Finite Group Databasebeta

← C5×#29613 #11952333

Subgroups — C5×#29613 #11952333

Order 1280.

1256 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
2C5×#3095C211yesmaximal, characteristic, complemented
3C5×#3007C211yesmaximal, characteristic
4C5×#2434C211yesmaximal, characteristic, complemented
5C5×#2038C214yesmaximal, complemented
6C5×#2005C214yesmaximal, complemented
7C5×#1826C211yesmaximal, characteristic, complemented
8C5×#1789C211yesmaximal, characteristic
9C5×#1784C211yesmaximal, characteristic, complemented
10C22×C5×#41C2211yescharacteristic
11C23×C5×Q8C2211yescharacteristic
12C2×C5×#137C2212yes—
13C5×#536—22no—
14C5×#536—22no—
15C5×#536—22no—
16C5×#536C2214yes—
17C2×C5×C4⋊Q8—21no—
18C2×C5×C4⋊Q8C2212yescomplemented
19C2×C5×#124—21no—
20C2×C5×#123—22no—
21C2×C5×C22⋊Q8—21no—
22C2×C5×C22⋊Q8—21no—
23C2×C5×C22⋊Q8C2211yescharacteristic
24C2×C5×#117C2211yescharacteristic
25C5×#485—22no—
26C5×#484—21no—
27C5×#483—21no—
28C5×#452—22no—
29C5×#451—21no—
30C5×#450—21no—
31C5×#428C2212yescomplemented
32C2×C5×#104C2214yes—
33C5×#419C2212yescomplemented
34C5×#417C2212yes—
35C2×C5×#101C2214yescomplemented
36C5×#407C2211yescharacteristic
37C5×#407C2211yescharacteristic
38C5×#395—22no—
39C5×#391C2212yescomplemented
40C5×#385—22no—
41C4×C5×C22⋊C4C2211yescharacteristic
42C5×#339C2214yescomplemented
43#29613C511yesmaximal, characteristic, direct product factor, Sylow
44C2×C5×#41—21no—
45C2×C5×#41—21no—
46C2×C5×#41—44no—
47C2×C5×#41D412yes—
48C2×C5×#41C2312yes—
49C22×C5×Q8—21no—
50C22×C5×Q8—21no—
51C22×C5×Q8—21no—
52C22×C5×Q8—21no—
53C22×C5×Q8—44no—
54C22×C5×Q8D411yescharacteristic
55C22×C5×Q8D412yes—
56C22×C5×Q8C2311yescharacteristic
57C22×C5×Q8C2312yes—
58C22×C5×D4—21no—
59C22×C5×D4D411yescharacteristic
60C23×C4×C5—21no—
61C23×C4×C5D412yes—
62C23×C4×C5C2311yescharacteristic
63C5×#137—44no—
64C2×C5×Q16—22no—
65C2×C5×SD16—22no—
66C2×C5×M16C2312yes—
67C2×C5×M16C2312yes—
68C5×C4⋊Q8—24no—
69C5×C4⋊Q8—24no—
70C5×C4⋊Q8—44no—
71C5×#124—24no—
72C5×#124—24no—
73C5×#123—28no—
74C5×#123—28no—
75C5×C22⋊Q8—24no—
76C5×C22⋊Q8—24no—
77C5×C22⋊Q8—24no—
78C5×C22⋊Q8—24no—
79C5×C22⋊Q8—21no—
80C5×C22⋊Q8—21no—
81C5×C22⋊Q8—21no—
82C5×C22⋊Q8—21no—
83C5×#117—21no—
84C5×#117—21no—
85C5×#117D412yes—
86C5×#117C2312yes—
87C2×C5×C4⋊C4—22no—
88C2×C5×C4⋊C4—22no—
89C2×C5×C4⋊C4—24no—
90C2×C5×C4⋊C4—21no—
91C2×C5×C4⋊C4D411yescharacteristic
92C2×C5×C4⋊C4D411yescharacteristic
93C2×C5×C4⋊C4C2311yescharacteristic
94C2×C5×C22⋊C4—22no—
95C2×C5×C22⋊C4—22no—
96C2×C5×C22⋊C4—21no—
97C2×C5×C22⋊C4—21no—
98C2×C5×C22⋊C4—21no—
99C2×C42×C5—21no—
100C2×C42×C5C2312yes—
101C5×#107—22no—
102C5×#106—22no—
103C5×#104—44no—
104C5×#103—22no—
105C5×#103—22no—
106C5×#103—22no—
107C5×#102—22no—
108C5×#101—44no—
109C5×C22⋊C8—21no—
110C5×C22⋊C8—21no—
111C5×C22⋊C8—21no—
112C5×C22⋊C8—21no—
113C5×#95—21no—
114C5×#95—24no—
115C2×#536C2×C511yescharacteristic
116#3095C2×C511yescharacteristic, complemented
117#3007C2×C511yescharacteristic
118#2434C2×C511yescharacteristic, complemented
119#2038C2×C514yescomplemented
120#2005C2×C514yescomplemented
121#1826C2×C511yescharacteristic, complemented
122#1789C2×C511yescharacteristic
123#1784C2×C511yescharacteristic, complemented
124C24×C5—21no—
125C24×C5C2×D411yescharacteristic
126C5×#41—24no—
127C5×#41—24no—
128C5×#41—42no—
129C5×#41—42no—
130C5×#41—44no—
131C5×#41—44no—
132C2×C5×Q8—22no—
133C2×C5×Q8—22no—
134C2×C5×Q8—21no—
135C2×C5×Q8—22no—
136C2×C5×Q8—22no—
137C2×C5×Q8—24no—
138C2×C5×Q8—24no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8—24no—
142C2×C5×Q8—44no—
143C2×C5×Q8—42no—
144C2×C5×Q8—44no—
145C2×C5×Q8—44no—
146C2×C5×Q8—44no—
147C2×C5×Q8—44no—
148C2×C5×Q8C2×D412yes—
149C2×C5×D4—22no—
150C2×C5×D4—22no—
151C2×C5×D4—21no—
152C2×C5×D4—42no—
153C2×C5×D4—44no—
154C2×C5×D4C2×D412yes—
155C22×C4×C5—24no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—22no—
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5—22no—
163C22×C4×C5—22no—
164C22×C4×C5—21no—
165C22×C4×C5—21no—
166C22×C4×C5—44no—
167C22×C4×C5—44no—
168C22×C4×C5C2×D414yes—
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2×D411yescharacteristic
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2×D411yescharacteristic
174C22×C4×C5C2411yescharacteristic
175C5×Q16—44no—
176C5×SD16—44no—
177C5×M16—24no—
178C5×M16—24no—
179C2×C5×C8—22no—
180C2×C5×C8—22no—
181C5×C4⋊C4—28no—
182C5×C4⋊C4—28no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—48no—
188C5×C4⋊C4—41no—
189C5×C4⋊C4—41no—
190C5×C4⋊C4C2×D412yes—
191C5×C4⋊C4C2×D412yes—
192C5×C22⋊C4—24no—
193C5×C22⋊C4—28no—
194C5×C22⋊C4—28no—
195C5×C22⋊C4—22no—
196C5×C22⋊C4—22no—
197C5×C22⋊C4—41no—
198C5×C22⋊C4—41no—
199C42×C5—22no—
200C42×C5—22no—
201C42×C5—24no—
202C22×#41C22×C511yescharacteristic
203C23×Q8C22×C511yescharacteristic
204C2×#137C22×C512yes—
205#536—22no—
206#536—22no—
207#536—22no—
208#536C22×C514yes—
209C2×C4⋊Q8—21no—
210C2×C4⋊Q8C22×C512yescomplemented
211C2×#124—21no—
212C2×#123—22no—
213C2×C22⋊Q8—21no—
214C2×C22⋊Q8—21no—
215C2×C22⋊Q8C22×C511yescharacteristic
216C2×#117C22×C511yescharacteristic
217#485—22no—
218#484—21no—
219#483—21no—
220#452—22no—
221#451—21no—
222#450—21no—
223#428C22×C512yescomplemented
224C2×#104C22×C514yes—
225#419C22×C512yescomplemented
226#417C22×C512yes—
227C2×#101C22×C514yescomplemented
228#407C22×C511yescharacteristic
229#407C22×C511yescharacteristic
230#395—22no—
231#391C22×C512yescomplemented
232#385—22no—
233C4×C22⋊C4C22×C511yescharacteristic
234#339C22×C514yescomplemented
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—41no—
239C23×C5—42no—
240C23×C5—42no—
241C23×C5—42no—
242C23×C5#12011yescharacteristic
243C23×C5#12011yescharacteristic
244C23×C5C22×D411yescharacteristic
245C5×Q8—22no—
246C5×Q8—22no—
247C5×Q8—44no—
248C5×Q8—44no—
249C5×Q8—42no—
250C5×Q8—44no—
251C5×Q8—44no—
252C5×Q8—42no—
253C5×Q8—42no—
254C5×D4—22no—
255C5×D4—22no—
256C5×D4—42no—
257C5×D4—44no—
258C5×D4—44no—
259C2×C4×C5—22no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—24no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—24no—
268C2×C4×C5—22no—
269C2×C4×C5—24no—
270C2×C4×C5—24no—
271C2×C4×C5—44no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—44no—
276C2×C4×C5—44no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—44no—
280C2×C4×C5—42no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—42no—
284C2×C4×C5#12011yescharacteristic
285C2×C4×C5#12011yescharacteristic
286C2×C4×C5C22×D411yescharacteristic
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5H(5,𝔽2)14yes—
289C5×C8—42no—
290C5×C8—42no—
291C2×#41—21no—
292C2×#41—21no—
293C2×#41—44no—
294C2×#41C5×D412yes—
295C2×#41C23×C512yes—
296C22×Q8—21no—
297C22×Q8—21no—
298C22×Q8—21no—
299C22×Q8—21no—
300C22×Q8—44no—
301C22×Q8C5×D411yescharacteristic
302C22×Q8C5×D412yes—
303C22×Q8C23×C511yescharacteristic
304C22×Q8C23×C512yes—
305C22×D4—21no—
306C22×D4C5×D411yescharacteristic
307C23×C4—21no—
308C23×C4C5×D412yes—
309C23×C4C23×C511yescharacteristic
310#137—44no—
311C2×Q16—22no—
312C2×SD16—22no—
313C2×M16C23×C512yes—
314C2×M16C23×C512yes—
315C4⋊Q8—24no—
316C4⋊Q8—24no—
317C4⋊Q8—44no—
318#124—24no—
319#124—24no—
320#123—28no—
321#123—28no—
322C22⋊Q8—24no—
323C22⋊Q8—24no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330#117—21no—
331#117—21no—
332#117C5×D412yes—
333#117C23×C512yes—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4C5×D411yescharacteristic
339C2×C4⋊C4C5×D411yescharacteristic
340C2×C4⋊C4C23×C511yescharacteristic
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C42—21no—
347C2×C42C23×C512yes—
348#107—22no—
349#106—22no—
350#104—44no—
351#103—22no—
352#103—22no—
353#103—22no—
354#102—22no—
355#101—44no—
356C22⋊C8—21no—
357C22⋊C8—21no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360#95—21no—
361#95—24no—
362C22×C5—21no—
363C22×C5—21no—
364C22×C5—21no—
365C22×C5—21no—
366C22×C5—21no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5—44no—
370C22×C5—42no—
371C22×C5—42no—
372C22×C5—42no—
373C22×C5—84no—
374C22×C5#45612yes—
375C22×C5C2×#12011yescharacteristic
376C22×C5#53412yes—
377C4×C5—21no—
378C4×C5—21no—
379C4×C5—44no—
380C4×C5—44no—
381C4×C5—42no—
382C4×C5—42no—
383C4×C5—42no—
384C4×C5—81no—
385C4×C5—81no—
386C4×C5#53412yes—
387C4×C5#53512yes—
388C24—21no—
389C24C2×C5×D411yescharacteristic
390#41—24no—
391#41—24no—
392#41—42no—
393#41—42no—
394#41—44no—
395#41—44no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8C2×C5×D412yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4C2×C5×D412yes—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4C2×C5×D414yes—
433C22×C4C2×C5×D411yescharacteristic
434C22×C4C2×C5×D411yescharacteristic
435C22×C4C2×C5×D411yescharacteristic
436C22×C4C2×C5×D411yescharacteristic
437C22×C4C2×C5×D411yescharacteristic
438C22×C4C24×C511yescharacteristic
439Q16—44no—
440SD16—44no—
441M16—24no—
442M16—24no—
443C2×C8—22no—
444C2×C8—22no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—48no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4C2×C5×D412yes—
455C4⋊C4C2×C5×D412yes—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—22no—
464C42—22no—
465C42—24no—
466C2×C5—22no—
467C2×C5—22no—
468C2×C5—41no—
469C2×C5—82no—
470C2×C5#261311yescharacteristic
471C2×C5C2×#45611yescharacteristic
472C2×C5#302011yescharacteristic
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23C5×#12011yescharacteristic
481C23C5×#12011yescharacteristic
482C23C22×C5×D411yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—44no—
486Q8—44no—
487Q8—42no—
488Q8—44no—
489Q8—44no—
490Q8—42no—
491Q8—42no—
492D4—22no—
493D4—22no—
494D4—42no—
495D4—44no—
496D4—44no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4C5×#12011yescharacteristic
523C2×C4C5×#12011yescharacteristic
524C2×C4C22×C5×D411yescharacteristic
525C2×C4C22×C5×D411yescharacteristic
526C2×C4C5×H(5,𝔽2)14yes—
527C8—42no—
528C8—42no—
529C5#2961311yescharacteristic, direct product factor, Sylow
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—44no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—84no—
542C22C5×#45612yes—
543C22C2×C5×#12011yescharacteristic
544C22C5×#53412yes—
545C4—21no—
546C4—21no—
547C4—44no—
548C4—42no—
549C4—42no—
550C4—44no—
551C4—42no—
552C4—81no—
553C4—81no—
554C4C5×#53412yes—
555C4C5×#53512yes—
556C2—22no—
557C2—22no—
558C2—41no—
559C2—82no—
560C2C5×#261311yescharacteristic
561C2C2×C5×#45611yescharacteristic
562C2C5×#302011yescharacteristic
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC22×C4C24×C5
prime radicalC22×C5×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#3095C2semidirect
4. C5×#2434C2semidirect
5. C5×#2038C2semidirect
6. C5×#2005C2semidirect
7. C5×#1826C2semidirect
9. C5×#1784C2semidirect
18. C2×C5×C4⋊Q8C22semidirect
31. C5×#428C22semidirect
33. C5×#419C22semidirect
35. C2×C5×#101C22semidirect
39. C5×#391C22semidirect
42. C5×#339C22semidirect
43. #29613C5direct (N × H)
116. #3095C2×C5semidirect
118. #2434C2×C5semidirect
119. #2038C2×C5semidirect
120. #2005C2×C5semidirect
121. #1826C2×C5semidirect
123. #1784C2×C5semidirect
210. C2×C4⋊Q8C22×C5semidirect
223. #428C22×C5semidirect
225. #419C22×C5semidirect
227. C2×#101C22×C5semidirect
231. #391C22×C5semidirect
234. #339C22×C5semidirect
529. C5#29613direct (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
C5×#30952
C5×#30072
C5×#24342
C5×#20382
C5×#20052
C5×#18262
C5×#17892
C5×#17842
#296135