Finite Group Databasebeta

← C5×#25996 #11948716

Subgroups — C5×#25996 #11948716

Order 1280.

758 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2442C211yesmaximal, characteristic
2C5×#2344C211yesmaximal, characteristic
3C5×#2101C211yesmaximal, characteristic, complemented
4C5×#2088C211yesmaximal, characteristic, complemented
5C5×#2087C211yesmaximal, characteristic, complemented
6C5×#2059C211yesmaximal, characteristic, complemented
7C5×#1989C211yesmaximal, characteristic, complemented
8C5×#1949C211yesmaximal, characteristic, complemented
9C5×#1921C211yesmaximal, characteristic, complemented
10C5×#1919C211yesmaximal, characteristic, complemented
11C5×#1853C211yesmaximal, characteristic
12C5×#1850C211yesmaximal, characteristic
13C5×#1823C211yesmaximal, characteristic
14C5×#1808C211yesmaximal, characteristic
15C2×C5×#340C211yesmaximal, characteristic
16C23×C5×C8C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C5×#117—21no—
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C4⋊C4C2211yescharacteristic
21C2×C5×#107C2211yescharacteristic
22C2×C5×#107C2211yescharacteristic
23C2×C5×#106C2211yescharacteristic
24C2×C5×#106C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C2×C5×C4⋊C8C2211yescharacteristic
27C2×C5×C22⋊C8C2211yescharacteristic
28C5×#400—21no—
29C5×#398—21no—
30C5×#389C2211yescharacteristic
31C5×#389C2211yescharacteristic
32C5×#389C2211yescharacteristic
33C5×#389C2211yescharacteristic
34C5×#387—21no—
35C5×#387—21no—
36C5×#385—21no—
37C5×#382C2211yescharacteristic
38C5×#381—21no—
39C5×#380—21no—
40C5×#380C2211yescharacteristic
41C5×#380C2211yescharacteristic
42C4×C5×C4⋊C4C2211yescharacteristic
43C4×C5×C4⋊C4C2211yescharacteristic
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#25996C511yesmaximal, characteristic, direct product factor, Sylow
60C23×C4×C5—21no—
61C23×C4×C5—21no—
62C23×C4×C5C2311yescharacteristic
63C22×C5×C8—22no—
64C22×C5×C8—22no—
65C22×C5×C8—41no—
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C5×#125—22no—
71C5×#125—41no—
72C5×#117—44no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—22no—
75C2×C5×C4⋊C4—22no—
76C2×C5×C4⋊C4—21no—
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—41no—
84C2×C5×C4⋊C4Q814yes—
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4D411yescharacteristic
96C2×C5×C22⋊C4D411yescharacteristic
97C2×C42×C5—21no—
98C2×C42×C5—21no—
99C2×C42×C5C2311yescharacteristic
100C2×C42×C5C2311yescharacteristic
101C5×#107—22no—
102C5×#107—41no—
103C5×#106—22no—
104C5×#106—41no—
105C5×C4⋊C8—22no—
106C5×C4⋊C8—41no—
107C5×C22⋊C8—22no—
108C5×#95—21no—
109C5×#95—21no—
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#2442C2×C511yescharacteristic
121#2344C2×C511yescharacteristic
122#2101C2×C511yescharacteristic, complemented
123#2088C2×C511yescharacteristic, complemented
124#2087C2×C511yescharacteristic, complemented
125#2059C2×C511yescharacteristic, complemented
126#1989C2×C511yescharacteristic, complemented
127#1949C2×C511yescharacteristic, complemented
128#1921C2×C511yescharacteristic, complemented
129#1919C2×C511yescharacteristic, complemented
130#1853C2×C511yescharacteristic
131#1850C2×C511yescharacteristic
132#1823C2×C511yescharacteristic
133#1808C2×C511yescharacteristic
134C2×#340C2×C511yescharacteristic
135C24×C5C2×D411yescharacteristic
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—24no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—42no—
150C22×C4×C5—42no—
151C22×C4×C5—42no—
152C22×C4×C5—42no—
153C22×C4×C5—81no—
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×Q812yes—
160C22×C4×C5#4112yes—
161C22×C4×C5C2411yescharacteristic
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
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—44no—
174C2×C5×C8#4112yes—
175C5×C4⋊C4—24no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
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—42no—
190C5×C4⋊C4—42no—
191C5×C4⋊C4—42no—
192C5×C4⋊C4—42no—
193C5×C4⋊C4—42no—
194C5×C4⋊C4—81no—
195C5×C4⋊C4—81no—
196C5×C4⋊C4C2×Q812yes—
197C5×C4⋊C4C2×Q812yes—
198C5×C4⋊C4#4112yes—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—42no—
202C5×C22⋊C4—42no—
203C42×C5—21no—
204C42×C5—21no—
205C42×C5—21no—
206C42×C5—21no—
207C42×C5—42no—
208C42×C5—42no—
209C23×C8C22×C511yescharacteristic
210C2×#125C22×C511yescharacteristic
211C2×#117—21no—
212C22×C4⋊C4C22×C511yescharacteristic
213C22×C4⋊C4C22×C511yescharacteristic
214C2×#107C22×C511yescharacteristic
215C2×#107C22×C511yescharacteristic
216C2×#106C22×C511yescharacteristic
217C2×#106C22×C511yescharacteristic
218C2×C4⋊C8C22×C511yescharacteristic
219C2×C4⋊C8C22×C511yescharacteristic
220C2×C22⋊C8C22×C511yescharacteristic
221#400—21no—
222#398—21no—
223#389C22×C511yescharacteristic
224#389C22×C511yescharacteristic
225#389C22×C511yescharacteristic
226#389C22×C511yescharacteristic
227#387—21no—
228#387—21no—
229#385—21no—
230#382C22×C511yescharacteristic
231#381—21no—
232#380—21no—
233#380C22×C511yescharacteristic
234#380C22×C511yescharacteristic
235C4×C4⋊C4C22×C511yescharacteristic
236C4×C4⋊C4C22×C511yescharacteristic
237C4×C22⋊C4C22×C511yescharacteristic
238#340C22×C511yescharacteristic
239#340C22×C511yescharacteristic
240#340C22×C511yescharacteristic
241#340C22×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#336C22×C511yescharacteristic
251#336C22×C511yescharacteristic
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5C22⋊Q812yes—
257C23×C5#13512yes—
258C23×C5AGL(1,ℤ/8)12yes—
259C23×C5C22×D411yescharacteristic
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—22no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—48no—
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—84no—
304C2×C4×C5C22⋊Q812yes—
305C2×C4×C5C22×Q811yescharacteristic
306C2×C4×C5C2×#4111yescharacteristic
307C2×C4×C5C2×#4111yescharacteristic
308C2×C4×C5C2×#4111yescharacteristic
309C2×C4×C5H(5,𝔽2)11yescharacteristic
310C2×C4×C52−1+411yescharacteristic
311C5×C8—22no—
312C5×C8—41no—
313C5×C8—41no—
314C5×C8—41no—
315C23×C4—21no—
316C23×C4—21no—
317C23×C4C23×C511yescharacteristic
318C22×C8—22no—
319C22×C8—22no—
320C22×C8—41no—
321C22×C8C23×C511yescharacteristic
322C22×C8C23×C511yescharacteristic
323C22×C8C23×C511yescharacteristic
324C22×C8C23×C511yescharacteristic
325#125—22no—
326#125—41no—
327#117—44no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—41no—
339C2×C4⋊C4C5×Q814yes—
340C2×C4⋊C4C23×C511yescharacteristic
341C2×C4⋊C4C23×C511yescharacteristic
342C2×C4⋊C4C23×C511yescharacteristic
343C2×C4⋊C4C23×C511yescharacteristic
344C2×C4⋊C4C23×C511yescharacteristic
345C2×C4⋊C4C23×C511yescharacteristic
346C2×C4⋊C4C23×C511yescharacteristic
347C2×C4⋊C4C23×C511yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4C5×D411yescharacteristic
351C2×C22⋊C4C5×D411yescharacteristic
352C2×C42—21no—
353C2×C42—21no—
354C2×C42C23×C511yescharacteristic
355C2×C42C23×C511yescharacteristic
356#107—22no—
357#107—41no—
358#106—22no—
359#106—41no—
360C4⋊C8—22no—
361C4⋊C8—41no—
362C22⋊C8—22no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95C5×D411yescharacteristic
374#95C5×D411yescharacteristic
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—22no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5#47814yes—
382C22×C5C2×C22⋊Q811yescharacteristic
383C22×C5#54611yescharacteristic
384C22×C5#54711yescharacteristic
385C22×C5C2×#13511yescharacteristic
386C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C5#57711yescharacteristic
388C22×C5#57811yescharacteristic
389C4×C5—22no—
390C4×C5—42no—
391C4×C5—42no—
392C4×C5—41no—
393C4×C5—41no—
394C4×C5—41no—
395C4×C5—42no—
396C4×C5—41no—
397C4×C5—81no—
398C4×C5—81no—
399C4×C5—81no—
400C4×C5—81no—
401C4×C5—81no—
402C4×C5#53311yescharacteristic
403C4×C5#54411yescharacteristic
404C4×C5#55111yescharacteristic
405C4×C5#55211yescharacteristic
406C24C2×C5×D411yescharacteristic
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—81no—
425C22×C4C2×C5×D411yescharacteristic
426C22×C4C2×C5×D411yescharacteristic
427C22×C4C2×C5×D411yescharacteristic
428C22×C4C2×C5×D411yescharacteristic
429C22×C4C2×C5×D411yescharacteristic
430C22×C4C2×C5×Q812yes—
431C22×C4C5×#4112yes—
432C22×C4C24×C511yescharacteristic
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—44no—
445C2×C8C5×#4112yes—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4C2×C5×Q812yes—
468C4⋊C4C2×C5×Q812yes—
469C4⋊C4C5×#4112yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—42no—
479C42—42no—
480C2×C5—24no—
481C2×C5#243811yescharacteristic
482C2×C5C2×#47811yescharacteristic
483C2×C5#308211yescharacteristic
484C2×C5#329611yescharacteristic
485C2×C5#330811yescharacteristic
486C2×C5#333611yescharacteristic
487C2×C5#334511yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C5×C22⋊Q812yes—
493C23C5×#13512yes—
494C23C5×AGL(1,ℤ/8)12yes—
495C23C22×C5×D411yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—48no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
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—84no—
540C2×C4C5×C22⋊Q812yes—
541C2×C4C22×C5×Q811yescharacteristic
542C2×C4C2×C5×#4111yescharacteristic
543C2×C4C2×C5×#4111yescharacteristic
544C2×C4C2×C5×#4111yescharacteristic
545C2×C4C5×H(5,𝔽2)11yescharacteristic
546C2×C4C5×2−1+411yescharacteristic
547C8—22no—
548C8—41no—
549C8—41no—
550C8—41no—
551C5#2599611yescharacteristic, direct product factor, Sylow
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22C5×#47814yes—
559C22C2×C5×C22⋊Q811yescharacteristic
560C22C5×#54611yescharacteristic
561C22C5×#54711yescharacteristic
562C22C2×C5×#13511yescharacteristic
563C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
564C22C5×#57711yescharacteristic
565C22C5×#57811yescharacteristic
566C4—22no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4C5×#53311yescharacteristic
580C4C5×#54411yescharacteristic
581C4C5×#55111yescharacteristic
582C4C5×#55211yescharacteristic
583C2—24no—
584C2C5×#243811yescharacteristic
585C2C2×C5×#47811yescharacteristic
586C2C5×#308211yescharacteristic
587C2C5×#329611yescharacteristic
588C2C5×#330811yescharacteristic
589C2C5×#333611yescharacteristic
590C2C5×#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×#2101C2semidirect
4. C5×#2088C2semidirect
5. C5×#2087C2semidirect
6. C5×#2059C2semidirect
7. C5×#1989C2semidirect
8. C5×#1949C2semidirect
9. C5×#1921C2semidirect
10. C5×#1919C2semidirect
59. #25996C5direct (N × H)
122. #2101C2×C5semidirect
123. #2088C2×C5semidirect
124. #2087C2×C5semidirect
125. #2059C2×C5semidirect
126. #1989C2×C5semidirect
127. #1949C2×C5semidirect
128. #1921C2×C5semidirect
129. #1919C2×C5semidirect
551. C5#25996direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24422
C5×#23442
C5×#21012
C5×#20882
C5×#20872
C5×#20592
C5×#19892
C5×#19492
C5×#19212
C5×#19192
C5×#18532
C5×#18502
C5×#18232
C5×#18082
C2×C5×#3402
#259965