Finite Group Databasebeta

← C5×#29682 #11952402

Subgroups — C5×#29682 #11952402

Order 1280.

1152 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3460C211yesmaximal, characteristic, complemented
2C5×#3040C212yesmaximal, complemented
3C5×#2544C211yesmaximal, characteristic, complemented
4C5×#2028C211yesmaximal, characteristic, complemented
5C5×#2027C211yesmaximal, characteristic, complemented
6C5×#2007C211yesmaximal, characteristic, complemented
7C5×#2006C211yesmaximal, characteristic, complemented
8C5×#2004C212yesmaximal, complemented
9C5×#2002C212yesmaximal, complemented
10C5×#1806C212yesmaximal, complemented
11C5×#1800C211yesmaximal, characteristic, complemented
12C2×C5×#137C2212yes—
13C2×C5×AGL(1,ℤ/8)C2212yes—
14C5×#544C2211yescharacteristic
15C5×#542—22no—
16C5×#540—22no—
17C5×#538—22no—
18C5×#537C2211yescharacteristic
19C5×#535C2212yes—
20C5×#532—22no—
21C2×C5×#127C2211yescharacteristic
22C5×#529—22no—
23C2×C5×#124C2211yescharacteristic
24C2×C5×#124C2211yescharacteristic
25C5×#525—22no—
26C4×C5×#41C2212yes—
27C2×C4×C5×D4C2211yescharacteristic
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×#100C2212yes—
38C5×#394—22no—
39C5×#391C2211yescharacteristic
40C5×#390C2211yescharacteristic
41C5×#386—22no—
42C4×C5×C4⋊C4C2211yescharacteristic
43C5×#343C2211yescharacteristic
44C5×#343C2211yescharacteristic
45C5×#339C2212yes—
46C5×#339C2212yes—
47#29682C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41—22no—
49C2×C5×#41C2312yes—
50C22×C5×Q8C2311yescharacteristic
51C22×C5×D4—22no—
52C22×C5×D4C2311yescharacteristic
53C22×C5×D4C2311yescharacteristic
54C23×C4×C5D412yes—
55C5×#137—42no—
56C5×#137—42no—
57C5×AGL(1,ℤ/8)—42no—
58C5×AGL(1,ℤ/8)—42no—
59C2×C5×Q16—22no—
60C2×C5×SD16—22no—
61C2×C5×SD16—22no—
62C2×C5×D8—22no—
63C2×C5×M16C2312yes—
64C2×C5×M16C2312yes—
65C5×C4⋊Q8—22no—
66C5×#127—22no—
67C5×#127—42no—
68C5×#126—24no—
69C5×#125—22no—
70C5×#124—24no—
71C5×#124—22no—
72C5×#124—22no—
73C5×#124—22no—
74C5×#124—42no—
75C5×#123—24no—
76C5×#123—24no—
77C5×C22⋊Q8—22no—
78C5×C22⋊Q8—22no—
79C5×C22⋊Q8—22no—
80C5×#121—22no—
81C5×#121—24no—
82C5×#121—22no—
83C5×#121—22no—
84C5×#121—22no—
85C5×#121—22no—
86C5×#120—24no—
87C4×C5×Q8D412yescomplemented
88C4×C5×Q8D412yescomplemented
89C4×C5×D4—24no—
90C4×C5×D4—22no—
91C4×C5×D4—22no—
92C4×C5×D4—22no—
93C4×C5×D4—22no—
94C4×C5×D4D412yescomplemented
95C4×C5×D4D412yescomplemented
96C5×#117—22no—
97C5×#117—22no—
98C5×#117C2312yes—
99C2×C5×C4⋊C4—21no—
100C2×C5×C4⋊C4C2311yescharacteristic
101C2×C5×C22⋊C4—22no—
102C2×C5×C22⋊C4—21no—
103C2×C5×C22⋊C4—21no—
104C2×C5×C22⋊C4—22no—
105C2×C5×C22⋊C4D412yes—
106C2×C42×C5—22no—
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×#100—42no—
119C5×#95—21no—
120C5×#95—22no—
121#3460C2×C511yescharacteristic, complemented
122#3040C2×C512yescomplemented
123#2544C2×C511yescharacteristic, complemented
124#2028C2×C511yescharacteristic, complemented
125#2027C2×C511yescharacteristic, complemented
126#2007C2×C511yescharacteristic, complemented
127#2006C2×C511yescharacteristic, complemented
128#2004C2×C512yescomplemented
129#2002C2×C512yescomplemented
130#1806C2×C512yescomplemented
131#1800C2×C511yescharacteristic, complemented
132C24×C5—21no—
133C24×C5C2×D412yes—
134C5×#41—22no—
135C5×#41—22no—
136C5×#41—22no—
137C5×#41—22no—
138C5×#41—84no—
139C2×C5×Q8—22no—
140C2×C5×Q8—22no—
141C2×C5×Q8—22no—
142C2×C5×Q8—21no—
143C2×C5×Q8—21no—
144C2×C5×Q8C2×D412yes—
145C2×C5×D4—22no—
146C2×C5×D4—24no—
147C2×C5×D4—24no—
148C2×C5×D4—22no—
149C2×C5×D4—22no—
150C2×C5×D4—22no—
151C2×C5×D4—22no—
152C2×C5×D4—21no—
153C2×C5×D4—22no—
154C2×C5×D4—21no—
155C2×C5×D4—21no—
156C2×C5×D4—21no—
157C2×C5×D4—44no—
158C2×C5×D4—44no—
159C2×C5×D4—82no—
160C2×C5×D4C2×D412yes—
161C22×C4×C5—21no—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5—22no—
166C22×C4×C5—22no—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—82no—
170C22×C4×C5C2×D412yes—
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2411yescharacteristic
174C5×Q16—44no—
175C5×SD16—44no—
176C5×SD16—44no—
177C5×D8—44no—
178C5×M16—22no—
179C5×M16—22no—
180C5×M16—22no—
181C5×M16—22no—
182C2×C5×C8—22no—
183C2×C5×C8—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—22no—
190C5×C4⋊C4—42no—
191C5×C4⋊C4C2×D412yes—
192C5×C4⋊C4C2×D412yes—
193C5×C22⋊C4—24no—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—24no—
196C5×C22⋊C4—22no—
197C5×C22⋊C4—24no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—44no—
201C5×C22⋊C4—41no—
202C5×C22⋊C4—42no—
203C5×C22⋊C4—41no—
204C5×C22⋊C4—82no—
205C42×C5—22no—
206C42×C5—22no—
207C42×C5—22no—
208C42×C5—22no—
209C42×C5—22no—
210C42×C5C2×D412yes—
211C42×C5C2×D412yes—
212C2×#137C22×C512yes—
213C2×AGL(1,ℤ/8)C22×C512yes—
214#544C22×C511yescharacteristic
215#542—22no—
216#540—22no—
217#538—22no—
218#537C22×C511yescharacteristic
219#535C22×C512yes—
220#532—22no—
221C2×#127C22×C511yescharacteristic
222#529—22no—
223C2×#124C22×C511yescharacteristic
224C2×#124C22×C511yescharacteristic
225#525—22no—
226C4×#41C22×C512yes—
227C2×C4×D4C22×C511yescharacteristic
228#464—22no—
229#464—22no—
230#463—22no—
231#463—22no—
232#423C22×C512yes—
233C2×#104C22×C512yes—
234C2×#104C22×C512yes—
235#419C22×C512yes—
236#418C22×C512yes—
237C2×#100C22×C512yes—
238#394—22no—
239#391C22×C511yescharacteristic
240#390C22×C511yescharacteristic
241#386—22no—
242C4×C4⋊C4C22×C511yescharacteristic
243#343C22×C511yescharacteristic
244#343C22×C511yescharacteristic
245#339C22×C512yes—
246#339C22×C512yes—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5—42no—
252C23×C5—42no—
253C23×C5—42no—
254C23×C5—42no—
255C23×C5—41no—
256C23×C5—41no—
257C23×C5—41no—
258C23×C5C22×D411yescharacteristic
259C5×Q8—22no—
260C5×Q8—22no—
261C5×Q8—42no—
262C5×Q8—42no—
263C5×Q8—82no—
264C5×D4—22no—
265C5×D4—22no—
266C5×D4—42no—
267C5×D4—42no—
268C5×D4—42no—
269C5×D4—42no—
270C5×D4—42no—
271C5×D4—42no—
272C5×D4—84no—
273C5×D4—82no—
274C5×D4—84no—
275C5×D4—88no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—44no—
284C2×C4×C5—44no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—82no—
297C2×C4×C5—82no—
298C2×C4×C5#12011yescharacteristic
299C2×C4×C5#12012yes—
300C2×C4×C5#12011yescharacteristic
301C2×C4×C5C22×D412yes—
302C2×C4×C5H(5,𝔽2)11yescharacteristic
303C2×C4×C5H(5,𝔽2)12yes—
304C2×C4×C52−1+411yescharacteristic
305C5×C8—42no—
306C5×C8—42no—
307C2×#41—22no—
308C2×#41C23×C512yes—
309C22×Q8C23×C511yescharacteristic
310C22×D4—22no—
311C22×D4C23×C511yescharacteristic
312C22×D4C23×C511yescharacteristic
313C23×C4C5×D412yes—
314#137—42no—
315#137—42no—
316AGL(1,ℤ/8)—42no—
317AGL(1,ℤ/8)—42no—
318C2×Q16—22no—
319C2×SD16—22no—
320C2×SD16—22no—
321C2×D8—22no—
322C2×M16C23×C512yes—
323C2×M16C23×C512yes—
324C4⋊Q8—22no—
325#127—22no—
326#127—42no—
327#126—24no—
328#125—22no—
329#124—24no—
330#124—22no—
331#124—22no—
332#124—22no—
333#124—42no—
334#123—24no—
335#123—24no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339#121—22no—
340#121—24no—
341#121—22no—
342#121—22no—
343#121—22no—
344#121—22no—
345#120—24no—
346C4×Q8C5×D412yescomplemented
347C4×Q8C5×D412yescomplemented
348C4×D4—24no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4C5×D412yescomplemented
354C4×D4C5×D412yescomplemented
355#117—22no—
356#117—22no—
357#117C23×C512yes—
358C2×C4⋊C4—21no—
359C2×C4⋊C4C23×C511yescharacteristic
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4C5×D412yes—
365C2×C42—22no—
366C2×C42C23×C511yescharacteristic
367C2×C42C23×C511yescharacteristic
368C2×C42C23×C511yescharacteristic
369C4⋊C8—22no—
370C4⋊C8—22no—
371#104—42no—
372#104—42no—
373#103—22no—
374#103—22no—
375#102—22no—
376#102—22no—
377#100—42no—
378#95—21no—
379#95—22no—
380C22×C5—21no—
381C22×C5—21no—
382C22×C5—44no—
383C22×C5—42no—
384C22×C5—42no—
385C22×C5—42no—
386C22×C5—42no—
387C22×C5—42no—
388C22×C5—42no—
389C22×C5—82no—
390C22×C5—82no—
391C22×C5—82no—
392C22×C5—81no—
393C22×C5—81no—
394C22×C5—81no—
395C22×C5C2×#12011yescharacteristic
396C22×C5#53511yescharacteristic
397C22×C5#53711yescharacteristic
398C4×C5—22no—
399C4×C5—21no—
400C4×C5—21no—
401C4×C5—42no—
402C4×C5—42no—
403C4×C5—42no—
404C4×C5—42no—
405C4×C5—42no—
406C4×C5—81no—
407C4×C5—81no—
408C4×C5#53512yes—
409C4×C5#53712yes—
410C24—21no—
411C24C2×C5×D412yes—
412#41—22no—
413#41—22no—
414#41—22no—
415#41—22no—
416#41—84no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8C2×C5×D412yes—
423C2×D4—22no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—82no—
438C2×D4C2×C5×D412yes—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—82no—
448C22×C4C2×C5×D412yes—
449C22×C4C2×C5×D411yescharacteristic
450C22×C4C2×C5×D411yescharacteristic
451C22×C4C24×C511yescharacteristic
452Q16—44no—
453SD16—44no—
454SD16—44no—
455D8—44no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460C2×C8—22no—
461C2×C8—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4C2×C5×D412yes—
470C4⋊C4C2×C5×D412yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—44no—
479C22⋊C4—41no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—82no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42C2×C5×D412yes—
489C42C2×C5×D412yes—
490C2×C5—21no—
491C2×C5—21no—
492C2×C5—82no—
493C2×C5—82no—
494C2×C5—161no—
495C2×C5#261511yescharacteristic
496C2×C5#301811yescharacteristic
497C2×C5#301911yescharacteristic
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C5×D411yescharacteristic
510Q8—22no—
511Q8—22no—
512Q8—42no—
513Q8—42no—
514Q8—82no—
515D4—22no—
516D4—22no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—82no—
524D4—84no—
525D4—84no—
526D4—88no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4C5×#12011yescharacteristic
550C2×C4C5×#12012yes—
551C2×C4C5×#12011yescharacteristic
552C2×C4C22×C5×D412yes—
553C2×C4C5×H(5,𝔽2)11yescharacteristic
554C2×C4C5×H(5,𝔽2)12yes—
555C2×C4C5×2−1+411yescharacteristic
556C8—42no—
557C8—42no—
558C5#2968211yescharacteristic, direct product factor, Sylow
559C22—21no—
560C22—21no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×C5×#12011yescharacteristic
575C22C5×#53511yescharacteristic
576C22C5×#53711yescharacteristic
577C4—22no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4C5×#53512yes—
588C4C5×#53712yes—
589C2—21no—
590C2—21no—
591C2—82no—
592C2—82no—
593C2—161no—
594C2C5×#261511yescharacteristic
595C2C5×#301811yescharacteristic
596C2C5×#301911yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
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×#3460C2semidirect
2. C5×#3040C2semidirect
3. C5×#2544C2semidirect
4. C5×#2028C2semidirect
5. C5×#2027C2semidirect
6. C5×#2007C2semidirect
7. C5×#2006C2semidirect
8. C5×#2004C2semidirect
9. C5×#2002C2semidirect
10. C5×#1806C2semidirect
11. C5×#1800C2semidirect
47. #29682C5direct (N × H)
87. C4×C5×Q8D4semidirect
88. C4×C5×Q8D4semidirect
94. C4×C5×D4D4semidirect
95. C4×C5×D4D4semidirect
121. #3460C2×C5semidirect
122. #3040C2×C5semidirect
123. #2544C2×C5semidirect
124. #2028C2×C5semidirect
125. #2027C2×C5semidirect
126. #2007C2×C5semidirect
127. #2006C2×C5semidirect
128. #2004C2×C5semidirect
129. #2002C2×C5semidirect
130. #1806C2×C5semidirect
131. #1800C2×C5semidirect
346. C4×Q8C5×D4semidirect
347. C4×Q8C5×D4semidirect
353. C4×D4C5×D4semidirect
354. C4×D4C5×D4semidirect
558. C5#29682direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34602
C5×#30402
C5×#25442
C5×#20282
C5×#20272
C5×#20072
C5×#20062
C5×#20042
C5×#20022
C5×#18062
C5×#18002
#296825