Finite Group Databasebeta

← C5×#25876 #11948596

Subgroups — C5×#25876 #11948596

Order 1280.

1222 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
1C5×#3499C211yesmaximal, characteristic
2C5×#3141C211yesmaximal, characteristic, complemented
3C5×#3079C211yesmaximal, characteristic, complemented
4C5×#2862C211yesmaximal, characteristic, complemented
5C5×#2038C214yesmaximal, complemented
6C5×#1966C211yesmaximal, characteristic, complemented
7C5×#1962C211yesmaximal, characteristic, complemented
8C5×#1793C214yesmaximal, complemented
9C4×C5×#104C211yesmaximal, characteristic, complemented
10C2×C5×2−1+4C2212yes—
11C5×D4×Q8—21no—
12C5×D4×Q8—21no—
13C5×#547—21no—
14C5×#547—21no—
15C5×#547—22no—
16C5×#547—22no—
17C5×#536—22no—
18C5×#536C2212yes—
19C5×#532—21no—
20C5×#532C2211yescharacteristic
21C2×C5×C4⋊Q8—21no—
22C2×C5×C4⋊Q8C2211yescharacteristic
23C2×C5×C4⋊Q8C2211yescharacteristic
24C4×C5×#41C2211yescharacteristic
25C5×#489—22no—
26C5×#487—21no—
27C5×#486—21no—
28C5×#477—21no—
29C5×#476—21no—
30C5×#475—21no—
31C5×#474—21no—
32C5×#428C2212yes—
33C5×#423C2211yescharacteristic
34C2×C5×#104C2212yes—
35C2×C5×#104C2214yes—
36C5×#417C2214yes—
37C5×#417C2212yes—
38C2×C5×#101C2214yes—
39C4×C5×M16C2211yescharacteristic
40C5×#391—22no—
41C5×#391C2214yes—
42C5×#382—22no—
43C43×C5C2211yescharacteristic
44C5×#339C2212yes—
45#25876C511yesmaximal, characteristic, direct product factor, Sylow
46C5×2−1+4—28no—
47C5×2−1+4—44no—
48C2×C5×#41—22no—
49C2×C5×#41—22no—
50C2×C5×#41—24no—
51C2×C5×#41C2312yes—
52C2×C5×#41C2311yescharacteristic
53C22×C5×Q8—22no—
54C22×C5×Q8—22no—
55C22×C5×Q8—21no—
56C22×C5×Q8C2312yes—
57C2×C5×M16C2312yes—
58C2×C5×M16C2312yes—
59C5×C4⋊Q8—22no—
60C5×C4⋊Q8—22no—
61C5×C4⋊Q8—22no—
62C5×C4⋊Q8—21no—
63C5×C4⋊Q8—21no—
64C5×C4⋊Q8—21no—
65C5×C4⋊Q8—21no—
66C5×C4⋊Q8—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8—21no—
69C5×C4⋊Q8—21no—
70C5×C4⋊Q8D411yescharacteristic
71C5×#127D411yescharacteristic
72C5×#124—22no—
73C5×#124D412yes—
74C5×#123—24no—
75C5×#123—24no—
76C5×#123—24no—
77C5×C22⋊Q8—24no—
78C5×C22⋊Q8—24no—
79C5×C22⋊Q8—22no—
80C5×C22⋊Q8—22no—
81C5×#121—22no—
82C5×#121—22no—
83C5×#121—21no—
84C5×#121—21no—
85C4×C5×Q8D411yescharacteristic
86C4×C5×Q8D411yescharacteristic
87C4×C5×D4—22no—
88C4×C5×D4—22no—
89C4×C5×D4—21no—
90C4×C5×D4—21no—
91C4×C5×D4—21no—
92C4×C5×D4—21no—
93C4×C5×D4D411yescharacteristic
94C4×C5×D4D411yescharacteristic
95C5×#117—21no—
96C5×#117C2311yescharacteristic
97C2×C5×C4⋊C4—22no—
98C2×C5×C4⋊C4—22no—
99C2×C5×C4⋊C4—21no—
100C2×C5×C4⋊C4D412yes—
101C2×C5×C4⋊C4D412yes—
102C2×C5×C4⋊C4C2312yes—
103C2×C42×C5—21no—
104C2×C42×C5—21no—
105C2×C42×C5—21no—
106C2×C42×C5C2312yes—
107C2×C42×C5C2311yescharacteristic
108C5×#107—22no—
109C5×#106—22no—
110C5×C4⋊C8—21no—
111C5×C4⋊C8—21no—
112C5×#104—24no—
113C5×#104—44no—
114C5×#103—22no—
115C5×#103—24no—
116C5×#102—22no—
117C5×#102—24no—
118C5×#101—44no—
119C5×#97—21no—
120C4×C5×C8—21no—
121C5×#95—28no—
122#3499C2×C511yescharacteristic
123#3141C2×C511yescharacteristic, complemented
124#3079C2×C511yescharacteristic, complemented
125#2862C2×C511yescharacteristic, complemented
126#2038C2×C514yescomplemented
127#1966C2×C511yescharacteristic, complemented
128#1962C2×C511yescharacteristic, complemented
129#1793C2×C514yescomplemented
130C4×#104C2×C511yescharacteristic, complemented
131C5×#41—22no—
132C5×#41—24no—
133C5×#41—24no—
134C5×#41—48no—
135C5×#41—416no—
136C5×#41—44no—
137C5×#41—44no—
138C5×#41#4114yes—
139C2×C5×Q8—24no—
140C2×C5×Q8—22no—
141C2×C5×Q8—22no—
142C2×C5×Q8—24no—
143C2×C5×Q8—24no—
144C2×C5×Q8—48no—
145C2×C5×Q8—44no—
146C2×C5×Q8—44no—
147C2×C5×Q8—44no—
148C2×C5×Q8—41no—
149C2×C5×Q8—41no—
150C2×C5×Q8C2×D412yes—
151C2×C5×Q8C2×D411yescharacteristic
152C2×C5×D4—21no—
153C2×C5×D4—24no—
154C2×C5×D4—22no—
155C2×C5×D4—21no—
156C2×C5×D4—21no—
157C2×C5×D4C2×D412yes—
158C2×C5×D4C2×D411yescharacteristic
159C22×C4×C5—21no—
160C22×C4×C5—24no—
161C22×C4×C5—24no—
162C22×C4×C5—22no—
163C22×C4×C5—21no—
164C22×C4×C5—21no—
165C22×C4×C5—22no—
166C22×C4×C5—22no—
167C22×C4×C5C2×D414yes—
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2411yescharacteristic
171C5×M16—22no—
172C5×M16—22no—
173C5×M16—24no—
174C2×C5×C8—22no—
175C2×C5×C8—22no—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—24no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—22no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—41no—
189C5×C4⋊C4—41no—
190C5×C4⋊C4C2×D411yescharacteristic
191C5×C4⋊C4C2×D411yescharacteristic
192C5×C22⋊C4—24no—
193C5×C22⋊C4—24no—
194C5×C22⋊C4—21no—
195C5×C22⋊C4—21no—
196C5×C22⋊C4—21no—
197C5×C22⋊C4—22no—
198C5×C22⋊C4—22no—
199C42×C5—24no—
200C42×C5—21no—
201C42×C5—21no—
202C42×C5—21no—
203C42×C5—22no—
204C42×C5—22no—
205C42×C5—42no—
206C42×C5C2×D411yescharacteristic
207C42×C5C2×D411yescharacteristic
208C42×C5C2×D411yescharacteristic
209C42×C5C2×D411yescharacteristic
210C2×2−1+4C22×C512yes—
211D4×Q8—21no—
212D4×Q8—21no—
213#547—21no—
214#547—21no—
215#547—22no—
216#547—22no—
217#536—22no—
218#536C22×C512yes—
219#532—21no—
220#532C22×C511yescharacteristic
221C2×C4⋊Q8—21no—
222C2×C4⋊Q8C22×C511yescharacteristic
223C2×C4⋊Q8C22×C511yescharacteristic
224C4×#41C22×C511yescharacteristic
225#489—22no—
226#487—21no—
227#486—21no—
228#477—21no—
229#476—21no—
230#475—21no—
231#474—21no—
232#428C22×C512yes—
233#423C22×C511yescharacteristic
234C2×#104C22×C512yes—
235C2×#104C22×C514yes—
236#417C22×C512yes—
237#417C22×C514yes—
238C2×#101C22×C514yes—
239C4×M16C22×C511yescharacteristic
240#391—22no—
241#391C22×C514yes—
242#382—22no—
243C43C22×C511yescharacteristic
244#339C22×C512yes—
245C23×C5—21no—
246C23×C5—22no—
247C23×C5C22×D411yescharacteristic
248C5×Q8—21no—
249C5×Q8—24no—
250C5×Q8—48no—
251C5×Q8—42no—
252C5×Q8—42no—
253C5×Q8—44no—
254C5×Q8—82no—
255C5×Q8—82no—
256C5×Q8#12412yes—
257C5×D4—24no—
258C5×D4—21no—
259C5×D4—24no—
260C5×D4—48no—
261C5×D4—44no—
262C5×D4—42no—
263C5×D4—42no—
264C5×D4#12412yes—
265C2×C4×C5—21no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—24no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—22no—
273C2×C4×C5—24no—
274C2×C4×C5—28no—
275C2×C4×C5—48no—
276C2×C4×C5—42no—
277C2×C4×C5—48no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—42no—
281C2×C4×C5—42no—
282C2×C4×C5—42no—
283C2×C4×C5—44no—
284C2×C4×C5—42no—
285C2×C4×C5#12011yescharacteristic
286C2×C4×C5#12011yescharacteristic
287C2×C4×C5#12011yescharacteristic
288C2×C4×C5#12011yescharacteristic
289C2×C4×C5C22×D411yescharacteristic
290C2×C4×C5C22×D411yescharacteristic
291C2×C4×C5C2×#4112yes—
292C2×C4×C5H(5,𝔽2)12yes—
293C5×C8—42no—
294C5×C8—42no—
2952−1+4—28no—
2962−1+4—44no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—24no—
300C2×#41C23×C512yes—
301C2×#41C23×C511yescharacteristic
302C22×Q8—22no—
303C22×Q8—22no—
304C22×Q8—21no—
305C22×Q8C23×C512yes—
306C2×M16C23×C512yes—
307C2×M16C23×C512yes—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8C5×D411yescharacteristic
320#127C5×D411yescharacteristic
321#124—22no—
322#124C5×D412yes—
323#123—24no—
324#123—24no—
325#123—24no—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330#121—22no—
331#121—22no—
332#121—21no—
333#121—21no—
334C4×Q8C5×D411yescharacteristic
335C4×Q8C5×D411yescharacteristic
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4C5×D411yescharacteristic
343C4×D4C5×D411yescharacteristic
344#117—21no—
345#117C23×C511yescharacteristic
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C5×D412yes—
350C2×C4⋊C4C5×D412yes—
351C2×C4⋊C4C23×C512yes—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42C23×C512yes—
356C2×C42C23×C511yescharacteristic
357#107—22no—
358#106—22no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361#104—24no—
362#104—44no—
363#103—22no—
364#103—24no—
365#102—22no—
366#102—24no—
367#101—44no—
368#97—21no—
369C4×C8—21no—
370#95—28no—
371C22×C5—21no—
372C22×C5—21no—
373C22×C5—22no—
374C22×C5—41no—
375C22×C5—41no—
376C22×C5—42no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5C2×#12011yescharacteristic
380C22×C5#54612yes—
381C4×C5—22no—
382C4×C5—21no—
383C4×C5—21no—
384C4×C5—44no—
385C4×C5—44no—
386C4×C5—42no—
387C4×C5—41no—
388C4×C5—41no—
389C4×C5—82no—
390C4×C5#45412yes—
391C4×C5#45612yes—
392C4×C5C2×#12411yescharacteristic
393C4×C5#53511yescharacteristic
394C4×C5#54612yes—
395#41—22no—
396#41—24no—
397#41—24no—
398#41—48no—
399#41—416no—
400#41—44no—
401#41—44no—
402#41C5×#4114yes—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—48no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8C2×C5×D412yes—
415C2×Q8C2×C5×D411yescharacteristic
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×C5×D412yes—
422C2×D4C2×C5×D411yescharacteristic
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4C2×C5×D414yes—
432C22×C4C2×C5×D411yescharacteristic
433C22×C4C2×C5×D411yescharacteristic
434C22×C4C24×C511yescharacteristic
435M16—22no—
436M16—22no—
437M16—24no—
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4C2×C5×D411yescharacteristic
455C4⋊C4C2×C5×D411yescharacteristic
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C42—24no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—42no—
470C42C2×C5×D411yescharacteristic
471C42C2×C5×D411yescharacteristic
472C42C2×C5×D411yescharacteristic
473C42C2×C5×D411yescharacteristic
474C2×C5—22no—
475C2×C5—42no—
476C2×C5—82no—
477C2×C5#243511yescharacteristic
478C2×C5C2×#45411yescharacteristic
479C2×C5C2×#45611yescharacteristic
480C23—21no—
481C23—22no—
482C23C22×C5×D411yescharacteristic
483Q8—21no—
484Q8—24no—
485Q8—48no—
486Q8—42no—
487Q8—42no—
488Q8—44no—
489Q8—82no—
490Q8—82no—
491Q8C5×#12412yes—
492D4—24no—
493D4—21no—
494D4—24no—
495D4—48no—
496D4—44no—
497D4—42no—
498D4—42no—
499D4C5×#12412yes—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4C5×#12011yescharacteristic
521C2×C4C5×#12011yescharacteristic
522C2×C4C5×#12011yescharacteristic
523C2×C4C5×#12011yescharacteristic
524C2×C4C22×C5×D411yescharacteristic
525C2×C4C22×C5×D411yescharacteristic
526C2×C4C2×C5×#4112yes—
527C2×C4C5×H(5,𝔽2)12yes—
528C8—42no—
529C8—42no—
530C5#2587611yescharacteristic, direct product factor, Sylow
531C22—21no—
532C22—21no—
533C22—22no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×C5×#12011yescharacteristic
540C22C5×#54612yes—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—44no—
545C4—41no—
546C4—41no—
547C4—44no—
548C4—42no—
549C4—82no—
550C4C5×#45412yes—
551C4C5×#45612yes—
552C4C2×C5×#12411yescharacteristic
553C4C5×#53511yescharacteristic
554C4C5×#54612yes—
555C2—22no—
556C2—42no—
557C2—82no—
558C2C5×#243511yescharacteristic
559C2C2×C5×#45411yescharacteristic
560C2C2×C5×#45611yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC5×#3499C2
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
2. C5×#3141C2semidirect
3. C5×#3079C2semidirect
4. C5×#2862C2semidirect
5. C5×#2038C2semidirect
6. C5×#1966C2semidirect
7. C5×#1962C2semidirect
8. C5×#1793C2semidirect
9. C4×C5×#104C2semidirect
45. #25876C5direct (N × H)
123. #3141C2×C5semidirect
124. #3079C2×C5semidirect
125. #2862C2×C5semidirect
126. #2038C2×C5semidirect
127. #1966C2×C5semidirect
128. #1962C2×C5semidirect
129. #1793C2×C5semidirect
130. C4×#104C2×C5semidirect
530. C5#25876direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34992
C5×#31412
C5×#30792
C5×#28622
C5×#20382
C5×#19662
C5×#19622
C5×#17932
C4×C5×#1042
#258765