Finite Group Databasebeta

← C5×#24105 #11946825

Subgroups — C5×#24105 #11946825

Order 1280.

1126 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#555C211yesmaximal, characteristic
2C2×C5×#480C211yesmaximal, characteristic, complemented
3C2×C5×#428C211yesmaximal, characteristic, complemented
4C5×#2487C211yesmaximal, characteristic, complemented
5C5×#2057C212yesmaximal, complemented
6C5×#2033C212yesmaximal, complemented
7C5×#1895C211yesmaximal, characteristic, complemented
8C5×#1870C211yesmaximal, characteristic, complemented
9C5×#1864C212yesmaximal, complemented
10C5×#1854C212yesmaximal, complemented
11C5×#1825C211yesmaximal, characteristic, complemented
12C22×C5×M16C2211yescharacteristic
13C5×#555—24no—
14C5×#555—24no—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×#124—21no—
17C2×C5×#124—21no—
18C2×C5×#121—21no—
19C2×C5×#121C2211yescharacteristic
20C2×C5×#121C2211yescharacteristic
21C2×C4×C5×D4C2211yescharacteristic
22C2×C4×C5×D4C2211yescharacteristic
23C2×C5×#117—21no—
24C2×C5×#117C2211yescharacteristic
25C2×C5×#117C2211yescharacteristic
26C22×C5×C4⋊C4C2211yescharacteristic
27C5×#480—24no—
28C5×#428—24no—
29C2×C5×#107C2212yes—
30C2×C5×#107C2212yes—
31C2×C5×#106C2212yes—
32C2×C5×#102C2212yes—
33C2×C5×#102C2212yes—
34C2×C5×C22⋊C8C2212yes—
35C2×C5×C22⋊C8C2211yescharacteristic
36C5×#400—21no—
37C5×#399—21no—
38C5×#389C2212yes—
39C5×#386—21no—
40C5×#385—21no—
41C5×#380C2212yes—
42C5×#380C2211yescharacteristic
43C5×#340C2212yes—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#24105C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×D4—21no—
50C22×C5×D4—21no—
51C22×C5×D4D411yescharacteristic
52C22×C5×D4C2311yescharacteristic
53C23×C4×C5—21no—
54C23×C4×C5—21no—
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C2×C5×M16—22no—
58C2×C5×M16—24no—
59C22×C5×C8C2312yes—
60C22×C5×C8C2312yes—
61C5×C4⋊Q8—22no—
62C5×C4⋊Q8—22no—
63C5×#124—28no—
64C5×#124—28no—
65C5×#121—28no—
66C5×#121—22no—
67C5×#121—22no—
68C5×#121—22no—
69C5×#121D414yes—
70C4×C5×D4—22no—
71C4×C5×D4—22no—
72C4×C5×D4—22no—
73C4×C5×D4—22no—
74C5×#117—28no—
75C5×#117—22no—
76C5×#117—22no—
77C5×#117—22no—
78C5×#117D414yes—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—22no—
81C2×C5×C4⋊C4—22no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—22no—
84C2×C5×C4⋊C4D411yescharacteristic
85C2×C5×C4⋊C4C2312yes—
86C2×C5×C4⋊C4C2312yes—
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4—21no—
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—22no—
97C2×C5×C22⋊C4D411yescharacteristic
98C2×C5×C22⋊C4D411yescharacteristic
99C2×C42×C5—21no—
100C2×C42×C5—21no—
101C2×C42×C5C2311yescharacteristic
102C5×#107—24no—
103C5×#107—24no—
104C5×#106—24no—
105C5×#102—24no—
106C5×#102—24no—
107C5×C22⋊C8—22no—
108C5×C22⋊C8—24no—
109C5×#95—22no—
110C5×#95—21no—
111C5×#95—22no—
112C2×#555C2×C511yescharacteristic
113C2×#480C2×C511yescharacteristic, complemented
114C2×#428C2×C511yescharacteristic, complemented
115#2487C2×C511yescharacteristic, complemented
116#2057C2×C512yescomplemented
117#2033C2×C512yescomplemented
118#1895C2×C511yescharacteristic, complemented
119#1870C2×C511yescharacteristic, complemented
120#1864C2×C512yescomplemented
121#1854C2×C512yescomplemented
122#1825C2×C511yescharacteristic, complemented
123C24×C5—21no—
124C24×C5C2×D411yescharacteristic
125C24×C5C2×D411yescharacteristic
126C2×C5×Q8—24no—
127C2×C5×Q8—82no—
128C2×C5×D4—22no—
129C2×C5×D4—24no—
130C2×C5×D4—24no—
131C2×C5×D4—22no—
132C2×C5×D4—21no—
133C2×C5×D4—22no—
134C2×C5×D4—22no—
135C2×C5×D4—41no—
136C2×C5×D4—82no—
137C2×C5×D4—82no—
138C2×C5×D4C2×D412yes—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—24no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—22no—
148C22×C4×C5—22no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—42no—
152C22×C4×C5—42no—
153C22×C4×C5—42no—
154C22×C4×C5—82no—
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D412yes—
159C22×C4×C5C2×D412yes—
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2411yescharacteristic
162C5×M16—28no—
163C2×C5×C8—22no—
164C2×C5×C8—22no—
165C2×C5×C8—22no—
166C2×C5×C8—22no—
167C2×C5×C8—22no—
168C2×C5×C8#4114yes—
169C5×C4⋊C4—24no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—42no—
179C5×C4⋊C4—82no—
180C5×C4⋊C4C2×D412yes—
181C5×C4⋊C4C2×D412yes—
182C5×C4⋊C4C2×D412yes—
183C5×C22⋊C4—24no—
184C5×C22⋊C4—24no—
185C5×C22⋊C4—24no—
186C5×C22⋊C4—24no—
187C5×C22⋊C4—22no—
188C5×C22⋊C4—24no—
189C5×C22⋊C4—22no—
190C5×C22⋊C4—42no—
191C5×C22⋊C4—82no—
192C42×C5—24no—
193C42×C5—24no—
194C42×C5—21no—
195C42×C5—21no—
196C22×M16C22×C511yescharacteristic
197#555—24no—
198#555—24no—
199C2×C4⋊Q8C22×C511yescharacteristic
200C2×#124—21no—
201C2×#124—21no—
202C2×#121—21no—
203C2×#121C22×C511yescharacteristic
204C2×#121C22×C511yescharacteristic
205C2×C4×D4C22×C511yescharacteristic
206C2×C4×D4C22×C511yescharacteristic
207C2×#117—21no—
208C2×#117C22×C511yescharacteristic
209C2×#117C22×C511yescharacteristic
210C22×C4⋊C4C22×C511yescharacteristic
211#480—24no—
212#428—24no—
213C2×#107C22×C512yes—
214C2×#107C22×C512yes—
215C2×#106C22×C512yes—
216C2×#102C22×C512yes—
217C2×#102C22×C512yes—
218C2×C22⋊C8C22×C511yescharacteristic
219C2×C22⋊C8C22×C512yes—
220#400—21no—
221#399—21no—
222#389C22×C512yes—
223#386—21no—
224#385—21no—
225#380C22×C511yescharacteristic
226#380C22×C512yes—
227#340C22×C512yes—
228#340C22×C512yes—
229#340C22×C512yes—
230#336C22×C512yes—
231C23×C5—22no—
232C23×C5—21no—
233C23×C5—21no—
234C23×C5—22no—
235C23×C5—21no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—22no—
241C23×C5—42no—
242C23×C5—41no—
243C23×C5—41no—
244C23×C5—42no—
245C23×C5#12012yes—
246C23×C5C22×D411yescharacteristic
247C5×Q8—84no—
248C5×D4—24no—
249C5×D4—44no—
250C5×D4—41no—
251C5×D4—41no—
252C5×D4—84no—
253C5×D4—84no—
254C2×C4×C5—21no—
255C2×C4×C5—22no—
256C2×C4×C5—21no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—24no—
260C2×C4×C5—21no—
261C2×C4×C5—22no—
262C2×C4×C5—21no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—48no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—42no—
281C2×C4×C5—84no—
282C2×C4×C5#12012yes—
283C2×C4×C5#13712yes—
284C2×C4×C5C22×D411yescharacteristic
285C2×C4×C5C22×D411yescharacteristic
286C2×C4×C5C2×#4112yes—
287C2×C4×C52−1+412yes—
288C5×C8—24no—
289C5×C8—42no—
290C22×Q8—21no—
291C22×D4—21no—
292C22×D4—21no—
293C22×D4C5×D411yescharacteristic
294C22×D4C23×C511yescharacteristic
295C23×C4—21no—
296C23×C4—21no—
297C23×C4C23×C511yescharacteristic
298C23×C4C23×C511yescharacteristic
299C2×M16—22no—
300C2×M16—24no—
301C22×C8C23×C512yes—
302C22×C8C23×C512yes—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305#124—28no—
306#124—28no—
307#121—28no—
308#121—22no—
309#121—22no—
310#121—22no—
311#121C5×D414yes—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316#117—28no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117C5×D414yes—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4C5×D411yescharacteristic
327C2×C4⋊C4C23×C512yes—
328C2×C4⋊C4C23×C512yes—
329C2×C4⋊C4C23×C511yescharacteristic
330C2×C4⋊C4C23×C511yescharacteristic
331C2×C4⋊C4C23×C511yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4C5×D411yescharacteristic
340C2×C22⋊C4C5×D411yescharacteristic
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C23×C511yescharacteristic
344#107—24no—
345#107—24no—
346#106—24no—
347#102—24no—
348#102—24no—
349C22⋊C8—22no—
350C22⋊C8—24no—
351#95—22no—
352#95—21no—
353#95—22no—
354C22×C5—22no—
355C22×C5—22no—
356C22×C5—22no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—42no—
360C22×C5—42no—
361C22×C5—41no—
362C22×C5—42no—
363C22×C5—42no—
364C22×C5—41no—
365C22×C5—41no—
366C22×C5—42no—
367C22×C5—82no—
368C22×C5—82no—
369C22×C5C2×#12011yescharacteristic
370C22×C5#54712yes—
371C22×C5C2×#13711yescharacteristic
372C22×C5#57511yescharacteristic
373C22×C5#57612yes—
374C4×C5—22no—
375C4×C5—22no—
376C4×C5—41no—
377C4×C5—42no—
378C4×C5—41no—
379C4×C5—42no—
380C4×C5—42no—
381C4×C5—82no—
382C4×C5—82no—
383C4×C5#52511yescharacteristic
384C4×C5#53611yescharacteristic
385C4×C5#54712yes—
386C24—21no—
387C24C2×C5×D411yescharacteristic
388C24C2×C5×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—82no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—41no—
399C2×D4—82no—
400C2×D4—82no—
401C2×D4C2×C5×D412yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4C2×C5×D411yescharacteristic
419C22×C4C2×C5×D411yescharacteristic
420C22×C4C2×C5×D411yescharacteristic
421C22×C4C2×C5×D412yes—
422C22×C4C2×C5×D412yes—
423C22×C4C2×C5×D411yescharacteristic
424C22×C4C24×C511yescharacteristic
425M16—28no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8C5×#4114yes—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—82no—
443C4⋊C4C2×C5×D412yes—
444C4⋊C4C2×C5×D412yes—
445C4⋊C4C2×C5×D412yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—22no—
451C22⋊C4—24no—
452C22⋊C4—22no—
453C22⋊C4—42no—
454C22⋊C4—82no—
455C42—24no—
456C42—24no—
457C42—21no—
458C42—21no—
459C2×C5—42no—
460C2×C5—81no—
461C2×C5—82no—
462C2×C5#240611yescharacteristic
463C2×C5#300711yescharacteristic
464C2×C5#300811yescharacteristic
465C2×C5#334712yes—
466C2×C5#335712yes—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23C5×#12012yes—
482C23C22×C5×D411yescharacteristic
483Q8—84no—
484D4—24no—
485D4—44no—
486D4—41no—
487D4—41no—
488D4—84no—
489D4—84no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—48no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4C5×#12012yes—
519C2×C4C5×#13712yes—
520C2×C4C22×C5×D411yescharacteristic
521C2×C4C22×C5×D411yescharacteristic
522C2×C4C2×C5×#4112yes—
523C2×C4C5×2−1+412yes—
524C8—24no—
525C8—42no—
526C5#2410511yescharacteristic, direct product factor, Sylow
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22C2×C5×#12011yescharacteristic
543C22C5×#54712yes—
544C22C2×C5×#13711yescharacteristic
545C22C5×#57511yescharacteristic
546C22C5×#57612yes—
547C4—22no—
548C4—22no—
549C4—42no—
550C4—41no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—82no—
555C4—82no—
556C4C5×#52511yescharacteristic
557C4C5×#53611yescharacteristic
558C4C5×#54712yes—
559C2—42no—
560C2—81no—
561C2—82no—
562C2C5×#240611yescharacteristic
563C2C5×#300711yescharacteristic
564C2C5×#300811yescharacteristic
565C2C5×#334712yes—
566C2C5×#335712yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#555C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C4×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#480C2semidirect
3. C2×C5×#428C2semidirect
4. C5×#2487C2semidirect
5. C5×#2057C2semidirect
6. C5×#2033C2semidirect
7. C5×#1895C2semidirect
8. C5×#1870C2semidirect
9. C5×#1864C2semidirect
10. C5×#1854C2semidirect
11. C5×#1825C2semidirect
47. #24105C5direct (N × H)
113. C2×#480C2×C5semidirect
114. C2×#428C2×C5semidirect
115. #2487C2×C5semidirect
116. #2057C2×C5semidirect
117. #2033C2×C5semidirect
118. #1895C2×C5semidirect
119. #1870C2×C5semidirect
120. #1864C2×C5semidirect
121. #1854C2×C5semidirect
122. #1825C2×C5semidirect
526. C5#24105direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5552
C2×C5×#4802
C2×C5×#4282
C5×#24872
C5×#20572
C5×#20332
C5×#18952
C5×#18702
C5×#18642
C5×#18542
C5×#18252
#241055