Finite Group Databasebeta

← #14425752

Subgroups — #14425752

Order 1920.

1544 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11757146C211yesmaximal, characteristic, complemented
2#11756589C211yesmaximal, characteristic, complemented
3#11756284C212yesmaximal, complemented
4C3×C5×#410C211yesmaximal, characteristic, complemented
5#11751107C212yesmaximal, complemented
6#11750842C218yesmaximal, complemented
7#10610422—31nomaximal, Hall
8C2×C3×C5×#41C2211yescharacteristic
9#92101—22no—
10#92100—22no—
11#41×D15—24no—
12#92098—21no—
13#92095—21no—
14#92095C2214yes—
15C2×#6817—22no—
16C2×#6817C2212yes—
17C22×D60C2211yescharacteristic, complemented
18C3×C5×#117C2212yescomplemented
19#91836—21no—
20#91829—21no—
21#91826C2214yescomplemented
22#91819—22no—
23#91815C2211yescharacteristic, complemented
24#91784—24no—
25#91771C2218yescomplemented
26#91764—22no—
27#91763—22no—
28C3×C5×#99C2214yescomplemented
29#91090C2218yescomplemented
30#78691—51nomaximal, Hall
31#66283—31no—
32#66164—31no—
33#65859—32no—
34C5×#410D311yescharacteristic, complemented
35#65295—32no—
36#65030—38no—
37C23×D15C2312yes—
38C3×C5×#41—22no—
39C3×C5×#41—22no—
40C2×C3×C5×Q8D411yescharacteristic, complemented
41C2×C3×C5×D4D411yescharacteristic, complemented
42C2×C3×C5×D4C2312yes—
43C22×C3×C4×C5D412yescomplemented
44C22×C3×C4×C5C2311yescharacteristic
45C2×#1220—24no—
46C2×#1220—24no—
47C2×#1220C2314yes—
48#6821—24no—
49Q8×D15—24no—
50#6819—24no—
51#6819—28no—
52D4×D15—24no—
53D4×D15—28no—
54#6817—24no—
55#6817—24no—
56#6817—22no—
57#6817—24no—
58#6817—22no—
59#6817—22no—
60#6817—22no—
61C2×D60—22no—
62C2×D60—22no—
63C2×D60—22no—
64C2×D60—21no—
65C2×D60—21no—
66C2×D60—28no—
67C2×D60D412yescomplemented
68C2×C4×D15—24no—
69C2×C4×D15D414yescomplemented
70C2×Dic30—22no—
71C2×Dic30D412yescomplemented
72C3×C5×C4⋊C4—22no—
73C3×C5×C22⋊C4C2314yes—
74C3×C42×C5—22no—
75#6719C2312yes—
76#6714—24no—
77#6714—28no—
78C4⋊Dic15—21no—
79C4×Dic15—21no—
80#5964—51no—
81#5830—51no—
82#5525—52no—
83C3×#410D511yescharacteristic, complemented
84#5006—52no—
85#4741—58no—
86C2×C5×#41C2×D311yescharacteristic
87#4150—62no—
88#4149—62no—
89D5×#41—64no—
90#4147—61no—
91#4144—34no—
92#4144—61no—
93C2×#709—32no—
94C2×#709—62no—
95C22×D20—31no—
96C5×#117C2×D312yescomplemented
97#4093—61no—
98#4086—61no—
99#4083—34no—
100#4076—62no—
101#4072—31no—
102#4041—64no—
103#4028—38no—
104#4021—62no—
105#4020—62no—
106C5×#99C2×D314yescomplemented
107#3938—38no—
108#3027—151noSylow
109C23×C3×C5C2×D412yes—
110C23×C3×C5C2411yescharacteristic
111C22×D15—24no—
112C22×D15—24no—
113C22×D15—22no—
114C22×D15—82no—
115C22×D15C2×D414yes—
116C3×C5×Q8—22no—
117C3×C5×D4—24no—
118C3×C5×D4—22no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—22no—
121C2×C3×C4×C5—24no—
122C2×C3×C4×C5C2×D412yes—
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D412yes—
126C2×C3×C4×C5C2×D412yes—
127#1220—28no—
128#1220—24no—
129#1220—24no—
130#1220—84no—
131C2×Dic15—24no—
132C2×Dic15—22no—
133C2×Dic15C2×D414yes—
134D60—24no—
135D60—22no—
136D60—22no—
137D60—41no—
138D60—42no—
139D60—41no—
140D60—88no—
141C4×D15—28no—
142C4×D15—24no—
143C4×D15—24no—
144Dic30—24no—
145Dic30—22no—
146Dic30—22no—
147C2×C3×#41C2×D511yescharacteristic
148#1010—102no—
149#1009—102no—
150D3×#41—104no—
151#1007—101no—
152#1004—54no—
153#1004—101no—
154C2×#235—52no—
155C2×#235—102no—
156C22×D12—51no—
157C3×#117C2×D512yescomplemented
158#947—101no—
159#940—101no—
160#937—54no—
161#930—102no—
162#926—51no—
163#895—104no—
164#882—58no—
165#875—102no—
166#874—102no—
167C3×#99C2×D514yescomplemented
168#806—58no—
169C23×D5—32no—
170C5×#41—22no—
171C5×#41—22no—
172C2×C5×Q8D1211yescharacteristic, complemented
173C2×C5×D4D1211yescharacteristic, complemented
174C2×C5×D4C22×D312yes—
175C22×C4×C5D1212yescomplemented
176C22×C4×C5C22×D311yescharacteristic
177C2×C22⋊D5—34no—
178C2×C22⋊D5—64no—
179C2×C22⋊D5—64no—
180#713—64no—
181Q8×D5—64no—
182#711—68no—
183#711—64no—
184D4×D5—68no—
185D4×D5—64no—
186#709—64no—
187#709—64no—
188#709—62no—
189#709—62no—
190#709—64no—
191#709—62no—
192#709—62no—
193C2×D20—32no—
194C2×D20—62no—
195C2×D20—61no—
196C2×D20—61no—
197C2×D20—62no—
198C2×D20—62no—
199C2×D20—68no—
200C2×C4×D5—34no—
201C2×C4×D5—64no—
202C2×Dic10—32no—
203C2×Dic10—62no—
204C5×C4⋊C4—22no—
205C5×C22⋊C4C22×D314yes—
206C42×C5—22no—
207C22⋊Dic5—32no—
208#685—68no—
209#685—64no—
210C4⋊Dic5—61no—
211C4×Dic5—61no—
212#585—151no—
213#535—151no—
214#535—152no—
215B(4,𝔽2)—158no—
216#410—152no—
217#410D1511yescharacteristic, complemented
218C22×C3×C5—22no—
219C22×C3×C5—41no—
220C22×C3×C5—42no—
221C22×C3×C5C22×D411yescharacteristic
222C22×C3×C5C22×D412yes—
223C2×D15—24no—
224C2×D15—44no—
225C2×D15—48no—
226C2×D15—44no—
227C2×D15—42no—
228C2×D15—84no—
229C3×C4×C5—22no—
230C3×C4×C5—44no—
231C3×C4×C5#12011yescharacteristic
232C3×C4×C5#12012yes—
233C3×C4×C5#12011yescharacteristic
234Dic15—24no—
235Dic15—42no—
236C23×D3—52no—
237C3×#41—22no—
238C3×#41—22no—
239C2×C3×Q8D2011yescharacteristic, complemented
240C2×C3×D4D2011yescharacteristic, complemented
241C2×C3×D4C22×D512yes—
242C22×C3×C4D2012yescomplemented
243C22×C3×C4C22×D511yescharacteristic
244C2×#67—54no—
245C2×#67—104no—
246C2×#67—104no—
247#239—104no—
248D3×Q8—104no—
249#237—108no—
250#237—104no—
251D3×D4—108no—
252D3×D4—104no—
253#235—104no—
254#235—104no—
255#235—102no—
256#235—102no—
257#235—104no—
258#235—102no—
259#235—102no—
260C2×D12—52no—
261C2×D12—102no—
262C2×D12—101no—
263C2×D12—101no—
264C2×D12—102no—
265C2×D12—102no—
266C2×D12—108no—
267C2×C4×D3—54no—
268C2×C4×D3—104no—
269C2×Dic6—52no—
270C2×Dic6—102no—
271C3×C4⋊C4—22no—
272C3×C22⋊C4C22×D514yes—
273C3×C42—22no—
274#217—52no—
275(C2×D3)⋊C4—108no—
276(C2×D3)⋊C4—104no—
277C4⋊Dic3—101no—
278C4×Dic3—101no—
279C23×C5C2×D1212yes—
280C23×C5C23×D311yescharacteristic
281C22×D5—34no—
282C22×D5—64no—
283C22×D5—62no—
284C22×D5—64no—
285C22×D5—242no—
286C5×Q8—22no—
287C5×D4—24no—
288C5×D4—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—24no—
292C2×C4×C5C2×D1212yes—
293C2×C4×C5C2×D1211yescharacteristic
294C2×C4×C5C2×D1211yescharacteristic
295C2×C4×C5D3×D412yes—
296C2×C4×C5D3×D412yes—
297C22⋊D5—68no—
298C22⋊D5—64no—
299C22⋊D5—64no—
300C22⋊D5—244no—
301C2×Dic5—34no—
302C2×Dic5—64no—
303C2×Dic5—62no—
304D20—64no—
305D20—62no—
306D20—62no—
307D20—121no—
308D20—122no—
309D20—121no—
310D20—248no—
311C4×D5—68no—
312C4×D5—64no—
313C4×D5—64no—
314Dic10—64no—
315Dic10—62no—
316Dic10—62no—
3172−1+4—302no—
3182−1+4—301no—
319H(5,𝔽2)—154no—
320H(5,𝔽2)—302no—
321H(5,𝔽2)—301no—
322C2×#41—152no—
323C2×#41—304no—
324C2×#41—302no—
325C2×#41C2×D1511yescharacteristic
326C22×D4—151no—
327#127—302no—
328#127—301no—
329#124—302no—
330#124—301no—
331#121—304no—
332#121—302no—
333#120—154no—
334#120—158no—
335#117—151no—
336#117C2×D1512yescomplemented
337#99—158no—
338#99C2×D1514yescomplemented
339C2×C3×C5—21no—
340C2×C3×C5—22no—
341C2×C3×C5—42no—
342C2×C3×C5C2×#12011yescharacteristic
343D15—44no—
344D15—82no—
345C23×C3C2×D2012yes—
346C23×C3C23×D511yescharacteristic
347C22×D3—54no—
348C22×D3—104no—
349C22×D3—102no—
350C22×D3—104no—
351C22×D3—402no—
352C3×Q8—22no—
353C3×D4—24no—
354C3×D4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4C2×D2012yes—
359C2×C3×C4C2×D2011yescharacteristic
360C2×C3×C4C2×D2011yescharacteristic
361C2×C3×C4D4×D512yes—
362C2×C3×C4D4×D512yes—
363#67—108no—
364#67—104no—
365#67—104no—
366#67—404no—
367C2×Dic3—54no—
368C2×Dic3—104no—
369C2×Dic3—102no—
370D12—104no—
371D12—102no—
372D12—102no—
373D12—201no—
374D12—202no—
375D12—201no—
376D12—408no—
377C4×D3—108no—
378C4×D3—104no—
379C4×D3—104no—
380Dic6—104no—
381Dic6—102no—
382Dic6—102no—
383C22×C5—22no—
384C22×C5—41no—
385C22×C5—42no—
386C22×C5C22×D1211yescharacteristic
387C22×C5C2×D3×D412yes—
388C2×D5—64no—
389C2×D5—124no—
390C2×D5—128no—
391C2×D5—124no—
392C2×D5—122no—
393C2×D5—244no—
394C4×C5—22no—
395C4×C5—44no—
396C4×C5#88211yescharacteristic
397C4×C5#88212yes—
398C4×C5#88211yescharacteristic
399Dic5—64no—
400Dic5—122no—
401C24—152no—
402#41—22no—
403#41—22no—
404#41—304no—
405#41—304no—
406#41—308no—
407#41—304no—
408#41—304no—
409#41—302no—
410#41—302no—
411#41—304no—
412#41—302no—
413#41—302no—
414C2×Q8—152no—
415C2×Q8—302no—
416C2×Q8—304no—
417C2×Q8D6011yescharacteristic, complemented
418C2×D4—154no—
419C2×D4—152no—
420C2×D4—302no—
421C2×D4—308no—
422C2×D4—308no—
423C2×D4—304no—
424C2×D4—304no—
425C2×D4—304no—
426C2×D4—301no—
427C2×D4—301no—
428C2×D4—302no—
429C2×D4—302no—
430C2×D4D6011yescharacteristic, complemented
431C2×D4C22×D1512yes—
432C22×C4—154no—
433C22×C4—304no—
434C22×C4D6012yescomplemented
435C22×C4C22×D1511yescharacteristic
436C4⋊C4—22no—
437C4⋊C4—301no—
438C22⋊C4—152no—
439C22⋊C4—308no—
440C22⋊C4—304no—
441C22⋊C4C22×D1514yes—
442C42—22no—
443C42—301no—
444C3×C5#302711yescharacteristic, complemented, Hall
445C22×C3—22no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3C22×D2011yescharacteristic
449C22×C3C2×D4×D512yes—
450C2×D3—104no—
451C2×D3—204no—
452C2×D3—208no—
453C2×D3—204no—
454C2×D3—202no—
455C2×D3—404no—
456C3×C4—22no—
457C3×C4—44no—
458C3×C4#402811yescharacteristic
459C3×C4#402812yes—
460C3×C4#402811yescharacteristic
461Dic3—104no—
462Dic3—202no—
463C2×C5—21no—
464C2×C5—22no—
465C2×C5—42no—
466C2×C5C2×#88211yescharacteristic
467D5—124no—
468D5—242no—
469C23—154no—
470C23—304no—
471C23—302no—
472C23—304no—
473C23—1202no—
474C23C2×D6012yes—
475C23C23×D1511yescharacteristic
476Q8—22no—
477Q8—304no—
478Q8—302no—
479Q8—302no—
480D4—24no—
481D4—22no—
482D4—304no—
483D4—308no—
484D4—304no—
485D4—304no—
486D4—302no—
487D4—302no—
488D4—601no—
489D4—601no—
490D4—602no—
491D4—1208no—
492D4—1204no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—154no—
497C2×C4—308no—
498C2×C4—304no—
499C2×C4—304no—
500C2×C4—304no—
501C2×C4—302no—
502C2×C4C2×D6012yes—
503C2×C4C2×D6011yescharacteristic
504C2×C4C2×D6011yescharacteristic
505C2×C4D4×D1512yes—
506C2×C4D4×D1512yes—
507C2×C3—21no—
508C2×C3—22no—
509C2×C3—42no—
510C2×C3C2×#402811yescharacteristic
511D3—204no—
512D3—402no—
513C5#7869111yescharacteristic, complemented, Sylow
514C22—22no—
515C22—41no—
516C22—42no—
517C22—304no—
518C22—604no—
519C22—608no—
520C22—604no—
521C22—602no—
522C22—1204no—
523C22C22×D6011yescharacteristic
524C22C2×D4×D1512yes—
525C4—22no—
526C4—44no—
527C4—304no—
528C4—602no—
529C4#9177111yescharacteristic
530C4#9177111yescharacteristic
531C4#9177112yes—
532C3#1061042211yescharacteristic, complemented, Sylow
533C2—22no—
534C2—21no—
535C2—42no—
536C2—604no—
537C2—1202no—
538C2C2×#9177111yescharacteristic
subgroupG/H
centerC2C2×#91771
commutator, cosocleC23×C3×C5C24
FrattiniC23C23×D15
FittingC3×C5×#410C2
socleC2×C3×C5C2×#120
hypercenter#410D15
hypocenterC3×C5#3027
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #11757146C2semidirect
2. #11756589C2semidirect
3. #11756284C2semidirect
4. C3×C5×#410C2semidirect
5. #11751107C2semidirect
6. #11750842C2semidirect
17. C22×D60C22semidirect
18. C3×C5×#117C22semidirect
21. #91826C22semidirect
23. #91815C22semidirect
25. #91771C22semidirect
28. C3×C5×#99C22semidirect
29. #91090C22semidirect
34. C5×#410D3semidirect
40. C2×C3×C5×Q8D4semidirect
41. C2×C3×C5×D4D4semidirect
43. C22×C3×C4×C5D4semidirect
67. C2×D60D4semidirect
69. C2×C4×D15D4semidirect
71. C2×Dic30D4semidirect
83. C3×#410D5semidirect
96. C5×#117C2×D3semidirect
106. C5×#99C2×D3semidirect
157. C3×#117C2×D5semidirect
167. C3×#99C2×D5semidirect
172. C2×C5×Q8D12semidirect
173. C2×C5×D4D12semidirect
175. C22×C4×C5D12semidirect
217. #410D15semidirect
239. C2×C3×Q8D20semidirect
240. C2×C3×D4D20semidirect
242. C22×C3×C4D20semidirect
336. #117C2×D15semidirect
338. #99C2×D15semidirect
417. C2×Q8D60semidirect
430. C2×D4D60semidirect
434. C22×C4D60semidirect
444. C3×C5#3027semidirect
513. C5#78691semidirect
532. C3#10610422semidirect

Maximal subgroups

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

subgroupindex
#117571462
#117565892
#117562842
C3×C5×#4102
#117511072
#117508422
#106104223
#786915