Finite Group Databasebeta

← C5×#19017 #11941737

Subgroups — C5×#19017 #11941737

Order 1280.

944 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3559C211yesmaximal, characteristic
2C5×#3238C211yesmaximal, characteristic, complemented
3C2×C5×#501C211yesmaximal, characteristic, complemented
4C5×#3152C211yesmaximal, characteristic, complemented
5C5×#1661C214yesmaximal, complemented
6C5×#1660C214yesmaximal, complemented
7C5×#1530C211yesmaximal, characteristic, complemented
8C5×#1528C211yesmaximal, characteristic, complemented
9C5×#1524C211yesmaximal, characteristic, complemented
10C2×C5×#137—22no—
11C5×#563C2212yes—
12C5×#562C2212yes—
13C5×#554—21no—
14C5×#553—21no—
15C5×#553—21no—
16C5×#547—22no—
17C5×#547—21no—
18C5×#544—21no—
19C5×#540—21no—
20C5×#539—22no—
21C5×#539—21no—
22C5×#537—21no—
23C5×#532C2211yescharacteristic
24C2×C5×C4⋊Q8C2211yescharacteristic
25C5×#529C2211yescharacteristic
26C2×C5×#125C2211yescharacteristic
27C5×#501—22no—
28C5×#501C2212yes—
29C5×#501C2214yes—
30C5×#497C2214yes—
31C5×#494—21no—
32C5×#492—21no—
33C5×#489C2212yes—
34C5×#483—21no—
35C5×#480—21no—
36C5×#477—21no—
37C5×#476—21no—
38C5×#432C2211yescharacteristic
39C2×C5×#107—21no—
40C2×C5×#106—21no—
41C4×C5×M16C2211yescharacteristic
42C2×C5×#97C2211yescharacteristic
43C5×#330C2212yes—
44C5×#330C2214yes—
45C5×#329C2212yes—
46C5×#329C2214yes—
47#19017C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41—22no—
49C2×C5×#41D412yes—
50C22×C5×Q8—21no—
51C5×#137—48no—
52C2×C5×Q16—24no—
53C2×C5×SD16—24no—
54C2×C5×M16—21no—
55C22×C5×C8—21no—
56C5×C4⋊Q8—21no—
57C5×C4⋊Q8—22no—
58C5×C4⋊Q8—21no—
59C5×C4⋊Q8—21no—
60C5×C4⋊Q8—21no—
61C5×C4⋊Q8D411yescharacteristic
62C5×C4⋊Q8C2312yes—
63C5×#127D411yescharacteristic
64C5×#126—22no—
65C5×#126—22no—
66C5×#126—21no—
67C5×#125—22no—
68C5×#125—21no—
69C5×#125—21no—
70C5×#125—21no—
71C5×#125D412yes—
72C5×#125C2312yes—
73C5×#125C2311yescharacteristic
74C5×#124C2311yescharacteristic
75C5×#124C2312yes—
76C5×#123—22no—
77C5×#123—22no—
78C5×#123—22no—
79C5×#123—22no—
80C5×#123—21no—
81C5×#123—22no—
82C5×#123—22no—
83C5×C22⋊Q8—22no—
84C5×C22⋊Q8—22no—
85C5×C22⋊Q8—22no—
86C5×C22⋊Q8—22no—
87C5×C22⋊Q8—22no—
88C5×C22⋊Q8—22no—
89C5×C22⋊Q8D411yescharacteristic
90C5×#121—22no—
91C5×#121—22no—
92C5×#121—21no—
93C5×#121—21no—
94C5×#121—22no—
95C5×#121—22no—
96C5×#121—21no—
97C5×#121—21no—
98C5×#121D411yescharacteristic
99C4×C5×Q8D411yescharacteristic
100C4×C5×D4—22no—
101C4×C5×D4—22no—
102C4×C5×D4—21no—
103C4×C5×D4—21no—
104C4×C5×D4—21no—
105C4×C5×D4—21no—
106C4×C5×D4—21no—
107C4×C5×D4D411yescharacteristic
108C5×#117—21no—
109C5×#117—21no—
110C2×C5×C4⋊C4—22no—
111C2×C5×C4⋊C4—21no—
112C2×C5×C4⋊C4—21no—
113C2×C5×C4⋊C4—21no—
114C2×C5×C4⋊C4—21no—
115C2×C5×C4⋊C4—21no—
116C2×C5×C4⋊C4D411yescharacteristic
117C2×C5×C4⋊C4D411yescharacteristic
118C2×C42×C5C2311yescharacteristic
119C5×#107—24no—
120C5×#107—22no—
121C5×#106—24no—
122C5×#106—22no—
123C5×C4⋊C8—21no—
124C5×#103—22no—
125C5×#102—22no—
126C5×C22⋊C8—21no—
127C5×#97—21no—
128C5×#97C2312yes—
129C5×#97C2312yes—
130C5×#97C2312yes—
131C4×C5×C8—21no—
132#3559C2×C511yescharacteristic
133#3238C2×C511yescharacteristic, complemented
134C2×#501C2×C511yescharacteristic, complemented
135#3152C2×C511yescharacteristic, complemented
136#1661C2×C514yescomplemented
137#1660C2×C514yescomplemented
138#1530C2×C511yescharacteristic, complemented
139#1528C2×C511yescharacteristic, complemented
140#1524C2×C511yescharacteristic, complemented
141C5×#41—44no—
142C5×#41—84no—
143C2×C5×Q8—22no—
144C2×C5×Q8—22no—
145C2×C5×Q8—44no—
146C2×C5×Q8—41no—
147C2×C5×Q8C2×D411yescharacteristic
148C2×C5×Q8C2×D412yes—
149C2×C5×D4—22no—
150C2×C5×D4—21no—
151C2×C5×D4—22no—
152C2×C5×D4—22no—
153C2×C5×D4—21no—
154C2×C5×D4—21no—
155C2×C5×D4C2×D411yescharacteristic
156C2×C5×D4C2×D412yes—
157C22×C4×C5—21no—
158C22×C4×C5—22no—
159C22×C4×C5—22no—
160C22×C4×C5—22no—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—22no—
164C22×C4×C5—41no—
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C5×Q16—48no—
169C5×SD16—48no—
170C5×M16—24no—
171C2×C5×C8—24no—
172C2×C5×C8—22no—
173C2×C5×C8—22no—
174C2×C5×C8—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—22no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—22no—
193C5×C4⋊C4—41no—
194C5×C4⋊C4—41no—
195C5×C4⋊C4C2×D412yes—
196C5×C4⋊C4C2×D412yes—
197C5×C4⋊C4C2×D411yescharacteristic
198C5×C4⋊C4C2×D411yescharacteristic
199C5×C22⋊C4—22no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—22no—
202C5×C22⋊C4—22no—
203C5×C22⋊C4—22no—
204C5×C22⋊C4—21no—
205C5×C22⋊C4—21no—
206C5×C22⋊C4—21no—
207C5×C22⋊C4—21no—
208C5×C22⋊C4—22no—
209C5×C22⋊C4—22no—
210C5×C22⋊C4—21no—
211C42×C5—22no—
212C42×C5—21no—
213C42×C5C2×D411yescharacteristic
214C42×C5C2×D411yescharacteristic
215C42×C5C2×D411yescharacteristic
216C42×C5C2411yescharacteristic
217C2×#137—22no—
218#563C22×C512yes—
219#562C22×C512yes—
220#554—21no—
221#553—21no—
222#553—21no—
223#547—22no—
224#547—21no—
225#544—21no—
226#540—21no—
227#539—22no—
228#539—21no—
229#537—21no—
230#532C22×C511yescharacteristic
231C2×C4⋊Q8C22×C511yescharacteristic
232#529C22×C511yescharacteristic
233C2×#125C22×C511yescharacteristic
234#501—22no—
235#501C22×C514yes—
236#501C22×C512yes—
237#497C22×C514yes—
238#494—21no—
239#492—21no—
240#489C22×C512yes—
241#483—21no—
242#480—21no—
243#477—21no—
244#476—21no—
245#432C22×C511yescharacteristic
246C2×#107—21no—
247C2×#106—21no—
248C4×M16C22×C511yescharacteristic
249C2×#97C22×C511yescharacteristic
250#330C22×C512yes—
251#330C22×C514yes—
252#329C22×C514yes—
253#329C22×C512yes—
254C23×C5—22no—
255C23×C5—21no—
256C23×C5#12011yescharacteristic
257C5×Q8—42no—
258C5×Q8—41no—
259C5×Q8—44no—
260C5×Q8—82no—
261C5×Q8—82no—
262C5×D4—42no—
263C5×D4—41no—
264C5×D4—44no—
265C5×D4—42no—
266C5×D4—42no—
267C5×D4—81no—
268C5×D4—82no—
269C5×D4—82no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—21no—
277C2×C4×C5—22no—
278C2×C4×C5—41no—
279C2×C4×C5—42no—
280C2×C4×C5—42no—
281C2×C4×C5—42no—
282C2×C4×C5—82no—
283C2×C4×C5—81no—
284C2×C4×C5—82no—
285C2×C4×C5—82no—
286C2×C4×C5—82no—
287C2×C4×C5—82no—
288C2×C4×C5#12011yescharacteristic
289C2×C4×C5#12011yescharacteristic
290C2×C4×C5#12011yescharacteristic
291C2×C4×C5C22×D411yescharacteristic
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C22×D411yescharacteristic
294C5×C8—24no—
295C5×C8—24no—
296C5×C8—42no—
297C2×#41—22no—
298C2×#41C5×D412yes—
299C22×Q8—21no—
300#137—48no—
301C2×Q16—24no—
302C2×SD16—24no—
303C2×M16—21no—
304C22×C8—21no—
305C4⋊Q8—21no—
306C4⋊Q8—22no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8C5×D411yescharacteristic
311C4⋊Q8C23×C512yes—
312#127C5×D411yescharacteristic
313#126—22no—
314#126—22no—
315#126—21no—
316#125—22no—
317#125—21no—
318#125—21no—
319#125—21no—
320#125C5×D412yes—
321#125C23×C512yes—
322#125C23×C511yescharacteristic
323#124C23×C511yescharacteristic
324#124C23×C512yes—
325#123—22no—
326#123—22no—
327#123—22no—
328#123—22no—
329#123—21no—
330#123—22no—
331#123—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8C5×D411yescharacteristic
339#121—22no—
340#121—22no—
341#121—21no—
342#121—21no—
343#121—22no—
344#121—22no—
345#121—21no—
346#121—21no—
347#121C5×D411yescharacteristic
348C4×Q8C5×D411yescharacteristic
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4C5×D411yescharacteristic
357#117—21no—
358#117—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4C5×D411yescharacteristic
366C2×C4⋊C4C5×D411yescharacteristic
367C2×C42C23×C511yescharacteristic
368#107—24no—
369#107—22no—
370#106—24no—
371#106—22no—
372C4⋊C8—21no—
373#103—22no—
374#102—22no—
375C22⋊C8—21no—
376#97—21no—
377#97C23×C512yes—
378#97C23×C512yes—
379#97C23×C512yes—
380C4×C8—21no—
381C22×C5—21no—
382C22×C5—21no—
383C22×C5—42no—
384C22×C5—41no—
385C22×C5—41no—
386C22×C5—42no—
387C22×C5—42no—
388C22×C5—41no—
389C22×C5#45312yes—
390C22×C5C2×#12011yescharacteristic
391C4×C5—21no—
392C4×C5—21no—
393C4×C5—21no—
394C4×C5—21no—
395C4×C5—21no—
396C4×C5—42no—
397C4×C5—42no—
398C4×C5—41no—
399C4×C5—41no—
400C4×C5—42no—
401C4×C5—41no—
402C4×C5—82no—
403C4×C5#45612yes—
404#41—44no—
405#41—84no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—41no—
410C2×Q8C2×C5×D411yescharacteristic
411C2×Q8C2×C5×D412yes—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C5×D411yescharacteristic
419C2×D4C2×C5×D412yes—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4C2×C5×D411yescharacteristic
429C22×C4C2×C5×D411yescharacteristic
430C22×C4C2×C5×D411yescharacteristic
431Q16—48no—
432SD16—48no—
433M16—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4C2×C5×D412yes—
459C4⋊C4C2×C5×D411yescharacteristic
460C4⋊C4C2×C5×D412yes—
461C4⋊C4C2×C5×D411yescharacteristic
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—21no—
474C42—22no—
475C42—21no—
476C42C2×C5×D411yescharacteristic
477C42C2×C5×D411yescharacteristic
478C42C2×C5×D411yescharacteristic
479C42C24×C511yescharacteristic
480C2×C5—22no—
481C2×C5—82no—
482C2×C5—81no—
483C2×C5C2×#45311yescharacteristic
484C2×C5C2×#45611yescharacteristic
485C2×C5#302211yescharacteristic
486C23—22no—
487C23—21no—
488C23C5×#12011yescharacteristic
489Q8—42no—
490Q8—41no—
491Q8—44no—
492Q8—82no—
493Q8—82no—
494D4—42no—
495D4—41no—
496D4—44no—
497D4—42no—
498D4—42no—
499D4—81no—
500D4—82no—
501D4—82no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4C5×#12011yescharacteristic
521C2×C4C5×#12011yescharacteristic
522C2×C4C5×#12011yescharacteristic
523C2×C4C22×C5×D411yescharacteristic
524C2×C4C22×C5×D411yescharacteristic
525C2×C4C22×C5×D411yescharacteristic
526C8—24no—
527C8—24no—
528C8—42no—
529C5#1901711yescharacteristic, direct product factor, Sylow
530C22—21no—
531C22—21no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22C5×#45312yes—
539C22C2×C5×#12011yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—82no—
552C4C5×#45612yes—
553C2—22no—
554C2—82no—
555C2—81no—
556C2C2×C5×#45311yescharacteristic
557C2C2×C5×#45611yescharacteristic
558C2C5×#302211yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3559C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3238C2semidirect
3. C2×C5×#501C2semidirect
4. C5×#3152C2semidirect
5. C5×#1661C2semidirect
6. C5×#1660C2semidirect
7. C5×#1530C2semidirect
8. C5×#1528C2semidirect
9. C5×#1524C2semidirect
47. #19017C5direct (N × H)
133. #3238C2×C5semidirect
134. C2×#501C2×C5semidirect
135. #3152C2×C5semidirect
136. #1661C2×C5semidirect
137. #1660C2×C5semidirect
138. #1530C2×C5semidirect
139. #1528C2×C5semidirect
140. #1524C2×C5semidirect
529. C5#19017direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35592
C5×#32382
C2×C5×#5012
C5×#31522
C5×#16612
C5×#16602
C5×#15302
C5×#15282
C5×#15242
#190175