Finite Group Databasebeta

← #14329035

Subgroups — #14329035

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91784C211yesmaximal, characteristic
2C2×#91781C211yesmaximal, characteristic, complemented
3C2×C4×D60C211yesmaximal, characteristic, complemented
4C3×C5×#389C211yesmaximal, characteristic, complemented
5#11751091C212yesmaximal, complemented
6#11751089C211yesmaximal, characteristic, complemented
7#11751089C212yesmaximal, complemented
8C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
9C4×#6714C211yesmaximal, characteristic, complemented
10#11750767C212yesmaximal, complemented
11#11750766C212yesmaximal, complemented
12#10607716—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×D60C2212yes—
15C22×C4×D15C2212yes—
16C22×C4×D15C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C5×C4⋊C4C2211yescharacteristic
20C2×C3×C42×C5C2211yescharacteristic
21C2×#6714C2211yescharacteristic
22C2×#6714C2212yes—
23C2×#6714C2212yes—
24C2×#6714C2211yescharacteristic
25C2×#6714C2212yes—
26C2×#6714C2212yes—
27C2×#6714C2212yes—
28C2×C4⋊Dic15C2211yescharacteristic
29C2×C4×Dic15C2212yes—
30C2×C4×Dic15C2211yescharacteristic
31C2×C4×Dic15C2212yes—
32#91784C418yescomplemented
33#91781—24no—
34C4×D60—24no—
35C3×C5×#95C2212yes—
36#91115C2212yes—
37#91115C2211yescharacteristic
38#91115C2211yescharacteristic
39#75985—51nomaximal, Hall
40C2×#4041—31no—
41C2×#4038—31no—
42C2×C4×D20—31no—
43C5×#389D311yescharacteristic, complemented
44#65279—32no—
45#65277—31no—
46#65277—32no—
47C4⋊C4×Dic5—31no—
48C4×#685—31no—
49#64955—32no—
50#64954—32no—
51C23×D15C2311yescharacteristic
52C23×D15C2312yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C22×Dic15C2312yes—
61C22×Dic15C2311yescharacteristic
62C2×D60—22no—
63C2×D60—22no—
64C2×D60—42no—
65C2×D60—42no—
66C2×D60C2×C414yes—
67C2×D60C2×C418yescomplemented
68C2×C4×D15—22no—
69C2×C4×D15—22no—
70C2×C4×D15—24no—
71C2×C4×D15—24no—
72C2×C4×D15—42no—
73C2×C4×D15C2×C414yescomplemented
74C3×C5×C4⋊C4—22no—
75C3×C5×C4⋊C4—42no—
76C3×C5×C4⋊C4C2×C414yescomplemented
77C3×C42×C5—22no—
78#6714—22no—
79#6714—24no—
80#6714—24no—
81#6714—24no—
82#6714—22no—
83#6714—42no—
84#6714C2×C418yescomplemented
85C4⋊Dic15D414yescomplemented
86C4×Dic15—22no—
87C4×Dic15—24no—
88C4×Dic15—42no—
89C2×#895—51no—
90C2×#892—51no—
91C2×C4×D12—51no—
92C3×#389D511yescharacteristic, complemented
93#4990—52no—
94#4988—51no—
95#4988—52no—
96Dic3×C4⋊C4—51no—
97C4×(C2×D3)⋊C4—51no—
98#4666—52no—
99#4665—52no—
100C22×D20—31no—
101C22×D20—32no—
102C22×C4×D5—31no—
103C22×C4×D5—32no—
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C5×C4⋊C4C2×D312yes—
106C2×C5×C4⋊C4C2×D311yescharacteristic
107C2×C42×C5C2×D311yescharacteristic
108C2×#685—32no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—31no—
112C2×#685—32no—
113C2×#685—31no—
114C2×#685—32no—
115C2×C4⋊Dic5—31no—
116C2×C4×Dic5—32no—
117C2×C4×Dic5—31no—
118C2×C4×Dic5—32no—
119#4041—38no—
120#4038—64no—
121C4×D20—64no—
122C5×#95C2×D312yes—
123#3963—32no—
124#3963—31no—
125#3963—31no—
126#2363—151noSylow
127C23×C3×C5C2411yescharacteristic
128C22×D15—21no—
129C22×D15—22no—
130C22×D15—21no—
131C22×D15—21no—
132C22×D15—22no—
133C22×D15—21no—
134C22×D15—22no—
135C22×D15—22no—
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—21no—
139C22×D15—21no—
140C22×D15C22×C414yes—
141C22×D15C22×C412yes—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—21no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—21no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—21no—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5C22×C414yes—
153C2×C3×C4×C5C22×C412yes—
154C2×C3×C4×C5C2×D412yes—
155C2×C3×C4×C5#4112yes—
156C2×C3×C4×C5#4112yes—
157C2×Dic15—21no—
158C2×Dic15—21no—
159C2×Dic15—21no—
160C2×Dic15—22no—
161C2×Dic15—21no—
162C2×Dic15—22no—
163C2×Dic15—21no—
164C2×Dic15—21no—
165C2×Dic15—21no—
166C2×Dic15C22×C412yes—
167C2×Dic15C2×D414yes—
168C2×Dic15#4112yes—
169D60—28no—
170D60—48no—
171C4×D15—28no—
172C4×D15—48no—
173C22×D12—51no—
174C22×D12—52no—
175C22×C4×D3—51no—
176C22×C4×D3—52no—
177C2×C3×C4⋊C4C2×D511yescharacteristic
178C2×C3×C4⋊C4C2×D512yes—
179C2×C3×C4⋊C4C2×D511yescharacteristic
180C2×C3×C42C2×D511yescharacteristic
181C2×(C2×D3)⋊C4—52no—
182C2×(C2×D3)⋊C4—51no—
183C2×(C2×D3)⋊C4—52no—
184C2×(C2×D3)⋊C4—52no—
185C2×(C2×D3)⋊C4—51no—
186C2×(C2×D3)⋊C4—52no—
187C2×(C2×D3)⋊C4—52no—
188C2×C4⋊Dic3—51no—
189C2×C4×Dic3—52no—
190C2×C4×Dic3—51no—
191C2×C4×Dic3—52no—
192#895—58no—
193#892—104no—
194C4×D12—104no—
195C3×#95C2×D512yes—
196C3⋊#95—52no—
197C3⋊#95—51no—
198C3⋊#95—51no—
199C23×D5—32no—
200C23×D5—31no—
201C22×C4×C5C22×D311yescharacteristic
202C22×C4×C5C22×D312yes—
203C22×C4×C5C22×D312yes—
204C22×C4×C5C22×D311yescharacteristic
205C22×C4×C5C22×D311yescharacteristic
206C22×Dic5—31no—
207C22×Dic5—32no—
208C22×Dic5—31no—
209C22×Dic5—31no—
210C2×D20—34no—
211C2×D20—38no—
212C2×D20—62no—
213C2×D20—62no—
214C2×D20—122no—
215C2×D20—122no—
216C2×C4×D5—34no—
217C2×C4×D5—62no—
218C2×C4×D5—62no—
219C2×C4×D5—64no—
220C2×C4×D5—64no—
221C2×C4×D5—122no—
222C5×C4⋊C4—22no—
223C5×C4⋊C4—42no—
224C5×C4⋊C4C4×D314yescomplemented
225C42×C5—22no—
226#685—38no—
227#685—64no—
228#685—64no—
229#685—64no—
230#685—62no—
231#685—62no—
232#685—122no—
233C4⋊Dic5—34no—
234C4×Dic5—64no—
235C4×Dic5—62no—
236C4×Dic5—122no—
237C2×#121—151no—
238C2×C4×D4—151no—
239C2×#117—151no—
240#390—151no—
241#390—152no—
242#389D1511yescharacteristic, complemented
243#388—152no—
244#386—152no—
245C4×C4⋊C4—151no—
246C4×C22⋊C4—151no—
247C4×C22⋊C4—152no—
248C22×C3×C5C23×C411yescharacteristic
249C22×C3×C5C22×D411yescharacteristic
250C22×C3×C5C2×#4111yescharacteristic
251C22×C3×C5C2×#4111yescharacteristic
252C22×C3×C5C2×#4111yescharacteristic
253C22×C3×C5H(5,𝔽2)11yescharacteristic
254C22×C3×C5H(5,𝔽2)11yescharacteristic
255C2×D15—22no—
256C2×D15—24no—
257C2×D15—22no—
258C2×D15—24no—
259C2×D15—22no—
260C2×D15—24no—
261C2×D15—42no—
262C2×D15—41no—
263C2×D15—41no—
264C2×D15—42no—
265C2×D15—41no—
266C2×D15—42no—
267C2×D15—41no—
268C2×D15—42no—
269C3×C4×C5—24no—
270C3×C4×C5—22no—
271C3×C4×C5—42no—
272C3×C4×C5—41no—
273C3×C4×C5C4×D414yes—
274Dic15—22no—
275Dic15—24no—
276Dic15—22no—
277Dic15—41no—
278C23×D3—52no—
279C23×D3—51no—
280C22×C3×C4C22×D511yescharacteristic
281C22×C3×C4C22×D512yes—
282C22×C3×C4C22×D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C22×C3×C4C22×D511yescharacteristic
285C22×Dic3—51no—
286C22×Dic3—51no—
287C22×Dic3—52no—
288C22×Dic3—51no—
289C2×D12—54no—
290C2×D12—58no—
291C2×D12—102no—
292C2×D12—102no—
293C2×D12—202no—
294C2×D12—202no—
295C2×C4×D3—54no—
296C2×C4×D3—102no—
297C2×C4×D3—102no—
298C2×C4×D3—104no—
299C2×C4×D3—104no—
300C2×C4×D3—202no—
301C3×C4⋊C4—22no—
302C3×C4⋊C4—42no—
303C3×C4⋊C4C4×D514yescomplemented
304C3×C42—22no—
305(C2×D3)⋊C4—58no—
306(C2×D3)⋊C4—104no—
307(C2×D3)⋊C4—104no—
308(C2×D3)⋊C4—104no—
309(C2×D3)⋊C4—102no—
310(C2×D3)⋊C4—102no—
311(C2×D3)⋊C4—202no—
312C4⋊Dic3—54no—
313C4×Dic3—104no—
314C4×Dic3—102no—
315C4×Dic3—202no—
316C23×C5C23×D311yescharacteristic
317C22×D5—34no—
318C22×D5—32no—
319C22×D5—61no—
320C22×D5—62no—
321C22×D5—62no—
322C22×D5—61no—
323C22×D5—61no—
324C22×D5—61no—
325C22×D5—62no—
326C22×D5—62no—
327C22×D5—62no—
328C22×D5—61no—
329C22×D5—62no—
330C22×D5—61no—
331C2×C4×C5—21no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5—21no—
335C2×C4×C5—22no—
336C2×C4×C5—21no—
337C2×C4×C5—22no—
338C2×C4×C5—21no—
339C2×C4×C5—21no—
340C2×C4×C5—22no—
341C2×C4×C5C2×C4×D314yes—
342C2×C4×C5C2×C4×D312yes—
343C2×C4×C5D3×D412yes—
344C2×C4×C5#23912yes—
345C2×C4×C5#23912yes—
346C2×Dic5—32no—
347C2×Dic5—34no—
348C2×Dic5—32no—
349C2×Dic5—61no—
350C2×Dic5—61no—
351C2×Dic5—61no—
352C2×Dic5—62no—
353C2×Dic5—61no—
354C2×Dic5—62no—
355C2×Dic5—61no—
356C2×Dic5—61no—
357C2×Dic5—61no—
358D20—68no—
359D20—128no—
360C4×D5—68no—
361C4×D5—128no—
362C22×D4—151no—
363C22×D4—152no—
364C23×C4—151no—
365C23×C4—152no—
366#121—158no—
367C4×D4—304no—
368#117—304no—
369C2×C4⋊C4—151no—
370C2×C4⋊C4C2×D1511yescharacteristic
371C2×C4⋊C4C2×D1512yes—
372C2×C4⋊C4C2×D1511yescharacteristic
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—151no—
378C2×C22⋊C4—152no—
379C2×C22⋊C4—152no—
380C2×C42—152no—
381C2×C42—151no—
382C2×C42—152no—
383C2×C42C2×D1511yescharacteristic
384#95—151no—
385#95—151no—
386#95—152no—
387#95C2×D1512yes—
388C2×C3×C5C2×C4×D411yescharacteristic
389C2×C3×C5C4×#4111yescharacteristic
390C2×C3×C5#51811yescharacteristic
391C2×C3×C5#54611yescharacteristic
392C2×C3×C5#55011yescharacteristic
393C2×C3×C5#55311yescharacteristic
394C2×C3×C5#55511yescharacteristic
395D15—42no—
396D15—44no—
397C23×C3C23×D511yescharacteristic
398C22×D3—54no—
399C22×D3—52no—
400C22×D3—101no—
401C22×D3—102no—
402C22×D3—102no—
403C22×D3—101no—
404C22×D3—101no—
405C22×D3—101no—
406C22×D3—102no—
407C22×D3—102no—
408C22×D3—102no—
409C22×D3—101no—
410C22×D3—102no—
411C22×D3—101no—
412C2×C3×C4—21no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—21no—
418C2×C3×C4—22no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—22no—
422C2×C3×C4C2×C4×D514yes—
423C2×C3×C4C2×C4×D512yes—
424C2×C3×C4D4×D512yes—
425C2×C3×C4#71312yes—
426C2×C3×C4#71312yes—
427C2×Dic3—52no—
428C2×Dic3—54no—
429C2×Dic3—52no—
430C2×Dic3—101no—
431C2×Dic3—101no—
432C2×Dic3—101no—
433C2×Dic3—102no—
434C2×Dic3—101no—
435C2×Dic3—102no—
436C2×Dic3—101no—
437C2×Dic3—101no—
438C2×Dic3—101no—
439D12—108no—
440D12—208no—
441C4×D3—108no—
442C4×D3—208no—
443C22×C5C22×C4×D311yescharacteristic
444C22×C5C2×D3×D411yescharacteristic
445C22×C5#100411yescharacteristic
446C22×C5C2×#23911yescharacteristic
447C22×C5C2×#23911yescharacteristic
448C22×C5D3×#4111yescharacteristic
449C22×C5#100911yescharacteristic
450C2×D5—62no—
451C2×D5—64no—
452C2×D5—62no—
453C2×D5—64no—
454C2×D5—64no—
455C2×D5—62no—
456C2×D5—121no—
457C2×D5—122no—
458C2×D5—122no—
459C2×D5—121no—
460C2×D5—121no—
461C2×D5—122no—
462C2×D5—122no—
463C2×D5—121no—
464C4×C5—24no—
465C4×C5—22no—
466C4×C5—42no—
467C4×C5—41no—
468C4×C5#89314yes—
469Dic5—62no—
470Dic5—62no—
471Dic5—64no—
472Dic5—121no—
473C24—152no—
474C24—151no—
475C2×D4—154no—
476C2×D4—158no—
477C2×D4—302no—
478C2×D4—302no—
479C2×D4—602no—
480C2×D4—602no—
481C22×C4—151no—
482C22×C4—151no—
483C22×C4—152no—
484C22×C4—154no—
485C22×C4—151no—
486C22×C4—304no—
487C22×C4—302no—
488C22×C4—302no—
489C22×C4—304no—
490C22×C4—602no—
491C22×C4C22×D1511yescharacteristic
492C22×C4C22×D1512yes—
493C22×C4C22×D1512yes—
494C22×C4C22×D1511yescharacteristic
495C22×C4C22×D1511yescharacteristic
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—154no—
499C4⋊C4C4×D1514yescomplemented
500C22⋊C4—158no—
501C22⋊C4—304no—
502C22⋊C4—304no—
503C22⋊C4—302no—
504C22⋊C4—304no—
505C22⋊C4—302no—
506C22⋊C4—602no—
507C42—22no—
508C42—304no—
509C42—302no—
510C42—602no—
511C3×C5#236311yescharacteristic, complemented, Hall
512C22×C3C22×C4×D511yescharacteristic
513C22×C3C2×D4×D511yescharacteristic
514C22×C3#414411yescharacteristic
515C22×C3C2×#71311yescharacteristic
516C22×C3C2×#71311yescharacteristic
517C22×C3D5×#4111yescharacteristic
518C22×C3#414911yescharacteristic
519C2×D3—102no—
520C2×D3—104no—
521C2×D3—102no—
522C2×D3—104no—
523C2×D3—104no—
524C2×D3—102no—
525C2×D3—202no—
526C2×D3—201no—
527C2×D3—201no—
528C2×D3—202no—
529C2×D3—202no—
530C2×D3—201no—
531C2×D3—201no—
532C2×D3—202no—
533C3×C4—24no—
534C3×C4—22no—
535C3×C4—42no—
536C3×C4—41no—
537C3×C4#403914yes—
538Dic3—102no—
539Dic3—104no—
540Dic3—102no—
541Dic3—201no—
542C2×C5C2×#89311yescharacteristic
543C2×C5#554511yescharacteristic
544C2×C5C4×#23911yescharacteristic
545C2×C5#559911yescharacteristic
546C2×C5#563011yescharacteristic
547C2×C5#569511yescharacteristic
548C2×C5#573411yescharacteristic
549D5—122no—
550D5—124no—
551C23—154no—
552C23—152no—
553C23—301no—
554C23—302no—
555C23—302no—
556C23—301no—
557C23—301no—
558C23—302no—
559C23—301no—
560C23—302no—
561C23—302no—
562C23—301no—
563C23—302no—
564C23—301no—
565C23C23×D1511yescharacteristic
566D4—308no—
567D4—608no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—152no—
579C2×C4—154no—
580C2×C4—152no—
581C2×C4—301no—
582C2×C4—301no—
583C2×C4—302no—
584C2×C4—301no—
585C2×C4—301no—
586C2×C4—302no—
587C2×C4—308no—
588C2×C4—301no—
589C2×C4—301no—
590C2×C4—301no—
591C2×C4—608no—
592C2×C4C2×C4×D1514yes—
593C2×C4C2×C4×D1512yes—
594C2×C4D4×D1512yes—
595C2×C4#682112yes—
596C2×C4#682112yes—
597C2×C3C2×#403911yescharacteristic
598C2×C3#6587911yescharacteristic
599C2×C3C4×#71311yescharacteristic
600C2×C3#6593311yescharacteristic
601C2×C3#6596411yescharacteristic
602C2×C3#6602911yescharacteristic
603C2×C3#6606811yescharacteristic
604D3—202no—
605D3—204no—
606C5#7598511yescharacteristic, complemented, Sylow
607C22—302no—
608C22—304no—
609C22—304no—
610C22—302no—
611C22—302no—
612C22—304no—
613C22—601no—
614C22—602no—
615C22—602no—
616C22—601no—
617C22—601no—
618C22—602no—
619C22—602no—
620C22—601no—
621C22C22×C4×D1511yescharacteristic
622C22C2×D4×D1511yescharacteristic
623C22#9209511yescharacteristic
624C22C2×#682111yescharacteristic
625C22C2×#682111yescharacteristic
626C22#41×D1511yescharacteristic
627C22#9210011yescharacteristic
628C4—24no—
629C4—22no—
630C4—41no—
631C4—42no—
632C4—304no—
633C4—302no—
634C4—302no—
635C4—601no—
636C4#9178214yes—
637C3#1060771611yescharacteristic, complemented, Sylow
638C2—602no—
639C2—604no—
640C2C2×#9178211yescharacteristic
641C2#1175630411yescharacteristic
642C2C4×#682111yescharacteristic
643C2#1175635811yescharacteristic
644C2#1175638911yescharacteristic
645C2#1175645411yescharacteristic
646C2#1175649311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#389C2
socle, cosocleC23×C3×C5C24
prime radicalC2×#91784C2
hypercenter#389D15
hypocenterC3×C5#2363
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×#91781C2semidirect
3. C2×C4×D60C2semidirect
4. C3×C5×#389C2semidirect
5. #11751091C2semidirect
6. #11751089C2semidirect
7. #11751089C2semidirect
8. C4⋊C4×Dic15C2semidirect
9. C4×#6714C2semidirect
10. #11750767C2semidirect
11. #11750766C2semidirect
32. #91784C4semidirect
43. C5×#389D3semidirect
67. C2×D60C2×C4semidirect
73. C2×C4×D15C2×C4semidirect
76. C3×C5×C4⋊C4C2×C4semidirect
84. #6714C2×C4semidirect
85. C4⋊Dic15D4semidirect
92. C3×#389D5semidirect
224. C5×C4⋊C4C4×D3semidirect
242. #389D15semidirect
303. C3×C4⋊C4C4×D5semidirect
499. C4⋊C4C4×D15semidirect
511. C3×C5#2363semidirect
606. C5#75985semidirect
637. C3#10607716semidirect

Maximal subgroups

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

subgroupindex
C2×#917842
C2×#917812
C2×C4×D602
C3×C5×#3892
#117510912
#117510892
C4⋊C4×Dic152
C4×#67142
#117507672
#117507662
#106077163
#759855