Finite Group Databasebeta

← C5×#70055 #14302552

Subgroups — C5×#70055 #14302552

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5822C211yesmaximal, characteristic
2C5×#5522C211yesmaximal, characteristic, complemented
3C3×C5×#410C211yesmaximal, characteristic, complemented
4C2×C5×#837C211yesmaximal, characteristic, complemented
5C5×#5005C211yesmaximal, characteristic, complemented
6C5×#4533C211yesmaximal, characteristic, complemented
7C5×#4532C211yesmaximal, characteristic, complemented
8C5×#1906—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C5×#951—21no—
14C5×#950—22no—
15C5×#949—21no—
16C5×#942C412yescomplemented
17C5×#932C412yescomplemented
18C2×C4×C5×Dic3C2211yescharacteristic
19C2×C5×#208C2211yescharacteristic
20C2×C5×#208C2211yescharacteristic
21C5×#893—21no—
22C5×#887—21no—
23C5×#67⋊C4—22no—
24C5×#872—21no—
25C42×C5×D3—21no—
26C3×C5×#99—22no—
27C5×#837—21no—
28C5×#837—21no—
29C5×#805—22no—
30#70055C511yesmaximal, characteristic, direct product factor, Hall
31C5×#575—31no—
32C4×C5×#41—31no—
33C5×#421—31no—
34C2×C5×#104—31no—
35C5×#410D311yescharacteristic, complemented
36C5×#344—31no—
37C5×#339—31no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C2×C3×C5×Q8C2×C411yescharacteristic, complemented
43C2×C3×C5×D4—21no—
44C2×C3×C5×D4C2×C411yescharacteristic, complemented
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5C2311yescharacteristic
47C2×C5×#67—21no—
48C22×C5×Dic3—21no—
49C5×#235—21no—
50C5×#235—21no—
51C5×#235C2×C412yes—
52C5×#235D412yes—
53C2×C5×D12D411yescharacteristic
54C2×C4×C5×D3—21no—
55C2×C4×C5×D3—22no—
56C2×C5×Dic6D411yescharacteristic
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4D412yes—
61C3×C42×C5—21no—
62C3×C42×C5—21no—
63C5×#216—22no—
64C5×#215—22no—
65C5×#214—22no—
66C5×#213—22no—
67C5×(C2×D3)⋊C4—22no—
68C5×Dic3⋊C4—22no—
69C4×C5×Dic3—22no—
70C4×C5×Dic3—21no—
71C4×C5×Dic3D411yescharacteristic
72C4×C5×Dic3D411yescharacteristic
73C5×#208—21no—
74C5×#208—21no—
75C5×#208—21no—
76C5×#208C2×C412yescomplemented
77C2×C5×C3⋊C8—21no—
78C2×C5×C3⋊C8—21no—
79#5822C2×C511yescharacteristic
80#5522C2×C511yescharacteristic, complemented
81C3×#410C2×C511yescharacteristic, complemented
82C2×#837C2×C511yescharacteristic, complemented
83#5005C2×C511yescharacteristic, complemented
84#4533C2×C511yescharacteristic, complemented
85#4532C2×C511yescharacteristic, complemented
86C2×C5×#41—31no—
87C2×C5×#41C2×D311yescharacteristic
88C5×#137—32no—
89C5×#137—61no—
90C5×AGL(1,ℤ/8)—32no—
91C5×AGL(1,ℤ/8)—61no—
92C5×#135—62no—
93C2×C5×M16—31no—
94C2×C5×M16—31no—
95C4×C5×Q8—61no—
96C4×C5×D4—61no—
97C4×C5×D4—62no—
98C5×#117—61no—
99C5×#117C2×D311yescharacteristic
100C5×#117C2×D311yescharacteristic
101C2×C42×C5—31no—
102C2×C42×C5—61no—
103C5×#104—62no—
104C5×#104—61no—
105C5×#104—61no—
106C5×#99—22no—
107#1906—31noSylow
108C23×C3×C5—21no—
109C23×C3×C5C2×D411yescharacteristic
110C22×C5×D3—21no—
111C3×C5×Q8—21no—
112C3×C5×Q8—21no—
113C3×C5×D4—21no—
114C3×C5×D4—21no—
115C3×C5×D4—41no—
116C3×C5×D4—41no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—41no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5C22×C411yescharacteristic
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5#4111yescharacteristic
127C2×C3×C4×C5#4111yescharacteristic
128C5×#67—22no—
129C5×#67—22no—
130C2×C5×Dic3—21no—
131C2×C5×Dic3—21no—
132C2×C5×Dic3—21no—
133C2×C5×Dic3—42no—
134C5×D12—21no—
135C5×D12C22⋊C412yes—
136C4×C5×D3—22no—
137C4×C5×D3—22no—
138C4×C5×D3—44no—
139C5×Dic6—21no—
140C5×Dic6C22⋊C412yes—
141C5×C3⋊C8—22no—
142C5×C3⋊C8—41no—
143C2×C3×#41C22×C511yescharacteristic
144C2×#235C22×C511yescharacteristic
145C3×#117C22×C511yescharacteristic
146C3×#117C22×C511yescharacteristic
147#951—21no—
148#950—22no—
149#949—21no—
150#942C4×C512yescomplemented
151#932C4×C512yescomplemented
152C2×C4×Dic3C22×C511yescharacteristic
153C2×#208C22×C511yescharacteristic
154C2×#208C22×C511yescharacteristic
155#893—21no—
156#887—21no—
157#67⋊C4—22no—
158#872—21no—
159C42×D3—21no—
160C3×#99—22no—
161#837—21no—
162#837—21no—
163#805—22no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41—21no—
167C5×#41—21no—
168C5×#41—32no—
169C5×#41—32no—
170C5×#41—61no—
171C5×#41—61no—
172C2×C5×Q8—31no—
173C2×C5×Q8C4×D311yescharacteristic, complemented
174C2×C5×D4—21no—
175C2×C5×D4—31no—
176C2×C5×D4—61no—
177C2×C5×D4C4×D311yescharacteristic, complemented
178C22×C4×C5—21no—
179C22×C4×C5—62no—
180C22×C4×C5—61no—
181C22×C4×C5—61no—
182C22×C4×C5C22×D311yescharacteristic
183C5×Q16—62no—
184C5×SD16—62no—
185C5×SD16—62no—
186C5×D8—62no—
187C5×M16—32no—
188C5×M16—61no—
189C5×M16—61no—
190C5×M16—61no—
191C2×C5×C8—61no—
192C2×C5×C8—61no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4—62no—
196C5×C22⋊C4—21no—
197C5×C22⋊C4—62no—
198C5×C22⋊C4#6712yes—
199C42×C5—21no—
200C42×C5—21no—
201C42×C5—31no—
202C42×C5—31no—
203C42×C5—62no—
204C42×C5—61no—
205#575—31no—
206C4×#41—31no—
207#421—31no—
208C2×#104—31no—
209#410C5×D311yescharacteristic, complemented
210#344—31no—
211#339—31no—
212C22×C3×C5—41no—
213C22×C3×C5—41no—
214C22×C3×C5—81no—
215C22×C3×C5C4×D411yescharacteristic
216C22×C3×C5#12011yescharacteristic
217C22×C3×C5#12111yescharacteristic
218C2×C5×D3—22no—
219C2×C5×D3—42no—
220C3×C4×C5—42no—
221C3×C4×C5—41no—
222C3×C4×C5—81no—
223C3×C4×C5C2×C22⋊C411yescharacteristic
224C3×C4×C5C4×D411yescharacteristic
225C3×C4×C5#12111yescharacteristic
226C3×C4×C5#12311yescharacteristic
227C5×Dic3—22no—
228C5×Dic3—22no—
229C5×Dic3—41no—
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C2×C3×Q8C2×C4×C511yescharacteristic, complemented
235C2×C3×D4—21no—
236C2×C3×D4C2×C4×C511yescharacteristic, complemented
237C22×C3×C4—21no—
238C22×C3×C4C23×C511yescharacteristic
239C2×#67—21no—
240C22×Dic3—21no—
241#235—21no—
242#235—21no—
243#235C2×C4×C512yes—
244#235C5×D412yes—
245C2×D12C5×D411yescharacteristic
246C2×C4×D3—21no—
247C2×C4×D3—22no—
248C2×Dic6C5×D411yescharacteristic
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4C5×D412yes—
253C3×C42—21no—
254C3×C42—21no—
255#216—22no—
256#215—22no—
257#214—22no—
258#213—22no—
259(C2×D3)⋊C4—22no—
260Dic3⋊C4—22no—
261C4×Dic3—22no—
262C4×Dic3—21no—
263C4×Dic3C5×D411yescharacteristic
264C4×Dic3C5×D411yescharacteristic
265#208—21no—
266#208—21no—
267#208—21no—
268#208C2×C4×C512yescomplemented
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C23×C5—21no—
272C23×C5—61no—
273C23×C5C2×#6711yescharacteristic
274C5×Q8—21no—
275C5×Q8—21no—
276C5×Q8—32no—
277C5×Q8—61no—
278C5×D4—21no—
279C5×D4—21no—
280C5×D4—32no—
281C5×D4—41no—
282C5×D4—41no—
283C5×D4—62no—
284C5×D4—62no—
285C5×D4—61no—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—61no—
292C2×C4×C5—61no—
293C2×C4×C5—62no—
294C2×C4×C5—62no—
295C2×C4×C5—61no—
296C2×C4×C5—124no—
297C2×C4×C5—122no—
298C2×C4×C5C2×C4×D311yescharacteristic
299C2×C4×C5#23511yescharacteristic
300C2×C4×C5D3×D411yescharacteristic
301C2×C4×C5D3×D411yescharacteristic
302C2×C4×C5D3×D411yescharacteristic
303C2×C4×C5#23711yescharacteristic
304C5×C8—62no—
305C5×C8—121no—
306C2×#41—31no—
307C2×#41C2×C5×D311yescharacteristic
308#137—32no—
309#137—61no—
310AGL(1,ℤ/8)—32no—
311AGL(1,ℤ/8)—61no—
312#135—62no—
313C2×M16—31no—
314C2×M16—31no—
315C4×Q8—61no—
316C4×D4—61no—
317C4×D4—62no—
318#117—61no—
319#117C2×C5×D311yescharacteristic
320#117C2×C5×D311yescharacteristic
321C2×C42—31no—
322C2×C42—61no—
323#104—62no—
324#104—61no—
325#104—61no—
326#99—22no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—81no—
331C2×C3×C5#38611yescharacteristic
332C5×D3—42no—
333C23×C3—21no—
334C23×C3C2×C5×D411yescharacteristic
335C22×D3—21no—
336C3×Q8—21no—
337C3×Q8—21no—
338C3×D4—21no—
339C3×D4—21no—
340C3×D4—41no—
341C3×D4—41no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4C22×C4×C511yescharacteristic
348C2×C3×C4C2×C5×D411yescharacteristic
349C2×C3×C4C2×C5×D411yescharacteristic
350C2×C3×C4C2×C5×D411yescharacteristic
351C2×C3×C4C5×#4111yescharacteristic
352C2×C3×C4C5×#4111yescharacteristic
353#67—22no—
354#67—22no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—42no—
359D12—21no—
360D12C5×C22⋊C412yes—
361C4×D3—22no—
362C4×D3—22no—
363C4×D3—44no—
364Dic6—21no—
365Dic6C5×C22⋊C412yes—
366C3⋊C8—22no—
367C3⋊C8—41no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—62no—
371C22×C5—81no—
372C22×C5—122no—
373C22×C5C4×#6711yescharacteristic
374C22×C5#93711yescharacteristic
375C22×C5#93811yescharacteristic
376C4×C5—42no—
377C4×C5—41no—
378C4×C5—62no—
379C4×C5—62no—
380C4×C5—81no—
381C4×C5—121no—
382C4×C5D3×C22⋊C411yescharacteristic
383C4×C5#67⋊C411yescharacteristic
384C4×C5#88311yescharacteristic
385C4×C5#88411yescharacteristic
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—32no—
391#41—32no—
392#41—61no—
393#41—61no—
394C2×Q8—31no—
395C2×Q8C4×C5×D311yescharacteristic, complemented
396C2×D4—21no—
397C2×D4—31no—
398C2×D4—61no—
399C2×D4C4×C5×D311yescharacteristic, complemented
400C22×C4—21no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4C22×C5×D311yescharacteristic
405Q16—62no—
406SD16—62no—
407SD16—62no—
408D8—62no—
409M16—32no—
410M16—61no—
411M16—61no—
412M16—61no—
413C2×C8—61no—
414C2×C8—61no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—62no—
418C22⋊C4—21no—
419C22⋊C4—62no—
420C22⋊C4C5×#6712yes—
421C42—21no—
422C42—21no—
423C42—31no—
424C42—31no—
425C42—62no—
426C42—61no—
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—81no—
431C22×C3C4×C5×D411yescharacteristic
432C22×C3C5×#12011yescharacteristic
433C22×C3C5×#12111yescharacteristic
434C2×D3—22no—
435C2×D3—42no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—81no—
439C3×C4C2×C5×C22⋊C411yescharacteristic
440C3×C4C4×C5×D411yescharacteristic
441C3×C4C5×#12111yescharacteristic
442C3×C4C5×#12311yescharacteristic
443Dic3—22no—
444Dic3—22no—
445Dic3—41no—
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—81no—
450C2×C5—122no—
451C2×C5#495611yescharacteristic
452C23—21no—
453C23—61no—
454C23C2×C5×#6711yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—32no—
458Q8—61no—
459D4—21no—
460D4—21no—
461D4—32no—
462D4—41no—
463D4—41no—
464D4—62no—
465D4—62no—
466D4—61no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—61no—
477C2×C4—124no—
478C2×C4—122no—
479C2×C4C2×C4×C5×D311yescharacteristic
480C2×C4C5×#23511yescharacteristic
481C2×C4C5×D3×D411yescharacteristic
482C2×C4C5×D3×D411yescharacteristic
483C2×C4C5×D3×D411yescharacteristic
484C2×C4C5×#23711yescharacteristic
485C8—62no—
486C8—121no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—81no—
491C2×C3C5×#38611yescharacteristic
492D3—42no—
493C5#7005511yescharacteristic, direct product factor, Sylow
494C22—41no—
495C22—41no—
496C22—62no—
497C22—81no—
498C22—122no—
499C22C4×C5×#6711yescharacteristic
500C22C5×#93711yescharacteristic
501C22C5×#93811yescharacteristic
502C4—42no—
503C4—41no—
504C4—62no—
505C4—62no—
506C4—81no—
507C4—121no—
508C4C5×D3×C22⋊C411yescharacteristic
509C4C5×#67⋊C411yescharacteristic
510C4C5×#88311yescharacteristic
511C4C5×#88411yescharacteristic
512C3C5×#190611yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—81no—
517C2—122no—
518C2C5×#495611yescharacteristic
subgroupG/H
centerC4×C5#884
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#410C2
socleC2×C3×C5#386
prime radicalC5×#5822C2
cosocleC22×C3×C4C23×C5
hypercenterC5×#410D3
hypocenterC3C5×#1906
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#410

Complements

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

NH ≅ G/Ntype
2. C5×#5522C2semidirect
3. C3×C5×#410C2semidirect
4. C2×C5×#837C2semidirect
5. C5×#5005C2semidirect
6. C5×#4533C2semidirect
7. C5×#4532C2semidirect
16. C5×#942C4semidirect
17. C5×#932C4semidirect
30. #70055C5direct (N × H)
35. C5×#410D3semidirect
42. C2×C3×C5×Q8C2×C4semidirect
44. C2×C3×C5×D4C2×C4semidirect
76. C5×#208C2×C4semidirect
80. #5522C2×C5semidirect
81. C3×#410C2×C5semidirect
82. C2×#837C2×C5semidirect
83. #5005C2×C5semidirect
84. #4533C2×C5semidirect
85. #4532C2×C5semidirect
150. #942C4×C5semidirect
151. #932C4×C5semidirect
173. C2×C5×Q8C4×D3semidirect
177. C2×C5×D4C4×D3semidirect
209. #410C5×D3semidirect
234. C2×C3×Q8C2×C4×C5semidirect
236. C2×C3×D4C2×C4×C5semidirect
268. #208C2×C4×C5semidirect
395. C2×Q8C4×C5×D3semidirect
399. C2×D4C4×C5×D3semidirect
427. C3×C5#1906semidirect
493. C5#70055direct (N × H)
512. C3C5×#1906semidirect

Maximal subgroups

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

subgroupindex
C5×#58222
C5×#55222
C3×C5×#4102
C2×C5×#8372
C5×#50052
C5×#45332
C5×#45322
C5×#19063
#700555