Finite Group Databasebeta

← C3×#10610422 #14425751

Subgroups — C3×#10610422 #14425751

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
1C3×#66283C211yesmaximal, characteristic, complemented
2C3×#66164C211yesmaximal, characteristic, complemented
3C3×#65859C212yesmaximal, complemented
4C3×C5×#410C211yesmaximal, characteristic, complemented
5C3×#65295C212yesmaximal, complemented
6C3×#65030C218yesmaximal, complemented
7#10610422C311yesmaximal, characteristic, direct product factor, Hall
8C2×C3×C5×#41C2211yescharacteristic
9C3×#4150—22no—
10C3×#4149—22no—
11C3×D5×#41—24no—
12C3×#4147—21no—
13C3×#4144—21no—
14C3×#4144C2214yes—
15C2×C3×#709—22no—
16C2×C3×#709C2212yes—
17C22×C3×D20C2211yescharacteristic, complemented
18C3×C5×#117C2212yescomplemented
19C3×#4093—21no—
20C3×#4086—21no—
21C3×#4083C2214yescomplemented
22C3×#4076—22no—
23C3×#4072C2211yescharacteristic, complemented
24C3×#4041—24no—
25C3×#4028C2218yescomplemented
26C3×#4021—22no—
27C3×#4020—22no—
28C3×C5×#99C2214yescomplemented
29C3×#3938C2218yescomplemented
30C3×#3027—51nomaximal, Hall
31#66283C2×C311yescharacteristic, complemented
32#66164C2×C311yescharacteristic, complemented
33#65859C2×C312yescomplemented
34C5×#410C2×C311yescharacteristic, complemented
35#65295C2×C312yescomplemented
36#65030C2×C318yescomplemented
37C23×C3×D5C2312yes—
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×C3×C22⋊D5—24no—
46C2×C3×C22⋊D5—24no—
47C2×C3×C22⋊D5C2314yes—
48C3×#713—24no—
49C3×Q8×D5—24no—
50C3×#711—24no—
51C3×#711—28no—
52C3×D4×D5—24no—
53C3×D4×D5—28no—
54C3×#709—24no—
55C3×#709—24no—
56C3×#709—22no—
57C3×#709—24no—
58C3×#709—22no—
59C3×#709—22no—
60C3×#709—22no—
61C2×C3×D20—22no—
62C2×C3×D20—21no—
63C2×C3×D20—22no—
64C2×C3×D20—22no—
65C2×C3×D20—21no—
66C2×C3×D20—28no—
67C2×C3×D20D412yescomplemented
68C2×C3×C4×D5—24no—
69C2×C3×C4×D5D414yescomplemented
70C2×C3×Dic10—22no—
71C2×C3×Dic10D412yescomplemented
72C3×C5×C4⋊C4—22no—
73C3×C5×C22⋊C4C2314yes—
74C3×C42×C5—22no—
75C3×C22⋊Dic5C2312yes—
76C3×#685—24no—
77C3×#685—28no—
78C3×C4⋊Dic5—21no—
79C3×C4×Dic5—21no—
80C3×#585—51no—
81C3×#535—51no—
82C3×#535—52no—
83C3×B(4,𝔽2)—58no—
84C3×#410—52no—
85C3×#410D511yescharacteristic, complemented
86C2×C5×#41C22×C311yescharacteristic
87#4150—22no—
88#4149—22no—
89D5×#41—24no—
90#4147—21no—
91#4144—21no—
92#4144C22×C314yes—
93C2×#709—22no—
94C2×#709C22×C312yes—
95C22×D20C22×C311yescharacteristic, complemented
96C5×#117C22×C312yescomplemented
97#4093—21no—
98#4086—21no—
99#4083C22×C314yescomplemented
100#4076—22no—
101#4072C22×C311yescharacteristic, complemented
102#4041—24no—
103#4028C22×C318yescomplemented
104#4021—22no—
105#4020—22no—
106C5×#99C22×C314yescomplemented
107#3938C22×C318yescomplemented
108#3027—51noSylow
109C23×C3×C5C2×D412yes—
110C23×C3×C5C2411yescharacteristic
111C22×C3×D5—24no—
112C22×C3×D5—24no—
113C22×C3×D5—22no—
114C22×C3×D5—82no—
115C22×C3×D5C2×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—
127C3×C22⋊D5—28no—
128C3×C22⋊D5—24no—
129C3×C22⋊D5—24no—
130C3×C22⋊D5—84no—
131C2×C3×Dic5—24no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5C2×D414yes—
134C3×D20—24no—
135C3×D20—22no—
136C3×D20—22no—
137C3×D20—41no—
138C3×D20—42no—
139C3×D20—41no—
140C3×D20—88no—
141C3×C4×D5—28no—
142C3×C4×D5—24no—
143C3×C4×D5—24no—
144C3×Dic10—24no—
145C3×Dic10—22no—
146C3×Dic10—22no—
147C3×2−1+4—102no—
148C3×2−1+4—101no—
149C3×H(5,𝔽2)—54no—
150C3×H(5,𝔽2)—102no—
151C3×H(5,𝔽2)—101no—
152C2×C3×#41—52no—
153C2×C3×#41—104no—
154C2×C3×#41—102no—
155C2×C3×#41C2×D511yescharacteristic
156C22×C3×D4—51no—
157C3×#127—102no—
158C3×#127—101no—
159C3×#124—102no—
160C3×#124—101no—
161C3×#121—104no—
162C3×#121—102no—
163C3×#120—54no—
164C3×#120—58no—
165C3×#117—51no—
166C3×#117C2×D512yescomplemented
167C3×#99—58no—
168C3×#99C2×D514yescomplemented
169C23×D5C23×C312yes—
170C5×#41—22no—
171C5×#41—22no—
172C2×C5×Q8C3×D411yescharacteristic, complemented
173C2×C5×D4C3×D411yescharacteristic, complemented
174C2×C5×D4C23×C312yes—
175C22×C4×C5C3×D412yescomplemented
176C22×C4×C5C23×C311yescharacteristic
177C2×C22⋊D5—24no—
178C2×C22⋊D5—24no—
179C2×C22⋊D5C23×C314yes—
180#713—24no—
181Q8×D5—24no—
182#711—24no—
183#711—28no—
184D4×D5—24no—
185D4×D5—28no—
186#709—24no—
187#709—24no—
188#709—22no—
189#709—24no—
190#709—22no—
191#709—22no—
192#709—22no—
193C2×D20—22no—
194C2×D20—22no—
195C2×D20—22no—
196C2×D20—21no—
197C2×D20—21no—
198C2×D20—28no—
199C2×D20C3×D412yescomplemented
200C2×C4×D5—24no—
201C2×C4×D5C3×D414yescomplemented
202C2×Dic10—22no—
203C2×Dic10C3×D412yescomplemented
204C5×C4⋊C4—22no—
205C5×C22⋊C4C23×C314yes—
206C42×C5—22no—
207C22⋊Dic5C23×C312yes—
208#685—24no—
209#685—28no—
210C4⋊Dic5—21no—
211C4×Dic5—21no—
212#585—51no—
213#535—51no—
214#535—52no—
215B(4,𝔽2)—58no—
216#410—52no—
217#410C3×D511yescharacteristic, complemented
218C22×C3×C5—22no—
219C22×C3×C5—41no—
220C22×C3×C5—42no—
221C22×C3×C5C22×D411yescharacteristic
222C22×C3×C5C22×D412yes—
223C2×C3×D5—24no—
224C2×C3×D5—44no—
225C2×C3×D5—48no—
226C2×C3×D5—44no—
227C2×C3×D5—42no—
228C2×C3×D5—84no—
229C3×C4×C5—22no—
230C3×C4×C5—44no—
231C3×C4×C5#12011yescharacteristic
232C3×C4×C5#12012yes—
233C3×C4×C5#12011yescharacteristic
234C3×Dic5—24no—
235C3×Dic5—42no—
236C24×C3—52no—
237C3×#41—22no—
238C3×#41—22no—
239C3×#41—104no—
240C3×#41—104no—
241C3×#41—108no—
242C3×#41—104no—
243C3×#41—104no—
244C3×#41—102no—
245C3×#41—102no—
246C3×#41—104no—
247C3×#41—102no—
248C3×#41—102no—
249C2×C3×Q8—52no—
250C2×C3×Q8—102no—
251C2×C3×Q8—104no—
252C2×C3×Q8D2011yescharacteristic, complemented
253C2×C3×D4—54no—
254C2×C3×D4—52no—
255C2×C3×D4—102no—
256C2×C3×D4—108no—
257C2×C3×D4—108no—
258C2×C3×D4—104no—
259C2×C3×D4—104no—
260C2×C3×D4—104no—
261C2×C3×D4—101no—
262C2×C3×D4—101no—
263C2×C3×D4—102no—
264C2×C3×D4—102no—
265C2×C3×D4D2011yescharacteristic, complemented
266C2×C3×D4C22×D512yes—
267C22×C3×C4—54no—
268C22×C3×C4—104no—
269C22×C3×C4D2012yescomplemented
270C22×C3×C4C22×D511yescharacteristic
271C3×C4⋊C4—22no—
272C3×C4⋊C4—101no—
273C3×C22⋊C4—52no—
274C3×C22⋊C4—108no—
275C3×C22⋊C4—104no—
276C3×C22⋊C4C22×D514yes—
277C3×C42—22no—
278C3×C42—101no—
279C23×C5C2×C3×D412yes—
280C23×C5C24×C311yescharacteristic
281C22×D5—24no—
282C22×D5—24no—
283C22×D5—22no—
284C22×D5—82no—
285C22×D5C2×C3×D414yes—
286C5×Q8—22no—
287C5×D4—24no—
288C5×D4—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—24no—
292C2×C4×C5C2×C3×D412yes—
293C2×C4×C5C2×C3×D411yescharacteristic
294C2×C4×C5C2×C3×D411yescharacteristic
295C2×C4×C5C2×C3×D412yes—
296C2×C4×C5C2×C3×D412yes—
297C22⋊D5—28no—
298C22⋊D5—24no—
299C22⋊D5—24no—
300C22⋊D5—84no—
301C2×Dic5—24no—
302C2×Dic5—22no—
303C2×Dic5C2×C3×D414yes—
304D20—24no—
305D20—22no—
306D20—22no—
307D20—41no—
308D20—42no—
309D20—41no—
310D20—88no—
311C4×D5—28no—
312C4×D5—24no—
313C4×D5—24no—
314Dic10—24no—
315Dic10—22no—
316Dic10—22no—
3172−1+4—102no—
3182−1+4—101no—
319H(5,𝔽2)—54no—
320H(5,𝔽2)—102no—
321H(5,𝔽2)—101no—
322C2×#41—52no—
323C2×#41—104no—
324C2×#41—102no—
325C2×#41C2×C3×D511yescharacteristic
326C22×D4—51no—
327#127—102no—
328#127—101no—
329#124—102no—
330#124—101no—
331#121—104no—
332#121—102no—
333#120—54no—
334#120—58no—
335#117—51no—
336#117C2×C3×D512yescomplemented
337#99—58no—
338#99C2×C3×D514yescomplemented
339C2×C3×C5—21no—
340C2×C3×C5—22no—
341C2×C3×C5—42no—
342C2×C3×C5C2×#12011yescharacteristic
343C3×D5—44no—
344C3×D5—82no—
345C23×C3—54no—
346C23×C3—104no—
347C23×C3—102no—
348C23×C3—104no—
349C23×C3—402no—
350C23×C3C2×D2012yes—
351C23×C3C23×D511yescharacteristic
352C3×Q8—22no—
353C3×Q8—104no—
354C3×Q8—102no—
355C3×Q8—102no—
356C3×D4—24no—
357C3×D4—22no—
358C3×D4—104no—
359C3×D4—108no—
360C3×D4—104no—
361C3×D4—104no—
362C3×D4—102no—
363C3×D4—102no—
364C3×D4—201no—
365C3×D4—201no—
366C3×D4—202no—
367C3×D4—408no—
368C3×D4—404no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—24no—
372C2×C3×C4—54no—
373C2×C3×C4—108no—
374C2×C3×C4—104no—
375C2×C3×C4—104no—
376C2×C3×C4—104no—
377C2×C3×C4—102no—
378C2×C3×C4C2×D2012yes—
379C2×C3×C4C2×D2011yescharacteristic
380C2×C3×C4C2×D2011yescharacteristic
381C2×C3×C4D4×D512yes—
382C2×C3×C4D4×D512yes—
383C22×C5—22no—
384C22×C5—41no—
385C22×C5—42no—
386C22×C5C22×C3×D411yescharacteristic
387C22×C5C22×C3×D412yes—
388C2×D5—24no—
389C2×D5—44no—
390C2×D5—48no—
391C2×D5—44no—
392C2×D5—42no—
393C2×D5—84no—
394C4×C5—22no—
395C4×C5—44no—
396C4×C5C3×#12011yescharacteristic
397C4×C5C3×#12012yes—
398C4×C5C3×#12011yescharacteristic
399Dic5—24no—
400Dic5—42no—
401C24—52no—
402#41—22no—
403#41—22no—
404#41—104no—
405#41—104no—
406#41—108no—
407#41—104no—
408#41—104no—
409#41—102no—
410#41—102no—
411#41—104no—
412#41—102no—
413#41—102no—
414C2×Q8—52no—
415C2×Q8—102no—
416C2×Q8—104no—
417C2×Q8C3×D2011yescharacteristic, complemented
418C2×D4—54no—
419C2×D4—52no—
420C2×D4—102no—
421C2×D4—108no—
422C2×D4—108no—
423C2×D4—104no—
424C2×D4—104no—
425C2×D4—104no—
426C2×D4—101no—
427C2×D4—101no—
428C2×D4—102no—
429C2×D4—102no—
430C2×D4C3×D2011yescharacteristic, complemented
431C2×D4C22×C3×D512yes—
432C22×C4—54no—
433C22×C4—104no—
434C22×C4C3×D2012yescomplemented
435C22×C4C22×C3×D511yescharacteristic
436C4⋊C4—22no—
437C4⋊C4—101no—
438C22⋊C4—52no—
439C22⋊C4—108no—
440C22⋊C4—104no—
441C22⋊C4C22×C3×D514yes—
442C42—22no—
443C42—101no—
444C3×C5#302711yescharacteristic, complemented, Hall
445C22×C3—22no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3—104no—
449C22×C3—204no—
450C22×C3—208no—
451C22×C3—204no—
452C22×C3—202no—
453C22×C3—404no—
454C22×C3C22×D2011yescharacteristic
455C22×C3C2×D4×D512yes—
456C3×C4—22no—
457C3×C4—44no—
458C3×C4—104no—
459C3×C4—202no—
460C3×C4#402811yescharacteristic
461C3×C4#402811yescharacteristic
462C3×C4#402812yes—
463C2×C5—21no—
464C2×C5—22no—
465C2×C5—42no—
466C2×C5C2×C3×#12011yescharacteristic
467D5—44no—
468D5—82no—
469C23—54no—
470C23—104no—
471C23—102no—
472C23—104no—
473C23—402no—
474C23C2×C3×D2012yes—
475C23C23×C3×D511yescharacteristic
476Q8—22no—
477Q8—104no—
478Q8—102no—
479Q8—102no—
480D4—24no—
481D4—22no—
482D4—104no—
483D4—108no—
484D4—104no—
485D4—104no—
486D4—102no—
487D4—102no—
488D4—201no—
489D4—201no—
490D4—202no—
491D4—408no—
492D4—404no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—54no—
497C2×C4—108no—
498C2×C4—104no—
499C2×C4—104no—
500C2×C4—104no—
501C2×C4—102no—
502C2×C4C2×C3×D2012yes—
503C2×C4C2×C3×D2011yescharacteristic
504C2×C4C2×C3×D2011yescharacteristic
505C2×C4C3×D4×D512yes—
506C2×C4C3×D4×D512yes—
507C2×C3—22no—
508C2×C3—21no—
509C2×C3—42no—
510C2×C3—204no—
511C2×C3—402no—
512C2×C3C2×#402811yescharacteristic
513C5C3×#302711yescharacteristic, complemented, Sylow
514C22—22no—
515C22—41no—
516C22—42no—
517C22—104no—
518C22—204no—
519C22—208no—
520C22—204no—
521C22—202no—
522C22—404no—
523C22C22×C3×D2011yescharacteristic
524C22C2×C3×D4×D512yes—
525C4—22no—
526C4—44no—
527C4—104no—
528C4—202no—
529C4C3×#402811yescharacteristic
530C4C3×#402811yescharacteristic
531C4C3×#402812yes—
532C3#1061042211yescharacteristic, direct product factor, Sylow
533C2—22no—
534C2—21no—
535C2—42no—
536C2—204no—
537C2—402no—
538C2C2×C3×#402811yescharacteristic
subgroupG/H
centerC2×C3C2×#4028
commutator, cosocleC23×C5C24×C3
FrattiniC23C23×C3×D5
FittingC3×C5×#410C2
socleC2×C3×C5C2×#120
hypercenterC3×#410D5
hypocenterC5C3×#3027
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
1. C3×#66283C2semidirect
2. C3×#66164C2semidirect
3. C3×#65859C2semidirect
4. C3×C5×#410C2semidirect
5. C3×#65295C2semidirect
6. C3×#65030C2semidirect
7. #10610422C3direct (N × H)
17. C22×C3×D20C22semidirect
18. C3×C5×#117C22semidirect
21. C3×#4083C22semidirect
23. C3×#4072C22semidirect
25. C3×#4028C22semidirect
28. C3×C5×#99C22semidirect
29. C3×#3938C22semidirect
31. #66283C2×C3semidirect
32. #66164C2×C3semidirect
33. #65859C2×C3semidirect
34. C5×#410C2×C3semidirect
35. #65295C2×C3semidirect
36. #65030C2×C3semidirect
40. C2×C3×C5×Q8D4semidirect
41. C2×C3×C5×D4D4semidirect
43. C22×C3×C4×C5D4semidirect
67. C2×C3×D20D4semidirect
69. C2×C3×C4×D5D4semidirect
71. C2×C3×Dic10D4semidirect
85. C3×#410D5semidirect
95. C22×D20C22×C3semidirect
96. C5×#117C22×C3semidirect
99. #4083C22×C3semidirect
101. #4072C22×C3semidirect
103. #4028C22×C3semidirect
106. C5×#99C22×C3semidirect
107. #3938C22×C3semidirect
166. C3×#117C2×D5semidirect
168. C3×#99C2×D5semidirect
172. C2×C5×Q8C3×D4semidirect
173. C2×C5×D4C3×D4semidirect
175. C22×C4×C5C3×D4semidirect
199. C2×D20C3×D4semidirect
201. C2×C4×D5C3×D4semidirect
203. C2×Dic10C3×D4semidirect
217. #410C3×D5semidirect
252. C2×C3×Q8D20semidirect
265. C2×C3×D4D20semidirect
269. C22×C3×C4D20semidirect
336. #117C2×C3×D5semidirect
338. #99C2×C3×D5semidirect
417. C2×Q8C3×D20semidirect
430. C2×D4C3×D20semidirect
434. C22×C4C3×D20semidirect
444. C3×C5#3027semidirect
513. C5C3×#3027semidirect
532. C3#10610422direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662832
C3×#661642
C3×#658592
C3×C5×#4102
C3×#652952
C3×#650302
#106104223
C3×#30275