Finite Group Databasebeta

← #14416672

Subgroups — #14416672

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×Dic15C211yesmaximal, characteristic, complemented
2C2×#91838C211yesmaximal, characteristic
3C2×#91782C211yesmaximal, characteristic, complemented
4C3×C5×#417C211yesmaximal, characteristic, complemented
5#11751232C211yesmaximal, characteristic, complemented
6M16×Dic15C211yesmaximal, characteristic, complemented
7#11751065C211yesmaximal, characteristic, complemented
8#11750918C212yesmaximal, complemented
9#11750889C212yesmaximal, complemented
10#11750886C212yesmaximal, complemented
11#11750857C212yesmaximal, complemented
12#10610597—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C22×C4×D15—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18#91838C418yescomplemented
19Q8×Dic15C2211yescharacteristic, complemented
20Q8×Dic15C2211yescharacteristic, complemented
21C2×#6717C2212yes—
22D4×Dic15—21no—
23D4×Dic15—21no—
24D4×Dic15C2211yescharacteristic, complemented
25D4×Dic15C2211yescharacteristic, complemented
26C2×#6715C2212yes—
27C2×#6714—21no—
28#91815—21no—
29#91815C2211yescharacteristic
30C2×C4×Dic15—21no—
31C2×C4×Dic15C2211yescharacteristic
32C2×#6710C2211yescharacteristic
33#91782—21no—
34#91782—21no—
35#91782C2212yes—
36#91782C2212yes—
37C3×C5×#103C2212yescomplemented
38C3×C5×#102C2212yescomplemented
39#91105C2212yes—
40#91105C2212yes—
41#91099C2212yescomplemented
42C8×Dic15C2212yescomplemented
43#91092C2212yescomplemented
44#91091C2212yescomplemented
45#78866—51nomaximal, Hall
46#41×Dic5—31no—
47C2×#4095—31no—
48C2×#4039—31no—
49C5×#417D311yescharacteristic, complemented
50#65420—31no—
51M16×Dic5—31no—
52#65253—31no—
53#65106—32no—
54#65077—32no—
55#65074—32no—
56#65045—32no—
57C23×D15—21no—
58C3×C5×#41—22no—
59C3×C5×#41C2×C414yescomplemented
60C2×C3×C5×Q8C2311yescharacteristic
61C2×C3×C5×D4—21no—
62C2×C3×C5×D4C2311yescharacteristic
63C22×C3×C4×C5—21no—
64C22×C3×C4×C5—21no—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15—21no—
67C22×Dic15—21no—
68C22×Dic15D411yescharacteristic
69C22×Dic15D411yescharacteristic
70C2×D60—21no—
71C2×D60—22no—
72C2×D60C2×C414yescomplemented
73C2×D60C2312yes—
74C2×C4×D15—22no—
75C2×C4×D15—22no—
76C2×C4×D15—44no—
77C3×C5×M16—22no—
78C2×C3×C5×C8C2312yes—
79C3×C5×C4⋊C4—21no—
80C3×C5×C4⋊C4C2312yes—
81#6719—21no—
82#6719—21no—
83#6719—21no—
84#6717C2×C418yescomplemented
85#6715C2×C418yescomplemented
86#6714—22no—
87#6714—22no—
88C4⋊Dic15—21no—
89C4⋊Dic15—21no—
90C4⋊Dic15C2311yescharacteristic
91C4⋊Dic15C2311yescharacteristic
92C4×Dic15—21no—
93C4×Dic15—21no—
94C4×Dic15—21no—
95C4×Dic15D411yescharacteristic
96C4×Dic15D411yescharacteristic
97C4×Dic15C2311yescharacteristic
98C4×Dic15C2311yescharacteristic
99#6710C2×C414yescomplemented
100C2×C3⋊#168C2312yes—
101Dic3×#41—51no—
102C2×#949—51no—
103C2×#893—51no—
104C3×#417D511yescharacteristic, complemented
105#5131—51no—
106Dic3×M16—51no—
107#4964—51no—
108#4817—52no—
109#4788—52no—
110#4785—52no—
111#4756—52no—
112C2×C5×#41C2×D311yescharacteristic
113C22×D20—31no—
114C22×C4×D5—61no—
115C2×C5×M16C2×D311yescharacteristic
116C2×C5×C4⋊C4C2×D311yescharacteristic
117#4095—38no—
118Q8×Dic5—31no—
119Q8×Dic5—31no—
120C2×#688—32no—
121D4×Dic5—31no—
122D4×Dic5—31no—
123D4×Dic5—61no—
124D4×Dic5—61no—
125C2×#686—32no—
126C2×#685—61no—
127#4072—31no—
128#4072—61no—
129C2×C4×Dic5—31no—
130C2×C4×Dic5—61no—
131C2×#681—31no—
132#4039—32no—
133#4039—32no—
134#4039—61no—
135#4039—61no—
136C5×#103C2×D312yescomplemented
137C5×#102C2×D312yescomplemented
138#3953—32no—
139#3953—32no—
140#3947—32no—
141C8×Dic5—32no—
142#3940—32no—
143#3939—32no—
144C4×AGL(1,ℤ/8)—151noSylow
145C23×C3×C5—21no—
146C23×C3×C5C2×D411yescharacteristic
147C22×D15—24no—
148C22×D15—22no—
149C22×D15—42no—
150C22×D15—42no—
151C3×C5×Q8—21no—
152C3×C5×Q8C22×C412yescomplemented
153C3×C5×D4—24no—
154C3×C5×D4—21no—
155C3×C5×D4C22×C412yescomplemented
156C2×C3×C4×C5—21no—
157C2×C3×C4×C5—24no—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—42no—
160C2×C3×C4×C5C22×C412yes—
161C2×C3×C4×C5C2×D411yescharacteristic
162C2×C3×C4×C5#4112yes—
163C2×C3×C4×C5C2411yescharacteristic
164C2×Dic15—22no—
165C2×Dic15—21no—
166C2×Dic15—21no—
167C2×Dic15—41no—
168C2×Dic15—42no—
169C2×Dic15—42no—
170C2×Dic15C2×D411yescharacteristic
171C2×Dic15C2×D411yescharacteristic
172C2×Dic15C2×D411yescharacteristic
173C2×Dic15C2×D411yescharacteristic
174D60—24no—
175D60—22no—
176D60C22×C414yes—
177C4×D15—44no—
178C4×D15—44no—
179C3×C5×C8—22no—
180C3⋊#168C22×C414yescomplemented
181C2×C3×#41C2×D511yescharacteristic
182C22×D12—51no—
183C22×C4×D3—101no—
184C2×C3×M16C2×D511yescharacteristic
185C2×C3×C4⋊C4C2×D511yescharacteristic
186#949—58no—
187Q8×Dic3—51no—
188Q8×Dic3—51no—
189C2×#215—52no—
190D4×Dic3—51no—
191D4×Dic3—51no—
192D4×Dic3—101no—
193D4×Dic3—101no—
194C2×#213—52no—
195C2×(C2×D3)⋊C4—101no—
196#926—51no—
197#926—101no—
198C2×C4×Dic3—51no—
199C2×C4×Dic3—101no—
200C2×#208—51no—
201#893—52no—
202#893—52no—
203#893—101no—
204#893—101no—
205C3×#103C2×D512yescomplemented
206C3×#102C2×D512yescomplemented
207#821—52no—
208#821—52no—
209#815—52no—
210C8×Dic3—52no—
211#808—52no—
212#807—52no—
213C23×D5—61no—
214C5×#41—22no—
215C5×#41C4×D314yescomplemented
216C2×C5×Q8C22×D311yescharacteristic
217C2×C5×D4—21no—
218C2×C5×D4C22×D311yescharacteristic
219C22×C4×C5—21no—
220C22×C4×C5—21no—
221C22×C4×C5C22×D311yescharacteristic
222C22×Dic5—31no—
223C22×Dic5—31no—
224C22×Dic5—61no—
225C22×Dic5—61no—
226C2×D20—34no—
227C2×D20—32no—
228C2×D20—61no—
229C2×D20—62no—
230C2×C4×D5—62no—
231C2×C4×D5—62no—
232C2×C4×D5—124no—
233C5×M16—22no—
234C2×C5×C8C22×D312yes—
235C5×C4⋊C4—21no—
236C5×C4⋊C4C22×D312yes—
237C22⋊Dic5—61no—
238C22⋊Dic5—61no—
239C22⋊Dic5—61no—
240#688—38no—
241#686—38no—
242#685—62no—
243#685—62no—
244C4⋊Dic5—31no—
245C4⋊Dic5—31no—
246C4⋊Dic5—61no—
247C4⋊Dic5—61no—
248C4×Dic5—31no—
249C4×Dic5—31no—
250C4×Dic5—31no—
251C4×Dic5—31no—
252C4×Dic5—61no—
253C4×Dic5—61no—
254C4×Dic5—61no—
255#681—34no—
256C2×#168—32no—
257C2×AGL(1,ℤ/8)—151no—
258C4×#41—151no—
259C2×C4×D4—151no—
260#442—152no—
261#440—152no—
262C4×SD16—152no—
263C4×D8—152no—
264#428—151no—
265#418—151no—
266#417D1511yescharacteristic, complemented
267C4×M16—151no—
268C22×C3×C5—21no—
269C22×C3×C5—21no—
270C22×C3×C5—22no—
271C22×C3×C5—41no—
272C22×C3×C5—41no—
273C22×C3×C5C4×D412yes—
274C22×C3×C5C22×D411yescharacteristic
275C2×D15—24no—
276C2×D15—48no—
277C2×D15—42no—
278C2×D15—42no—
279C3×C4×C5—22no—
280C3×C4×C5—42no—
281C3×C4×C5C4×D412yes—
282C3×C4×C5C23×C411yescharacteristic
283C3×C4×C5C2×#4111yescharacteristic
284Dic15—21no—
285Dic15—21no—
286Dic15—21no—
287Dic15—41no—
288Dic15—41no—
289Dic15AGL(1,ℤ/8)12yescomplemented
290C23×D3—101no—
291C3×#41—22no—
292C3×#41C4×D514yescomplemented
293C2×C3×Q8C22×D511yescharacteristic
294C2×C3×D4—21no—
295C2×C3×D4C22×D511yescharacteristic
296C22×C3×C4—21no—
297C22×C3×C4—21no—
298C22×C3×C4C22×D511yescharacteristic
299C22×Dic3—51no—
300C22×Dic3—51no—
301C22×Dic3—101no—
302C22×Dic3—101no—
303C2×D12—54no—
304C2×D12—52no—
305C2×D12—101no—
306C2×D12—102no—
307C2×C4×D3—102no—
308C2×C4×D3—102no—
309C2×C4×D3—204no—
310C3×M16—22no—
311C2×C3×C8C22×D512yes—
312C3×C4⋊C4—21no—
313C3×C4⋊C4C22×D512yes—
314#217—101no—
315#217—101no—
316#217—101no—
317#215—58no—
318#213—58no—
319(C2×D3)⋊C4—102no—
320(C2×D3)⋊C4—102no—
321C4⋊Dic3—51no—
322C4⋊Dic3—51no—
323C4⋊Dic3—101no—
324C4⋊Dic3—101no—
325C4×Dic3—51no—
326C4×Dic3—51no—
327C4×Dic3—51no—
328C4×Dic3—51no—
329C4×Dic3—101no—
330C4×Dic3—101no—
331C4×Dic3—101no—
332#208—54no—
333C2×C3⋊C8—52no—
334C23×C5—21no—
335C23×C5D3×D411yescharacteristic
336C22×D5—64no—
337C22×D5—62no—
338C22×D5—122no—
339C22×D5—122no—
340C5×Q8—21no—
341C5×Q8C2×C4×D312yescomplemented
342C5×D4—24no—
343C5×D4—21no—
344C5×D4C2×C4×D312yescomplemented
345C2×C4×C5—21no—
346C2×C4×C5—24no—
347C2×C4×C5—22no—
348C2×C4×C5—42no—
349C2×C4×C5C2×C4×D312yes—
350C2×C4×C5D3×D411yescharacteristic
351C2×C4×C5#23712yes—
352C2×C4×C5C23×D311yescharacteristic
353C2×Dic5—31no—
354C2×Dic5—31no—
355C2×Dic5—31no—
356C2×Dic5—31no—
357C2×Dic5—62no—
358C2×Dic5—61no—
359C2×Dic5—61no—
360C2×Dic5—122no—
361C2×Dic5—121no—
362C2×Dic5—122no—
363D20—34no—
364D20—64no—
365D20—62no—
366C4×D5—124no—
367C4×D5—124no—
368C5×C8—22no—
369#168—34no—
370C2×#41C2×D1511yescharacteristic
371C22×D4—151no—
372C23×C4—301no—
373AGL(1,ℤ/8)—158no—
374C2×SD16—152no—
375C2×D8—152no—
376C2×M16—151no—
377C2×M16C2×D1511yescharacteristic
378C4×Q8—151no—
379C4×Q8—151no—
380C4×D4—151no—
381C4×D4—151no—
382C4×D4—152no—
383C4×D4—152no—
384C4×D4—301no—
385C4×D4—301no—
386C4×D4—301no—
387C4×D4—301no—
388#117—151no—
389#117—301no—
390C2×C4⋊C4C2×D1511yescharacteristic
391C2×C22⋊C4—301no—
392C2×C42—151no—
393C2×C42—301no—
394#107—152no—
395#106—152no—
396#103C2×D1512yescomplemented
397#102—152no—
398#102—152no—
399#102C2×D1512yescomplemented
400#97—152no—
401C4×C8—152no—
402C2×C3×C5—22no—
403C2×C3×C5—42no—
404C2×C3×C5C2×C4×D411yescharacteristic
405C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
406C2×C3×C5#57511yescharacteristic
407D15—44no—
408C23×C3—21no—
409C23×C3D4×D511yescharacteristic
410C22×D3—102no—
411C22×D3—104no—
412C22×D3—202no—
413C22×D3—202no—
414C3×Q8—21no—
415C3×Q8C2×C4×D512yescomplemented
416C3×D4—24no—
417C3×D4—21no—
418C3×D4C2×C4×D512yescomplemented
419C2×C3×C4—21no—
420C2×C3×C4—24no—
421C2×C3×C4—22no—
422C2×C3×C4—42no—
423C2×C3×C4C2×C4×D512yes—
424C2×C3×C4D4×D511yescharacteristic
425C2×C3×C4#71112yes—
426C2×C3×C4C23×D511yescharacteristic
427C2×Dic3—51no—
428C2×Dic3—51no—
429C2×Dic3—51no—
430C2×Dic3—51no—
431C2×Dic3—102no—
432C2×Dic3—101no—
433C2×Dic3—101no—
434C2×Dic3—202no—
435C2×Dic3—201no—
436C2×Dic3—202no—
437D12—54no—
438D12—102no—
439D12—104no—
440C4×D3—204no—
441C4×D3—204no—
442C3×C8—22no—
443C3⋊C8—54no—
444C22×C5—21no—
445C22×C5—21no—
446C22×C5—22no—
447C22×C5—41no—
448C22×C5—41no—
449C22×C5#67⋊C412yes—
450C22×C5C2×D3×D411yescharacteristic
451C2×D5—64no—
452C2×D5—128no—
453C2×D5—122no—
454C2×D5—122no—
455C4×C5—22no—
456C4×C5—42no—
457C4×C5#67⋊C412yes—
458C4×C5C22×C4×D311yescharacteristic
459C4×C5C2×#23711yescharacteristic
460Dic5—32no—
461Dic5—61no—
462Dic5—61no—
463Dic5—61no—
464Dic5—121no—
465Dic5—121no—
466C24—301no—
467#41—22no—
468#41C4×D1514yescomplemented
469C2×Q8C22×D1511yescharacteristic
470C2×D4—21no—
471C2×D4—154no—
472C2×D4—152no—
473C2×D4—301no—
474C2×D4—302no—
475C2×D4C22×D1511yescharacteristic
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—151no—
479C22×C4—151no—
480C22×C4—302no—
481C22×C4—302no—
482C22×C4—301no—
483C22×C4—301no—
484C22×C4—604no—
485C22×C4C22×D1511yescharacteristic
486SD16—158no—
487D8—158no—
488M16—22no—
489M16—154no—
490C2×C8—152no—
491C2×C8C22×D1512yes—
492C4⋊C4—21no—
493C4⋊C4—151no—
494C4⋊C4—151no—
495C4⋊C4—301no—
496C4⋊C4—301no—
497C4⋊C4C22×D1512yes—
498C22⋊C4—302no—
499C22⋊C4—302no—
500C22⋊C4—301no—
501C22⋊C4—301no—
502C22⋊C4—301no—
503C42—151no—
504C42—151no—
505C42—151no—
506C42—151no—
507C42—301no—
508C42—301no—
509C42—301no—
510C3×C5C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—22no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3#402712yes—
517C22×C3C2×D4×D511yescharacteristic
518C2×D3—104no—
519C2×D3—208no—
520C2×D3—202no—
521C2×D3—202no—
522C3×C4—22no—
523C3×C4—42no—
524C3×C4#402712yes—
525C3×C4C22×C4×D511yescharacteristic
526C3×C4C2×#71111yescharacteristic
527Dic3—52no—
528Dic3—101no—
529Dic3—101no—
530Dic3—101no—
531Dic3—201no—
532Dic3—201no—
533C2×C5—22no—
534C2×C5—42no—
535C2×C5C2×#67⋊C411yescharacteristic
536C2×C5D3×AGL(1,ℤ/8)11yescharacteristic
537C2×C5#577711yescharacteristic
538D5—124no—
539C23—21no—
540C23—304no—
541C23—302no—
542C23—602no—
543C23—602no—
544C23D4×D1511yescharacteristic
545Q8—21no—
546Q8C2×C4×D1512yescomplemented
547D4—24no—
548D4—21no—
549D4—154no—
550D4—304no—
551D4—302no—
552D4C2×C4×D1512yescomplemented
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—151no—
558C2×C4—151no—
559C2×C4—151no—
560C2×C4—151no—
561C2×C4—302no—
562C2×C4—301no—
563C2×C4—301no—
564C2×C4—604no—
565C2×C4—604no—
566C2×C4—602no—
567C2×C4—601no—
568C2×C4—602no—
569C2×C4C2×C4×D1512yes—
570C2×C4D4×D1511yescharacteristic
571C2×C4#681912yes—
572C2×C4C23×D1511yescharacteristic
573C8—22no—
574C8—154no—
575C2×C3—22no—
576C2×C3—42no—
577C2×C3C2×#402711yescharacteristic
578C2×C3D5×AGL(1,ℤ/8)11yescharacteristic
579C2×C3#6611111yescharacteristic
580D3—204no—
581C5#7886611yescharacteristic, complemented, Sylow
582C22—21no—
583C22—21no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—304no—
588C22—608no—
589C22—602no—
590C22—602no—
591C22#9177012yes—
592C22C2×D4×D1511yescharacteristic
593C4—22no—
594C4—42no—
595C4—152no—
596C4—301no—
597C4—301no—
598C4—301no—
599C4—601no—
600C4—601no—
601C4#9177012yes—
602C4C22×C4×D1511yescharacteristic
603C4C2×#681911yescharacteristic
604C3#1061059711yescharacteristic, complemented, Sylow
605C2—22no—
606C2—42no—
607C2—604no—
608C2C2×#9177011yescharacteristic
609C2D15×AGL(1,ℤ/8)11yescharacteristic
610C2#1175653611yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#417C2
socleC22×C3×C5C22×D4
prime radicalC2×#91838C2
cosocleC2×C3×C4×C5C24
hypercenter#417D15
hypocenterC3×C5C4×AGL(1,ℤ/8)
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
1. #41×Dic15C2semidirect
3. C2×#91782C2semidirect
4. C3×C5×#417C2semidirect
5. #11751232C2semidirect
6. M16×Dic15C2semidirect
7. #11751065C2semidirect
8. #11750918C2semidirect
9. #11750889C2semidirect
10. #11750886C2semidirect
11. #11750857C2semidirect
18. #91838C4semidirect
19. Q8×Dic15C22semidirect
20. Q8×Dic15C22semidirect
24. D4×Dic15C22semidirect
25. D4×Dic15C22semidirect
37. C3×C5×#103C22semidirect
38. C3×C5×#102C22semidirect
41. #91099C22semidirect
42. C8×Dic15C22semidirect
43. #91092C22semidirect
44. #91091C22semidirect
49. C5×#417D3semidirect
59. C3×C5×#41C2×C4semidirect
72. C2×D60C2×C4semidirect
84. #6717C2×C4semidirect
85. #6715C2×C4semidirect
99. #6710C2×C4semidirect
104. C3×#417D5semidirect
136. C5×#103C2×D3semidirect
137. C5×#102C2×D3semidirect
152. C3×C5×Q8C22×C4semidirect
155. C3×C5×D4C22×C4semidirect
180. C3⋊#168C22×C4semidirect
205. C3×#103C2×D5semidirect
206. C3×#102C2×D5semidirect
215. C5×#41C4×D3semidirect
266. #417D15semidirect
289. Dic15AGL(1,ℤ/8)semidirect
292. C3×#41C4×D5semidirect
341. C5×Q8C2×C4×D3semidirect
344. C5×D4C2×C4×D3semidirect
396. #103C2×D15semidirect
399. #102C2×D15semidirect
415. C3×Q8C2×C4×D5semidirect
418. C3×D4C2×C4×D5semidirect
468. #41C4×D15semidirect
510. C3×C5C4×AGL(1,ℤ/8)semidirect
546. Q8C2×C4×D15semidirect
552. D4C2×C4×D15semidirect
581. C5#78866semidirect
604. C3#10610597semidirect

Maximal subgroups

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

subgroupindex
#41×Dic152
C2×#918382
C2×#917822
C3×C5×#4172
#117512322
M16×Dic152
#117510652
#117509182
#117508892
#117508862
#117508572
#106105973
#788665