Finite Group Databasebeta

← C5×#68122 #14281170

Subgroups — C5×#68122 #14281170

Order 1920.

560 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
1C2×C5×#883C211yesmaximal, characteristic
2C5×#4726C211yesmaximal, characteristic, complemented
3C5×D3×C22⋊C8C211yesmaximal, characteristic, complemented
4C3×C5×#323C211yesmaximal, characteristic, complemented
5C5×#4525C211yesmaximal, characteristic, complemented
6C5×#4469C211yesmaximal, characteristic, complemented
7C5×#4469C211yesmaximal, characteristic, complemented
8C5×#1468—31nomaximal, Hall
9C22×C5×#67C411yescharacteristic
10C22×C4×C5×D3C2211yescharacteristic
11C2×C3×C5×C22⋊C4C2211yescharacteristic
12C2×C5×#217C411yescharacteristic
13C2×C5×(C2×D3)⋊C4C411yescharacteristic
14C2×C5×C4⋊Dic3C411yescharacteristic
15C2×C5×Dic3⋊C4C2211yescharacteristic
16C2×C5×ΓL(1,𝔽25)—21no—
17C2×C5×D3×C8—21no—
18C5×#883—21no—
19C5×#883—21no—
20C5×#883—21no—
21C5×#883—21no—
22C3×C5×C22⋊C8C2211yescharacteristic
23C3×C5×C22⋊C8C2211yescharacteristic
24C5×#830C2211yescharacteristic
25C5×#830C2211yescharacteristic
26C5×(C2×D3)⋊C8—21no—
27C5×(C2×D3)⋊C8—21no—
28#68122C511yesmaximal, characteristic, direct product factor, Hall
29C2×C5×#123—31no—
30C5×#407—31no—
31C2×C5×C22⋊C8—31no—
32C5×#324—31no—
33C5×#324—31no—
34C5×#323—31no—
35C5×#323D311yescharacteristic, complemented
36C24×C3×C5C2×C411yescharacteristic
37C23×C5×D3C2×C411yescharacteristic
38C22×C3×C4×C5C2×C411yescharacteristic
39C22×C3×C4×C5C2311yescharacteristic
40C2×C5×#67—21no—
41C2×C5×#67—21no—
42C2×C5×#67—42no—
43C22×C5×Dic3C2×C411yescharacteristic
44C22×C5×Dic3C2×C411yescharacteristic
45C22×C5×Dic3C2×C411yescharacteristic
46C2×C4×C5×D3—42no—
47C2×C4×C5×D3D411yescharacteristic
48C2×C4×C5×D3D411yescharacteristic
49C2×C4×C5×D3D411yescharacteristic
50C2×C4×C5×D3D411yescharacteristic
51C2×C3×C5×C8—21no—
52C2×C3×C5×C8—21no—
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C5×#217—21no—
56C5×#217—21no—
57C5×(C2×D3)⋊C4—21no—
58C5×(C2×D3)⋊C4—21no—
59C5×C4⋊Dic3—21no—
60C5×C4⋊Dic3—21no—
61C5×Dic3⋊C4—21no—
62C5×Dic3⋊C4—21no—
63C2×C5×C3⋊C8—21no—
64C2×C5×C3⋊C8—21no—
65C5×ΓL(1,𝔽25)—42no—
66C5×D3×C8—42no—
67C2×#883C2×C511yescharacteristic
68#4726C2×C511yescharacteristic, complemented
69D3×C22⋊C8C2×C511yescharacteristic, complemented
70C3×#323C2×C511yescharacteristic, complemented
71#4525C2×C511yescharacteristic, complemented
72#4469C2×C511yescharacteristic, complemented
73#4469C2×C511yescharacteristic, complemented
74C22×C5×D4—31no—
75C23×C4×C5—31no—
76C2×C5×M16—61no—
77C22×C5×C8—61no—
78C5×#123—61no—
79C5×#123—61no—
80C5×#123—61no—
81C5×#123—61no—
82C2×C5×C4⋊C4—31no—
83C2×C5×C4⋊C4—31no—
84C2×C5×C22⋊C4—31no—
85C2×C5×C22⋊C4—31no—
86C2×C5×C22⋊C4C2×D311yescharacteristic
87C5×C22⋊C8—31no—
88C5×C22⋊C8—31no—
89C5×C22⋊C8—61no—
90C5×C22⋊C8—61no—
91C5×C22⋊C8C2×D311yescharacteristic
92C5×C22⋊C8C2×D311yescharacteristic
93#1468—31noSylow
94C23×C3×C5—21no—
95C23×C3×C5—21no—
96C23×C3×C5—21no—
97C23×C3×C5—41no—
98C23×C3×C5—41no—
99C23×C3×C5C22×C411yescharacteristic
100C22×C5×D3—21no—
101C22×C5×D3—21no—
102C22×C5×D3—22no—
103C22×C5×D3—41no—
104C22×C5×D3C22⋊C411yescharacteristic
105C22×C5×D3C22⋊C411yescharacteristic
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C5×#67—42no—
112C5×#67—42no—
113C2×C5×Dic3—21no—
114C2×C5×Dic3—21no—
115C2×C5×Dic3—21no—
116C2×C5×Dic3—21no—
117C2×C5×Dic3—41no—
118C2×C5×Dic3—41no—
119C2×C5×Dic3C22⋊C411yescharacteristic
120C2×C5×Dic3C22⋊C411yescharacteristic
121C4×C5×D3—22no—
122C4×C5×D3—22no—
123C4×C5×D3—41no—
124C4×C5×D3—41no—
125C3×C5×C8—41no—
126C3×C5×C8—41no—
127C5×C3⋊C8—41no—
128C5×C3⋊C8—41no—
129C22×#67C4×C511yescharacteristic
130C22×C4×D3C22×C511yescharacteristic
131C2×C3×C22⋊C4C22×C511yescharacteristic
132C2×#217C4×C511yescharacteristic
133C2×(C2×D3)⋊C4C4×C511yescharacteristic
134C2×C4⋊Dic3C4×C511yescharacteristic
135C2×Dic3⋊C4C22×C511yescharacteristic
136C2×ΓL(1,𝔽25)—21no—
137C2×D3×C8—21no—
138#883—21no—
139#883—21no—
140#883—21no—
141#883—21no—
142C3×C22⋊C8C22×C511yescharacteristic
143C3×C22⋊C8C22×C511yescharacteristic
144#830C22×C511yescharacteristic
145#830C22×C511yescharacteristic
146(C2×D3)⋊C8—21no—
147(C2×D3)⋊C8—21no—
148C24×C5—31no—
149C24×C5C4×D311yescharacteristic
150C2×C5×D4—61no—
151C2×C5×D4—61no—
152C2×C5×D4—122no—
153C22×C4×C5—31no—
154C22×C4×C5—31no—
155C22×C4×C5—31no—
156C22×C4×C5—31no—
157C22×C4×C5—31no—
158C22×C4×C5—31no—
159C22×C4×C5—31no—
160C22×C4×C5—122no—
161C22×C4×C5C4×D311yescharacteristic
162C22×C4×C5C22×D311yescharacteristic
163C5×M16—122no—
164C2×C5×C8—21no—
165C2×C5×C8—21no—
166C2×C5×C8—61no—
167C2×C5×C8—61no—
168C2×C5×C8—122no—
169C5×C4⋊C4—61no—
170C5×C4⋊C4—61no—
171C5×C4⋊C4—61no—
172C5×C4⋊C4—61no—
173C5×C22⋊C4—21no—
174C5×C22⋊C4—21no—
175C5×C22⋊C4—61no—
176C5×C22⋊C4—61no—
177C5×C22⋊C4—61no—
178C5×C22⋊C4—61no—
179C2×#123—31no—
180#407—31no—
181C2×C22⋊C8—31no—
182#324—31no—
183#324—31no—
184#323—31no—
185#323C5×D311yescharacteristic, complemented
186C22×C3×C5—21no—
187C22×C3×C5—21no—
188C22×C3×C5—41no—
189C22×C3×C5—41no—
190C22×C3×C5—41no—
191C22×C3×C5—81no—
192C22×C3×C5—81no—
193C22×C3×C5C2×C22⋊C411yescharacteristic
194C22×C3×C5#13111yescharacteristic
195C22×C3×C5#13111yescharacteristic
196C2×C5×D3—21no—
197C2×C5×D3—21no—
198C2×C5×D3—21no—
199C2×C5×D3—21no—
200C2×C5×D3—21no—
201C2×C5×D3—42no—
202C2×C5×D3—42no—
203C2×C5×D3#9912yes—
204C3×C4×C5—21no—
205C3×C4×C5—21no—
206C3×C4×C5—41no—
207C5×Dic3—21no—
208C5×Dic3—21no—
209C5×Dic3—41no—
210C5×Dic3—41no—
211C24×C3C2×C4×C511yescharacteristic
212C23×D3C2×C4×C511yescharacteristic
213C22×C3×C4C2×C4×C511yescharacteristic
214C22×C3×C4C23×C511yescharacteristic
215C2×#67—21no—
216C2×#67—21no—
217C2×#67—42no—
218C22×Dic3C2×C4×C511yescharacteristic
219C22×Dic3C2×C4×C511yescharacteristic
220C22×Dic3C2×C4×C511yescharacteristic
221C2×C4×D3—42no—
222C2×C4×D3C5×D411yescharacteristic
223C2×C4×D3C5×D411yescharacteristic
224C2×C4×D3C5×D411yescharacteristic
225C2×C4×D3C5×D411yescharacteristic
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—21no—
230#217—21no—
231#217—21no—
232(C2×D3)⋊C4—21no—
233(C2×D3)⋊C4—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236Dic3⋊C4—21no—
237Dic3⋊C4—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240ΓL(1,𝔽25)—42no—
241D3×C8—42no—
242C23×C5—21no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—31no—
246C23×C5—31no—
247C23×C5—41no—
248C23×C5—41no—
249C23×C5—62no—
250C23×C5—61no—
251C23×C5—61no—
252C23×C5—121no—
253C23×C5C2×C4×D311yescharacteristic
254C5×D4—122no—
255C5×D4—122no—
256C2×C4×C5—21no—
257C2×C4×C5—31no—
258C2×C4×C5—31no—
259C2×C4×C5—41no—
260C2×C4×C5—41no—
261C2×C4×C5—62no—
262C2×C4×C5—62no—
263C2×C4×C5—61no—
264C2×C4×C5—61no—
265C2×C4×C5—61no—
266C2×C4×C5—61no—
267C2×C4×C5—121no—
268C2×C4×C5—121no—
269C2×C4×C5—121no—
270C2×C4×C5—121no—
271C2×C4×C5D3×D411yescharacteristic
272C2×C4×C5D3×D411yescharacteristic
273C5×C8—41no—
274C5×C8—41no—
275C5×C8—121no—
276C5×C8—121no—
277C22×D4—31no—
278C23×C4—31no—
279C2×M16—61no—
280C22×C8—61no—
281#123—61no—
282#123—61no—
283#123—61no—
284#123—61no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C22⋊C4—31no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4C2×C5×D311yescharacteristic
290C22⋊C8—31no—
291C22⋊C8—31no—
292C22⋊C8—61no—
293C22⋊C8—61no—
294C22⋊C8C2×C5×D311yescharacteristic
295C22⋊C8C2×C5×D311yescharacteristic
296C2×C3×C5—21no—
297C2×C3×C5—21no—
298C2×C3×C5—81no—
299C2×C3×C5#40811yescharacteristic
300C2×C3×C5C2×#9911yescharacteristic
301C2×C3×C5#41311yescharacteristic
302C5×D3—22no—
303C5×D3—41no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3C22×C4×C511yescharacteristic
310C22×D3—21no—
311C22×D3—21no—
312C22×D3—22no—
313C22×D3—41no—
314C22×D3C5×C22⋊C411yescharacteristic
315C22×D3C5×C22⋊C411yescharacteristic
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4C2×C5×D411yescharacteristic
320C2×C3×C4C2×C5×D411yescharacteristic
321#67—42no—
322#67—42no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3C5×C22⋊C411yescharacteristic
330C2×Dic3C5×C22⋊C411yescharacteristic
331C4×D3—22no—
332C4×D3—22no—
333C4×D3—41no—
334C4×D3—41no—
335C3×C8—41no—
336C3×C8—41no—
337C3⋊C8—41no—
338C3⋊C8—41no—
339C22×C5—21no—
340C22×C5—21no—
341C22×C5—32no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5—41no—
345C22×C5—61no—
346C22×C5—61no—
347C22×C5—61no—
348C22×C5—61no—
349C22×C5—61no—
350C22×C5—81no—
351C22×C5—81no—
352C22×C5—122no—
353C22×C5—122no—
354C22×C5D3×C22⋊C411yescharacteristic
355C22×C5#90111yescharacteristic
356C22×C5#90711yescharacteristic
357C4×C5—21no—
358C4×C5—21no—
359C4×C5—41no—
360C4×C5—61no—
361C4×C5—61no—
362C4×C5—121no—
363C4×C5—121no—
364C24—31no—
365C24C4×C5×D311yescharacteristic
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—122no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—31no—
373C22×C4—31no—
374C22×C4—31no—
375C22×C4—31no—
376C22×C4—122no—
377C22×C4C4×C5×D311yescharacteristic
378C22×C4C22×C5×D311yescharacteristic
379M16—122no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—122no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—61no—
388C4⋊C4—61no—
389C22⋊C4—21no—
390C22⋊C4—21no—
391C22⋊C4—61no—
392C22⋊C4—61no—
393C22⋊C4—61no—
394C22⋊C4—61no—
395C3×C5#146811yescharacteristic, complemented, Hall
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3C2×C5×C22⋊C411yescharacteristic
404C22×C3C5×#13111yescharacteristic
405C22×C3C5×#13111yescharacteristic
406C2×D3—21no—
407C2×D3—21no—
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—21no—
411C2×D3—42no—
412C2×D3—42no—
413C2×D3C5×#9912yes—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—41no—
417Dic3—21no—
418Dic3—21no—
419Dic3—41no—
420Dic3—41no—
421C2×C5—21no—
422C2×C5—21no—
423C2×C5—62no—
424C2×C5—81no—
425C2×C5—121no—
426C2×C5#472711yescharacteristic
427C2×C5D3×#9911yescharacteristic
428C2×C5#474511yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—31no—
433C23—31no—
434C23—41no—
435C23—41no—
436C23—62no—
437C23—61no—
438C23—61no—
439C23—121no—
440C23C2×C4×C5×D311yescharacteristic
441D4—122no—
442D4—122no—
443C2×C4—21no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4C5×D3×D411yescharacteristic
459C2×C4C5×D3×D411yescharacteristic
460C8—41no—
461C8—41no—
462C8—121no—
463C8—121no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—81no—
467C2×C3C5×#40811yescharacteristic
468C2×C3C2×C5×#9911yescharacteristic
469C2×C3C5×#41311yescharacteristic
470D3—22no—
471D3—41no—
472C5#6812211yescharacteristic, direct product factor, Sylow
473C22—21no—
474C22—21no—
475C22—32no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—61no—
480C22—61no—
481C22—61no—
482C22—61no—
483C22—61no—
484C22—81no—
485C22—81no—
486C22—122no—
487C22—122no—
488C22C5×D3×C22⋊C411yescharacteristic
489C22C5×#90111yescharacteristic
490C22C5×#90711yescharacteristic
491C4—21no—
492C4—21no—
493C4—41no—
494C4—61no—
495C4—61no—
496C4—121no—
497C4—121no—
498C3C5×#146811yescharacteristic, complemented, Sylow
499C2—21no—
500C2—21no—
501C2—62no—
502C2—81no—
503C2—121no—
504C2C5×#472711yescharacteristic
505C2C5×D3×#9911yescharacteristic
506C2C5×#474511yescharacteristic
subgroupG/H
centerC22×C5D3×C22⋊C4
commutatorC23×C3C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#323C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C5×#67C4
cosocleC22×C3×C4C23×C5
hypercenterC5×#323D3
hypocenterC3C5×#1468
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#323

Complements

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

NH ≅ G/Ntype
2. C5×#4726C2semidirect
3. C5×D3×C22⋊C8C2semidirect
4. C3×C5×#323C2semidirect
5. C5×#4525C2semidirect
6. C5×#4469C2semidirect
7. C5×#4469C2semidirect
28. #68122C5direct (N × H)
35. C5×#323D3semidirect
68. #4726C2×C5semidirect
69. D3×C22⋊C8C2×C5semidirect
70. C3×#323C2×C5semidirect
71. #4525C2×C5semidirect
72. #4469C2×C5semidirect
73. #4469C2×C5semidirect
185. #323C5×D3semidirect
395. C3×C5#1468semidirect
472. C5#68122direct (N × H)
498. C3C5×#1468semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8832
C5×#47262
C5×D3×C22⋊C82
C3×C5×#3232
C5×#45252
C5×#44692
C5×#14683
#681225