Finite Group Databasebeta

← C5×#29672 #11952392

Subgroups — C5×#29672 #11952392

Order 1280.

1096 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#542C211yesmaximal, characteristic
2C2×C5×#462C211yesmaximal, characteristic, complemented
3C2×C5×#461C211yesmaximal, characteristic, complemented
4C5×#2621C211yesmaximal, characteristic, complemented
5C5×#2024C212yesmaximal, complemented
6C5×#2015C212yesmaximal, complemented
7C5×#2001C212yesmaximal, complemented
8C5×#2000C212yesmaximal
9C5×#1805C211yesmaximal, characteristic, complemented
10C5×#1804C211yesmaximal, characteristic, complemented
11C5×#1798C211yesmaximal, characteristic, complemented
12C22×C5×Q16C2212yes—
13C22×C5×SD16C2212yes—
14C5×#542—24no—
15C5×#542—24no—
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×C4⋊Q8C2211yescharacteristic
18C2×C5×#126—21no—
19C2×C5×#124—21no—
20C2×C5×#124C2211yescharacteristic
21C2×C5×#123—21no—
22C2×C5×C22⋊Q8—21no—
23C2×C5×C22⋊Q8C2211yescharacteristic
24C2×C5×C22⋊Q8C2212yes—
25C2×C5×#121C2211yescharacteristic
26C2×C4×C5×Q8C2211yescharacteristic
27C2×C4×C5×D4C2211yescharacteristic
28C2×C5×#117C2211yescharacteristic
29C5×#462—24no—
30C5×#461—24no—
31C2×C5×C4⋊C8C2211yescharacteristic
32C2×C5×C4⋊C8C2211yescharacteristic
33C2×C5×#103C2212yes—
34C2×C5×#103C2212yes—
35C2×C5×#103C2212yes—
36C2×C5×#102C2212yes—
37C2×C5×C22⋊C8C2212yes—
38C5×#398—21no—
39C5×#398—21no—
40C5×#397—21no—
41C5×#397—21no—
42C5×#391C2212yes—
43C5×#382C2211yescharacteristic
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#29672C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8C2312yes—
50C22×C5×Q8C2311yescharacteristic
51C22×C5×D4—21no—
52C22×C5×D4C2311yescharacteristic
53C23×C4×C5—21no—
54C23×C4×C5C2311yescharacteristic
55C2×C5×Q16—24no—
56C2×C5×Q16—42no—
57C2×C5×Q16—42no—
58C2×C5×SD16—24no—
59C2×C5×SD16—42no—
60C2×C5×SD16—42no—
61C22×C5×C8C2312yes—
62C22×C5×C8C2312yes—
63C5×C4⋊Q8—22no—
64C5×C4⋊Q8—22no—
65C5×C4⋊Q8—22no—
66C5×C4⋊Q8—22no—
67C5×#126—28no—
68C5×#124—28no—
69C5×#124—22no—
70C5×#124—22no—
71C5×#123—28no—
72C5×C22⋊Q8—28no—
73C5×C22⋊Q8—22no—
74C5×C22⋊Q8—22no—
75C5×C22⋊Q8—42no—
76C5×C22⋊Q8—42no—
77C5×#121—22no—
78C5×#121—22no—
79C4×C5×Q8—22no—
80C4×C5×Q8D414yes—
81C4×C5×D4—22no—
82C4×C5×D4D414yes—
83C5×#117—22no—
84C5×#117—22no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4—21no—
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4—21no—
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4—22no—
91C2×C5×C4⋊C4D411yescharacteristic
92C2×C5×C4⋊C4D411yescharacteristic
93C2×C5×C4⋊C4C2312yes—
94C2×C5×C4⋊C4C2311yescharacteristic
95C2×C5×C4⋊C4C2311yescharacteristic
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—21no—
99C2×C5×C22⋊C4—21no—
100C2×C5×C22⋊C4—21no—
101C2×C5×C22⋊C4—22no—
102C2×C5×C22⋊C4D411yescharacteristic
103C2×C5×C22⋊C4D411yescharacteristic
104C2×C42×C5—21no—
105C2×C42×C5C2311yescharacteristic
106C2×C42×C5C2311yescharacteristic
107C5×C4⋊C8—22no—
108C5×C4⋊C8—22no—
109C5×#103—24no—
110C5×#103—24no—
111C5×#103—42no—
112C5×#102—42no—
113C5×C22⋊C8—42no—
114C5×#95—21no—
115C5×#95—21no—
116C5×#95—22no—
117C5×#95—22no—
118C2×#542C2×C511yescharacteristic
119C2×#462C2×C511yescharacteristic, complemented
120C2×#461C2×C511yescharacteristic, complemented
121#2621C2×C511yescharacteristic, complemented
122#2024C2×C512yescomplemented
123#2015C2×C512yescomplemented
124#2001C2×C512yescomplemented
125#2000C2×C512yes—
126#1805C2×C511yescharacteristic, complemented
127#1804C2×C511yescharacteristic, complemented
128#1798C2×C511yescharacteristic, complemented
129C24×C5—21no—
130C24×C5C2×D411yescharacteristic
131C2×C5×Q8—24no—
132C2×C5×Q8—21no—
133C2×C5×Q8—22no—
134C2×C5×Q8—22no—
135C2×C5×Q8—21no—
136C2×C5×Q8—21no—
137C2×C5×Q8—21no—
138C2×C5×Q8—22no—
139C2×C5×Q8—22no—
140C2×C5×Q8—22no—
141C2×C5×Q8—21no—
142C2×C5×Q8—22no—
143C2×C5×Q8—82no—
144C2×C5×Q8C2×D412yes—
145C2×C5×D4—24no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—21no—
149C2×C5×D4—21no—
150C2×C5×D4—21no—
151C2×C5×D4—82no—
152C2×C5×D4C2×D412yes—
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—21no—
162C22×C4×C5—24no—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5—82no—
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2411yescharacteristic
172C5×Q16—48no—
173C5×SD16—48no—
174C2×C5×C8—22no—
175C2×C5×C8—22no—
176C2×C5×C8—22no—
177C2×C5×C8—22no—
178C2×C5×C8—22no—
179C2×C5×C8—22no—
180C5×C4⋊C4—24no—
181C5×C4⋊C4—24no—
182C5×C4⋊C4—24no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—44no—
190C5×C4⋊C4—42no—
191C5×C4⋊C4—42no—
192C5×C4⋊C4C2×D412yes—
193C5×C4⋊C4C2×D412yes—
194C5×C22⋊C4—24no—
195C5×C22⋊C4—24no—
196C5×C22⋊C4—24no—
197C5×C22⋊C4—24no—
198C5×C22⋊C4—24no—
199C5×C22⋊C4—22no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—44no—
202C42×C5—24no—
203C42×C5—21no—
204C42×C5—21no—
205C42×C5C2×D412yes—
206C42×C5C2×D412yes—
207C22×Q16C22×C512yes—
208C22×SD16C22×C512yes—
209#542—24no—
210#542—24no—
211C2×C4⋊Q8C22×C511yescharacteristic
212C2×C4⋊Q8C22×C511yescharacteristic
213C2×#126—21no—
214C2×#124—21no—
215C2×#124C22×C511yescharacteristic
216C2×#123—21no—
217C2×C22⋊Q8—21no—
218C2×C22⋊Q8C22×C511yescharacteristic
219C2×C22⋊Q8C22×C512yes—
220C2×#121C22×C511yescharacteristic
221C2×C4×Q8C22×C511yescharacteristic
222C2×C4×D4C22×C511yescharacteristic
223C2×#117C22×C511yescharacteristic
224#462—24no—
225#461—24no—
226C2×C4⋊C8C22×C511yescharacteristic
227C2×C4⋊C8C22×C511yescharacteristic
228C2×#103C22×C512yes—
229C2×#103C22×C512yes—
230C2×#103C22×C512yes—
231C2×#102C22×C512yes—
232C2×C22⋊C8C22×C512yes—
233#398—21no—
234#398—21no—
235#397—21no—
236#397—21no—
237#391C22×C512yes—
238#382C22×C511yescharacteristic
239#340C22×C512yes—
240#340C22×C512yes—
241#336C22×C512yes—
242C23×C5—21no—
243C23×C5—22no—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—41no—
249C23×C5—41no—
250C23×C5—42no—
251C23×C5—42no—
252C23×C5C22×D411yescharacteristic
253C5×Q8—22no—
254C5×Q8—22no—
255C5×Q8—41no—
256C5×Q8—41no—
257C5×Q8—42no—
258C5×Q8—42no—
259C5×Q8—42no—
260C5×Q8—42no—
261C5×Q8—84no—
262C5×D4—22no—
263C5×D4—22no—
264C5×D4—41no—
265C5×D4—41no—
266C5×D4—84no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—22no—
271C2×C4×C5—21no—
272C2×C4×C5—22no—
273C2×C4×C5—21no—
274C2×C4×C5—22no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—42no—
282C2×C4×C5—42no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—42no—
293C2×C4×C5—84no—
294C2×C4×C5#12012yes—
295C2×C4×C5#12012yes—
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D411yescharacteristic
298C2×C4×C5H(5,𝔽2)11yescharacteristic
299C2×C4×C5H(5,𝔽2)11yescharacteristic
300C2×C4×C5H(5,𝔽2)11yescharacteristic
301C2×C4×C52−1+411yescharacteristic
302C5×C8—42no—
303C5×C8—42no—
304C22×Q8—21no—
305C22×Q8C23×C512yes—
306C22×Q8C23×C511yescharacteristic
307C22×D4—21no—
308C22×D4C23×C511yescharacteristic
309C23×C4—21no—
310C23×C4C23×C511yescharacteristic
311C2×Q16—24no—
312C2×Q16—42no—
313C2×Q16—42no—
314C2×SD16—24no—
315C2×SD16—42no—
316C2×SD16—42no—
317C22×C8C23×C512yes—
318C22×C8C23×C512yes—
319C4⋊Q8—22no—
320C4⋊Q8—22no—
321C4⋊Q8—22no—
322C4⋊Q8—22no—
323#126—28no—
324#124—28no—
325#124—22no—
326#124—22no—
327#123—28no—
328C22⋊Q8—28no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—42no—
332C22⋊Q8—42no—
333#121—22no—
334#121—22no—
335C4×Q8—22no—
336C4×Q8C5×D414yes—
337C4×D4—22no—
338C4×D4C5×D414yes—
339#117—22no—
340#117—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4C5×D411yescharacteristic
348C2×C4⋊C4C5×D411yescharacteristic
349C2×C4⋊C4C23×C512yes—
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×C22⋊C4—22no—
358C2×C22⋊C4C5×D411yescharacteristic
359C2×C22⋊C4C5×D411yescharacteristic
360C2×C42—21no—
361C2×C42C23×C511yescharacteristic
362C2×C42C23×C511yescharacteristic
363C4⋊C8—22no—
364C4⋊C8—22no—
365#103—24no—
366#103—24no—
367#103—42no—
368#102—42no—
369C22⋊C8—42no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—22no—
374C22×C5—41no—
375C22×C5—42no—
376C22×C5—42no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—41no—
380C22×C5—41no—
381C22×C5—42no—
382C22×C5—82no—
383C22×C5—82no—
384C22×C5C2×#12011yescharacteristic
385C22×C5#53511yescharacteristic
386C22×C5#53711yescharacteristic
387C22×C5#57712yes—
388C22×C5#57812yes—
389C4×C5—22no—
390C4×C5—22no—
391C4×C5—41no—
392C4×C5—42no—
393C4×C5—41no—
394C4×C5—42no—
395C4×C5—42no—
396C4×C5—41no—
397C4×C5—82no—
398C4×C5—82no—
399C4×C5#53511yescharacteristic
400C4×C5#53511yescharacteristic
401C4×C5#53711yescharacteristic
402C4×C5#53711yescharacteristic
403C24—21no—
404C24C2×C5×D411yescharacteristic
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—82no—
418C2×Q8C2×C5×D412yes—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—82no—
426C2×D4C2×C5×D412yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—82no—
440C22×C4C2×C5×D411yescharacteristic
441C22×C4C2×C5×D411yescharacteristic
442C22×C4C2×C5×D411yescharacteristic
443C22×C4C2×C5×D411yescharacteristic
444C22×C4C2×C5×D411yescharacteristic
445C22×C4C24×C511yescharacteristic
446Q16—48no—
447SD16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4C2×C5×D412yes—
467C4⋊C4C2×C5×D412yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—44no—
476C42—24no—
477C42—21no—
478C42—21no—
479C42C2×C5×D412yes—
480C42C2×C5×D412yes—
481C2×C5—81no—
482C2×C5—82no—
483C2×C5#261511yescharacteristic
484C2×C5#300911yescharacteristic
485C2×C5#301111yescharacteristic
486C2×C5#322512yes—
487C2×C5#325812yes—
488C23—21no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23C22×C5×D411yescharacteristic
499Q8—22no—
500Q8—22no—
501Q8—41no—
502Q8—41no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—84no—
508D4—22no—
509D4—22no—
510D4—41no—
511D4—41no—
512D4—84no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4C5×#12012yes—
541C2×C4C5×#12012yes—
542C2×C4C22×C5×D411yescharacteristic
543C2×C4C22×C5×D411yescharacteristic
544C2×C4C5×H(5,𝔽2)11yescharacteristic
545C2×C4C5×H(5,𝔽2)11yescharacteristic
546C2×C4C5×H(5,𝔽2)11yescharacteristic
547C2×C4C5×2−1+411yescharacteristic
548C8—42no—
549C8—42no—
550C5#2967211yescharacteristic, direct product factor, Sylow
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—82no—
560C22—82no—
561C22C2×C5×#12011yescharacteristic
562C22C5×#53511yescharacteristic
563C22C5×#53711yescharacteristic
564C22C5×#57712yes—
565C22C5×#57812yes—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—42no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—82no—
575C4—82no—
576C4C5×#53511yescharacteristic
577C4C5×#53511yescharacteristic
578C4C5×#53711yescharacteristic
579C4C5×#53711yescharacteristic
580C2—81no—
581C2—82no—
582C2C5×#261511yescharacteristic
583C2C5×#300911yescharacteristic
584C2C5×#301111yescharacteristic
585C2C5×#322512yes—
586C2C5×#325812yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#121C22
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
2. C2×C5×#462C2semidirect
3. C2×C5×#461C2semidirect
4. C5×#2621C2semidirect
5. C5×#2024C2semidirect
6. C5×#2015C2semidirect
7. C5×#2001C2semidirect
9. C5×#1805C2semidirect
10. C5×#1804C2semidirect
11. C5×#1798C2semidirect
47. #29672C5direct (N × H)
119. C2×#462C2×C5semidirect
120. C2×#461C2×C5semidirect
121. #2621C2×C5semidirect
122. #2024C2×C5semidirect
123. #2015C2×C5semidirect
124. #2001C2×C5semidirect
126. #1805C2×C5semidirect
127. #1804C2×C5semidirect
128. #1798C2×C5semidirect
550. C5#29672direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5422
C2×C5×#4622
C2×C5×#4612
C5×#26212
C5×#20242
C5×#20152
C5×#20012
C5×#20002
C5×#18052
C5×#18042
C5×#17982
#296725