Finite Group Databasebeta

← C5×#29605 #11952325

Subgroups — C5×#29605 #11952325

Order 1280.

1268 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
1C2×C5×#536C211yesmaximal, characteristic
2C2×C5×#485C211yesmaximal, characteristic, complemented
3C2×C5×#451C211yesmaximal, characteristic, complemented
4C5×#2616C211yesmaximal, characteristic
5C5×#2036C212yesmaximal, complemented
6C5×#2035C212yesmaximal, complemented
7C5×#2001C212yesmaximal, complemented
8C5×#2001C212yesmaximal, complemented
9C5×#1825C211yesmaximal, characteristic, complemented
10C5×#1787C211yesmaximal, characteristic, complemented
11C5×#1780C211yesmaximal, characteristic
12C22×C5×#41C2211yescharacteristic
13C23×C5×Q8C2211yescharacteristic
14C22×C5×Q16C2212yes—
15C5×#536—24no—
16C5×#536—24no—
17C2×C5×C4⋊Q8—21no—
18C2×C5×#124—21no—
19C2×C5×#123—22no—
20C2×C5×C22⋊Q8C2211yescharacteristic
21C2×C5×C22⋊Q8C2212yes—
22C2×C5×C22⋊Q8C2212yes—
23C2×C5×C22⋊Q8C2212yes—
24C2×C5×#117C2211yescharacteristic
25C5×#485—24no—
26C5×#451—24no—
27C2×C5×#107C2211yescharacteristic
28C2×C5×#106C2211yescharacteristic
29C2×C5×#103C2212yes—
30C2×C5×#103C2211yescharacteristic
31C2×C5×#103C2211yescharacteristic
32C2×C5×#103C2212yes—
33C2×C5×#103C2212yes—
34C2×C5×C22⋊C8C2211yescharacteristic
35C2×C5×C22⋊C8C2211yescharacteristic
36C2×C5×C22⋊C8C2212yes—
37C2×C5×C22⋊C8C2212yes—
38C5×#395—21no—
39C5×#394—22no—
40C5×#394—21no—
41C5×#380C2212yes—
42C5×#380C2211yescharacteristic
43C5×#340C2212yes—
44C5×#340C2212yes—
45#29605C511yesmaximal, characteristic, direct product factor, Sylow
46C2×C5×#41—22no—
47C2×C5×#41—22no—
48C2×C5×#41—42no—
49C2×C5×#41—42no—
50C22×C5×Q8—21no—
51C22×C5×Q8—22no—
52C22×C5×Q8—42no—
53C22×C5×Q8—42no—
54C22×C5×Q8D414yes—
55C22×C5×Q8D411yescharacteristic
56C22×C5×Q8C2311yescharacteristic
57C22×C5×Q8C2312yes—
58C22×C5×D4D412yes—
59C22×C5×D4D411yescharacteristic
60C23×C4×C5—21no—
61C23×C4×C5C2312yes—
62C23×C4×C5C2311yescharacteristic
63C2×C5×Q16—24no—
64C2×C5×Q16—42no—
65C2×C5×Q16—42no—
66C22×C5×C8C2312yes—
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C5×C4⋊Q8—28no—
70C5×#124—28no—
71C5×#123—216no—
72C5×C22⋊Q8—24no—
73C5×C22⋊Q8—24no—
74C5×C22⋊Q8—24no—
75C5×C22⋊Q8—24no—
76C5×C22⋊Q8—22no—
77C5×C22⋊Q8—22no—
78C5×C22⋊Q8—42no—
79C5×C22⋊Q8—42no—
80C5×#117—22no—
81C5×#117D414yes—
82C2×C5×C4⋊C4—22no—
83C2×C5×C4⋊C4—22no—
84C2×C5×C4⋊C4—22no—
85C2×C5×C4⋊C4—21no—
86C2×C5×C4⋊C4C2312yes—
87C2×C5×C4⋊C4C2311yescharacteristic
88C2×C5×C4⋊C4C2311yescharacteristic
89C2×C5×C4⋊C4C2311yescharacteristic
90C2×C5×C22⋊C4—21no—
91C2×C5×C22⋊C4—22no—
92C2×C5×C22⋊C4—22no—
93C2×C5×C22⋊C4—22no—
94C2×C5×C22⋊C4—21no—
95C2×C5×C22⋊C4—22no—
96C2×C5×C22⋊C4—21no—
97C2×C42×C5—21no—
98C5×#107—22no—
99C5×#106—22no—
100C5×#103—24no—
101C5×#103—22no—
102C5×#103—22no—
103C5×#103—42no—
104C5×#103—42no—
105C5×C22⋊C8—22no—
106C5×C22⋊C8—22no—
107C5×C22⋊C8—42no—
108C5×C22⋊C8—42no—
109C5×#95—22no—
110C5×#95—21no—
111C2×#536C2×C511yescharacteristic
112C2×#485C2×C511yescharacteristic, complemented
113C2×#451C2×C511yescharacteristic, complemented
114#2616C2×C511yescharacteristic
115#2036C2×C512yescomplemented
116#2035C2×C512yescomplemented
117#2001C2×C512yescomplemented
118#2001C2×C512yescomplemented
119#1825C2×C511yescharacteristic, complemented
120#1787C2×C511yescharacteristic, complemented
121#1780C2×C511yescharacteristic
122C24×C5C2×D412yes—
123C24×C5C2×D411yescharacteristic
124C5×#41—48no—
125C5×#41—44no—
126C5×#41—44no—
127C2×C5×Q8—24no—
128C2×C5×Q8—22no—
129C2×C5×Q8—22no—
130C2×C5×Q8—21no—
131C2×C5×Q8—21no—
132C2×C5×Q8—22no—
133C2×C5×Q8—22no—
134C2×C5×Q8—22no—
135C2×C5×Q8—22no—
136C2×C5×Q8—22no—
137C2×C5×Q8—22no—
138C2×C5×Q8—42no—
139C2×C5×Q8—48no—
140C2×C5×Q8—44no—
141C2×C5×Q8—44no—
142C2×C5×Q8—44no—
143C2×C5×Q8C2×D414yes—
144C2×C5×D4—24no—
145C2×C5×D4—22no—
146C2×C5×D4—42no—
147C2×C5×D4—42no—
148C2×C5×D4—42no—
149C22×C4×C5—22no—
150C22×C4×C5—22no—
151C22×C4×C5—24no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—22no—
155C22×C4×C5—22no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
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—44no—
165C22×C4×C5C2×D412yes—
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D412yes—
168C22×C4×C5C2×D412yes—
169C22×C4×C5C2411yescharacteristic
170C5×Q16—48no—
171C2×C5×C8—22no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—22no—
175C2×C5×C8—22no—
176C2×C5×C8—22no—
177C2×C5×C8—22no—
178C5×C4⋊C4—28no—
179C5×C4⋊C4—28no—
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—44no—
187C5×C4⋊C4—42no—
188C5×C4⋊C4C2×D412yes—
189C5×C4⋊C4C2×D412yes—
190C5×C22⋊C4—28no—
191C5×C22⋊C4—28no—
192C5×C22⋊C4—24no—
193C5×C22⋊C4—44no—
194C5×C22⋊C4—42no—
195C5×C22⋊C4—81no—
196C5×C22⋊C4—82no—
197C42×C5—24no—
198C22×#41C22×C511yescharacteristic
199C23×Q8C22×C511yescharacteristic
200C22×Q16C22×C512yes—
201#536—24no—
202#536—24no—
203C2×C4⋊Q8—21no—
204C2×#124—21no—
205C2×#123—22no—
206C2×C22⋊Q8C22×C511yescharacteristic
207C2×C22⋊Q8C22×C512yes—
208C2×C22⋊Q8C22×C512yes—
209C2×C22⋊Q8C22×C512yes—
210C2×#117C22×C511yescharacteristic
211#485—24no—
212#451—24no—
213C2×#107C22×C511yescharacteristic
214C2×#106C22×C511yescharacteristic
215C2×#103C22×C511yescharacteristic
216C2×#103C22×C511yescharacteristic
217C2×#103C22×C512yes—
218C2×#103C22×C512yes—
219C2×#103C22×C512yes—
220C2×C22⋊C8C22×C511yescharacteristic
221C2×C22⋊C8C22×C511yescharacteristic
222C2×C22⋊C8C22×C512yes—
223C2×C22⋊C8C22×C512yes—
224#395—21no—
225#394—22no—
226#394—21no—
227#380C22×C511yescharacteristic
228#380C22×C512yes—
229#340C22×C512yes—
230#340C22×C512yes—
231C23×C5—22no—
232C23×C5—21no—
233C23×C5—22no—
234C23×C5—21no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5—22no—
242C23×C5#12012yes—
243C23×C5C22×D411yescharacteristic
244C5×Q8—24no—
245C5×Q8—24no—
246C5×Q8—48no—
247C5×Q8—44no—
248C5×Q8—42no—
249C5×Q8—42no—
250C5×Q8—42no—
251C5×Q8—41no—
252C5×Q8—41no—
253C5×D4—44no—
254C5×D4—48no—
255C2×C4×C5—22no—
256C2×C4×C5—24no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—24no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—48no—
266C2×C4×C5—42no—
267C2×C4×C5—42no—
268C2×C4×C5—41no—
269C2×C4×C5—42no—
270C2×C4×C5—41no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—41no—
281C2×C4×C5—42no—
282C2×C4×C5—41no—
283C2×C4×C5#12012yes—
284C2×C4×C5C22×D411yescharacteristic
285C2×C4×C5C22×D411yescharacteristic
286C2×C4×C5H(5,𝔽2)12yes—
287C2×C4×C5H(5,𝔽2)12yes—
288C5×C8—42no—
289C5×C8—41no—
290C5×C8—41no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—42no—
294C2×#41—42no—
295C22×Q8—21no—
296C22×Q8—22no—
297C22×Q8—42no—
298C22×Q8—42no—
299C22×Q8C5×D414yes—
300C22×Q8C5×D411yescharacteristic
301C22×Q8C23×C511yescharacteristic
302C22×Q8C23×C512yes—
303C22×D4C5×D412yes—
304C22×D4C5×D411yescharacteristic
305C23×C4—21no—
306C23×C4C23×C512yes—
307C23×C4C23×C511yescharacteristic
308C2×Q16—24no—
309C2×Q16—42no—
310C2×Q16—42no—
311C22×C8C23×C512yes—
312C22×C8C23×C511yescharacteristic
313C22×C8C23×C511yescharacteristic
314C4⋊Q8—28no—
315#124—28no—
316#123—216no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—42no—
324C22⋊Q8—42no—
325#117—22no—
326#117C5×D414yes—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4C23×C512yes—
332C2×C4⋊C4C23×C511yescharacteristic
333C2×C4⋊C4C23×C511yescharacteristic
334C2×C4⋊C4C23×C511yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C42—21no—
343#107—22no—
344#106—22no—
345#103—24no—
346#103—22no—
347#103—22no—
348#103—42no—
349#103—42no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—42no—
353C22⋊C8—42no—
354#95—22no—
355#95—21no—
356C22×C5—22no—
357C22×C5—22no—
358C22×C5—22no—
359C22×C5—42no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5—42no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5C2×#12011yescharacteristic
369C22×C5#53412yes—
370C22×C5#57711yescharacteristic
371C22×C5#57812yes—
372C22×C5#57811yescharacteristic
373C4×C5—22no—
374C4×C5—42no—
375C4×C5—44no—
376C4×C5—42no—
377C4×C5—42no—
378C4×C5—81no—
379C4×C5—82no—
380C4×C5—81no—
381C4×C5#53412yes—
382C4×C5#53511yescharacteristic
383C4×C5#53511yescharacteristic
384C24C2×C5×D412yes—
385C24C2×C5×D411yescharacteristic
386#41—48no—
387#41—44no—
388#41—44no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8C2×C5×D414yes—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—44no—
427C22×C4C2×C5×D412yes—
428C22×C4C2×C5×D411yescharacteristic
429C22×C4C2×C5×D412yes—
430C22×C4C2×C5×D412yes—
431C22×C4C24×C511yescharacteristic
432Q16—48no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4C2×C5×D412yes—
451C4⋊C4C2×C5×D412yes—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—24no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—81no—
458C22⋊C4—82no—
459C42—24no—
460C2×C5—42no—
461C2×C5—82no—
462C2×C5#261311yescharacteristic
463C2×C5#301111yescharacteristic
464C2×C5#301311yescharacteristic
465C2×C5#320112yes—
466C2×C5#320212yes—
467C23—22no—
468C23—21no—
469C23—22no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23C5×#12012yes—
479C23C22×C5×D411yescharacteristic
480Q8—24no—
481Q8—24no—
482Q8—48no—
483Q8—44no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—41no—
488Q8—41no—
489D4—44no—
490D4—48no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4C5×#12012yes—
520C2×C4C22×C5×D411yescharacteristic
521C2×C4C22×C5×D411yescharacteristic
522C2×C4C5×H(5,𝔽2)12yes—
523C2×C4C5×H(5,𝔽2)12yes—
524C8—42no—
525C8—41no—
526C8—41no—
527C5#2960511yescharacteristic, direct product factor, Sylow
528C22—22no—
529C22—22no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22C2×C5×#12011yescharacteristic
541C22C5×#53412yes—
542C22C5×#57711yescharacteristic
543C22C5×#57812yes—
544C22C5×#57811yescharacteristic
545C4—22no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—44no—
550C4—81no—
551C4—81no—
552C4—82no—
553C4C5×#53412yes—
554C4C5×#53511yescharacteristic
555C4C5×#53511yescharacteristic
556C2—42no—
557C2—82no—
558C2C5×#261311yescharacteristic
559C2C5×#301111yescharacteristic
560C2C5×#301311yescharacteristic
561C2C5×#320112yes—
562C2C5×#320212yes—
subgroupG/H
center, socleC23×C5C22×D4
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C5 ◁ C22×C5×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#485C2semidirect
3. C2×C5×#451C2semidirect
5. C5×#2036C2semidirect
6. C5×#2035C2semidirect
7. C5×#2001C2semidirect
8. C5×#2001C2semidirect
9. C5×#1825C2semidirect
10. C5×#1787C2semidirect
45. #29605C5direct (N × H)
112. C2×#485C2×C5semidirect
113. C2×#451C2×C5semidirect
115. #2036C2×C5semidirect
116. #2035C2×C5semidirect
117. #2001C2×C5semidirect
118. #2001C2×C5semidirect
119. #1825C2×C5semidirect
120. #1787C2×C5semidirect
527. C5#29605direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5362
C2×C5×#4852
C2×C5×#4512
C5×#26162
C5×#20362
C5×#20352
C5×#20012
C5×#18252
C5×#17872
C5×#17802
#296055