Finite Group Databasebeta

← #14326283

Subgroups — #14326283

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
1C2×C5×#877C211yesmaximal, characteristic, complemented
2C2×C3×D5×C22⋊C4C211yesmaximal, characteristic
3C22×#6665C211yesmaximal, characteristic
4C22⋊C4×Dic15C211yesmaximal, characteristic, complemented
5#11749516C211yesmaximal, characteristic, complemented
6#11749488C211yesmaximal, characteristic
7#11749472C211yesmaximal, characteristic, complemented
8#11748709C212yesmaximal, complemented
9#11748705C212yesmaximal, complemented
10#11748703C212yesmaximal, complemented
11C4×#6665C212yesmaximal, complemented
12#10614002—31nomaximal, Hall
13C24×C3×D5C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×C5×Dic3C2211yescharacteristic
16C22×C3×C4×D5C2212yes—
17C2×C3×C5×C22⋊C4C2211yescharacteristic
18C2×#6719C2211yescharacteristic
19C2×C4×Dic15C2212yescomplemented
20C2×C5×#217C2211yescharacteristic
21C2×C5×Dic3⋊C4C2212yescomplemented
22C2×C4×C5×Dic3C2212yescomplemented
23C5×#877—24no—
24C2×C3×C22⋊Dic5C2211yescharacteristic
25C2×C3×#685C2212yes—
26C3×D5×C22⋊C4—24no—
27C2×#6669C2212yescomplemented
28C2×#6665C418yescomplemented
29C2×#6665C2212yes—
30C2×#6665C2212yes—
31C2×#6665C2211yescharacteristic
32C2×#6665C2211yescharacteristic
33C2×#6665C2211yescharacteristic
34C2×#6665C2211yescharacteristic
35#91115C2212yescomplemented
36#90994C2212yescomplemented
37#90994C2212yescomplemented
38#90994C2212yescomplemented
39#82098—51nomaximal, Hall
40C2×C5×#117—31no—
41C22×#685—31no—
42C2×D5×C22⋊C4D311yescharacteristic
43#65500—31no—
44#65279—32no—
45#65278—32no—
46#65244—31no—
47#65231—31no—
48C22⋊C4×Dic5—31no—
49#65229—32no—
50C4×#685—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C3×D5—22no—
53C23×C3×D5—22no—
54C23×C3×D5—24no—
55C23×C3×D5—42no—
56C23×C3×D5C2×C414yes—
57C23×C3×D5C2311yescharacteristic
58C23×C3×D5C2311yescharacteristic
59C22×C3×C4×C5C2312yes—
60C22×Dic15—42no—
61C22×Dic15C2×C414yescomplemented
62C22×Dic15C2312yes—
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2311yescharacteristic
65C22×C5×Dic3—22no—
66C22×C5×Dic3—22no—
67C22×C5×Dic3C2×C414yescomplemented
68C22×C5×Dic3C2312yes—
69C22×C5×Dic3C2311yescharacteristic
70C22×C5×Dic3C2311yescharacteristic
71C22×C3×Dic5C2312yes—
72C2×C3×C4×D5—24no—
73C2×C3×C4×D5—24no—
74C2×C3×C4×D5—42no—
75C3×C5×C22⋊C4D414yes—
76#6719—22no—
77C4×Dic15—42no—
78C5×#217—22no—
79C5×Dic3⋊C4—24no—
80C4×C5×Dic3—24no—
81C3×C22⋊Dic5—22no—
82C3×#685—24no—
83#6669—42no—
84#6665—24no—
85#6665—42no—
86#6665C2×C414yes—
87#6665C2×C414yes—
88#6665C2×C414yescomplemented
89#6665C2×C414yescomplemented
90C22×C3×C22⋊C4—51no—
91C22×#217—51no—
92C2×#877D511yescharacteristic, complemented
93#5247—51no—
94#5210—52no—
95C4×#217—52no—
96#4945—52no—
97#4944—52no—
98#4943—51no—
99#4942—51no—
100Dic3×C22⋊C4—51no—
101C24×D5C2×D311yescharacteristic
102C23×C4×C5—31no—
103C23×Dic5—31no—
104C22×C4×D5C2×D312yes—
105C5×#117—64no—
106C2×C5×C4⋊C4—32no—
107C2×C5×C22⋊C4—31no—
108C2×C5×C22⋊C4C2×D311yescharacteristic
109C2×C42×C5—32no—
110C2×C22⋊Dic5—31no—
111C2×C22⋊Dic5C2×D311yescharacteristic
112C2×#685—38no—
113C2×#685—31no—
114C2×#685—31no—
115C2×#685—31no—
116C2×#685—31no—
117C2×#685—32no—
118C2×#685—32no—
119C2×#685C2×D312yes—
120C2×#683—32no—
121C2×C4×Dic5—32no—
122D5×C22⋊C4—24no—
123#3963—32no—
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×C3×D5—21no—
135C22×C3×D5—21no—
136C22×C3×D5—22no—
137C22×C3×D5—28no—
138C22×C3×D5—21no—
139C22×C3×D5—22no—
140C22×C3×D5—21no—
141C22×C3×D5—22no—
142C22×C3×D5—28no—
143C22×C3×D5—44no—
144C22×C3×D5—48no—
145C22×C3×D5—48no—
146C22×C3×D5C22×C412yes—
147C22×C3×D5C22×C412yes—
148C22×C3×D5#4114yes—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5C2×D414yes—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15—22no—
155C2×Dic15—22no—
156C2×Dic15—22no—
157C2×Dic15—44no—
158C2×Dic15C22×C412yes—
159C2×Dic15C22×C412yes—
160C2×C5×Dic3—28no—
161C2×C5×Dic3—21no—
162C2×C5×Dic3—21no—
163C2×C5×Dic3—21no—
164C2×C5×Dic3—21no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—22no—
168C2×C5×Dic3C22×C412yes—
169C2×C5×Dic3C22×C412yes—
170C2×C3×Dic5—22no—
171C2×C3×Dic5—22no—
172C2×C3×Dic5—22no—
173C3×C4×D5—48no—
174C25×C3—51no—
175C23×C3×C4—52no—
176C23×Dic3—51no—
177C23×Dic3C2×D511yescharacteristic
178C2×C3×C22⋊C4—51no—
179C2×C3×C22⋊C4—52no—
180C2×C3×C22⋊C4—104no—
181C2×C3×C22⋊C4C2×D511yescharacteristic
182C2×#217—58no—
183C2×#217—51no—
184C2×#217—51no—
185C2×#217—51no—
186C2×#217—51no—
187C2×#217—51no—
188C2×#217—52no—
189C2×#217—52no—
190C2×#217C2×D511yescharacteristic
191C2×Dic3⋊C4—52no—
192C2×Dic3⋊C4C2×D512yescomplemented
193C2×C4×Dic3—52no—
194C2×C4×Dic3C2×D512yescomplemented
195#877—24no—
196C3⋊#95—52no—
197C3⋊#95—52no—
198C3⋊#95—52no—
199C3⋊#95—52no—
200C24×C5C22×D311yescharacteristic
201C23×D5—22no—
202C23×D5—22no—
203C23×D5—24no—
204C23×D5—42no—
205C23×D5C4×D314yes—
206C23×D5C22×D311yescharacteristic
207C23×D5C22×D311yescharacteristic
208C22×C4×C5—34no—
209C22×C4×C5—31no—
210C22×C4×C5—31no—
211C22×C4×C5—32no—
212C22×C4×C5—62no—
213C22×C4×C5—62no—
214C22×C4×C5C22×D312yes—
215C22×Dic5—31no—
216C22×Dic5—31no—
217C22×Dic5—34no—
218C22×Dic5—32no—
219C22×Dic5—122no—
220C22×Dic5C22×D312yes—
221C2×C4×D5—24no—
222C2×C4×D5—24no—
223C2×C4×D5—42no—
224C5×C4⋊C4—64no—
225C5×C22⋊C4—62no—
226C5×C22⋊C4#6714yes—
227C42×C5—64no—
228C22⋊Dic5—22no—
229C22⋊Dic5—62no—
230#685—24no—
231#685—34no—
232#685—34no—
233#685—34no—
234#685—34no—
235#685—64no—
236#685—122no—
237#683—122no—
238C4×Dic5—122no—
239C2×#117—31no—
240C22×C22⋊C4—51no—
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⋊C4—151no—
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×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—24no—
263C2×C3×D5—24no—
264C2×C3×D5—22no—
265C2×C3×D5—22no—
266C2×C3×D5—22no—
267C2×C3×D5—416no—
268C2×C3×D5—42no—
269C2×C3×D5—42no—
270C2×C3×D5#11714yescomplemented
271C3×C4×C5—24no—
272Dic15—22no—
273Dic15—22no—
274Dic15—42no—
275C3×Dic5—42no—
276C5×Dic3—22no—
277C5×Dic3—22no—
278C5×Dic3—42no—
279C24×C3—51no—
280C24×C3—51no—
281C24×C3—54no—
282C24×C3—104no—
283C24×C3—102no—
284C24×C3—102no—
285C24×C3—202no—
286C24×C3C22×D511yescharacteristic
287C22×C3×C4—52no—
288C22×C3×C4—104no—
289C22×C3×C4—104no—
290C22×C3×C4—202no—
291C22×C3×C4C22×D512yes—
292C22×Dic3—22no—
293C22×Dic3—22no—
294C22×Dic3—51no—
295C22×Dic3—51no—
296C22×Dic3—54no—
297C22×Dic3—52no—
298C22×Dic3—202no—
299C22×Dic3C4×D514yescomplemented
300C22×Dic3C22×D512yes—
301C22×Dic3C22×D511yescharacteristic
302C22×Dic3C22×D511yescharacteristic
303C3×C22⋊C4—102no—
304C3×C22⋊C4—104no—
305C3×C22⋊C4C22⋊D514yes—
306#217—22no—
307#217—54no—
308#217—54no—
309#217—54no—
310#217—54no—
311#217—104no—
312#217—102no—
313#217—202no—
314Dic3⋊C4—24no—
315Dic3⋊C4—202no—
316C4×Dic3—24no—
317C4×Dic3—202no—
318C23×C5—22no—
319C23×C5—22no—
320C23×C5C2×C4×D312yes—
321C23×C5#23512yes—
322C23×C5C2×#6712yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—21no—
325C22×D5—22no—
326C22×D5—21no—
327C22×D5—28no—
328C22×D5—21no—
329C22×D5—22no—
330C22×D5—22no—
331C22×D5—21no—
332C22×D5—28no—
333C22×D5—44no—
334C22×D5—48no—
335C22×D5—48no—
336C22×D5C2×C4×D312yes—
337C22×D5C2×C4×D312yes—
338C22×D5#23714yes—
339C2×C4×C5—22no—
340C2×C4×C5—22no—
341C2×C4×C5—32no—
342C2×C4×C5—32no—
343C2×C4×C5—68no—
344C2×C4×C5—61no—
345C2×C4×C5—61no—
346C2×C4×C5—61no—
347C2×C4×C5—61no—
348C2×C4×C5—62no—
349C2×C4×C5—62no—
350C2×C4×C5—62no—
351C2×C4×C5C2×#6714yes—
352C2×Dic5—22no—
353C2×Dic5—22no—
354C2×Dic5—22no—
355C2×Dic5—32no—
356C2×Dic5—32no—
357C2×Dic5—62no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5—62no—
362C2×Dic5—124no—
363C4×D5—48no—
364C25—51no—
365C23×C4—31no—
366C23×C4—52no—
367C23×C4—151no—
368#117—64no—
369C2×C4⋊C4—32no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—31no—
372C2×C22⋊C4—51no—
373C2×C22⋊C4—52no—
374C2×C22⋊C4—104no—
375C2×C22⋊C4—151no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—151no—
378C2×C22⋊C4—151no—
379C2×C22⋊C4—158no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4—151no—
382C2×C22⋊C4—152no—
383C2×C22⋊C4D3×D511yescharacteristic
384C2×C42—32no—
385C2×C42—152no—
386#95—152no—
387#95—152no—
388#95—152no—
389#95—152no—
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—
396C3×D5—24no—
397C3×D5—42no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3—52no—
401C23×C3—54no—
402C23×C3—52no—
403C23×C3—108no—
404C23×C3—101no—
405C23×C3—102no—
406C23×C3—101no—
407C23×C3—101no—
408C23×C3—102no—
409C23×C3—102no—
410C23×C3—101no—
411C23×C3—108no—
412C23×C3—204no—
413C23×C3—208no—
414C23×C3—208no—
415C23×C3C2×C4×D512yes—
416C23×C3#70912yes—
417C23×C3C2×C22⋊D512yes—
418C23×C3C23×D511yescharacteristic
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—102no—
422C2×C3×C4—102no—
423C2×C3×C4—102no—
424C2×C3×C4—208no—
425C2×C3×C4C2×C22⋊D514yes—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—28no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—52no—
435C2×Dic3—52no—
436C2×Dic3—102no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3—102no—
441C2×Dic3—204no—
442C2×Dic3C2×C4×D512yes—
443C2×Dic3C2×C4×D512yes—
444C22×C5—22no—
445C22×C5—22no—
446C22×C5—24no—
447C22×C5—24no—
448C22×C5C4×#6714yes—
449C22×C5C22×C4×D311yescharacteristic
450C22×C5C2×#23511yescharacteristic
451C22×C5C2×#23712yes—
452C22×C5#100412yes—
453C22×C5C22×#6711yescharacteristic
454C2×D5—22no—
455C2×D5—24no—
456C2×D5—22no—
457C2×D5—24no—
458C2×D5—22no—
459C2×D5—22no—
460C2×D5—22no—
461C2×D5—416no—
462C2×D5—42no—
463C2×D5—42no—
464C2×D5#87714yescomplemented
465C4×C5—24no—
466C4×C5—62no—
467C4×C5—62no—
468C4×C5—122no—
469Dic5—42no—
470Dic5—62no—
471Dic5—62no—
472Dic5—122no—
473C24—51no—
474C24—51no—
475C24—54no—
476C24—104no—
477C24—102no—
478C24—102no—
479C24—202no—
480C24C2×D3×D511yescharacteristic
481C22×C4—34no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—32no—
485C22×C4—52no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—104no—
489C22×C4—104no—
490C22×C4—151no—
491C22×C4—151no—
492C22×C4—154no—
493C22×C4—152no—
494C22×C4—202no—
495C22×C4—602no—
496C22×C4C2×D3×D512yes—
497C4⋊C4—64no—
498C4⋊C4—602no—
499C22⋊C4—62no—
500C22⋊C4—102no—
501C22⋊C4—104no—
502C22⋊C4—154no—
503C22⋊C4—154no—
504C22⋊C4—154no—
505C22⋊C4—154no—
506C22⋊C4—304no—
507C22⋊C4—302no—
508C22⋊C4—602no—
509C22⋊C4#120114yes—
510C42—64no—
511C42—602no—
512C3×C5#234211yescharacteristic, complemented, Hall
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—24no—
516C22×C3—24no—
517C22×C3—54no—
518C22×C3—102no—
519C22×C3—104no—
520C22×C3—102no—
521C22×C3—104no—
522C22×C3—102no—
523C22×C3—102no—
524C22×C3—102no—
525C22×C3—2016no—
526C22×C3—202no—
527C22×C3—202no—
528C22×C3C4×C22⋊D514yes—
529C22×C3C22×C4×D511yescharacteristic
530C22×C3C2×#70911yescharacteristic
531C22×C3D5×#4112yes—
532C22×C3#414912yes—
533C22×C3C22×C22⋊D511yescharacteristic
534C3×C4—24no—
535C3×C4—202no—
536Dic3—22no—
537Dic3—22no—
538Dic3—42no—
539Dic3—102no—
540Dic3—102no—
541Dic3—202no—
542C2×C5—24no—
543C2×C5C2×#87711yescharacteristic
544C2×C5#548611yescharacteristic
545C2×C5#549512yes—
546C2×C5C2×C4×#6711yescharacteristic
547C2×C5#580612yes—
548D5—24no—
549D5—42no—
550C23—22no—
551C23—22no—
552C23—52no—
553C23—54no—
554C23—52no—
555C23—108no—
556C23—101no—
557C23—102no—
558C23—101no—
559C23—108no—
560C23—101no—
561C23—102no—
562C23—102no—
563C23—101no—
564C23—204no—
565C23—208no—
566C23—208no—
567C23#677112yes—
568C23C4×D3×D512yes—
569C23C2×#120112yes—
570C23C22×D3×D511yescharacteristic
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—68no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—102no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—152no—
587C2×C4—152no—
588C2×C4—208no—
589C2×C4—302no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—302no—
593C2×C4—302no—
594C2×C4—604no—
595C2×C4C2×#120114yes—
596C2×C3—24no—
597C2×C3—104no—
598C2×C3—202no—
599C2×C3D5×#11711yescharacteristic
600C2×C3#6585511yescharacteristic
601C2×C3#6586112yes—
602C2×C3C2×C4×C22⋊D511yescharacteristic
603C2×C3#6616312yes—
604C5#8209811yescharacteristic, complemented, Sylow
605C22—24no—
606C22—22no—
607C22—22no—
608C22—24no—
609C22—54no—
610C22—102no—
611C22—104no—
612C22—102no—
613C22—104no—
614C22—102no—
615C22—102no—
616C22—102no—
617C22—2016no—
618C22—202no—
619C22—202no—
620C22C4×#120114yes—
621C22C2×#677111yescharacteristic
622C22C2×C4×D3×D511yescharacteristic
623C22D5×#23712yes—
624C22#9202612yes—
625C22C22×#120111yescharacteristic
626C4—24no—
627C4—62no—
628C4—62no—
629C4—122no—
630C4—202no—
631C4—302no—
632C4—302no—
633C4—602no—
634C3#1061400211yescharacteristic, complemented, Sylow
635C2—24no—
636C2—104no—
637C2—202no—
638C2#1175280012yes—
639C2D5×#87711yescharacteristic
640C2#1175306611yescharacteristic
641C2C2×C4×#120111yescharacteristic
642C2#1175496112yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×D5C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2342
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#877C2semidirect
4. C22⋊C4×Dic15C2semidirect
5. #11749516C2semidirect
7. #11749472C2semidirect
8. #11748709C2semidirect
9. #11748705C2semidirect
10. #11748703C2semidirect
11. C4×#6665C2semidirect
19. C2×C4×Dic15C22semidirect
21. C2×C5×Dic3⋊C4C22semidirect
22. C2×C4×C5×Dic3C22semidirect
27. C2×#6669C22semidirect
28. C2×#6665C4semidirect
35. #91115C22semidirect
36. #90994C22semidirect
37. #90994C22semidirect
38. #90994C22semidirect
61. C22×Dic15C2×C4semidirect
67. C22×C5×Dic3C2×C4semidirect
88. #6665C2×C4semidirect
89. #6665C2×C4semidirect
92. C2×#877D5semidirect
192. C2×Dic3⋊C4C2×D5semidirect
194. C2×C4×Dic3C2×D5semidirect
270. C2×C3×D5#117semidirect
299. C22×Dic3C4×D5semidirect
464. C2×D5#877semidirect
512. C3×C5#2342semidirect
604. C5#82098semidirect
634. C3#10614002semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#8772
C2×C3×D5×C22⋊C42
C22×#66652
C22⋊C4×Dic152
#117495162
#117494882
#117494722
#117487092
#117487052
#117487032
C4×#66652
#106140023
#820985