Finite Group Databasebeta

← C5×#74664 #14323546

Subgroups — C5×#74664 #14323546

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#935C211yesmaximal, characteristic
2C2×C5×D4×Dic3C211yesmaximal, characteristic
3C22×C5×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C3×C5×#386C211yesmaximal, characteristic
5C5×#5210C211yesmaximal, characteristic, complemented
6C5×#4946C212yesmaximal, complemented
7C5×#4944C212yesmaximal, complemented
8C5×#4943C211yesmaximal, characteristic, complemented
9C5×Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
10C5×#4652C212yesmaximal, complemented
11C5×#4647C212yesmaximal, complemented
12C5×#2322—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×C5×Dic3C2211yescharacteristic
16C23×C5×Dic3C2211yescharacteristic
17C23×C5×Dic3C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2212yes—
20C2×C5×#217C2211yescharacteristic
21C2×C5×#217C2211yescharacteristic
22C2×C5×#217C2211yescharacteristic
23C2×C5×#217C2212yes—
24C5×#935C418yescomplemented
25C5×D4×Dic3—24no—
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C2×C5×Dic3⋊C4—24no—
28C2×C5×Dic3⋊C4C2212yes—
29C2×C5×Dic3⋊C4C2212yes—
30C2×C5×Dic3⋊C4C2212yes—
31C2×C5×Dic3⋊C4C2212yes—
32C2×C4×C5×Dic3C2211yescharacteristic
33C2×C4×C5×Dic3C2212yes—
34C3×C5×#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37C5×C3⋊#95C2212yes—
38C5×C3⋊#95C2212yes—
39#74664C511yesmaximal, characteristic, direct product factor, Hall
40C2×C5×#123—31no—
41C2×C4×C5×D4—31no—
42C22×C5×C4⋊C4—31no—
43C5×#389—32no—
44C5×#388—32no—
45C5×#387—32no—
46C5×#386D311yescharacteristic
47C5×#385—31no—
48C5×#385—32no—
49C5×#381—31no—
50C4×C5×C22⋊C4—31no—
51C24×C3×C5C2311yescharacteristic
52C24×C3×C5C2311yescharacteristic
53C2×C3×C5×D4—22no—
54C2×C3×C5×D4—22no—
55C2×C3×C5×D4C2×C414yes—
56C22×C3×C4×C5—22no—
57C22×C3×C4×C5—42no—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C5×Dic3—22no—
62C22×C5×Dic3—22no—
63C22×C5×Dic3—22no—
64C22×C5×Dic3—42no—
65C22×C5×Dic3—42no—
66C22×C5×Dic3C2×C414yescomplemented
67C22×C5×Dic3D414yes—
68C22×C5×Dic3C2311yescharacteristic
69C22×C5×Dic3C2311yescharacteristic
70C22×C5×Dic3C2312yes—
71C22×C5×Dic3C2312yes—
72C22×C5×Dic3C2311yescharacteristic
73C22×C5×Dic3C2311yescharacteristic
74C3×C5×C22⋊C4—22no—
75C3×C5×C22⋊C4—42no—
76C5×#217—22no—
77C5×#217—22no—
78C5×#217C2×C414yescomplemented
79C5×#217C2×C418yescomplemented
80C5×C4⋊Dic3—22no—
81C5×Dic3⋊C4—24no—
82C5×Dic3⋊C4—24no—
83C5×Dic3⋊C4—42no—
84C5×Dic3⋊C4C2×C418yescomplemented
85C4×C5×Dic3—22no—
86C4×C5×Dic3—42no—
87C2×#935C2×C511yescharacteristic
88C2×D4×Dic3C2×C511yescharacteristic
89C22×Dic3⋊C4C2×C511yescharacteristic, complemented
90C3×#386C2×C511yescharacteristic
91#5210C2×C511yescharacteristic, complemented
92#4946C2×C512yescomplemented
93#4944C2×C512yescomplemented
94#4943C2×C511yescharacteristic, complemented
95Dic3×C22⋊C4C2×C511yescharacteristic, complemented
96#4652C2×C512yescomplemented
97#4647C2×C512yescomplemented
98C22×C5×D4C2×D311yescharacteristic
99C23×C4×C5—31no—
100C23×C4×C5—31no—
101C23×C4×C5—31no—
102C23×C4×C5C2×D311yescharacteristic
103C5×#123—38no—
104C4×C5×D4—64no—
105C2×C5×C4⋊C4—32no—
106C2×C5×C4⋊C4—32no—
107C2×C5×C4⋊C4—32no—
108C2×C5×C4⋊C4—31no—
109C2×C5×C4⋊C4—32no—
110C2×C5×C4⋊C4—64no—
111C2×C5×C22⋊C4—31no—
112C2×C5×C22⋊C4—32no—
113C2×C5×C22⋊C4—31no—
114C2×C5×C22⋊C4—31no—
115C2×C5×C22⋊C4C2×D311yescharacteristic
116C2×C5×C22⋊C4C2×D312yes—
117C2×C42×C5—31no—
118C2×C42×C5—32no—
119C5×#95—32no—
120C5×#95—32no—
121C5×#95—32no—
122C5×#95—32no—
123C5×#95C2×D312yes—
124#2322—31noSylow
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—22no—
130C23×C3×C5—22no—
131C23×C3×C5C22×C412yes—
132C23×C3×C5C22×C412yes—
133C23×C3×C5C2×D412yes—
134C23×C3×C5#4112yes—
135C23×C3×C5C2411yescharacteristic
136C3×C5×D4—28no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—22no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—44no—
146C2×C3×C4×C5C22×C412yes—
147C2×C5×Dic3—28no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3—21no—
155C2×C5×Dic3—21no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—44no—
159C2×C5×Dic3—44no—
160C2×C5×Dic3C22×C414yes—
161C2×C5×Dic3C22×C414yes—
162C2×C5×Dic3C2×D412yes—
163C2×C5×Dic3C2×D412yes—
164C22×C3×D4C22×C511yescharacteristic
165C23×C3×C4C22×C511yescharacteristic
166C23×Dic3C22×C511yescharacteristic
167C23×Dic3C22×C511yescharacteristic
168C23×Dic3C22×C511yescharacteristic
169C2×C3×C22⋊C4C22×C511yescharacteristic
170C2×C3×C22⋊C4C22×C512yes—
171C2×#217C22×C511yescharacteristic
172C2×#217C22×C511yescharacteristic
173C2×#217C22×C511yescharacteristic
174C2×#217C22×C512yes—
175#935C4×C518yescomplemented
176D4×Dic3—24no—
177C2×C4⋊Dic3C22×C511yescharacteristic
178C2×Dic3⋊C4—24no—
179C2×Dic3⋊C4C22×C512yes—
180C2×Dic3⋊C4C22×C512yes—
181C2×Dic3⋊C4C22×C512yes—
182C2×Dic3⋊C4C22×C512yes—
183C2×C4×Dic3C22×C511yescharacteristic
184C2×C4×Dic3C22×C512yes—
185C3×#95C22×C512yes—
186C3⋊#95C22×C512yes—
187C3⋊#95C22×C512yes—
188C3⋊#95C22×C512yes—
189C3⋊#95C22×C512yes—
190C24×C5C22×D311yescharacteristic
191C24×C5C22×D311yescharacteristic
192C2×C5×D4—22no—
193C2×C5×D4—22no—
194C2×C5×D4C4×D314yes—
195C22×C4×C5—22no—
196C22×C4×C5—34no—
197C22×C4×C5—31no—
198C22×C4×C5—32no—
199C22×C4×C5—31no—
200C22×C4×C5—32no—
201C22×C4×C5—34no—
202C22×C4×C5—31no—
203C22×C4×C5—31no—
204C22×C4×C5—42no—
205C22×C4×C5—62no—
206C22×C4×C5—62no—
207C22×C4×C5—62no—
208C22×C4×C5—122no—
209C22×C4×C5—122no—
210C22×C4×C5C22×D312yes—
211C22×C4×C5C22×D312yes—
212C22×C4×C5C22×D311yescharacteristic
213C5×C4⋊C4—38no—
214C5×C4⋊C4—64no—
215C5×C4⋊C4—64no—
216C5×C4⋊C4—62no—
217C5×C4⋊C4—122no—
218C5×C22⋊C4—22no—
219C5×C22⋊C4—34no—
220C5×C22⋊C4—38no—
221C5×C22⋊C4—42no—
222C5×C22⋊C4—62no—
223C5×C22⋊C4—62no—
224C42×C5—62no—
225C42×C5—122no—
226C2×#123—31no—
227C2×C4×D4—31no—
228C22×C4⋊C4—31no—
229#389—32no—
230#388—32no—
231#387—32no—
232#386C5×D311yescharacteristic
233#385—31no—
234#385—32no—
235#381—31no—
236C4×C22⋊C4—31no—
237C22×C3×C5—24no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—24no—
241C22×C3×C5—22no—
242C22×C3×C5—24no—
243C22×C3×C5—41no—
244C22×C3×C5—41no—
245C22×C3×C5—42no—
246C22×C3×C5C4×D414yes—
247C22×C3×C5C23×C411yescharacteristic
248C22×C3×C5C22×D411yescharacteristic
249C22×C3×C5C2×#4111yescharacteristic
250C22×C3×C5H(5,𝔽2)12yes—
251C22×C3×C52−1+412yes—
252C3×C4×C5—22no—
253C3×C4×C5—42no—
254C3×C4×C5—42no—
255C5×Dic3—24no—
256C5×Dic3—24no—
257C5×Dic3—22no—
258C5×Dic3—22no—
259C5×Dic3—41no—
260C5×Dic3—41no—
261C24×C3C23×C511yescharacteristic
262C24×C3C23×C511yescharacteristic
263C2×C3×D4—22no—
264C2×C3×D4—22no—
265C2×C3×D4C2×C4×C514yes—
266C22×C3×C4—22no—
267C22×C3×C4—42no—
268C22×C3×C4C23×C512yes—
269C22×C3×C4C23×C512yes—
270C22×C3×C4C23×C511yescharacteristic
271C22×Dic3—22no—
272C22×Dic3—22no—
273C22×Dic3—22no—
274C22×Dic3—42no—
275C22×Dic3—42no—
276C22×Dic3C2×C4×C514yescomplemented
277C22×Dic3C5×D414yes—
278C22×Dic3C23×C511yescharacteristic
279C22×Dic3C23×C511yescharacteristic
280C22×Dic3C23×C512yes—
281C22×Dic3C23×C512yes—
282C22×Dic3C23×C511yescharacteristic
283C22×Dic3C23×C511yescharacteristic
284C3×C22⋊C4—22no—
285C3×C22⋊C4—42no—
286#217—22no—
287#217—22no—
288#217C2×C4×C514yescomplemented
289#217C2×C4×C518yescomplemented
290C4⋊Dic3—22no—
291Dic3⋊C4—24no—
292Dic3⋊C4—24no—
293Dic3⋊C4—42no—
294Dic3⋊C4C2×C4×C518yescomplemented
295C4×Dic3—22no—
296C4×Dic3—42no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5—21no—
300C23×C5—21no—
301C23×C5—22no—
302C23×C5—22no—
303C23×C5C2×C4×D312yes—
304C23×C5C2×C4×D312yes—
305C23×C5D3×D412yes—
306C23×C5#23712yes—
307C23×C5C23×D311yescharacteristic
308C5×D4—28no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—22no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—22no—
317C2×C4×C5—34no—
318C2×C4×C5—34no—
319C2×C4×C5—32no—
320C2×C4×C5—32no—
321C2×C4×C5—44no—
322C2×C4×C5—68no—
323C2×C4×C5—62no—
324C2×C4×C5—62no—
325C2×C4×C5—62no—
326C2×C4×C5—62no—
327C2×C4×C5—62no—
328C2×C4×C5—62no—
329C2×C4×C5—61no—
330C2×C4×C5—61no—
331C2×C4×C5—62no—
332C2×C4×C5—62no—
333C2×C4×C5—124no—
334C2×C4×C5—124no—
335C2×C4×C5C2×C4×D312yes—
336C22×D4C2×C5×D311yescharacteristic
337C23×C4—31no—
338C23×C4—31no—
339C23×C4—31no—
340C23×C4C2×C5×D311yescharacteristic
341#123—38no—
342C4×D4—64no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—64no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—32no—
351C2×C22⋊C4—31no—
352C2×C22⋊C4—31no—
353C2×C22⋊C4C2×C5×D311yescharacteristic
354C2×C22⋊C4C2×C5×D312yes—
355C2×C42—31no—
356C2×C42—32no—
357#95—32no—
358#95—32no—
359#95—32no—
360#95—32no—
361#95C2×C5×D312yes—
362C2×C3×C5—24no—
363C2×C3×C5—42no—
364C2×C3×C5C2×C4×D411yescharacteristic
365C2×C3×C5#52012yes—
366C2×C3×C5#53411yescharacteristic
367C2×C3×C5#53611yescharacteristic
368C2×C3×C5#53912yes—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—22no—
374C23×C3—22no—
375C23×C3C22×C4×C512yes—
376C23×C3C22×C4×C512yes—
377C23×C3C2×C5×D412yes—
378C23×C3C5×#4112yes—
379C23×C3C24×C511yescharacteristic
380C3×D4—28no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—44no—
390C2×C3×C4C22×C4×C512yes—
391C2×Dic3—28no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—44no—
403C2×Dic3—44no—
404C2×Dic3C22×C4×C514yes—
405C2×Dic3C22×C4×C514yes—
406C2×Dic3C2×C5×D412yes—
407C2×Dic3C2×C5×D412yes—
408C22×C5—22no—
409C22×C5—24no—
410C22×C5—22no—
411C22×C5—24no—
412C22×C5—22no—
413C22×C5—24no—
414C22×C5—41no—
415C22×C5—41no—
416C22×C5—42no—
417C22×C5#67⋊C414yes—
418C22×C5C22×C4×D311yescharacteristic
419C22×C5C2×D3×D411yescharacteristic
420C22×C5C2×#23711yescharacteristic
421C22×C5#100412yes—
422C22×C5#101012yes—
423C4×C5—22no—
424C4×C5—42no—
425C4×C5—42no—
426C4×C5—64no—
427C4×C5—64no—
428C4×C5—62no—
429C4×C5—62no—
430C4×C5—121no—
431C4×C5—121no—
432C24C22×C5×D311yescharacteristic
433C24C22×C5×D311yescharacteristic
434C2×D4—22no—
435C2×D4—22no—
436C2×D4C4×C5×D314yes—
437C22×C4—22no—
438C22×C4—34no—
439C22×C4—32no—
440C22×C4—31no—
441C22×C4—32no—
442C22×C4—31no—
443C22×C4—34no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—42no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4C22×C5×D312yes—
453C22×C4C22×C5×D312yes—
454C22×C4C22×C5×D311yescharacteristic
455C4⋊C4—38no—
456C4⋊C4—64no—
457C4⋊C4—64no—
458C4⋊C4—62no—
459C4⋊C4—122no—
460C22⋊C4—22no—
461C22⋊C4—34no—
462C22⋊C4—38no—
463C22⋊C4—42no—
464C22⋊C4—62no—
465C22⋊C4—62no—
466C42—62no—
467C42—122no—
468C3×C5#232211yescharacteristic, complemented, Hall
469C22×C3—24no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—24no—
473C22×C3—22no—
474C22×C3—24no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—42no—
478C22×C3C4×C5×D414yes—
479C22×C3C23×C4×C511yescharacteristic
480C22×C3C22×C5×D411yescharacteristic
481C22×C3C2×C5×#4111yescharacteristic
482C22×C3C5×H(5,𝔽2)12yes—
483C22×C3C5×2−1+412yes—
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—42no—
487Dic3—24no—
488Dic3—24no—
489Dic3—22no—
490Dic3—22no—
491Dic3—41no—
492Dic3—41no—
493C2×C5—24no—
494C2×C5—42no—
495C2×C5C2×#67⋊C411yescharacteristic
496C2×C5#554612yes—
497C2×C5#558911yescharacteristic
498C2×C5#560212yes—
499C2×C5#564811yescharacteristic
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—22no—
506C23C2×C4×C5×D312yes—
507C23C2×C4×C5×D312yes—
508C23C5×D3×D412yes—
509C23C5×#23712yes—
510C23C23×C5×D311yescharacteristic
511D4—28no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—34no—
521C2×C4—34no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—44no—
525C2×C4—68no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—124no—
537C2×C4—124no—
538C2×C4C2×C4×C5×D312yes—
539C2×C3—24no—
540C2×C3—42no—
541C2×C3C2×C4×C5×D411yescharacteristic
542C2×C3C5×#52012yes—
543C2×C3C5×#53411yescharacteristic
544C2×C3C5×#53611yescharacteristic
545C2×C3C5×#53912yes—
546C5#7466411yescharacteristic, direct product factor, Sylow
547C22—22no—
548C22—24no—
549C22—22no—
550C22—24no—
551C22—22no—
552C22—24no—
553C22—41no—
554C22—41no—
555C22—42no—
556C22C5×#67⋊C414yes—
557C22C22×C4×C5×D311yescharacteristic
558C22C2×C5×D3×D411yescharacteristic
559C22C2×C5×#23711yescharacteristic
560C22C5×#100412yes—
561C22C5×#101012yes—
562C4—22no—
563C4—42no—
564C4—42no—
565C4—64no—
566C4—64no—
567C4—62no—
568C4—62no—
569C4—121no—
570C4—121no—
571C3C5×#232211yescharacteristic, complemented, Sylow
572C2—24no—
573C2—42no—
574C2C2×C5×#67⋊C411yescharacteristic
575C2C5×#554612yes—
576C2C5×#558911yescharacteristic
577C2C5×#560212yes—
578C2C5×#564811yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#386C2
socleC23×C3×C5C24
prime radicalC22×C3×C5×D4C22
cosocleC23×C3C24×C5
hypercenterC5×#386D3
hypocenterC3C5×#2322
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#386

Complements

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

NH ≅ G/Ntype
3. C22×C5×Dic3⋊C4C2semidirect
5. C5×#5210C2semidirect
6. C5×#4946C2semidirect
7. C5×#4944C2semidirect
8. C5×#4943C2semidirect
9. C5×Dic3×C22⋊C4C2semidirect
10. C5×#4652C2semidirect
11. C5×#4647C2semidirect
24. C5×#935C4semidirect
39. #74664C5direct (N × H)
66. C22×C5×Dic3C2×C4semidirect
78. C5×#217C2×C4semidirect
79. C5×#217C2×C4semidirect
84. C5×Dic3⋊C4C2×C4semidirect
89. C22×Dic3⋊C4C2×C5semidirect
91. #5210C2×C5semidirect
92. #4946C2×C5semidirect
93. #4944C2×C5semidirect
94. #4943C2×C5semidirect
95. Dic3×C22⋊C4C2×C5semidirect
96. #4652C2×C5semidirect
97. #4647C2×C5semidirect
175. #935C4×C5semidirect
276. C22×Dic3C2×C4×C5semidirect
288. #217C2×C4×C5semidirect
289. #217C2×C4×C5semidirect
294. Dic3⋊C4C2×C4×C5semidirect
468. C3×C5#2322semidirect
546. C5#74664direct (N × H)
571. C3C5×#2322semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9352
C2×C5×D4×Dic32
C22×C5×Dic3⋊C42
C3×C5×#3862
C5×#52102
C5×#49462
C5×#49442
C5×#49432
C5×Dic3×C22⋊C42
C5×#46522
C5×#46472
C5×#23223
#746645