Finite Group Databasebeta

← #14287905

Subgroups — #14287905

Order 1920.

640 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#538C211yesmaximal, characteristic, complemented
2#11751147C211yesmaximal, characteristic, complemented
3#11751137C211yesmaximal, characteristic, complemented
4#11751061C211yesmaximal, characteristic, complemented
5#11746359C211yesmaximal, characteristic, complemented
6#11746358C211yesmaximal, characteristic, complemented
7#11746346C211yesmaximal, characteristic, complemented
8#10602677—31nomaximal, Hall
9C22×#1220—21no—
10C22×D60C2211yescharacteristic
11C3×C5×#126—21no—
12C3×C5×#124—21no—
13C3×C5×#123—21no—
14C3×C5×C22⋊Q8C2211yescharacteristic
15C3×C5×#121—21no—
16C3×C5×#121C2211yescharacteristic
17C3×C5×#120—21no—
18C3×C4×C5×D4—21no—
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×#6717—21no—
21C2×#6715—21no—
22C2×#6714—21no—
23#91115C2211yescharacteristic
24#91114C2211yescharacteristic
25#91114C2211yescharacteristic
26#91093—21no—
27#91093—21no—
28#71884—51nomaximal, Hall
29C5×#538D311yescharacteristic, complemented
30#65335—31no—
31#65325—31no—
32#65249—31no—
33#64758—31no—
34#64757—31no—
35#64745—31no—
36C24×C3×C5D411yescharacteristic
37C23×D15—21no—
38C2×C3×C5×Q8D411yescharacteristic, complemented
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4D411yescharacteristic, complemented
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5D411yescharacteristic
45C22×C3×C4×C5C2311yescharacteristic
46C2×#1220—42no—
47C2×#1220—42no—
48C2×#1220—42no—
49C22×Dic15—21no—
50C2×D60—21no—
51C2×D60—21no—
52C2×D60—42no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4—21no—
55C3×C5×C4⋊C4D411yescharacteristic, complemented
56C3×C5×C4⋊C4D411yescharacteristic, complemented
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C5×C22⋊C4—21no—
63C3×C5×C22⋊C4—21no—
64C3×C42×C5—21no—
65#6717—42no—
66#6715—42no—
67#6714—42no—
68C2×C3⋊#168—21no—
69C2×C3⋊#168—21no—
70C3×#538D511yescharacteristic, complemented
71#5046—51no—
72#5036—51no—
73#4960—51no—
74#4538—51no—
75#4537—51no—
76#4525—51no—
77C22×C22⋊D5—61no—
78C22×D20—31no—
79C5×#126—21no—
80C5×#124—21no—
81C5×#123—21no—
82C5×C22⋊Q8C2×D311yescharacteristic
83C5×#121—21no—
84C5×#121C2×D311yescharacteristic
85C5×#120—21no—
86C4×C5×D4—21no—
87C2×C5×C22⋊C4C2×D311yescharacteristic
88C2×#688—61no—
89C2×#686—61no—
90C2×#685—61no—
91#3963—31no—
92C22⋊#168—31no—
93C22⋊#168—31no—
94#3941—61no—
95#3941—61no—
96#1600—151noSylow
97C23×C3×C5—21no—
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5—21no—
101C23×C3×C5—41no—
102C23×C3×C5—41no—
103C23×C3×C5C2×D411yescharacteristic
104C22×D15—22no—
105C22×D15—41no—
106C22×D15—41no—
107C22×D15—41no—
108C22×D15—41no—
109C22×D15—41no—
110C22×D15—41no—
111C3×C5×Q8—41no—
112C3×C5×D4—41no—
113C3×C5×D4—41no—
114C3×C5×D4—41no—
115C3×C5×D4—81no—
116C3×C5×D4—81no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5—41no—
123C2×C3×C4×C5—81no—
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126#1220—48no—
127C2×Dic15—22no—
128C2×Dic15—41no—
129C2×Dic15—41no—
130D60—22no—
131D60—22no—
132D60—41no—
133D60—41no—
134C3⋊#168—41no—
135C3⋊#168—41no—
136C22×#67—101no—
137C22×D12—51no—
138C3×#126—21no—
139C3×#124—21no—
140C3×#123—21no—
141C3×C22⋊Q8C2×D511yescharacteristic
142C3×#121—21no—
143C3×#121C2×D511yescharacteristic
144C3×#120—21no—
145C3×C4×D4—21no—
146C2×C3×C22⋊C4C2×D511yescharacteristic
147C2×#215—101no—
148C2×#213—101no—
149C2×(C2×D3)⋊C4—101no—
150C3⋊#95—51no—
151#830—51no—
152#830—51no—
153#809—101no—
154#809—101no—
155C24×C5#6711yescharacteristic
156C23×D5—61no—
157C2×C5×Q8#6711yescharacteristic, complemented
158C2×C5×D4—21no—
159C2×C5×D4—21no—
160C2×C5×D4—21no—
161C2×C5×D4#6711yescharacteristic, complemented
162C22×C4×C5—21no—
163C22×C4×C5#6711yescharacteristic
164C22×C4×C5C22×D311yescharacteristic
165C2×C22⋊D5—122no—
166C2×C22⋊D5—122no—
167C2×C22⋊D5—122no—
168C22×Dic5—61no—
169C2×D20—61no—
170C2×D20—61no—
171C2×D20—122no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4#6711yescharacteristic, complemented
175C5×C4⋊C4#6711yescharacteristic, complemented
176C5×C22⋊C4—21no—
177C5×C22⋊C4—21no—
178C5×C22⋊C4—21no—
179C5×C22⋊C4—21no—
180C5×C22⋊C4—21no—
181C5×C22⋊C4—21no—
182C5×C22⋊C4—21no—
183C42×C5—21no—
184#688—122no—
185#686—122no—
186#685—122no—
187C2×#168—61no—
188C2×#168—61no—
189#538D1511yescharacteristic, complemented
190#450—151no—
191#447—151no—
192#392—151no—
193#328—151no—
194#327—151no—
195#323—151no—
196C22×C3×C5—21no—
197C22×C3×C5—21no—
198C22×C3×C5—42no—
199C22×C3×C5—41no—
200C22×C3×C5—41no—
201C22×C3×C5—41no—
202C22×C3×C5—41no—
203C22×C3×C5—41no—
204C22×C3×C5—41no—
205C22×C3×C5—81no—
206C22×C3×C5#12011yescharacteristic
207C22×C3×C5#13511yescharacteristic
208C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
209C2×D15—42no—
210C2×D15—42no—
211C2×D15—42no—
212C2×D15—81no—
213C2×D15—81no—
214C2×D15—81no—
215C2×D15—81no—
216C3×C4×C5—21no—
217C3×C4×C5—21no—
218C3×C4×C5—41no—
219C3×C4×C5—41no—
220C3×C4×C5—41no—
221C3×C4×C5—41no—
222Dic15—42no—
223C24×C3C22⋊D511yescharacteristic
224C23×D3—101no—
225C2×C3×Q8C22⋊D511yescharacteristic, complemented
226C2×C3×D4—21no—
227C2×C3×D4—21no—
228C2×C3×D4—21no—
229C2×C3×D4C22⋊D511yescharacteristic, complemented
230C22×C3×C4—21no—
231C22×C3×C4C22⋊D511yescharacteristic
232C22×C3×C4C22×D511yescharacteristic
233C2×#67—202no—
234C2×#67—202no—
235C2×#67—202no—
236C22×Dic3—101no—
237C2×D12—101no—
238C2×D12—101no—
239C2×D12—202no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C22⋊D511yescharacteristic, complemented
243C3×C4⋊C4C22⋊D511yescharacteristic, complemented
244C3×C22⋊C4—21no—
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C42—21no—
252#215—202no—
253#213—202no—
254(C2×D3)⋊C4—202no—
255C2×C3⋊C8—101no—
256C2×C3⋊C8—101no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5—41no—
262C23×C5—41no—
263C23×C5C2×#6711yescharacteristic
264C22×D5—62no—
265C22×D5—121no—
266C22×D5—121no—
267C22×D5—121no—
268C22×D5—121no—
269C22×D5—121no—
270C22×D5—121no—
271C5×Q8—41no—
272C5×D4—41no—
273C5×D4—41no—
274C5×D4—41no—
275C5×D4—81no—
276C5×D4—81no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—81no—
284C2×C4×C5C2×#6711yescharacteristic
285C2×C4×C5C2×#6711yescharacteristic
286C22⋊D5—128no—
287C2×Dic5—62no—
288C2×Dic5—121no—
289C2×Dic5—121no—
290D20—62no—
291D20—62no—
292D20—121no—
293D20—121no—
294#168—121no—
295#168—121no—
296C22×D4—151no—
297C22×D4—301no—
298C2×SD16—301no—
299C2×D8—301no—
300#126—21no—
301#124—21no—
302#123—21no—
303C22⋊Q8C2×D1511yescharacteristic
304#121—21no—
305#121C2×D1511yescharacteristic
306#120—21no—
307C4×D4—21no—
308C2×C22⋊C4—301no—
309C2×C22⋊C4C2×D1511yescharacteristic
310#102—301no—
311#102—301no—
312C22⋊C8—151no—
313C22⋊C8—151no—
314#95—151no—
315C2×C3×C5—21no—
316C2×C3×C5—21no—
317C2×C3×C5—81no—
318C2×C3×C5—81no—
319C2×C3×C5#44911yescharacteristic
320C2×C3×C5#45311yescharacteristic
321C2×C3×C5B(4,𝔽2)11yescharacteristic
322D15—82no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C2×C22⋊D511yescharacteristic
330C22×D3—102no—
331C22×D3—201no—
332C22×D3—201no—
333C22×D3—201no—
334C22×D3—201no—
335C22×D3—201no—
336C22×D3—201no—
337C3×Q8—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—81no—
342C3×D4—81no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—81no—
350C2×C3×C4C2×C22⋊D511yescharacteristic
351C2×C3×C4C2×C22⋊D511yescharacteristic
352#67—208no—
353C2×Dic3—102no—
354C2×Dic3—201no—
355C2×Dic3—201no—
356D12—102no—
357D12—102no—
358D12—201no—
359D12—201no—
360C3⋊C8—201no—
361C3⋊C8—201no—
362C22×C5—21no—
363C22×C5—21no—
364C22×C5—42no—
365C22×C5—41no—
366C22×C5—41no—
367C22×C5—41no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—41no—
371C22×C5—81no—
372C22×C5#94911yescharacteristic
373C22×C5#95011yescharacteristic
374C22×C5#95311yescharacteristic
375C2×D5—122no—
376C2×D5—122no—
377C2×D5—122no—
378C2×D5—241no—
379C2×D5—241no—
380C2×D5—241no—
381C2×D5—241no—
382C4×C5—21no—
383C4×C5—21no—
384C4×C5—41no—
385C4×C5—41no—
386C4×C5—41no—
387C4×C5—41no—
388Dic5—122no—
389C24—301no—
390C24#122011yescharacteristic
391C2×Q8#122011yescharacteristic, complemented
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—301no—
396C2×D4—301no—
397C2×D4—602no—
398C2×D4—602no—
399C2×D4—602no—
400C2×D4—602no—
401C2×D4#122011yescharacteristic, complemented
402C22×C4—21no—
403C22×C4—301no—
404C22×C4#122011yescharacteristic
405C22×C4C22×D1511yescharacteristic
406SD16—602no—
407D8—602no—
408C2×C8—301no—
409C2×C8—301no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4#122011yescharacteristic, complemented
413C4⋊C4#122011yescharacteristic, complemented
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—602no—
422C42—21no—
423C3×C5#160011yescharacteristic, complemented, Hall
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—42no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—81no—
434C22×C3#409511yescharacteristic
435C22×C3#409611yescharacteristic
436C22×C3#409911yescharacteristic
437C2×D3—202no—
438C2×D3—202no—
439C2×D3—202no—
440C2×D3—401no—
441C2×D3—401no—
442C2×D3—401no—
443C2×D3—401no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—41no—
450Dic3—202no—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—81no—
454C2×C5—81no—
455C2×C5#523811yescharacteristic
456C2×C5#524111yescharacteristic
457C2×C5#524411yescharacteristic
458D5—242no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—41no—
464C23—41no—
465C23—302no—
466C23—601no—
467C23—601no—
468C23—601no—
469C23—601no—
470C23—601no—
471C23—601no—
472C23C2×#122011yescharacteristic
473Q8—41no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—81no—
478D4—81no—
479D4—302no—
480D4—302no—
481D4—601no—
482D4—601no—
483D4—608no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—302no—
492C2×C4—601no—
493C2×C4—601no—
494C2×C4C2×#122011yescharacteristic
495C2×C4C2×#122011yescharacteristic
496C8—601no—
497C8—601no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—81no—
501C2×C3—81no—
502C2×C3#6552711yescharacteristic
503C2×C3#6553011yescharacteristic
504C2×C3#6553311yescharacteristic
505D3—402no—
506C5#7188411yescharacteristic, complemented, Sylow
507C22—21no—
508C22—21no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—81no—
517C22—602no—
518C22—602no—
519C22—602no—
520C22—1201no—
521C22—1201no—
522C22—1201no—
523C22—1201no—
524C22#9183811yescharacteristic
525C22#9183911yescharacteristic
526C22#9184211yescharacteristic
527C4—21no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—602no—
534C3#1060267711yescharacteristic, complemented, Sylow
535C2—21no—
536C2—21no—
537C2—81no—
538C2—81no—
539C2—1202no—
540C2#1175133911yescharacteristic
541C2#1175134211yescharacteristic
542C2#1175134511yescharacteristic
subgroupG/H
centerC22#91842
commutator, cosocleC22×C3×C4×C5C23
FrattiniC22×C4C22×D15
FittingC3×C5×#538C2
socleC22×C3×C5#120
hypercenter#538D15
hypocenterC3×C5#1600
LCSG ▷ C22×C3×C4×C5 ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. C3×C5×#538C2semidirect
2. #11751147C2semidirect
3. #11751137C2semidirect
4. #11751061C2semidirect
5. #11746359C2semidirect
6. #11746358C2semidirect
7. #11746346C2semidirect
29. C5×#538D3semidirect
38. C2×C3×C5×Q8D4semidirect
42. C2×C3×C5×D4D4semidirect
55. C3×C5×C4⋊C4D4semidirect
56. C3×C5×C4⋊C4D4semidirect
70. C3×#538D5semidirect
157. C2×C5×Q8#67semidirect
161. C2×C5×D4#67semidirect
174. C5×C4⋊C4#67semidirect
175. C5×C4⋊C4#67semidirect
189. #538D15semidirect
225. C2×C3×Q8C22⋊D5semidirect
229. C2×C3×D4C22⋊D5semidirect
242. C3×C4⋊C4C22⋊D5semidirect
243. C3×C4⋊C4C22⋊D5semidirect
391. C2×Q8#1220semidirect
401. C2×D4#1220semidirect
412. C4⋊C4#1220semidirect
413. C4⋊C4#1220semidirect
423. C3×C5#1600semidirect
506. C5#71884semidirect
534. C3#10602677semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5382
#117511472
#117511372
#117510612
#117463592
#117463582
#117463462
#106026773
#718845