Finite Group Databasebeta

← C5×#32757 #11955477

Subgroups — C5×#32757 #11955477

Order 1280.

606 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2629C211yesmaximal, characteristic
2C5×#2625C211yesmaximal, characteristic
3C5×#2088C211yesmaximal, characteristic, complemented
4C5×#2086C211yesmaximal, characteristic, complemented
5C5×#2083C211yesmaximal, characteristic, complemented
6C5×#2076C211yesmaximal, characteristic
7C5×#2075C211yesmaximal, characteristic
8C5×#2072C211yesmaximal, characteristic, complemented
9C5×#2068C211yesmaximal, characteristic, complemented
10C5×#2068C211yesmaximal, characteristic, complemented
11C5×#2059C211yesmaximal, characteristic, complemented
12C5×#2036C211yesmaximal, characteristic
13C5×#2035C211yesmaximal, characteristic
14C5×#2026C211yesmaximal, characteristic, complemented
15C5×#1798C211yesmaximal, characteristic
16C2×C5×#125C2211yescharacteristic
17C2×C5×C22⋊Q8C2211yescharacteristic
18C2×C5×#117C2211yescharacteristic
19C2×C5×#107C2211yescharacteristic
20C2×C5×#107C2211yescharacteristic
21C2×C5×#107C2211yescharacteristic
22C2×C5×#106C2211yescharacteristic
23C2×C5×C4⋊C8C2211yescharacteristic
24C2×C5×C4⋊C8C2211yescharacteristic
25C2×C5×#103C2211yescharacteristic
26C2×C5×#103C2211yescharacteristic
27C2×C5×#103C2211yescharacteristic
28C2×C5×#103C2211yescharacteristic
29C2×C5×C22⋊C8C2211yescharacteristic
30C2×C5×C22⋊C8C2211yescharacteristic
31C5×#400—21no—
32C5×#398—21no—
33C5×#398—21no—
34C5×#398—21no—
35C5×#397—21no—
36C5×#397—21no—
37C5×#396—21no—
38C5×#396—21no—
39C5×#391C2211yescharacteristic
40C5×#391C2211yescharacteristic
41C5×#389C2211yescharacteristic
42C5×#389C2211yescharacteristic
43C5×#389C2211yescharacteristic
44C5×#389C2211yescharacteristic
45C5×#384C2211yescharacteristic
46C5×#380C2211yescharacteristic
47C5×#380C2211yescharacteristic
48C5×#380C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54C5×#340C2211yescharacteristic
55C5×#340C2211yescharacteristic
56C5×#340C2211yescharacteristic
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#32757C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×Q8C2311yescharacteristic
61C23×C4×C5C2311yescharacteristic
62C22×C5×C8C2311yescharacteristic
63C22×C5×C8C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C5×#125—41no—
67C5×#125—41no—
68C5×C22⋊Q8—41no—
69C5×C22⋊Q8—41no—
70C5×#117—41no—
71C5×#117—41no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
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⋊C4D411yescharacteristic
82C2×C5×C4⋊C4D411yescharacteristic
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⋊C4—21no—
94C2×C5×C22⋊C4D411yescharacteristic
95C2×C5×C22⋊C4D411yescharacteristic
96C2×C42×C5C2311yescharacteristic
97C2×C42×C5C2311yescharacteristic
98C5×#107—41no—
99C5×#107—41no—
100C5×#107—41no—
101C5×#106—41no—
102C5×C4⋊C8—41no—
103C5×C4⋊C8—41no—
104C5×#103—41no—
105C5×#103—41no—
106C5×#103—41no—
107C5×#103—41no—
108C5×C22⋊C8—41no—
109C5×C22⋊C8—41no—
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×#95—21no—
119C5×#95—21no—
120C5×#95—21no—
121#2629C2×C511yescharacteristic
122#2625C2×C511yescharacteristic
123#2088C2×C511yescharacteristic, complemented
124#2086C2×C511yescharacteristic, complemented
125#2083C2×C511yescharacteristic, complemented
126#2076C2×C511yescharacteristic
127#2075C2×C511yescharacteristic
128#2072C2×C511yescharacteristic, complemented
129#2068C2×C511yescharacteristic, complemented
130#2068C2×C511yescharacteristic, complemented
131#2059C2×C511yescharacteristic, complemented
132#2036C2×C511yescharacteristic
133#2035C2×C511yescharacteristic
134#2026C2×C511yescharacteristic, complemented
135#1798C2×C511yescharacteristic
136C24×C5C2×D411yescharacteristic
137C2×C5×Q8—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—21no—
142C2×C5×Q8—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—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2411yescharacteristic
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—21no—
174C2×C5×C8—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—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C4⋊C4—41no—
190C5×C4⋊C4—41no—
191C5×C4⋊C4—42no—
192C5×C4⋊C4—42no—
193C5×C4⋊C4—42no—
194C5×C4⋊C4—81no—
195C5×C4⋊C4—81no—
196C5×C4⋊C4—81no—
197C5×C4⋊C4—81no—
198C5×C4⋊C4—81no—
199C5×C4⋊C4—81no—
200C5×C22⋊C4—41no—
201C5×C22⋊C4—41no—
202C5×C22⋊C4—42no—
203C5×C22⋊C4—81no—
204C5×C22⋊C4—81no—
205C5×C22⋊C4—81no—
206C42×C5—21no—
207C42×C5—21no—
208C42×C5—21no—
209C42×C5—21no—
210C2×#125C22×C511yescharacteristic
211C2×C22⋊Q8C22×C511yescharacteristic
212C2×#117C22×C511yescharacteristic
213C2×#107C22×C511yescharacteristic
214C2×#107C22×C511yescharacteristic
215C2×#107C22×C511yescharacteristic
216C2×#106C22×C511yescharacteristic
217C2×C4⋊C8C22×C511yescharacteristic
218C2×C4⋊C8C22×C511yescharacteristic
219C2×#103C22×C511yescharacteristic
220C2×#103C22×C511yescharacteristic
221C2×#103C22×C511yescharacteristic
222C2×#103C22×C511yescharacteristic
223C2×C22⋊C8C22×C511yescharacteristic
224C2×C22⋊C8C22×C511yescharacteristic
225#400—21no—
226#398—21no—
227#398—21no—
228#398—21no—
229#397—21no—
230#397—21no—
231#396—21no—
232#396—21no—
233#391C22×C511yescharacteristic
234#391C22×C511yescharacteristic
235#389C22×C511yescharacteristic
236#389C22×C511yescharacteristic
237#389C22×C511yescharacteristic
238#389C22×C511yescharacteristic
239#384C22×C511yescharacteristic
240#380C22×C511yescharacteristic
241#380C22×C511yescharacteristic
242#380C22×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#336C22×C511yescharacteristic
252#336C22×C511yescharacteristic
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5C22×D411yescharacteristic
261C5×Q8—41no—
262C5×Q8—41no—
263C5×Q8—41no—
264C5×Q8—41no—
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—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—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—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5—41no—
303C2×C4×C5—41no—
304C2×C4×C5—41no—
305C2×C4×C5—41no—
306C2×C4×C5—41no—
307C2×C4×C5—41no—
308C2×C4×C5H(5,𝔽2)11yescharacteristic
309C2×C4×C5H(5,𝔽2)11yescharacteristic
310C2×C4×C5H(5,𝔽2)11yescharacteristic
311C2×C4×C52−1+411yescharacteristic
312C2×C4×C52−1+411yescharacteristic
313C2×C4×C52−1+411yescharacteristic
314C5×C8—41no—
315C5×C8—41no—
316C5×C8—41no—
317C5×C8—41no—
318C22×Q8C23×C511yescharacteristic
319C23×C4C23×C511yescharacteristic
320C22×C8C23×C511yescharacteristic
321C22×C8C23×C511yescharacteristic
322C22×C8C23×C511yescharacteristic
323C22×C8C23×C511yescharacteristic
324#125—41no—
325#125—41no—
326C22⋊Q8—41no—
327C22⋊Q8—41no—
328#117—41no—
329#117—41no—
330C2×C4⋊C4—21no—
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—21no—
339C2×C4⋊C4C5×D411yescharacteristic
340C2×C4⋊C4C5×D411yescharacteristic
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⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4C5×D411yescharacteristic
353C2×C22⋊C4C5×D411yescharacteristic
354C2×C42C23×C511yescharacteristic
355C2×C42C23×C511yescharacteristic
356#107—41no—
357#107—41no—
358#107—41no—
359#106—41no—
360C4⋊C8—41no—
361C4⋊C8—41no—
362#103—41no—
363#103—41no—
364#103—41no—
365#103—41no—
366C22⋊C8—41no—
367C22⋊C8—41no—
368#95—21no—
369#95—21no—
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#95—21no—
379C22×C5—41no—
380C22×C5—41no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5—41no—
385C22×C5—41no—
386C22×C5#53411yescharacteristic
387C22×C5#53611yescharacteristic
388C22×C5#53711yescharacteristic
389C22×C5#57611yescharacteristic
390C22×C5#57711yescharacteristic
391C22×C5#57811yescharacteristic
392C22×C5#57811yescharacteristic
393C4×C5—41no—
394C4×C5—41no—
395C4×C5—41no—
396C4×C5—41no—
397C4×C5—41no—
398C4×C5—81no—
399C4×C5—81no—
400C4×C5—81no—
401C4×C5—81no—
402C4×C5—81no—
403C4×C5—81no—
404C4×C5—81no—
405C4×C5—81no—
406C4×C5#56211yescharacteristic
407C4×C5#56211yescharacteristic
408C4×C5#56311yescharacteristic
409C4×C5#56311yescharacteristic
410C24C2×C5×D411yescharacteristic
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4C2×C5×D411yescharacteristic
432C22×C4C2×C5×D411yescharacteristic
433C22×C4C2×C5×D411yescharacteristic
434C22×C4C2×C5×D411yescharacteristic
435C22×C4C2×C5×D411yescharacteristic
436C22×C4C24×C511yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—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—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C2×C5—81no—
485C2×C5#281411yescharacteristic
486C2×C5#320111yescharacteristic
487C2×C5#320311yescharacteristic
488C2×C5#324211yescharacteristic
489C2×C5#324311yescharacteristic
490C2×C5#325711yescharacteristic
491C2×C5#326111yescharacteristic
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C22×C5×D411yescharacteristic
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
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—21no—
518C2×C4—21no—
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—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4C5×H(5,𝔽2)11yescharacteristic
548C2×C4C5×H(5,𝔽2)11yescharacteristic
549C2×C4C5×H(5,𝔽2)11yescharacteristic
550C2×C4C5×2−1+411yescharacteristic
551C2×C4C5×2−1+411yescharacteristic
552C2×C4C5×2−1+411yescharacteristic
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C5#3275711yescharacteristic, direct product factor, Sylow
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C5×#53411yescharacteristic
566C22C5×#53611yescharacteristic
567C22C5×#53711yescharacteristic
568C22C5×#57611yescharacteristic
569C22C5×#57711yescharacteristic
570C22C5×#57811yescharacteristic
571C22C5×#57811yescharacteristic
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4C5×#56211yescharacteristic
586C4C5×#56211yescharacteristic
587C4C5×#56311yescharacteristic
588C4C5×#56311yescharacteristic
589C2—81no—
590C2C5×#281411yescharacteristic
591C2C5×#320111yescharacteristic
592C2C5×#320311yescharacteristic
593C2C5×#324211yescharacteristic
594C2C5×#324311yescharacteristic
595C2C5×#325711yescharacteristic
596C2C5×#326111yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC24×C5C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C5×#2088C2semidirect
4. C5×#2086C2semidirect
5. C5×#2083C2semidirect
8. C5×#2072C2semidirect
9. C5×#2068C2semidirect
10. C5×#2068C2semidirect
11. C5×#2059C2semidirect
14. C5×#2026C2semidirect
59. #32757C5direct (N × H)
123. #2088C2×C5semidirect
124. #2086C2×C5semidirect
125. #2083C2×C5semidirect
128. #2072C2×C5semidirect
129. #2068C2×C5semidirect
130. #2068C2×C5semidirect
131. #2059C2×C5semidirect
134. #2026C2×C5semidirect
557. C5#32757direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#26292
C5×#26252
C5×#20882
C5×#20862
C5×#20832
C5×#20762
C5×#20752
C5×#20722
C5×#20682
C5×#20592
C5×#20362
C5×#20352
C5×#20262
C5×#17982
#327575