Finite Group Databasebeta

← C5×#73463 #14328002

Subgroups — C5×#73463 #14328002

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×#67C211yesmaximal, characteristic
2C3×C4×C5×C22⋊C4C211yesmaximal, characteristic, complemented
3C5×#5211C211yesmaximal, characteristic, complemented
4C5×#4957C212yesmaximal
5C5×#4945C212yesmaximal, complemented
6C5×#4940C212yesmaximal, complemented
7C4×C5×Dic3⋊C4C212yesmaximal, complemented
8C5×#4666C211yesmaximal, characteristic, complemented
9C5×#4664C211yesmaximal, characteristic, complemented
10C5×#4650C211yesmaximal, characteristic, complemented
11C5×#4649C211yesmaximal, characteristic, complemented
12C5×#2357—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×#67C2211yescharacteristic
15C22×C4×C5×D3C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C3×C42×C5C2212yes—
18C2×C5×#217C2212yes—
19C2×C5×#217C2211yescharacteristic
20C4×C5×#67C418yescomplemented
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×(C2×D3)⋊C4C2211yescharacteristic
24C2×C5×Dic3⋊C4C2212yes—
25C2×C5×Dic3⋊C4C2212yes—
26C2×C5×Dic3⋊C4C2211yescharacteristic
27C2×C4×C5×Dic3C2212yes—
28C2×C4×C5×Dic3C2212yes—
29C2×C4×C5×Dic3C2211yescharacteristic
30C3×C5×#95C2211yescharacteristic
31C3×C5×#95C2211yescharacteristic
32C5×C3⋊#95C2211yescharacteristic
33C5×C3⋊#95C2211yescharacteristic
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37#73463C511yesmaximal, characteristic, direct product factor, Hall
38C2×C4×C5×D4—31no—
39C5×#390—32no—
40C5×#388—32no—
41C5×#388—32no—
42C5×#382—31no—
43C5×#381—31no—
44C5×#381—31no—
45C4×C5×C4⋊C4—31no—
46C4×C5×C4⋊C4—32no—
47C4×C5×C22⋊C4—31no—
48C4×C5×C22⋊C4D311yescharacteristic, complemented
49C24×C3×C5C2311yescharacteristic
50C23×C5×D3C2311yescharacteristic
51C22×C3×C4×C5—42no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2312yes—
57C2×C5×#67—22no—
58C2×C5×#67—22no—
59C2×C5×#67C2×C414yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2311yescharacteristic
63C22×C5×Dic3C2311yescharacteristic
64C22×C5×Dic3C2311yescharacteristic
65C2×C4×C5×D3—42no—
66C2×C4×C5×D3C2×C414yes—
67C3×C5×C22⋊C4—24no—
68C3×C42×C5—42no—
69C5×#217—42no—
70C5×#217C2×C414yes—
71C5×(C2×D3)⋊C4—24no—
72C5×(C2×D3)⋊C4—42no—
73C5×(C2×D3)⋊C4C2×C414yes—
74C5×Dic3⋊C4—24no—
75C5×Dic3⋊C4—24no—
76C5×Dic3⋊C4C2×C414yes—
77C4×C5×Dic3—24no—
78C4×C5×Dic3—42no—
79C4×C5×Dic3C2×C414yes—
80C2×C4×#67C2×C511yescharacteristic
81C3×C4×C22⋊C4C2×C511yescharacteristic, complemented
82#5211C2×C511yescharacteristic, complemented
83#4957C2×C512yes—
84#4945C2×C512yescomplemented
85#4940C2×C512yescomplemented
86C4×Dic3⋊C4C2×C512yescomplemented
87#4666C2×C511yescharacteristic, complemented
88#4664C2×C511yescharacteristic, complemented
89#4650C2×C511yescharacteristic, complemented
90#4649C2×C511yescharacteristic, complemented
91C22×C5×D4—31no—
92C23×C4×C5—31no—
93C23×C4×C5C2×D311yescharacteristic
94C4×C5×D4—38no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C4⋊C4—32no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C22⋊C4—31no—
99C2×C5×C22⋊C4—31no—
100C2×C5×C22⋊C4—32no—
101C2×C5×C22⋊C4—32no—
102C2×C5×C22⋊C4—32no—
103C2×C5×C22⋊C4C2×D312yes—
104C2×C42×C5—31no—
105C2×C42×C5—32no—
106C2×C42×C5—32no—
107C2×C42×C5C2×D312yes—
108C5×#95—31no—
109C5×#95—31no—
110C5×#95—32no—
111C5×#95—32no—
112C5×#95—32no—
113C5×#95C2×D311yescharacteristic
114C5×#95C2×D311yescharacteristic
115#2357—31noSylow
116C23×C3×C5—22no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5—22no—
120C23×C3×C5C22×C412yes—
121C23×C3×C5C2411yescharacteristic
122C22×C5×D3—22no—
123C22×C5×D3—21no—
124C22×C5×D3—21no—
125C22×C5×D3—22no—
126C22×C5×D3C22×C412yes—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—44no—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5#4114yes—
138C5×#67—28no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3C22×C412yes—
148C2×C5×Dic3C22×C412yes—
149C2×C5×Dic3C22×C412yes—
150C2×C5×Dic3#4112yes—
151C2×C5×Dic3#4112yes—
152C4×C5×D3—44no—
153C23×C3×C4C22×C511yescharacteristic
154C22×#67C22×C511yescharacteristic
155C22×C4×D3C22×C511yescharacteristic
156C2×C3×C22⋊C4C22×C512yes—
157C2×C3×C42C22×C512yes—
158C2×#217C22×C512yes—
159C2×#217C22×C511yescharacteristic
160C4×#67C4×C518yescomplemented
161C2×(C2×D3)⋊C4C22×C512yes—
162C2×(C2×D3)⋊C4C22×C512yes—
163C2×(C2×D3)⋊C4C22×C511yescharacteristic
164C2×Dic3⋊C4C22×C512yes—
165C2×Dic3⋊C4C22×C512yes—
166C2×Dic3⋊C4C22×C511yescharacteristic
167C2×C4×Dic3C22×C512yes—
168C2×C4×Dic3C22×C512yes—
169C2×C4×Dic3C22×C511yescharacteristic
170C3×#95C22×C511yescharacteristic
171C3×#95C22×C511yescharacteristic
172C3⋊#95C22×C511yescharacteristic
173C3⋊#95C22×C511yescharacteristic
174C3⋊#95C22×C512yes—
175C3⋊#95C22×C512yes—
176C3⋊#95C22×C512yes—
177C24×C5—31no—
178C24×C5C22×D311yescharacteristic
179C2×C5×D4—34no—
180C2×C5×D4—62no—
181C2×C5×D4—62no—
182C22×C4×C5—31no—
183C22×C4×C5—31no—
184C22×C4×C5—31no—
185C22×C4×C5—34no—
186C22×C4×C5—32no—
187C22×C4×C5—32no—
188C22×C4×C5—42no—
189C22×C4×C5—122no—
190C22×C4×C5C4×D314yes—
191C22×C4×C5C22×D312yes—
192C22×C4×C5C22×D311yescharacteristic
193C22×C4×C5C22×D311yescharacteristic
194C22×C4×C5C22×D312yes—
195C5×C4⋊C4—34no—
196C5×C4⋊C4—64no—
197C5×C4⋊C4—64no—
198C5×C22⋊C4—24no—
199C5×C22⋊C4—34no—
200C5×C22⋊C4—34no—
201C5×C22⋊C4—64no—
202C5×C22⋊C4—122no—
203C5×C22⋊C4—122no—
204C42×C5—34no—
205C42×C5—42no—
206C42×C5—64no—
207C42×C5—122no—
208C2×C4×D4—31no—
209#390—32no—
210#388—32no—
211#388—32no—
212#382—31no—
213#381—31no—
214#381—31no—
215C4×C4⋊C4—32no—
216C4×C4⋊C4—31no—
217C4×C22⋊C4—31no—
218C4×C22⋊C4C5×D311yescharacteristic, complemented
219C22×C3×C5—22no—
220C22×C3×C5—22no—
221C22×C3×C5—22no—
222C22×C3×C5—42no—
223C22×C3×C5—42no—
224C22×C3×C5C23×C411yescharacteristic
225C22×C3×C5C2×#4112yes—
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5H(5,𝔽2)12yes—
229C2×C5×D3—22no—
230C2×C5×D3—22no—
231C2×C5×D3—22no—
232C2×C5×D3—42no—
233C2×C5×D3—42no—
234C3×C4×C5—22no—
235C3×C4×C5—22no—
236C3×C4×C5—24no—
237C3×C4×C5—42no—
238C5×Dic3—22no—
239C5×Dic3—22no—
240C5×Dic3—42no—
241C5×Dic3—42no—
242C5×Dic3#11714yes—
243C24×C3C23×C511yescharacteristic
244C23×D3C23×C511yescharacteristic
245C22×C3×C4—42no—
246C22×C3×C4C2×C4×C514yes—
247C22×C3×C4C23×C512yes—
248C22×C3×C4C23×C511yescharacteristic
249C22×C3×C4C23×C511yescharacteristic
250C22×C3×C4C23×C512yes—
251C2×#67—22no—
252C2×#67—22no—
253C2×#67C2×C4×C514yes—
254C22×Dic3C23×C512yes—
255C22×Dic3C23×C512yes—
256C22×Dic3C23×C511yescharacteristic
257C22×Dic3C23×C511yescharacteristic
258C22×Dic3C23×C511yescharacteristic
259C2×C4×D3—42no—
260C2×C4×D3C2×C4×C514yes—
261C3×C22⋊C4—24no—
262C3×C42—42no—
263#217—42no—
264#217C2×C4×C514yes—
265(C2×D3)⋊C4—24no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4C2×C4×C514yes—
268Dic3⋊C4—24no—
269Dic3⋊C4—24no—
270Dic3⋊C4C2×C4×C514yes—
271C4×Dic3—24no—
272C4×Dic3—42no—
273C4×Dic3C2×C4×C514yes—
274C23×C5—22no—
275C23×C5—22no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5—32no—
279C23×C5—62no—
280C23×C5—62no—
281C23×C5—61no—
282C23×C5—61no—
283C23×C5C2×C4×D312yes—
284C23×C5C23×D311yescharacteristic
285C5×D4—68no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—32no—
294C2×C4×C5—32no—
295C2×C4×C5—32no—
296C2×C4×C5—32no—
297C2×C4×C5—32no—
298C2×C4×C5—44no—
299C2×C4×C5—62no—
300C2×C4×C5—62no—
301C2×C4×C5—62no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5—62no—
305C2×C4×C5—62no—
306C2×C4×C5—62no—
307C2×C4×C5—124no—
308C2×C4×C5C2×C4×D312yes—
309C2×C4×C5C2×C4×D312yes—
310C2×C4×C5#23514yes—
311C22×D4—31no—
312C23×C4—31no—
313C23×C4C2×C5×D311yescharacteristic
314C4×D4—38no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—32no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—31no—
320C2×C22⋊C4—32no—
321C2×C22⋊C4—32no—
322C2×C22⋊C4—32no—
323C2×C22⋊C4C2×C5×D312yes—
324C2×C42—31no—
325C2×C42—32no—
326C2×C42—32no—
327C2×C42C2×C5×D312yes—
328#95—31no—
329#95—31no—
330#95—32no—
331#95—32no—
332#95—32no—
333#95C2×C5×D311yescharacteristic
334#95C2×C5×D311yescharacteristic
335C2×C3×C5—42no—
336C2×C3×C5C2×#11711yescharacteristic
337C2×C3×C5C4×#4111yescharacteristic
338C2×C3×C5#51811yescharacteristic
339C2×C3×C5#55312yes—
340C2×C3×C5#55912yes—
341C5×D3—42no—
342C23×C3—22no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3—22no—
346C23×C3C22×C4×C512yes—
347C23×C3C24×C511yescharacteristic
348C22×D3—22no—
349C22×D3—21no—
350C22×D3—21no—
351C22×D3—22no—
352C22×D3C22×C4×C512yes—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—44no—
361C2×C3×C4C22×C4×C512yes—
362C2×C3×C4C22×C4×C512yes—
363C2×C3×C4C5×#4114yes—
364#67—28no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—22no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3C22×C4×C512yes—
374C2×Dic3C22×C4×C512yes—
375C2×Dic3C22×C4×C512yes—
376C2×Dic3C5×#4112yes—
377C2×Dic3C5×#4112yes—
378C4×D3—44no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—42no—
383C22×C5—42no—
384C22×C5—62no—
385C22×C5—62no—
386C22×C5—62no—
387C22×C5—122no—
388C22×C5—122no—
389C22×C5C22×C4×D311yescharacteristic
390C22×C5C2×#23512yes—
391C22×C5#100412yes—
392C22×C5D3×#4111yescharacteristic
393C22×C5D3×#4111yescharacteristic
394C4×C5—22no—
395C4×C5—22no—
396C4×C5—24no—
397C4×C5—34no—
398C4×C5—42no—
399C4×C5—62no—
400C4×C5—62no—
401C4×C5—122no—
402C4×C5—122no—
403C24—31no—
404C24C22×C5×D311yescharacteristic
405C2×D4—34no—
406C2×D4—62no—
407C2×D4—62no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—34no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—42no—
415C22×C4—122no—
416C22×C4C4×C5×D314yes—
417C22×C4C22×C5×D312yes—
418C22×C4C22×C5×D311yescharacteristic
419C22×C4C22×C5×D311yescharacteristic
420C22×C4C22×C5×D312yes—
421C4⋊C4—34no—
422C4⋊C4—64no—
423C4⋊C4—64no—
424C22⋊C4—24no—
425C22⋊C4—34no—
426C22⋊C4—34no—
427C22⋊C4—64no—
428C22⋊C4—122no—
429C22⋊C4—122no—
430C42—34no—
431C42—42no—
432C42—64no—
433C42—122no—
434C3×C5#235711yescharacteristic, complemented, Hall
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3C23×C4×C511yescharacteristic
441C22×C3C2×C5×#4112yes—
442C22×C3C2×C5×#4111yescharacteristic
443C22×C3C2×C5×#4111yescharacteristic
444C22×C3C5×H(5,𝔽2)12yes—
445C2×D3—22no—
446C2×D3—22no—
447C2×D3—22no—
448C2×D3—42no—
449C2×D3—42no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—24no—
453C3×C4—42no—
454Dic3—22no—
455Dic3—22no—
456Dic3—42no—
457Dic3—42no—
458Dic3C5×#11714yes—
459C2×C5—42no—
460C2×C5—122no—
461C2×C5C4×#23511yescharacteristic
462C2×C5#548611yescharacteristic
463C2×C5D3×#11711yescharacteristic
464C2×C5#554012yes—
465C2×C5#555912yes—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—32no—
471C23—62no—
472C23—62no—
473C23—61no—
474C23—61no—
475C23C2×C4×C5×D312yes—
476C23C23×C5×D311yescharacteristic
477D4—68no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—32no—
488C2×C4—32no—
489C2×C4—32no—
490C2×C4—44no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—62no—
499C2×C4—124no—
500C2×C4C2×C4×C5×D312yes—
501C2×C4C2×C4×C5×D312yes—
502C2×C4C5×#23514yes—
503C2×C3—42no—
504C2×C3C2×C5×#11711yescharacteristic
505C2×C3C4×C5×#4111yescharacteristic
506C2×C3C5×#51811yescharacteristic
507C2×C3C5×#55312yes—
508C2×C3C5×#55912yes—
509D3—42no—
510C5#7346311yescharacteristic, direct product factor, Sylow
511C22—22no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—122no—
520C22—122no—
521C22C22×C4×C5×D311yescharacteristic
522C22C2×C5×#23512yes—
523C22C5×#100412yes—
524C22C5×D3×#4111yescharacteristic
525C22C5×D3×#4111yescharacteristic
526C4—22no—
527C4—22no—
528C4—24no—
529C4—34no—
530C4—42no—
531C4—62no—
532C4—62no—
533C4—122no—
534C4—122no—
535C3C5×#235711yescharacteristic, complemented, Sylow
536C2—42no—
537C2—122no—
538C2C4×C5×#23511yescharacteristic
539C2C5×#548611yescharacteristic
540C2C5×D3×#11711yescharacteristic
541C2C5×#554012yes—
542C2C5×#555912yes—
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C4×C5×C22⋊C4C2
socleC23×C3×C5C24
prime radicalC22×C5×#67C22
cosocleC23×C3C24×C5
hypercenterC4×C5×C22⋊C4D3
hypocenterC3C5×#2357
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C4×C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×C22⋊C4C2semidirect
3. C5×#5211C2semidirect
5. C5×#4945C2semidirect
6. C5×#4940C2semidirect
7. C4×C5×Dic3⋊C4C2semidirect
8. C5×#4666C2semidirect
9. C5×#4664C2semidirect
10. C5×#4650C2semidirect
11. C5×#4649C2semidirect
20. C4×C5×#67C4semidirect
37. #73463C5direct (N × H)
48. C4×C5×C22⋊C4D3semidirect
81. C3×C4×C22⋊C4C2×C5semidirect
82. #5211C2×C5semidirect
84. #4945C2×C5semidirect
85. #4940C2×C5semidirect
86. C4×Dic3⋊C4C2×C5semidirect
87. #4666C2×C5semidirect
88. #4664C2×C5semidirect
89. #4650C2×C5semidirect
90. #4649C2×C5semidirect
160. C4×#67C4×C5semidirect
218. C4×C22⋊C4C5×D3semidirect
434. C3×C5#2357semidirect
510. C5#73463direct (N × H)
535. C3C5×#2357semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×#672
C3×C4×C5×C22⋊C42
C5×#52112
C5×#49572
C5×#49452
C5×#49402
C4×C5×Dic3⋊C42
C5×#46662
C5×#46642
C5×#46502
C5×#46492
C5×#23573
#734635