Finite Group Databasebeta

← C5×#24351 #11947071

Subgroups — C5×#24351 #11947071

Order 1280.

1122 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3501C211yesmaximal, characteristic
2C5×#2855C212yesmaximal, complemented
3C4×C5×#123C211yesmaximal, characteristic
4C5×#2062C212yesmaximal, complemented
5C5×#2030C212yesmaximal, complemented
6C5×#1996C212yesmaximal, complemented
7C5×#1890C212yesmaximal
8C5×#1867C212yesmaximal
9C5×#1799C211yesmaximal, characteristic
10C5×#554—24no—
11C5×#554—22no—
12C5×D4×Q8—22no—
13C5×#544—21no—
14C5×#543—21no—
15C5×#543C2212yes—
16C5×#533—21no—
17C2×C5×#123C2211yescharacteristic
18C2×C5×C22⋊Q8—21no—
19C2×C5×C22⋊Q8C2212yes—
20C5×#518C2212yes—
21C2×C4×C5×D4C2211yescharacteristic
22C22×C42×C5C2211yescharacteristic
23C2×C5×#99C2212yes—
24C5×#391—22no—
25C5×#389—22no—
26C5×#388—21no—
27C5×#387—21no—
28C5×#387—24no—
29C5×#386C2211yescharacteristic
30C5×#385C2211yescharacteristic
31C5×#385C2214yes—
32C5×#381C2211yescharacteristic
33C5×#381C2214yes—
34C5×#380C2212yes—
35C4×C5×C4⋊C4—21no—
36C4×C5×C22⋊C4—21no—
37C4×C5×C22⋊C4C2211yescharacteristic
38C5×#342C2212yes—
39C5×#342C2214yes—
40C5×#342C2214yes—
41#24351C511yesmaximal, characteristic, direct product factor, Sylow
42C22×C5×Q8—21no—
43C22×C5×Q8D412yes—
44C22×C5×D4C2311yescharacteristic
45C23×C4×C5D411yescharacteristic
46C23×C4×C5C2311yescharacteristic
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C5×C4⋊Q8—21no—
50C5×C4⋊Q8—21no—
51C5×C4⋊Q8—24no—
52C5×#125—22no—
53C5×#125—24no—
54C5×#123—22no—
55C5×#123Q814yescomplemented
56C5×C22⋊Q8—22no—
57C5×C22⋊Q8—22no—
58C5×C22⋊Q8—24no—
59C5×C22⋊Q8—24no—
60C5×C22⋊Q8—22no—
61C5×C22⋊Q8—22no—
62C5×C22⋊Q8—22no—
63C5×C22⋊Q8—22no—
64C5×C22⋊Q8—22no—
65C5×C22⋊Q8—22no—
66C5×C22⋊Q8—24no—
67C5×C22⋊Q8—24no—
68C4×C5×Q8—22no—
69C4×C5×Q8—22no—
70C4×C5×D4—22no—
71C4×C5×D4—22no—
72C4×C5×D4—24no—
73C4×C5×D4—24no—
74C5×#117—22no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—22no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—22no—
80C2×C5×C4⋊C4—24no—
81C2×C5×C4⋊C4D411yescharacteristic
82C2×C5×C4⋊C4D414yes—
83C2×C5×C4⋊C4D412yes—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4—22no—
86C2×C5×C22⋊C4—21no—
87C2×C5×C22⋊C4D411yescharacteristic
88C2×C5×C22⋊C4C2311yescharacteristic
89C2×C5×C22⋊C4C2314yes—
90C2×C5×C22⋊C4C2311yescharacteristic
91C2×C5×C22⋊C4C2314yes—
92C2×C5×C22⋊C4C2311yescharacteristic
93C2×C42×C5—21no—
94C2×C42×C5—21no—
95C2×C42×C5—22no—
96C2×C42×C5—21no—
97C2×C42×C5—22no—
98C2×C42×C5D411yescharacteristic
99C5×#99—24no—
100C5×#95—21no—
101C5×#95—21no—
102C5×#95—21no—
103C5×#95—21no—
104C5×#95—21no—
105C5×#95—22no—
106C5×#95—24no—
107C5×#95—24no—
108C5×#95—24no—
109#3501C2×C511yescharacteristic
110#2855C2×C512yescomplemented
111C4×#123C2×C511yescharacteristic
112#2062C2×C512yescomplemented
113#2030C2×C512yescomplemented
114#1996C2×C512yescomplemented
115#1890C2×C512yes—
116#1867C2×C512yes—
117#1799C2×C511yescharacteristic
118C24×C5C2×D411yescharacteristic
119C24×C5C2411yescharacteristic
120C2×C5×Q8—22no—
121C2×C5×Q8—22no—
122C2×C5×Q8—22no—
123C2×C5×Q8—22no—
124C2×C5×Q8—44no—
125C2×C5×Q8—82no—
126C2×C5×D4—22no—
127C2×C5×D4—22no—
128C2×C5×D4C2×Q812yes—
129C2×C5×D4#4112yes—
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—21no—
135C22×C4×C5—21no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—24no—
143C22×C4×C5—42no—
144C22×C4×C5—42no—
145C22×C4×C5—42no—
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D412yes—
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D414yes—
151C22×C4×C5C2×D414yes—
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D412yes—
155C22×C4×C5C2×Q812yescomplemented
156C5×C4⋊C4—22no—
157C5×C4⋊C4—24no—
158C5×C4⋊C4—22no—
159C5×C4⋊C4—22no—
160C5×C4⋊C4—24no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—24no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—22no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—24no—
168C5×C4⋊C4—44no—
169C5×C4⋊C4—42no—
170C5×C4⋊C4—48no—
171C5×C22⋊C4—24no—
172C5×C22⋊C4—24no—
173C5×C22⋊C4—24no—
174C5×C22⋊C4—22no—
175C5×C22⋊C4—24no—
176C5×C22⋊C4—21no—
177C5×C22⋊C4—22no—
178C5×C22⋊C4—24no—
179C5×C22⋊C4—22no—
180C5×C22⋊C4—24no—
181C5×C22⋊C4—42no—
182C5×C22⋊C4C2×Q812yescomplemented
183C42×C5—24no—
184C42×C5—21no—
185C42×C5—21no—
186C42×C5—42no—
187C42×C5—41no—
188C42×C5—42no—
189C42×C5—41no—
190#554—24no—
191#554—22no—
192D4×Q8—22no—
193#544—21no—
194#543—21no—
195#543C22×C512yes—
196#533—21no—
197C2×#123C22×C511yescharacteristic
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8C22×C512yes—
200#518C22×C512yes—
201C2×C4×D4C22×C511yescharacteristic
202C22×C42C22×C511yescharacteristic
203C2×#99C22×C512yes—
204#391—22no—
205#389—22no—
206#388—21no—
207#387—24no—
208#387—21no—
209#386C22×C511yescharacteristic
210#385C22×C511yescharacteristic
211#385C22×C514yes—
212#381C22×C514yes—
213#381C22×C511yescharacteristic
214#380C22×C512yes—
215C4×C4⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4C22×C511yescharacteristic
218#342C22×C512yes—
219#342C22×C514yes—
220#342C22×C514yes—
221C23×C5—21no—
222C23×C5—21no—
223C23×C5—41no—
224C23×C5—41no—
225C23×C5—41no—
226C23×C5—41no—
227C23×C5C22⋊Q812yes—
228C23×C5C22×D411yescharacteristic
229C23×C5C22×D412yes—
230C23×C5C22×Q811yescharacteristic
231C23×C5C2×#4111yescharacteristic
232C23×C52−1+412yes—
233C5×Q8—44no—
234C5×Q8—82no—
235C5×Q8—82no—
236C5×Q8—82no—
237C5×D4—24no—
238C5×D4—24no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—22no—
242C2×C4×C5—24no—
243C2×C4×C5—22no—
244C2×C4×C5—24no—
245C2×C4×C5—21no—
246C2×C4×C5—24no—
247C2×C4×C5—24no—
248C2×C4×C5—21no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—24no—
252C2×C4×C5—42no—
253C2×C4×C5—41no—
254C2×C4×C5—41no—
255C2×C4×C5—41no—
256C2×C4×C5—42no—
257C2×C4×C5—41no—
258C2×C4×C5—41no—
259C2×C4×C5—41no—
260C2×C4×C5—41no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—44no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—44no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—44no—
270C2×C4×C5—44no—
271C2×C4×C5—41no—
272C2×C4×C5—44no—
273C2×C4×C5—81no—
274C2×C4×C5—82no—
275C2×C4×C5—82no—
276C2×C4×C5#12011yescharacteristic
277C2×C4×C5#12012yes—
278C2×C4×C5#12011yescharacteristic
279C2×C4×C5C22⋊Q812yes—
280C22×Q8—21no—
281C22×Q8C5×D412yes—
282C22×D4C23×C511yescharacteristic
283C23×C4C5×D411yescharacteristic
284C23×C4C23×C511yescharacteristic
285C23×C4C23×C511yescharacteristic
286C23×C4C23×C511yescharacteristic
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—24no—
290#125—22no—
291#125—24no—
292#123—22no—
293#123C5×Q814yescomplemented
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—24no—
311C4×D4—24no—
312#117—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4C5×D411yescharacteristic
320C2×C4⋊C4C5×D414yes—
321C2×C4⋊C4C5×D412yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4C5×D411yescharacteristic
326C2×C22⋊C4C23×C511yescharacteristic
327C2×C22⋊C4C23×C514yes—
328C2×C22⋊C4C23×C511yescharacteristic
329C2×C22⋊C4C23×C514yes—
330C2×C22⋊C4C23×C511yescharacteristic
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42C5×D411yescharacteristic
337#99—24no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—24no—
347C22×C5—22no—
348C22×C5—21no—
349C22×C5—22no—
350C22×C5—21no—
351C22×C5—22no—
352C22×C5—41no—
353C22×C5—41no—
354C22×C5—42no—
355C22×C5—42no—
356C22×C5—41no—
357C22×C5—41no—
358C22×C5—81no—
359C22×C5C2×#12011yescharacteristic
360C22×C5C2×C22⋊Q811yescharacteristic
361C22×C5#53611yescharacteristic
362C22×C5#54812yes—
363C22×C5D4×Q812yes—
364C4×C5—22no—
365C4×C5—21no—
366C4×C5—22no—
367C4×C5—42no—
368C4×C5—42no—
369C4×C5—41no—
370C4×C5—44no—
371C4×C5—42no—
372C4×C5—44no—
373C4×C5—41no—
374C4×C5—84no—
375C4×C5—81no—
376C4×C5#45812yes—
377C24C2×C5×D411yescharacteristic
378C24C24×C511yescharacteristic
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—44no—
384C2×Q8—82no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4C2×C5×Q812yes—
388C2×D4C5×#4112yes—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C2×C5×D411yescharacteristic
406C22×C4C2×C5×D412yes—
407C22×C4C2×C5×D411yescharacteristic
408C22×C4C2×C5×D414yes—
409C22×C4C2×C5×D411yescharacteristic
410C22×C4C2×C5×D411yescharacteristic
411C22×C4C2×C5×D414yes—
412C22×C4C2×C5×D411yescharacteristic
413C22×C4C2×C5×D412yes—
414C22×C4C2×C5×Q812yescomplemented
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—48no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—24no—
435C22⋊C4—21no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—24no—
440C22⋊C4—42no—
441C22⋊C4C2×C5×Q812yescomplemented
442C42—24no—
443C42—21no—
444C42—21no—
445C42—42no—
446C42—41no—
447C42—42no—
448C42—41no—
449C2×C5—21no—
450C2×C5—22no—
451C2×C5—21no—
452C2×C5—22no—
453C2×C5—42no—
454C2×C5#240911yescharacteristic
455C2×C5C2×#45811yescharacteristic
456C2×C5#302511yescharacteristic
457C23—21no—
458C23—21no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23C5×C22⋊Q812yes—
464C23C22×C5×D411yescharacteristic
465C23C22×C5×D412yes—
466C23C22×C5×Q811yescharacteristic
467C23C2×C5×#4111yescharacteristic
468C23C5×2−1+412yes—
469Q8—44no—
470Q8—82no—
471Q8—82no—
472Q8—82no—
473D4—24no—
474D4—24no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4C5×#12011yescharacteristic
513C2×C4C5×#12012yes—
514C2×C4C5×#12011yescharacteristic
515C2×C4C5×C22⋊Q812yes—
516C5#2435111yescharacteristic, direct product factor, Sylow
517C22—22no—
518C22—21no—
519C22—22no—
520C22—21no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—41no—
528C22—81no—
529C22C2×C5×#12011yescharacteristic
530C22C2×C5×C22⋊Q811yescharacteristic
531C22C5×#53611yescharacteristic
532C22C5×#54812yes—
533C22C5×D4×Q812yes—
534C4—21no—
535C4—22no—
536C4—22no—
537C4—42no—
538C4—41no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—41no—
543C4—42no—
544C4—84no—
545C4—81no—
546C4C5×#45812yes—
547C2—21no—
548C2—22no—
549C2—21no—
550C2—22no—
551C2—42no—
552C2C5×#240911yescharacteristic
553C2C2×C5×#45811yescharacteristic
554C2C5×#302511yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC24C24×C5
prime radicalC22×C5×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2855C2semidirect
4. C5×#2062C2semidirect
5. C5×#2030C2semidirect
6. C5×#1996C2semidirect
41. #24351C5direct (N × H)
55. C5×#123Q8semidirect
110. #2855C2×C5semidirect
112. #2062C2×C5semidirect
113. #2030C2×C5semidirect
114. #1996C2×C5semidirect
155. C22×C4×C5C2×Q8semidirect
182. C5×C22⋊C4C2×Q8semidirect
293. #123C5×Q8semidirect
414. C22×C4C2×C5×Q8semidirect
441. C22⋊C4C2×C5×Q8semidirect
516. C5#24351direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35012
C5×#28552
C4×C5×#1232
C5×#20622
C5×#20302
C5×#19962
C5×#18902
C5×#18672
C5×#17992
#243515