Finite Group Databasebeta

← #14323359

Subgroups — #14323359

Order 1920.

1616 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6714C211yesmaximal, characteristic, complemented
2C2×#91771C211yesmaximal, characteristic
3C2×#91770C211yesmaximal, characteristic, complemented
4C3×C5×#385C211yesmaximal, characteristic, complemented
5#11751314C211yesmaximal, characteristic, complemented
6#11751091C212yesmaximal, complemented
7#11751089C212yesmaximal, complemented
8#11751056C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic15C211yesmaximal, characteristic, complemented
10#11750770C212yesmaximal, complemented
11#11750766C212yesmaximal, complemented
12#10606323—31nomaximal, Hall
13C24×D15C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C22×#1220C2211yescharacteristic
16C23×Dic15C2211yescharacteristic
17C22×D60C2212yes—
18C22×C4×D15C2211yescharacteristic, complemented
19C2×C3×C5×C4⋊C4C2212yescomplemented
20C2×C3×C5×C22⋊C4C2211yescharacteristic
21C2×C3×C5×C22⋊C4C2211yescharacteristic
22C2×#6719C2211yescharacteristic
23C2×#6714—24no—
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×#6714C2212yes—
27C2×#6714C2212yes—
28C2×#6714C2212yes—
29C2×#6714C2211yescharacteristic, complemented
30C2×#6714C2212yescomplemented
31C2×#6712C2211yescharacteristic, complemented
32C2×C4×Dic15C2211yescharacteristic, complemented
33C2×C4×Dic15C2212yescomplemented
34#91771C418yescomplemented
35#91770—24no—
36C3×C5×#95C2212yescomplemented
37#91115C2212yescomplemented
38#91115C2212yescomplemented
39#74592—51nomaximal, Hall
40C22×#685—31no—
41C2×#4028—31no—
42C2×#4027—31no—
43C5×#385D311yescharacteristic, complemented
44#65502—31no—
45#65279—32no—
46#65277—32no—
47#65244—31no—
48C22⋊C4×Dic5—31no—
49#64958—32no—
50#64954—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15—22no—
53C23×D15—22no—
54C23×D15—42no—
55C23×D15—42no—
56C23×D15C2×C414yescomplemented
57C23×D15C2312yes—
58C23×D15C2311yescharacteristic
59C22×C3×C4×C5—22no—
60C22×C3×C4×C5—42no—
61C22×C3×C4×C5C2311yescharacteristic
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2312yes—
64C22×C3×C4×C5C2312yes—
65C2×#1220—22no—
66C2×#1220—22no—
67C2×#1220C2×C414yes—
68C22×Dic15—42no—
69C22×Dic15D414yescomplemented
70C22×Dic15C2312yes—
71C22×Dic15C2311yescharacteristic
72C22×Dic15C2311yescharacteristic
73C22×Dic15C2311yescharacteristic
74C2×D60—42no—
75C2×D60—42no—
76C2×D60C2×C418yescomplemented
77C2×C4×D15—22no—
78C2×C4×D15—42no—
79C3×C5×C4⋊C4—42no—
80C3×C5×C22⋊C4—22no—
81C3×C5×C22⋊C4C2×C414yescomplemented
82#6719—22no—
83#6714—22no—
84#6714—24no—
85#6714—24no—
86#6714—42no—
87#6714—42no—
88#6714—42no—
89#6714C2×C418yescomplemented
90#6712—22no—
91C4×Dic15—22no—
92C4×Dic15—42no—
93C22×(C2×D3)⋊C4—51no—
94C2×#882—51no—
95C2×#67⋊C4—51no—
96C3×#385D511yescharacteristic, complemented
97#5213—51no—
98#4990—52no—
99#4988—52no—
100#4955—51no—
101Dic3×C22⋊C4—51no—
102#4669—52no—
103#4665—52no—
104C24×D5—31no—
105C23×C4×C5C2×D311yescharacteristic
106C22×C22⋊D5—31no—
107C23×Dic5—31no—
108C22×D20—32no—
109C22×C4×D5—31no—
110C2×C5×C4⋊C4C2×D312yescomplemented
111C2×C5×C22⋊C4C2×D311yescharacteristic
112C2×C5×C22⋊C4C2×D311yescharacteristic
113C2×C22⋊Dic5—31no—
114C2×#685—32no—
115C2×#685—31no—
116C2×#685—32no—
117C2×#685—32no—
118C2×#685—32no—
119C2×#685—32no—
120C2×#685—32no—
121C2×#685—64no—
122C2×#683—31no—
123C2×C4×Dic5—31no—
124C2×C4×Dic5—32no—
125#4028—38no—
126#4027—64no—
127C5×#95C2×D312yescomplemented
128#3963—32no—
129#3963—32no—
130#2321—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5C22×C412yes—
134C23×C3×C5C2×D412yes—
135C23×C3×C5#4112yes—
136C23×C3×C5C2411yescharacteristic
137C22×D15—28no—
138C22×D15—22no—
139C22×D15—21no—
140C22×D15—21no—
141C22×D15—22no—
142C22×D15—22no—
143C22×D15—22no—
144C22×D15—22no—
145C22×D15—22no—
146C22×D15—22no—
147C22×D15—22no—
148C22×D15—48no—
149C22×D15—44no—
150C22×D15—44no—
151C22×D15—48no—
152C22×D15C22×C414yes—
153C22×D15C22×C412yes—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—21no—
157C2×C3×C4×C5—21no—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—22no—
160C2×C3×C4×C5—22no—
161C2×C3×C4×C5—22no—
162C2×C3×C4×C5—22no—
163C2×C3×C4×C5—44no—
164C2×C3×C4×C5C22×C414yes—
165#1220—28no—
166C2×Dic15—21no—
167C2×Dic15—21no—
168C2×Dic15—22no—
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15—22no—
172C2×Dic15—22no—
173C2×Dic15—44no—
174C2×Dic15C22×C412yes—
175C2×Dic15C2×D412yes—
176C2×Dic15C2×D412yes—
177D60—48no—
178C4×D15—44no—
179C24×D3—51no—
180C23×C3×C4C2×D511yescharacteristic
181C22×#67—51no—
182C23×Dic3—51no—
183C22×D12—52no—
184C22×C4×D3—51no—
185C2×C3×C4⋊C4C2×D512yescomplemented
186C2×C3×C22⋊C4C2×D511yescharacteristic
187C2×C3×C22⋊C4C2×D511yescharacteristic
188C2×#217—51no—
189C2×(C2×D3)⋊C4—52no—
190C2×(C2×D3)⋊C4—52no—
191C2×(C2×D3)⋊C4—52no—
192C2×(C2×D3)⋊C4—51no—
193C2×(C2×D3)⋊C4—52no—
194C2×(C2×D3)⋊C4—52no—
195C2×(C2×D3)⋊C4—52no—
196C2×(C2×D3)⋊C4—104no—
197C2×Dic3⋊C4—51no—
198C2×C4×Dic3—51no—
199C2×C4×Dic3—52no—
200#882—58no—
201#67⋊C4—104no—
202C3×#95C2×D512yescomplemented
203C3⋊#95—52no—
204C3⋊#95—52no—
205C24×C5C22×D311yescharacteristic
206C23×D5—31no—
207C23×D5—32no—
208C23×D5—34no—
209C23×D5—62no—
210C23×D5—62no—
211C23×D5—122no—
212C23×D5—122no—
213C22×C4×C5—22no—
214C22×C4×C5—42no—
215C22×C4×C5C22×D311yescharacteristic
216C22×C4×C5C22×D311yescharacteristic
217C22×C4×C5C22×D312yes—
218C22×C4×C5C22×D312yes—
219C2×C22⋊D5—34no—
220C2×C22⋊D5—62no—
221C2×C22⋊D5—62no—
222C22×Dic5—31no—
223C22×Dic5—31no—
224C22×Dic5—31no—
225C22×Dic5—34no—
226C22×Dic5—32no—
227C22×Dic5—122no—
228C2×D20—38no—
229C2×D20—122no—
230C2×D20—122no—
231C2×C4×D5—62no—
232C2×C4×D5—122no—
233C5×C4⋊C4—42no—
234C5×C22⋊C4—22no—
235C5×C22⋊C4C4×D314yescomplemented
236C22⋊Dic5—62no—
237#685—38no—
238#685—62no—
239#685—64no—
240#685—64no—
241#685—122no—
242#685—122no—
243#685—122no—
244#683—62no—
245C4×Dic5—62no—
246C4×Dic5—122no—
247C2×#120—151no—
248C2×C4×D4—151no—
249C22×C22⋊C4—151no—
250#390—152no—
251#388—152no—
252#388—152no—
253#386—151no—
254#386—152no—
255#385D1511yescharacteristic, complemented
256#379—151no—
257C4×C22⋊C4—151no—
258C22×C3×C5—24no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—24no—
262C22×C3×C5C4×D414yes—
263C22×C3×C5C23×C411yescharacteristic
264C22×C3×C5C22×D411yescharacteristic
265C22×C3×C5C2×#4111yescharacteristic
266C22×C3×C5H(5,𝔽2)12yes—
267C22×C3×C5H(5,𝔽2)12yes—
268C2×D15—22no—
269C2×D15—24no—
270C2×D15—24no—
271C2×D15—24no—
272C2×D15—24no—
273C2×D15—416no—
274C2×D15—41no—
275C2×D15—42no—
276C2×D15—41no—
277C2×D15—42no—
278C2×D15—42no—
279C2×D15—42no—
280C2×D15—42no—
281C3×C4×C5—24no—
282C3×C4×C5—41no—
283C3×C4×C5—41no—
284C3×C4×C5—42no—
285Dic15—22no—
286Dic15—22no—
287Dic15—22no—
288Dic15—42no—
289C24×C3C22×D511yescharacteristic
290C23×D3—52no—
291C23×D3—51no—
292C23×D3—54no—
293C23×D3—102no—
294C23×D3—102no—
295C23×D3—202no—
296C23×D3—202no—
297C22×C3×C4—22no—
298C22×C3×C4—42no—
299C22×C3×C4C22×D512yes—
300C22×C3×C4C22×D511yescharacteristic
301C22×C3×C4C22×D511yescharacteristic
302C22×C3×C4C22×D512yes—
303C2×#67—54no—
304C2×#67—102no—
305C2×#67—102no—
306C22×Dic3—51no—
307C22×Dic3—51no—
308C22×Dic3—51no—
309C22×Dic3—54no—
310C22×Dic3—52no—
311C22×Dic3—202no—
312C2×D12—58no—
313C2×D12—202no—
314C2×D12—202no—
315C2×C4×D3—102no—
316C2×C4×D3—202no—
317C3×C4⋊C4—42no—
318C3×C22⋊C4—22no—
319C3×C22⋊C4C4×D514yescomplemented
320#217—102no—
321(C2×D3)⋊C4—58no—
322(C2×D3)⋊C4—102no—
323(C2×D3)⋊C4—104no—
324(C2×D3)⋊C4—104no—
325(C2×D3)⋊C4—202no—
326(C2×D3)⋊C4—202no—
327(C2×D3)⋊C4—202no—
328Dic3⋊C4—102no—
329C4×Dic3—102no—
330C4×Dic3—202no—
331C23×C5—22no—
332C23×C5—22no—
333C23×C5C2×C4×D312yes—
334C23×C5D3×D412yes—
335C23×C5#23912yes—
336C23×C5C23×D311yescharacteristic
337C22×D5—34no—
338C22×D5—32no—
339C22×D5—68no—
340C22×D5—62no—
341C22×D5—61no—
342C22×D5—62no—
343C22×D5—62no—
344C22×D5—62no—
345C22×D5—61no—
346C22×D5—62no—
347C22×D5—62no—
348C22×D5—62no—
349C22×D5—62no—
350C22×D5—128no—
351C22×D5—128no—
352C22×D5—124no—
353C22×D5—124no—
354C2×C4×C5—22no—
355C2×C4×C5—22no—
356C2×C4×C5—21no—
357C2×C4×C5—21no—
358C2×C4×C5—22no—
359C2×C4×C5—22no—
360C2×C4×C5—22no—
361C2×C4×C5—22no—
362C2×C4×C5—22no—
363C2×C4×C5—44no—
364C2×C4×C5C2×C4×D314yes—
365C22⋊D5—68no—
366C2×Dic5—32no—
367C2×Dic5—32no—
368C2×Dic5—32no—
369C2×Dic5—61no—
370C2×Dic5—61no—
371C2×Dic5—62no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—62no—
375C2×Dic5—62no—
376C2×Dic5—124no—
377D20—128no—
378C4×D5—124no—
379C25—151no—
380C22×D4—151no—
381C22×D4—152no—
382C23×C4—151no—
383C23×C4—151no—
384C23×C4C2×D1511yescharacteristic
385#120—158no—
386C4×D4—304no—
387C2×C4⋊C4—151no—
388C2×C4⋊C4C2×D1512yescomplemented
389C2×C22⋊C4—152no—
390C2×C22⋊C4—151no—
391C2×C22⋊C4—152no—
392C2×C22⋊C4—152no—
393C2×C22⋊C4—151no—
394C2×C22⋊C4—152no—
395C2×C22⋊C4—152no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—304no—
398C2×C22⋊C4C2×D1511yescharacteristic
399C2×C22⋊C4C2×D1511yescharacteristic
400C2×C42—151no—
401C2×C42—152no—
402#95—152no—
403#95—152no—
404#95C2×D1512yescomplemented
405C2×C3×C5—24no—
406C2×C3×C5C2×C4×D411yescharacteristic
407C2×C3×C5#51812yes—
408C2×C3×C5#53411yescharacteristic
409C2×C3×C5#53511yescharacteristic
410C2×C3×C5#53812yes—
411D15—42no—
412D15—44no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3C2×C4×D512yes—
416C23×C3D4×D512yes—
417C23×C3#71312yes—
418C23×C3C23×D511yescharacteristic
419C22×D3—54no—
420C22×D3—52no—
421C22×D3—102no—
422C22×D3—101no—
423C22×D3—102no—
424C22×D3—102no—
425C22×D3—102no—
426C22×D3—101no—
427C22×D3—102no—
428C22×D3—102no—
429C22×D3—102no—
430C22×D3—102no—
431C22×D3—108no—
432C22×D3—208no—
433C22×D3—204no—
434C22×D3—208no—
435C22×D3—204no—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—21no—
439C2×C3×C4—21no—
440C2×C3×C4—22no—
441C2×C3×C4—22no—
442C2×C3×C4—22no—
443C2×C3×C4—22no—
444C2×C3×C4—22no—
445C2×C3×C4—44no—
446C2×C3×C4C2×C4×D514yes—
447#67—108no—
448C2×Dic3—52no—
449C2×Dic3—52no—
450C2×Dic3—52no—
451C2×Dic3—101no—
452C2×Dic3—101no—
453C2×Dic3—102no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3—102no—
458C2×Dic3—204no—
459D12—208no—
460C4×D3—204no—
461C22×C5—24no—
462C22×C5—22no—
463C22×C5—24no—
464C22×C5—22no—
465C22×C5#89314yes—
466C22×C5C22×C4×D311yescharacteristic
467C22×C5C2×D3×D411yescharacteristic
468C22×C5#100412yes—
469C22×C5C2×#23911yescharacteristic
470C22×C5#100912yes—
471C2×D5—62no—
472C2×D5—64no—
473C2×D5—64no—
474C2×D5—64no—
475C2×D5—64no—
476C2×D5—1216no—
477C2×D5—122no—
478C2×D5—122no—
479C2×D5—122no—
480C2×D5—121no—
481C2×D5—122no—
482C2×D5—121no—
483C2×D5—122no—
484C4×C5—24no—
485C4×C5—41no—
486C4×C5—41no—
487C4×C5—42no—
488Dic5—62no—
489Dic5—62no—
490Dic5—62no—
491Dic5—122no—
492C24—152no—
493C24—151no—
494C24—154no—
495C24—302no—
496C24—302no—
497C24—602no—
498C24—602no—
499C24C22×D1511yescharacteristic
500C2×D4—154no—
501C2×D4—158no—
502C2×D4—302no—
503C2×D4—302no—
504C2×D4—602no—
505C2×D4—602no—
506C22×C4—22no—
507C22×C4—42no—
508C22×C4—151no—
509C22×C4—151no—
510C22×C4—151no—
511C22×C4—154no—
512C22×C4—152no—
513C22×C4—302no—
514C22×C4—602no—
515C22×C4—602no—
516C22×C4C22×D1512yes—
517C22×C4C22×D1511yescharacteristic
518C22×C4C22×D1511yescharacteristic
519C22×C4C22×D1512yes—
520C4⋊C4—42no—
521C4⋊C4—302no—
522C22⋊C4—22no—
523C22⋊C4—158no—
524C22⋊C4—304no—
525C22⋊C4—304no—
526C22⋊C4—302no—
527C22⋊C4—302no—
528C22⋊C4—602no—
529C22⋊C4—602no—
530C22⋊C4—602no—
531C22⋊C4C4×D1514yescomplemented
532C42—302no—
533C42—602no—
534C3×C5#232111yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—22no—
537C22×C3—24no—
538C22×C3—22no—
539C22×C3#403914yes—
540C22×C3C22×C4×D511yescharacteristic
541C22×C3C2×D4×D511yescharacteristic
542C22×C3#414412yes—
543C22×C3C2×#71311yescharacteristic
544C22×C3#414912yes—
545C2×D3—102no—
546C2×D3—104no—
547C2×D3—104no—
548C2×D3—104no—
549C2×D3—104no—
550C2×D3—2016no—
551C2×D3—201no—
552C2×D3—202no—
553C2×D3—202no—
554C2×D3—202no—
555C2×D3—202no—
556C2×D3—201no—
557C2×D3—202no—
558C3×C4—24no—
559C3×C4—41no—
560C3×C4—42no—
561C3×C4—41no—
562Dic3—102no—
563Dic3—102no—
564Dic3—102no—
565Dic3—202no—
566C2×C5—24no—
567C2×C5C2×#89311yescharacteristic
568C2×C5#554512yes—
569C2×C5#559211yescharacteristic
570C2×C5#564412yes—
571C2×C5#565311yescharacteristic
572D5—124no—
573D5—122no—
574C23—22no—
575C23—22no—
576C23—154no—
577C23—152no—
578C23—308no—
579C23—302no—
580C23—301no—
581C23—302no—
582C23—302no—
583C23—302no—
584C23—301no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23—302no—
589C23—608no—
590C23—604no—
591C23—608no—
592C23—604no—
593C23C2×C4×D1512yes—
594C23D4×D1512yes—
595C23#682112yes—
596C23C23×D1511yescharacteristic
597D4—308no—
598D4—608no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—152no—
610C2×C4—152no—
611C2×C4—152no—
612C2×C4—301no—
613C2×C4—301no—
614C2×C4—302no—
615C2×C4—302no—
616C2×C4—302no—
617C2×C4—302no—
618C2×C4—302no—
619C2×C4—604no—
620C2×C4—604no—
621C2×C4C2×C4×D1514yes—
622C2×C3—24no—
623C2×C3C2×#403911yescharacteristic
624C2×C3#6587912yes—
625C2×C3#6592611yescharacteristic
626C2×C3#6597812yes—
627C2×C3#6598711yescharacteristic
628D3—204no—
629D3—202no—
630C5#7459211yescharacteristic, complemented, Sylow
631C22—24no—
632C22—22no—
633C22—24no—
634C22—22no—
635C22—304no—
636C22—304no—
637C22—304no—
638C22—302no—
639C22—304no—
640C22—6016no—
641C22—602no—
642C22—601no—
643C22—602no—
644C22—602no—
645C22—602no—
646C22—601no—
647C22—602no—
648C22#9178214yes—
649C22C22×C4×D1511yescharacteristic
650C22C2×D4×D1511yescharacteristic
651C22#9209512yes—
652C22C2×#682111yescharacteristic
653C22#9210012yes—
654C4—24no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—302no—
659C4—302no—
660C4—302no—
661C4—602no—
662C3#1060632311yescharacteristic, complemented, Sylow
663C2—24no—
664C2—604no—
665C2—602no—
666C2C2×#9178211yescharacteristic
667C2#1175630412yes—
668C2#1175635111yescharacteristic
669C2#1175640312yes—
670C2#1175641211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#385C2
socle, cosocleC23×C3×C5C24
prime radicalC2×#91771C2
hypercenter#385D15
hypocenterC3×C5#2321
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×#6714C2semidirect
3. C2×#91770C2semidirect
4. C3×C5×#385C2semidirect
5. #11751314C2semidirect
6. #11751091C2semidirect
7. #11751089C2semidirect
8. #11751056C2semidirect
9. C22⋊C4×Dic15C2semidirect
10. #11750770C2semidirect
11. #11750766C2semidirect
18. C22×C4×D15C22semidirect
19. C2×C3×C5×C4⋊C4C22semidirect
29. C2×#6714C22semidirect
30. C2×#6714C22semidirect
31. C2×#6712C22semidirect
32. C2×C4×Dic15C22semidirect
33. C2×C4×Dic15C22semidirect
34. #91771C4semidirect
36. C3×C5×#95C22semidirect
37. #91115C22semidirect
38. #91115C22semidirect
43. C5×#385D3semidirect
56. C23×D15C2×C4semidirect
69. C22×Dic15D4semidirect
76. C2×D60C2×C4semidirect
81. C3×C5×C22⋊C4C2×C4semidirect
89. #6714C2×C4semidirect
96. C3×#385D5semidirect
110. C2×C5×C4⋊C4C2×D3semidirect
127. C5×#95C2×D3semidirect
185. C2×C3×C4⋊C4C2×D5semidirect
202. C3×#95C2×D5semidirect
235. C5×C22⋊C4C4×D3semidirect
255. #385D15semidirect
319. C3×C22⋊C4C4×D5semidirect
388. C2×C4⋊C4C2×D15semidirect
404. #95C2×D15semidirect
531. C22⋊C4C4×D15semidirect
534. C3×C5#2321semidirect
630. C5#74592semidirect
662. C3#10606323semidirect

Maximal subgroups

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

subgroupindex
C22×#67142
C2×#917712
C2×#917702
C3×C5×#3852
#117513142
#117510912
#117510892
#117510562
C22⋊C4×Dic152
#117507702
#117507662
#106063233
#745925