Finite Group Databasebeta

← C5×#73880 #14326668

Subgroups — C5×#73880 #14326668

Order 1920.

1096 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
1C22×C5×Dic3⋊C4C211yesmaximal, characteristic
2C3×C5×#380C211yesmaximal, characteristic
3C4×C5×#217C211yesmaximal, characteristic
4C5×#4978C212yesmaximal, complemented
5C5×#4976C212yesmaximal, complemented
6C5×#4942C211yesmaximal, characteristic
7C5×#4942C211yesmaximal, characteristic
8C5×Dic3×C22⋊C4C211yesmaximal, characteristic
9C5×Dic3×C22⋊C4C211yesmaximal, characteristic
10C5×#4652C212yesmaximal, complemented
11C5×#4649C212yesmaximal, complemented
12C5×#2345—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×C5×Dic3C2211yescharacteristic
15C23×C5×Dic3C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C22⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×C5×#217C2211yescharacteristic
20C2×C5×#217C2211yescharacteristic
21C2×C5×Dic3⋊C4C418yescomplemented
22C2×C5×Dic3⋊C4C2212yes—
23C2×C5×Dic3⋊C4C2212yes—
24C2×C5×Dic3⋊C4C2211yescharacteristic
25C2×C5×Dic3⋊C4C2211yescharacteristic
26C2×C5×Dic3⋊C4C2211yescharacteristic
27C2×C5×Dic3⋊C4C2211yescharacteristic
28C2×C4×C5×Dic3C2212yes—
29C2×C4×C5×Dic3C2212yes—
30C2×C4×C5×Dic3C2212yes—
31C3×C5×#95C2212yes—
32C5×C3⋊#95C2212yes—
33C5×C3⋊#95C2212yes—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37#73880C511yesmaximal, characteristic, direct product factor, Hall
38C22×C5×C4⋊C4—31no—
39C5×#389—32no—
40C5×#387—32no—
41C5×#387—32no—
42C5×#380—31no—
43C5×#380—31no—
44C5×#380D311yescharacteristic
45C4×C5×C4⋊C4—32no—
46C4×C5×C22⋊C4—31no—
47C4×C5×C22⋊C4—31no—
48C4×C5×C22⋊C4—31no—
49C24×C3×C5C2311yescharacteristic
50C22×C3×C4×C5—42no—
51C22×C3×C4×C5C2×C414yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C5×Dic3—24no—
57C22×C5×Dic3—42no—
58C22×C5×Dic3C2×C414yes—
59C22×C5×Dic3C2×C414yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2311yescharacteristic
64C22×C5×Dic3C2312yes—
65C22×C5×Dic3C2311yescharacteristic
66C3×C5×C4⋊C4—42no—
67C3×C5×C22⋊C4—22no—
68C3×C5×C22⋊C4—22no—
69C5×#217—22no—
70C5×#217Q814yes—
71C5×Dic3⋊C4—24no—
72C5×Dic3⋊C4—42no—
73C5×Dic3⋊C4C2×C414yes—
74C5×Dic3⋊C4C2×C414yes—
75C5×Dic3⋊C4C2×C414yes—
76C5×Dic3⋊C4C2×C414yes—
77C4×C5×Dic3—24no—
78C4×C5×Dic3—42no—
79C4×C5×Dic3—42no—
80C22×Dic3⋊C4C2×C511yescharacteristic
81C3×#380C2×C511yescharacteristic
82C4×#217C2×C511yescharacteristic
83#4978C2×C512yescomplemented
84#4976C2×C512yescomplemented
85#4942C2×C511yescharacteristic
86#4942C2×C511yescharacteristic
87Dic3×C22⋊C4C2×C511yescharacteristic
88Dic3×C22⋊C4C2×C511yescharacteristic
89#4652C2×C512yescomplemented
90#4649C2×C512yescomplemented
91C23×C4×C5—31no—
92C23×C4×C5—31no—
93C23×C4×C5C2×D311yescharacteristic
94C2×C5×C4⋊C4—38no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C4⋊C4—31no—
97C2×C5×C4⋊C4—31no—
98C2×C5×C4⋊C4—31no—
99C2×C5×C4⋊C4—32no—
100C2×C5×C4⋊C4—32no—
101C2×C5×C4⋊C4C2×D312yes—
102C2×C5×C22⋊C4—31no—
103C2×C5×C22⋊C4—31no—
104C2×C5×C22⋊C4C2×D311yescharacteristic
105C2×C5×C22⋊C4C2×D311yescharacteristic
106C2×C42×C5—32no—
107C2×C42×C5—32no—
108C2×C42×C5—32no—
109C5×#95—32no—
110C5×#95—32no—
111C5×#95—32no—
112C5×#95—32no—
113C5×#95—32no—
114C5×#95C2×D312yes—
115#2345—31noSylow
116C23×C3×C5—22no—
117C23×C3×C5—22no—
118C23×C3×C5C22×C412yes—
119C23×C3×C5C2×Q812yes—
120C23×C3×C5#4112yes—
121C23×C3×C5C2411yescharacteristic
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—44no—
131C2×C3×C4×C5C22×C412yes—
132C2×C3×C4×C5C22×C412yes—
133C2×C5×Dic3—22no—
134C2×C5×Dic3—22no—
135C2×C5×Dic3—22no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—22no—
138C2×C5×Dic3—28no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—44no—
143C2×C5×Dic3C22×C412yes—
144C2×C5×Dic3C22×C412yes—
145C2×C5×Dic3C22×C412yes—
146C2×C5×Dic3C22×C412yes—
147C2×C5×Dic3C2×Q814yes—
148C2×C5×Dic3#4114yes—
149C23×C3×C4C22×C511yescharacteristic
150C23×Dic3C22×C511yescharacteristic
151C23×Dic3C22×C511yescharacteristic
152C2×C3×C4⋊C4C22×C512yes—
153C2×C3×C22⋊C4C22×C511yescharacteristic
154C2×C3×C22⋊C4C22×C511yescharacteristic
155C2×#217C22×C511yescharacteristic
156C2×#217C22×C511yescharacteristic
157C2×Dic3⋊C4C4×C518yescomplemented
158C2×Dic3⋊C4C22×C512yes—
159C2×Dic3⋊C4C22×C512yes—
160C2×Dic3⋊C4C22×C511yescharacteristic
161C2×Dic3⋊C4C22×C511yescharacteristic
162C2×Dic3⋊C4C22×C511yescharacteristic
163C2×Dic3⋊C4C22×C511yescharacteristic
164C2×C4×Dic3C22×C512yes—
165C2×C4×Dic3C22×C512yes—
166C2×C4×Dic3C22×C512yes—
167C3×#95C22×C512yes—
168C3⋊#95C22×C512yes—
169C3⋊#95C22×C512yes—
170C3⋊#95C22×C512yes—
171C3⋊#95C22×C512yes—
172C3⋊#95C22×C512yes—
173C24×C5C22×D311yescharacteristic
174C22×C4×C5—31no—
175C22×C4×C5—34no—
176C22×C4×C5—32no—
177C22×C4×C5—31no—
178C22×C4×C5—31no—
179C22×C4×C5—31no—
180C22×C4×C5—34no—
181C22×C4×C5—32no—
182C22×C4×C5—42no—
183C22×C4×C5—64no—
184C22×C4×C5—122no—
185C22×C4×C5C4×D314yes—
186C22×C4×C5C22×D312yes—
187C22×C4×C5C22×D312yes—
188C22×C4×C5C22×D311yescharacteristic
189C22×C4×C5C22×D311yescharacteristic
190C5×C4⋊C4—34no—
191C5×C4⋊C4—34no—
192C5×C4⋊C4—34no—
193C5×C4⋊C4—34no—
194C5×C4⋊C4—42no—
195C5×C4⋊C4—64no—
196C5×C4⋊C4—122no—
197C5×C22⋊C4—22no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—34no—
200C5×C22⋊C4—62no—
201C42×C5—64no—
202C42×C5—122no—
203C42×C5—122no—
204C22×C4⋊C4—31no—
205#389—32no—
206#387—32no—
207#387—32no—
208#380—31no—
209#380—31no—
210#380C5×D311yescharacteristic
211C4×C4⋊C4—32no—
212C4×C22⋊C4—31no—
213C4×C22⋊C4—31no—
214C4×C22⋊C4—31no—
215C22×C3×C5—22no—
216C22×C3×C5—22no—
217C22×C3×C5—24no—
218C22×C3×C5—24no—
219C22×C3×C5C4×Q814yes—
220C22×C3×C5C23×C411yescharacteristic
221C22×C3×C5C22×Q811yescharacteristic
222C22×C3×C5C2×#4111yescharacteristic
223C22×C3×C5C2×#4112yes—
224C22×C3×C5H(5,𝔽2)12yes—
225C3×C4×C5—22no—
226C3×C4×C5—22no—
227C3×C4×C5—42no—
228C3×C4×C5—42no—
229C5×Dic3—22no—
230C5×Dic3—22no—
231C5×Dic3—22no—
232C5×Dic3—24no—
233C5×Dic3—42no—
234C5×Dic3#11714yes—
235C24×C3C23×C511yescharacteristic
236C22×C3×C4—42no—
237C22×C3×C4C2×C4×C514yes—
238C22×C3×C4C23×C512yes—
239C22×C3×C4C23×C512yes—
240C22×C3×C4C23×C511yescharacteristic
241C22×C3×C4C23×C511yescharacteristic
242C22×Dic3—24no—
243C22×Dic3—42no—
244C22×Dic3C2×C4×C514yes—
245C22×Dic3C2×C4×C514yes—
246C22×Dic3C23×C512yes—
247C22×Dic3C23×C511yescharacteristic
248C22×Dic3C23×C511yescharacteristic
249C22×Dic3C23×C511yescharacteristic
250C22×Dic3C23×C512yes—
251C22×Dic3C23×C511yescharacteristic
252C3×C4⋊C4—42no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255#217—22no—
256#217C5×Q814yes—
257Dic3⋊C4—24no—
258Dic3⋊C4—42no—
259Dic3⋊C4C2×C4×C514yes—
260Dic3⋊C4C2×C4×C514yes—
261Dic3⋊C4C2×C4×C514yes—
262Dic3⋊C4C2×C4×C514yes—
263C4×Dic3—24no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C23×C5—22no—
267C23×C5—22no—
268C23×C5C2×C4×D312yes—
269C23×C5#23712yes—
270C23×C5D3×Q812yes—
271C23×C5C23×D311yescharacteristic
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—22no—
280C2×C4×C5—32no—
281C2×C4×C5—34no—
282C2×C4×C5—32no—
283C2×C4×C5—32no—
284C2×C4×C5—32no—
285C2×C4×C5—34no—
286C2×C4×C5—44no—
287C2×C4×C5—68no—
288C2×C4×C5—62no—
289C2×C4×C5—62no—
290C2×C4×C5—62no—
291C2×C4×C5—62no—
292C2×C4×C5—62no—
293C2×C4×C5—62no—
294C2×C4×C5—62no—
295C2×C4×C5—62no—
296C2×C4×C5—124no—
297C2×C4×C5C2×C4×D312yes—
298C2×C4×C5C2×C4×D312yes—
299C23×C4—31no—
300C23×C4—31no—
301C23×C4C2×C5×D311yescharacteristic
302C2×C4⋊C4—38no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—31no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—32no—
308C2×C4⋊C4—32no—
309C2×C4⋊C4C2×C5×D312yes—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—31no—
312C2×C22⋊C4C2×C5×D311yescharacteristic
313C2×C22⋊C4C2×C5×D311yescharacteristic
314C2×C42—32no—
315C2×C42—32no—
316C2×C42—32no—
317#95—32no—
318#95—32no—
319#95—32no—
320#95—32no—
321#95—32no—
322#95C2×C5×D312yes—
323C2×C3×C5—24no—
324C2×C3×C5C2×#11711yescharacteristic
325C2×C3×C5C2×C4×Q811yescharacteristic
326C2×C3×C5#51811yescharacteristic
327C2×C3×C5#55312yes—
328C2×C3×C5#55412yes—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3C22×C4×C512yes—
332C23×C3C2×C5×Q812yes—
333C23×C3C5×#4112yes—
334C23×C3C24×C511yescharacteristic
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—22no—
343C2×C3×C4—44no—
344C2×C3×C4C22×C4×C512yes—
345C2×C3×C4C22×C4×C512yes—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—22no—
350C2×Dic3—22no—
351C2×Dic3—28no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—44no—
356C2×Dic3C22×C4×C512yes—
357C2×Dic3C22×C4×C512yes—
358C2×Dic3C22×C4×C512yes—
359C2×Dic3C22×C4×C512yes—
360C2×Dic3C2×C5×Q814yes—
361C2×Dic3C5×#4114yes—
362C22×C5—24no—
363C22×C5—22no—
364C22×C5—22no—
365C22×C5—24no—
366C22×C5#88714yes—
367C22×C5C22×C4×D311yescharacteristic
368C22×C5C2×#23711yescharacteristic
369C22×C5#100412yes—
370C22×C5C2×D3×Q811yescharacteristic
371C22×C5D3×#4112yes—
372C4×C5—22no—
373C4×C5—22no—
374C4×C5—34no—
375C4×C5—42no—
376C4×C5—42no—
377C4×C5—64no—
378C4×C5—62no—
379C4×C5—62no—
380C4×C5—62no—
381C4×C5—122no—
382C24C22×C5×D311yescharacteristic
383C22×C4—31no—
384C22×C4—34no—
385C22×C4—31no—
386C22×C4—32no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—34no—
390C22×C4—32no—
391C22×C4—42no—
392C22×C4—64no—
393C22×C4—122no—
394C22×C4C4×C5×D314yes—
395C22×C4C22×C5×D312yes—
396C22×C4C22×C5×D312yes—
397C22×C4C22×C5×D311yescharacteristic
398C22×C4C22×C5×D311yescharacteristic
399C4⋊C4—34no—
400C4⋊C4—34no—
401C4⋊C4—34no—
402C4⋊C4—34no—
403C4⋊C4—42no—
404C4⋊C4—64no—
405C4⋊C4—122no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—34no—
409C22⋊C4—62no—
410C42—64no—
411C42—122no—
412C42—122no—
413C3×C5#234511yescharacteristic, complemented, Hall
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—24no—
417C22×C3—24no—
418C22×C3C4×C5×Q814yes—
419C22×C3C23×C4×C511yescharacteristic
420C22×C3C22×C5×Q811yescharacteristic
421C22×C3C2×C5×#4111yescharacteristic
422C22×C3C2×C5×#4112yes—
423C22×C3C5×H(5,𝔽2)12yes—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—42no—
427C3×C4—42no—
428Dic3—22no—
429Dic3—22no—
430Dic3—22no—
431Dic3—24no—
432Dic3—42no—
433Dic3C5×#11714yes—
434C2×C5—24no—
435C2×C5#548611yescharacteristic
436C2×C5C2×#88711yescharacteristic
437C2×C5D3×#11711yescharacteristic
438C2×C5#560112yes—
439C2×C5#563612yes—
440C23—22no—
441C23—22no—
442C23C2×C4×C5×D312yes—
443C23C5×#23712yes—
444C23C5×D3×Q812yes—
445C23C23×C5×D311yescharacteristic
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—32no—
455C2×C4—34no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—32no—
459C2×C4—34no—
460C2×C4—44no—
461C2×C4—68no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—124no—
471C2×C4C2×C4×C5×D312yes—
472C2×C4C2×C4×C5×D312yes—
473C2×C3—24no—
474C2×C3C2×C5×#11711yescharacteristic
475C2×C3C2×C4×C5×Q811yescharacteristic
476C2×C3C5×#51811yescharacteristic
477C2×C3C5×#55312yes—
478C2×C3C5×#55412yes—
479C5#7388011yescharacteristic, direct product factor, Sylow
480C22—22no—
481C22—22no—
482C22—24no—
483C22—24no—
484C22C5×#88714yes—
485C22C22×C4×C5×D311yescharacteristic
486C22C2×C5×#23711yescharacteristic
487C22C5×#100412yes—
488C22C2×C5×D3×Q811yescharacteristic
489C22C5×D3×#4112yes—
490C4—22no—
491C4—22no—
492C4—34no—
493C4—42no—
494C4—42no—
495C4—62no—
496C4—62no—
497C4—62no—
498C4—64no—
499C4—122no—
500C3C5×#234511yescharacteristic, complemented, Sylow
501C2—24no—
502C2C5×#548611yescharacteristic
503C2C2×C5×#88711yescharacteristic
504C2C5×D3×#11711yescharacteristic
505C2C5×#560112yes—
506C2C5×#563612yes—
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#380C2
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
cosocleC23×C3C24×C5
hypercenterC5×#380D3
hypocenterC3C5×#2345
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#380

Complements

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

NH ≅ G/Ntype
4. C5×#4978C2semidirect
5. C5×#4976C2semidirect
10. C5×#4652C2semidirect
11. C5×#4649C2semidirect
21. C2×C5×Dic3⋊C4C4semidirect
37. #73880C5direct (N × H)
83. #4978C2×C5semidirect
84. #4976C2×C5semidirect
89. #4652C2×C5semidirect
90. #4649C2×C5semidirect
157. C2×Dic3⋊C4C4×C5semidirect
413. C3×C5#2345semidirect
479. C5#73880direct (N × H)
500. C3C5×#2345semidirect

Maximal subgroups

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

subgroupindex
C22×C5×Dic3⋊C42
C3×C5×#3802
C4×C5×#2172
C5×#49782
C5×#49762
C5×#49422
C5×Dic3×C22⋊C42
C5×#46522
C5×#46492
C5×#23453
#738805