Finite Group Databasebeta

← #14323137

Subgroups — #14323137

Order 1920.

1080 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×#91782C211yesmaximal, characteristic
2C3×C5×#389C211yesmaximal, characteristic, complemented
3#11751091C212yesmaximal, complemented
4#11751089C211yesmaximal, characteristic
5#11751088C212yesmaximal, complemented
6#11751085C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
8#11751040C211yesmaximal, characteristic
9#11751033C211yesmaximal, characteristic, complemented
10#11750771C212yesmaximal, complemented
11#11750765C212yesmaximal, complemented
12#10607700—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C42×C5C2211yescharacteristic
19C2×#6714C2212yes—
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25C2×C4⋊Dic15C2212yes—
26C2×C4×Dic15C2211yescharacteristic
27C2×C4×Dic15C2212yes—
28C2×C4×Dic15C2211yescharacteristic
29#91782C418yescomplemented
30C3×C5×#95C2212yes—
31#91115C2211yescharacteristic
32#91115C2212yes—
33#91115C2212yes—
34#91115C2211yescharacteristic
35#91115C2211yescharacteristic
36#91115C2211yescharacteristic
37#75969—51nomaximal, Hall
38C2×#4039—31no—
39C5×#389D311yescharacteristic, complemented
40#65279—32no—
41#65277—31no—
42#65276—32no—
43#65274—31no—
44C4⋊C4×Dic5—31no—
45#65228—31no—
46#65221—31no—
47#64959—32no—
48#64953—32no—
49C23×D15C2312yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2311yescharacteristic
52C22×C3×C4×C5C2311yescharacteristic
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×Dic15C2311yescharacteristic
56C22×Dic15C2312yes—
57C22×Dic15C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C2×D60—22no—
61C2×D60—22no—
62C2×D60C2×C414yes—
63C2×C4×D15—42no—
64C2×C4×D15—42no—
65C2×C4×D15C2×C418yes—
66C3×C5×C4⋊C4—22no—
67C3×C5×C4⋊C4—42no—
68C3×C5×C4⋊C4C2×C414yes—
69C3×C42×C5—22no—
70#6714—42no—
71#6714—42no—
72#6714—42no—
73#6714—42no—
74#6714C2×C418yes—
75C4⋊Dic15—22no—
76C4⋊Dic15—42no—
77C4×Dic15—22no—
78C4×Dic15—42no—
79C4×Dic15C2×C414yes—
80C2×#893—51no—
81C3×#389D511yescharacteristic, complemented
82#4990—52no—
83#4988—51no—
84#4987—52no—
85#4984—51no—
86Dic3×C4⋊C4—51no—
87#4939—51no—
88#4932—51no—
89#4670—52no—
90#4664—52no—
91C22×D20—31no—
92C22×C4×D5—32no—
93C2×C5×C4⋊C4C2×D311yescharacteristic
94C2×C5×C4⋊C4C2×D312yes—
95C2×C5×C4⋊C4C2×D311yescharacteristic
96C2×C42×C5C2×D311yescharacteristic
97C2×#685—32no—
98C2×#685—32no—
99C2×#685—32no—
100C2×#685—32no—
101C2×#685—32no—
102C2×C4⋊Dic5—31no—
103C2×C4⋊Dic5—32no—
104C2×C4×Dic5—31no—
105C2×C4×Dic5—31no—
106C2×C4×Dic5—32no—
107#4039—38no—
108C5×#95C2×D312yes—
109#3963—31no—
110#3963—32no—
111#3963—32no—
112#3963—31no—
113#3963—31no—
114#3963—31no—
115#2319—151noSylow
116C23×C3×C5C2411yescharacteristic
117C22×D15—22no—
118C22×D15—22no—
119C22×D15—22no—
120C22×D15—22no—
121C22×D15—22no—
122C22×D15—22no—
123C22×D15C22×C414yes—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5C22×C414yes—
136C2×C3×C4×C5C22×C412yes—
137C2×C3×C4×C5#4112yes—
138C2×C3×C4×C5#4112yes—
139C2×Dic15—21no—
140C2×Dic15—21no—
141C2×Dic15—21no—
142C2×Dic15—21no—
143C2×Dic15—21no—
144C2×Dic15—21no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15C22×C412yes—
153C2×Dic15C22×C412yes—
154D60—28no—
155C4×D15—48no—
156C22×D12—51no—
157C22×C4×D3—52no—
158C2×C3×C4⋊C4C2×D511yescharacteristic
159C2×C3×C4⋊C4C2×D512yes—
160C2×C3×C4⋊C4C2×D511yescharacteristic
161C2×C3×C42C2×D511yescharacteristic
162C2×(C2×D3)⋊C4—52no—
163C2×(C2×D3)⋊C4—52no—
164C2×(C2×D3)⋊C4—52no—
165C2×(C2×D3)⋊C4—52no—
166C2×(C2×D3)⋊C4—52no—
167C2×C4⋊Dic3—51no—
168C2×C4⋊Dic3—52no—
169C2×C4×Dic3—51no—
170C2×C4×Dic3—51no—
171C2×C4×Dic3—52no—
172#893—58no—
173C3×#95C2×D512yes—
174C3⋊#95—51no—
175C3⋊#95—52no—
176C3⋊#95—52no—
177C3⋊#95—51no—
178C3⋊#95—51no—
179C3⋊#95—51no—
180C23×D5—32no—
181C22×C4×C5C22×D311yescharacteristic
182C22×C4×C5C22×D311yescharacteristic
183C22×C4×C5C22×D312yes—
184C22×C4×C5C22×D312yes—
185C22×C4×C5C22×D311yescharacteristic
186C22×Dic5—31no—
187C22×Dic5—31no—
188C22×Dic5—31no—
189C22×Dic5—31no—
190C22×Dic5—32no—
191C2×D20—34no—
192C2×D20—62no—
193C2×D20—62no—
194C2×C4×D5—38no—
195C2×C4×D5—122no—
196C2×C4×D5—122no—
197C5×C4⋊C4—22no—
198C5×C4⋊C4—42no—
199C5×C4⋊C4C4×D314yes—
200C42×C5—22no—
201#685—38no—
202#685—122no—
203#685—122no—
204#685—122no—
205#685—122no—
206C4⋊Dic5—62no—
207C4⋊Dic5—122no—
208C4×Dic5—34no—
209C4×Dic5—62no—
210C4×Dic5—122no—
211C2×C4×D4—151no—
212#390—151no—
213#390—151no—
214#389—151no—
215#389D1511yescharacteristic, complemented
216#388—152no—
217#388—152no—
218#381—152no—
219#380—152no—
220C4×C4⋊C4—151no—
221#376—151no—
222C22×C3×C5C23×C411yescharacteristic
223C22×C3×C5C2×#4111yescharacteristic
224C22×C3×C5C2×#4111yescharacteristic
225C22×C3×C5H(5,𝔽2)11yescharacteristic
226C22×C3×C5H(5,𝔽2)11yescharacteristic
227C22×C3×C5H(5,𝔽2)11yescharacteristic
228C22×C3×C52−1+411yescharacteristic
229C2×D15—24no—
230C2×D15—24no—
231C2×D15—24no—
232C2×D15—42no—
233C2×D15—42no—
234C2×D15—42no—
235C2×D15—42no—
236C3×C4×C5—24no—
237C3×C4×C5—41no—
238C3×C4×C5—42no—
239C3×C4×C5—41no—
240C3×C4×C5#11714yes—
241Dic15—22no—
242Dic15—22no—
243Dic15—41no—
244Dic15—42no—
245Dic15—41no—
246C23×D3—52no—
247C22×C3×C4C22×D511yescharacteristic
248C22×C3×C4C22×D511yescharacteristic
249C22×C3×C4C22×D512yes—
250C22×C3×C4C22×D512yes—
251C22×C3×C4C22×D511yescharacteristic
252C22×Dic3—51no—
253C22×Dic3—51no—
254C22×Dic3—51no—
255C22×Dic3—51no—
256C22×Dic3—52no—
257C2×D12—54no—
258C2×D12—102no—
259C2×D12—102no—
260C2×C4×D3—58no—
261C2×C4×D3—202no—
262C2×C4×D3—202no—
263C3×C4⋊C4—22no—
264C3×C4⋊C4—42no—
265C3×C4⋊C4C4×D514yes—
266C3×C42—22no—
267(C2×D3)⋊C4—58no—
268(C2×D3)⋊C4—202no—
269(C2×D3)⋊C4—202no—
270(C2×D3)⋊C4—202no—
271(C2×D3)⋊C4—202no—
272C4⋊Dic3—102no—
273C4⋊Dic3—202no—
274C4×Dic3—54no—
275C4×Dic3—102no—
276C4×Dic3—202no—
277C23×C5C23×D311yescharacteristic
278C22×D5—34no—
279C22×D5—62no—
280C22×D5—62no—
281C22×D5—62no—
282C22×D5—62no—
283C22×D5—62no—
284C22×D5—62no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5C2×C4×D314yes—
297C2×C4×C5C2×C4×D312yes—
298C2×C4×C5#23712yes—
299C2×C4×C5#23912yes—
300C2×Dic5—32no—
301C2×Dic5—32no—
302C2×Dic5—61no—
303C2×Dic5—61no—
304C2×Dic5—61no—
305C2×Dic5—61no—
306C2×Dic5—61no—
307C2×Dic5—61no—
308C2×Dic5—61no—
309C2×Dic5—62no—
310C2×Dic5—61no—
311C2×Dic5—62no—
312C2×Dic5—61no—
313C2×Dic5—62no—
314C2×Dic5—61no—
315D20—68no—
316C4×D5—128no—
317C22×D4—151no—
318C23×C4—152no—
319C4×D4—158no—
320C2×C4⋊C4—151no—
321C2×C4⋊C4—152no—
322C2×C4⋊C4C2×D1511yescharacteristic
323C2×C4⋊C4C2×D1512yes—
324C2×C4⋊C4C2×D1511yescharacteristic
325C2×C22⋊C4—152no—
326C2×C22⋊C4—152no—
327C2×C22⋊C4—152no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—152no—
330C2×C42—151no—
331C2×C42—151no—
332C2×C42—152no—
333C2×C42C2×D1511yescharacteristic
334#95—151no—
335#95—151no—
336#95—151no—
337#95—151no—
338#95—152no—
339#95—152no—
340#95C2×D1512yes—
341C2×C3×C5C2×#11711yescharacteristic
342C2×C3×C5#51811yescharacteristic
343C2×C3×C5#52011yescharacteristic
344C2×C3×C5#54011yescharacteristic
345C2×C3×C5#54011yescharacteristic
346C2×C3×C5#54211yescharacteristic
347C2×C3×C5#54211yescharacteristic
348D15—44no—
349C23×C3C23×D511yescharacteristic
350C22×D3—54no—
351C22×D3—102no—
352C22×D3—102no—
353C22×D3—102no—
354C22×D3—102no—
355C22×D3—102no—
356C22×D3—102no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4C2×C4×D514yes—
369C2×C3×C4C2×C4×D512yes—
370C2×C3×C4#71112yes—
371C2×C3×C4#71312yes—
372C2×Dic3—52no—
373C2×Dic3—52no—
374C2×Dic3—101no—
375C2×Dic3—101no—
376C2×Dic3—101no—
377C2×Dic3—101no—
378C2×Dic3—101no—
379C2×Dic3—101no—
380C2×Dic3—101no—
381C2×Dic3—102no—
382C2×Dic3—101no—
383C2×Dic3—102no—
384C2×Dic3—101no—
385C2×Dic3—102no—
386C2×Dic3—101no—
387D12—108no—
388C4×D3—208no—
389C22×C5C22×C4×D311yescharacteristic
390C22×C5C2×#23711yescharacteristic
391C22×C5#100411yescharacteristic
392C22×C5C2×#23911yescharacteristic
393C22×C5#100711yescharacteristic
394C22×C5#100911yescharacteristic
395C22×C5#100911yescharacteristic
396C2×D5—64no—
397C2×D5—64no—
398C2×D5—64no—
399C2×D5—122no—
400C2×D5—122no—
401C2×D5—122no—
402C2×D5—122no—
403C4×C5—24no—
404C4×C5—41no—
405C4×C5—41no—
406C4×C5—42no—
407C4×C5#89214yes—
408Dic5—62no—
409Dic5—62no—
410Dic5—121no—
411Dic5—122no—
412Dic5—121no—
413C24—152no—
414C2×D4—154no—
415C2×D4—302no—
416C2×D4—302no—
417C22×C4—151no—
418C22×C4—151no—
419C22×C4—158no—
420C22×C4—151no—
421C22×C4—151no—
422C22×C4—152no—
423C22×C4—602no—
424C22×C4—602no—
425C22×C4C22×D1511yescharacteristic
426C22×C4C22×D1511yescharacteristic
427C22×C4C22×D1512yes—
428C22×C4C22×D1512yes—
429C22×C4C22×D1511yescharacteristic
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—302no—
433C4⋊C4—602no—
434C4⋊C4C4×D1514yes—
435C22⋊C4—158no—
436C22⋊C4—602no—
437C22⋊C4—602no—
438C22⋊C4—602no—
439C22⋊C4—602no—
440C42—22no—
441C42—154no—
442C42—302no—
443C42—602no—
444C3×C5#231911yescharacteristic, complemented, Hall
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×#71111yescharacteristic
447C22×C3#414411yescharacteristic
448C22×C3C2×#71311yescharacteristic
449C22×C3#414711yescharacteristic
450C22×C3#414911yescharacteristic
451C22×C3#414911yescharacteristic
452C2×D3—104no—
453C2×D3—104no—
454C2×D3—104no—
455C2×D3—202no—
456C2×D3—202no—
457C2×D3—202no—
458C2×D3—202no—
459C3×C4—24no—
460C3×C4—41no—
461C3×C4—42no—
462C3×C4—41no—
463C3×C4#403814yes—
464Dic3—102no—
465Dic3—102no—
466Dic3—201no—
467Dic3—201no—
468Dic3—202no—
469C2×C5C2×#89211yescharacteristic
470C2×C5#554511yescharacteristic
471C2×C5#557411yescharacteristic
472C2×C5#561911yescharacteristic
473C2×C5#564311yescharacteristic
474C2×C5#569711yescharacteristic
475C2×C5#573311yescharacteristic
476D5—124no—
477C23—154no—
478C23—302no—
479C23—302no—
480C23—302no—
481C23—302no—
482C23—302no—
483C23—302no—
484C23C23×D1511yescharacteristic
485D4—308no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—152no—
498C2×C4—152no—
499C2×C4—301no—
500C2×C4—301no—
501C2×C4—301no—
502C2×C4—301no—
503C2×C4—301no—
504C2×C4—301no—
505C2×C4—301no—
506C2×C4—302no—
507C2×C4—301no—
508C2×C4—302no—
509C2×C4—301no—
510C2×C4—302no—
511C2×C4—301no—
512C2×C4—608no—
513C2×C4C2×C4×D1514yes—
514C2×C4C2×C4×D1512yes—
515C2×C4#681912yes—
516C2×C4#682112yes—
517C2×C3C2×#403811yescharacteristic
518C2×C3#6587911yescharacteristic
519C2×C3#6590811yescharacteristic
520C2×C3#6595311yescharacteristic
521C2×C3#6597711yescharacteristic
522C2×C3#6603211yescharacteristic
523C2×C3#6606711yescharacteristic
524D3—204no—
525C5#7596911yescharacteristic, complemented, Sylow
526C22—304no—
527C22—304no—
528C22—304no—
529C22—602no—
530C22—602no—
531C22—602no—
532C22—602no—
533C22C22×C4×D1511yescharacteristic
534C22C2×#681911yescharacteristic
535C22#9209511yescharacteristic
536C22C2×#682111yescharacteristic
537C22#9209811yescharacteristic
538C22#9210011yescharacteristic
539C22#9210011yescharacteristic
540C4—24no—
541C4—41no—
542C4—42no—
543C4—41no—
544C4—302no—
545C4—302no—
546C4—601no—
547C4—602no—
548C4—601no—
549C4#9178114yes—
550C3#1060770011yescharacteristic, complemented, Sylow
551C2—604no—
552C2C2×#9178111yescharacteristic
553C2#1175630411yescharacteristic
554C2#1175633311yescharacteristic
555C2#1175637811yescharacteristic
556C2#1175640211yescharacteristic
557C2#1175645611yescharacteristic
558C2#1175649211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#389C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenter#389D15
hypocenterC3×C5#2319
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C3×C5×#389C2semidirect
3. #11751091C2semidirect
5. #11751088C2semidirect
6. #11751085C2semidirect
7. C4⋊C4×Dic15C2semidirect
9. #11751033C2semidirect
10. #11750771C2semidirect
11. #11750765C2semidirect
29. #91782C4semidirect
39. C5×#389D3semidirect
81. C3×#389D5semidirect
215. #389D15semidirect
444. C3×C5#2319semidirect
525. C5#75969semidirect
550. C3#10607700semidirect

Maximal subgroups

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

subgroupindex
C2×#917822
C3×C5×#3892
#117510912
#117510892
#117510882
#117510852
C4⋊C4×Dic152
#117510402
#117510332
#117507712
#117507652
#106077003
#759695