Finite Group Databasebeta

← C5×#73768 #14326671

Subgroups — C5×#73768 #14326671

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D3×C4⋊C4C211yesmaximal, characteristic
2C3×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
3C5×#4987C211yesmaximal, characteristic
4C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
5C5×#4972C211yesmaximal, characteristic, complemented
6C4×C5×(C2×D3)⋊C4C211yesmaximal, characteristic
7C4×C5×(C2×D3)⋊C4C212yesmaximal
8C5×#4933C212yesmaximal, complemented
9C5×#4928C211yesmaximal, characteristic, complemented
10C5×#4667C212yesmaximal
11C5×#4652C212yesmaximal, complemented
12C5×#2345—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C4×C5×D3C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C42×C5C2211yescharacteristic
18C2×C3×C42×C5C2212yes—
19C2×C5×(C2×D3)⋊C4C2211yescharacteristic
20C2×C5×(C2×D3)⋊C4C2212yes—
21C2×C5×(C2×D3)⋊C4C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×Dic3⋊C4C2212yes—
25C2×C5×Dic3⋊C4C2212yes—
26C2×C5×Dic3⋊C4C2212yes—
27C2×C4×C5×Dic3C2211yescharacteristic
28C2×C4×C5×Dic3C2212yes—
29C5×D3×C4⋊C4C418yescomplemented
30C3×C5×#95C2212yes—
31C5×C3⋊#95C2211yescharacteristic
32C5×C3⋊#95C2212yes—
33C5×C3⋊#95C2212yes—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2211yescharacteristic
37#73768C511yesmaximal, characteristic, direct product factor, Hall
38C22×C5×C4⋊C4—31no—
39C5×#389—31no—
40C5×#389—31no—
41C5×#387—32no—
42C5×#387—32no—
43C5×#380—32no—
44C5×#380—31no—
45C4×C5×C4⋊C4—31no—
46C4×C5×C4⋊C4D311yescharacteristic, complemented
47C4×C5×C22⋊C4—31no—
48C4×C5×C22⋊C4—32no—
49C23×C5×D3C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C5×Dic3C2311yescharacteristic
56C22×C5×Dic3C2312yes—
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2311yescharacteristic
59C22×C5×Dic3C2311yescharacteristic
60C2×C4×C5×D3—22no—
61C2×C4×C5×D3—22no—
62C2×C4×C5×D3—42no—
63C2×C4×C5×D3—42no—
64C2×C4×C5×D3C2×C414yes—
65C2×C4×C5×D3C2×C418yes—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4C2×C414yes—
68C3×C42×C5—22no—
69C3×C42×C5—42no—
70C5×(C2×D3)⋊C4—22no—
71C5×(C2×D3)⋊C4—24no—
72C5×(C2×D3)⋊C4Q814yes—
73C5×C4⋊Dic3—22no—
74C5×C4⋊Dic3C2×C414yes—
75C5×Dic3⋊C4—42no—
76C5×Dic3⋊C4—42no—
77C5×Dic3⋊C4C2×C418yes—
78C4×C5×Dic3—22no—
79C4×C5×Dic3—42no—
80C2×D3×C4⋊C4C2×C511yescharacteristic
81C3×C4×C4⋊C4C2×C511yescharacteristic, complemented
82#4987C2×C511yescharacteristic
83Dic3×C4⋊C4C2×C511yescharacteristic, complemented
84#4972C2×C511yescharacteristic, complemented
85C4×(C2×D3)⋊C4C2×C511yescharacteristic
86C4×(C2×D3)⋊C4C2×C512yes—
87#4933C2×C512yescomplemented
88#4928C2×C511yescharacteristic, complemented
89#4667C2×C512yes—
90#4652C2×C512yescomplemented
91C23×C4×C5—31no—
92C23×C4×C5—32no—
93C2×C5×C4⋊C4—32no—
94C2×C5×C4⋊C4—31no—
95C2×C5×C4⋊C4—38no—
96C2×C5×C4⋊C4—31no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C4⋊C4—32no—
99C2×C5×C4⋊C4C2×D311yescharacteristic
100C2×C5×C4⋊C4C2×D311yescharacteristic
101C2×C5×C22⋊C4—31no—
102C2×C5×C22⋊C4—31no—
103C2×C5×C22⋊C4—32no—
104C2×C42×C5—31no—
105C2×C42×C5—32no—
106C2×C42×C5C2×D311yescharacteristic
107C2×C42×C5C2×D312yes—
108C5×#95—31no—
109C5×#95—31no—
110C5×#95—32no—
111C5×#95—32no—
112C5×#95—32no—
113C5×#95—32no—
114C5×#95C2×D312yes—
115#2345—31noSylow
116C23×C3×C5C2411yescharacteristic
117C22×C5×D3—21no—
118C22×C5×D3—21no—
119C22×C5×D3—21no—
120C22×C5×D3—21no—
121C22×C5×D3C22×C412yes—
122C22×C5×D3C2×Q812yes—
123C22×C5×D3#4112yes—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5C22×C414yes—
135C2×C3×C4×C5C22×C412yes—
136C2×C3×C4×C5C2×Q812yes—
137C2×C3×C4×C5#4112yes—
138C2×C3×C4×C5#4112yes—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3C22×C412yes—
152C2×C5×Dic3C22×C414yes—
153C2×C5×Dic3C2×Q812yes—
154C4×C5×D3—28no—
155C4×C5×D3—48no—
156C22×C4×D3C22×C511yescharacteristic
157C22×C4×D3C22×C512yes—
158C2×C3×C4⋊C4C22×C511yescharacteristic
159C2×C3×C4⋊C4C22×C511yescharacteristic
160C2×C3×C42C22×C511yescharacteristic
161C2×C3×C42C22×C512yes—
162C2×(C2×D3)⋊C4C22×C511yescharacteristic
163C2×(C2×D3)⋊C4C22×C512yes—
164C2×(C2×D3)⋊C4C22×C511yescharacteristic
165C2×C4⋊Dic3C22×C511yescharacteristic
166C2×C4⋊Dic3C22×C511yescharacteristic
167C2×Dic3⋊C4C22×C512yes—
168C2×Dic3⋊C4C22×C512yes—
169C2×Dic3⋊C4C22×C512yes—
170C2×C4×Dic3C22×C511yescharacteristic
171C2×C4×Dic3C22×C512yes—
172D3×C4⋊C4C4×C518yescomplemented
173C3×#95C22×C512yes—
174C3⋊#95C22×C511yescharacteristic
175C3⋊#95C22×C512yes—
176C3⋊#95C22×C512yes—
177C3⋊#95C22×C512yes—
178C3⋊#95C22×C512yes—
179C3⋊#95C22×C511yescharacteristic
180C24×C5—31no—
181C22×C4×C5—31no—
182C22×C4×C5—32no—
183C22×C4×C5—31no—
184C22×C4×C5—34no—
185C22×C4×C5—38no—
186C22×C4×C5—31no—
187C22×C4×C5—32no—
188C22×C4×C5—62no—
189C22×C4×C5—62no—
190C22×C4×C5—122no—
191C22×C4×C5—122no—
192C22×C4×C5C22×D311yescharacteristic
193C22×C4×C5C22×D312yes—
194C22×C4×C5C22×D312yes—
195C22×C4×C5C22×D311yescharacteristic
196C22×C4×C5C22×D311yescharacteristic
197C5×C4⋊C4—22no—
198C5×C4⋊C4—38no—
199C5×C4⋊C4—34no—
200C5×C4⋊C4—62no—
201C5×C4⋊C4—122no—
202C5×C4⋊C4—122no—
203C5×C4⋊C4C4×D314yes—
204C5×C22⋊C4—34no—
205C5×C22⋊C4—62no—
206C5×C22⋊C4—64no—
207C42×C5—22no—
208C42×C5—42no—
209C42×C5—62no—
210C42×C5—122no—
211C22×C4⋊C4—31no—
212#389—31no—
213#389—31no—
214#387—32no—
215#387—32no—
216#380—31no—
217#380—32no—
218C4×C4⋊C4—31no—
219C4×C4⋊C4C5×D311yescharacteristic, complemented
220C4×C22⋊C4—31no—
221C4×C22⋊C4—32no—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C22×Q811yescharacteristic
224C22×C3×C5C2×#4111yescharacteristic
225C22×C3×C5C2×#4111yescharacteristic
226C22×C3×C5C2×#4111yescharacteristic
227C22×C3×C5H(5,𝔽2)11yescharacteristic
228C22×C3×C5H(5,𝔽2)11yescharacteristic
229C2×C5×D3—22no—
230C2×C5×D3—22no—
231C2×C5×D3—22no—
232C2×C5×D3—22no—
233C2×C5×D3—22no—
234C2×C5×D3—22no—
235C2×C5×D3C4×Q814yes—
236C3×C4×C5—24no—
237C3×C4×C5—22no—
238C3×C4×C5—41no—
239C3×C4×C5—42no—
240C3×C4×C5#11714yes—
241C5×Dic3—22no—
242C5×Dic3—24no—
243C5×Dic3—22no—
244C5×Dic3—42no—
245C5×Dic3—41no—
246C23×D3C23×C511yescharacteristic
247C22×C3×C4C23×C512yes—
248C22×C3×C4C23×C511yescharacteristic
249C22×C3×C4C23×C512yes—
250C22×C3×C4C23×C511yescharacteristic
251C22×C3×C4C23×C511yescharacteristic
252C22×Dic3C23×C511yescharacteristic
253C22×Dic3C23×C512yes—
254C22×Dic3C23×C511yescharacteristic
255C22×Dic3C23×C512yes—
256C22×Dic3C23×C511yescharacteristic
257C2×C4×D3—22no—
258C2×C4×D3—22no—
259C2×C4×D3—42no—
260C2×C4×D3—42no—
261C2×C4×D3C2×C4×C514yes—
262C2×C4×D3C2×C4×C518yes—
263C3×C4⋊C4—22no—
264C3×C4⋊C4C2×C4×C514yes—
265C3×C42—22no—
266C3×C42—42no—
267(C2×D3)⋊C4—22no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4C5×Q814yes—
270C4⋊Dic3—22no—
271C4⋊Dic3C2×C4×C514yes—
272Dic3⋊C4—42no—
273Dic3⋊C4—42no—
274Dic3⋊C4C2×C4×C518yes—
275C4×Dic3—22no—
276C4×Dic3—42no—
277C23×C5—32no—
278C23×C5—32no—
279C23×C5—32no—
280C23×C5—61no—
281C23×C5—61no—
282C23×C5—61no—
283C23×C5—61no—
284C23×C5C23×D311yescharacteristic
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—21no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—21no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—32no—
296C2×C4×C5—34no—
297C2×C4×C5—32no—
298C2×C4×C5—61no—
299C2×C4×C5—61no—
300C2×C4×C5—61no—
301C2×C4×C5—62no—
302C2×C4×C5—61no—
303C2×C4×C5—62no—
304C2×C4×C5—68no—
305C2×C4×C5—61no—
306C2×C4×C5—62no—
307C2×C4×C5—61no—
308C2×C4×C5—62no—
309C2×C4×C5—61no—
310C2×C4×C5—62no—
311C2×C4×C5—128no—
312C2×C4×C5C2×C4×D314yes—
313C2×C4×C5C2×C4×D312yes—
314C2×C4×C5#23512yes—
315C2×C4×C5#23512yes—
316C2×C4×C5D3×Q812yes—
317C23×C4—31no—
318C23×C4—32no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—38no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—32no—
325C2×C4⋊C4C2×C5×D311yescharacteristic
326C2×C4⋊C4C2×C5×D311yescharacteristic
327C2×C22⋊C4—31no—
328C2×C22⋊C4—32no—
329C2×C22⋊C4—31no—
330C2×C42—31no—
331C2×C42—32no—
332C2×C42C2×C5×D311yescharacteristic
333C2×C42C2×C5×D312yes—
334#95—31no—
335#95—31no—
336#95—32no—
337#95—32no—
338#95—32no—
339#95—32no—
340#95C2×C5×D312yes—
341C2×C3×C5C2×#11711yescharacteristic
342C2×C3×C5C2×C4×Q811yescharacteristic
343C2×C3×C5#51811yescharacteristic
344C2×C3×C5#55311yescharacteristic
345C2×C3×C5#55311yescharacteristic
346C2×C3×C5#55411yescharacteristic
347C2×C3×C5#55411yescharacteristic
348C5×D3—24no—
349C23×C3C24×C511yescharacteristic
350C22×D3—21no—
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3C22×C4×C512yes—
355C22×D3C2×C5×Q812yes—
356C22×D3C5×#4112yes—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4C22×C4×C514yes—
368C2×C3×C4C22×C4×C512yes—
369C2×C3×C4C2×C5×Q812yes—
370C2×C3×C4C5×#4112yes—
371C2×C3×C4C5×#4112yes—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—21no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—21no—
384C2×Dic3C22×C4×C512yes—
385C2×Dic3C22×C4×C514yes—
386C2×Dic3C2×C5×Q812yes—
387C4×D3—28no—
388C4×D3—48no—
389C22×C5—34no—
390C22×C5—62no—
391C22×C5—62no—
392C22×C5—62no—
393C22×C5—62no—
394C22×C5—62no—
395C22×C5—62no—
396C22×C5C22×C4×D311yescharacteristic
397C22×C5C2×#23511yescharacteristic
398C22×C5C2×#23511yescharacteristic
399C22×C5#100411yescharacteristic
400C22×C5C2×D3×Q811yescharacteristic
401C22×C5D3×#4111yescharacteristic
402C22×C5#100911yescharacteristic
403C4×C5—22no—
404C4×C5—24no—
405C4×C5—41no—
406C4×C5—42no—
407C4×C5—62no—
408C4×C5—64no—
409C4×C5—62no—
410C4×C5—122no—
411C4×C5—121no—
412C4×C5#87214yes—
413C24—31no—
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—32no—
417C22×C4—34no—
418C22×C4—38no—
419C22×C4—31no—
420C22×C4—32no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—122no—
424C22×C4—122no—
425C22×C4C22×C5×D312yes—
426C22×C4C22×C5×D311yescharacteristic
427C22×C4C22×C5×D312yes—
428C22×C4C22×C5×D311yescharacteristic
429C22×C4C22×C5×D311yescharacteristic
430C4⋊C4—22no—
431C4⋊C4—38no—
432C4⋊C4—34no—
433C4⋊C4—62no—
434C4⋊C4—122no—
435C4⋊C4—122no—
436C4⋊C4C4×C5×D314yes—
437C22⋊C4—34no—
438C22⋊C4—62no—
439C22⋊C4—64no—
440C42—22no—
441C42—42no—
442C42—62no—
443C42—122no—
444C3×C5#234511yescharacteristic, complemented, Hall
445C22×C3C23×C4×C511yescharacteristic
446C22×C3C22×C5×Q811yescharacteristic
447C22×C3C2×C5×#4111yescharacteristic
448C22×C3C2×C5×#4111yescharacteristic
449C22×C3C2×C5×#4111yescharacteristic
450C22×C3C5×H(5,𝔽2)11yescharacteristic
451C22×C3C5×H(5,𝔽2)11yescharacteristic
452C2×D3—22no—
453C2×D3—22no—
454C2×D3—22no—
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—22no—
458C2×D3C4×C5×Q814yes—
459C3×C4—24no—
460C3×C4—22no—
461C3×C4—41no—
462C3×C4—42no—
463C3×C4C5×#11714yes—
464Dic3—22no—
465Dic3—24no—
466Dic3—22no—
467Dic3—42no—
468Dic3—41no—
469C2×C5—64no—
470C2×C5C2×#87211yescharacteristic
471C2×C5#551111yescharacteristic
472C2×C5#553011yescharacteristic
473C2×C5#554511yescharacteristic
474C2×C5#555811yescharacteristic
475C2×C5C4×D3×Q811yescharacteristic
476C2×C5#557911yescharacteristic
477C23—32no—
478C23—32no—
479C23—32no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23—61no—
484C23C23×C5×D311yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—32no—
496C2×C4—34no—
497C2×C4—32no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—62no—
502C2×C4—61no—
503C2×C4—62no—
504C2×C4—68no—
505C2×C4—61no—
506C2×C4—62no—
507C2×C4—61no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—62no—
511C2×C4—128no—
512C2×C4C2×C4×C5×D314yes—
513C2×C4C2×C4×C5×D312yes—
514C2×C4C5×#23512yes—
515C2×C4C5×#23512yes—
516C2×C4C5×D3×Q812yes—
517C2×C3C2×C5×#11711yescharacteristic
518C2×C3C2×C4×C5×Q811yescharacteristic
519C2×C3C5×#51811yescharacteristic
520C2×C3C5×#55311yescharacteristic
521C2×C3C5×#55311yescharacteristic
522C2×C3C5×#55411yescharacteristic
523C2×C3C5×#55411yescharacteristic
524D3—24no—
525C5#7376811yescharacteristic, direct product factor, Sylow
526C22—34no—
527C22—62no—
528C22—62no—
529C22—62no—
530C22—62no—
531C22—62no—
532C22—62no—
533C22C22×C4×C5×D311yescharacteristic
534C22C2×C5×#23511yescharacteristic
535C22C2×C5×#23511yescharacteristic
536C22C5×#100411yescharacteristic
537C22C2×C5×D3×Q811yescharacteristic
538C22C5×D3×#4111yescharacteristic
539C22C5×#100911yescharacteristic
540C4—22no—
541C4—24no—
542C4—41no—
543C4—42no—
544C4—62no—
545C4—64no—
546C4—62no—
547C4—122no—
548C4—121no—
549C4C5×#87214yes—
550C3C5×#234511yescharacteristic, complemented, Sylow
551C2—64no—
552C2C2×C5×#87211yescharacteristic
553C2C5×#551111yescharacteristic
554C2C5×#553011yescharacteristic
555C2C5×#554511yescharacteristic
556C2C5×#555811yescharacteristic
557C2C4×C5×D3×Q811yescharacteristic
558C2C5×#557911yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C4×C5×C4⋊C4C2
socleC23×C3×C5C24
prime radicalC23×C5×D3C23
cosocleC23×C3C24×C5
hypercenterC4×C5×C4⋊C4D3
hypocenterC3C5×#2345
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C4×C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×C5×C4⋊C4C2semidirect
4. C5×Dic3×C4⋊C4C2semidirect
5. C5×#4972C2semidirect
8. C5×#4933C2semidirect
9. C5×#4928C2semidirect
11. C5×#4652C2semidirect
29. C5×D3×C4⋊C4C4semidirect
37. #73768C5direct (N × H)
46. C4×C5×C4⋊C4D3semidirect
81. C3×C4×C4⋊C4C2×C5semidirect
83. Dic3×C4⋊C4C2×C5semidirect
84. #4972C2×C5semidirect
87. #4933C2×C5semidirect
88. #4928C2×C5semidirect
90. #4652C2×C5semidirect
172. D3×C4⋊C4C4×C5semidirect
219. C4×C4⋊C4C5×D3semidirect
444. C3×C5#2345semidirect
525. C5#73768direct (N × H)
550. C3C5×#2345semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D3×C4⋊C42
C3×C4×C5×C4⋊C42
C5×#49872
C5×Dic3×C4⋊C42
C5×#49722
C4×C5×(C2×D3)⋊C42
C5×#49332
C5×#49282
C5×#46672
C5×#46522
C5×#23453
#737685