Finite Group Databasebeta

← #14326250

Subgroups — #14326250

Order 1920.

1584 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6714C211yesmaximal, characteristic
2C2×C22⋊C4×D15C211yesmaximal, characteristic
3C2×#91766C211yesmaximal, characteristic, complemented
4C3×C4×C5×C22⋊C4C211yesmaximal, characteristic, complemented
5#11751313C211yesmaximal, characteristic, complemented
6#11751056C211yesmaximal, characteristic
7#11751044C211yesmaximal, characteristic, complemented
8#11751041C212yesmaximal, complemented
9C4×#6714C212yesmaximal, complemented
10#11750771C212yesmaximal, complemented
11#11750769C212yesmaximal, complemented
12#10605306—31nomaximal, Hall
13C24×D15C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C22×C4×D15C2212yes—
17C2×C3×C5×C22⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×C3×C42×C5C2212yescomplemented
20C2×#6719C2211yescharacteristic
21C2×#6719C2211yescharacteristic
22C2×#6714C418yescomplemented
23C2×#6714C2212yes—
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×#6714C2211yescharacteristic
27C2×#6714C2211yescharacteristic
28C2×#6714C2211yescharacteristic
29C2×#6714C2211yescharacteristic
30C2×C4⋊Dic15C2212yescomplemented
31C2×#6712C2212yescomplemented
32C2×C4×Dic15C2212yescomplemented
33C22⋊C4×D15—24no—
34#91766—24no—
35C3×C5×#95C2212yescomplemented
36#91115C2212yescomplemented
37#91115C2212yescomplemented
38#91115C2212yescomplemented
39#73575—51nomaximal, Hall
40C22×#685—31no—
41C2×D5×C22⋊C4—31no—
42C2×#4023—31no—
43C4×C5×C22⋊C4D311yescharacteristic, complemented
44#65501—31no—
45#65244—31no—
46#65232—31no—
47#65229—32no—
48C4×#685—32no—
49#64959—32no—
50#64957—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15—22no—
53C23×D15—22no—
54C23×D15—24no—
55C23×D15—42no—
56C23×D15C2×C414yes—
57C23×D15C2311yescharacteristic
58C23×D15C2311yescharacteristic
59C22×C3×C4×C5—42no—
60C22×C3×C4×C5C2×C414yescomplemented
61C22×C3×C4×C5C2312yes—
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2312yes—
65C22×Dic15—22no—
66C22×Dic15—22no—
67C22×Dic15C2×C414yescomplemented
68C22×Dic15C2312yes—
69C22×Dic15C2312yes—
70C22×Dic15C2311yescharacteristic
71C22×Dic15C2311yescharacteristic
72C2×C4×D15—24no—
73C2×C4×D15—24no—
74C2×C4×D15—42no—
75C3×C5×C22⋊C4—22no—
76C3×C5×C22⋊C4D414yes—
77C3×C42×C5—42no—
78#6719—22no—
79#6719—22no—
80#6714—24no—
81#6714—24no—
82#6714—42no—
83#6714C2×C414yes—
84#6714C2×C414yes—
85#6714C2×C414yescomplemented
86#6714C2×C414yescomplemented
87C4⋊Dic15—42no—
88#6712—24no—
89C4×Dic15—24no—
90C22×(C2×D3)⋊C4—51no—
91C2×D3×C22⋊C4—51no—
92C2×#877—51no—
93C3×C4×C22⋊C4D511yescharacteristic, complemented
94#5212—51no—
95#4955—51no—
96#4943—51no—
97#4940—52no—
98C4×(C2×D3)⋊C4—52no—
99#4670—52no—
100#4668—52no—
101C24×D5—31no—
102C23×C4×C5C2×D311yescharacteristic
103C23×Dic5—31no—
104C22×C4×D5—32no—
105C2×C5×C22⋊C4C2×D311yescharacteristic
106C2×C5×C22⋊C4C2×D311yescharacteristic
107C2×C42×C5C2×D312yescomplemented
108C2×C22⋊Dic5—31no—
109C2×C22⋊Dic5—31no—
110C2×#685—38no—
111C2×#685—31no—
112C2×#685—31no—
113C2×#685—31no—
114C2×#685—31no—
115C2×#685—32no—
116C2×#685—32no—
117C2×#685—32no—
118C2×C4⋊Dic5—32no—
119C2×#683—32no—
120C2×C4×Dic5—32no—
121D5×C22⋊C4—64no—
122#4023—64no—
123C5×#95C2×D312yescomplemented
124#3963—32no—
125#3963—32no—
126#3963—32no—
127#2342—151noSylow
128C23×C3×C5—22no—
129C23×C3×C5—22no—
130C23×C3×C5C22×C412yes—
131C23×C3×C5C2×D412yes—
132C23×C3×C5#4112yes—
133C23×C3×C5C2411yescharacteristic
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—22no—
137C22×D15—28no—
138C22×D15—21no—
139C22×D15—22no—
140C22×D15—21no—
141C22×D15—22no—
142C22×D15—28no—
143C22×D15—44no—
144C22×D15—48no—
145C22×D15—48no—
146C22×D15C22×C412yes—
147C22×D15C22×C412yes—
148C22×D15#4114yes—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5—22no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—44no—
157C2×C3×C4×C5C22×C412yes—
158C2×C3×C4×C5C22×C412yes—
159C2×C3×C4×C5C2×D414yes—
160C2×Dic15—28no—
161C2×Dic15—21no—
162C2×Dic15—21no—
163C2×Dic15—21no—
164C2×Dic15—21no—
165C2×Dic15—22no—
166C2×Dic15—22no—
167C2×Dic15—22no—
168C2×Dic15—22no—
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15C22×C412yes—
172C2×Dic15C22×C412yes—
173C4×D15—48no—
174C24×D3—51no—
175C23×C3×C4C2×D511yescharacteristic
176C23×Dic3—51no—
177C22×C4×D3—52no—
178C2×C3×C22⋊C4C2×D511yescharacteristic
179C2×C3×C22⋊C4C2×D511yescharacteristic
180C2×C3×C42C2×D512yescomplemented
181C2×#217—51no—
182C2×#217—51no—
183C2×(C2×D3)⋊C4—58no—
184C2×(C2×D3)⋊C4—51no—
185C2×(C2×D3)⋊C4—51no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—51no—
188C2×(C2×D3)⋊C4—52no—
189C2×(C2×D3)⋊C4—52no—
190C2×(C2×D3)⋊C4—52no—
191C2×C4⋊Dic3—52no—
192C2×Dic3⋊C4—52no—
193C2×C4×Dic3—52no—
194D3×C22⋊C4—104no—
195#877—104no—
196C3×#95C2×D512yescomplemented
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95—52no—
200C24×C5C22×D311yescharacteristic
201C23×D5—31no—
202C23×D5—31no—
203C23×D5—34no—
204C23×D5—64no—
205C23×D5—62no—
206C23×D5—62no—
207C23×D5—122no—
208C22×C4×C5—42no—
209C22×C4×C5C4×D314yescomplemented
210C22×C4×C5C22×D312yes—
211C22×C4×C5C22×D311yescharacteristic
212C22×C4×C5C22×D311yescharacteristic
213C22×C4×C5C22×D312yes—
214C22×Dic5—34no—
215C22×Dic5—31no—
216C22×Dic5—31no—
217C22×Dic5—32no—
218C22×Dic5—32no—
219C22×Dic5—62no—
220C22×Dic5—62no—
221C2×C4×D5—64no—
222C2×C4×D5—64no—
223C2×C4×D5—122no—
224C5×C22⋊C4—22no—
225C5×C22⋊C4D1214yes—
226C42×C5—42no—
227C22⋊Dic5—62no—
228C22⋊Dic5—62no—
229#685—34no—
230#685—34no—
231#685—34no—
232#685—34no—
233#685—64no—
234#685—64no—
235#685—122no—
236C4⋊Dic5—122no—
237#683—64no—
238C4×Dic5—64no—
239C2×#117—151no—
240C22×C22⋊C4—151no—
241C22×C22⋊C4—151no—
242#388—152no—
243#388—152no—
244#385—152no—
245#381—151no—
246#380—151no—
247#379—151no—
248C4×C22⋊C4—152no—
249C4×C22⋊C4D1511yescharacteristic, complemented
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—24no—
253C22×C3×C5—24no—
254C22×C3×C5C4×D414yes—
255C22×C3×C5C23×C411yescharacteristic
256C22×C3×C5C22×D411yescharacteristic
257C22×C3×C5C2×#4111yescharacteristic
258C22×C3×C5C2×#4112yes—
259C22×C3×C5H(5,𝔽2)12yes—
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—24no—
263C2×D15—24no—
264C2×D15—22no—
265C2×D15—22no—
266C2×D15—22no—
267C2×D15—416no—
268C2×D15—42no—
269C2×D15—42no—
270C2×D15#11714yescomplemented
271C3×C4×C5—22no—
272C3×C4×C5—22no—
273C3×C4×C5—24no—
274C3×C4×C5—42no—
275Dic15—22no—
276Dic15—22no—
277Dic15—42no—
278Dic15—42no—
279C24×C3C22×D511yescharacteristic
280C23×D3—51no—
281C23×D3—51no—
282C23×D3—54no—
283C23×D3—104no—
284C23×D3—102no—
285C23×D3—102no—
286C23×D3—202no—
287C22×C3×C4—42no—
288C22×C3×C4C4×D514yescomplemented
289C22×C3×C4C22×D512yes—
290C22×C3×C4C22×D511yescharacteristic
291C22×C3×C4C22×D511yescharacteristic
292C22×C3×C4C22×D512yes—
293C22×Dic3—54no—
294C22×Dic3—51no—
295C22×Dic3—51no—
296C22×Dic3—52no—
297C22×Dic3—52no—
298C22×Dic3—102no—
299C22×Dic3—102no—
300C2×C4×D3—104no—
301C2×C4×D3—104no—
302C2×C4×D3—202no—
303C3×C22⋊C4—22no—
304C3×C22⋊C4D2014yes—
305C3×C42—42no—
306#217—102no—
307#217—102no—
308(C2×D3)⋊C4—54no—
309(C2×D3)⋊C4—54no—
310(C2×D3)⋊C4—54no—
311(C2×D3)⋊C4—54no—
312(C2×D3)⋊C4—104no—
313(C2×D3)⋊C4—104no—
314(C2×D3)⋊C4—202no—
315C4⋊Dic3—202no—
316Dic3⋊C4—104no—
317C4×Dic3—104no—
318C23×C5—22no—
319C23×C5—22no—
320C23×C5C2×C4×D312yes—
321C23×C5C2×D1212yes—
322C23×C5#23512yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—32no—
325C22×D5—34no—
326C22×D5—32no—
327C22×D5—68no—
328C22×D5—61no—
329C22×D5—61no—
330C22×D5—62no—
331C22×D5—68no—
332C22×D5—61no—
333C22×D5—61no—
334C22×D5—62no—
335C22×D5—62no—
336C22×D5—124no—
337C22×D5—128no—
338C22×D5—128no—
339C2×C4×C5—22no—
340C2×C4×C5—22no—
341C2×C4×C5—22no—
342C2×C4×C5—22no—
343C2×C4×C5—22no—
344C2×C4×C5—22no—
345C2×C4×C5—22no—
346C2×C4×C5—44no—
347C2×C4×C5C2×C4×D312yes—
348C2×C4×C5C2×C4×D312yes—
349C2×C4×C5C2×D1214yes—
350C2×Dic5—32no—
351C2×Dic5—32no—
352C2×Dic5—68no—
353C2×Dic5—61no—
354C2×Dic5—61no—
355C2×Dic5—61no—
356C2×Dic5—61no—
357C2×Dic5—62no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5—62no—
362C2×Dic5—62no—
363C4×D5—128no—
364C25—151no—
365C23×C4—151no—
366C23×C4—152no—
367C23×C4C2×D1511yescharacteristic
368#117—304no—
369C2×C4⋊C4—152no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—158no—
372C2×C22⋊C4—151no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—151no—
375C2×C22⋊C4—151no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—151no—
378C2×C22⋊C4—152no—
379C2×C22⋊C4—152no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4—304no—
382C2×C22⋊C4C2×D1511yescharacteristic
383C2×C22⋊C4C2×D1511yescharacteristic
384C2×C42—152no—
385C2×C42C2×D1512yescomplemented
386#95—152no—
387#95—152no—
388#95—152no—
389#95C2×D1512yescomplemented
390C2×C3×C5—24no—
391C2×C3×C5C2×#11711yescharacteristic
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5#51811yescharacteristic
394C2×C3×C5#54612yes—
395C2×C3×C5#55112yes—
396D15—24no—
397D15—42no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3C2×C4×D512yes—
401C23×C3C2×D2012yes—
402C23×C3#70912yes—
403C23×C3C23×D511yescharacteristic
404C22×D3—52no—
405C22×D3—54no—
406C22×D3—52no—
407C22×D3—108no—
408C22×D3—101no—
409C22×D3—101no—
410C22×D3—102no—
411C22×D3—101no—
412C22×D3—101no—
413C22×D3—102no—
414C22×D3—102no—
415C22×D3—108no—
416C22×D3—204no—
417C22×D3—208no—
418C22×D3—208no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—44no—
427C2×C3×C4C2×C4×D512yes—
428C2×C3×C4C2×C4×D512yes—
429C2×C3×C4C2×D2014yes—
430C2×Dic3—52no—
431C2×Dic3—52no—
432C2×Dic3—101no—
433C2×Dic3—101no—
434C2×Dic3—101no—
435C2×Dic3—101no—
436C2×Dic3—108no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3—102no—
441C2×Dic3—102no—
442C2×Dic3—102no—
443C4×D3—208no—
444C22×C5—22no—
445C22×C5—22no—
446C22×C5—24no—
447C22×C5—24no—
448C22×C5C4×D1214yes—
449C22×C5C22×C4×D311yescharacteristic
450C22×C5C22×D1211yescharacteristic
451C22×C5C2×#23511yescharacteristic
452C22×C5#100412yes—
453C22×C5D3×#4112yes—
454C2×D5—34no—
455C2×D5—62no—
456C2×D5—62no—
457C2×D5—64no—
458C2×D5—64no—
459C2×D5—62no—
460C2×D5—62no—
461C2×D5—62no—
462C2×D5—1216no—
463C2×D5—122no—
464C2×D5—122no—
465C4×C5—22no—
466C4×C5—22no—
467C4×C5—24no—
468C4×C5—42no—
469Dic5—62no—
470Dic5—62no—
471Dic5—122no—
472Dic5—122no—
473C24—151no—
474C24—151no—
475C24—154no—
476C24—304no—
477C24—302no—
478C24—302no—
479C24—602no—
480C24C22×D1511yescharacteristic
481C22×C4—42no—
482C22×C4—154no—
483C22×C4—151no—
484C22×C4—151no—
485C22×C4—152no—
486C22×C4—152no—
487C22×C4—304no—
488C22×C4—302no—
489C22×C4—302no—
490C22×C4—304no—
491C22×C4—602no—
492C22×C4C4×D1514yescomplemented
493C22×C4C22×D1512yes—
494C22×C4C22×D1511yescharacteristic
495C22×C4C22×D1511yescharacteristic
496C22×C4C22×D1512yes—
497C4⋊C4—304no—
498C4⋊C4—602no—
499C22⋊C4—22no—
500C22⋊C4—154no—
501C22⋊C4—154no—
502C22⋊C4—154no—
503C22⋊C4—154no—
504C22⋊C4—304no—
505C22⋊C4—302no—
506C22⋊C4—302no—
507C22⋊C4—304no—
508C22⋊C4—602no—
509C22⋊C4D6014yes—
510C42—42no—
511C42—304no—
512C3×C5#234211yescharacteristic, complemented, Hall
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—24no—
516C22×C3—24no—
517C22×C3C4×D2014yes—
518C22×C3C22×C4×D511yescharacteristic
519C22×C3C22×D2011yescharacteristic
520C22×C3C2×#70911yescharacteristic
521C22×C3#414412yes—
522C22×C3D5×#4112yes—
523C2×D3—54no—
524C2×D3—102no—
525C2×D3—102no—
526C2×D3—104no—
527C2×D3—104no—
528C2×D3—102no—
529C2×D3—102no—
530C2×D3—102no—
531C2×D3—2016no—
532C2×D3—202no—
533C2×D3—202no—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—24no—
537C3×C4—42no—
538Dic3—102no—
539Dic3—102no—
540Dic3—202no—
541Dic3—202no—
542C2×C5—24no—
543C2×C5C2×C4×D1211yescharacteristic
544C2×C5#548611yescharacteristic
545C2×C5D3×#11711yescharacteristic
546C2×C5#555412yes—
547C2×C5#555612yes—
548D5—64no—
549D5—122no—
550C23—22no—
551C23—22no—
552C23—152no—
553C23—154no—
554C23—152no—
555C23—308no—
556C23—301no—
557C23—301no—
558C23—302no—
559C23—308no—
560C23—301no—
561C23—301no—
562C23—302no—
563C23—302no—
564C23—604no—
565C23—608no—
566C23—608no—
567C23C2×C4×D1512yes—
568C23C2×D6012yes—
569C23#681712yes—
570C23C23×D1511yescharacteristic
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—152no—
580C2×C4—152no—
581C2×C4—308no—
582C2×C4—301no—
583C2×C4—301no—
584C2×C4—301no—
585C2×C4—301no—
586C2×C4—302no—
587C2×C4—302no—
588C2×C4—302no—
589C2×C4—302no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—608no—
593C2×C4C2×C4×D1512yes—
594C2×C4C2×C4×D1512yes—
595C2×C4C2×D6014yes—
596C2×C3—24no—
597C2×C3C2×C4×D2011yescharacteristic
598C2×C3#6582011yescharacteristic
599C2×C3D5×#11711yescharacteristic
600C2×C3#6588812yes—
601C2×C3#6589012yes—
602D3—104no—
603D3—202no—
604C5#7357511yescharacteristic, complemented, Sylow
605C22—24no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—154no—
610C22—304no—
611C22—304no—
612C22—302no—
613C22—302no—
614C22—302no—
615C22—302no—
616C22—302no—
617C22—6016no—
618C22—602no—
619C22—602no—
620C22C4×D6014yes—
621C22C22×C4×D1511yescharacteristic
622C22C22×D6011yescharacteristic
623C22C2×#681711yescharacteristic
624C22#9209512yes—
625C22#41×D1512yes—
626C4—24no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—302no—
631C4—302no—
632C4—602no—
633C4—602no—
634C3#1060530611yescharacteristic, complemented, Sylow
635C2—24no—
636C2—304no—
637C2—602no—
638C2C2×C4×D6011yescharacteristic
639C2#1175624511yescharacteristic
640C2D15×#11711yescharacteristic
641C2#1175631312yes—
642C2#1175631512yes—
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C4×C5×C22⋊C4C2
socle, cosocleC23×C3×C5C24
prime radicalC24×D15C22
hypercenterC4×C22⋊C4D15
hypocenterC3×C5#2342
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#91766C2semidirect
4. C3×C4×C5×C22⋊C4C2semidirect
5. #11751313C2semidirect
7. #11751044C2semidirect
8. #11751041C2semidirect
9. C4×#6714C2semidirect
10. #11750771C2semidirect
11. #11750769C2semidirect
19. C2×C3×C42×C5C22semidirect
22. C2×#6714C4semidirect
30. C2×C4⋊Dic15C22semidirect
31. C2×#6712C22semidirect
32. C2×C4×Dic15C22semidirect
35. C3×C5×#95C22semidirect
36. #91115C22semidirect
37. #91115C22semidirect
38. #91115C22semidirect
43. C4×C5×C22⋊C4D3semidirect
60. C22×C3×C4×C5C2×C4semidirect
67. C22×Dic15C2×C4semidirect
85. #6714C2×C4semidirect
86. #6714C2×C4semidirect
93. C3×C4×C22⋊C4D5semidirect
107. C2×C42×C5C2×D3semidirect
123. C5×#95C2×D3semidirect
180. C2×C3×C42C2×D5semidirect
196. C3×#95C2×D5semidirect
209. C22×C4×C5C4×D3semidirect
249. C4×C22⋊C4D15semidirect
270. C2×D15#117semidirect
288. C22×C3×C4C4×D5semidirect
385. C2×C42C2×D15semidirect
389. #95C2×D15semidirect
492. C22×C4C4×D15semidirect
512. C3×C5#2342semidirect
604. C5#73575semidirect
634. C3#10605306semidirect

Maximal subgroups

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

subgroupindex
C22×#67142
C2×C22⋊C4×D152
C2×#917662
C3×C4×C5×C22⋊C42
#117513132
#117510562
#117510442
#117510412
C4×#67142
#117507712
#117507692
#106053063
#735755