Finite Group Databasebeta

← C5×#29761 #11952481

Subgroups — C5×#29761 #11952481

Order 1280.

910 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#484C211yesmaximal, characteristic, complemented
2C2×C5×#481C211yesmaximal, characteristic
3C5×#2850C211yesmaximal, characteristic
4C2×C5×#389C211yesmaximal, characteristic, complemented
5C5×#2085C212yesmaximal, complemented
6C5×#2067C212yesmaximal, complemented
7C5×#2066C212yesmaximal, complemented
8C5×#2042C212yesmaximal, complemented
9C5×#1874C211yesmaximal, characteristic
10C5×#1872C211yesmaximal, characteristic, complemented
11C5×#1797C211yesmaximal, characteristic, complemented
12C2×C5×#124C2211yescharacteristic
13C2×C5×C22⋊Q8C2211yescharacteristic
14C2×C5×#121C2211yescharacteristic
15C22×C5×C4⋊C4—21no—
16C22×C5×C4⋊C4C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C42×C5C2211yescharacteristic
19C5×#484—24no—
20C5×#481—24no—
21C2×C5×#107C2212yes—
22C2×C5×#106C2212yes—
23C2×C5×C4⋊C8C2212yes—
24C2×C5×#103C2212yes—
25C2×C5×#103C2212yes—
26C2×C5×#102C2212yes—
27C2×C5×#102C2212yes—
28C2×C5×C22⋊C8C2211yescharacteristic
29C2×C5×C22⋊C8C2211yescharacteristic
30C5×#391C2211yescharacteristic
31C5×#390C2211yescharacteristic
32C5×#389—21no—
33C5×#389—21no—
34C5×#389C2212yes—
35C5×#389C2212yes—
36C5×#388—21no—
37C5×#388—21no—
38C5×#386—21no—
39C5×#386—21no—
40C5×#382C2211yescharacteristic
41C2×C5×#95—21no—
42C5×#340C2212yes—
43C5×#340C2212yes—
44C5×#336C2212yes—
45#29761C511yesmaximal, characteristic, direct product factor, Sylow
46C22×C5×Q8C2311yescharacteristic
47C22×C5×D4—21no—
48C22×C5×D4C2311yescharacteristic
49C23×C4×C5—21no—
50C23×C4×C5—21no—
51C23×C4×C5D411yescharacteristic
52C23×C4×C5D411yescharacteristic
53C23×C4×C5C2311yescharacteristic
54C22×C5×C8C2312yes—
55C22×C5×C8C2312yes—
56C5×#124—42no—
57C5×C22⋊Q8—22no—
58C5×C22⋊Q8—22no—
59C5×#121—22no—
60C5×#121—22no—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4—22no—
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—42no—
68C2×C5×C4⋊C4—44no—
69C2×C5×C4⋊C4D414yes—
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2312yes—
73C2×C5×C4⋊C4C2312yes—
74C2×C5×C22⋊C4—21no—
75C2×C5×C22⋊C4—21no—
76C2×C5×C22⋊C4—21no—
77C2×C5×C22⋊C4—21no—
78C2×C5×C22⋊C4—21no—
79C2×C5×C22⋊C4—21no—
80C2×C42×C5—42no—
81C2×C42×C5D411yescharacteristic
82C2×C42×C5D411yescharacteristic
83C2×C42×C5C2311yescharacteristic
84C2×C42×C5C2311yescharacteristic
85C5×#107—24no—
86C5×#106—24no—
87C5×C4⋊C8—42no—
88C5×#103—24no—
89C5×#103—42no—
90C5×#102—24no—
91C5×#102—42no—
92C5×C22⋊C8—22no—
93C5×C22⋊C8—22no—
94C5×#95—22no—
95C5×#95—22no—
96C5×#95—21no—
97C5×#95—21no—
98C5×#95—21no—
99C5×#95—21no—
100C2×#484C2×C511yescharacteristic, complemented
101C2×#481C2×C511yescharacteristic
102#2850C2×C511yescharacteristic
103C2×#389C2×C511yescharacteristic, complemented
104#2085C2×C512yescomplemented
105#2067C2×C512yescomplemented
106#2066C2×C512yescomplemented
107#2042C2×C512yescomplemented
108#1874C2×C511yescharacteristic
109#1872C2×C511yescharacteristic, complemented
110#1797C2×C511yescharacteristic, complemented
111C24×C5—21no—
112C24×C5C2×D411yescharacteristic
113C2×C5×Q8—21no—
114C2×C5×Q8—21no—
115C2×C5×Q8—22no—
116C2×C5×Q8—22no—
117C2×C5×D4—21no—
118C2×C5×D4—21no—
119C2×C5×D4—22no—
120C2×C5×D4—22no—
121C2×C5×D4—42no—
122C2×C5×D4—42no—
123C2×C5×D4—42no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—24no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—22no—
135C22×C4×C5—41no—
136C22×C4×C5—41no—
137C22×C4×C5—41no—
138C22×C4×C5—42no—
139C22×C4×C5—42no—
140C22×C4×C5—42no—
141C22×C4×C5—42no—
142C22×C4×C5—42no—
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D412yes—
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5#4112yes—
150C22×C4×C5C2411yescharacteristic
151C2×C5×C8—22no—
152C2×C5×C8—22no—
153C2×C5×C8—22no—
154C2×C5×C8—22no—
155C2×C5×C8—22no—
156C2×C5×C8—22no—
157C5×C4⋊C4—24no—
158C5×C4⋊C4—22no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—42no—
166C5×C4⋊C4—44no—
167C5×C4⋊C4—42no—
168C5×C4⋊C4—44no—
169C5×C4⋊C4C2×D414yes—
170C5×C22⋊C4—42no—
171C5×C22⋊C4—42no—
172C5×C22⋊C4—42no—
173C5×C22⋊C4—42no—
174C5×C22⋊C4—42no—
175C5×C22⋊C4—42no—
176C42×C5—22no—
177C42×C5—22no—
178C42×C5—41no—
179C42×C5—41no—
180C2×#124C22×C511yescharacteristic
181C2×C22⋊Q8C22×C511yescharacteristic
182C2×#121C22×C511yescharacteristic
183C22×C4⋊C4—21no—
184C22×C4⋊C4C22×C511yescharacteristic
185C22×C4⋊C4C22×C511yescharacteristic
186C22×C42C22×C511yescharacteristic
187#484—24no—
188#481—24no—
189C2×#107C22×C512yes—
190C2×#106C22×C512yes—
191C2×C4⋊C8C22×C512yes—
192C2×#103C22×C512yes—
193C2×#103C22×C512yes—
194C2×#102C22×C512yes—
195C2×#102C22×C512yes—
196C2×C22⋊C8C22×C511yescharacteristic
197C2×C22⋊C8C22×C511yescharacteristic
198#391C22×C511yescharacteristic
199#390C22×C511yescharacteristic
200#389—21no—
201#389—21no—
202#389C22×C512yes—
203#389C22×C512yes—
204#388—21no—
205#388—21no—
206#386—21no—
207#386—21no—
208#382C22×C511yescharacteristic
209C2×#95—21no—
210#340C22×C512yes—
211#340C22×C512yes—
212#336C22×C512yes—
213C23×C5—21no—
214C23×C5—21no—
215C23×C5—21no—
216C23×C5—21no—
217C23×C5—41no—
218C23×C5—41no—
219C23×C5—41no—
220C23×C5—41no—
221C23×C5—41no—
222C23×C5—41no—
223C23×C5—41no—
224C23×C5#12112yes—
225C23×C5#13512yes—
226C23×C5#13712yes—
227C23×C5C22×D411yescharacteristic
228C5×Q8—42no—
229C5×Q8—42no—
230C5×D4—42no—
231C5×D4—42no—
232C5×D4—84no—
233C2×C4×C5—21no—
234C2×C4×C5—21no—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—24no—
238C2×C4×C5—24no—
239C2×C4×C5—24no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—21no—
245C2×C4×C5—21no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—44no—
250C2×C4×C5—44no—
251C2×C4×C5—48no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—41no—
255C2×C4×C5—41no—
256C2×C4×C5—41no—
257C2×C4×C5—41no—
258C2×C4×C5—42no—
259C2×C4×C5—41no—
260C2×C4×C5—42no—
261C2×C4×C5—41no—
262C2×C4×C5—41no—
263C2×C4×C5—42no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—84no—
267C2×C4×C5#12112yes—
268C2×C4×C5AGL(1,ℤ/8)12yes—
269C2×C4×C5C22×D411yescharacteristic
270C2×C4×C5C2×#4111yescharacteristic
271C2×C4×C5H(5,𝔽2)12yes—
272C2×C4×C52−1+412yes—
273C5×C8—42no—
274C5×C8—42no—
275C22×Q8C23×C511yescharacteristic
276C22×D4—21no—
277C22×D4C23×C511yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4C5×D411yescharacteristic
281C23×C4C5×D411yescharacteristic
282C23×C4C23×C511yescharacteristic
283C22×C8C23×C512yes—
284C22×C8C23×C512yes—
285#124—42no—
286C22⋊Q8—22no—
287C22⋊Q8—22no—
288#121—22no—
289#121—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C5×D414yes—
299C2×C4⋊C4C23×C512yes—
300C2×C4⋊C4C23×C511yescharacteristic
301C2×C4⋊C4C23×C511yescharacteristic
302C2×C4⋊C4C23×C512yes—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C42—42no—
310C2×C42C5×D411yescharacteristic
311C2×C42C5×D411yescharacteristic
312C2×C42C23×C511yescharacteristic
313C2×C42C23×C511yescharacteristic
314#107—24no—
315#106—24no—
316C4⋊C8—42no—
317#103—24no—
318#103—42no—
319#102—24no—
320#102—42no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323#95—22no—
324#95—22no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329C22×C5—22no—
330C22×C5—22no—
331C22×C5—22no—
332C22×C5—22no—
333C22×C5—22no—
334C22×C5—22no—
335C22×C5—81no—
336C22×C5—81no—
337C22×C5—81no—
338C22×C5—81no—
339C22×C5—81no—
340C22×C5—81no—
341C22×C5—81no—
342C22×C5#46414yes—
343C22×C5C2×#12111yescharacteristic
344C22×C5#53411yescharacteristic
345C22×C5#53611yescharacteristic
346C22×C5C2×#13511yescharacteristic
347C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
348C22×C5C2×#13711yescharacteristic
349C22×C5#57511yescharacteristic
350C4×C5—22no—
351C4×C5—22no—
352C4×C5—41no—
353C4×C5—41no—
354C4×C5—41no—
355C4×C5—44no—
356C4×C5—81no—
357C4×C5—82no—
358C4×C5—81no—
359C4×C5—81no—
360C4×C5#53712yes—
361C4×C5#53912yes—
362C24—21no—
363C24C2×C5×D411yescharacteristic
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4C2×C5×D411yescharacteristic
395C22×C4C2×C5×D411yescharacteristic
396C22×C4C2×C5×D411yescharacteristic
397C22×C4C2×C5×D411yescharacteristic
398C22×C4C2×C5×D412yes—
399C22×C4C2×C5×D411yescharacteristic
400C22×C4C5×#4112yes—
401C22×C4C24×C511yescharacteristic
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4C2×C5×D414yes—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C42—22no—
428C42—22no—
429C42—41no—
430C42—41no—
431C2×C5—24no—
432C2×C5—161no—
433C2×C5#261811yescharacteristic
434C2×C5C2×#46411yescharacteristic
435C2×C5#303611yescharacteristic
436C2×C5#318811yescharacteristic
437C2×C5#319611yescharacteristic
438C2×C5#323211yescharacteristic
439C2×C5#323911yescharacteristic
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23C5×#12112yes—
452C23C5×#13512yes—
453C23C5×#13712yes—
454C23C22×C5×D411yescharacteristic
455Q8—42no—
456Q8—42no—
457D4—42no—
458D4—42no—
459D4—84no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—84no—
494C2×C4C5×#12112yes—
495C2×C4C5×AGL(1,ℤ/8)12yes—
496C2×C4C22×C5×D411yescharacteristic
497C2×C4C2×C5×#4111yescharacteristic
498C2×C4C5×H(5,𝔽2)12yes—
499C2×C4C5×2−1+412yes—
500C8—42no—
501C8—42no—
502C5#2976111yescharacteristic, direct product factor, Sylow
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22C5×#46414yes—
517C22C2×C5×#12111yescharacteristic
518C22C5×#53411yescharacteristic
519C22C5×#53611yescharacteristic
520C22C2×C5×#13511yescharacteristic
521C22C2×C5×AGL(1,ℤ/8)11yescharacteristic
522C22C2×C5×#13711yescharacteristic
523C22C5×#57511yescharacteristic
524C4—22no—
525C4—22no—
526C4—44no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—82no—
534C4C5×#53712yes—
535C4C5×#53912yes—
536C2—24no—
537C2—161no—
538C2C5×#261811yescharacteristic
539C2C2×C5×#46411yescharacteristic
540C2C5×#303611yescharacteristic
541C2C5×#318811yescharacteristic
542C2C5×#319611yescharacteristic
543C2C5×#323211yescharacteristic
544C2C5×#323911yescharacteristic
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC2×C5×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C5×#484C2semidirect
4. C2×C5×#389C2semidirect
5. C5×#2085C2semidirect
6. C5×#2067C2semidirect
7. C5×#2066C2semidirect
8. C5×#2042C2semidirect
10. C5×#1872C2semidirect
11. C5×#1797C2semidirect
45. #29761C5direct (N × H)
100. C2×#484C2×C5semidirect
103. C2×#389C2×C5semidirect
104. #2085C2×C5semidirect
105. #2067C2×C5semidirect
106. #2066C2×C5semidirect
107. #2042C2×C5semidirect
109. #1872C2×C5semidirect
110. #1797C2×C5semidirect
502. C5#29761direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4842
C2×C5×#4812
C5×#28502
C2×C5×#3892
C5×#20852
C5×#20672
C5×#20662
C5×#20422
C5×#18742
C5×#18722
C5×#17972
#297615