Finite Group Databasebeta

← C5×#29698 #11952418

Subgroups — C5×#29698 #11952418

Order 1280.

1076 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#540C211yesmaximal, characteristic
2C2×C5×#476C211yesmaximal, characteristic, complemented
3C2×C5×#474C211yesmaximal, characteristic, complemented
4C5×#2628C211yesmaximal, characteristic, complemented
5C5×#2059C212yesmaximal, complemented
6C5×#2041C212yesmaximal, complemented
7C5×#2034C212yesmaximal, complemented
8C5×#2033C212yesmaximal, complemented
9C5×#1804C211yesmaximal, characteristic, complemented
10C5×#1804C211yesmaximal, characteristic, complemented
11C5×#1798C211yesmaximal, characteristic, complemented
12C5×#540—24no—
13C5×#540—24no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#127C2211yescharacteristic
16C2×C5×#125C2211yescharacteristic
17C2×C5×#123—21no—
18C2×C5×#123—21no—
19C2×C5×#121—21no—
20C2×C5×#121—21no—
21C2×C5×#121C2211yescharacteristic
22C2×C5×#121C2211yescharacteristic
23C2×C4×C5×D4C2211yescharacteristic
24C2×C4×C5×D4C2211yescharacteristic
25C2×C5×#117C2211yescharacteristic
26C5×#476—24no—
27C5×#474—24no—
28C2×C5×#107C2212yes—
29C2×C5×#106C2212yes—
30C2×C5×C4⋊C8C2211yescharacteristic
31C2×C5×C4⋊C8C2211yescharacteristic
32C2×C5×#102C2212yes—
33C2×C5×#102C2212yes—
34C2×C5×#102C2212yes—
35C2×C5×#102C2212yes—
36C2×C5×C22⋊C8C2212yes—
37C5×#400—21no—
38C5×#399—21no—
39C5×#395—21no—
40C5×#393—21no—
41C5×#389C2212yes—
42C5×#382C2211yescharacteristic
43C5×#380C2212yes—
44C5×#340C2212yes—
45C5×#340C2212yes—
46C5×#336C2212yes—
47#29698C511yesmaximal, characteristic, direct product factor, Sylow
48C22×C5×Q8—21no—
49C22×C5×D4—21no—
50C22×C5×D4—21no—
51C22×C5×D4—21no—
52C22×C5×D4—21no—
53C22×C5×D4C2311yescharacteristic
54C22×C5×D4C2311yescharacteristic
55C23×C4×C5—21no—
56C23×C4×C5—21no—
57C23×C4×C5C2311yescharacteristic
58C22×C5×C8C2312yes—
59C22×C5×C8C2312yes—
60C5×C4⋊Q8—22no—
61C5×C4⋊Q8—22no—
62C5×#127—22no—
63C5×#127—22no—
64C5×#125—22no—
65C5×#125—22no—
66C5×#123—28no—
67C5×#123—28no—
68C5×#121—28no—
69C5×#121—28no—
70C5×#121—22no—
71C5×#121—22no—
72C5×#121—22no—
73C5×#121—22no—
74C4×C5×D4—22no—
75C4×C5×D4—22no—
76C4×C5×D4D414yes—
77C4×C5×D4D414yes—
78C5×#117—22no—
79C5×#117—22no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4—22no—
84C2×C5×C4⋊C4D411yescharacteristic
85C2×C5×C4⋊C4D411yescharacteristic
86C2×C5×C4⋊C4C2312yes—
87C2×C5×C4⋊C4C2312yes—
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⋊C4—22no—
95C2×C5×C22⋊C4D411yescharacteristic
96C2×C5×C22⋊C4D411yescharacteristic
97C2×C42×C5C2311yescharacteristic
98C2×C42×C5C2311yescharacteristic
99C5×#107—24no—
100C5×#106—24no—
101C5×C4⋊C8—22no—
102C5×C4⋊C8—22no—
103C5×#102—24no—
104C5×#102—24no—
105C5×#102—42no—
106C5×#102—42no—
107C5×C22⋊C8—42no—
108C5×#95—22no—
109C5×#95—21no—
110C5×#95—21no—
111C5×#95—22no—
112C2×#540C2×C511yescharacteristic
113C2×#476C2×C511yescharacteristic, complemented
114C2×#474C2×C511yescharacteristic, complemented
115#2628C2×C511yescharacteristic, complemented
116#2059C2×C512yescomplemented
117#2041C2×C512yescomplemented
118#2034C2×C512yescomplemented
119#2033C2×C512yescomplemented
120#1804C2×C511yescharacteristic, complemented
121#1804C2×C511yescharacteristic, complemented
122#1798C2×C511yescharacteristic, complemented
123C24×C5—21no—
124C24×C5—21no—
125C24×C5C2×D411yescharacteristic
126C2×C5×Q8—42no—
127C2×C5×Q8—82no—
128C2×C5×D4—24no—
129C2×C5×D4—24no—
130C2×C5×D4—24no—
131C2×C5×D4—24no—
132C2×C5×D4—21no—
133C2×C5×D4—21no—
134C2×C5×D4—21no—
135C2×C5×D4—21no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—21no—
141C2×C5×D4—21no—
142C2×C5×D4—82no—
143C2×C5×D4—82no—
144C2×C5×D4—82no—
145C2×C5×D4—82no—
146C2×C5×D4C2×D412yes—
147C2×C5×D4C2×D412yes—
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—24no—
157C22×C4×C5—24no—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5—82no—
161C22×C4×C5—82no—
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2411yescharacteristic
168C2×C5×C8—22no—
169C2×C5×C8—22no—
170C2×C5×C8—22no—
171C2×C5×C8—22no—
172C2×C5×C8—22no—
173C2×C5×C8—22no—
174C5×C4⋊C4—24no—
175C5×C4⋊C4—24no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—42no—
185C5×C4⋊C4—82no—
186C5×C4⋊C4C2×D412yes—
187C5×C4⋊C4C2×D412yes—
188C5×C22⋊C4—24no—
189C5×C22⋊C4—24no—
190C5×C22⋊C4—24no—
191C5×C22⋊C4—24no—
192C5×C22⋊C4—22no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—82no—
195C42×C5—21no—
196C42×C5—21no—
197C42×C5C2×D412yes—
198C42×C5C2×D412yes—
199#540—24no—
200#540—24no—
201C2×C4⋊Q8C22×C511yescharacteristic
202C2×#127C22×C511yescharacteristic
203C2×#125C22×C511yescharacteristic
204C2×#123—21no—
205C2×#123—21no—
206C2×#121—21no—
207C2×#121—21no—
208C2×#121C22×C511yescharacteristic
209C2×#121C22×C511yescharacteristic
210C2×C4×D4C22×C511yescharacteristic
211C2×C4×D4C22×C511yescharacteristic
212C2×#117C22×C511yescharacteristic
213#476—24no—
214#474—24no—
215C2×#107C22×C512yes—
216C2×#106C22×C512yes—
217C2×C4⋊C8C22×C511yescharacteristic
218C2×C4⋊C8C22×C511yescharacteristic
219C2×#102C22×C512yes—
220C2×#102C22×C512yes—
221C2×#102C22×C512yes—
222C2×#102C22×C512yes—
223C2×C22⋊C8C22×C512yes—
224#400—21no—
225#399—21no—
226#395—21no—
227#393—21no—
228#389C22×C512yes—
229#382C22×C511yescharacteristic
230#380C22×C512yes—
231#340C22×C512yes—
232#340C22×C512yes—
233#336C22×C512yes—
234C23×C5—21no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—21no—
238C23×C5—21no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—41no—
242C23×C5—41no—
243C23×C5—41no—
244C23×C5—41no—
245C23×C5—42no—
246C23×C5—42no—
247C23×C5—42no—
248C23×C5—42no—
249C23×C5C22×D411yescharacteristic
250C5×Q8—84no—
251C5×D4—22no—
252C5×D4—22no—
253C5×D4—22no—
254C5×D4—22no—
255C5×D4—41no—
256C5×D4—41no—
257C5×D4—41no—
258C5×D4—41no—
259C5×D4—84no—
260C5×D4—84no—
261C5×D4—84no—
262C5×D4—84no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—42no—
281C2×C4×C5—42no—
282C2×C4×C5—42no—
283C2×C4×C5—42no—
284C2×C4×C5—42no—
285C2×C4×C5—42no—
286C2×C4×C5—84no—
287C2×C4×C5—84no—
288C2×C4×C5#12012yes—
289C2×C4×C5#12012yes—
290C2×C4×C5C22×D411yescharacteristic
291C2×C4×C5C22×D411yescharacteristic
292C2×C4×C5H(5,𝔽2)11yescharacteristic
293C2×C4×C52−1+411yescharacteristic
294C2×C4×C52−1+411yescharacteristic
295C2×C4×C52−1+411yescharacteristic
296C5×C8—42no—
297C5×C8—42no—
298C22×Q8—21no—
299C22×D4—21no—
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303C22×D4C23×C511yescharacteristic
304C22×D4C23×C511yescharacteristic
305C23×C4—21no—
306C23×C4—21no—
307C23×C4C23×C511yescharacteristic
308C22×C8C23×C512yes—
309C22×C8C23×C512yes—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312#127—22no—
313#127—22no—
314#125—22no—
315#125—22no—
316#123—28no—
317#123—28no—
318#121—28no—
319#121—28no—
320#121—22no—
321#121—22no—
322#121—22no—
323#121—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4C5×D414yes—
327C4×D4C5×D414yes—
328#117—22no—
329#117—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4C5×D411yescharacteristic
335C2×C4⋊C4C5×D411yescharacteristic
336C2×C4⋊C4C23×C512yes—
337C2×C4⋊C4C23×C512yes—
338C2×C4⋊C4C23×C511yescharacteristic
339C2×C4⋊C4C23×C511yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C5×D411yescharacteristic
346C2×C22⋊C4C5×D411yescharacteristic
347C2×C42C23×C511yescharacteristic
348C2×C42C23×C511yescharacteristic
349#107—24no—
350#106—24no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353#102—24no—
354#102—24no—
355#102—42no—
356#102—42no—
357C22⋊C8—42no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—22no—
362C22×C5—41no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5—41no—
369C22×C5—42no—
370C22×C5—42no—
371C22×C5—41no—
372C22×C5—42no—
373C22×C5—82no—
374C22×C5—82no—
375C22×C5—82no—
376C22×C5—82no—
377C22×C5C2×#12011yescharacteristic
378C22×C5#53611yescharacteristic
379C22×C5#53711yescharacteristic
380C22×C5#57612yes—
381C22×C5#57712yes—
382C4×C5—22no—
383C4×C5—22no—
384C4×C5—41no—
385C4×C5—41no—
386C4×C5—42no—
387C4×C5—41no—
388C4×C5—42no—
389C4×C5—82no—
390C4×C5—82no—
391C4×C5#53611yescharacteristic
392C4×C5#53611yescharacteristic
393C4×C5#53711yescharacteristic
394C4×C5#53711yescharacteristic
395C24—21no—
396C24—21no—
397C24C2×C5×D411yescharacteristic
398C2×Q8—42no—
399C2×Q8—82no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4C2×C5×D412yes—
419C2×D4C2×C5×D412yes—
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—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C2×C5×D411yescharacteristic
435C22×C4C2×C5×D411yescharacteristic
436C22×C4C2×C5×D411yescharacteristic
437C22×C4C2×C5×D411yescharacteristic
438C22×C4C2×C5×D411yescharacteristic
439C22×C4C24×C511yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4C2×C5×D412yes—
459C4⋊C4C2×C5×D412yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—82no—
467C42—21no—
468C42—21no—
469C42C2×C5×D412yes—
470C42C2×C5×D412yes—
471C2×C5—81no—
472C2×C5—82no—
473C2×C5—82no—
474C2×C5#261611yescharacteristic
475C2×C5#300811yescharacteristic
476C2×C5#300911yescharacteristic
477C2×C5#324112yes—
478C2×C5#325712yes—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23C22×C5×D411yescharacteristic
495Q8—84no—
496D4—22no—
497D4—22no—
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—84no—
505D4—84no—
506D4—84no—
507D4—84no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4C5×#12012yes—
534C2×C4C5×#12012yes—
535C2×C4C22×C5×D411yescharacteristic
536C2×C4C22×C5×D411yescharacteristic
537C2×C4C5×H(5,𝔽2)11yescharacteristic
538C2×C4C5×2−1+411yescharacteristic
539C2×C4C5×2−1+411yescharacteristic
540C2×C4C5×2−1+411yescharacteristic
541C8—42no—
542C8—42no—
543C5#2969811yescharacteristic, direct product factor, Sylow
544C22—42no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—42no—
549C22—42no—
550C22—41no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—42no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22C2×C5×#12011yescharacteristic
560C22C5×#53611yescharacteristic
561C22C5×#53711yescharacteristic
562C22C5×#57612yes—
563C22C5×#57712yes—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—42no—
570C4—41no—
571C4—82no—
572C4—82no—
573C4C5×#53611yescharacteristic
574C4C5×#53611yescharacteristic
575C4C5×#53711yescharacteristic
576C4C5×#53711yescharacteristic
577C2—81no—
578C2—82no—
579C2—82no—
580C2C5×#261611yescharacteristic
581C2C5×#300811yescharacteristic
582C2C5×#300911yescharacteristic
583C2C5×#324112yes—
584C2C5×#325712yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#540C2
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×#476C2semidirect
3. C2×C5×#474C2semidirect
4. C5×#2628C2semidirect
5. C5×#2059C2semidirect
6. C5×#2041C2semidirect
7. C5×#2034C2semidirect
8. C5×#2033C2semidirect
9. C5×#1804C2semidirect
10. C5×#1804C2semidirect
11. C5×#1798C2semidirect
47. #29698C5direct (N × H)
113. C2×#476C2×C5semidirect
114. C2×#474C2×C5semidirect
115. #2628C2×C5semidirect
116. #2059C2×C5semidirect
117. #2041C2×C5semidirect
118. #2034C2×C5semidirect
119. #2033C2×C5semidirect
120. #1804C2×C5semidirect
121. #1804C2×C5semidirect
122. #1798C2×C5semidirect
543. C5#29698direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5402
C2×C5×#4762
C2×C5×#4742
C5×#26282
C5×#20592
C5×#20412
C5×#20342
C5×#20332
C5×#18042
C5×#17982
#296985