Finite Group Databasebeta

← C5×#27037 #11949757

Subgroups — C5×#27037 #11949757

Order 1280.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2494C211yesmaximal, characteristic
2C5×#2360C211yesmaximal, characteristic
3C5×#2074C211yesmaximal, characteristic, complemented
4C5×#2066C211yesmaximal, characteristic, complemented
5C5×#2036C211yesmaximal, characteristic
6C5×#2035C211yesmaximal, characteristic
7C5×#1949C211yesmaximal, characteristic, complemented
8C5×#1941C211yesmaximal, characteristic, complemented
9C5×#1923C211yesmaximal, characteristic
10C5×#1921C211yesmaximal, characteristic, complemented
11C5×#1920C211yesmaximal, characteristic, complemented
12C5×#1865C211yesmaximal, characteristic, complemented
13C5×#1853C211yesmaximal, characteristic
14C5×#1823C211yesmaximal, characteristic
15C4×C5×#103C211yesmaximal, characteristic, complemented
16C2×C5×C22⋊Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C2×C5×#107C2211yescharacteristic
20C2×C5×#106C2211yescharacteristic
21C2×C5×#106C2211yescharacteristic
22C2×C5×#106C2211yescharacteristic
23C2×C5×C4⋊C8C2211yescharacteristic
24C2×C5×#103C2211yescharacteristic
25C2×C5×#103C2211yescharacteristic
26C2×C5×#103C2211yescharacteristic
27C2×C5×#103C2211yescharacteristic
28C2×C5×C22⋊C8C2211yescharacteristic
29C2×C5×C22⋊C8C2211yescharacteristic
30C2×C4×C5×C8C2211yescharacteristic
31C5×#401—21no—
32C5×#397—21no—
33C5×#391C2211yescharacteristic
34C5×#389—21no—
35C5×#389C2211yescharacteristic
36C5×#389C2211yescharacteristic
37C5×#389C2211yescharacteristic
38C5×#388—21no—
39C5×#387—21no—
40C5×#387—21no—
41C5×#385—21no—
42C5×#382C2211yescharacteristic
43C5×#380C2211yescharacteristic
44C5×#380C2211yescharacteristic
45C4×C5×C4⋊C4C2211yescharacteristic
46C4×C5×C4⋊C4C2211yescharacteristic
47C4×C5×C22⋊C4—21no—
48C4×C5×C22⋊C4C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54C5×#340C2211yescharacteristic
55C5×#340C2211yescharacteristic
56C5×#340C2211yescharacteristic
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#27037C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×Q8C2311yescharacteristic
61C23×C4×C5—21no—
62C23×C4×C5C2311yescharacteristic
63C22×C5×C8C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C5×C22⋊Q8—22no—
68C5×C22⋊Q8—22no—
69C4×C5×Q8—22no—
70C4×C5×Q8—41no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—22no—
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—41no—
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C22⋊C4—21no—
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—21no—
89C2×C5×C22⋊C4D411yescharacteristic
90C2×C5×C22⋊C4D411yescharacteristic
91C2×C42×C5—21no—
92C2×C42×C5—21no—
93C2×C42×C5—21no—
94C2×C42×C5C2311yescharacteristic
95C2×C42×C5C2311yescharacteristic
96C5×#107—22no—
97C5×#106—22no—
98C5×#106—22no—
99C5×#106—41no—
100C5×C4⋊C8—22no—
101C5×#103—22no—
102C5×#103—22no—
103C5×#103—22no—
104C5×#103—41no—
105C5×C22⋊C8—22no—
106C5×C22⋊C8—41no—
107C4×C5×C8—41no—
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×#95—21no—
119C5×#95D411yescharacteristic
120C5×#95D411yescharacteristic
121#2494C2×C511yescharacteristic
122#2360C2×C511yescharacteristic
123#2074C2×C511yescharacteristic, complemented
124#2066C2×C511yescharacteristic, complemented
125#2036C2×C511yescharacteristic
126#2035C2×C511yescharacteristic
127#1949C2×C511yescharacteristic, complemented
128#1941C2×C511yescharacteristic, complemented
129#1923C2×C511yescharacteristic
130#1921C2×C511yescharacteristic, complemented
131#1920C2×C511yescharacteristic, complemented
132#1865C2×C511yescharacteristic, complemented
133#1853C2×C511yescharacteristic
134#1823C2×C511yescharacteristic
135C4×#103C2×C511yescharacteristic, complemented
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#4112yes—
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×C5—42no—
158C22×C4×C5—42no—
159C22×C4×C5—81no—
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2411yescharacteristic
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—21no—
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—42no—
192C5×C4⋊C4—42no—
193C5×C4⋊C4—42no—
194C5×C4⋊C4—81no—
195C5×C4⋊C4—81no—
196C5×C4⋊C4#4112yes—
197C5×C4⋊C4#4112yes—
198C5×C22⋊C4—24no—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—42no—
202C5×C22⋊C4—81no—
203C42×C5—21no—
204C42×C5—21no—
205C42×C5—21no—
206C42×C5—21no—
207C42×C5—42no—
208C42×C5—81no—
209C42×C5—81no—
210C2×C22⋊Q8C22×C511yescharacteristic
211C2×C4×Q8C22×C511yescharacteristic
212C22×C4⋊C4C22×C511yescharacteristic
213C2×#107C22×C511yescharacteristic
214C2×#106C22×C511yescharacteristic
215C2×#106C22×C511yescharacteristic
216C2×#106C22×C511yescharacteristic
217C2×C4⋊C8C22×C511yescharacteristic
218C2×#103C22×C511yescharacteristic
219C2×#103C22×C511yescharacteristic
220C2×#103C22×C511yescharacteristic
221C2×#103C22×C511yescharacteristic
222C2×C22⋊C8C22×C511yescharacteristic
223C2×C22⋊C8C22×C511yescharacteristic
224C2×C4×C8C22×C511yescharacteristic
225#401—21no—
226#397—21no—
227#391C22×C511yescharacteristic
228#389—21no—
229#389C22×C511yescharacteristic
230#389C22×C511yescharacteristic
231#389C22×C511yescharacteristic
232#388—21no—
233#387—21no—
234#387—21no—
235#385—21no—
236#382C22×C511yescharacteristic
237#380C22×C511yescharacteristic
238#380C22×C511yescharacteristic
239C4×C4⋊C4C22×C511yescharacteristic
240C4×C4⋊C4C22×C511yescharacteristic
241C4×C22⋊C4—21no—
242C4×C22⋊C4C22×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#13512yes—
260C23×C5C22×D411yescharacteristic
261C5×Q8—22no—
262C5×Q8—22no—
263C5×Q8—41no—
264C5×Q8—41no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—22no—
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—21no—
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—84no—
308C2×C4×C5#13712yes—
309C2×C4×C5C2×#4111yescharacteristic
310C2×C4×C5C2×#4111yescharacteristic
311C2×C4×C5C2×#4111yescharacteristic
312C2×C4×C5C2×#4111yescharacteristic
313C2×C4×C5H(5,𝔽2)11yescharacteristic
314C2×C4×C52−1+411yescharacteristic
315C5×C8—22no—
316C5×C8—41no—
317C5×C8—41no—
318C5×C8—41no—
319C22×Q8C23×C511yescharacteristic
320C23×C4—21no—
321C23×C4C23×C511yescharacteristic
322C22×C8C23×C511yescharacteristic
323C22×C8C23×C511yescharacteristic
324C22×C8C23×C511yescharacteristic
325C22×C8C23×C511yescharacteristic
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C4×Q8—22no—
329C4×Q8—41no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—41no—
338C2×C4⋊C4C23×C511yescharacteristic
339C2×C4⋊C4C23×C511yescharacteristic
340C2×C4⋊C4C23×C511yescharacteristic
341C2×C4⋊C4C23×C511yescharacteristic
342C2×C4⋊C4C23×C511yescharacteristic
343C2×C4⋊C4C23×C511yescharacteristic
344C2×C4⋊C4C23×C511yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4C5×D411yescharacteristic
349C2×C22⋊C4C5×D411yescharacteristic
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42C23×C511yescharacteristic
354C2×C42C23×C511yescharacteristic
355#107—22no—
356#106—22no—
357#106—22no—
358#106—41no—
359C4⋊C8—22no—
360#103—22no—
361#103—22no—
362#103—22no—
363#103—41no—
364C22⋊C8—22no—
365C22⋊C8—41no—
366C4×C8—41no—
367#95—21no—
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#95C5×D411yescharacteristic
379#95C5×D411yescharacteristic
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—41no—
384C22×C5—41no—
385C22×C5—41no—
386C22×C5—41no—
387C22×C5#52511yescharacteristic
388C22×C5#54611yescharacteristic
389C22×C5#54711yescharacteristic
390C22×C5C2×#13511yescharacteristic
391C22×C5C2×#13711yescharacteristic
392C22×C5#57711yescharacteristic
393C22×C5#57711yescharacteristic
394C4×C5—22no—
395C4×C5—41no—
396C4×C5—41no—
397C4×C5—42no—
398C4×C5—41no—
399C4×C5—42no—
400C4×C5—41no—
401C4×C5—42no—
402C4×C5—81no—
403C4×C5—81no—
404C4×C5—81no—
405C4×C5—81no—
406C4×C5—81no—
407C4×C5#52911yescharacteristic
408C4×C5#53911yescharacteristic
409C4×C5#55311yescharacteristic
410C4×C5#55611yescharacteristic
411C24C2×C5×D411yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8C5×#4112yes—
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×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—81no—
435C22×C4C2×C5×D411yescharacteristic
436C22×C4C2×C5×D411yescharacteristic
437C22×C4C2×C5×D411yescharacteristic
438C22×C4C2×C5×D411yescharacteristic
439C22×C4C2×C5×D411yescharacteristic
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×C8C5×#4112yes—
453C4⋊C4—24no—
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—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4C5×#4112yes—
472C4⋊C4C5×#4112yes—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—42no—
483C42—81no—
484C42—81no—
485C2×C5—42no—
486C2×C5#246611yescharacteristic
487C2×C5#310811yescharacteristic
488C2×C5#312511yescharacteristic
489C2×C5#329611yescharacteristic
490C2×C5#332011yescharacteristic
491C2×C5#333711yescharacteristic
492C2×C5#335611yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C5×#13512yes—
500C23C22×C5×D411yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
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×C4—84no—
548C2×C4C5×#13712yes—
549C2×C4C2×C5×#4111yescharacteristic
550C2×C4C2×C5×#4111yescharacteristic
551C2×C4C2×C5×#4111yescharacteristic
552C2×C4C2×C5×#4111yescharacteristic
553C2×C4C5×H(5,𝔽2)11yescharacteristic
554C2×C4C5×2−1+411yescharacteristic
555C8—22no—
556C8—41no—
557C8—41no—
558C8—41no—
559C5#2703711yescharacteristic, direct product factor, Sylow
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22C5×#52511yescharacteristic
568C22C5×#54611yescharacteristic
569C22C5×#54711yescharacteristic
570C22C2×C5×#13511yescharacteristic
571C22C2×C5×#13711yescharacteristic
572C22C5×#57711yescharacteristic
573C22C5×#57711yescharacteristic
574C4—22no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4C5×#52911yescharacteristic
588C4C5×#53911yescharacteristic
589C4C5×#55311yescharacteristic
590C4C5×#55611yescharacteristic
591C2—42no—
592C2C5×#246611yescharacteristic
593C2C5×#310811yescharacteristic
594C2C5×#312511yescharacteristic
595C2C5×#329611yescharacteristic
596C2C5×#332011yescharacteristic
597C2C5×#333711yescharacteristic
598C2C5×#335611yescharacteristic
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×#2074C2semidirect
4. C5×#2066C2semidirect
7. C5×#1949C2semidirect
8. C5×#1941C2semidirect
10. C5×#1921C2semidirect
11. C5×#1920C2semidirect
12. C5×#1865C2semidirect
15. C4×C5×#103C2semidirect
59. #27037C5direct (N × H)
123. #2074C2×C5semidirect
124. #2066C2×C5semidirect
127. #1949C2×C5semidirect
128. #1941C2×C5semidirect
130. #1921C2×C5semidirect
131. #1920C2×C5semidirect
132. #1865C2×C5semidirect
135. C4×#103C2×C5semidirect
559. C5#27037direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24942
C5×#23602
C5×#20742
C5×#20662
C5×#20362
C5×#20352
C5×#19492
C5×#19412
C5×#19232
C5×#19212
C5×#19202
C5×#18652
C5×#18532
C5×#18232
C4×C5×#1032
#270375