5×D4—24no—
264C5×D4—48no—
265C5×D4—42no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—24no—
271C2×C4×C5—24no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—48no—
280C2×C4×C5—48no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—42no—
293C2×C4×C5#12012yes—
294C2×C4×C5#12012yes—
295C2×C4×C5#12412yes—
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D411yescharacteristic
298C2×C4×C5C2×#4112yes—
299C2×C4×C5H(5,𝔽2)12yes—
300C5×C8—42no—
301C5×C8—42no—
302C2×#41—22no—
303C2×#41—22no—
304C2×#41—24no—
305C2×#41—42no—
306C22×Q8—22no—
307C22×Q8—22no—
308C22×Q8—21no—
309C22×Q8—21no—
310C22×Q8—24no—
311C22×Q8—42no—
312C22×Q8C23×C511yescharacteristic
313C22×Q8C23×C511yescharacteristic
314C22×Q8C23×C511yescharacteristic
315C22×D4—21no—
316C22×D4C23×C511yescharacteristic
317C23×C4—21no—
318C23×C4—21no—
319C23×C4C23×C511yescharacteristic
320C22×C8C23×C512yes—
321C22×C8C23×C512yes—
322C4⋊Q8—22no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326C4⋊Q8—22no—
327C4⋊Q8C5×D414yes—
328#124—22no—
329#124—22no—
330#124—22no—
331#124C5×D414yes—
332#123—28no—
333#123—28no—
334C22⋊Q8—28no—
335C22⋊Q8—28no—
336#117—22no—
337#117—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4C5×D411yescharacteristic
345C2×C4⋊C4C5×D411yescharacteristic
346C2×C4⋊C4C23×C512yes—
347C2×C4⋊C4C23×C512yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4C5×D411yescharacteristic
354C2×C22⋊C4C5×D411yescharacteristic
355C2×C42C23×C511yescharacteristic
356C2×C42C23×C511yescharacteristic
357#103—24no—
358#103—24no—
359#103—24no—
360#103—42no—
361#103—42no—
362#103—42no—
363#102—24no—
364#102—42no—
365C22⋊C8—24no—
366#97—22no—
367C4×C8—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372C22×C5—24no—
373C22×C5—22no—
374C22×C5—42no—
375C22×C5—41no—
376C22×C5—41no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—82no—
381C22×C5—82no—
382C22×C5C2×#12011yescharacteristic
383C22×C5C2×#12411yescharacteristic
384C22×C5#53511yescharacteristic
385C22×C5#57712yes—
386C22×C5#57812yes—
387C4×C5—22no—
388C4×C5—22no—
389C4×C5—22no—
390C4×C5—42no—
391C4×C5—42no—
392C4×C5—42no—
393C4×C5—41no—
394C4×C5—41no—
395C4×C5—82no—
396C4×C5—82no—
397C4×C5#54612yes—
398C4×C5#54612yes—
399C24—21no—
400C24C2×C5×D411yescharacteristic
401#41—48no—
402#41—48no—
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—82no—
419C2×Q8C2×C5×D412yes—
420C2×Q8C2×C5×D412yes—
421C2×Q8C2×C5×D412yes—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C2×C5×D412yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4C2×C5×D411yescharacteristic
444C22×C4C2×C5×D411yescharacteristic
445C22×C4C2×C5×D411yescharacteristic
446C22×C4C2×C5×D411yescharacteristic
447C22×C4C2×C5×D411yescharacteristic
448C22×C4C5×#4114yes—
449C22×C4C24×C511yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—82no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—82no—
475C42—21no—
476C42—21no—
477C42C2×C5×D412yes—
478C42C2×C5×D412yes—
479C2×C5—42no—
480C2×C5—82no—
481C2×C5#243511yescharacteristic
482C2×C5#301011yescharacteristic
483C2×C5#301111yescharacteristic
484C2×C5#313712yes—
485C2×C5#322512yes—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23C5×#12412yes—
496C23C22×C5×D411yescharacteristic
497Q8—24no—
498Q8—24no—
499Q8—24no—
500Q8—48no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504Q8—84no—
505D4—24no—
506D4—48no—
507D4—42no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4C5×#12012yes—
536C2×C4C5×#12012yes—
537C2×C4C5×#12412yes—
538C2×C4C22×C5×D411yescharacteristic
539C2×C4C22×C5×D411yescharacteristic
540C2×C4C2×C5×#4112yes—
541C2×C4C5×H(5,𝔽2)12yes—
542C8—42no—
543C8—42no—
544C5#2586611yescharacteristic, direct product factor, Sylow
545C22—24no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22C2×C5×#12011yescharacteristic
556C22C2×C5×#12411yescharacteristic
557C22C5×#53511yescharacteristic
558C22C5×#57712yes—
559C22C5×#57812yes—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—82no—
569C4—82no—
570C4C5×#54612yes—
571C4C5×#54612yes—
572C2—42no—
573C2—82no—
574C2C5×#243511yescharacteristic
575C2C5×#301011yescharacteristic
576C2C5×#301111yescharacteristic
577C2C5×#313712yes—
578C2C5×#322512yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#489C2semidirect
3. C2×C5×#487C2semidirect
4. C5×#2628C2semidirect
5. C5×#2043C2semidirect
6. C5×#2042C2semidirect
7. C5×#1960C2semidirect
8. C5×#1958C2semidirect
10. C5×#1787C2semidirect
11. C5×#1750C2semidirect
47. #25866C5direct (N × H)
119. C2×#489C2×C5semidirect
120. C2×#487C2×C5semidirect
121. #2628C2×C5semidirect
122. #2043C2×C5semidirect
123. #2042C2×C5semidirect
124. #1960C2×C5semidirect
125. #1958C2×C5semidirect
127. #1787C2×C5semidirect
128. #1750C2×C5semidirect
544. C5#25866direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5362
C2×C5×#4892
C2×C5×#4872
C5×#26282
C5×#20432
C5×#20422
C5×#19602
C5×#19582
C5×#17882
C5×#17872
C5×#17502
#258665