Finite Group Databasebeta

← C5×#24108 #11946828

Subgroups — C5×#24108 #11946828

Order 1280.

1062 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#556C211yesmaximal, characteristic
2C2×C5×#483C211yesmaximal, characteristic, complemented
3C2×C5×#428C211yesmaximal, characteristic, complemented
4C5×#2487C211yesmaximal, characteristic
5C5×#2058C212yesmaximal, complemented
6C5×#2035C212yesmaximal, complemented
7C5×#1894C211yesmaximal, characteristic, complemented
8C5×#1870C211yesmaximal, characteristic
9C5×#1863C212yesmaximal, complemented
10C5×#1854C212yesmaximal, complemented
11C5×#1825C211yesmaximal, characteristic, complemented
12C22×C5×M16C2211yescharacteristic
13C5×#556—24no—
14C5×#556—24no—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×#126—21no—
17C2×C5×#126—21no—
18C2×C5×#124—21no—
19C2×C5×C22⋊Q8C2211yescharacteristic
20C2×C5×C22⋊Q8C2211yescharacteristic
21C2×C4×C5×Q8—21no—
22C2×C4×C5×Q8C2211yescharacteristic
23C2×C4×C5×D4C2211yescharacteristic
24C2×C5×#117C2211yescharacteristic
25C2×C5×#117C2211yescharacteristic
26C22×C5×C4⋊C4C2211yescharacteristic
27C5×#483—24no—
28C5×#428—24no—
29C2×C5×#107C2212yes—
30C2×C5×#106C2212yes—
31C2×C5×#106C2212yes—
32C2×C5×#103C2212yes—
33C2×C5×#103C2212yes—
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#24108C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8C2311yescharacteristic
50C22×C5×D4D411yescharacteristic
51C23×C4×C5—21no—
52C23×C4×C5C2311yescharacteristic
53C23×C4×C5C2311yescharacteristic
54C2×C5×M16—22no—
55C2×C5×M16—24no—
56C22×C5×C8C2312yes—
57C22×C5×C8C2312yes—
58C5×C4⋊Q8—22no—
59C5×C4⋊Q8—22no—
60C5×#126—28no—
61C5×#126—28no—
62C5×#124—28no—
63C5×C22⋊Q8—22no—
64C5×C22⋊Q8—22no—
65C5×C22⋊Q8—22no—
66C5×C22⋊Q8D414yes—
67C4×C5×Q8—28no—
68C4×C5×Q8—22no—
69C4×C5×Q8—22no—
70C4×C5×D4—22no—
71C4×C5×D4—22no—
72C5×#117—22no—
73C5×#117—22no—
74C5×#117—22no—
75C5×#117D414yes—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
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—22no—
96C2×C5×C22⋊C4D411yescharacteristic
97C2×C5×C22⋊C4D411yescharacteristic
98C2×C42×C5—21no—
99C2×C42×C5—21no—
100C2×C42×C5—21no—
101C2×C42×C5C2311yescharacteristic
102C5×#107—24no—
103C5×#106—24no—
104C5×#106—24no—
105C5×#103—24no—
106C5×#103—24no—
107C5×C22⋊C8—22no—
108C5×C22⋊C8—24no—
109C5×#95—22no—
110C5×#95—21no—
111C5×#95—22no—
112C2×#556C2×C511yescharacteristic
113C2×#483C2×C511yescharacteristic, complemented
114C2×#428C2×C511yescharacteristic, complemented
115#2487C2×C511yescharacteristic
116#2058C2×C512yescomplemented
117#2035C2×C512yescomplemented
118#1894C2×C511yescharacteristic, complemented
119#1870C2×C511yescharacteristic
120#1863C2×C512yescomplemented
121#1854C2×C512yescomplemented
122#1825C2×C511yescharacteristic, complemented
123C24×C5C2×D411yescharacteristic
124C24×C5C2×D411yescharacteristic
125C2×C5×Q8—24no—
126C2×C5×Q8—21no—
127C2×C5×Q8—22no—
128C2×C5×Q8—22no—
129C2×C5×Q8—42no—
130C2×C5×Q8—42no—
131C2×C5×Q8C2×D412yes—
132C2×C5×D4—22no—
133C2×C5×D4—22no—
134C2×C5×D4—41no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—42no—
148C22×C4×C5—82no—
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D412yes—
153C22×C4×C5C2×D412yes—
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2411yescharacteristic
156C5×M16—28no—
157C2×C5×C8—22no—
158C2×C5×C8—22no—
159C2×C5×C8—22no—
160C2×C5×C8—22no—
161C2×C5×C8—22no—
162C2×C5×C8#4114yes—
163C5×C4⋊C4—24no—
164C5×C4⋊C4—24no—
165C5×C4⋊C4—24no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—22no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—42no—
176C5×C4⋊C4—82no—
177C5×C4⋊C4C2×D412yes—
178C5×C4⋊C4C2×D412yes—
179C5×C4⋊C4C2×D412yes—
180C5×C22⋊C4—24no—
181C5×C22⋊C4—24no—
182C5×C22⋊C4—24no—
183C5×C22⋊C4—24no—
184C5×C22⋊C4—22no—
185C5×C22⋊C4—22no—
186C5×C22⋊C4—42no—
187C5×C22⋊C4—82no—
188C42×C5—24no—
189C42×C5—24no—
190C42×C5—24no—
191C42×C5—21no—
192C42×C5—21no—
193C22×M16C22×C511yescharacteristic
194#556—24no—
195#556—24no—
196C2×C4⋊Q8C22×C511yescharacteristic
197C2×#126—21no—
198C2×#126—21no—
199C2×#124—21no—
200C2×C22⋊Q8C22×C511yescharacteristic
201C2×C22⋊Q8C22×C511yescharacteristic
202C2×C4×Q8—21no—
203C2×C4×Q8C22×C511yescharacteristic
204C2×C4×D4C22×C511yescharacteristic
205C2×#117C22×C511yescharacteristic
206C2×#117C22×C511yescharacteristic
207C22×C4⋊C4C22×C511yescharacteristic
208#483—24no—
209#428—24no—
210C2×#107C22×C512yes—
211C2×#106C22×C512yes—
212C2×#106C22×C512yes—
213C2×#103C22×C512yes—
214C2×#103C22×C512yes—
215C2×C22⋊C8C22×C511yescharacteristic
216C2×C22⋊C8C22×C512yes—
217#400—21no—
218#399—21no—
219#389C22×C512yes—
220#386—21no—
221#385—21no—
222#380C22×C511yescharacteristic
223#380C22×C512yes—
224#340C22×C512yes—
225#340C22×C512yes—
226#340C22×C512yes—
227#336C22×C512yes—
228C23×C5—22no—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5—22no—
232C23×C5—21no—
233C23×C5—22no—
234C23×C5—21no—
235C23×C5—21no—
236C23×C5—22no—
237C23×C5#12012yes—
238C23×C5C22×D411yescharacteristic
239C5×Q8—24no—
240C5×Q8—48no—
241C5×Q8—41no—
242C5×Q8—41no—
243C5×D4—44no—
244C2×C4×C5—21no—
245C2×C4×C5—22no—
246C2×C4×C5—21no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—24no—
250C2×C4×C5—22no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—42no—
259C2×C4×C5—41no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5—41no—
263C2×C4×C5—42no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—84no—
274C2×C4×C5#12012yes—
275C2×C4×C5#13712yes—
276C2×C4×C5C22×D411yescharacteristic
277C2×C4×C5C22×D411yescharacteristic
278C2×C4×C5C2×#4112yes—
279C2×C4×C52−1+412yes—
280C5×C8—24no—
281C5×C8—42no—
282C22×Q8—21no—
283C22×Q8C23×C511yescharacteristic
284C22×D4C5×D411yescharacteristic
285C23×C4—21no—
286C23×C4C23×C511yescharacteristic
287C23×C4C23×C511yescharacteristic
288C2×M16—22no—
289C2×M16—24no—
290C22×C8C23×C512yes—
291C22×C8C23×C512yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294#126—28no—
295#126—28no—
296#124—28no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8C5×D414yes—
301C4×Q8—28no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×D4—22no—
305C4×D4—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117C5×D414yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4C5×D411yescharacteristic
319C2×C4⋊C4C23×C512yes—
320C2×C4⋊C4C23×C512yes—
321C2×C4⋊C4C23×C511yescharacteristic
322C2×C4⋊C4C23×C511yescharacteristic
323C2×C4⋊C4C23×C511yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C5×D411yescharacteristic
331C2×C22⋊C4C5×D411yescharacteristic
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C23×C511yescharacteristic
336#107—24no—
337#106—24no—
338#106—24no—
339#103—24no—
340#103—24no—
341C22⋊C8—22no—
342C22⋊C8—24no—
343#95—22no—
344#95—21no—
345#95—22no—
346C22×C5—22no—
347C22×C5—22no—
348C22×C5—22no—
349C22×C5—42no—
350C22×C5—42no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—41no—
354C22×C5—41no—
355C22×C5—41no—
356C22×C5C2×#12011yescharacteristic
357C22×C5#54712yes—
358C22×C5C2×#13711yescharacteristic
359C22×C5#57511yescharacteristic
360C22×C5#57712yes—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—41no—
364C4×C5—42no—
365C4×C5—42no—
366C4×C5—42no—
367C4×C5—41no—
368C4×C5—42no—
369C4×C5—82no—
370C4×C5—82no—
371C4×C5#52511yescharacteristic
372C4×C5#53611yescharacteristic
373C4×C5#54712yes—
374C24C2×C5×D411yescharacteristic
375C24C2×C5×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8C2×C5×D412yes—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—41no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—42no—
399C22×C4—82no—
400C22×C4C2×C5×D411yescharacteristic
401C22×C4C2×C5×D411yescharacteristic
402C22×C4C2×C5×D411yescharacteristic
403C22×C4C2×C5×D412yes—
404C22×C4C2×C5×D412yes—
405C22×C4C2×C5×D411yescharacteristic
406C22×C4C24×C511yescharacteristic
407M16—28no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8C5×#4114yes—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—42no—
427C4⋊C4—82no—
428C4⋊C4C2×C5×D412yes—
429C4⋊C4C2×C5×D412yes—
430C4⋊C4C2×C5×D412yes—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—42no—
438C22⋊C4—82no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—21no—
443C42—21no—
444C2×C5—42no—
445C2×C5—81no—
446C2×C5#240611yescharacteristic
447C2×C5#300711yescharacteristic
448C2×C5#301011yescharacteristic
449C2×C5#335012yes—
450C2×C5#335612yes—
451C23—22no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—21no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—22no—
460C23C5×#12012yes—
461C23C22×C5×D411yescharacteristic
462Q8—24no—
463Q8—48no—
464Q8—41no—
465Q8—41no—
466D4—44no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4C5×#12012yes—
498C2×C4C5×#13712yes—
499C2×C4C22×C5×D411yescharacteristic
500C2×C4C22×C5×D411yescharacteristic
501C2×C4C2×C5×#4112yes—
502C2×C4C5×2−1+412yes—
503C8—24no—
504C8—42no—
505C5#2410811yescharacteristic, direct product factor, Sylow
506C22—22no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—41no—
512C22—41no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22C2×C5×#12011yescharacteristic
517C22C5×#54712yes—
518C22C2×C5×#13711yescharacteristic
519C22C5×#57511yescharacteristic
520C22C5×#57712yes—
521C4—22no—
522C4—22no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4C5×#52511yescharacteristic
532C4C5×#53611yescharacteristic
533C4C5×#54712yes—
534C2—42no—
535C2—81no—
536C2C5×#240611yescharacteristic
537C2C5×#300711yescharacteristic
538C2C5×#301011yescharacteristic
539C2C5×#335012yes—
540C2C5×#335612yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×D4D4
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×#483C2semidirect
3. C2×C5×#428C2semidirect
5. C5×#2058C2semidirect
6. C5×#2035C2semidirect
7. C5×#1894C2semidirect
9. C5×#1863C2semidirect
10. C5×#1854C2semidirect
11. C5×#1825C2semidirect
47. #24108C5direct (N × H)
113. C2×#483C2×C5semidirect
114. C2×#428C2×C5semidirect
116. #2058C2×C5semidirect
117. #2035C2×C5semidirect
118. #1894C2×C5semidirect
120. #1863C2×C5semidirect
121. #1854C2×C5semidirect
122. #1825C2×C5semidirect
505. C5#24108direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5562
C2×C5×#4832
C2×C5×#4282
C5×#24872
C5×#20582
C5×#20352
C5×#18942
C5×#18702
C5×#18632
C5×#18542
C5×#18252
#241085