Finite Group Databasebeta

← C5×#29683 #11952403

Subgroups — C5×#29683 #11952403

Order 1280.

1056 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3461C211yesmaximal, characteristic, complemented
2C5×#3040C212yesmaximal, complemented
3C5×#2545C211yesmaximal, characteristic, complemented
4C5×#2028C211yesmaximal, characteristic, complemented
5C5×#2027C211yesmaximal, characteristic, complemented
6C5×#2007C211yesmaximal, characteristic, complemented
7C5×#2006C211yesmaximal, characteristic
8C5×#2005C212yesmaximal, complemented
9C5×#2003C212yesmaximal, complemented
10C5×#1806C212yesmaximal, complemented
11C5×#1801C211yesmaximal, characteristic, complemented
12C2×C5×#137C2212yes—
13C2×C5×AGL(1,ℤ/8)C2212yes—
14C5×#544C2211yescharacteristic
15C5×#542—22no—
16C5×#541—22no—
17C5×#539—22no—
18C5×#537C2211yescharacteristic
19C5×#536C2212yes—
20C5×#533—22no—
21C2×C5×C4⋊Q8C2211yescharacteristic
22C5×#529—22no—
23C5×#529—22no—
24C2×C5×#124C2211yescharacteristic
25C2×C5×#124C2211yescharacteristic
26C4×C5×#41C2212yes—
27C2×C4×C5×Q8C2211yescharacteristic
28C5×#464—22no—
29C5×#464—22no—
30C5×#463—22no—
31C5×#463—22no—
32C5×#423C2212yes—
33C2×C5×#104C2212yes—
34C2×C5×#104C2212yes—
35C5×#419C2212yes—
36C5×#418C2212yes—
37C2×C5×#101C2212yes—
38C5×#396—22no—
39C5×#391C2211yescharacteristic
40C5×#390C2211yescharacteristic
41C5×#388—22no—
42C4×C5×C4⋊C4C2211yescharacteristic
43C5×#343C2211yescharacteristic
44C5×#343C2211yescharacteristic
45C5×#339C2212yes—
46C5×#339C2212yes—
47#29683C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41C2312yes—
49C22×C5×Q8C2311yescharacteristic
50C22×C5×Q8C2311yescharacteristic
51C22×C5×D4C2311yescharacteristic
52C5×#137—42no—
53C5×#137—42no—
54C5×AGL(1,ℤ/8)—42no—
55C5×AGL(1,ℤ/8)—42no—
56C2×C5×Q16—22no—
57C2×C5×SD16—22no—
58C2×C5×SD16—22no—
59C2×C5×D8—22no—
60C2×C5×M16C2312yes—
61C2×C5×M16C2312yes—
62C5×C4⋊Q8—22no—
63C5×C4⋊Q8—22no—
64C5×C4⋊Q8—42no—
65C5×#126—24no—
66C5×#126—24no—
67C5×#125—22no—
68C5×#125—24no—
69C5×#125—22no—
70C5×#124—22no—
71C5×#124—22no—
72C5×#124—22no—
73C5×#124—22no—
74C5×#124—42no—
75C5×#123—24no—
76C5×#123—24no—
77C5×C22⋊Q8—24no—
78C5×C22⋊Q8—22no—
79C5×C22⋊Q8—22no—
80C5×C22⋊Q8—22no—
81C5×C22⋊Q8—22no—
82C5×#121—22no—
83C5×#121—22no—
84C4×C5×Q8—22no—
85C4×C5×Q8—22no—
86C4×C5×Q8—22no—
87C4×C5×Q8D412yes—
88C4×C5×Q8D412yes—
89C4×C5×D4—22no—
90C4×C5×D4—22no—
91C4×C5×D4—22no—
92C4×C5×D4D412yes—
93C4×C5×D4D412yes—
94C5×#117—24no—
95C5×#117—22no—
96C5×#117C2312yes—
97C2×C5×C4⋊C4—21no—
98C2×C5×C4⋊C4—22no—
99C2×C5×C4⋊C4D412yes—
100C2×C5×C4⋊C4C2311yescharacteristic
101C2×C5×C22⋊C4—21no—
102C2×C5×C22⋊C4—22no—
103C2×C5×C22⋊C4—22no—
104C2×C5×C22⋊C4—21no—
105C2×C42×C5—22no—
106C2×C42×C5D412yes—
107C2×C42×C5C2311yescharacteristic
108C2×C42×C5C2311yescharacteristic
109C2×C42×C5C2311yescharacteristic
110C5×C4⋊C8—22no—
111C5×C4⋊C8—22no—
112C5×#104—42no—
113C5×#104—42no—
114C5×#103—22no—
115C5×#103—22no—
116C5×#102—22no—
117C5×#102—22no—
118C5×#101—42no—
119C5×#95—21no—
120C5×#95—22no—
121#3461C2×C511yescharacteristic, complemented
122#3040C2×C512yescomplemented
123#2545C2×C511yescharacteristic, complemented
124#2028C2×C511yescharacteristic, complemented
125#2027C2×C511yescharacteristic, complemented
126#2007C2×C511yescharacteristic, complemented
127#2006C2×C511yescharacteristic
128#2005C2×C512yescomplemented
129#2003C2×C512yescomplemented
130#1806C2×C512yescomplemented
131#1801C2×C511yescharacteristic, complemented
132C24×C5—21no—
133C5×#41—22no—
134C5×#41—22no—
135C5×#41—22no—
136C5×#41—22no—
137C2×C5×Q8—22no—
138C2×C5×Q8—22no—
139C2×C5×Q8—22no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—22no—
143C2×C5×Q8—22no—
144C2×C5×Q8—21no—
145C2×C5×Q8—21no—
146C2×C5×Q8C2×D412yes—
147C2×C5×D4—22no—
148C2×C5×D4—22no—
149C2×C5×D4—22no—
150C2×C5×D4—22no—
151C2×C5×D4—21no—
152C2×C5×D4—21no—
153C2×C5×D4C2×D412yes—
154C22×C4×C5—21no—
155C22×C4×C5—22no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5C2×D412yes—
161C22×C4×C5C2×D412yes—
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C5×Q16—44no—
166C5×SD16—44no—
167C5×SD16—44no—
168C5×D8—44no—
169C5×M16—22no—
170C5×M16—22no—
171C5×M16—22no—
172C5×M16—22no—
173C2×C5×C8—22no—
174C2×C5×C8—22no—
175C5×C4⋊C4—24no—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—24no—
180C5×C4⋊C4—24no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—42no—
185C5×C4⋊C4—44no—
186C5×C4⋊C4C2×D412yes—
187C5×C4⋊C4C2×D412yes—
188C5×C22⋊C4—24no—
189C5×C22⋊C4—24no—
190C5×C22⋊C4—24no—
191C5×C22⋊C4—22no—
192C5×C22⋊C4—22no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—44no—
195C5×C22⋊C4—44no—
196C5×C22⋊C4—41no—
197C5×C22⋊C4—41no—
198C5×C22⋊C4—42no—
199C42×C5—22no—
200C42×C5—22no—
201C42×C5—22no—
202C42×C5—22no—
203C42×C5—22no—
204C42×C5—22no—
205C42×C5—22no—
206C42×C5C2×D412yes—
207C42×C5C2×D412yes—
208C2×#137C22×C512yes—
209C2×AGL(1,ℤ/8)C22×C512yes—
210#544C22×C511yescharacteristic
211#542—22no—
212#541—22no—
213#539—22no—
214#537C22×C511yescharacteristic
215#536C22×C512yes—
216#533—22no—
217C2×C4⋊Q8C22×C511yescharacteristic
218#529—22no—
219#529—22no—
220C2×#124C22×C511yescharacteristic
221C2×#124C22×C511yescharacteristic
222C4×#41C22×C512yes—
223C2×C4×Q8C22×C511yescharacteristic
224#464—22no—
225#464—22no—
226#463—22no—
227#463—22no—
228#423C22×C512yes—
229C2×#104C22×C512yes—
230C2×#104C22×C512yes—
231#419C22×C512yes—
232#418C22×C512yes—
233C2×#101C22×C512yes—
234#396—22no—
235#391C22×C511yescharacteristic
236#390C22×C511yescharacteristic
237#388—22no—
238C4×C4⋊C4C22×C511yescharacteristic
239#343C22×C511yescharacteristic
240#343C22×C511yescharacteristic
241#339C22×C512yes—
242#339C22×C512yes—
243C23×C5—22no—
244C23×C5—42no—
245C23×C5—42no—
246C23×C5—41no—
247C23×C5—41no—
248C23×C5—41no—
249C23×C5C22×D411yescharacteristic
250C5×Q8—22no—
251C5×Q8—22no—
252C5×Q8—42no—
253C5×Q8—42no—
254C5×Q8—42no—
255C5×Q8—42no—
256C5×Q8—82no—
257C5×D4—22no—
258C5×D4—22no—
259C5×D4—42no—
260C5×D4—42no—
261C5×D4—42no—
262C5×D4—42no—
263C5×D4—82no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—44no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—82no—
287C2×C4×C5#12011yescharacteristic
288C2×C4×C5#12011yescharacteristic
289C2×C4×C5#12012yes—
290C2×C4×C5C22×D412yes—
291C2×C4×C5H(5,𝔽2)11yescharacteristic
292C2×C4×C5H(5,𝔽2)12yes—
293C2×C4×C52−1+411yescharacteristic
294C5×C8—42no—
295C5×C8—42no—
296C2×#41C23×C512yes—
297C22×Q8C23×C511yescharacteristic
298C22×Q8C23×C511yescharacteristic
299C22×D4C23×C511yescharacteristic
300#137—42no—
301#137—42no—
302AGL(1,ℤ/8)—42no—
303AGL(1,ℤ/8)—42no—
304C2×Q16—22no—
305C2×SD16—22no—
306C2×SD16—22no—
307C2×D8—22no—
308C2×M16C23×C512yes—
309C2×M16C23×C512yes—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—42no—
313#126—24no—
314#126—24no—
315#125—22no—
316#125—24no—
317#125—22no—
318#124—22no—
319#124—22no—
320#124—22no—
321#124—22no—
322#124—42no—
323#123—24no—
324#123—24no—
325C22⋊Q8—24no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330#121—22no—
331#121—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8C5×D412yes—
336C4×Q8C5×D412yes—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4C5×D412yes—
341C4×D4C5×D412yes—
342#117—24no—
343#117—22no—
344#117C23×C512yes—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4C5×D412yes—
348C2×C4⋊C4C23×C511yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C42—22no—
354C2×C42C5×D412yes—
355C2×C42C23×C511yescharacteristic
356C2×C42C23×C511yescharacteristic
357C2×C42C23×C511yescharacteristic
358C4⋊C8—22no—
359C4⋊C8—22no—
360#104—42no—
361#104—42no—
362#103—22no—
363#103—22no—
364#102—22no—
365#102—22no—
366#101—42no—
367#95—21no—
368#95—22no—
369C22×C5—21no—
370C22×C5—21no—
371C22×C5—42no—
372C22×C5—42no—
373C22×C5—42no—
374C22×C5—82no—
375C22×C5—82no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5—81no—
379C22×C5C2×#12011yescharacteristic
380C22×C5#53511yescharacteristic
381C22×C5#53711yescharacteristic
382C4×C5—22no—
383C4×C5—21no—
384C4×C5—21no—
385C4×C5—42no—
386C4×C5—42no—
387C4×C5—42no—
388C4×C5—42no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5—81no—
392C4×C5—81no—
393C4×C5#53512yes—
394C4×C5#53712yes—
395C24—21no—
396#41—22no—
397#41—22no—
398#41—22no—
399#41—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8C2×C5×D412yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4C2×C5×D412yes—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4C2×C5×D412yes—
424C22×C4C2×C5×D412yes—
425C22×C4C2×C5×D411yescharacteristic
426C22×C4C2×C5×D411yescharacteristic
427C22×C4C24×C511yescharacteristic
428Q16—44no—
429SD16—44no—
430SD16—44no—
431D8—44no—
432M16—22no—
433M16—22no—
434M16—22no—
435M16—22no—
436C2×C8—22no—
437C2×C8—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4C2×C5×D412yes—
450C4⋊C4C2×C5×D412yes—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42C2×C5×D412yes—
470C42C2×C5×D412yes—
471C2×C5—21no—
472C2×C5—21no—
473C2×C5—82no—
474C2×C5—161no—
475C2×C5#261511yescharacteristic
476C2×C5#301811yescharacteristic
477C2×C5#302011yescharacteristic
478C23—22no—
479C23—42no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C22×C5×D411yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—82no—
492D4—22no—
493D4—22no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—82no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4C5×#12011yescharacteristic
523C2×C4C5×#12011yescharacteristic
524C2×C4C5×#12012yes—
525C2×C4C22×C5×D412yes—
526C2×C4C5×H(5,𝔽2)11yescharacteristic
527C2×C4C5×H(5,𝔽2)12yes—
528C2×C4C5×2−1+411yescharacteristic
529C8—42no—
530C8—42no—
531C5#2968311yescharacteristic, direct product factor, Sylow
532C22—21no—
533C22—21no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22C2×C5×#12011yescharacteristic
543C22C5×#53511yescharacteristic
544C22C5×#53711yescharacteristic
545C4—22no—
546C4—21no—
547C4—21no—
548C4—42no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—81no—
555C4—81no—
556C4C5×#53512yes—
557C4C5×#53712yes—
558C2—21no—
559C2—21no—
560C2—82no—
561C2—161no—
562C2C5×#261511yescharacteristic
563C2C5×#301811yescharacteristic
564C2C5×#302011yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#2006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#3461C2semidirect
2. C5×#3040C2semidirect
3. C5×#2545C2semidirect
4. C5×#2028C2semidirect
5. C5×#2027C2semidirect
6. C5×#2007C2semidirect
8. C5×#2005C2semidirect
9. C5×#2003C2semidirect
10. C5×#1806C2semidirect
11. C5×#1801C2semidirect
47. #29683C5direct (N × H)
121. #3461C2×C5semidirect
122. #3040C2×C5semidirect
123. #2545C2×C5semidirect
124. #2028C2×C5semidirect
125. #2027C2×C5semidirect
126. #2007C2×C5semidirect
128. #2005C2×C5semidirect
129. #2003C2×C5semidirect
130. #1806C2×C5semidirect
131. #1801C2×C5semidirect
531. C5#29683direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34612
C5×#30402
C5×#25452
C5×#20282
C5×#20272
C5×#20072
C5×#20062
C5×#20052
C5×#20032
C5×#18062
C5×#18012
#296835