Finite Group Databasebeta

← C5×#74014 #14330840

Subgroups — C5×#74014 #14330840

Order 1920.

1152 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
1C2×C5×#930C211yesmaximal, characteristic
2C3×C5×#380C211yesmaximal, characteristic, complemented
3C5×#5211C211yesmaximal, characteristic, complemented
4C5×#4990C212yesmaximal, complemented
5C5×#4989C212yesmaximal, complemented
6C5×#4957C212yesmaximal, complemented
7C5×#4956C212yesmaximal, complemented
8C5×#4665C212yesmaximal, complemented
9C5×#4653C212yesmaximal, complemented
10C5×#2377—31nomaximal, Hall
11C23×C3×C4×C5C2211yescharacteristic
12C22×C5×#67C2212yes—
13C22×C5×D12C2211yescharacteristic
14C22×C4×C5×D3C2212yes—
15C2×C3×C5×C4⋊C4C2212yes—
16C2×C3×C5×C22⋊C4C2211yescharacteristic
17C2×C3×C5×C22⋊C4C2211yescharacteristic
18C2×C5×#217C2212yes—
19C5×#930C418yescomplemented
20C2×C5×(C2×D3)⋊C4C2212yes—
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×(C2×D3)⋊C4C2212yes—
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×Dic3⋊C4C2212yes—
26C2×C4×C5×Dic3C2212yes—
27C3×C5×#95C2212yes—
28C5×C3⋊#95C2212yes—
29C5×C3⋊#95C2212yes—
30C5×C3⋊#95C2212yes—
31C5×C3⋊#95C2211yescharacteristic
32C5×C3⋊#95C2211yescharacteristic
33#74014C511yesmaximal, characteristic, direct product factor, Hall
34C2×C5×#121—31no—
35C5×#390—32no—
36C5×#388—32no—
37C5×#387—32no—
38C5×#386—32no—
39C5×#386—32no—
40C5×#385—32no—
41C5×#381—31no—
42C5×#380D311yescharacteristic, complemented
43C24×C3×C5C2311yescharacteristic
44C23×C5×D3C2312yes—
45C22×C3×C4×C5—42no—
46C22×C3×C4×C5C2×C414yescomplemented
47C22×C3×C4×C5C2312yes—
48C22×C3×C4×C5C2312yes—
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C2×C5×#67—42no—
52C2×C5×#67—42no—
53C2×C5×#67C2×C418yescomplemented
54C22×C5×Dic3C2312yes—
55C22×C5×Dic3C2312yes—
56C22×C5×Dic3C2312yes—
57C2×C5×D12—42no—
58C2×C5×D12C2×C414yes—
59C2×C4×C5×D3—42no—
60C2×C4×C5×D3—42no—
61C2×C4×C5×D3—42no—
62C3×C5×C4⋊C4—42no—
63C3×C5×C22⋊C4—41no—
64C3×C5×C22⋊C4—41no—
65C5×#217—42no—
66C5×(C2×D3)⋊C4—42no—
67C5×(C2×D3)⋊C4—42no—
68C5×(C2×D3)⋊C4—42no—
69C5×(C2×D3)⋊C4C2×C418yescomplemented
70C5×C4⋊Dic3C2×C414yescomplemented
71C5×Dic3⋊C4—42no—
72C4×C5×Dic3—42no—
73C2×#930C2×C511yescharacteristic
74C3×#380C2×C511yescharacteristic, complemented
75#5211C2×C511yescharacteristic, complemented
76#4990C2×C512yescomplemented
77#4989C2×C512yescomplemented
78#4957C2×C512yescomplemented
79#4956C2×C512yescomplemented
80#4665C2×C512yescomplemented
81#4653C2×C512yescomplemented
82C22×C5×D4—32no—
83C22×C5×D4—31no—
84C23×C4×C5—32no—
85C23×C4×C5C2×D311yescharacteristic
86C5×#121—38no—
87C2×C5×C4⋊C4—31no—
88C2×C5×C4⋊C4—32no—
89C2×C5×C4⋊C4C2×D312yes—
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×C5×C22⋊C4C2×D311yescharacteristic
96C2×C5×C22⋊C4C2×D311yescharacteristic
97C2×C42×C5—32no—
98C5×#95—31no—
99C5×#95—32no—
100C5×#95—31no—
101C5×#95—32no—
102C5×#95—32no—
103C5×#95C2×D312yes—
104#2377—31noSylow
105C23×C3×C5—22no—
106C23×C3×C5—21no—
107C23×C3×C5—22no—
108C23×C3×C5—21no—
109C23×C3×C5C22×C412yes—
110C23×C3×C5C2411yescharacteristic
111C22×C5×D3—22no—
112C22×C5×D3—22no—
113C22×C5×D3—22no—
114C22×C5×D3—22no—
115C22×C5×D3—22no—
116C22×C5×D3—22no—
117C22×C5×D3C22×C414yes—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—22no—
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×C5—44no—
127C2×C3×C4×C5C22×C412yes—
128C2×C3×C4×C5C22×C412yes—
129C5×#67—48no—
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×Dic3—22no—
138C2×C5×Dic3C22×C414yes—
139C5×D12—44no—
140C4×C5×D3—48no—
141C23×C3×C4C22×C511yescharacteristic
142C22×#67C22×C512yes—
143C22×D12C22×C511yescharacteristic
144C22×C4×D3C22×C512yes—
145C2×C3×C4⋊C4C22×C512yes—
146C2×C3×C22⋊C4C22×C511yescharacteristic
147C2×C3×C22⋊C4C22×C511yescharacteristic
148C2×#217C22×C512yes—
149#930C4×C518yescomplemented
150C2×(C2×D3)⋊C4C22×C512yes—
151C2×(C2×D3)⋊C4C22×C512yes—
152C2×(C2×D3)⋊C4C22×C512yes—
153C2×(C2×D3)⋊C4C22×C512yes—
154C2×C4⋊Dic3C22×C511yescharacteristic
155C2×Dic3⋊C4C22×C512yes—
156C2×C4×Dic3C22×C512yes—
157C3×#95C22×C512yes—
158C3⋊#95C22×C512yes—
159C3⋊#95C22×C512yes—
160C3⋊#95C22×C512yes—
161C3⋊#95C22×C511yescharacteristic
162C3⋊#95C22×C511yescharacteristic
163C24×C5—32no—
164C24×C5C22×D311yescharacteristic
165C2×C5×D4—38no—
166C2×C5×D4—34no—
167C2×C5×D4—122no—
168C2×C5×D4—122no—
169C2×C5×D4—122no—
170C22×C4×C5—32no—
171C22×C4×C5—32no—
172C22×C4×C5—32no—
173C22×C4×C5—42no—
174C22×C4×C5—122no—
175C22×C4×C5—122no—
176C22×C4×C5—122no—
177C22×C4×C5C4×D314yescomplemented
178C22×C4×C5C22×D312yes—
179C22×C4×C5C22×D312yes—
180C22×C4×C5C22×D311yescharacteristic
181C22×C4×C5C22×D311yescharacteristic
182C5×C4⋊C4—34no—
183C5×C4⋊C4—42no—
184C5×C4⋊C4—122no—
185C5×C22⋊C4—38no—
186C5×C22⋊C4—41no—
187C5×C22⋊C4—41no—
188C5×C22⋊C4—122no—
189C5×C22⋊C4—122no—
190C5×C22⋊C4—122no—
191C5×C22⋊C4—122no—
192C42×C5—122no—
193C2×#121—31no—
194#390—32no—
195#388—32no—
196#387—32no—
197#386—32no—
198#386—32no—
199#385—32no—
200#381—31no—
201#380C5×D311yescharacteristic, complemented
202C22×C3×C5—22no—
203C22×C3×C5—22no—
204C22×C3×C5—22no—
205C22×C3×C5—42no—
206C22×C3×C5—42no—
207C22×C3×C5C23×C411yescharacteristic
208C22×C3×C5H(5,𝔽2)12yes—
209C22×C3×C5H(5,𝔽2)11yescharacteristic
210C22×C3×C5H(5,𝔽2)12yes—
211C22×C3×C52−1+411yescharacteristic
212C2×C5×D3—24no—
213C2×C5×D3—24no—
214C2×C5×D3—24no—
215C2×C5×D3—42no—
216C2×C5×D3—42no—
217C2×C5×D3—42no—
218C2×C5×D3—42no—
219C3×C4×C5—22no—
220C3×C4×C5—22no—
221C3×C4×C5—42no—
222C3×C4×C5—42no—
223C5×Dic3—24no—
224C5×Dic3—42no—
225C5×Dic3—42no—
226C24×C3C23×C511yescharacteristic
227C23×D3C23×C512yes—
228C22×C3×C4—42no—
229C22×C3×C4C2×C4×C514yescomplemented
230C22×C3×C4C23×C512yes—
231C22×C3×C4C23×C512yes—
232C22×C3×C4C23×C511yescharacteristic
233C22×C3×C4C23×C511yescharacteristic
234C2×#67—42no—
235C2×#67—42no—
236C2×#67C2×C4×C518yescomplemented
237C22×Dic3C23×C512yes—
238C22×Dic3C23×C512yes—
239C22×Dic3C23×C512yes—
240C2×D12—42no—
241C2×D12C2×C4×C514yes—
242C2×C4×D3—42no—
243C2×C4×D3—42no—
244C2×C4×D3—42no—
245C3×C4⋊C4—42no—
246C3×C22⋊C4—41no—
247C3×C22⋊C4—41no—
248#217—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4C2×C4×C518yescomplemented
253C4⋊Dic3C2×C4×C514yescomplemented
254Dic3⋊C4—42no—
255C4×Dic3—42no—
256C23×C5—21no—
257C23×C5—22no—
258C23×C5—21no—
259C23×C5—22no—
260C23×C5—34no—
261C23×C5—62no—
262C23×C5—62no—
263C23×C5—62no—
264C23×C5—62no—
265C23×C5—62no—
266C23×C5—62no—
267C23×C5C2×C4×D312yes—
268C23×C5C23×D311yescharacteristic
269C5×D4—124no—
270C5×D4—128no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—22no—
279C2×C4×C5—34no—
280C2×C4×C5—44no—
281C2×C4×C5—62no—
282C2×C4×C5—62no—
283C2×C4×C5—62no—
284C2×C4×C5—62no—
285C2×C4×C5—62no—
286C2×C4×C5—62no—
287C2×C4×C5—62no—
288C2×C4×C5—62no—
289C2×C4×C5—128no—
290C2×C4×C5C2×C4×D312yes—
291C2×C4×C5C2×C4×D312yes—
292C22×D4—32no—
293C22×D4—31no—
294C23×C4—32no—
295C23×C4C2×C5×D311yescharacteristic
296#121—38no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4C2×C5×D312yes—
300C2×C22⋊C4—32no—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—32no—
303C2×C22⋊C4—32no—
304C2×C22⋊C4—32no—
305C2×C22⋊C4C2×C5×D311yescharacteristic
306C2×C22⋊C4C2×C5×D311yescharacteristic
307C2×C42—32no—
308#95—31no—
309#95—32no—
310#95—31no—
311#95—32no—
312#95—32no—
313#95C2×C5×D312yes—
314C2×C3×C5—42no—
315C2×C3×C5#51811yescharacteristic
316C2×C3×C5#51811yescharacteristic
317C2×C3×C5#52011yescharacteristic
318C2×C3×C5#56012yes—
319C2×C3×C5#56212yes—
320C5×D3—44no—
321C23×C3—22no—
322C23×C3—21no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3C22×C4×C512yes—
326C23×C3C24×C511yescharacteristic
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—22no—
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3C22×C4×C514yes—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—44no—
343C2×C3×C4C22×C4×C512yes—
344C2×C3×C4C22×C4×C512yes—
345#67—48no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3C22×C4×C514yes—
355D12—44no—
356C4×D3—48no—
357C22×C5—22no—
358C22×C5—22no—
359C22×C5—22no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5—64no—
363C22×C5—64no—
364C22×C5—64no—
365C22×C5—122no—
366C22×C5—122no—
367C22×C5—122no—
368C22×C5—122no—
369C22×C5C22×C4×D311yescharacteristic
370C22×C5#100412yes—
371C22×C5#100411yescharacteristic
372C22×C5#100711yescharacteristic
373C22×C5#100912yes—
374C4×C5—22no—
375C4×C5—22no—
376C4×C5—42no—
377C4×C5—42no—
378C4×C5—64no—
379C4×C5—122no—
380C4×C5—122no—
381C24—32no—
382C24C22×C5×D311yescharacteristic
383C2×D4—38no—
384C2×D4—34no—
385C2×D4—122no—
386C2×D4—122no—
387C2×D4—122no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—32no—
391C22×C4—42no—
392C22×C4—122no—
393C22×C4—122no—
394C22×C4—122no—
395C22×C4C4×C5×D314yescomplemented
396C22×C4C22×C5×D312yes—
397C22×C4C22×C5×D312yes—
398C22×C4C22×C5×D311yescharacteristic
399C22×C4C22×C5×D311yescharacteristic
400C4⋊C4—34no—
401C4⋊C4—42no—
402C4⋊C4—122no—
403C22⋊C4—38no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C22⋊C4—122no—
409C22⋊C4—122no—
410C42—122no—
411C3×C5#237711yescharacteristic, complemented, Hall
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3C23×C4×C511yescharacteristic
418C22×C3C5×H(5,𝔽2)12yes—
419C22×C3C5×H(5,𝔽2)11yescharacteristic
420C22×C3C5×H(5,𝔽2)12yes—
421C22×C3C5×2−1+411yescharacteristic
422C2×D3—24no—
423C2×D3—24no—
424C2×D3—24no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—42no—
432C3×C4—42no—
433Dic3—24no—
434Dic3—42no—
435Dic3—42no—
436C2×C5—42no—
437C2×C5—124no—
438C2×C5#548611yescharacteristic
439C2×C5#550411yescharacteristic
440C2×C5#552111yescharacteristic
441C2×C5#562012yes—
442C2×C5#564712yes—
443C23—21no—
444C23—22no—
445C23—21no—
446C23—22no—
447C23—34no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23C2×C4×C5×D312yes—
455C23C23×C5×D311yescharacteristic
456D4—124no—
457D4—128no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—34no—
467C2×C4—44no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—128no—
477C2×C4C2×C4×C5×D312yes—
478C2×C4C2×C4×C5×D312yes—
479C2×C3—42no—
480C2×C3C5×#51811yescharacteristic
481C2×C3C5×#51811yescharacteristic
482C2×C3C5×#52011yescharacteristic
483C2×C3C5×#56012yes—
484C2×C3C5×#56212yes—
485D3—44no—
486C5#7401411yescharacteristic, direct product factor, Sylow
487C22—22no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—64no—
493C22—64no—
494C22—64no—
495C22—122no—
496C22—122no—
497C22—122no—
498C22—122no—
499C22C22×C4×C5×D311yescharacteristic
500C22C5×#100412yes—
501C22C5×#100411yescharacteristic
502C22C5×#100711yescharacteristic
503C22C5×#100912yes—
504C4—22no—
505C4—22no—
506C4—42no—
507C4—42no—
508C4—64no—
509C4—122no—
510C4—122no—
511C3C5×#237711yescharacteristic, complemented, Sylow
512C2—42no—
513C2—124no—
514C2C5×#548611yescharacteristic
515C2C5×#550411yescharacteristic
516C2C5×#552111yescharacteristic
517C2C5×#562012yes—
518C2C5×#564712yes—
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#380C2
socleC23×C3×C5C24
prime radicalC2×C5×#930C2
cosocleC23×C3C24×C5
hypercenterC5×#380D3
hypocenterC3C5×#2377
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#380

Complements

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

NH ≅ G/Ntype
2. C3×C5×#380C2semidirect
3. C5×#5211C2semidirect
4. C5×#4990C2semidirect
5. C5×#4989C2semidirect
6. C5×#4957C2semidirect
7. C5×#4956C2semidirect
8. C5×#4665C2semidirect
9. C5×#4653C2semidirect
19. C5×#930C4semidirect
33. #74014C5direct (N × H)
42. C5×#380D3semidirect
46. C22×C3×C4×C5C2×C4semidirect
53. C2×C5×#67C2×C4semidirect
69. C5×(C2×D3)⋊C4C2×C4semidirect
70. C5×C4⋊Dic3C2×C4semidirect
74. C3×#380C2×C5semidirect
75. #5211C2×C5semidirect
76. #4990C2×C5semidirect
77. #4989C2×C5semidirect
78. #4957C2×C5semidirect
79. #4956C2×C5semidirect
80. #4665C2×C5semidirect
81. #4653C2×C5semidirect
149. #930C4×C5semidirect
177. C22×C4×C5C4×D3semidirect
201. #380C5×D3semidirect
229. C22×C3×C4C2×C4×C5semidirect
236. C2×#67C2×C4×C5semidirect
252. (C2×D3)⋊C4C2×C4×C5semidirect
253. C4⋊Dic3C2×C4×C5semidirect
395. C22×C4C4×C5×D3semidirect
411. C3×C5#2377semidirect
486. C5#74014direct (N × H)
511. C3C5×#2377semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9302
C3×C5×#3802
C5×#52112
C5×#49902
C5×#49892
C5×#49572
C5×#49562
C5×#46652
C5×#46532
C5×#23773
#740145