Finite Group Databasebeta

← C5×#25994 #11948714

Subgroups — C5×#25994 #11948714

Order 1280.

774 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2440C211yesmaximal, characteristic
2C5×#2345C211yesmaximal, characteristic
3C5×#2100C211yesmaximal, characteristic, complemented
4C5×#2087C211yesmaximal, characteristic, complemented
5C5×#2087C211yesmaximal, characteristic, complemented
6C5×#2057C211yesmaximal, characteristic, complemented
7C5×#1987C211yesmaximal, characteristic, complemented
8C5×#1947C211yesmaximal, characteristic, complemented
9C5×#1921C211yesmaximal, characteristic, complemented
10C5×#1921C211yesmaximal, characteristic, complemented
11C5×#1852C211yesmaximal, characteristic
12C5×#1850C211yesmaximal, characteristic
13C5×#1824C211yesmaximal, characteristic
14C5×#1808C211yesmaximal, characteristic
15C2×C5×#340C211yesmaximal, characteristic
16C23×C5×C8C2211yescharacteristic
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×C22⋊Q8—21no—
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C2×C5×#107C2211yescharacteristic
22C2×C5×#107C2211yescharacteristic
23C2×C5×#107C2211yescharacteristic
24C2×C5×#107C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C2×C5×C4⋊C8C2211yescharacteristic
27C2×C5×C22⋊C8C2211yescharacteristic
28C5×#398—21no—
29C5×#395—21no—
30C5×#389C2211yescharacteristic
31C5×#389C2211yescharacteristic
32C5×#389C2211yescharacteristic
33C5×#389C2211yescharacteristic
34C5×#387—21no—
35C5×#387—21no—
36C5×#385—21no—
37C5×#384C2211yescharacteristic
38C5×#380—21no—
39C5×#380C2211yescharacteristic
40C5×#380C2211yescharacteristic
41C4×C5×C4⋊C4C2211yescharacteristic
42C4×C5×C4⋊C4C2211yescharacteristic
43C4×C5×C22⋊C4—21no—
44C4×C5×C22⋊C4C2211yescharacteristic
45C5×#340C2211yescharacteristic
46C5×#340C2211yescharacteristic
47C5×#340C2211yescharacteristic
48C5×#340C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54C5×#340C2211yescharacteristic
55C5×#340C2211yescharacteristic
56C5×#340C2211yescharacteristic
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#25994C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×Q8—21no—
61C23×C4×C5—21no—
62C23×C4×C5—21no—
63C23×C4×C5C2311yescharacteristic
64C22×C5×C8—22no—
65C22×C5×C8—22no—
66C22×C5×C8—41no—
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C22×C5×C8C2311yescharacteristic
71C5×C4⋊Q8—22no—
72C5×C4⋊Q8—41no—
73C5×C22⋊Q8—44no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—22no—
76C2×C5×C4⋊C4—22no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4—41no—
86C2×C5×C4⋊C4Q814yescomplemented
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C4⋊C4C2311yescharacteristic
91C2×C5×C4⋊C4C2311yescharacteristic
92C2×C5×C4⋊C4C2311yescharacteristic
93C2×C5×C4⋊C4C2311yescharacteristic
94C2×C5×C4⋊C4C2311yescharacteristic
95C2×C5×C22⋊C4—21no—
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4D411yescharacteristic
98C2×C5×C22⋊C4D411yescharacteristic
99C2×C42×C5—21no—
100C2×C42×C5—21no—
101C2×C42×C5C2311yescharacteristic
102C2×C42×C5C2311yescharacteristic
103C5×#107—22no—
104C5×#107—22no—
105C5×#107—41no—
106C5×#107—41no—
107C5×C4⋊C8—22no—
108C5×C4⋊C8—41no—
109C5×C22⋊C8—22no—
110C5×#95—21no—
111C5×#95—21no—
112C5×#95—21no—
113C5×#95—21no—
114C5×#95—21no—
115C5×#95—21no—
116C5×#95—21no—
117C5×#95—21no—
118C5×#95D411yescharacteristic
119C5×#95D411yescharacteristic
120#2440C2×C511yescharacteristic
121#2345C2×C511yescharacteristic
122#2100C2×C511yescharacteristic, complemented
123#2087C2×C511yescharacteristic, complemented
124#2087C2×C511yescharacteristic, complemented
125#2057C2×C511yescharacteristic, complemented
126#1987C2×C511yescharacteristic, complemented
127#1947C2×C511yescharacteristic, complemented
128#1921C2×C511yescharacteristic, complemented
129#1921C2×C511yescharacteristic, complemented
130#1852C2×C511yescharacteristic
131#1850C2×C511yescharacteristic
132#1824C2×C511yescharacteristic
133#1808C2×C511yescharacteristic
134C2×#340C2×C511yescharacteristic
135C24×C5C2×D411yescharacteristic
136C2×C5×Q8—42no—
137C2×C5×Q8—42no—
138C2×C5×Q8—81no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—24no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—42no—
153C22×C4×C5—42no—
154C22×C4×C5—81no—
155C22×C4×C5—81no—
156C22×C4×C5—81no—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×Q812yes—
163C22×C4×C5#4112yes—
164C22×C4×C5C2411yescharacteristic
165C2×C5×C8—21no—
166C2×C5×C8—21no—
167C2×C5×C8—21no—
168C2×C5×C8—21no—
169C2×C5×C8—21no—
170C2×C5×C8—21no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—44no—
177C2×C5×C8#4112yes—
178C5×C4⋊C4—24no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—42no—
193C5×C4⋊C4—42no—
194C5×C4⋊C4—42no—
195C5×C4⋊C4—42no—
196C5×C4⋊C4—42no—
197C5×C4⋊C4—81no—
198C5×C4⋊C4—81no—
199C5×C4⋊C4—81no—
200C5×C4⋊C4C2×Q812yes—
201C5×C4⋊C4C2×Q812yes—
202C5×C4⋊C4#4112yes—
203C5×C22⋊C4—22no—
204C5×C22⋊C4—42no—
205C5×C22⋊C4—42no—
206C5×C22⋊C4Q1614yes—
207C42×C5—21no—
208C42×C5—21no—
209C42×C5—21no—
210C42×C5—21no—
211C42×C5—81no—
212C42×C5—81no—
213C23×C8C22×C511yescharacteristic
214C2×C4⋊Q8C22×C511yescharacteristic
215C2×C22⋊Q8—21no—
216C22×C4⋊C4C22×C511yescharacteristic
217C22×C4⋊C4C22×C511yescharacteristic
218C2×#107C22×C511yescharacteristic
219C2×#107C22×C511yescharacteristic
220C2×#107C22×C511yescharacteristic
221C2×#107C22×C511yescharacteristic
222C2×C4⋊C8C22×C511yescharacteristic
223C2×C4⋊C8C22×C511yescharacteristic
224C2×C22⋊C8C22×C511yescharacteristic
225#398—21no—
226#395—21no—
227#389C22×C511yescharacteristic
228#389C22×C511yescharacteristic
229#389C22×C511yescharacteristic
230#389C22×C511yescharacteristic
231#387—21no—
232#387—21no—
233#385—21no—
234#384C22×C511yescharacteristic
235#380—21no—
236#380C22×C511yescharacteristic
237#380C22×C511yescharacteristic
238C4×C4⋊C4C22×C511yescharacteristic
239C4×C4⋊C4C22×C511yescharacteristic
240C4×C22⋊C4—21no—
241C4×C22⋊C4C22×C511yescharacteristic
242#340C22×C511yescharacteristic
243#340C22×C511yescharacteristic
244#340C22×C511yescharacteristic
245#340C22×C511yescharacteristic
246#340C22×C511yescharacteristic
247#340C22×C511yescharacteristic
248#340C22×C511yescharacteristic
249#340C22×C511yescharacteristic
250#340C22×C511yescharacteristic
251#340C22×C511yescharacteristic
252#340C22×C511yescharacteristic
253#340C22×C511yescharacteristic
254#336C22×C511yescharacteristic
255#336C22×C511yescharacteristic
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5C22⋊Q812yes—
261C23×C5C2×Q1612yes—
262C23×C5AGL(1,ℤ/8)12yes—
263C23×C5C22×D411yescharacteristic
264C5×Q8—84no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—21no—
278C2×C4×C5—22no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
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—41no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5—41no—
303C2×C4×C5—41no—
304C2×C4×C5—41no—
305C2×C4×C5—84no—
306C2×C4×C5—84no—
307C2×C4×C5C22⋊Q812yes—
308C2×C4×C5C2×Q1612yes—
309C2×C4×C5C2×Q1612yes—
310C2×C4×C5C22×Q811yescharacteristic
311C2×C4×C5C2×#4111yescharacteristic
312C2×C4×C5C2×#4111yescharacteristic
313C2×C4×C5C2×#4111yescharacteristic
314C2×C4×C5H(5,𝔽2)11yescharacteristic
315C2×C4×C52−1+411yescharacteristic
316C5×C8—22no—
317C5×C8—41no—
318C5×C8—41no—
319C5×C8—41no—
320C22×Q8—21no—
321C23×C4—21no—
322C23×C4—21no—
323C23×C4C23×C511yescharacteristic
324C22×C8—22no—
325C22×C8—22no—
326C22×C8—41no—
327C22×C8C23×C511yescharacteristic
328C22×C8C23×C511yescharacteristic
329C22×C8C23×C511yescharacteristic
330C22×C8C23×C511yescharacteristic
331C4⋊Q8—22no—
332C4⋊Q8—41no—
333C22⋊Q8—44no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—41no—
346C2×C4⋊C4C5×Q814yescomplemented
347C2×C4⋊C4C23×C511yescharacteristic
348C2×C4⋊C4C23×C511yescharacteristic
349C2×C4⋊C4C23×C511yescharacteristic
350C2×C4⋊C4C23×C511yescharacteristic
351C2×C4⋊C4C23×C511yescharacteristic
352C2×C4⋊C4C23×C511yescharacteristic
353C2×C4⋊C4C23×C511yescharacteristic
354C2×C4⋊C4C23×C511yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4C5×D411yescharacteristic
358C2×C22⋊C4C5×D411yescharacteristic
359C2×C42—21no—
360C2×C42—21no—
361C2×C42C23×C511yescharacteristic
362C2×C42C23×C511yescharacteristic
363#107—22no—
364#107—22no—
365#107—41no—
366#107—41no—
367C4⋊C8—22no—
368C4⋊C8—41no—
369C22⋊C8—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95C5×D411yescharacteristic
379#95C5×D411yescharacteristic
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—22no—
384C22×C5—22no—
385C22×C5—22no—
386C22×C5#47714yes—
387C22×C5C2×C22⋊Q811yescharacteristic
388C22×C5#54611yescharacteristic
389C22×C5#54711yescharacteristic
390C22×C5C22×Q1611yescharacteristic
391C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
392C22×C5#57611yescharacteristic
393C22×C5#57711yescharacteristic
394C4×C5—22no—
395C4×C5—22no—
396C4×C5—22no—
397C4×C5—42no—
398C4×C5—42no—
399C4×C5—41no—
400C4×C5—41no—
401C4×C5—42no—
402C4×C5—81no—
403C4×C5—81no—
404C4×C5—81no—
405C4×C5—81no—
406C4×C5—81no—
407C4×C5#53311yescharacteristic
408C4×C5#54411yescharacteristic
409C4×C5#55111yescharacteristic
410C4×C5#55211yescharacteristic
411C24C2×C5×D411yescharacteristic
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—81no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4C2×C5×D411yescharacteristic
434C22×C4C2×C5×D411yescharacteristic
435C22×C4C2×C5×D411yescharacteristic
436C22×C4C2×C5×D411yescharacteristic
437C22×C4C2×C5×D411yescharacteristic
438C22×C4C2×C5×Q812yes—
439C22×C4C5×#4112yes—
440C22×C4C24×C511yescharacteristic
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—44no—
453C2×C8C5×#4112yes—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4C2×C5×Q812yes—
477C4⋊C4C2×C5×Q812yes—
478C4⋊C4C5×#4112yes—
479C22⋊C4—22no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4C5×Q1614yes—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—81no—
488C42—81no—
489C2×C5—24no—
490C2×C5#243811yescharacteristic
491C2×C5C2×#47711yescharacteristic
492C2×C5#308111yescharacteristic
493C2×C5#329911yescharacteristic
494C2×C5#331811yescharacteristic
495C2×C5#333811yescharacteristic
496C2×C5#334511yescharacteristic
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C5×C22⋊Q812yes—
502C23C2×C5×Q1612yes—
503C23C5×AGL(1,ℤ/8)12yes—
504C23C22×C5×D411yescharacteristic
505Q8—84no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4C5×C22⋊Q812yes—
549C2×C4C2×C5×Q1612yes—
550C2×C4C2×C5×Q1612yes—
551C2×C4C22×C5×Q811yescharacteristic
552C2×C4C2×C5×#4111yescharacteristic
553C2×C4C2×C5×#4111yescharacteristic
554C2×C4C2×C5×#4111yescharacteristic
555C2×C4C5×H(5,𝔽2)11yescharacteristic
556C2×C4C5×2−1+411yescharacteristic
557C8—22no—
558C8—41no—
559C8—41no—
560C8—41no—
561C5#2599411yescharacteristic, direct product factor, Sylow
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22C5×#47714yes—
569C22C2×C5×C22⋊Q811yescharacteristic
570C22C5×#54611yescharacteristic
571C22C5×#54711yescharacteristic
572C22C22×C5×Q1611yescharacteristic
573C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
574C22C5×#57611yescharacteristic
575C22C5×#57711yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4C5×#53311yescharacteristic
590C4C5×#54411yescharacteristic
591C4C5×#55111yescharacteristic
592C4C5×#55211yescharacteristic
593C2—24no—
594C2C5×#243811yescharacteristic
595C2C2×C5×#47711yescharacteristic
596C2C5×#308111yescharacteristic
597C2C5×#329911yescharacteristic
598C2C5×#331811yescharacteristic
599C2C5×#333811yescharacteristic
600C2C5×#334511yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC24×C5C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#2100C2semidirect
4. C5×#2087C2semidirect
5. C5×#2087C2semidirect
6. C5×#2057C2semidirect
7. C5×#1987C2semidirect
8. C5×#1947C2semidirect
9. C5×#1921C2semidirect
10. C5×#1921C2semidirect
59. #25994C5direct (N × H)
86. C2×C5×C4⋊C4Q8semidirect
122. #2100C2×C5semidirect
123. #2087C2×C5semidirect
124. #2087C2×C5semidirect
125. #2057C2×C5semidirect
126. #1987C2×C5semidirect
127. #1947C2×C5semidirect
128. #1921C2×C5semidirect
129. #1921C2×C5semidirect
346. C2×C4⋊C4C5×Q8semidirect
561. C5#25994direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24402
C5×#23452
C5×#21002
C5×#20872
C5×#20572
C5×#19872
C5×#19472
C5×#19212
C5×#18522
C5×#18502
C5×#18242
C5×#18082
C2×C5×#3402
#259945