Finite Group Databasebeta

← C5×#18152 #11940872

Subgroups — C5×#18152 #11940872

Order 1280.

1240 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3449C211yesmaximal, characteristic, complemented
2C5×#3109C212yesmaximal, complemented
3C5×#2598C211yesmaximal, characteristic, complemented
4C5×#1616C212yesmaximal, complemented
5C5×#1613C212yesmaximal, complemented
6C5×#1598C212yesmaximal, complemented
7C5×#1595C212yesmaximal, complemented
8C5×#1514C212yesmaximal, complemented
9C5×#1464C211yesmaximal, characteristic, complemented
10C22×C5×#41—21no—
11C5×#537—21no—
12C5×#537C2212yes—
13C5×#536—22no—
14C5×#535—22no—
15C5×#534—21no—
16C5×#534C2212yes—
17C5×#532—21no—
18C5×#532C2212yes—
19C5×#525—24no—
20C2×C4×C5×D4C2211yescharacteristic, complemented
21C2×C5×#117C2211yescharacteristic
22C5×#481C2214yescomplemented
23C5×#476—22no—
24C5×#459—22no—
25C5×#447C2214yescomplemented
26C5×#442—24no—
27C5×#432C2212yescomplemented
28C5×#400C2211yescharacteristic, complemented
29C5×#399C2211yescharacteristic
30C5×#394C2211yescharacteristic, complemented
31C5×#392C2211yescharacteristic
32C5×#388—21no—
33C5×#386—21no—
34C5×#386—21no—
35C5×#385—21no—
36C5×#383C2211yescharacteristic, complemented
37C5×#327C2214yescomplemented
38C5×#327C2214yescomplemented
39C5×#324C2212yescomplemented
40C5×#323C2212yes—
41#18152C511yesmaximal, characteristic, direct product factor, Sylow
42C2×C5×#41—22no—
43C2×C5×#41—22no—
44C2×C5×#41—22no—
45C2×C5×#41—22no—
46C2×C5×#41—44no—
47C2×C5×#41—44no—
48C22×C5×Q8—21no—
49C22×C5×D4—21no—
50C22×C5×D4—22no—
51C22×C5×D4—21no—
52C22×C5×D4C2311yescharacteristic
53C23×C4×C5—22no—
54C23×C4×C5—21no—
55C23×C4×C5D411yescharacteristic
56C2×C5×D8—24no—
57C5×C4⋊Q8—21no—
58C5×C4⋊Q8D412yescomplemented
59C5×#127—21no—
60C5×#127D412yescomplemented
61C5×#124—22no—
62C5×#124—22no—
63C5×#123—24no—
64C5×#123—24no—
65C5×C22⋊Q8—24no—
66C5×C22⋊Q8—24no—
67C5×#121—24no—
68C5×#121—24no—
69C5×#121—22no—
70C5×#121—22no—
71C5×#121—22no—
72C5×#121C2314yes—
73C5×#120—24no—
74C5×#120—24no—
75C4×C5×D4—22no—
76C4×C5×D4—22no—
77C4×C5×D4—22no—
78C4×C5×D4—22no—
79C4×C5×D4—22no—
80C4×C5×D4D414yescomplemented
81C5×#117—22no—
82C5×#117—22no—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4D411yescharacteristic
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C5×C4⋊C4C2311yescharacteristic
87C2×C5×C22⋊C4—21no—
88C2×C5×C22⋊C4—21no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—21no—
92C2×C5×C22⋊C4D411yescharacteristic
93C2×C5×C22⋊C4C2311yescharacteristic
94C2×C42×C5D411yescharacteristic
95C2×C42×C5C2311yescharacteristic
96C5×#106—24no—
97C5×C4⋊C8—22no—
98C5×#102—24no—
99C5×#102—24no—
100C5×C22⋊C8C2314yes—
101C5×#97—22no—
102C5×#95—21no—
103C5×#95—21no—
104C5×#95—21no—
105C5×#95—21no—
106C5×#95C2311yescharacteristic
107C5×#95C2311yescharacteristic
108#3449C2×C511yescharacteristic, complemented
109#3109C2×C512yescomplemented
110#2598C2×C511yescharacteristic, complemented
111#1616C2×C512yescomplemented
112#1613C2×C512yescomplemented
113#1598C2×C512yescomplemented
114#1595C2×C512yescomplemented
115#1514C2×C512yescomplemented
116#1464C2×C511yescharacteristic, complemented
117C24×C5—22no—
118C24×C5—21no—
119C24×C5C2×D411yescharacteristic
120C5×#41—48no—
121C5×#41—48no—
122C5×#41—44no—
123C5×#41—44no—
124C5×#41—44no—
125C5×#41—44no—
126C2×C5×Q8—22no—
127C2×C5×Q8—22no—
128C2×C5×Q8—44no—
129C2×C5×D4—24no—
130C2×C5×D4—22no—
131C2×C5×D4—24no—
132C2×C5×D4—22no—
133C2×C5×D4—22no—
134C2×C5×D4—22no—
135C2×C5×D4—22no—
136C2×C5×D4—22no—
137C2×C5×D4—44no—
138C2×C5×D4—42no—
139C2×C5×D4—44no—
140C2×C5×D4—44no—
141C2×C5×D4—42no—
142C2×C5×D4—42no—
143C2×C5×D4—81no—
144C2×C5×D4C2×D414yes—
145C22×C4×C5—22no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—22no—
149C22×C4×C5—24no—
150C22×C4×C5—22no—
151C22×C4×C5—24no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—48no—
156C22×C4×C5—42no—
157C22×C4×C5—81no—
158C22×C4×C5—81no—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C5×D8—48no—
166C2×C5×C8—24no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—22no—
171C5×C4⋊C4—22no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—81no—
174C5×C4⋊C4C2×D414yes—
175C5×C22⋊C4—24no—
176C5×C22⋊C4—24no—
177C5×C22⋊C4—22no—
178C5×C22⋊C4—24no—
179C5×C22⋊C4—22no—
180C5×C22⋊C4—24no—
181C5×C22⋊C4—42no—
182C5×C22⋊C4—41no—
183C5×C22⋊C4—41no—
184C5×C22⋊C4—42no—
185C5×C22⋊C4—81no—
186C5×C22⋊C4—81no—
187C42×C5—21no—
188C42×C5—21no—
189C42×C5C2×D412yes—
190C42×C5C2×D412yes—
191C22×#41—21no—
192#537—21no—
193#537C22×C512yes—
194#536—22no—
195#535—22no—
196#534—21no—
197#534C22×C512yes—
198#532—21no—
199#532C22×C512yes—
200#525—24no—
201C2×C4×D4C22×C511yescharacteristic, complemented
202C2×#117C22×C511yescharacteristic
203#481C22×C514yescomplemented
204#476—22no—
205#459—22no—
206#447C22×C514yescomplemented
207#442—24no—
208#432C22×C512yescomplemented
209#400C22×C511yescharacteristic, complemented
210#399C22×C511yescharacteristic
211#394C22×C511yescharacteristic, complemented
212#392C22×C511yescharacteristic
213#388—21no—
214#386—21no—
215#386—21no—
216#385—21no—
217#383C22×C511yescharacteristic, complemented
218#327C22×C514yescomplemented
219#327C22×C514yescomplemented
220#324C22×C512yescomplemented
221#323C22×C512yes—
222C23×C5—21no—
223C23×C5—21no—
224C23×C5—22no—
225C23×C5—24no—
226C23×C5—41no—
227C23×C5—41no—
228C23×C5—41no—
229C23×C5—44no—
230C23×C5—41no—
231C23×C5—41no—
232C23×C5—41no—
233C23×C5—41no—
234C23×C5—41no—
235C23×C5—82no—
236C23×C5—82no—
237C23×C5AGL(1,ℤ/8)12yes—
238C23×C5C22×D411yescharacteristic
239C5×Q8—44no—
240C5×Q8—44no—
241C5×D4—24no—
242C5×D4—44no—
243C5×D4—42no—
244C5×D4—44no—
245C5×D4—44no—
246C5×D4—48no—
247C5×D4—44no—
248C5×D4—44no—
249C5×D4—82no—
250C5×D4—82no—
251C5×D4—82no—
252C5×D4—84no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—24no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—48no—
265C2×C4×C5—48no—
266C2×C4×C5—44no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—84no—
279C2×C4×C5#12011yescharacteristic
280C2×C4×C5#12011yescharacteristic
281C2×C4×C5#12012yes—
282C2×C4×C5AGL(1,ℤ/8)12yes—
283C2×C4×C5C22×D412yes—
284C5×C8—44no—
285C2×#41—22no—
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—22no—
289C2×#41—44no—
290C2×#41—44no—
291C22×Q8—21no—
292C22×D4—21no—
293C22×D4—22no—
294C22×D4—21no—
295C22×D4C23×C511yescharacteristic
296C23×C4—22no—
297C23×C4—21no—
298C23×C4C5×D411yescharacteristic
299C2×D8—24no—
300C4⋊Q8—21no—
301C4⋊Q8C5×D412yescomplemented
302#127—21no—
303#127C5×D412yescomplemented
304#124—22no—
305#124—22no—
306#123—24no—
307#123—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310#121—24no—
311#121—24no—
312#121—22no—
313#121—22no—
314#121—22no—
315#121C23×C514yes—
316#120—24no—
317#120—24no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4C5×D414yescomplemented
324#117—22no—
325#117—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C5×D411yescharacteristic
328C2×C4⋊C4C23×C511yescharacteristic
329C2×C4⋊C4C23×C511yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4C5×D411yescharacteristic
336C2×C22⋊C4C23×C511yescharacteristic
337C2×C42C5×D411yescharacteristic
338C2×C42C23×C511yescharacteristic
339#106—24no—
340C4⋊C8—22no—
341#102—24no—
342#102—24no—
343C22⋊C8C23×C514yes—
344#97—22no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95C23×C511yescharacteristic
350#95C23×C511yescharacteristic
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—22no—
354C22×C5—42no—
355C22×C5—42no—
356C22×C5—42no—
357C22×C5—41no—
358C22×C5—41no—
359C22×C5—44no—
360C22×C5—44no—
361C22×C5—42no—
362C22×C5—44no—
363C22×C5—81no—
364C22×C5—81no—
365C22×C5—81no—
366C22×C5—88no—
367C22×C5—81no—
368C22×C5—81no—
369C22×C5C2×#12011yescharacteristic
370C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
371C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
372C4×C5—21no—
373C4×C5—22no—
374C4×C5—21no—
375C4×C5—22no—
376C4×C5—22no—
377C4×C5—44no—
378C4×C5—41no—
379C4×C5—41no—
380C4×C5—81no—
381C4×C5—81no—
382C4×C5—81no—
383C24—22no—
384C24—21no—
385C24C2×C5×D411yescharacteristic
386#41—48no—
387#41—48no—
388#41—44no—
389#41—44no—
390#41—44no—
391#41—44no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—44no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C2×D4C2×C5×D414yes—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—48no—
422C22×C4—42no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4C2×C5×D411yescharacteristic
426C22×C4C2×C5×D411yescharacteristic
427C22×C4C2×C5×D411yescharacteristic
428C22×C4C2×C5×D411yescharacteristic
429C22×C4C2×C5×D411yescharacteristic
430C22×C4C24×C511yescharacteristic
431D8—48no—
432C2×C8—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—81no—
440C4⋊C4C2×C5×D414yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—42no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C42—21no—
454C42—21no—
455C42C2×C5×D412yes—
456C42C2×C5×D412yes—
457C2×C5—21no—
458C2×C5—21no—
459C2×C5—42no—
460C2×C5—82no—
461C2×C5—84no—
462C2×C5#300311yescharacteristic
463C2×C5#302111yescharacteristic
464C2×C5#302511yescharacteristic
465C23—21no—
466C23—21no—
467C23—22no—
468C23—24no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—44no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—82no—
479C23—82no—
480C23C5×AGL(1,ℤ/8)12yes—
481C23C22×C5×D411yescharacteristic
482Q8—44no—
483Q8—44no—
484D4—24no—
485D4—44no—
486D4—44no—
487D4—44no—
488D4—42no—
489D4—48no—
490D4—44no—
491D4—44no—
492D4—82no—
493D4—82no—
494D4—82no—
495D4—84no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4C5×#12011yescharacteristic
523C2×C4C5×#12011yescharacteristic
524C2×C4C5×#12012yes—
525C2×C4C5×AGL(1,ℤ/8)12yes—
526C2×C4C22×C5×D412yes—
527C8—44no—
528C5#1815211yescharacteristic, direct product factor, Sylow
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—44no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—88no—
545C22—81no—
546C22—81no—
547C22C2×C5×#12011yescharacteristic
548C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
549C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
550C4—21no—
551C4—22no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—81no—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—82no—
565C2—84no—
566C2C5×#300311yescharacteristic
567C2C5×#302111yescharacteristic
568C2C5×#302511yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#3449C2semidirect
2. C5×#3109C2semidirect
3. C5×#2598C2semidirect
4. C5×#1616C2semidirect
5. C5×#1613C2semidirect
6. C5×#1598C2semidirect
7. C5×#1595C2semidirect
8. C5×#1514C2semidirect
9. C5×#1464C2semidirect
20. C2×C4×C5×D4C22semidirect
22. C5×#481C22semidirect
25. C5×#447C22semidirect
27. C5×#432C22semidirect
28. C5×#400C22semidirect
30. C5×#394C22semidirect
36. C5×#383C22semidirect
37. C5×#327C22semidirect
38. C5×#327C22semidirect
39. C5×#324C22semidirect
41. #18152C5direct (N × H)
58. C5×C4⋊Q8D4semidirect
60. C5×#127D4semidirect
80. C4×C5×D4D4semidirect
108. #3449C2×C5semidirect
109. #3109C2×C5semidirect
110. #2598C2×C5semidirect
111. #1616C2×C5semidirect
112. #1613C2×C5semidirect
113. #1598C2×C5semidirect
114. #1595C2×C5semidirect
115. #1514C2×C5semidirect
116. #1464C2×C5semidirect
201. C2×C4×D4C22×C5semidirect
203. #481C22×C5semidirect
206. #447C22×C5semidirect
208. #432C22×C5semidirect
209. #400C22×C5semidirect
211. #394C22×C5semidirect
217. #383C22×C5semidirect
218. #327C22×C5semidirect
219. #327C22×C5semidirect
220. #324C22×C5semidirect
301. C4⋊Q8C5×D4semidirect
303. #127C5×D4semidirect
323. C4×D4C5×D4semidirect
528. C5#18152direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#34492
C5×#31092
C5×#25982
C5×#16162
C5×#16132
C5×#15982
C5×#15952
C5×#15142
C5×#14642
#181525