Finite Group Databasebeta

← C5×#26015 #11948735

Subgroups — C5×#26015 #11948735

Order 1280.

696 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2442C211yesmaximal, characteristic
2C5×#2347C211yesmaximal, characteristic
3C5×#2101C211yesmaximal, characteristic, complemented
4C5×#2087C211yesmaximal, characteristic, complemented
5C5×#2087C211yesmaximal, characteristic, complemented
6C5×#2059C211yesmaximal, characteristic, complemented
7C5×#1989C211yesmaximal, characteristic, complemented
8C5×#1949C211yesmaximal, characteristic, complemented
9C5×#1918C211yesmaximal, characteristic, complemented
10C5×#1918C211yesmaximal, characteristic, complemented
11C5×#1852C211yesmaximal, characteristic
12C5×#1851C211yesmaximal, characteristic
13C5×#1825C211yesmaximal, characteristic
14C5×#1809C211yesmaximal, characteristic
15C5×#1736C211yesmaximal, characteristic
16C22×C5×M16C2211yescharacteristic
17C2×C5×#125C2211yescharacteristic
18C2×C5×#117C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C2×C5×#107C2211yescharacteristic
21C2×C5×#107C2211yescharacteristic
22C2×C5×#107C2211yescharacteristic
23C2×C5×#106C2211yescharacteristic
24C2×C5×C4⋊C8C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C2×C5×C22⋊C8C2211yescharacteristic
27C5×#400—21no—
28C5×#398—21no—
29C5×#389C2211yescharacteristic
30C5×#389C2211yescharacteristic
31C5×#389C2211yescharacteristic
32C5×#389C2211yescharacteristic
33C5×#388—21no—
34C5×#388—21no—
35C5×#385—21no—
36C5×#383—21no—
37C5×#382C2211yescharacteristic
38C5×#380—21no—
39C5×#380C2211yescharacteristic
40C5×#380C2211yescharacteristic
41C4×C5×C4⋊C4—21no—
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#26015C511yesmaximal, characteristic, direct product factor, Sylow
60C23×C4×C5—21no—
61C23×C4×C5C2311yescharacteristic
62C2×C5×M16—22no—
63C2×C5×M16—22no—
64C2×C5×M16—41no—
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C5×#125—22no—
70C5×#125—41no—
71C5×#117—22no—
72C5×#117Q814yes—
73C2×C5×C4⋊C4—22no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
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—41no—
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4D411yescharacteristic
94C2×C5×C22⋊C4D411yescharacteristic
95C2×C42×C5—21no—
96C2×C42×C5—21no—
97C2×C42×C5—21no—
98C2×C42×C5C2311yescharacteristic
99C2×C42×C5C2311yescharacteristic
100C5×#107—22no—
101C5×#107—22no—
102C5×#107—41no—
103C5×#106—41no—
104C5×C4⋊C8—22no—
105C5×C4⋊C8—41no—
106C5×C22⋊C8—22no—
107C5×#95—21no—
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×#95D411yescharacteristic
118C5×#95D411yescharacteristic
119#2442C2×C511yescharacteristic
120#2347C2×C511yescharacteristic
121#2101C2×C511yescharacteristic, complemented
122#2087C2×C511yescharacteristic, complemented
123#2087C2×C511yescharacteristic, complemented
124#2059C2×C511yescharacteristic, complemented
125#1989C2×C511yescharacteristic, complemented
126#1949C2×C511yescharacteristic, complemented
127#1918C2×C511yescharacteristic, complemented
128#1918C2×C511yescharacteristic, complemented
129#1852C2×C511yescharacteristic
130#1851C2×C511yescharacteristic
131#1825C2×C511yescharacteristic
132#1809C2×C511yescharacteristic
133#1736C2×C511yescharacteristic
134C24×C5C2×D411yescharacteristic
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
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—21no—
147C22×C4×C5—42no—
148C22×C4×C5—42no—
149C22×C4×C5—81no—
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×Q812yes—
156C22×C4×C5#4112yes—
157C22×C4×C5C2411yescharacteristic
158C5×M16—44no—
159C2×C5×C8—21no—
160C2×C5×C8—21no—
161C2×C5×C8—21no—
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#4112yes—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
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—42no—
185C5×C4⋊C4—42no—
186C5×C4⋊C4—42no—
187C5×C4⋊C4—42no—
188C5×C4⋊C4—42no—
189C5×C4⋊C4—81no—
190C5×C4⋊C4—81no—
191C5×C4⋊C4C2×Q812yes—
192C5×C4⋊C4C2×Q812yes—
193C5×C4⋊C4#4112yes—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—41no—
196C5×C22⋊C4—41no—
197C5×C22⋊C4—42no—
198C5×C22⋊C4—81no—
199C42×C5—24no—
200C42×C5—21no—
201C42×C5—21no—
202C42×C5—21no—
203C42×C5—21no—
204C42×C5—81no—
205C42×C5—81no—
206C22×M16C22×C511yescharacteristic
207C2×#125C22×C511yescharacteristic
208C2×#117C22×C511yescharacteristic
209C22×C4⋊C4C22×C511yescharacteristic
210C2×#107C22×C511yescharacteristic
211C2×#107C22×C511yescharacteristic
212C2×#107C22×C511yescharacteristic
213C2×#106C22×C511yescharacteristic
214C2×C4⋊C8C22×C511yescharacteristic
215C2×C4⋊C8C22×C511yescharacteristic
216C2×C22⋊C8C22×C511yescharacteristic
217#400—21no—
218#398—21no—
219#389C22×C511yescharacteristic
220#389C22×C511yescharacteristic
221#389C22×C511yescharacteristic
222#389C22×C511yescharacteristic
223#388—21no—
224#388—21no—
225#385—21no—
226#383—21no—
227#382C22×C511yescharacteristic
228#380—21no—
229#380C22×C511yescharacteristic
230#380C22×C511yescharacteristic
231C4×C4⋊C4—21no—
232C4×C4⋊C4C22×C511yescharacteristic
233C4×C4⋊C4C22×C511yescharacteristic
234C4×C22⋊C4C22×C511yescharacteristic
235#340C22×C511yescharacteristic
236#340C22×C511yescharacteristic
237#340C22×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#336C22×C511yescharacteristic
248#336C22×C511yescharacteristic
249C23×C5—21no—
250C23×C5—21no—
251C23×C5—21no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5C22⋊Q812yes—
256C23×C5C22×D411yescharacteristic
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
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—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
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—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—84no—
300C2×C4×C5C22⋊Q812yes—
301C2×C4×C5C22×Q811yescharacteristic
302C2×C4×C5C2×#4111yescharacteristic
303C2×C4×C5C2×#4111yescharacteristic
304C2×C4×C5C2×#4111yescharacteristic
305C2×C4×C5H(5,𝔽2)11yescharacteristic
306C2×C4×C52−1+411yescharacteristic
307C5×C8—22no—
308C5×C8—41no—
309C5×C8—41no—
310C5×C8—41no—
311C23×C4—21no—
312C23×C4C23×C511yescharacteristic
313C2×M16—22no—
314C2×M16—22no—
315C2×M16—41no—
316C22×C8C23×C511yescharacteristic
317C22×C8C23×C511yescharacteristic
318C22×C8C23×C511yescharacteristic
319C22×C8C23×C511yescharacteristic
320#125—22no—
321#125—41no—
322#117—22no—
323#117C5×Q814yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4C23×C511yescharacteristic
334C2×C4⋊C4C23×C511yescharacteristic
335C2×C4⋊C4C23×C511yescharacteristic
336C2×C4⋊C4C23×C511yescharacteristic
337C2×C4⋊C4C23×C511yescharacteristic
338C2×C4⋊C4C23×C511yescharacteristic
339C2×C4⋊C4C23×C511yescharacteristic
340C2×C4⋊C4C23×C511yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4C5×D411yescharacteristic
345C2×C22⋊C4C5×D411yescharacteristic
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C23×C511yescharacteristic
350C2×C42C23×C511yescharacteristic
351#107—22no—
352#107—22no—
353#107—41no—
354#106—41no—
355C4⋊C8—22no—
356C4⋊C8—41no—
357C22⋊C8—22no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95C5×D411yescharacteristic
369#95C5×D411yescharacteristic
370C22×C5—22no—
371C22×C5—22no—
372C22×C5—22no—
373C22×C5—41no—
374C22×C5—41no—
375C22×C5—41no—
376C22×C5—41no—
377C22×C5C2×C22⋊Q811yescharacteristic
378C22×C5#54611yescharacteristic
379C22×C5#54711yescharacteristic
380C22×C5#57511yescharacteristic
381C22×C5#57511yescharacteristic
382C22×C5#57611yescharacteristic
383C22×C5#57811yescharacteristic
384C4×C5—22no—
385C4×C5—42no—
386C4×C5—42no—
387C4×C5—41no—
388C4×C5—41no—
389C4×C5—41no—
390C4×C5—41no—
391C4×C5—42no—
392C4×C5—81no—
393C4×C5—81no—
394C4×C5—81no—
395C4×C5—81no—
396C4×C5—81no—
397C4×C5#53311yescharacteristic
398C4×C5#54411yescharacteristic
399C4×C5#55111yescharacteristic
400C4×C5#55211yescharacteristic
401C24C2×C5×D411yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—81no—
417C22×C4C2×C5×D411yescharacteristic
418C22×C4C2×C5×D411yescharacteristic
419C22×C4C2×C5×D411yescharacteristic
420C22×C4C2×C5×D411yescharacteristic
421C22×C4C2×C5×D411yescharacteristic
422C22×C4C2×C5×Q812yes—
423C22×C4C5×#4112yes—
424C22×C4C24×C511yescharacteristic
425M16—44no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8C5×#4112yes—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4C2×C5×Q812yes—
459C4⋊C4C2×C5×Q812yes—
460C4⋊C4C5×#4112yes—
461C22⋊C4—24no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C42—24no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—81no—
472C42—81no—
473C2×C5—42no—
474C2×C5#243811yescharacteristic
475C2×C5#308011yescharacteristic
476C2×C5#308311yescharacteristic
477C2×C5#331311yescharacteristic
478C2×C5#332211yescharacteristic
479C2×C5#334911yescharacteristic
480C2×C5#335511yescharacteristic
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23C5×C22⋊Q812yes—
488C23C22×C5×D411yescharacteristic
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
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—84no—
532C2×C4C5×C22⋊Q812yes—
533C2×C4C22×C5×Q811yescharacteristic
534C2×C4C2×C5×#4111yescharacteristic
535C2×C4C2×C5×#4111yescharacteristic
536C2×C4C2×C5×#4111yescharacteristic
537C2×C4C5×H(5,𝔽2)11yescharacteristic
538C2×C4C5×2−1+411yescharacteristic
539C8—22no—
540C8—41no—
541C8—41no—
542C8—41no—
543C5#2601511yescharacteristic, direct product factor, Sylow
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×C5×C22⋊Q811yescharacteristic
552C22C5×#54611yescharacteristic
553C22C5×#54711yescharacteristic
554C22C5×#57511yescharacteristic
555C22C5×#57511yescharacteristic
556C22C5×#57611yescharacteristic
557C22C5×#57811yescharacteristic
558C4—22no—
559C4—41no—
560C4—41no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4C5×#53311yescharacteristic
572C4C5×#54411yescharacteristic
573C4C5×#55111yescharacteristic
574C4C5×#55211yescharacteristic
575C2—42no—
576C2C5×#243811yescharacteristic
577C2C5×#308011yescharacteristic
578C2C5×#308311yescharacteristic
579C2C5×#331311yescharacteristic
580C2C5×#332211yescharacteristic
581C2C5×#334911yescharacteristic
582C2C5×#335511yescharacteristic
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×#2087C2semidirect
5. C5×#2087C2semidirect
6. C5×#2059C2semidirect
7. C5×#1989C2semidirect
8. C5×#1949C2semidirect
9. C5×#1918C2semidirect
10. C5×#1918C2semidirect
59. #26015C5direct (N × H)
121. #2101C2×C5semidirect
122. #2087C2×C5semidirect
123. #2087C2×C5semidirect
124. #2059C2×C5semidirect
125. #1989C2×C5semidirect
126. #1949C2×C5semidirect
127. #1918C2×C5semidirect
128. #1918C2×C5semidirect
543. C5#26015direct (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×#23472
C5×#21012
C5×#20872
C5×#20592
C5×#19892
C5×#19492
C5×#19182
C5×#18522
C5×#18512
C5×#18252
C5×#18092
C5×#17362
#260155