Finite Group Databasebeta

← #14329510

Subgroups — #14329510

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C2×#91836C211yesmaximal, characteristic
3C2×#91782C211yesmaximal, characteristic
4#11751330C211yesmaximal, characteristic, complemented
5#11751091C212yesmaximal, complemented
6#11751090C212yesmaximal, complemented
7C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
8#11751041C212yesmaximal, complemented
9#11751040C211yesmaximal, characteristic
10C4×#6714C212yesmaximal, complemented
11C4×#6712C211yesmaximal, characteristic, complemented
12#10614197—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C22×D60C2211yescharacteristic
15C22×C4×D15C2212yes—
16C3×C4×C5×Q8—24no—
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C42×C5C2211yescharacteristic
20C2×C3×C42×C5C2212yes—
21#91836C418yescomplemented
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×#6714C2212yes—
27C2×C4⋊Dic15C2211yescharacteristic
28C2×#6712C2211yescharacteristic
29C2×#6712C2211yescharacteristic
30C2×C4×Dic15C2211yescharacteristic
31C2×C4×Dic15C2212yes—
32C2×C4×Dic15C2211yescharacteristic
33#91782—24no—
34#91115C2211yescharacteristic
35#91115C2211yescharacteristic
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#82466—51nomaximal, Hall
40C2×C4×C5×Q8D311yescharacteristic, complemented
41C2×#4093—31no—
42C2×#4039—31no—
43#65519—31no—
44#65279—32no—
45#65278—32no—
46C4⋊C4×Dic5—31no—
47#65229—32no—
48#65228—31no—
49C4×#685—32no—
50C4×#683—31no—
51C23×D15C2312yes—
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8—22no—
54C2×C3×C5×Q8C2×C414yescomplemented
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×Dic15C2311yescharacteristic
61C22×Dic15C2312yes—
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2311yescharacteristic
65C2×D60—22no—
66C2×D60—22no—
67C2×D60C2×C414yes—
68C2×C4×D15—24no—
69C2×C4×D15—24no—
70C2×C4×D15—42no—
71C3×C5×C4⋊C4—24no—
72C3×C5×C4⋊C4D414yes—
73C3×C42×C5—24no—
74C3×C42×C5—22no—
75#6714—24no—
76#6714—24no—
77#6714—42no—
78#6714C2×C418yescomplemented
79#6714C2×C418yescomplemented
80C4⋊Dic15—22no—
81#6712—22no—
82#6712—22no—
83C4×Dic15—22no—
84C4×Dic15—42no—
85C4×Dic15C2×C414yescomplemented
86C2×C3×C4×Q8D511yescharacteristic, complemented
87C2×#947—51no—
88C2×#893—51no—
89#5229—51no—
90#4990—52no—
91#4989—52no—
92Dic3×C4⋊C4—51no—
93#4940—52no—
94#4939—51no—
95C4×(C2×D3)⋊C4—52no—
96C4×Dic3⋊C4—51no—
97C22×C5×Q8C2×D311yescharacteristic
98C22×D20—31no—
99C22×C4×D5—32no—
100C4×C5×Q8—24no—
101C2×C5×C4⋊C4C2×D312yes—
102C2×C5×C4⋊C4C2×D311yescharacteristic
103C2×C42×C5C2×D311yescharacteristic
104C2×C42×C5C2×D312yes—
105#4093—38no—
106C2×#685—32no—
107C2×#685—32no—
108C2×#685—32no—
109C2×#685—32no—
110C2×#685—32no—
111C2×C4⋊Dic5—31no—
112C2×#683—31no—
113C2×#683—31no—
114C2×C4×Dic5—31no—
115C2×C4×Dic5—31no—
116C2×C4×Dic5—32no—
117#4039—64no—
118#3963—31no—
119#3963—31no—
120#3963—32no—
121#3963—32no—
122#3963—32no—
123#2367—151noSylow
124C23×C3×C5C2411yescharacteristic
125C22×D15—22no—
126C22×D15—22no—
127C22×D15—22no—
128C22×D15—22no—
129C22×D15—22no—
130C22×D15—22no—
131C22×D15C22×C414yes—
132C3×C5×Q8—28no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5C22×C414yes—
143C2×C3×C4×C5C22×C412yes—
144C2×C3×C4×C5C2×D414yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5#4112yes—
147C2×C3×C4×C5#4112yes—
148C2×Dic15—21no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15—21no—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15—21no—
160C2×Dic15C22×C412yes—
161C2×Dic15C22×C412yes—
162C2×Dic15#4112yes—
163D60—28no—
164C4×D15—48no—
165C22×C3×Q8C2×D511yescharacteristic
166C22×D12—51no—
167C22×C4×D3—52no—
168C3×C4×Q8—24no—
169C2×C3×C4⋊C4C2×D512yes—
170C2×C3×C4⋊C4C2×D511yescharacteristic
171C2×C3×C42C2×D511yescharacteristic
172C2×C3×C42C2×D512yes—
173#947—58no—
174C2×(C2×D3)⋊C4—52no—
175C2×(C2×D3)⋊C4—52no—
176C2×(C2×D3)⋊C4—52no—
177C2×(C2×D3)⋊C4—52no—
178C2×(C2×D3)⋊C4—52no—
179C2×C4⋊Dic3—51no—
180C2×Dic3⋊C4—51no—
181C2×Dic3⋊C4—51no—
182C2×C4×Dic3—51no—
183C2×C4×Dic3—51no—
184C2×C4×Dic3—52no—
185#893—104no—
186C3⋊#95—51no—
187C3⋊#95—51no—
188C3⋊#95—52no—
189C3⋊#95—52no—
190C3⋊#95—52no—
191C23×D5—32no—
192C2×C5×Q8—22no—
193C2×C5×Q8—22no—
194C2×C5×Q8C4×D314yescomplemented
195C22×C4×C5C22×D311yescharacteristic
196C22×C4×C5C22×D312yes—
197C22×C4×C5C22×D311yescharacteristic
198C22×C4×C5C22×D312yes—
199C22×C4×C5C22×D311yescharacteristic
200C22×Dic5—31no—
201C22×Dic5—31no—
202C22×Dic5—31no—
203C22×Dic5—31no—
204C22×Dic5—32no—
205C2×D20—34no—
206C2×D20—62no—
207C2×D20—62no—
208C2×C4×D5—64no—
209C2×C4×D5—64no—
210C2×C4×D5—122no—
211C5×C4⋊C4—24no—
212C5×C4⋊C4#6714yes—
213C42×C5—24no—
214C42×C5—22no—
215#685—38no—
216#685—38no—
217#685—64no—
218#685—64no—
219#685—122no—
220C4⋊Dic5—62no—
221#683—62no—
222#683—62no—
223C4×Dic5—34no—
224C4×Dic5—62no—
225C4×Dic5—122no—
226C2×#124—151no—
227C2×C4×Q8D1511yescharacteristic, complemented
228C2×C4×D4—151no—
229#391—151no—
230#390—151no—
231#388—152no—
232#388—152no—
233#385—152no—
234C4×C4⋊C4—151no—
235C4×C4⋊C4—151no—
236C4×C22⋊C4—152no—
237C22×C3×C5C23×C411yescharacteristic
238C22×C3×C5C22×D411yescharacteristic
239C22×C3×C5C2×#4111yescharacteristic
240C22×C3×C5C2×#4111yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C52−1+411yescharacteristic
244C2×D15—24no—
245C2×D15—24no—
246C2×D15—24no—
247C2×D15—42no—
248C2×D15—42no—
249C2×D15—42no—
250C2×D15—42no—
251C3×C4×C5—24no—
252C3×C4×C5—24no—
253C3×C4×C5—22no—
254C3×C4×C5—41no—
255C3×C4×C5C4×D414yes—
256Dic15—22no—
257Dic15—22no—
258Dic15—22no—
259Dic15—41no—
260Dic15—42no—
261C23×D3—52no—
262C2×C3×Q8—22no—
263C2×C3×Q8—22no—
264C2×C3×Q8C4×D514yescomplemented
265C22×C3×C4C22×D511yescharacteristic
266C22×C3×C4C22×D512yes—
267C22×C3×C4C22×D511yescharacteristic
268C22×C3×C4C22×D512yes—
269C22×C3×C4C22×D511yescharacteristic
270C22×Dic3—51no—
271C22×Dic3—51no—
272C22×Dic3—51no—
273C22×Dic3—51no—
274C22×Dic3—52no—
275C2×D12—54no—
276C2×D12—102no—
277C2×D12—102no—
278C2×C4×D3—104no—
279C2×C4×D3—104no—
280C2×C4×D3—202no—
281C3×C4⋊C4—24no—
282C3×C4⋊C4C22⋊D514yes—
283C3×C42—24no—
284C3×C42—22no—
285(C2×D3)⋊C4—58no—
286(C2×D3)⋊C4—58no—
287(C2×D3)⋊C4—104no—
288(C2×D3)⋊C4—104no—
289(C2×D3)⋊C4—202no—
290C4⋊Dic3—102no—
291Dic3⋊C4—102no—
292Dic3⋊C4—102no—
293C4×Dic3—54no—
294C4×Dic3—102no—
295C4×Dic3—202no—
296C23×C5C23×D311yescharacteristic
297C22×D5—34no—
298C22×D5—62no—
299C22×D5—62no—
300C22×D5—62no—
301C22×D5—62no—
302C22×D5—62no—
303C22×D5—62no—
304C5×Q8—28no—
305C2×C4×C5—22no—
306C2×C4×C5—21no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—21no—
310C2×C4×C5—22no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5C2×C4×D314yes—
315C2×C4×C5C2×C4×D312yes—
316C2×C4×C5#23512yes—
317C2×C4×C5#23912yes—
318C2×C4×C5C2×#6714yes—
319C2×C4×C5C2×#6712yes—
320C2×Dic5—32no—
321C2×Dic5—32no—
322C2×Dic5—32no—
323C2×Dic5—61no—
324C2×Dic5—61no—
325C2×Dic5—61no—
326C2×Dic5—61no—
327C2×Dic5—61no—
328C2×Dic5—61no—
329C2×Dic5—62no—
330C2×Dic5—61no—
331C2×Dic5—62no—
332C2×Dic5—61no—
333C2×Dic5—62no—
334C2×Dic5—61no—
335D20—68no—
336C4×D5—128no—
337C22×Q8C2×D1511yescharacteristic
338C22×D4—151no—
339C23×C4—152no—
340#124—158no—
341C4×Q8—24no—
342C4×D4—304no—
343C2×C4⋊C4—151no—
344C2×C4⋊C4—151no—
345C2×C4⋊C4—151no—
346C2×C4⋊C4C2×D1512yes—
347C2×C4⋊C4C2×D1511yescharacteristic
348C2×C22⋊C4—152no—
349C2×C22⋊C4—152no—
350C2×C22⋊C4—152no—
351C2×C22⋊C4—152no—
352C2×C22⋊C4—152no—
353C2×C42—151no—
354C2×C42—151no—
355C2×C42—152no—
356C2×C42C2×D1512yes—
357C2×C42C2×D1511yescharacteristic
358#95—151no—
359#95—152no—
360#95—152no—
361#95—151no—
362#95—152no—
363C2×C3×C5C2×C4×D411yescharacteristic
364C2×C3×C5C4×#4111yescharacteristic
365C2×C3×C5#52011yescharacteristic
366C2×C3×C5#54611yescharacteristic
367C2×C3×C5#54811yescharacteristic
368C2×C3×C5#55611yescharacteristic
369C2×C3×C5#55911yescharacteristic
370D15—44no—
371C23×C3C23×D511yescharacteristic
372C22×D3—54no—
373C22×D3—102no—
374C22×D3—102no—
375C22×D3—102no—
376C22×D3—102no—
377C22×D3—102no—
378C22×D3—102no—
379C3×Q8—28no—
380C2×C3×C4—22no—
381C2×C3×C4—21no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4C2×C4×D514yes—
390C2×C3×C4C2×C4×D512yes—
391C2×C3×C4#70912yes—
392C2×C3×C4#71312yes—
393C2×C3×C4C2×C22⋊D514yes—
394C2×C3×C4C2×C22⋊D512yes—
395C2×Dic3—52no—
396C2×Dic3—52no—
397C2×Dic3—52no—
398C2×Dic3—101no—
399C2×Dic3—101no—
400C2×Dic3—101no—
401C2×Dic3—101no—
402C2×Dic3—101no—
403C2×Dic3—101no—
404C2×Dic3—102no—
405C2×Dic3—101no—
406C2×Dic3—102no—
407C2×Dic3—101no—
408C2×Dic3—102no—
409C2×Dic3—101no—
410D12—108no—
411C4×D3—208no—
412C22×C5C22×C4×D311yescharacteristic
413C22×C5C2×#23511yescharacteristic
414C22×C5C2×#23911yescharacteristic
415C22×C5#100711yescharacteristic
416C22×C5D3×#4111yescharacteristic
417C22×C5#100911yescharacteristic
418C22×C5C22×#6711yescharacteristic
419C2×D5—64no—
420C2×D5—64no—
421C2×D5—64no—
422C2×D5—122no—
423C2×D5—122no—
424C2×D5—122no—
425C2×D5—122no—
426C4×C5—24no—
427C4×C5—22no—
428C4×C5—24no—
429C4×C5—41no—
430C4×C5C4×#6714yes—
431Dic5—62no—
432Dic5—62no—
433Dic5—62no—
434Dic5—121no—
435Dic5—122no—
436C24—152no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8C4×D1514yescomplemented
440C2×D4—154no—
441C2×D4—302no—
442C2×D4—302no—
443C22×C4—151no—
444C22×C4—151no—
445C22×C4—151no—
446C22×C4—152no—
447C22×C4—151no—
448C22×C4—304no—
449C22×C4—304no—
450C22×C4—602no—
451C22×C4C22×D1511yescharacteristic
452C22×C4C22×D1512yes—
453C22×C4C22×D1511yescharacteristic
454C22×C4C22×D1512yes—
455C22×C4C22×D1511yescharacteristic
456C4⋊C4—24no—
457C4⋊C4—302no—
458C4⋊C4—302no—
459C4⋊C4—302no—
460C4⋊C4#122014yes—
461C22⋊C4—158no—
462C22⋊C4—158no—
463C22⋊C4—304no—
464C22⋊C4—304no—
465C22⋊C4—602no—
466C42—24no—
467C42—22no—
468C42—154no—
469C42—302no—
470C42—602no—
471C3×C5#236711yescharacteristic, complemented, Hall
472C22×C3C22×C4×D511yescharacteristic
473C22×C3C2×#70911yescharacteristic
474C22×C3C2×#71311yescharacteristic
475C22×C3#414711yescharacteristic
476C22×C3D5×#4111yescharacteristic
477C22×C3#414911yescharacteristic
478C22×C3C22×C22⋊D511yescharacteristic
479C2×D3—104no—
480C2×D3—104no—
481C2×D3—104no—
482C2×D3—202no—
483C2×D3—202no—
484C2×D3—202no—
485C2×D3—202no—
486C3×C4—24no—
487C3×C4—22no—
488C3×C4—24no—
489C3×C4—41no—
490C3×C4C4×C22⋊D514yes—
491Dic3—102no—
492Dic3—102no—
493Dic3—102no—
494Dic3—201no—
495Dic3—202no—
496C2×C5#556811yescharacteristic
497C2×C5C4×#23911yescharacteristic
498C2×C5#557411yescharacteristic
499C2×C5#558511yescharacteristic
500C2×C5C2×C4×#6711yescharacteristic
501C2×C5#581711yescharacteristic
502C2×C5#582911yescharacteristic
503D5—124no—
504C23—154no—
505C23—302no—
506C23—302no—
507C23—302no—
508C23—302no—
509C23—302no—
510C23—302no—
511C23C23×D1511yescharacteristic
512Q8—28no—
513D4—308no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—152no—
524C2×C4—152no—
525C2×C4—152no—
526C2×C4—301no—
527C2×C4—301no—
528C2×C4—301no—
529C2×C4—301no—
530C2×C4—301no—
531C2×C4—301no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—301no—
535C2×C4—301no—
536C2×C4—302no—
537C2×C4—301no—
538C2×C4—608no—
539C2×C4C2×C4×D1514yes—
540C2×C4C2×C4×D1512yes—
541C2×C4#681712yes—
542C2×C4#682112yes—
543C2×C4C2×#122014yes—
544C2×C4C2×#122012yes—
545C2×C3#6590211yescharacteristic
546C2×C3C4×#71311yescharacteristic
547C2×C3#6590811yescharacteristic
548C2×C3#6591911yescharacteristic
549C2×C3C2×C4×C22⋊D511yescharacteristic
550C2×C3#6615111yescharacteristic
551C2×C3#6616311yescharacteristic
552D3—204no—
553C5#8246611yescharacteristic, complemented, Sylow
554C22—304no—
555C22—304no—
556C22—304no—
557C22—602no—
558C22—602no—
559C22—602no—
560C22—602no—
561C22C22×C4×D1511yescharacteristic
562C22C2×#681711yescharacteristic
563C22C2×#682111yescharacteristic
564C22#9209811yescharacteristic
565C22#41×D1511yescharacteristic
566C22#9210011yescharacteristic
567C22C22×#122011yescharacteristic
568C4—24no—
569C4—22no—
570C4—24no—
571C4—41no—
572C4—302no—
573C4—302no—
574C4—302no—
575C4—601no—
576C4—602no—
577C4C4×#122014yes—
578C3#1061419711yescharacteristic, complemented, Sylow
579C2—604no—
580C2#1175632711yescharacteristic
581C2C4×#682111yescharacteristic
582C2#1175633311yescharacteristic
583C2#1175634411yescharacteristic
584C2C2×C4×#122011yescharacteristic
585C2#1175657611yescharacteristic
586C2#1175658811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC2×C3×C4×C5×Q8C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C4×Q8D15
hypocenterC3×C5#2367
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
4. #11751330C2semidirect
5. #11751091C2semidirect
6. #11751090C2semidirect
7. C4⋊C4×Dic15C2semidirect
8. #11751041C2semidirect
10. C4×#6714C2semidirect
11. C4×#6712C2semidirect
21. #91836C4semidirect
40. C2×C4×C5×Q8D3semidirect
54. C2×C3×C5×Q8C2×C4semidirect
78. #6714C2×C4semidirect
79. #6714C2×C4semidirect
85. C4×Dic15C2×C4semidirect
86. C2×C3×C4×Q8D5semidirect
194. C2×C5×Q8C4×D3semidirect
227. C2×C4×Q8D15semidirect
264. C2×C3×Q8C4×D5semidirect
439. C2×Q8C4×D15semidirect
471. C3×C5#2367semidirect
553. C5#82466semidirect
578. C3#10614197semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C2×#918362
C2×#917822
#117513302
#117510912
#117510902
C4⋊C4×Dic152
#117510412
#117510402
C4×#67142
C4×#67122
#106141973
#824665