Finite Group Databasebeta

← #14416843

Subgroups — #14416843

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic15C211yesmaximal, characteristic
2C2×#91840C211yesmaximal, characteristic
3C2×#91776C211yesmaximal, characteristic, complemented
4C3×C5×#417C211yesmaximal, characteristic
5#11751221C211yesmaximal, characteristic, complemented
6M16×Dic15C211yesmaximal, characteristic, complemented
7#11751065C211yesmaximal, characteristic, complemented
8#11750891C212yesmaximal, complemented
9#11750890C212yesmaximal, complemented
10#11750859C212yesmaximal, complemented
11#11750858C212yesmaximal, complemented
12#10610598—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×Dic30C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17#91840C418yescomplemented
18Q8×Dic15C2211yescharacteristic
19Q8×Dic15C2211yescharacteristic
20C2×#6718C2212yes—
21D4×Dic15—21no—
22D4×Dic15—21no—
23D4×Dic15C2211yescharacteristic
24D4×Dic15C2211yescharacteristic
25C2×#6716C2212yes—
26#91815—21no—
27#91815C2211yescharacteristic
28C2×#6712—21no—
29C2×C4×Dic15—21no—
30C2×C4×Dic15—21no—
31C2×C4×Dic15C2211yescharacteristic
32C2×#6710C2211yescharacteristic
33#91776—21no—
34#91776—21no—
35#91776C2212yes—
36#91776C2212yes—
37C3×C5×#103C2212yes—
38C3×C5×#102C2212yes—
39#91100C2212yes—
40#91100C2212yes—
41#91099C2212yes—
42C8×Dic15C2212yes—
43#91092C2212yes—
44#91091C2212yes—
45#78867—51nomaximal, Hall
46#41×Dic5—31no—
47C2×#4097—31no—
48C2×#4033—31no—
49C5×#417D311yescharacteristic
50#65409—31no—
51M16×Dic5—31no—
52#65253—31no—
53#65079—32no—
54#65078—32no—
55#65047—32no—
56#65046—32no—
57C3×C5×#41—22no—
58C3×C5×#41C2×C414yes—
59C2×C3×C5×Q8C2311yescharacteristic
60C2×C3×C5×D4—21no—
61C2×C3×C5×D4C2311yescharacteristic
62C22×C3×C4×C5—21no—
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5C2311yescharacteristic
65C22×Dic15—21no—
66C22×Dic15—21no—
67C22×Dic15—21no—
68C22×Dic15D411yescharacteristic
69C22×Dic15D411yescharacteristic
70C2×Dic30—21no—
71C2×Dic30—22no—
72C2×Dic30C2×C414yescomplemented
73C2×Dic30C2312yes—
74C3×C5×M16—22no—
75C2×C3×C5×C8C2312yes—
76C3×C5×C4⋊C4—21no—
77C3×C5×C4⋊C4C2312yes—
78#6719—21no—
79#6719—21no—
80#6719—21no—
81#6718C2×C418yescomplemented
82#6716C2×C418yescomplemented
83C4⋊Dic15—21no—
84C4⋊Dic15—21no—
85C4⋊Dic15C2311yescharacteristic
86C4⋊Dic15C2311yescharacteristic
87#6712—22no—
88#6712—22no—
89C4×Dic15—21no—
90C4×Dic15—22no—
91C4×Dic15—22no—
92C4×Dic15—21no—
93C4×Dic15—21no—
94C4×Dic15D411yescharacteristic
95C4×Dic15D411yescharacteristic
96C4×Dic15C2311yescharacteristic
97C4×Dic15C2311yescharacteristic
98#6710C2×C414yescomplemented
99C2×C3⋊#168C2312yes—
100Dic3×#41—51no—
101C2×#951—51no—
102C2×#887—51no—
103C3×#417D511yescharacteristic
104#5120—51no—
105Dic3×M16—51no—
106#4964—51no—
107#4790—52no—
108#4789—52no—
109#4758—52no—
110#4757—52no—
111C2×C5×#41C2×D311yescharacteristic
112C22×Dic10—31no—
113C2×C5×M16C2×D311yescharacteristic
114C2×C5×C4⋊C4C2×D311yescharacteristic
115#4097—38no—
116Q8×Dic5—31no—
117Q8×Dic5—31no—
118C2×#689—32no—
119D4×Dic5—31no—
120D4×Dic5—31no—
121D4×Dic5—61no—
122D4×Dic5—61no—
123C2×#687—32no—
124#4072—31no—
125#4072—61no—
126C2×#683—61no—
127C2×C4×Dic5—31no—
128C2×C4×Dic5—61no—
129C2×C4×Dic5—61no—
130C2×#681—31no—
131#4033—32no—
132#4033—32no—
133#4033—61no—
134#4033—61no—
135C5×#103C2×D312yes—
136C5×#102C2×D312yes—
137#3948—32no—
138#3948—32no—
139#3947—32no—
140C8×Dic5—32no—
141#3940—32no—
142#3939—32no—
143C4×#137—151noSylow
144C23×C3×C5—21no—
145C23×C3×C5C2×D411yescharacteristic
146C3×C5×Q8—21no—
147C3×C5×Q8C22×C412yes—
148C3×C5×D4—24no—
149C3×C5×D4—21no—
150C3×C5×D4C22×C412yes—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5—24no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—42no—
155C2×C3×C4×C5C22×C412yes—
156C2×C3×C4×C5C2×D411yescharacteristic
157C2×C3×C4×C5#4112yes—
158C2×C3×C4×C5C2411yescharacteristic
159C2×Dic15—24no—
160C2×Dic15—22no—
161C2×Dic15—22no—
162C2×Dic15—21no—
163C2×Dic15—21no—
164C2×Dic15—41no—
165C2×Dic15—42no—
166C2×Dic15—42no—
167C2×Dic15C2×D411yescharacteristic
168C2×Dic15C2×D411yescharacteristic
169C2×Dic15C2×D411yescharacteristic
170C2×Dic15C2×D411yescharacteristic
171Dic30—24no—
172Dic30—22no—
173Dic30C22×C414yes—
174C3×C5×C8—22no—
175C3⋊#168C22×C414yes—
176C2×C3×#41C2×D511yescharacteristic
177C22×Dic6—51no—
178C2×C3×M16C2×D511yescharacteristic
179C2×C3×C4⋊C4C2×D511yescharacteristic
180#951—58no—
181Q8×Dic3—51no—
182Q8×Dic3—51no—
183C2×#216—52no—
184D4×Dic3—51no—
185D4×Dic3—51no—
186D4×Dic3—101no—
187D4×Dic3—101no—
188C2×#214—52no—
189#926—51no—
190#926—101no—
191C2×Dic3⋊C4—101no—
192C2×C4×Dic3—51no—
193C2×C4×Dic3—101no—
194C2×C4×Dic3—101no—
195C2×#208—51no—
196#887—52no—
197#887—52no—
198#887—101no—
199#887—101no—
200C3×#103C2×D512yes—
201C3×#102C2×D512yes—
202#816—52no—
203#816—52no—
204#815—52no—
205C8×Dic3—52no—
206#808—52no—
207#807—52no—
208C5×#41—22no—
209C5×#41C4×D314yes—
210C2×C5×Q8C22×D311yescharacteristic
211C2×C5×D4—21no—
212C2×C5×D4C22×D311yescharacteristic
213C22×C4×C5—21no—
214C22×C4×C5—21no—
215C22×C4×C5C22×D311yescharacteristic
216C22×Dic5—31no—
217C22×Dic5—31no—
218C22×Dic5—61no—
219C22×Dic5—61no—
220C22×Dic5—61no—
221C2×Dic10—34no—
222C2×Dic10—32no—
223C2×Dic10—61no—
224C2×Dic10—62no—
225C5×M16—22no—
226C2×C5×C8C22×D312yes—
227C5×C4⋊C4—21no—
228C5×C4⋊C4C22×D312yes—
229C22⋊Dic5—61no—
230C22⋊Dic5—61no—
231C22⋊Dic5—61no—
232#689—38no—
233#687—38no—
234C4⋊Dic5—31no—
235C4⋊Dic5—31no—
236C4⋊Dic5—61no—
237C4⋊Dic5—61no—
238#683—62no—
239#683—62no—
240C4×Dic5—31no—
241C4×Dic5—31no—
242C4×Dic5—31no—
243C4×Dic5—31no—
244C4×Dic5—62no—
245C4×Dic5—62no—
246C4×Dic5—61no—
247C4×Dic5—61no—
248C4×Dic5—61no—
249#681—34no—
250C2×#168—32no—
251C2×#137—151no—
252C4×#41—151no—
253C2×C4×Q8—151no—
254#441—152no—
255#440—152no—
256C4×Q16—152no—
257C4×SD16—152no—
258#428—151no—
259#419—151no—
260#417D1511yescharacteristic
261C4×M16—151no—
262C22×C3×C5—21no—
263C22×C3×C5—21no—
264C22×C3×C5—22no—
265C22×C3×C5—41no—
266C22×C3×C5—41no—
267C22×C3×C5C4×D412yes—
268C22×C3×C5C22×D411yescharacteristic
269C3×C4×C5—22no—
270C3×C4×C5—42no—
271C3×C4×C5C4×D412yes—
272C3×C4×C5C23×C411yescharacteristic
273C3×C4×C5C2×#4111yescharacteristic
274Dic15—24no—
275Dic15—21no—
276Dic15—21no—
277Dic15—21no—
278Dic15—41no—
279Dic15—41no—
280Dic15#13712yescomplemented
281C3×#41—22no—
282C3×#41C4×D514yes—
283C2×C3×Q8C22×D511yescharacteristic
284C2×C3×D4—21no—
285C2×C3×D4C22×D511yescharacteristic
286C22×C3×C4—21no—
287C22×C3×C4—21no—
288C22×C3×C4C22×D511yescharacteristic
289C22×Dic3—51no—
290C22×Dic3—51no—
291C22×Dic3—101no—
292C22×Dic3—101no—
293C22×Dic3—101no—
294C2×Dic6—54no—
295C2×Dic6—52no—
296C2×Dic6—101no—
297C2×Dic6—102no—
298C3×M16—22no—
299C2×C3×C8C22×D512yes—
300C3×C4⋊C4—21no—
301C3×C4⋊C4C22×D512yes—
302#217—101no—
303#217—101no—
304#217—101no—
305#216—58no—
306#214—58no—
307C4⋊Dic3—51no—
308C4⋊Dic3—51no—
309C4⋊Dic3—101no—
310C4⋊Dic3—101no—
311Dic3⋊C4—102no—
312Dic3⋊C4—102no—
313C4×Dic3—51no—
314C4×Dic3—51no—
315C4×Dic3—51no—
316C4×Dic3—51no—
317C4×Dic3—102no—
318C4×Dic3—102no—
319C4×Dic3—101no—
320C4×Dic3—101no—
321C4×Dic3—101no—
322#208—54no—
323C2×C3⋊C8—52no—
324C23×C5—21no—
325C23×C5D3×D411yescharacteristic
326C5×Q8—21no—
327C5×Q8C2×C4×D312yes—
328C5×D4—24no—
329C5×D4—21no—
330C5×D4C2×C4×D312yes—
331C2×C4×C5—21no—
332C2×C4×C5—24no—
333C2×C4×C5—22no—
334C2×C4×C5—42no—
335C2×C4×C5C2×C4×D312yes—
336C2×C4×C5D3×D411yescharacteristic
337C2×C4×C5#23712yes—
338C2×C4×C5C23×D311yescharacteristic
339C2×Dic5—31no—
340C2×Dic5—31no—
341C2×Dic5—31no—
342C2×Dic5—31no—
343C2×Dic5—62no—
344C2×Dic5—64no—
345C2×Dic5—62no—
346C2×Dic5—61no—
347C2×Dic5—61no—
348C2×Dic5—122no—
349C2×Dic5—121no—
350C2×Dic5—122no—
351Dic10—34no—
352Dic10—64no—
353Dic10—62no—
354C5×C8—22no—
355#168—34no—
356C2×#41C2×D1511yescharacteristic
357C22×Q8—151no—
358#137—158no—
359C2×Q16—152no—
360C2×SD16—152no—
361C2×M16—151no—
362C2×M16C2×D1511yescharacteristic
363C4×Q8—151no—
364C4×Q8—151no—
365C4×Q8—152no—
366C4×Q8—152no—
367C4×Q8—301no—
368C4×Q8—301no—
369C4×D4—151no—
370C4×D4—151no—
371C4×D4—301no—
372C4×D4—301no—
373#117—151no—
374#117—301no—
375C2×C4⋊C4—301no—
376C2×C4⋊C4C2×D1511yescharacteristic
377C2×C42—151no—
378C2×C42—301no—
379C2×C42—301no—
380#107—152no—
381#106—152no—
382#103—152no—
383#103—152no—
384#103C2×D1512yes—
385#102C2×D1512yes—
386#97—152no—
387C4×C8—152no—
388C2×C3×C5—22no—
389C2×C3×C5—42no—
390C2×C3×C5C2×C4×D411yescharacteristic
391C2×C3×C5C2×#13711yescharacteristic
392C2×C3×C5#57511yescharacteristic
393C23×C3—21no—
394C23×C3D4×D511yescharacteristic
395C3×Q8—21no—
396C3×Q8C2×C4×D512yes—
397C3×D4—24no—
398C3×D4—21no—
399C3×D4C2×C4×D512yes—
400C2×C3×C4—21no—
401C2×C3×C4—24no—
402C2×C3×C4—22no—
403C2×C3×C4—42no—
404C2×C3×C4C2×C4×D512yes—
405C2×C3×C4D4×D511yescharacteristic
406C2×C3×C4#71112yes—
407C2×C3×C4C23×D511yescharacteristic
408C2×Dic3—51no—
409C2×Dic3—51no—
410C2×Dic3—51no—
411C2×Dic3—51no—
412C2×Dic3—102no—
413C2×Dic3—102no—
414C2×Dic3—104no—
415C2×Dic3—101no—
416C2×Dic3—101no—
417C2×Dic3—202no—
418C2×Dic3—201no—
419C2×Dic3—202no—
420Dic6—54no—
421Dic6—102no—
422Dic6—104no—
423C3×C8—22no—
424C3⋊C8—54no—
425C22×C5—21no—
426C22×C5—21no—
427C22×C5—22no—
428C22×C5—41no—
429C22×C5—41no—
430C22×C5#67⋊C412yes—
431C22×C5C2×D3×D411yescharacteristic
432C4×C5—22no—
433C4×C5—42no—
434C4×C5#67⋊C412yes—
435C4×C5C22×C4×D311yescharacteristic
436C4×C5C2×#23711yescharacteristic
437Dic5—32no—
438Dic5—64no—
439Dic5—61no—
440Dic5—61no—
441Dic5—61no—
442Dic5—121no—
443Dic5—121no—
444#41—22no—
445#41C4×D1514yes—
446C2×Q8—154no—
447C2×Q8—152no—
448C2×Q8—301no—
449C2×Q8—302no—
450C2×Q8C22×D1511yescharacteristic
451C2×D4—21no—
452C2×D4C22×D1511yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—151no—
456C22×C4—151no—
457C22×C4—301no—
458C22×C4—301no—
459C22×C4—301no—
460C22×C4C22×D1511yescharacteristic
461Q16—158no—
462SD16—158no—
463M16—22no—
464M16—154no—
465C2×C8—152no—
466C2×C8C22×D1512yes—
467C4⋊C4—21no—
468C4⋊C4—151no—
469C4⋊C4—151no—
470C4⋊C4—302no—
471C4⋊C4—302no—
472C4⋊C4—301no—
473C4⋊C4—301no—
474C4⋊C4C22×D1512yes—
475C22⋊C4—301no—
476C22⋊C4—301no—
477C22⋊C4—301no—
478C42—151no—
479C42—151no—
480C42—151no—
481C42—151no—
482C42—301no—
483C42—301no—
484C42—302no—
485C42—302no—
486C42—301no—
487C3×C5C4×#13711yescharacteristic, complemented, Hall
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3#402712yes—
494C22×C3C2×D4×D511yescharacteristic
495C3×C4—22no—
496C3×C4—42no—
497C3×C4#402712yes—
498C3×C4C22×C4×D511yescharacteristic
499C3×C4C2×#71111yescharacteristic
500Dic3—52no—
501Dic3—104no—
502Dic3—101no—
503Dic3—101no—
504Dic3—101no—
505Dic3—201no—
506Dic3—201no—
507C2×C5—22no—
508C2×C5—42no—
509C2×C5C2×#67⋊C411yescharacteristic
510C2×C5#577311yescharacteristic
511C2×C5D3×#13711yescharacteristic
512C23—21no—
513C23D4×D1511yescharacteristic
514Q8—21no—
515Q8—154no—
516Q8—304no—
517Q8—302no—
518Q8C2×C4×D1512yes—
519D4—24no—
520D4—21no—
521D4C2×C4×D1512yes—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—151no—
527C2×C4—151no—
528C2×C4—151no—
529C2×C4—151no—
530C2×C4—302no—
531C2×C4—304no—
532C2×C4—302no—
533C2×C4—301no—
534C2×C4—301no—
535C2×C4—602no—
536C2×C4—601no—
537C2×C4—602no—
538C2×C4C2×C4×D1512yes—
539C2×C4D4×D1511yescharacteristic
540C2×C4#681912yes—
541C2×C4C23×D1511yescharacteristic
542C8—22no—
543C8—154no—
544C2×C3—22no—
545C2×C3—42no—
546C2×C3C2×#402711yescharacteristic
547C2×C3#6610711yescharacteristic
548C2×C3D5×#13711yescharacteristic
549C5#7886711yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22#9177012yes—
556C22C2×D4×D1511yescharacteristic
557C4—22no—
558C4—42no—
559C4—152no—
560C4—304no—
561C4—301no—
562C4—301no—
563C4—301no—
564C4—601no—
565C4—601no—
566C4#9177012yes—
567C4C22×C4×D1511yescharacteristic
568C4C2×#681911yescharacteristic
569C3#1061059811yescharacteristic, complemented, Sylow
570C2—22no—
571C2—42no—
572C2C2×#9177011yescharacteristic
573C2#1175653211yescharacteristic
574C2D15×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#417C2
socleC22×C3×C5C22×D4
prime radicalC2×C3×C5×#41C22
cosocleC2×C3×C4×C5C24
hypercenter#417D15
hypocenterC3×C5C4×#137
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
3. C2×#91776C2semidirect
5. #11751221C2semidirect
6. M16×Dic15C2semidirect
7. #11751065C2semidirect
8. #11750891C2semidirect
9. #11750890C2semidirect
10. #11750859C2semidirect
11. #11750858C2semidirect
17. #91840C4semidirect
72. C2×Dic30C2×C4semidirect
81. #6718C2×C4semidirect
82. #6716C2×C4semidirect
98. #6710C2×C4semidirect
280. Dic15#137semidirect
487. C3×C5C4×#137semidirect
549. C5#78867semidirect
569. C3#10610598semidirect

Maximal subgroups

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

subgroupindex
#41×Dic152
C2×#918402
C2×#917762
C3×C5×#4172
#117512212
M16×Dic152
#117510652
#117508912
#117508902
#117508592
#117508582
#106105983
#788675