Finite Group Databasebeta

← C5×#76496 #14331130

Subgroups — C5×#76496 #14331130

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#947C211yesmaximal, characteristic
2C3×C5×#391C211yesmaximal, characteristic, complemented
3C5×#5229C211yesmaximal, characteristic, complemented
4C5×#4989C212yesmaximal, complemented
5C5×#4940C212yesmaximal, complemented
6C5×#4669C212yesmaximal, complemented
7C5×#4668C212yesmaximal, complemented
8C5×#4665C212yesmaximal
9C5×#4650C212yesmaximal, complemented
10C5×#2379—31nomaximal, Hall
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C5×D12C2211yescharacteristic
13C22×C4×C5×D3C2212yes—
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C42×C5C2211yescharacteristic
16C5×#947C418yescomplemented
17C2×C5×(C2×D3)⋊C4C2212yes—
18C2×C5×(C2×D3)⋊C4C2212yes—
19C2×C5×(C2×D3)⋊C4C2212yes—
20C2×C5×(C2×D3)⋊C4C2212yes—
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C5×Dic3⋊C4C2212yes—
24C2×C5×Dic3⋊C4C2212yes—
25C2×C4×C5×Dic3C2211yescharacteristic
26C2×C4×C5×Dic3C2211yescharacteristic
27C3×C5×#95C2212yes—
28C3×C5×#95C2212yes—
29C5×C3⋊#95C2212yes—
30C5×C3⋊#95C2212yes—
31C5×C3⋊#95C2212yes—
32C5×C3⋊#95C2212yes—
33#76496C511yesmaximal, characteristic, direct product factor, Hall
34C2×C5×#124—31no—
35C5×#391—31no—
36C5×#391D311yescharacteristic, complemented
37C5×#388—32no—
38C5×#388—32no—
39C5×#386—32no—
40C5×#385—32no—
41C5×#385—32no—
42C5×#382—32no—
43C23×C5×D3C2312yes—
44C2×C3×C5×Q8—41no—
45C2×C3×C5×Q8—41no—
46C2×C3×C5×Q8C2×C414yescomplemented
47C22×C3×C4×C5C2312yes—
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C5×Dic3C2312yes—
53C22×C5×Dic3C2312yes—
54C22×C5×Dic3C2312yes—
55C2×C5×D12—41no—
56C2×C5×D12—41no—
57C2×C5×D12C2×C414yes—
58C2×C4×C5×D3—42no—
59C2×C4×C5×D3—42no—
60C2×C4×C5×D3—42no—
61C3×C5×C4⋊C4—41no—
62C3×C42×C5—41no—
63C5×(C2×D3)⋊C4—42no—
64C5×(C2×D3)⋊C4—42no—
65C5×(C2×D3)⋊C4—42no—
66C5×(C2×D3)⋊C4C2×C418yescomplemented
67C5×(C2×D3)⋊C4C2×C418yescomplemented
68C5×C4⋊Dic3—41no—
69C5×Dic3⋊C4—42no—
70C5×Dic3⋊C4—42no—
71C4×C5×Dic3—41no—
72C4×C5×Dic3C2×C414yescomplemented
73C2×#947C2×C511yescharacteristic
74C3×#391C2×C511yescharacteristic, complemented
75#5229C2×C511yescharacteristic, complemented
76#4989C2×C512yescomplemented
77#4940C2×C512yescomplemented
78#4669C2×C512yescomplemented
79#4668C2×C512yescomplemented
80#4665C2×C512yes—
81#4650C2×C512yescomplemented
82C22×C5×Q8C2×D311yescharacteristic
83C22×C5×D4—31no—
84C23×C4×C5—32no—
85C5×#124—38no—
86C2×C5×C4⋊C4—32no—
87C2×C5×C4⋊C4—32no—
88C2×C5×C4⋊C4—31no—
89C2×C5×C4⋊C4C2×D311yescharacteristic
90C2×C5×C22⋊C4—32no—
91C2×C5×C22⋊C4—32no—
92C2×C5×C22⋊C4—32no—
93C2×C5×C22⋊C4—32no—
94C2×C5×C22⋊C4—32no—
95C2×C42×C5—31no—
96C2×C42×C5—31no—
97C2×C42×C5C2×D311yescharacteristic
98C5×#95—32no—
99C5×#95—32no—
100C5×#95—32no—
101C5×#95—32no—
102C5×#95C2×D312yes—
103C5×#95C2×D312yes—
104#2379—31noSylow
105C23×C3×C5C2411yescharacteristic
106C22×C5×D3—22no—
107C22×C5×D3—22no—
108C22×C5×D3—22no—
109C22×C5×D3—22no—
110C22×C5×D3—22no—
111C22×C5×D3—22no—
112C22×C5×D3C22×C414yes—
113C3×C5×Q8—44no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5C22×C412yes—
129C2×C5×Dic3—22no—
130C2×C5×Dic3—22no—
131C2×C5×Dic3—22no—
132C2×C5×Dic3—22no—
133C2×C5×Dic3—22no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3C22×C414yes—
138C5×D12—44no—
139C4×C5×D3—48no—
140C22×C3×Q8C22×C511yescharacteristic
141C22×D12C22×C511yescharacteristic
142C22×C4×D3C22×C512yes—
143C2×C3×C4⋊C4C22×C511yescharacteristic
144C2×C3×C42C22×C511yescharacteristic
145#947C4×C518yescomplemented
146C2×(C2×D3)⋊C4C22×C512yes—
147C2×(C2×D3)⋊C4C22×C512yes—
148C2×(C2×D3)⋊C4C22×C512yes—
149C2×(C2×D3)⋊C4C22×C512yes—
150C2×(C2×D3)⋊C4C22×C512yes—
151C2×C4⋊Dic3C22×C511yescharacteristic
152C2×Dic3⋊C4C22×C512yes—
153C2×Dic3⋊C4C22×C512yes—
154C2×C4×Dic3C22×C511yescharacteristic
155C2×C4×Dic3C22×C511yescharacteristic
156C3×#95C22×C512yes—
157C3×#95C22×C512yes—
158C3⋊#95C22×C512yes—
159C3⋊#95C22×C512yes—
160C3⋊#95C22×C512yes—
161C3⋊#95C22×C512yes—
162C24×C5—32no—
163C2×C5×Q8—41no—
164C2×C5×Q8—41no—
165C2×C5×Q8C4×D314yescomplemented
166C2×C5×D4—34no—
167C2×C5×D4—121no—
168C2×C5×D4—121no—
169C22×C4×C5—32no—
170C22×C4×C5—32no—
171C22×C4×C5—32no—
172C22×C4×C5—122no—
173C22×C4×C5—122no—
174C22×C4×C5—122no—
175C22×C4×C5C22×D312yes—
176C22×C4×C5C22×D312yes—
177C22×C4×C5C22×D311yescharacteristic
178C22×C4×C5C22×D311yescharacteristic
179C22×C4×C5C22×D311yescharacteristic
180C5×C4⋊C4—41no—
181C5×C4⋊C4—122no—
182C5×C4⋊C4—122no—
183C5×C4⋊C4—121no—
184C5×C22⋊C4—38no—
185C5×C22⋊C4—38no—
186C5×C22⋊C4—122no—
187C5×C22⋊C4—122no—
188C5×C22⋊C4—122no—
189C42×C5—34no—
190C42×C5—41no—
191C42×C5—121no—
192C2×#124—31no—
193#391—31no—
194#391C5×D311yescharacteristic, complemented
195#388—32no—
196#388—32no—
197#386—32no—
198#385—32no—
199#385—32no—
200#382—32no—
201C22×C3×C5C23×C411yescharacteristic
202C22×C3×C5H(5,𝔽2)11yescharacteristic
203C22×C3×C5H(5,𝔽2)11yescharacteristic
204C22×C3×C5H(5,𝔽2)12yes—
205C22×C3×C52−1+412yes—
206C2×C5×D3—24no—
207C2×C5×D3—24no—
208C2×C5×D3—24no—
209C2×C5×D3—42no—
210C2×C5×D3—42no—
211C2×C5×D3—42no—
212C2×C5×D3—42no—
213C3×C4×C5—22no—
214C3×C4×C5—22no—
215C3×C4×C5—22no—
216C3×C4×C5—42no—
217C3×C4×C5—42no—
218C5×Dic3—24no—
219C5×Dic3—42no—
220C5×Dic3—42no—
221C23×D3C23×C512yes—
222C2×C3×Q8—41no—
223C2×C3×Q8—41no—
224C2×C3×Q8C2×C4×C514yescomplemented
225C22×C3×C4C23×C512yes—
226C22×C3×C4C23×C512yes—
227C22×C3×C4C23×C511yescharacteristic
228C22×C3×C4C23×C511yescharacteristic
229C22×C3×C4C23×C511yescharacteristic
230C22×Dic3C23×C512yes—
231C22×Dic3C23×C512yes—
232C22×Dic3C23×C512yes—
233C2×D12—41no—
234C2×D12—41no—
235C2×D12C2×C4×C514yes—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C3×C4⋊C4—41no—
240C3×C42—41no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4C2×C4×C518yescomplemented
245(C2×D3)⋊C4C2×C4×C518yescomplemented
246C4⋊Dic3—41no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249C4×Dic3—41no—
250C4×Dic3C2×C4×C514yescomplemented
251C23×C5—34no—
252C23×C5—62no—
253C23×C5—62no—
254C23×C5—62no—
255C23×C5—62no—
256C23×C5—62no—
257C23×C5—62no—
258C23×C5C23×D311yescharacteristic
259C5×Q8—44no—
260C5×D4—124no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—34no—
274C2×C4×C5—62no—
275C2×C4×C5—62no—
276C2×C4×C5—62no—
277C2×C4×C5—62no—
278C2×C4×C5—62no—
279C2×C4×C5—62no—
280C2×C4×C5—62no—
281C2×C4×C5—62no—
282C2×C4×C5—128no—
283C2×C4×C5C2×C4×D312yes—
284C2×C4×C5C2×C4×D312yes—
285C2×C4×C5C2×C4×D312yes—
286C22×Q8C2×C5×D311yescharacteristic
287C22×D4—31no—
288C23×C4—32no—
289#124—38no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4C2×C5×D311yescharacteristic
294C2×C22⋊C4—32no—
295C2×C22⋊C4—32no—
296C2×C22⋊C4—32no—
297C2×C22⋊C4—32no—
298C2×C22⋊C4—32no—
299C2×C42—31no—
300C2×C42—31no—
301C2×C42C2×C5×D311yescharacteristic
302#95—32no—
303#95—32no—
304#95—32no—
305#95—32no—
306#95C2×C5×D312yes—
307#95C2×C5×D312yes—
308C2×C3×C5#51811yescharacteristic
309C2×C3×C5#52012yes—
310C2×C3×C5H(3,𝔽4)11yescharacteristic
311C2×C3×C5#56212yes—
312C2×C3×C5#56311yescharacteristic
313C5×D3—44no—
314C23×C3C24×C511yescharacteristic
315C22×D3—22no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3C22×C4×C514yes—
322C3×Q8—44no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4C22×C4×C512yes—
336C2×C3×C4C22×C4×C512yes—
337C2×C3×C4C22×C4×C512yes—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3C22×C4×C514yes—
347D12—44no—
348C4×D3—48no—
349C22×C5—64no—
350C22×C5—64no—
351C22×C5—64no—
352C22×C5—122no—
353C22×C5—122no—
354C22×C5—122no—
355C22×C5—122no—
356C22×C5C22×C4×D311yescharacteristic
357C22×C5#100411yescharacteristic
358C22×C5#100411yescharacteristic
359C22×C5#100712yes—
360C22×C5#100912yes—
361C4×C5—22no—
362C4×C5—22no—
363C4×C5—22no—
364C4×C5—42no—
365C4×C5—42no—
366C4×C5—64no—
367C4×C5—122no—
368C4×C5—122no—
369C24—32no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8C4×C5×D314yescomplemented
373C2×D4—34no—
374C2×D4—121no—
375C2×D4—121no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—32no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4C22×C5×D312yes—
383C22×C4C22×C5×D312yes—
384C22×C4C22×C5×D311yescharacteristic
385C22×C4C22×C5×D311yescharacteristic
386C22×C4C22×C5×D311yescharacteristic
387C4⋊C4—41no—
388C4⋊C4—122no—
389C4⋊C4—122no—
390C4⋊C4—121no—
391C22⋊C4—38no—
392C22⋊C4—38no—
393C22⋊C4—122no—
394C22⋊C4—122no—
395C22⋊C4—122no—
396C42—34no—
397C42—41no—
398C42—121no—
399C3×C5#237911yescharacteristic, complemented, Hall
400C22×C3C23×C4×C511yescharacteristic
401C22×C3C5×H(5,𝔽2)11yescharacteristic
402C22×C3C5×H(5,𝔽2)11yescharacteristic
403C22×C3C5×H(5,𝔽2)12yes—
404C22×C3C5×2−1+412yes—
405C2×D3—24no—
406C2×D3—24no—
407C2×D3—24no—
408C2×D3—42no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—42no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4—42no—
417Dic3—24no—
418Dic3—42no—
419Dic3—42no—
420C2×C5—124no—
421C2×C5#548611yescharacteristic
422C2×C5#557412yes—
423C2×C5#564712yes—
424C2×C5#568311yescharacteristic
425C2×C5#573511yescharacteristic
426C23—34no—
427C23—62no—
428C23—62no—
429C23—62no—
430C23—62no—
431C23—62no—
432C23—62no—
433C23C23×C5×D311yescharacteristic
434Q8—44no—
435D4—124no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—34no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—128no—
458C2×C4C2×C4×C5×D312yes—
459C2×C4C2×C4×C5×D312yes—
460C2×C4C2×C4×C5×D312yes—
461C2×C3C5×#51811yescharacteristic
462C2×C3C5×#52012yes—
463C2×C3C5×H(3,𝔽4)11yescharacteristic
464C2×C3C5×#56212yes—
465C2×C3C5×#56311yescharacteristic
466D3—44no—
467C5#7649611yescharacteristic, direct product factor, Sylow
468C22—64no—
469C22—64no—
470C22—64no—
471C22—122no—
472C22—122no—
473C22—122no—
474C22—122no—
475C22C22×C4×C5×D311yescharacteristic
476C22C5×#100411yescharacteristic
477C22C5×#100411yescharacteristic
478C22C5×#100712yes—
479C22C5×#100912yes—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—42no—
484C4—42no—
485C4—64no—
486C4—122no—
487C4—122no—
488C3C5×#237911yescharacteristic, complemented, Sylow
489C2—124no—
490C2C5×#548611yescharacteristic
491C2C5×#557412yes—
492C2C5×#564712yes—
493C2C5×#568311yescharacteristic
494C2C5×#573511yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#391C2
socleC23×C3×C5C24
prime radicalC22×C5×D12C22
cosocleC23×C3C24×C5
hypercenterC5×#391D3
hypocenterC3C5×#2379
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#391

Complements

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

NH ≅ G/Ntype
2. C3×C5×#391C2semidirect
3. C5×#5229C2semidirect
4. C5×#4989C2semidirect
5. C5×#4940C2semidirect
6. C5×#4669C2semidirect
7. C5×#4668C2semidirect
9. C5×#4650C2semidirect
16. C5×#947C4semidirect
33. #76496C5direct (N × H)
36. C5×#391D3semidirect
46. C2×C3×C5×Q8C2×C4semidirect
66. C5×(C2×D3)⋊C4C2×C4semidirect
67. C5×(C2×D3)⋊C4C2×C4semidirect
72. C4×C5×Dic3C2×C4semidirect
74. C3×#391C2×C5semidirect
75. #5229C2×C5semidirect
76. #4989C2×C5semidirect
77. #4940C2×C5semidirect
78. #4669C2×C5semidirect
79. #4668C2×C5semidirect
81. #4650C2×C5semidirect
145. #947C4×C5semidirect
165. C2×C5×Q8C4×D3semidirect
194. #391C5×D3semidirect
224. C2×C3×Q8C2×C4×C5semidirect
244. (C2×D3)⋊C4C2×C4×C5semidirect
245. (C2×D3)⋊C4C2×C4×C5semidirect
250. C4×Dic3C2×C4×C5semidirect
372. C2×Q8C4×C5×D3semidirect
399. C3×C5#2379semidirect
467. C5#76496direct (N × H)
488. C3C5×#2379semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9472
C3×C5×#3912
C5×#52292
C5×#49892
C5×#49402
C5×#46692
C5×#46682
C5×#46652
C5×#46502
C5×#23793
#764965