Finite Group Databasebeta

← C5×#70356 #14305984

Subgroups — C5×#70356 #14305984

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5832C211yesmaximal, characteristic
2C2×C3×C5×#104C211yesmaximal, characteristic, complemented
3C2×C5×#837C211yesmaximal, characteristic, complemented
4C2×C5×#825C211yesmaximal, characteristic, complemented
5C5×#5118C211yesmaximal, characteristic, complemented
6C42×C5×Dic3C211yesmaximal, characteristic, complemented
7C2×C5×#805C211yesmaximal, characteristic, complemented
8C5×#1962—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C2×C3×C42×C5C2211yescharacteristic
13C5×#947C411yescharacteristic, complemented
14C5×#944C411yescharacteristic, complemented
15C5×#940C411yescharacteristic, complemented
16C5×#939—22no—
17C5×#936C411yescharacteristic, complemented
18C4×C5×#67—22no—
19C2×C4×C5×Dic3—22no—
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C4×C5×Dic3C2211yescharacteristic
22C2×C5×#208C2211yescharacteristic
23C3×C5×#104—22no—
24C5×#837—22no—
25C5×#825—22no—
26C5×Dic3⋊C8—22no—
27C5×#805—22no—
28#70356C511yesmaximal, characteristic, direct product factor, Hall
29C5×#532—31no—
30C5×#423—31no—
31C2×C5×#104—31no—
32C2×C5×#104—31no—
33C2×C5×#104—31no—
34C2×C5×#104D311yescharacteristic, complemented
35C43×C5—31no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C2×C3×C5×Q8C2×C411yescharacteristic
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4C2×C411yescharacteristic
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5—21no—
45C22×C3×C4×C5C2311yescharacteristic
46C2×C5×#67—21no—
47C2×C5×#67—22no—
48C22×C5×Dic3—21no—
49C22×C5×Dic3D412yes—
50C5×#235—21no—
51C5×#235—21no—
52C5×#235—21no—
53C5×#235—21no—
54C2×C5×D12C2×C411yescharacteristic
55C2×C4×C5×D3—21no—
56C2×C5×Dic6C2×C411yescharacteristic
57C3×C5×M16—21no—
58C3×C5×M16—21no—
59C2×C3×C5×C8—21no—
60C3×C42×C5—21no—
61C3×C42×C5D412yescomplemented
62C5×#217—22no—
63C5×(C2×D3)⋊C4—22no—
64C5×Dic3⋊C4—22no—
65C4×C5×Dic3—28no—
66C4×C5×Dic3—21no—
67C4×C5×Dic3C2×C411yescharacteristic
68C4×C5×Dic3C2×C411yescharacteristic
69C4×C5×Dic3D412yes—
70C4×C5×Dic3D412yescomplemented
71C5×#208—21no—
72C5×#208—21no—
73C2×C5×C3⋊C8—21no—
74#5832C2×C511yescharacteristic
75C2×C3×#104C2×C511yescharacteristic, complemented
76C2×#837C2×C511yescharacteristic, complemented
77C2×#825C2×C511yescharacteristic, complemented
78#5118C2×C511yescharacteristic, complemented
79C42×Dic3C2×C511yescharacteristic, complemented
80C2×#805C2×C511yescharacteristic, complemented
81C2×C5×#41—31no—
82C2×C5×#41C2×D311yescharacteristic
83C2×C5×M16—31no—
84C2×C5×M16C2×D311yescharacteristic
85C5×C4⋊Q8—31no—
86C5×#127—31no—
87C5×#124—31no—
88C5×#124—31no—
89C5×#121—62no—
90C4×C5×D4—62no—
91C2×C42×C5—31no—
92C2×C42×C5—31no—
93C2×C42×C5—62no—
94C2×C42×C5C2×D311yescharacteristic
95C5×C4⋊C8—62no—
96C5×#104—22no—
97C5×#104—62no—
98C5×#104—62no—
99C5×#104—62no—
100#1962—31noSylow
101C23×C3×C5—21no—
102C23×C3×C5C2×D411yescharacteristic
103C22×C5×D3—21no—
104C3×C5×Q8—21no—
105C3×C5×Q8—21no—
106C3×C5×D4—21no—
107C3×C5×D4—21no—
108C3×C5×D4—41no—
109C3×C5×D4—41no—
110C2×C3×C4×C5—24no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—41no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5C22×C411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C2×C3×C4×C5#4111yescharacteristic
121C5×#67—41no—
122C5×#67—41no—
123C5×#67—44no—
124C2×C5×Dic3—24no—
125C2×C5×Dic3—22no—
126C2×C5×Dic3—24no—
127C2×C5×Dic3—21no—
128C2×C5×Dic3C22⋊C412yes—
129C2×C5×Dic3C22⋊C412yes—
130C5×D12—21no—
131C5×D12—21no—
132C4×C5×D3—41no—
133C4×C5×D3—41no—
134C5×Dic6—21no—
135C5×Dic6—21no—
136C3×C5×C8—41no—
137C5×C3⋊C8—41no—
138C2×C3×#41C22×C511yescharacteristic
139C2×#235C22×C511yescharacteristic
140C2×C3×M16C22×C511yescharacteristic
141C2×C3×C42C22×C511yescharacteristic
142#947C4×C511yescharacteristic, complemented
143#944C4×C511yescharacteristic, complemented
144#940C4×C511yescharacteristic, complemented
145#939—22no—
146#936C4×C511yescharacteristic, complemented
147C4×#67—22no—
148C2×C4×Dic3—22no—
149C2×C4×Dic3C22×C511yescharacteristic
150C2×C4×Dic3C22×C511yescharacteristic
151C2×#208C22×C511yescharacteristic
152C3×#104—22no—
153#837—22no—
154#825—22no—
155Dic3⋊C8—22no—
156#805—22no—
157C5×#41—21no—
158C5×#41—21no—
159C5×#41—21no—
160C5×#41—21no—
161C5×#41—61no—
162C5×#41—61no—
163C5×#41—61no—
164C5×#41—61no—
165C2×C5×Q8—31no—
166C2×C5×Q8C4×D311yescharacteristic
167C2×C5×D4—21no—
168C2×C5×D4—31no—
169C2×C5×D4—62no—
170C2×C5×D4—61no—
171C2×C5×D4C4×D311yescharacteristic
172C22×C4×C5—21no—
173C22×C4×C5—21no—
174C22×C4×C5—32no—
175C22×C4×C5—61no—
176C22×C4×C5—61no—
177C22×C4×C5C22×D311yescharacteristic
178C5×M16—21no—
179C5×M16—21no—
180C5×M16—61no—
181C5×M16—61no—
182C2×C5×C8—21no—
183C2×C5×C8—61no—
184C5×C4⋊C4—62no—
185C5×C22⋊C4—62no—
186C5×C22⋊C4—62no—
187C42×C5—21no—
188C42×C5—31no—
189C42×C5—31no—
190C42×C5—32no—
191C42×C5—32no—
192C42×C5—68no—
193C42×C5—61no—
194C42×C5#6712yescomplemented
195#532—31no—
196#423—31no—
197C2×#104—31no—
198C2×#104—31no—
199C2×#104—31no—
200C2×#104C5×D311yescharacteristic, complemented
201C43—31no—
202C22×C3×C5—22no—
203C22×C3×C5—41no—
204C22×C3×C5—42no—
205C22×C3×C5C2×C22⋊C411yescharacteristic
206C22×C3×C5#12111yescharacteristic
207C22×C3×C5#12111yescharacteristic
208C2×C5×D3—41no—
209C2×C5×D3—42no—
210C3×C4×C5—24no—
211C3×C4×C5—41no—
212C3×C4×C5C4×D411yescharacteristic
213C3×C4×C5C4×D411yescharacteristic
214C3×C4×C5#12411yescharacteristic
215C3×C4×C5#12711yescharacteristic
216C5×Dic3—24no—
217C5×Dic3—22no—
218C5×Dic3—41no—
219C5×Dic3#10414yescomplemented
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C2×C3×Q8C2×C4×C511yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4C2×C4×C511yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4C23×C511yescharacteristic
230C2×#67—21no—
231C2×#67—22no—
232C22×Dic3—21no—
233C22×Dic3C5×D412yes—
234#235—21no—
235#235—21no—
236#235—21no—
237#235—21no—
238C2×D12C2×C4×C511yescharacteristic
239C2×C4×D3—21no—
240C2×Dic6C2×C4×C511yescharacteristic
241C3×M16—21no—
242C3×M16—21no—
243C2×C3×C8—21no—
244C3×C42—21no—
245C3×C42C5×D412yescomplemented
246#217—22no—
247(C2×D3)⋊C4—22no—
248Dic3⋊C4—22no—
249C4×Dic3—28no—
250C4×Dic3—21no—
251C4×Dic3C2×C4×C511yescharacteristic
252C4×Dic3C2×C4×C511yescharacteristic
253C4×Dic3C5×D412yes—
254C4×Dic3C5×D412yescomplemented
255#208—21no—
256#208—21no—
257C2×C3⋊C8—21no—
258C23×C5—21no—
259C23×C5—61no—
260C23×C5D3×D411yescharacteristic
261C5×Q8—21no—
262C5×Q8—21no—
263C5×Q8—61no—
264C5×Q8—61no—
265C5×D4—21no—
266C5×D4—21no—
267C5×D4—41no—
268C5×D4—41no—
269C5×D4—61no—
270C5×D4—61no—
271C5×D4—124no—
272C5×D4—121no—
273C5×D4—121no—
274C2×C4×C5—24no—
275C2×C4×C5—22no—
276C2×C4×C5—21no—
277C2×C4×C5—32no—
278C2×C4×C5—32no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—64no—
282C2×C4×C5—64no—
283C2×C4×C5—62no—
284C2×C4×C5—61no—
285C2×C4×C5—121no—
286C2×C4×C5—121no—
287C2×C4×C5C2×C4×D311yescharacteristic
288C2×C4×C5#23511yescharacteristic
289C2×C4×C5D3×D411yescharacteristic
290C2×C4×C5D3×D411yescharacteristic
291C2×C4×C5#23711yescharacteristic
292C2×C4×C5C2×#6711yescharacteristic
293C5×C8—41no—
294C5×C8—121no—
295C2×#41—31no—
296C2×#41C2×C5×D311yescharacteristic
297C2×M16—31no—
298C2×M16C2×C5×D311yescharacteristic
299C4⋊Q8—31no—
300#127—31no—
301#124—31no—
302#124—31no—
303#121—62no—
304C4×D4—62no—
305C2×C42—31no—
306C2×C42—31no—
307C2×C42—62no—
308C2×C42C2×C5×D311yescharacteristic
309C4⋊C8—62no—
310#104—22no—
311#104—62no—
312#104—62no—
313#104—62no—
314C2×C3×C5—21no—
315C2×C3×C5—21no—
316C2×C3×C5—81no—
317C2×C3×C5#39011yescharacteristic
318C2×C3×C5C2×#10412yes—
319C5×D3—81no—
320C23×C3—21no—
321C23×C3C2×C5×D411yescharacteristic
322C22×D3—21no—
323C3×Q8—21no—
324C3×Q8—21no—
325C3×D4—21no—
326C3×D4—21no—
327C3×D4—41no—
328C3×D4—41no—
329C2×C3×C4—24no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4C22×C4×C511yescharacteristic
335C2×C3×C4C2×C5×D411yescharacteristic
336C2×C3×C4C2×C5×D411yescharacteristic
337C2×C3×C4C2×C5×D411yescharacteristic
338C2×C3×C4C5×#4111yescharacteristic
339C2×C3×C4C5×#4111yescharacteristic
340#67—41no—
341#67—41no—
342#67—44no—
343C2×Dic3—24no—
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—21no—
347C2×Dic3C5×C22⋊C412yes—
348C2×Dic3C5×C22⋊C412yes—
349D12—21no—
350D12—21no—
351C4×D3—41no—
352C4×D3—41no—
353Dic6—21no—
354Dic6—21no—
355C3×C8—41no—
356C3⋊C8—41no—
357C22×C5—22no—
358C22×C5—41no—
359C22×C5—42no—
360C22×C5—121no—
361C22×C5—122no—
362C22×C5D3×C22⋊C411yescharacteristic
363C22×C5#88411yescharacteristic
364C22×C5#93911yescharacteristic
365C4×C5—24no—
366C4×C5—34no—
367C4×C5—41no—
368C4×C5—64no—
369C4×C5—62no—
370C4×C5—121no—
371C4×C5#67⋊C411yescharacteristic
372C4×C5#88511yescharacteristic
373C4×C5C4×#6711yescharacteristic
374C4×C5#94011yescharacteristic
375#41—21no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—61no—
380#41—61no—
381#41—61no—
382#41—61no—
383C2×Q8—31no—
384C2×Q8C4×C5×D311yescharacteristic
385C2×D4—21no—
386C2×D4—31no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4C4×C5×D311yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—32no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4C22×C5×D311yescharacteristic
396M16—21no—
397M16—21no—
398M16—61no—
399M16—61no—
400C2×C8—21no—
401C2×C8—61no—
402C4⋊C4—62no—
403C22⋊C4—62no—
404C22⋊C4—62no—
405C42—21no—
406C42—31no—
407C42—31no—
408C42—32no—
409C42—32no—
410C42—68no—
411C42—61no—
412C42C5×#6712yescomplemented
413C3×C5#196211yescharacteristic, complemented, Hall
414C22×C3—22no—
415C22×C3—41no—
416C22×C3—42no—
417C22×C3C2×C5×C22⋊C411yescharacteristic
418C22×C3C5×#12111yescharacteristic
419C22×C3C5×#12111yescharacteristic
420C2×D3—41no—
421C2×D3—42no—
422C3×C4—24no—
423C3×C4—41no—
424C3×C4C4×C5×D411yescharacteristic
425C3×C4C4×C5×D411yescharacteristic
426C3×C4C5×#12411yescharacteristic
427C3×C4C5×#12711yescharacteristic
428Dic3—24no—
429Dic3—22no—
430Dic3—41no—
431Dic3C5×#10414yescomplemented
432C2×C5—21no—
433C2×C5—21no—
434C2×C5—81no—
435C2×C5—241no—
436C2×C5D3×#10412yes—
437C2×C5#495711yescharacteristic
438C23—21no—
439C23—61no—
440C23C5×D3×D411yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—61no—
444Q8—61no—
445D4—21no—
446D4—21no—
447D4—41no—
448D4—41no—
449D4—61no—
450D4—61no—
451D4—124no—
452D4—121no—
453D4—121no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—62no—
464C2×C4—61no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4C2×C4×C5×D311yescharacteristic
468C2×C4C5×#23511yescharacteristic
469C2×C4C5×D3×D411yescharacteristic
470C2×C4C5×D3×D411yescharacteristic
471C2×C4C5×#23711yescharacteristic
472C2×C4C2×C5×#6711yescharacteristic
473C8—41no—
474C8—121no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—81no—
478C2×C3C5×#39011yescharacteristic
479C2×C3C2×C5×#10412yes—
480D3—81no—
481C5#7035611yescharacteristic, direct product factor, Sylow
482C22—22no—
483C22—41no—
484C22—42no—
485C22—121no—
486C22—122no—
487C22C5×D3×C22⋊C411yescharacteristic
488C22C5×#88411yescharacteristic
489C22C5×#93911yescharacteristic
490C4—24no—
491C4—34no—
492C4—41no—
493C4—64no—
494C4—62no—
495C4—121no—
496C4C5×#67⋊C411yescharacteristic
497C4C5×#88511yescharacteristic
498C4C4×C5×#6711yescharacteristic
499C4C5×#94011yescharacteristic
500C3C5×#196211yescharacteristic, complemented, Sylow
501C2—21no—
502C2—21no—
503C2—81no—
504C2—241no—
505C2C5×D3×#10412yes—
506C2C5×#495711yescharacteristic
subgroupG/H
centerC2×C4×C5D3×D4
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC2×C3×C5×#104C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC5×#5832C2
cosocleC22×C3×C4C23×C5
hypercenterC2×C5×#104D3
hypocenterC3C5×#1962
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#104

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#104C2semidirect
3. C2×C5×#837C2semidirect
4. C2×C5×#825C2semidirect
5. C5×#5118C2semidirect
6. C42×C5×Dic3C2semidirect
7. C2×C5×#805C2semidirect
13. C5×#947C4semidirect
14. C5×#944C4semidirect
15. C5×#940C4semidirect
17. C5×#936C4semidirect
28. #70356C5direct (N × H)
34. C2×C5×#104D3semidirect
61. C3×C42×C5D4semidirect
70. C4×C5×Dic3D4semidirect
75. C2×C3×#104C2×C5semidirect
76. C2×#837C2×C5semidirect
77. C2×#825C2×C5semidirect
78. #5118C2×C5semidirect
79. C42×Dic3C2×C5semidirect
80. C2×#805C2×C5semidirect
142. #947C4×C5semidirect
143. #944C4×C5semidirect
144. #940C4×C5semidirect
146. #936C4×C5semidirect
194. C42×C5#67semidirect
200. C2×#104C5×D3semidirect
219. C5×Dic3#104semidirect
245. C3×C42C5×D4semidirect
254. C4×Dic3C5×D4semidirect
412. C42C5×#67semidirect
413. C3×C5#1962semidirect
431. Dic3C5×#104semidirect
481. C5#70356direct (N × H)
500. C3C5×#1962semidirect

Maximal subgroups

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

subgroupindex
C5×#58322
C2×C3×C5×#1042
C2×C5×#8372
C2×C5×#8252
C5×#51182
C42×C5×Dic32
C2×C5×#8052
C5×#19623
#703565