Finite Group Databasebeta

← C5×#18993 #11941713

Subgroups — C5×#18993 #11941713

Order 1280.

1012 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#562C211yesmaximal, characteristic, complemented
2C5×#3234C211yesmaximal, characteristic, complemented
3C5×#3213C211yesmaximal, characteristic, complemented
4C5×#3153C211yesmaximal, characteristic, complemented
5C5×#1661C214yesmaximal, complemented
6C5×#1656C214yesmaximal, complemented
7C5×#1533C211yesmaximal, characteristic, complemented
8C5×#1528C211yesmaximal, characteristic, complemented
9C5×#1527C211yesmaximal, characteristic, complemented
10C2×C5×#137—21no—
11C2×C5×AGL(1,ℤ/8)—21no—
12C5×#562—22no—
13C5×#562—22no—
14C5×#562—22no—
15C5×#562C2214yes—
16C5×#532C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C5×#124C2211yescharacteristic
19C2×C5×#124C2211yescharacteristic
20C2×C5×#123—21no—
21C2×C5×#123—21no—
22C2×C5×C22⋊Q8—21no—
23C2×C5×C22⋊Q8—21no—
24C2×C5×#121—21no—
25C2×C5×#121—21no—
26C5×#501C2212yescomplemented
27C5×#497C2212yescomplemented
28C5×#496C2212yescomplemented
29C5×#496C2212yescomplemented
30C5×#495—22no—
31C5×#490C2212yescomplemented
32C5×#489C2212yescomplemented
33C5×#481—21no—
34C5×#480—21no—
35C5×#478—22no—
36C5×#463—22no—
37C5×#448—21no—
38C5×#447—21no—
39C5×#432C2211yescharacteristic
40C5×#432C2211yescharacteristic
41C4×C5×M16C2211yescharacteristic
42C5×#330C2214yescomplemented
43C5×#329C2214yescomplemented
44C5×#329C2214yescomplemented
45#18993C511yesmaximal, characteristic, direct product factor, Sylow
46C2×C5×#41—21no—
47C22×C5×Q8D411yescharacteristic
48C22×C5×Q8D411yescharacteristic
49C22×C5×D4—21no—
50C22×C5×D4—21no—
51C22×C5×D4D411yescharacteristic
52C22×C5×D4D411yescharacteristic
53C23×C4×C5—21no—
54C23×C4×C5—21no—
55C5×#137—44no—
56C5×AGL(1,ℤ/8)—44no—
57C2×C5×Q16—22no—
58C2×C5×SD16—22no—
59C2×C5×SD16—22no—
60C2×C5×SD16—22no—
61C2×C5×D8—22no—
62C2×C5×D8—22no—
63C2×C5×M16—21no—
64C5×C4⋊Q8C2311yescharacteristic
65C5×#127C2311yescharacteristic
66C5×#125—21no—
67C5×#125—21no—
68C5×#125D412yes—
69C5×#125C2312yes—
70C5×#124—21no—
71C5×#124—21no—
72C5×#124—21no—
73C5×#124—21no—
74C5×#124—21no—
75C5×#124D412yes—
76C5×#124D412yes—
77C5×#124C2312yes—
78C5×#124C2312yes—
79C5×#123—24no—
80C5×#123—24no—
81C5×#123—24no—
82C5×#123—24no—
83C5×C22⋊Q8—24no—
84C5×C22⋊Q8—24no—
85C5×C22⋊Q8—24no—
86C5×C22⋊Q8—24no—
87C5×#121—24no—
88C5×#121—24no—
89C5×#121—24no—
90C5×#121—24no—
91C5×#121—21no—
92C5×#121—21no—
93C4×C5×D4—21no—
94C4×C5×D4—21no—
95C2×C5×C4⋊C4—21no—
96C2×C5×C4⋊C4—21no—
97C2×C5×C4⋊C4—21no—
98C2×C5×C4⋊C4—21no—
99C2×C5×C4⋊C4D411yescharacteristic
100C2×C5×C4⋊C4D411yescharacteristic
101C2×C5×C22⋊C4—21no—
102C2×C5×C22⋊C4—21no—
103C2×C5×C22⋊C4—21no—
104C2×C5×C22⋊C4—21no—
105C2×C5×C22⋊C4—21no—
106C2×C5×C22⋊C4—21no—
107C2×C42×C5C2311yescharacteristic
108C5×#107—22no—
109C5×#106—22no—
110C5×C4⋊C8—21no—
111C5×C4⋊C8—21no—
112C5×#103—22no—
113C5×#102—22no—
114C5×#102—22no—
115C5×#102—22no—
116C5×C22⋊C8—21no—
117C5×C22⋊C8—21no—
118C5×#97C2312yes—
119C5×#97C2312yes—
120C5×#97C2312yes—
121C4×C5×C8—21no—
122C2×#562C2×C511yescharacteristic, complemented
123#3234C2×C511yescharacteristic, complemented
124#3213C2×C511yescharacteristic, complemented
125#3153C2×C511yescharacteristic, complemented
126#1661C2×C514yescomplemented
127#1656C2×C514yescomplemented
128#1533C2×C511yescharacteristic, complemented
129#1528C2×C511yescharacteristic, complemented
130#1527C2×C511yescharacteristic, complemented
131C24×C5—21no—
132C24×C5—21no—
133C5×#41—42no—
134C5×#41—42no—
135C2×C5×Q8—21no—
136C2×C5×Q8—21no—
137C2×C5×Q8—21no—
138C2×C5×Q8—42no—
139C2×C5×Q8—81no—
140C2×C5×Q8C2×D412yes—
141C2×C5×Q8C2×D412yes—
142C2×C5×D4—24no—
143C2×C5×D4—24no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4—42no—
150C2×C5×D4—41no—
151C2×C5×D4—81no—
152C2×C5×D4—82no—
153C2×C5×D4—82no—
154C2×C5×D4C2×D412yes—
155C2×C5×D4C2×D412yes—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—24no—
161C22×C4×C5—24no—
162C22×C4×C5—41no—
163C22×C4×C5—82no—
164C22×C4×C5—82no—
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C5×Q16—44no—
169C5×SD16—44no—
170C5×SD16—44no—
171C5×SD16—82no—
172C5×D8—44no—
173C5×D8—82no—
174C5×M16—24no—
175C2×C5×C8—22no—
176C2×C5×C8—22no—
177C2×C5×C8—22no—
178C5×C4⋊C4—24no—
179C5×C4⋊C4—24no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4C2×D412yes—
189C5×C4⋊C4C2×D412yes—
190C5×C22⋊C4—24no—
191C5×C22⋊C4—24no—
192C5×C22⋊C4—22no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—22no—
195C5×C22⋊C4—22no—
196C5×C22⋊C4—22no—
197C5×C22⋊C4—22no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—41no—
201C5×C22⋊C4—41no—
202C42×C5C2×D411yescharacteristic
203C42×C5C2×D411yescharacteristic
204C42×C5C2×D411yescharacteristic
205C42×C5C2411yescharacteristic
206C2×#137—21no—
207C2×AGL(1,ℤ/8)—21no—
208#562—22no—
209#562—22no—
210#562—22no—
211#562C22×C514yes—
212#532C22×C511yescharacteristic
213C2×#125C22×C511yescharacteristic
214C2×#124C22×C511yescharacteristic
215C2×#124C22×C511yescharacteristic
216C2×#123—21no—
217C2×#123—21no—
218C2×C22⋊Q8—21no—
219C2×C22⋊Q8—21no—
220C2×#121—21no—
221C2×#121—21no—
222#501C22×C512yescomplemented
223#497C22×C512yescomplemented
224#496C22×C512yescomplemented
225#496C22×C512yescomplemented
226#495—22no—
227#490C22×C512yescomplemented
228#489C22×C512yescomplemented
229#481—21no—
230#480—21no—
231#478—22no—
232#463—22no—
233#448—21no—
234#447—21no—
235#432C22×C511yescharacteristic
236#432C22×C511yescharacteristic
237C4×M16C22×C511yescharacteristic
238#330C22×C514yescomplemented
239#329C22×C514yescomplemented
240#329C22×C514yescomplemented
241C23×C5—22no—
242C23×C5—22no—
243C23×C5—42no—
244C23×C5—42no—
245C23×C5—42no—
246C23×C5—42no—
247C23×C5—42no—
248C23×C5—42no—
249C23×C5—41no—
250C23×C5#12011yescharacteristic
251C5×Q8—42no—
252C5×Q8—42no—
253C5×Q8—41no—
254C5×Q8—41no—
255C5×Q8—81no—
256C5×Q8—81no—
257C5×D4—42no—
258C5×D4—42no—
259C5×D4—41no—
260C5×D4—42no—
261C5×D4—41no—
262C5×D4—42no—
263C5×D4—81no—
264C5×D4—81no—
265C5×D4—84no—
266C5×D4—84no—
267C5×D4—82no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—41no—
280C2×C4×C5—82no—
281C2×C4×C5—84no—
282C2×C4×C5—84no—
283C2×C4×C5#12011yescharacteristic
284C2×C4×C5#12011yescharacteristic
285C2×C4×C5#12011yescharacteristic
286C2×C4×C5C22×D411yescharacteristic
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5C22×D411yescharacteristic
289C5×C8—24no—
290C5×C8—42no—
291C5×C8—42no—
292C2×#41—21no—
293C22×Q8C5×D411yescharacteristic
294C22×Q8C5×D411yescharacteristic
295C22×D4—21no—
296C22×D4—21no—
297C22×D4C5×D411yescharacteristic
298C22×D4C5×D411yescharacteristic
299C23×C4—21no—
300C23×C4—21no—
301#137—44no—
302AGL(1,ℤ/8)—44no—
303C2×Q16—22no—
304C2×SD16—22no—
305C2×SD16—22no—
306C2×SD16—22no—
307C2×D8—22no—
308C2×D8—22no—
309C2×M16—21no—
310C4⋊Q8C23×C511yescharacteristic
311#127C23×C511yescharacteristic
312#125—21no—
313#125—21no—
314#125C5×D412yes—
315#125C23×C512yes—
316#124—21no—
317#124—21no—
318#124—21no—
319#124—21no—
320#124—21no—
321#124C5×D412yes—
322#124C5×D412yes—
323#124C23×C512yes—
324#124C23×C512yes—
325#123—24no—
326#123—24no—
327#123—24no—
328#123—24no—
329C22⋊Q8—24no—
330C22⋊Q8—24no—
331C22⋊Q8—24no—
332C22⋊Q8—24no—
333#121—24no—
334#121—24no—
335#121—24no—
336#121—24no—
337#121—21no—
338#121—21no—
339C4×D4—21no—
340C4×D4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C5×D411yescharacteristic
346C2×C4⋊C4C5×D411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C42C23×C511yescharacteristic
354#107—22no—
355#106—22no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358#103—22no—
359#102—22no—
360#102—22no—
361#102—22no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364#97C23×C512yes—
365#97C23×C512yes—
366#97C23×C512yes—
367C4×C8—21no—
368C22×C5—21no—
369C22×C5—21no—
370C22×C5—42no—
371C22×C5—42no—
372C22×C5—42no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—42no—
376C22×C5—84no—
377C22×C5—84no—
378C22×C5—81no—
379C22×C5—81no—
380C22×C5—81no—
381C22×C5#45312yes—
382C22×C5C2×#12011yescharacteristic
383C4×C5—21no—
384C4×C5—21no—
385C4×C5—21no—
386C4×C5—21no—
387C4×C5—21no—
388C4×C5—21no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5—42no—
392C4×C5—42no—
393C4×C5—81no—
394C24—21no—
395C24—21no—
396#41—42no—
397#41—42no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×Q8—81no—
403C2×Q8C2×C5×D412yes—
404C2×Q8C2×C5×D412yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—41no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4C2×C5×D412yes—
418C2×D4C2×C5×D412yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—41no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4C2×C5×D411yescharacteristic
429C22×C4C2×C5×D411yescharacteristic
430C22×C4C2×C5×D411yescharacteristic
431Q16—44no—
432SD16—44no—
433SD16—44no—
434SD16—82no—
435D8—44no—
436D8—82no—
437M16—24no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4C2×C5×D412yes—
452C4⋊C4C2×C5×D412yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C42C2×C5×D411yescharacteristic
466C42C2×C5×D411yescharacteristic
467C42C2×C5×D411yescharacteristic
468C42C24×C511yescharacteristic
469C2×C5—22no—
470C2×C5—82no—
471C2×C5—82no—
472C2×C5—161no—
473C2×C5C2×#45311yescharacteristic
474C2×C5#301911yescharacteristic
475C2×C5#302011yescharacteristic
476C23—22no—
477C23—22no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23C5×#12011yescharacteristic
486Q8—42no—
487Q8—42no—
488Q8—41no—
489Q8—41no—
490Q8—81no—
491Q8—81no—
492D4—42no—
493D4—42no—
494D4—41no—
495D4—42no—
496D4—41no—
497D4—42no—
498D4—81no—
499D4—81no—
500D4—84no—
501D4—84no—
502D4—82no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4C5×#12011yescharacteristic
519C2×C4C5×#12011yescharacteristic
520C2×C4C5×#12011yescharacteristic
521C2×C4C22×C5×D411yescharacteristic
522C2×C4C22×C5×D411yescharacteristic
523C2×C4C22×C5×D411yescharacteristic
524C8—24no—
525C8—42no—
526C8—42no—
527C5#1899311yescharacteristic, direct product factor, Sylow
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22—84no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C5×#45312yes—
542C22C2×C5×#12011yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—81no—
554C2—22no—
555C2—82no—
556C2—82no—
557C2—161no—
558C2C2×C5×#45311yescharacteristic
559C2C5×#301911yescharacteristic
560C2C5×#302011yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
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
1. C2×C5×#562C2semidirect
2. C5×#3234C2semidirect
3. C5×#3213C2semidirect
4. C5×#3153C2semidirect
5. C5×#1661C2semidirect
6. C5×#1656C2semidirect
7. C5×#1533C2semidirect
8. C5×#1528C2semidirect
9. C5×#1527C2semidirect
26. C5×#501C22semidirect
27. C5×#497C22semidirect
28. C5×#496C22semidirect
29. C5×#496C22semidirect
31. C5×#490C22semidirect
32. C5×#489C22semidirect
42. C5×#330C22semidirect
43. C5×#329C22semidirect
44. C5×#329C22semidirect
45. #18993C5direct (N × H)
122. C2×#562C2×C5semidirect
123. #3234C2×C5semidirect
124. #3213C2×C5semidirect
125. #3153C2×C5semidirect
126. #1661C2×C5semidirect
127. #1656C2×C5semidirect
128. #1533C2×C5semidirect
129. #1528C2×C5semidirect
130. #1527C2×C5semidirect
222. #501C22×C5semidirect
223. #497C22×C5semidirect
224. #496C22×C5semidirect
225. #496C22×C5semidirect
227. #490C22×C5semidirect
228. #489C22×C5semidirect
238. #330C22×C5semidirect
239. #329C22×C5semidirect
240. #329C22×C5semidirect
527. C5#18993direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5622
C5×#32342
C5×#32132
C5×#31532
C5×#16612
C5×#16562
C5×#15332
C5×#15282
C5×#15272
#189935