Finite Group Databasebeta

← C5×#23481 #11946201

Subgroups — C5×#23481 #11946201

Order 1280.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×C22⋊Q8C211yesmaximal, characteristic
2C2×C5×#428C211yesmaximal, characteristic, complemented
3C2×C5×#419C211yesmaximal, characteristic, complemented
4C2×C5×#419C211yesmaximal, characteristic, complemented
5C2×C5×#407C211yesmaximal, characteristic
6C5×#2280C211yesmaximal, characteristic
7C5×#2036C214yesmaximal, complemented
8C5×#2035C214yesmaximal, complemented
9C5×#1736C211yesmaximal, characteristic, complemented
10C23×C5×Q8C2211yescharacteristic
11C24×C4×C5C2211yescharacteristic
12C22×C5×M16C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C2×C5×C22⋊Q8—24no—
15C2×C5×C22⋊Q8—22no—
16C2×C5×C22⋊Q8—24no—
17C2×C5×C22⋊Q8C2214yes—
18C2×C5×#117C2211yescharacteristic
19C2×C5×#117C2211yescharacteristic
20C22×C5×C4⋊C4—21no—
21C22×C5×C4⋊C4C2211yescharacteristic
22C22×C5×C22⋊C4—21no—
23C5×#428—24no—
24C2×C5×#107C2212yescomplemented
25C2×C5×#106C2212yescomplemented
26C5×#419—24no—
27C5×#419—24no—
28C2×C5×#103C2212yescomplemented
29C2×C5×#103C2214yescomplemented
30C2×C5×#103C2212yescomplemented
31C5×#407—24no—
32C2×C5×C22⋊C8C2214yes—
33C5×#381—21no—
34C5×#380C2212yes—
35C5×#380C2212yes—
36C5×#379—21no—
37C4×C5×C22⋊C4—22no—
38C5×#340C2214yescomplemented
39#23481C511yesmaximal, characteristic, direct product factor, Sylow
40C25×C5D411yescharacteristic
41C22×C5×Q8—22no—
42C22×C5×Q8—21no—
43C22×C5×Q8—22no—
44C22×C5×Q8—22no—
45C22×C5×Q8—22no—
46C22×C5×Q8—22no—
47C22×C5×Q8D414yes—
48C22×C5×Q8C2312yes—
49C23×C4×C5—21no—
50C23×C4×C5—21no—
51C23×C4×C5—22no—
52C23×C4×C5—22no—
53C23×C4×C5—22no—
54C23×C4×C5—22no—
55C23×C4×C5—22no—
56C23×C4×C5—22no—
57C23×C4×C5D412yes—
58C23×C4×C5D411yescharacteristic
59C23×C4×C5C2312yes—
60C23×C4×C5C2311yescharacteristic
61C2×C5×M16—22no—
62C2×C5×M16—22no—
63C2×C5×M16—22no—
64C2×C5×M16—22no—
65C2×C5×M16—22no—
66C2×C5×M16D414yes—
67C22×C5×C8C2312yes—
68C22×C5×C8C2312yes—
69C5×C22⋊Q8—216no—
70C5×C22⋊Q8—28no—
71C5×C22⋊Q8—28no—
72C5×#117—22no—
73C5×#117—22no—
74C5×#117—22no—
75C5×#117—22no—
76C2×C5×C4⋊C4—28no—
77C2×C5×C4⋊C4—24no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—22no—
80C2×C5×C4⋊C4D414yes—
81C2×C5×C4⋊C4C2312yes—
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2312yes—
85C2×C5×C22⋊C4—28no—
86C2×C5×C22⋊C4—24no—
87C2×C5×C22⋊C4—22no—
88C2×C5×C22⋊C4—22no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—21no—
91C2×C42×C5—21no—
92C2×C42×C5—21no—
93C5×#107—24no—
94C5×#106—24no—
95C5×#103—24no—
96C5×#103—24no—
97C5×#103—28no—
98C5×C22⋊C8—28no—
99C5×#95—22no—
100C5×#95—22no—
101C22×C22⋊Q8C2×C511yescharacteristic
102C2×#428C2×C511yescharacteristic, complemented
103C2×#419C2×C511yescharacteristic, complemented
104C2×#419C2×C511yescharacteristic, complemented
105C2×#407C2×C511yescharacteristic
106#2280C2×C511yescharacteristic
107#2036C2×C514yescomplemented
108#2035C2×C514yescomplemented
109#1736C2×C511yescharacteristic, complemented
110C24×C5—22no—
111C24×C5—24no—
112C24×C5—22no—
113C24×C5—22no—
114C24×C5—42no—
115C24×C5C2×D412yes—
116C24×C5C2×D411yescharacteristic
117C2×C5×Q8—24no—
118C2×C5×Q8—22no—
119C2×C5×Q8—28no—
120C2×C5×Q8—28no—
121C2×C5×Q8—22no—
122C2×C5×Q8—22no—
123C2×C5×Q8—22no—
124C2×C5×Q8—22no—
125C2×C5×Q8—42no—
126C2×C5×Q8—44no—
127C2×C5×Q8—42no—
128C2×C5×Q8—44no—
129C2×C5×Q8C2×D414yes—
130C22×C4×C5—24no—
131C22×C4×C5—24no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—24no—
138C22×C4×C5—24no—
139C22×C4×C5—24no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—22no—
149C22×C4×C5—44no—
150C22×C4×C5—42no—
151C22×C4×C5—42no—
152C22×C4×C5—44no—
153C22×C4×C5—44no—
154C22×C4×C5—44no—
155C22×C4×C5—44no—
156C22×C4×C5C2×D412yes—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D412yes—
159C22×C4×C5C2×D412yes—
160C22×C4×C5C2×D412yes—
161C22×C4×C5#4112yes—
162C22×C4×C5#4112yes—
163C22×C4×C5#4112yes—
164C22×C4×C5C2411yescharacteristic
165C5×M16—28no—
166C5×M16—44no—
167C2×C5×C8—22no—
168C2×C5×C8—22no—
169C2×C5×C8—22no—
170C2×C5×C8—22no—
171C2×C5×C8—22no—
172C2×C5×C8C2×D414yes—
173C5×C4⋊C4—216no—
174C5×C4⋊C4—24no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4C2×D414yes—
183C5×C22⋊C4—216no—
184C5×C22⋊C4—44no—
185C5×C22⋊C4—42no—
186C5×C22⋊C4—42no—
187C5×C22⋊C4—44no—
188C42×C5—42no—
189C42×C5—42no—
190C23×Q8C22×C511yescharacteristic
191C24×C4C22×C511yescharacteristic
192C22×M16C22×C511yescharacteristic
193C22×M16C22×C511yescharacteristic
194C2×C22⋊Q8—24no—
195C2×C22⋊Q8—22no—
196C2×C22⋊Q8—24no—
197C2×C22⋊Q8C22×C514yes—
198C2×#117C22×C511yescharacteristic
199C2×#117C22×C511yescharacteristic
200C22×C4⋊C4—21no—
201C22×C4⋊C4C22×C511yescharacteristic
202C22×C22⋊C4—21no—
203#428—24no—
204C2×#107C22×C512yescomplemented
205C2×#106C22×C512yescomplemented
206#419—24no—
207#419—24no—
208C2×#103C22×C512yescomplemented
209C2×#103C22×C514yescomplemented
210C2×#103C22×C512yescomplemented
211#407—24no—
212C2×C22⋊C8C22×C514yes—
213#381—21no—
214#380C22×C512yes—
215#380C22×C512yes—
216#379—21no—
217C4×C22⋊C4—22no—
218#340C22×C514yescomplemented
219C23×C5—22no—
220C23×C5—22no—
221C23×C5—28no—
222C23×C5—22no—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—22no—
228C23×C5—48no—
229C23×C5—48no—
230C23×C5—44no—
231C23×C5—44no—
232C23×C5#12012yes—
233C23×C5#12112yes—
234C23×C5#12312yes—
235C23×C5#13714yes—
236C23×C5C22×D411yescharacteristic
237C5×Q8—24no—
238C5×Q8—24no—
239C5×Q8—42no—
240C5×Q8—42no—
241C5×Q8—48no—
242C2×C4×C5—24no—
243C2×C4×C5—24no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—48no—
248C2×C4×C5—42no—
249C2×C4×C5—42no—
250C2×C4×C5—44no—
251C2×C4×C5—44no—
252C2×C4×C5—48no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—48no—
258C2×C4×C5—41no—
259C2×C4×C5—41no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—42no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5#12012yes—
266C2×C4×C5#12114yes—
267C2×C4×C5#12114yes—
268C2×C4×C5#12112yes—
269C2×C4×C5#12314yes—
270C2×C4×C5#12312yes—
271C2×C4×C5#12414yes—
272C2×C4×C5C22×D411yescharacteristic
273C2×C4×C5C22×D411yescharacteristic
274C2×C4×C5C22×D411yescharacteristic
275C2×C4×C5C2×#4111yescharacteristic
276C2×C4×C5C2×#4111yescharacteristic
277C2×C4×C5C2×#4111yescharacteristic
278C5×C8—24no—
279C5×C8—42no—
280C25C5×D411yescharacteristic
281C22×Q8—22no—
282C22×Q8—21no—
283C22×Q8—22no—
284C22×Q8—22no—
285C22×Q8—22no—
286C22×Q8—22no—
287C22×Q8C5×D414yes—
288C22×Q8C23×C512yes—
289C23×C4—21no—
290C23×C4—21no—
291C23×C4—22no—
292C23×C4—22no—
293C23×C4—22no—
294C23×C4—22no—
295C23×C4—22no—
296C23×C4—22no—
297C23×C4C5×D411yescharacteristic
298C23×C4C5×D412yes—
299C23×C4C23×C512yes—
300C23×C4C23×C511yescharacteristic
301C2×M16—22no—
302C2×M16—22no—
303C2×M16—22no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16C5×D414yes—
307C22×C8C23×C512yes—
308C22×C8C23×C512yes—
309C22⋊Q8—216no—
310C22⋊Q8—28no—
311C22⋊Q8—28no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4C5×D414yes—
321C2×C4⋊C4C23×C512yes—
322C2×C4⋊C4C23×C511yescharacteristic
323C2×C4⋊C4C23×C511yescharacteristic
324C2×C4⋊C4C23×C512yes—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C42—21no—
332C2×C42—21no—
333#107—24no—
334#106—24no—
335#103—24no—
336#103—24no—
337#103—28no—
338C22⋊C8—28no—
339#95—22no—
340#95—22no—
341C22×C5—24no—
342C22×C5—22no—
343C22×C5—22no—
344C22×C5—24no—
345C22×C5—24no—
346C22×C5—24no—
347C22×C5—24no—
348C22×C5—416no—
349C22×C5—42no—
350C22×C5—42no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5#39414yes—
354C22×C5C2×#12011yescharacteristic
355C22×C5C2×#12111yescharacteristic
356C22×C5C2×#12311yescharacteristic
357C22×C5C2×#13712yes—
358C22×C5#57512yes—
359C4×C5—42no—
360C4×C5—44no—
361C4×C5—44no—
362C4×C5—81no—
363C4×C5—81no—
364C4×C5—82no—
365C4×C5#39414yes—
366C4×C5C2×#12111yescharacteristic
367C4×C5C2×#12111yescharacteristic
368C4×C5C2×#12311yescharacteristic
369C4×C5C2×#12411yescharacteristic
370C24—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—42no—
375C24C2×C5×D412yes—
376C24C2×C5×D411yescharacteristic
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8C2×C5×D414yes—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4C2×C5×D412yes—
417C22×C4C2×C5×D411yescharacteristic
418C22×C4C2×C5×D412yes—
419C22×C4C2×C5×D412yes—
420C22×C4C2×C5×D412yes—
421C22×C4C5×#4112yes—
422C22×C4C5×#4112yes—
423C22×C4C5×#4112yes—
424C22×C4C24×C511yescharacteristic
425M16—28no—
426M16—44no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8C2×C5×D414yes—
433C4⋊C4—216no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4C2×C5×D414yes—
443C22⋊C4—216no—
444C22⋊C4—44no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C42—42no—
449C42—42no—
450C2×C5—24no—
451C2×C5—44no—
452C2×C5C2×#39411yescharacteristic
453C2×C5#300411yescharacteristic
454C2×C5#300511yescharacteristic
455C2×C5#305612yes—
456C2×C5#309512yes—
457C23—22no—
458C23—22no—
459C23—28no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—48no—
467C23—48no—
468C23—44no—
469C23—44no—
470C23C5×#12012yes—
471C23C5×#12112yes—
472C23C5×#12312yes—
473C23C5×#13714yes—
474C23C22×C5×D411yescharacteristic
475Q8—24no—
476Q8—24no—
477Q8—48no—
478Q8—42no—
479Q8—42no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—48no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4C5×#12012yes—
504C2×C4C5×#12112yes—
505C2×C4C5×#12114yes—
506C2×C4C5×#12114yes—
507C2×C4C5×#12312yes—
508C2×C4C5×#12314yes—
509C2×C4C5×#12414yes—
510C2×C4C22×C5×D411yescharacteristic
511C2×C4C22×C5×D411yescharacteristic
512C2×C4C22×C5×D411yescharacteristic
513C2×C4C2×C5×#4111yescharacteristic
514C2×C4C2×C5×#4111yescharacteristic
515C2×C4C2×C5×#4111yescharacteristic
516C8—24no—
517C8—42no—
518C5#2348111yescharacteristic, direct product factor, Sylow
519C22—22no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22C5×#39414yes—
532C22C2×C5×#12011yescharacteristic
533C22C2×C5×#12111yescharacteristic
534C22C2×C5×#12311yescharacteristic
535C22C2×C5×#13712yes—
536C22C5×#57512yes—
537C4—44no—
538C4—42no—
539C4—44no—
540C4—81no—
541C4—81no—
542C4—82no—
543C4C5×#39414yes—
544C4C2×C5×#12111yescharacteristic
545C4C2×C5×#12111yescharacteristic
546C4C2×C5×#12311yescharacteristic
547C4C2×C5×#12411yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×C5×#39411yescharacteristic
551C2C5×#300411yescharacteristic
552C2C5×#300511yescharacteristic
553C2C5×#305612yes—
554C2C5×#309512yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC25×C5D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C24×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#428C2semidirect
3. C2×C5×#419C2semidirect
4. C2×C5×#419C2semidirect
7. C5×#2036C2semidirect
8. C5×#2035C2semidirect
9. C5×#1736C2semidirect
24. C2×C5×#107C22semidirect
25. C2×C5×#106C22semidirect
28. C2×C5×#103C22semidirect
29. C2×C5×#103C22semidirect
30. C2×C5×#103C22semidirect
38. C5×#340C22semidirect
39. #23481C5direct (N × H)
102. C2×#428C2×C5semidirect
103. C2×#419C2×C5semidirect
104. C2×#419C2×C5semidirect
107. #2036C2×C5semidirect
108. #2035C2×C5semidirect
109. #1736C2×C5semidirect
204. C2×#107C22×C5semidirect
205. C2×#106C22×C5semidirect
208. C2×#103C22×C5semidirect
209. C2×#103C22×C5semidirect
210. C2×#103C22×C5semidirect
218. #340C22×C5semidirect
518. C5#23481direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×C22⋊Q82
C2×C5×#4282
C2×C5×#4192
C2×C5×#4072
C5×#22802
C5×#20362
C5×#20352
C5×#17362
#234815