Finite Group Databasebeta

← C5×#70357 #14306027

Subgroups — C5×#70357 #14306027

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 504 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
6C5×#4932C211yesmaximal, characteristic, complemented
7C2×C5×#805C211yesmaximal, characteristic, complemented
8C5×#1963—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×Dic3C2211yescharacteristic
20C2×C4×C5×Dic3C2211yescharacteristic
21C2×C5×#208C2211yescharacteristic
22C3×C5×#104—22no—
23C5×#837—22no—
24C5×C3⋊#95—22no—
25C5×#825—22no—
26C5×Dic3⋊C8—22no—
27C5×#805—22no—
28#70357C511yesmaximal, 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
35C5×#376—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—21no—
66C4×C5×Dic3C2×C411yescharacteristic
67C4×C5×Dic3C2×C411yescharacteristic
68C4×C5×Dic3D412yes—
69C4×C5×Dic3D412yescomplemented
70C5×#208—21no—
71C5×#208—21no—
72C2×C5×C3⋊C8—21no—
73#5832C2×C511yescharacteristic
74C2×C3×#104C2×C511yescharacteristic, complemented
75C2×#837C2×C511yescharacteristic, complemented
76C2×#825C2×C511yescharacteristic, complemented
77#5118C2×C511yescharacteristic, complemented
78#4932C2×C511yescharacteristic, complemented
79C2×#805C2×C511yescharacteristic, complemented
80C2×C5×#41—31no—
81C2×C5×#41C2×D311yescharacteristic
82C2×C5×M16—31no—
83C2×C5×M16C2×D311yescharacteristic
84C5×C4⋊Q8—31no—
85C5×#127—31no—
86C5×#124—31no—
87C5×#124—31no—
88C5×#121—62no—
89C4×C5×D4—62no—
90C2×C42×C5—31no—
91C2×C42×C5—31no—
92C2×C42×C5C2×D311yescharacteristic
93C5×C4⋊C8—62no—
94C5×#104—22no—
95C5×#104—62no—
96C5×#104—62no—
97C5×#104—62no—
98C5×#95—62no—
99#1963—31noSylow
100C23×C3×C5—21no—
101C23×C3×C5C2×D411yescharacteristic
102C22×C5×D3—21no—
103C3×C5×Q8—21no—
104C3×C5×Q8—21no—
105C3×C5×D4—21no—
106C3×C5×D4—21no—
107C3×C5×D4—41no—
108C3×C5×D4—41no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—22no—
111C2×C3×C4×C5—42no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5—41no—
114C2×C3×C4×C5C22×C411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C5×#67—44no—
121C5×#67—41no—
122C5×#67—41no—
123C2×C5×Dic3—21no—
124C2×C5×Dic3—22no—
125C2×C5×Dic3—42no—
126C2×C5×Dic3—42no—
127C2×C5×Dic3C22⋊C412yes—
128C2×C5×Dic3C22⋊C412yes—
129C5×D12—21no—
130C5×D12—21no—
131C4×C5×D3—41no—
132C4×C5×D3—41no—
133C5×Dic6—21no—
134C5×Dic6—21no—
135C3×C5×C8—41no—
136C5×C3⋊C8—41no—
137C2×C3×#41C22×C511yescharacteristic
138C2×#235C22×C511yescharacteristic
139C2×C3×M16C22×C511yescharacteristic
140C2×C3×C42C22×C511yescharacteristic
141#947C4×C511yescharacteristic, complemented
142#944C4×C511yescharacteristic, complemented
143#940C4×C511yescharacteristic, complemented
144#939—22no—
145#936C4×C511yescharacteristic, complemented
146C4×#67—22no—
147C2×C4×Dic3C22×C511yescharacteristic
148C2×C4×Dic3C22×C511yescharacteristic
149C2×#208C22×C511yescharacteristic
150C3×#104—22no—
151#837—22no—
152C3⋊#95—22no—
153#825—22no—
154Dic3⋊C8—22no—
155#805—22no—
156C5×#41—21no—
157C5×#41—21no—
158C5×#41—21no—
159C5×#41—21no—
160C5×#41—61no—
161C5×#41—61no—
162C5×#41—61no—
163C5×#41—61no—
164C2×C5×Q8—31no—
165C2×C5×Q8C4×D311yescharacteristic
166C2×C5×D4—21no—
167C2×C5×D4—31no—
168C2×C5×D4—62no—
169C2×C5×D4—61no—
170C2×C5×D4C4×D311yescharacteristic
171C22×C4×C5—21no—
172C22×C4×C5—21no—
173C22×C4×C5—32no—
174C22×C4×C5—61no—
175C22×C4×C5—61no—
176C22×C4×C5C22×D311yescharacteristic
177C5×M16—21no—
178C5×M16—21no—
179C5×M16—61no—
180C5×M16—61no—
181C2×C5×C8—21no—
182C2×C5×C8—61no—
183C5×C4⋊C4—62no—
184C5×C22⋊C4—62no—
185C5×C22⋊C4—62no—
186C42×C5—21no—
187C42×C5—31no—
188C42×C5—31no—
189C42×C5—32no—
190C42×C5—32no—
191C42×C5—61no—
192C42×C5#6712yescomplemented
193#532—31no—
194#423—31no—
195C2×#104—31no—
196C2×#104—31no—
197C2×#104—31no—
198C2×#104C5×D311yescharacteristic, complemented
199#376—31no—
200C22×C3×C5—22no—
201C22×C3×C5—41no—
202C22×C3×C5—42no—
203C22×C3×C5C2×C22⋊C411yescharacteristic
204C22×C3×C5#12111yescharacteristic
205C22×C3×C5#12111yescharacteristic
206C2×C5×D3—41no—
207C2×C5×D3—42no—
208C3×C4×C5—42no—
209C3×C4×C5—41no—
210C3×C4×C5C4×D411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5#12411yescharacteristic
213C3×C4×C5#12711yescharacteristic
214C5×Dic3—22no—
215C5×Dic3—22no—
216C5×Dic3—42no—
217C5×Dic3—41no—
218C3×#41—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41—21no—
222C2×C3×Q8C2×C4×C511yescharacteristic
223C2×C3×D4—21no—
224C2×C3×D4C2×C4×C511yescharacteristic
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4C23×C511yescharacteristic
228C2×#67—21no—
229C2×#67—22no—
230C22×Dic3—21no—
231C22×Dic3C5×D412yes—
232#235—21no—
233#235—21no—
234#235—21no—
235#235—21no—
236C2×D12C2×C4×C511yescharacteristic
237C2×C4×D3—21no—
238C2×Dic6C2×C4×C511yescharacteristic
239C3×M16—21no—
240C3×M16—21no—
241C2×C3×C8—21no—
242C3×C42—21no—
243C3×C42C5×D412yescomplemented
244#217—22no—
245(C2×D3)⋊C4—22no—
246Dic3⋊C4—22no—
247C4×Dic3—21no—
248C4×Dic3C2×C4×C511yescharacteristic
249C4×Dic3C2×C4×C511yescharacteristic
250C4×Dic3C5×D412yes—
251C4×Dic3C5×D412yescomplemented
252#208—21no—
253#208—21no—
254C2×C3⋊C8—21no—
255C23×C5—21no—
256C23×C5—61no—
257C23×C5D3×D411yescharacteristic
258C5×Q8—21no—
259C5×Q8—21no—
260C5×Q8—61no—
261C5×Q8—61no—
262C5×D4—21no—
263C5×D4—21no—
264C5×D4—41no—
265C5×D4—41no—
266C5×D4—61no—
267C5×D4—61no—
268C5×D4—124no—
269C5×D4—121no—
270C5×D4—121no—
271C2×C4×C5—21no—
272C2×C4×C5—22no—
273C2×C4×C5—32no—
274C2×C4×C5—32no—
275C2×C4×C5—42no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—61no—
279C2×C4×C5—62no—
280C2×C4×C5—122no—
281C2×C4×C5—122no—
282C2×C4×C5—121no—
283C2×C4×C5—121no—
284C2×C4×C5C2×C4×D311yescharacteristic
285C2×C4×C5#23511yescharacteristic
286C2×C4×C5D3×D411yescharacteristic
287C2×C4×C5D3×D411yescharacteristic
288C2×C4×C5#23711yescharacteristic
289C2×C4×C5C2×#6711yescharacteristic
290C5×C8—41no—
291C5×C8—121no—
292C2×#41—31no—
293C2×#41C2×C5×D311yescharacteristic
294C2×M16—31no—
295C2×M16C2×C5×D311yescharacteristic
296C4⋊Q8—31no—
297#127—31no—
298#124—31no—
299#124—31no—
300#121—62no—
301C4×D4—62no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42C2×C5×D311yescharacteristic
305C4⋊C8—62no—
306#104—22no—
307#104—62no—
308#104—62no—
309#104—62no—
310#95—62no—
311C2×C3×C5—21no—
312C2×C3×C5—21no—
313C2×C3×C5—81no—
314C2×C3×C5#39011yescharacteristic
315C2×C3×C5#42112yes—
316C5×D3—81no—
317C23×C3—21no—
318C23×C3C2×C5×D411yescharacteristic
319C22×D3—21no—
320C3×Q8—21no—
321C3×Q8—21no—
322C3×D4—21no—
323C3×D4—21no—
324C3×D4—41no—
325C3×D4—41no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4C22×C4×C511yescharacteristic
332C2×C3×C4C2×C5×D411yescharacteristic
333C2×C3×C4C2×C5×D411yescharacteristic
334C2×C3×C4C2×C5×D411yescharacteristic
335C2×C3×C4C5×#4111yescharacteristic
336C2×C3×C4C5×#4111yescharacteristic
337#67—44no—
338#67—41no—
339#67—41no—
340C2×Dic3—21no—
341C2×Dic3—22no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3C5×C22⋊C412yes—
345C2×Dic3C5×C22⋊C412yes—
346D12—21no—
347D12—21no—
348C4×D3—41no—
349C4×D3—41no—
350Dic6—21no—
351Dic6—21no—
352C3×C8—41no—
353C3⋊C8—41no—
354C22×C5—22no—
355C22×C5—41no—
356C22×C5—42no—
357C22×C5—121no—
358C22×C5—122no—
359C22×C5D3×C22⋊C411yescharacteristic
360C22×C5#88411yescharacteristic
361C22×C5#93911yescharacteristic
362C4×C5—42no—
363C4×C5—41no—
364C4×C5—62no—
365C4×C5—62no—
366C4×C5—122no—
367C4×C5—121no—
368C4×C5#67⋊C411yescharacteristic
369C4×C5#88511yescharacteristic
370C4×C5C4×#6711yescharacteristic
371C4×C5#94011yescharacteristic
372#41—21no—
373#41—21no—
374#41—21no—
375#41—21no—
376#41—61no—
377#41—61no—
378#41—61no—
379#41—61no—
380C2×Q8—31no—
381C2×Q8C4×C5×D311yescharacteristic
382C2×D4—21no—
383C2×D4—31no—
384C2×D4—62no—
385C2×D4—61no—
386C2×D4C4×C5×D311yescharacteristic
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—32no—
390C22×C4—61no—
391C22×C4—61no—
392C22×C4C22×C5×D311yescharacteristic
393M16—21no—
394M16—21no—
395M16—61no—
396M16—61no—
397C2×C8—21no—
398C2×C8—61no—
399C4⋊C4—62no—
400C22⋊C4—62no—
401C22⋊C4—62no—
402C42—21no—
403C42—31no—
404C42—31no—
405C42—32no—
406C42—32no—
407C42—61no—
408C42C5×#6712yescomplemented
409C3×C5#196311yescharacteristic, complemented, Hall
410C22×C3—22no—
411C22×C3—41no—
412C22×C3—42no—
413C22×C3C2×C5×C22⋊C411yescharacteristic
414C22×C3C5×#12111yescharacteristic
415C22×C3C5×#12111yescharacteristic
416C2×D3—41no—
417C2×D3—42no—
418C3×C4—42no—
419C3×C4—41no—
420C3×C4C4×C5×D411yescharacteristic
421C3×C4C4×C5×D411yescharacteristic
422C3×C4C5×#12411yescharacteristic
423C3×C4C5×#12711yescharacteristic
424Dic3—22no—
425Dic3—22no—
426Dic3—42no—
427Dic3—41no—
428C2×C5—21no—
429C2×C5—21no—
430C2×C5—81no—
431C2×C5—241no—
432C2×C5#482112yes—
433C2×C5#495711yescharacteristic
434C23—21no—
435C23—61no—
436C23C5×D3×D411yescharacteristic
437Q8—21no—
438Q8—21no—
439Q8—61no—
440Q8—61no—
441D4—21no—
442D4—21no—
443D4—41no—
444D4—41no—
445D4—61no—
446D4—61no—
447D4—124no—
448D4—121no—
449D4—121no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—32no—
453C2×C4—32no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—61no—
458C2×C4—62no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4C2×C4×C5×D311yescharacteristic
464C2×C4C5×#23511yescharacteristic
465C2×C4C5×D3×D411yescharacteristic
466C2×C4C5×D3×D411yescharacteristic
467C2×C4C5×#23711yescharacteristic
468C2×C4C2×C5×#6711yescharacteristic
469C8—41no—
470C8—121no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—81no—
474C2×C3C5×#39011yescharacteristic
475C2×C3C5×#42112yes—
476D3—81no—
477C5#7035711yescharacteristic, direct product factor, Sylow
478C22—22no—
479C22—41no—
480C22—42no—
481C22—121no—
482C22—122no—
483C22C5×D3×C22⋊C411yescharacteristic
484C22C5×#88411yescharacteristic
485C22C5×#93911yescharacteristic
486C4—42no—
487C4—41no—
488C4—62no—
489C4—62no—
490C4—122no—
491C4—121no—
492C4C5×#67⋊C411yescharacteristic
493C4C5×#88511yescharacteristic
494C4C4×C5×#6711yescharacteristic
495C4C5×#94011yescharacteristic
496C3C5×#196311yescharacteristic, complemented, Sylow
497C2—21no—
498C2—21no—
499C2—81no—
500C2—241no—
501C2C5×#482112yes—
502C2C5×#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×#1963
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. C5×#4932C2semidirect
7. C2×C5×#805C2semidirect
13. C5×#947C4semidirect
14. C5×#944C4semidirect
15. C5×#940C4semidirect
17. C5×#936C4semidirect
28. #70357C5direct (N × H)
34. C2×C5×#104D3semidirect
61. C3×C42×C5D4semidirect
69. C4×C5×Dic3D4semidirect
74. C2×C3×#104C2×C5semidirect
75. C2×#837C2×C5semidirect
76. C2×#825C2×C5semidirect
77. #5118C2×C5semidirect
78. #4932C2×C5semidirect
79. C2×#805C2×C5semidirect
141. #947C4×C5semidirect
142. #944C4×C5semidirect
143. #940C4×C5semidirect
145. #936C4×C5semidirect
192. C42×C5#67semidirect
198. C2×#104C5×D3semidirect
243. C3×C42C5×D4semidirect
251. C4×Dic3C5×D4semidirect
408. C42C5×#67semidirect
409. C3×C5#1963semidirect
477. C5#70357direct (N × H)
496. C3C5×#1963semidirect

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
C5×#49322
C2×C5×#8052
C5×#19633
#703575