Finite Group Databasebeta

← C5×#19027 #11941747

Subgroups — C5×#19027 #11941747

Order 1280.

884 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3561C211yesmaximal, characteristic
2C5×#3238C211yesmaximal, characteristic, complemented
3C5×#3139C211yesmaximal, characteristic, complemented
4C2×C5×#489C211yesmaximal, characteristic, complemented
5C5×#1664C214yesmaximal, complemented
6C5×#1662C214yesmaximal, complemented
7C5×#1531C211yesmaximal, characteristic, complemented
8C5×#1527C211yesmaximal, characteristic, complemented
9C5×#1524C211yesmaximal, characteristic, complemented
10C5×#563C2212yes—
11C5×#562C2212yes—
12C5×#556—21no—
13C5×#551—22no—
14C5×D4×Q8—21no—
15C5×#548—21no—
16C5×#546—21no—
17C5×#543—21no—
18C5×#542—21no—
19C5×#542—22no—
20C5×#538—21no—
21C5×#536—21no—
22C5×#533C2211yescharacteristic
23C2×C5×C4⋊Q8C2211yescharacteristic
24C5×#529C2211yescharacteristic
25C2×C5×#124C2211yescharacteristic
26C5×#501C2212yescomplemented
27C5×#491C2214yescomplemented
28C5×#489—22no—
29C5×#489C2214yes—
30C5×#489C2212yescomplemented
31C5×#487—21no—
32C5×#487—21no—
33C5×#483—21no—
34C5×#480—21no—
35C5×#477—21no—
36C5×#476—21no—
37C5×#432C2211yescharacteristic
38C5×#419—22no—
39C2×C5×#103—21no—
40C2×C5×#102—21no—
41C4×C5×M16C2211yescharacteristic
42C2×C5×#97C2211yescharacteristic
43C5×#330C2212yescomplemented
44C5×#330C2214yescomplemented
45C5×#329C2214yes—
46C5×#329C2212yescomplemented
47#19027C511yesmaximal, characteristic, direct product factor, Sylow
48C2×C5×#41—21no—
49C22×C5×Q8—21no—
50C22×C5×Q8—21no—
51C22×C5×Q8D411yescharacteristic
52C22×C5×D4D411yescharacteristic
53C2×C5×M16—21no—
54C22×C5×C8—21no—
55C5×C4⋊Q8—21no—
56C5×C4⋊Q8—22no—
57C5×C4⋊Q8—21no—
58C5×C4⋊Q8—21no—
59C5×C4⋊Q8—21no—
60C5×C4⋊Q8D411yescharacteristic
61C5×C4⋊Q8D411yescharacteristic
62C5×C4⋊Q8C2312yes—
63C5×#126—22no—
64C5×#126—22no—
65C5×#126—22no—
66C5×#126—21no—
67C5×#125C2312yes—
68C5×#125C2311yescharacteristic
69C5×#124—21no—
70C5×#124—22no—
71C5×#124—22no—
72C5×#124—21no—
73C5×#124—21no—
74C5×#124D412yes—
75C5×#124C2312yes—
76C5×#124C2311yescharacteristic
77C5×#123—22no—
78C5×#123—22no—
79C5×#123—21no—
80C5×#123—22no—
81C5×#123—22no—
82C5×C22⋊Q8—22no—
83C5×C22⋊Q8—22no—
84C5×C22⋊Q8—21no—
85C5×C22⋊Q8—22no—
86C5×C22⋊Q8—22no—
87C5×C22⋊Q8—22no—
88C5×C22⋊Q8—21no—
89C5×C22⋊Q8—21no—
90C5×C22⋊Q8—22no—
91C5×C22⋊Q8—22no—
92C5×C22⋊Q8D411yescharacteristic
93C5×#121—21no—
94C5×#121—22no—
95C5×#121—22no—
96C5×#121D411yescharacteristic
97C5×#120—22no—
98C4×C5×Q8—21no—
99C4×C5×Q8—21no—
100C4×C5×Q8—21no—
101C4×C5×Q8D411yescharacteristic
102C4×C5×D4—22no—
103C4×C5×D4—21no—
104C4×C5×D4—22no—
105C4×C5×D4—21no—
106C4×C5×D4D411yescharacteristic
107C5×#117—22no—
108C5×#117—21no—
109C5×#117D412yes—
110C2×C5×C4⋊C4—21no—
111C2×C5×C4⋊C4—21no—
112C2×C5×C22⋊C4—22no—
113C2×C5×C22⋊C4—21no—
114C2×C5×C22⋊C4—21no—
115C2×C42×C5C2311yescharacteristic
116C5×#107—22no—
117C5×#106—22no—
118C5×C4⋊C8—21no—
119C5×#103—24no—
120C5×#103—22no—
121C5×#103—24no—
122C5×#103—24no—
123C5×#102—24no—
124C5×#102—22no—
125C5×C22⋊C8—21no—
126C5×#97—21no—
127C5×#97C2312yes—
128C5×#97C2312yes—
129C5×#97C2312yes—
130C4×C5×C8—21no—
131#3561C2×C511yescharacteristic
132#3238C2×C511yescharacteristic, complemented
133#3139C2×C511yescharacteristic, complemented
134C2×#489C2×C511yescharacteristic, complemented
135#1664C2×C514yescomplemented
136#1662C2×C514yescomplemented
137#1531C2×C511yescharacteristic, complemented
138#1527C2×C511yescharacteristic, complemented
139#1524C2×C511yescharacteristic, complemented
140C24×C5—21no—
141C5×#41—82no—
142C2×C5×Q8—22no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8—21no—
146C2×C5×Q8—22no—
147C2×C5×Q8—42no—
148C2×C5×Q8—44no—
149C2×C5×Q8—41no—
150C2×C5×Q8—82no—
151C2×C5×Q8C2×D411yescharacteristic
152C2×C5×Q8C2×D412yes—
153C2×C5×D4—21no—
154C2×C5×D4—22no—
155C2×C5×D4—22no—
156C2×C5×D4—21no—
157C2×C5×D4—42no—
158C2×C5×D4C2×D411yescharacteristic
159C2×C5×D4C2×D412yes—
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5—22no—
163C22×C4×C5—21no—
164C22×C4×C5—21no—
165C22×C4×C5—41no—
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C5×M16—42no—
170C2×C5×C8—22no—
171C2×C5×C8—22no—
172C2×C5×C8—22no—
173C2×C5×C8—42no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—22no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—41no—
189C5×C4⋊C4—41no—
190C5×C4⋊C4C2×D412yes—
191C5×C4⋊C4C2×D412yes—
192C5×C4⋊C4C2×D411yescharacteristic
193C5×C4⋊C4C2×D411yescharacteristic
194C5×C22⋊C4—22no—
195C5×C22⋊C4—22no—
196C5×C22⋊C4—22no—
197C5×C22⋊C4—22no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—21no—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—22no—
202C5×C22⋊C4—22no—
203C5×C22⋊C4—21no—
204C5×C22⋊C4—21no—
205C5×C22⋊C4—22no—
206C5×C22⋊C4—22no—
207C5×C22⋊C4—21no—
208C5×C22⋊C4—21no—
209C42×C5—22no—
210C42×C5—22no—
211C42×C5—21no—
212C42×C5C2×D411yescharacteristic
213C42×C5C2×D411yescharacteristic
214C42×C5C2×D411yescharacteristic
215C42×C5C2411yescharacteristic
216#563C22×C512yes—
217#562C22×C512yes—
218#556—21no—
219#551—22no—
220D4×Q8—21no—
221#548—21no—
222#546—21no—
223#543—21no—
224#542—21no—
225#542—22no—
226#538—21no—
227#536—21no—
228#533C22×C511yescharacteristic
229C2×C4⋊Q8C22×C511yescharacteristic
230#529C22×C511yescharacteristic
231C2×#124C22×C511yescharacteristic
232#501C22×C512yescomplemented
233#491C22×C514yescomplemented
234#489—22no—
235#489C22×C514yes—
236#489C22×C512yescomplemented
237#487—21no—
238#487—21no—
239#483—21no—
240#480—21no—
241#477—21no—
242#476—21no—
243#432C22×C511yescharacteristic
244#419—22no—
245C2×#103—21no—
246C2×#102—21no—
247C4×M16C22×C511yescharacteristic
248C2×#97C22×C511yescharacteristic
249#330C22×C514yescomplemented
250#330C22×C512yescomplemented
251#329C22×C514yes—
252#329C22×C512yescomplemented
253C23×C5—22no—
254C23×C5—21no—
255C23×C5—42no—
256C23×C5—81no—
257C23×C5—81no—
258C23×C5#12011yescharacteristic
259C5×Q8—42no—
260C5×Q8—41no—
261C5×Q8—44no—
262C5×Q8—42no—
263C5×Q8—42no—
264C5×Q8—81no—
265C5×Q8—82no—
266C5×Q8—82no—
267C5×D4—42no—
268C5×D4—41no—
269C5×D4—42no—
270C5×D4—42no—
271C5×D4—82no—
272C5×D4—82no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—21no—
276C2×C4×C5—22no—
277C2×C4×C5—21no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—21no—
281C2×C4×C5—41no—
282C2×C4×C5—42no—
283C2×C4×C5—42no—
284C2×C4×C5—81no—
285C2×C4×C5—82no—
286C2×C4×C5—82no—
287C2×C4×C5—82no—
288C2×C4×C5—82no—
289C2×C4×C5#12011yescharacteristic
290C2×C4×C5#12011yescharacteristic
291C2×C4×C5#12011yescharacteristic
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C22×D411yescharacteristic
294C2×C4×C5C22×D411yescharacteristic
295C5×C8—42no—
296C5×C8—42no—
297C5×C8—42no—
298C2×#41—21no—
299C22×Q8—21no—
300C22×Q8—21no—
301C22×Q8C5×D411yescharacteristic
302C22×D4C5×D411yescharacteristic
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⋊Q8C5×D411yescharacteristic
312C4⋊Q8C23×C512yes—
313#126—22no—
314#126—22no—
315#126—22no—
316#126—21no—
317#125C23×C511yescharacteristic
318#125C23×C512yes—
319#124—21no—
320#124—22no—
321#124—22no—
322#124—21no—
323#124—21no—
324#124C5×D412yes—
325#124C23×C512yes—
326#124C23×C511yescharacteristic
327#123—22no—
328#123—22no—
329#123—21no—
330#123—22no—
331#123—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—21no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8C5×D411yescharacteristic
343#121—21no—
344#121—22no—
345#121—22no—
346#121C5×D411yescharacteristic
347#120—22no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8C5×D411yescharacteristic
352C4×D4—22no—
353C4×D4—21no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4C5×D411yescharacteristic
357#117—22no—
358#117—21no—
359#117C5×D412yes—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C42C23×C511yescharacteristic
366#107—22no—
367#106—22no—
368C4⋊C8—21no—
369#103—24no—
370#103—22no—
371#103—24no—
372#103—24no—
373#102—24no—
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—41no—
384C22×C5—42no—
385C22×C5—41no—
386C22×C5—42no—
387C22×C5—42no—
388C22×C5—41no—
389C22×C5—162no—
390C22×C5#45412yes—
391C22×C5C2×#12011yescharacteristic
392C4×C5—21no—
393C4×C5—21no—
394C4×C5—21no—
395C4×C5—21no—
396C4×C5—21no—
397C4×C5—41no—
398C4×C5—42no—
399C4×C5—42no—
400C4×C5—41no—
401C4×C5—41no—
402C4×C5—42no—
403C4×C5—82no—
404C4×C5#45512yes—
405C24—21no—
406#41—82no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—41no—
415C2×Q8—82no—
416C2×Q8C2×C5×D411yescharacteristic
417C2×Q8C2×C5×D412yes—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4C2×C5×D411yescharacteristic
424C2×D4C2×C5×D412yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4C2×C5×D411yescharacteristic
432C22×C4C2×C5×D411yescharacteristic
433C22×C4C2×C5×D411yescharacteristic
434M16—42no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—42no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4C2×C5×D412yes—
456C4⋊C4C2×C5×D412yes—
457C4⋊C4C2×C5×D411yescharacteristic
458C4⋊C4C2×C5×D411yescharacteristic
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—21no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C42—22no—
475C42—22no—
476C42—21no—
477C42C2×C5×D411yescharacteristic
478C42C2×C5×D411yescharacteristic
479C42C2×C5×D411yescharacteristic
480C42C24×C511yescharacteristic
481C2×C5—22no—
482C2×C5—81no—
483C2×C5—82no—
484C2×C5C2×#45511yescharacteristic
485C2×C5C2×#45411yescharacteristic
486C2×C5#302211yescharacteristic
487C23—22no—
488C23—21no—
489C23—42no—
490C23—81no—
491C23—81no—
492C23C5×#12011yescharacteristic
493Q8—42no—
494Q8—41no—
495Q8—44no—
496Q8—42no—
497Q8—42no—
498Q8—81no—
499Q8—82no—
500Q8—82no—
501D4—42no—
502D4—41no—
503D4—42no—
504D4—42no—
505D4—82no—
506D4—82no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4C5×#12011yescharacteristic
524C2×C4C5×#12011yescharacteristic
525C2×C4C5×#12011yescharacteristic
526C2×C4C22×C5×D411yescharacteristic
527C2×C4C22×C5×D411yescharacteristic
528C2×C4C22×C5×D411yescharacteristic
529C8—42no—
530C8—42no—
531C8—42no—
532C5#1902711yescharacteristic, direct product factor, Sylow
533C22—21no—
534C22—21no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—42no—
539C22—42no—
540C22—41no—
541C22—162no—
542C22C5×#45412yes—
543C22C2×C5×#12011yescharacteristic
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—42no—
551C4—42no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—82no—
556C4C5×#45512yes—
557C2—22no—
558C2—82no—
559C2—81no—
560C2C2×C5×#45511yescharacteristic
561C2C2×C5×#45411yescharacteristic
562C2C5×#302211yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
prime radicalC5×#3561C2
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. C5×#3139C2semidirect
4. C2×C5×#489C2semidirect
5. C5×#1664C2semidirect
6. C5×#1662C2semidirect
7. C5×#1531C2semidirect
8. C5×#1527C2semidirect
9. C5×#1524C2semidirect
26. C5×#501C22semidirect
27. C5×#491C22semidirect
30. C5×#489C22semidirect
43. C5×#330C22semidirect
44. C5×#330C22semidirect
46. C5×#329C22semidirect
47. #19027C5direct (N × H)
132. #3238C2×C5semidirect
133. #3139C2×C5semidirect
134. C2×#489C2×C5semidirect
135. #1664C2×C5semidirect
136. #1662C2×C5semidirect
137. #1531C2×C5semidirect
138. #1527C2×C5semidirect
139. #1524C2×C5semidirect
232. #501C22×C5semidirect
233. #491C22×C5semidirect
236. #489C22×C5semidirect
249. #330C22×C5semidirect
250. #330C22×C5semidirect
252. #329C22×C5semidirect
532. C5#19027direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35612
C5×#32382
C5×#31392
C2×C5×#4892
C5×#16642
C5×#16622
C5×#15312
C5×#15272
C5×#15242
#190275