Finite Group Databasebeta

← C5×#27015 #11949735

Subgroups — C5×#27015 #11949735

Order 1280.

770 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2452C211yesmaximal, characteristic
2C5×#2365C211yesmaximal, characteristic, complemented
3C5×#2067C211yesmaximal, characteristic
4C5×#2058C211yesmaximal, characteristic, complemented
5C5×#2057C211yesmaximal, characteristic, complemented
6C5×#2023C211yesmaximal, characteristic
7C5×#1987C211yesmaximal, characteristic, complemented
8C5×#1943C211yesmaximal, characteristic, complemented
9C5×#1926C211yesmaximal, characteristic
10C5×#1925C211yesmaximal, characteristic
11C5×#1921C211yesmaximal, characteristic, complemented
12C5×#1919C211yesmaximal, characteristic, complemented
13C5×#1917C211yesmaximal, characteristic, complemented
14C5×#1864C211yesmaximal, characteristic
15C5×#1762C211yesmaximal, characteristic
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×C22⋊Q8—21no—
19C2×C4×C5×Q8—21no—
20C2×C4×C5×D4C2211yescharacteristic
21C2×C5×#107C2211yescharacteristic
22C2×C5×#107C2211yescharacteristic
23C2×C5×#106C2211yescharacteristic
24C2×C5×#106C2211yescharacteristic
25C2×C5×C4⋊C8C2211yescharacteristic
26C2×C5×#102C2211yescharacteristic
27C2×C5×#102C2211yescharacteristic
28C2×C5×#102C2211yescharacteristic
29C2×C5×#102C2211yescharacteristic
30C2×C5×#97C2211yescharacteristic
31C5×#396—21no—
32C5×#393—21no—
33C5×#391—21no—
34C5×#390C2211yescharacteristic
35C5×#390C2211yescharacteristic
36C5×#389C2211yescharacteristic
37C5×#389C2211yescharacteristic
38C5×#389C2211yescharacteristic
39C5×#389C2211yescharacteristic
40C5×#387—21no—
41C5×#387—21no—
42C5×#385—21no—
43C4×C5×C4⋊C4C2211yescharacteristic
44C4×C5×C4⋊C4C2211yescharacteristic
45C4×C5×C4⋊C4C2211yescharacteristic
46C4×C5×C4⋊C4C2211yescharacteristic
47C5×#340C2211yescharacteristic
48C5×#340C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52C5×#340C2211yescharacteristic
53C5×#340C2211yescharacteristic
54C5×#340C2211yescharacteristic
55C5×#336C2211yescharacteristic
56C5×#336C2211yescharacteristic
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#27015C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×Q8—21no—
61C22×C5×Q8—21no—
62C22×C5×D4C2311yescharacteristic
63C23×C4×C5—21no—
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8—22no—
70C5×C4⋊Q8—22no—
71C5×C4⋊Q8—41no—
72C5×C22⋊Q8—44no—
73C4×C5×Q8—44no—
74C4×C5×D4—22no—
75C4×C5×D4—41no—
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⋊C4D411yescharacteristic
83C2×C5×C4⋊C4D411yescharacteristic
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×C4⋊C4C2311yescharacteristic
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4—21no—
93C2×C5×C22⋊C4—21no—
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—21no—
96C2×C42×C5—21no—
97C2×C42×C5—21no—
98C2×C42×C5—21no—
99C2×C42×C5C2311yescharacteristic
100C2×C42×C5C2311yescharacteristic
101C2×C42×C5C2311yescharacteristic
102C5×#107—22no—
103C5×#107—22no—
104C5×#106—22no—
105C5×#106—41no—
106C5×C4⋊C8—41no—
107C5×#102—22no—
108C5×#102—22no—
109C5×#102—41no—
110C5×#102Q814yescomplemented
111C5×#97—41no—
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#2452C2×C511yescharacteristic
122#2365C2×C511yescharacteristic, complemented
123#2067C2×C511yescharacteristic
124#2058C2×C511yescharacteristic, complemented
125#2057C2×C511yescharacteristic, complemented
126#2023C2×C511yescharacteristic
127#1987C2×C511yescharacteristic, complemented
128#1943C2×C511yescharacteristic, complemented
129#1926C2×C511yescharacteristic
130#1925C2×C511yescharacteristic
131#1921C2×C511yescharacteristic, complemented
132#1919C2×C511yescharacteristic, complemented
133#1917C2×C511yescharacteristic, complemented
134#1864C2×C511yescharacteristic
135#1762C2×C511yescharacteristic
136C24×C5—21no—
137C2×C5×Q8—24no—
138C2×C5×Q8—42no—
139C2×C5×Q8—42no—
140C2×C5×Q8—42no—
141C2×C5×Q8—42no—
142C2×C5×Q8—81no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4C2×Q812yes—
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—42no—
157C22×C4×C5—42no—
158C22×C4×C5—81no—
159C22×C4×C5C2×D411yescharacteristic
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×C8C2×Q812yes—
178C5×C4⋊C4—24no—
179C5×C4⋊C4—22no—
180C5×C4⋊C4—22no—
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—42no—
195C5×C4⋊C4—42no—
196C5×C4⋊C4C2×Q812yes—
197C5×C4⋊C4#4112yes—
198C5×C4⋊C4#4112yes—
199C5×C4⋊C4#4112yes—
200C5×C22⋊C4—42no—
201C5×C22⋊C4—42no—
202C5×C22⋊C4—42no—
203C5×C22⋊C4—42no—
204C5×C22⋊C4—81no—
205C42×C5—21no—
206C42×C5—21no—
207C42×C5—21no—
208C42×C5—21no—
209C42×C5—21no—
210C42×C5—21no—
211C42×C5—42no—
212C42×C5—42no—
213C42×C5—42no—
214C2×C4⋊Q8C22×C511yescharacteristic
215C2×C4⋊Q8C22×C511yescharacteristic
216C2×C22⋊Q8—21no—
217C2×C4×Q8—21no—
218C2×C4×D4C22×C511yescharacteristic
219C2×#107C22×C511yescharacteristic
220C2×#107C22×C511yescharacteristic
221C2×#106C22×C511yescharacteristic
222C2×#106C22×C511yescharacteristic
223C2×C4⋊C8C22×C511yescharacteristic
224C2×#102C22×C511yescharacteristic
225C2×#102C22×C511yescharacteristic
226C2×#102C22×C511yescharacteristic
227C2×#102C22×C511yescharacteristic
228C2×#97C22×C511yescharacteristic
229#396—21no—
230#393—21no—
231#391—21no—
232#390C22×C511yescharacteristic
233#390C22×C511yescharacteristic
234#389C22×C511yescharacteristic
235#389C22×C511yescharacteristic
236#389C22×C511yescharacteristic
237#389C22×C511yescharacteristic
238#387—21no—
239#387—21no—
240#385—21no—
241C4×C4⋊C4C22×C511yescharacteristic
242C4×C4⋊C4C22×C511yescharacteristic
243C4×C4⋊C4C22×C511yescharacteristic
244C4×C4⋊C4C22×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#336C22×C511yescharacteristic
254#336C22×C511yescharacteristic
255#336C22×C511yescharacteristic
256#336C22×C511yescharacteristic
257C23×C5—22no—
258C23×C5—41no—
259C23×C5—41no—
260C23×C5—41no—
261C23×C5—41no—
262C23×C5—41no—
263C23×C5—41no—
264C23×C5C22×D411yescharacteristic
265C5×Q8—48no—
266C5×Q8—84no—
267C5×D4—22no—
268C5×D4—22no—
269C5×D4—41no—
270C5×D4—41no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
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×C5—84no—
309C2×C4×C5AGL(1,ℤ/8)12yes—
310C2×C4×C5AGL(1,ℤ/8)12yes—
311C2×C4×C5C22×Q811yescharacteristic
312C2×C4×C5C2×#4111yescharacteristic
313C2×C4×C5C2×#4111yescharacteristic
314C2×C4×C5C2×#4111yescharacteristic
315C2×C4×C5H(5,𝔽2)11yescharacteristic
316C2×C4×C52−1+411yescharacteristic
317C5×C8—22no—
318C5×C8—41no—
319C5×C8—41no—
320C5×C8—41no—
321C22×Q8—21no—
322C22×Q8—21no—
323C22×D4C23×C511yescharacteristic
324C23×C4—21no—
325C22×C8C23×C511yescharacteristic
326C22×C8C23×C511yescharacteristic
327C22×C8C23×C511yescharacteristic
328C22×C8C23×C511yescharacteristic
329C4⋊Q8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332C4⋊Q8—41no—
333C22⋊Q8—44no—
334C4×Q8—44no—
335C4×D4—22no—
336C4×D4—41no—
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⋊C4C5×D411yescharacteristic
344C2×C4⋊C4C5×D411yescharacteristic
345C2×C4⋊C4C23×C511yescharacteristic
346C2×C4⋊C4C23×C511yescharacteristic
347C2×C4⋊C4C23×C511yescharacteristic
348C2×C4⋊C4C23×C511yescharacteristic
349C2×C4⋊C4C23×C511yescharacteristic
350C2×C4⋊C4C23×C511yescharacteristic
351C2×C4⋊C4C23×C511yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42C23×C511yescharacteristic
361C2×C42C23×C511yescharacteristic
362C2×C42C23×C511yescharacteristic
363#107—22no—
364#107—22no—
365#106—22no—
366#106—41no—
367C4⋊C8—41no—
368#102—22no—
369#102—22no—
370#102—41no—
371#102C5×Q814yescomplemented
372#97—41no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95C5×D411yescharacteristic
381#95C5×D411yescharacteristic
382C22×C5—42no—
383C22×C5—42no—
384C22×C5—42no—
385C22×C5—81no—
386C22×C5—81no—
387C22×C5—81no—
388C22×C5—81no—
389C22×C5#52511yescharacteristic
390C22×C5#54611yescharacteristic
391C22×C5D4×Q811yescharacteristic
392C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
393C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C5#57711yescharacteristic
395C22×C5#57811yescharacteristic
396C4×C5—22no—
397C4×C5—22no—
398C4×C5—42no—
399C4×C5—42no—
400C4×C5—41no—
401C4×C5—42no—
402C4×C5—41no—
403C4×C5—41no—
404C4×C5—41no—
405C4×C5—42no—
406C4×C5—81no—
407C4×C5—81no—
408C4×C5—81no—
409C4×C5#53311yescharacteristic
410C4×C5#54211yescharacteristic
411C4×C5#55411yescharacteristic
412C4×C5#55611yescharacteristic
413C24—21no—
414C2×Q8—24no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—81no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4C2×C5×Q812yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—81no—
436C22×C4C2×C5×D411yescharacteristic
437C22×C4C2×C5×D411yescharacteristic
438C22×C4C2×C5×D411yescharacteristic
439C22×C4C2×C5×D411yescharacteristic
440C22×C4C2×C5×D411yescharacteristic
441C22×C4C2×C5×D411yescharacteristic
442C22×C4C24×C511yescharacteristic
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—21no—
453C2×C8—21no—
454C2×C8C2×C5×Q812yes—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
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⋊C4C2×C5×Q812yes—
474C4⋊C4C5×#4112yes—
475C4⋊C4C5×#4112yes—
476C4⋊C4C5×#4112yes—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—81no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—42no—
490C42—42no—
491C2×C5—82no—
492C2×C5#246511yescharacteristic
493C2×C5#310911yescharacteristic
494C2×C5#312111yescharacteristic
495C2×C5#330811yescharacteristic
496C2×C5#332411yescharacteristic
497C2×C5#338411yescharacteristic
498C2×C5#338611yescharacteristic
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C5×D411yescharacteristic
507Q8—48no—
508Q8—84no—
509D4—22no—
510D4—22no—
511D4—41no—
512D4—41no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
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—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4C5×AGL(1,ℤ/8)12yes—
552C2×C4C5×AGL(1,ℤ/8)12yes—
553C2×C4C22×C5×Q811yescharacteristic
554C2×C4C2×C5×#4111yescharacteristic
555C2×C4C2×C5×#4111yescharacteristic
556C2×C4C2×C5×#4111yescharacteristic
557C2×C4C5×H(5,𝔽2)11yescharacteristic
558C2×C4C5×2−1+411yescharacteristic
559C8—22no—
560C8—41no—
561C8—41no—
562C8—41no—
563C5#2701511yescharacteristic, direct product factor, Sylow
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C5×#52511yescharacteristic
572C22C5×#54611yescharacteristic
573C22C5×D4×Q811yescharacteristic
574C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
575C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
576C22C5×#57711yescharacteristic
577C22C5×#57811yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4C5×#53311yescharacteristic
592C4C5×#54211yescharacteristic
593C4C5×#55411yescharacteristic
594C4C5×#55611yescharacteristic
595C2—82no—
596C2C5×#246511yescharacteristic
597C2C5×#310911yescharacteristic
598C2C5×#312111yescharacteristic
599C2C5×#330811yescharacteristic
600C2C5×#332411yescharacteristic
601C2C5×#338411yescharacteristic
602C2C5×#338611yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#2365C2semidirect
4. C5×#2058C2semidirect
5. C5×#2057C2semidirect
7. C5×#1987C2semidirect
8. C5×#1943C2semidirect
11. C5×#1921C2semidirect
12. C5×#1919C2semidirect
13. C5×#1917C2semidirect
59. #27015C5direct (N × H)
110. C5×#102Q8semidirect
122. #2365C2×C5semidirect
124. #2058C2×C5semidirect
125. #2057C2×C5semidirect
127. #1987C2×C5semidirect
128. #1943C2×C5semidirect
131. #1921C2×C5semidirect
132. #1919C2×C5semidirect
133. #1917C2×C5semidirect
371. #102C5×Q8semidirect
563. C5#27015direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24522
C5×#23652
C5×#20672
C5×#20582
C5×#20572
C5×#20232
C5×#19872
C5×#19432
C5×#19262
C5×#19252
C5×#19212
C5×#19192
C5×#19172
C5×#18642
C5×#17622
#270155