Finite Group Databasebeta

← C5×#27031 #11949751

Subgroups — C5×#27031 #11949751

Order 1280.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2495C211yesmaximal, characteristic, complemented
2C5×#2361C211yesmaximal, characteristic
3C5×#2073C211yesmaximal, characteristic, complemented
4C5×#2065C211yesmaximal, characteristic, complemented
5C5×#2034C211yesmaximal, characteristic
6C5×#2033C211yesmaximal, characteristic
7C5×#1949C211yesmaximal, characteristic, complemented
8C5×#1941C211yesmaximal, characteristic, complemented
9C5×#1923C211yesmaximal, characteristic, complemented
10C5×#1919C211yesmaximal, characteristic, complemented
11C5×#1918C211yesmaximal, characteristic, complemented
12C5×#1864C211yesmaximal, characteristic, complemented
13C5×#1853C211yesmaximal, characteristic, complemented
14C5×#1824C211yesmaximal, characteristic, complemented
15C4×C5×#102C211yesmaximal, characteristic, complemented
16C2×C5×#121C2211yescharacteristic
17C2×C4×C5×D4—21no—
18C2×C4×C5×D4C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C2×C5×#107C2211yescharacteristic
21C2×C5×#106C2211yescharacteristic
22C2×C5×#106C2211yescharacteristic
23C2×C5×#106C2211yescharacteristic
24C2×C5×C4⋊C8C2211yescharacteristic
25C2×C5×#102C2211yescharacteristic
26C2×C5×#102C2211yescharacteristic
27C2×C5×#102C2211yescharacteristic
28C2×C5×#102C2211yescharacteristic
29C2×C5×C22⋊C8C2211yescharacteristic
30C2×C5×C22⋊C8C2211yescharacteristic
31C2×C4×C5×C8C2211yescharacteristic
32C5×#400—21no—
33C5×#397—21no—
34C5×#390C2211yescharacteristic
35C5×#389C2211yescharacteristic
36C5×#389C2211yescharacteristic
37C5×#389C2211yescharacteristic
38C5×#388—21no—
39C5×#388—21no—
40C5×#386—21no—
41C5×#385—21no—
42C5×#383—21no—
43C5×#382C2211yescharacteristic
44C5×#380C2211yescharacteristic
45C5×#380C2211yescharacteristic
46C4×C5×C4⋊C4C2211yescharacteristic
47C4×C5×C4⋊C4C2211yescharacteristic
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#27031C511yesmaximal, characteristic, direct product factor, Sylow
60C22×C5×D4—21no—
61C22×C5×D4C2311yescharacteristic
62C23×C4×C5—21no—
63C23×C4×C5—21no—
64C23×C4×C5C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C5×#121—22no—
70C5×#121—22no—
71C4×C5×D4—22no—
72C4×C5×D4—41no—
73C4×C5×D4—44no—
74C2×C5×C4⋊C4—22no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—41no—
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×C4⋊C4C2311yescharacteristic
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—21no—
89C2×C5×C22⋊C4—21no—
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×C5C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C5×#107—22no—
100C5×#106—22no—
101C5×#106—22no—
102C5×#106—41no—
103C5×C4⋊C8—22no—
104C5×#102—22no—
105C5×#102—22no—
106C5×#102—22no—
107C5×#102—41no—
108C5×C22⋊C8—22no—
109C5×C22⋊C8—41no—
110C4×C5×C8—41no—
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×#95D411yescharacteristic
121C5×#95D411yescharacteristic
122#2495C2×C511yescharacteristic, complemented
123#2361C2×C511yescharacteristic
124#2073C2×C511yescharacteristic, complemented
125#2065C2×C511yescharacteristic, complemented
126#2034C2×C511yescharacteristic
127#2033C2×C511yescharacteristic
128#1949C2×C511yescharacteristic, complemented
129#1941C2×C511yescharacteristic, complemented
130#1923C2×C511yescharacteristic, complemented
131#1919C2×C511yescharacteristic, complemented
132#1918C2×C511yescharacteristic, complemented
133#1864C2×C511yescharacteristic, complemented
134#1853C2×C511yescharacteristic, complemented
135#1824C2×C511yescharacteristic, complemented
136C4×#102C2×C511yescharacteristic, complemented
137C24×C5—21no—
138C24×C5C2×D411yescharacteristic
139C2×C5×D4—24no—
140C2×C5×D4—21no—
141C2×C5×D4—21no—
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—42no—
146C2×C5×D4—42no—
147C2×C5×D4#4112yes—
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—21no—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5—42no—
162C22×C4×C5—42no—
163C22×C4×C5—42no—
164C22×C4×C5—42no—
165C22×C4×C5—81no—
166C22×C4×C5—81no—
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2411yescharacteristic
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—21no—
177C2×C5×C8—21no—
178C2×C5×C8—21no—
179C2×C5×C8—21no—
180C2×C5×C8—21no—
181C2×C5×C8—21no—
182C2×C5×C8—21no—
183C2×C5×C8—21no—
184C2×C5×C8#4112yes—
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—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4—21no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—42no—
198C5×C4⋊C4—42no—
199C5×C4⋊C4—81no—
200C5×C4⋊C4—81no—
201C5×C4⋊C4#4112yes—
202C5×C4⋊C4#4112yes—
203C5×C22⋊C4—24no—
204C5×C22⋊C4—22no—
205C5×C22⋊C4—22no—
206C5×C22⋊C4—42no—
207C5×C22⋊C4—42no—
208C5×C22⋊C4—42no—
209C5×C22⋊C4—81no—
210C5×C22⋊C4—81no—
211C42×C5—21no—
212C42×C5—21no—
213C42×C5—21no—
214C42×C5—21no—
215C42×C5—42no—
216C42×C5—42no—
217C2×#121C22×C511yescharacteristic
218C2×C4×D4—21no—
219C2×C4×D4C22×C511yescharacteristic
220C22×C4⋊C4C22×C511yescharacteristic
221C2×#107C22×C511yescharacteristic
222C2×#106C22×C511yescharacteristic
223C2×#106C22×C511yescharacteristic
224C2×#106C22×C511yescharacteristic
225C2×C4⋊C8C22×C511yescharacteristic
226C2×#102C22×C511yescharacteristic
227C2×#102C22×C511yescharacteristic
228C2×#102C22×C511yescharacteristic
229C2×#102C22×C511yescharacteristic
230C2×C22⋊C8C22×C511yescharacteristic
231C2×C22⋊C8C22×C511yescharacteristic
232C2×C4×C8C22×C511yescharacteristic
233#400—21no—
234#397—21no—
235#390C22×C511yescharacteristic
236#389C22×C511yescharacteristic
237#389C22×C511yescharacteristic
238#389C22×C511yescharacteristic
239#388—21no—
240#388—21no—
241#386—21no—
242#385—21no—
243#383—21no—
244#382C22×C511yescharacteristic
245#380C22×C511yescharacteristic
246#380C22×C511yescharacteristic
247C4×C4⋊C4C22×C511yescharacteristic
248C4×C4⋊C4C22×C511yescharacteristic
249C4×C22⋊C4C22×C511yescharacteristic
250#340C22×C511yescharacteristic
251#340C22×C511yescharacteristic
252#340C22×C511yescharacteristic
253#340C22×C511yescharacteristic
254#340C22×C511yescharacteristic
255#340C22×C511yescharacteristic
256#340C22×C511yescharacteristic
257#340C22×C511yescharacteristic
258#336C22×C511yescharacteristic
259#336C22×C511yescharacteristic
260C23×C5—21no—
261C23×C5—21no—
262C23×C5—21no—
263C23×C5—21no—
264C23×C5—21no—
265C23×C5—21no—
266C23×C5—22no—
267C23×C5—41no—
268C23×C5—41no—
269C23×C5—41no—
270C23×C5—41no—
271C23×C5—41no—
272C23×C5—41no—
273C23×C5#13512yes—
274C23×C5C22×D411yescharacteristic
275C5×D4—22no—
276C5×D4—22no—
277C5×D4—48no—
278C5×D4—41no—
279C5×D4—41no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—22no—
284C2×C4×C5—21no—
285C2×C4×C5—22no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
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—41no—
309C2×C4×C5—41no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5—41no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—84no—
320C2×C4×C5—84no—
321C2×C4×C5AGL(1,ℤ/8)12yes—
322C2×C4×C5C2×#4111yescharacteristic
323C2×C4×C5C2×#4111yescharacteristic
324C2×C4×C5C2×#4111yescharacteristic
325C2×C4×C5C2×#4111yescharacteristic
326C2×C4×C5H(5,𝔽2)11yescharacteristic
327C2×C4×C52−1+411yescharacteristic
328C5×C8—22no—
329C5×C8—41no—
330C5×C8—41no—
331C5×C8—41no—
332C22×D4—21no—
333C22×D4C23×C511yescharacteristic
334C23×C4—21no—
335C23×C4—21no—
336C23×C4C23×C511yescharacteristic
337C22×C8C23×C511yescharacteristic
338C22×C8C23×C511yescharacteristic
339C22×C8C23×C511yescharacteristic
340C22×C8C23×C511yescharacteristic
341#121—22no—
342#121—22no—
343C4×D4—22no—
344C4×D4—41no—
345C4×D4—44no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4C23×C511yescharacteristic
353C2×C4⋊C4C23×C511yescharacteristic
354C2×C4⋊C4C23×C511yescharacteristic
355C2×C4⋊C4C23×C511yescharacteristic
356C2×C4⋊C4C23×C511yescharacteristic
357C2×C4⋊C4C23×C511yescharacteristic
358C2×C4⋊C4C23×C511yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4C5×D411yescharacteristic
366C2×C22⋊C4C5×D411yescharacteristic
367C2×C42—21no—
368C2×C42—21no—
369C2×C42C23×C511yescharacteristic
370C2×C42C23×C511yescharacteristic
371#107—22no—
372#106—22no—
373#106—22no—
374#106—41no—
375C4⋊C8—22no—
376#102—22no—
377#102—22no—
378#102—22no—
379#102—41no—
380C22⋊C8—22no—
381C22⋊C8—41no—
382C4×C8—41no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95C5×D411yescharacteristic
393#95C5×D411yescharacteristic
394C22×C5—22no—
395C22×C5—22no—
396C22×C5—22no—
397C22×C5—41no—
398C22×C5—41no—
399C22×C5—41no—
400C22×C5—41no—
401C22×C5—42no—
402C22×C5—42no—
403C22×C5—42no—
404C22×C5—81no—
405C22×C5—81no—
406C22×C5—81no—
407C22×C5—81no—
408C22×C5#52511yescharacteristic
409C22×C5#54611yescharacteristic
410C22×C5#54711yescharacteristic
411C22×C5C2×#13511yescharacteristic
412C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C5#57711yescharacteristic
414C22×C5#57711yescharacteristic
415C4×C5—22no—
416C4×C5—41no—
417C4×C5—41no—
418C4×C5—42no—
419C4×C5—41no—
420C4×C5—41no—
421C4×C5—42no—
422C4×C5—81no—
423C4×C5—81no—
424C4×C5—81no—
425C4×C5—81no—
426C4×C5—81no—
427C4×C5#52911yescharacteristic
428C4×C5#53911yescharacteristic
429C4×C5#55311yescharacteristic
430C4×C5#55611yescharacteristic
431C24—21no—
432C24C2×C5×D411yescharacteristic
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4C5×#4112yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4C2×C5×D411yescharacteristic
462C22×C4C2×C5×D411yescharacteristic
463C22×C4C2×C5×D411yescharacteristic
464C22×C4C2×C5×D411yescharacteristic
465C22×C4C2×C5×D411yescharacteristic
466C22×C4C24×C511yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8C5×#4112yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4C5×#4112yes—
496C4⋊C4C5×#4112yes—
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—42no—
511C2×C5—42no—
512C2×C5—82no—
513C2×C5#246611yescharacteristic
514C2×C5#310511yescharacteristic
515C2×C5#312511yescharacteristic
516C2×C5#330111yescharacteristic
517C2×C5#331011yescharacteristic
518C2×C5#333711yescharacteristic
519C2×C5#335411yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C5×#13512yes—
534C23C22×C5×D411yescharacteristic
535D4—22no—
536D4—22no—
537D4—48no—
538D4—41no—
539D4—41no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4C5×AGL(1,ℤ/8)12yes—
582C2×C4C2×C5×#4111yescharacteristic
583C2×C4C2×C5×#4111yescharacteristic
584C2×C4C2×C5×#4111yescharacteristic
585C2×C4C2×C5×#4111yescharacteristic
586C2×C4C5×H(5,𝔽2)11yescharacteristic
587C2×C4C5×2−1+411yescharacteristic
588C8—22no—
589C8—41no—
590C8—41no—
591C8—41no—
592C5#2703111yescharacteristic, direct product factor, Sylow
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C5×#52511yescharacteristic
608C22C5×#54611yescharacteristic
609C22C5×#54711yescharacteristic
610C22C2×C5×#13511yescharacteristic
611C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
612C22C5×#57711yescharacteristic
613C22C5×#57711yescharacteristic
614C4—22no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4C5×#52911yescharacteristic
627C4C5×#53911yescharacteristic
628C4C5×#55311yescharacteristic
629C4C5×#55611yescharacteristic
630C2—42no—
631C2—82no—
632C2C5×#246611yescharacteristic
633C2C5×#310511yescharacteristic
634C2C5×#312511yescharacteristic
635C2C5×#330111yescharacteristic
636C2C5×#331011yescharacteristic
637C2C5×#333711yescharacteristic
638C2C5×#335411yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#121C22
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
1. C5×#2495C2semidirect
3. C5×#2073C2semidirect
4. C5×#2065C2semidirect
7. C5×#1949C2semidirect
8. C5×#1941C2semidirect
9. C5×#1923C2semidirect
10. C5×#1919C2semidirect
11. C5×#1918C2semidirect
12. C5×#1864C2semidirect
13. C5×#1853C2semidirect
14. C5×#1824C2semidirect
15. C4×C5×#102C2semidirect
59. #27031C5direct (N × H)
122. #2495C2×C5semidirect
124. #2073C2×C5semidirect
125. #2065C2×C5semidirect
128. #1949C2×C5semidirect
129. #1941C2×C5semidirect
130. #1923C2×C5semidirect
131. #1919C2×C5semidirect
132. #1918C2×C5semidirect
133. #1864C2×C5semidirect
134. #1853C2×C5semidirect
135. #1824C2×C5semidirect
136. C4×#102C2×C5semidirect
592. C5#27031direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#24952
C5×#23612
C5×#20732
C5×#20652
C5×#20342
C5×#20332
C5×#19492
C5×#19412
C5×#19232
C5×#19192
C5×#19182
C5×#18642
C5×#18532
C5×#18242
C4×C5×#1022
#270315