Finite Group Databasebeta

← #14495119

Subgroups — #14495119

Order 1920.

2060 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6821C211yesmaximal, characteristic, complemented
2C5×#5572C211yesmaximal, characteristic, complemented
3C3×#65908C211yesmaximal, characteristic, complemented
4#11755267C211yesmaximal, characteristic, complemented
5#11755251C211yesmaximal, characteristic, complemented
6#11755247C211yesmaximal, characteristic, complemented
7#11755131C211yesmaximal, characteristic, complemented
8#11753781C213yesmaximal, complemented
9#11753740C213yesmaximal, complemented
10#11753736C213yesmaximal, complemented
11#11753347C213yesmaximal, complemented
12#11752709C213yesmaximal, complemented
13#11752703C213yesmaximal, complemented
14#11752504C213yesmaximal, complemented
15#11752501C213yesmaximal, complemented
16#10617780—31nomaximal, Hall
17C2×#6821C2211yescharacteristic
18C2×C5×D3×Q8C2211yescharacteristic
19C2×C3×#713C2211yescharacteristic
20C3×C4×C5×Q8C2211yescharacteristic
21Q8×Dic15C2211yescharacteristic
22#91782C2213yes—
23#91781C2213yes—
24C4×D60C2213yes—
25C42×D15C2213yes—
26C5×Q8×Dic3C2211yescharacteristic
27C5×#892C2213yes—
28C5×#887C2213yes—
29C5×#872C2213yes—
30C4×C5×Dic6C2213yes—
31C3×Q8×Dic5C2211yescharacteristic
32C3×#4039C2213yes—
33C3×D5×C4⋊C4C2213yes—
34C3×C4×D20C2213yes—
35C3×#4018C2213yes—
36#91468C2213yes—
37C4×#1204C2213yes—
38#91465C2213yes—
39#91464C2213yes—
40#91461C2213yes—
41#91460C2213yes—
42#91458C2213yes—
43#91456C2213yes—
44#91450C2213yes—
45#91448C2216yes—
46C4×#1203C2213yes—
47C4×#1202C2216yes—
48C4×#1202C2213yes—
49C4×#1201C2213yes—
50#91384C2213yes—
51#91383C2213yes—
52#91382C2213yes—
53#91381C2213yes—
54#91380C2216yes—
55#91377C2213yes—
56#91375C2213yes—
57#91368C2213yes—
58#91367C2211yescharacteristic
59#91366C2213yes—
60#91359C2213yes—
61#91357C2216yes—
62#91355C2213yes—
63#91354C2216yes—
64#91354C2213yes—
65#91353C2216yes—
66#91348C2216yes—
67#85898—51nomaximal, Hall
68C5×#519—31no—
69#66151—31no—
70#66151—31no—
71#65919—33no—
72#65916—33no—
73#65915—33no—
74#65914—33no—
75#65911—31no—
76#65908D311yescharacteristic, complemented
77C4×#713—31no—
78#65903—31no—
79#65869—33no—
80#65866—33no—
81#65864—33no—
82#65862—33no—
83C2×C3×C5×Q8C2311yescharacteristic
84#6821—24no—
85C2×D60C2313yes—
86C2×C4×D15C2313yes—
87C2×C4×D15C2313yes—
88C2×C4×D15C2313yes—
89C5×D3×Q8—24no—
90C2×C4×C5×D3C2313yes—
91C2×C5×Dic6C2313yes—
92C3×#713—24no—
93C2×C3×D20C2313yes—
94C2×C3×C4×D5C2313yes—
95C2×C3×C4×D5C2316yes—
96C2×#1204C2313yes—
97C2×#1203C2313yes—
98C2×#1202C2316yes—
99C2×#1202C2313yes—
100C2×#1201C2313yes—
101C3×C5×C4⋊C4C2313yes—
102C3×C42×C5C2313yes—
103#6714C2316yes—
104C4⋊Dic15C2313yes—
105C4×Dic15C2313yes—
106C4×Dic15C2313yes—
107C4×Dic15C2311yescharacteristic
108C5×(C2×D3)⋊C4C2311yescharacteristic
109C5×(C2×D3)⋊C4C2313yes—
110C5×C4⋊Dic3C2313yes—
111C5×Dic3⋊C4C2313yes—
112C5×Dic3⋊C4C2313yes—
113C4×C5×Dic3C2316yes—
114C4×C5×Dic3C2313yes—
115C3×#685C2313yes—
116C3×#685C2313yes—
117C3×C4⋊Dic5C2313yes—
118C3×#683C2311yescharacteristic
119C3×#683C2313yes—
120C3×C4×Dic5C2313yes—
121Dic3⋊Dic5C2311yescharacteristic
122Dic3⋊Dic5C2313yes—
123#6669C2313yes—
124#6669C2313yes—
125#6668C2313yes—
126#6668C2316yes—
127#6668C2311yescharacteristic
128#6667C2313yes—
129#6667C2316yes—
130#6667C2313yes—
131(C2×D3)⋊Dic5C2311yescharacteristic
132(C2×D3)⋊Dic5C2311yescharacteristic
133#6665C2313yes—
134#6665C2313yes—
135#6665C2316yes—
136C3×#520—51no—
137#5834—51no—
138#5831—51no—
139#5576—51no—
140C4×#239—51no—
141#5572D511yescharacteristic, complemented
142#5570—51no—
143#5563—53no—
144#5562—53no—
145#5559—53no—
146#5557—53no—
147#5534—53no—
148#5528—53no—
149#5514—53no—
150#5513—53no—
151C22×C5×Q8—31no—
152C2×#713—31no—
153C2×#713C2×D311yescharacteristic
154C4×C5×Q8—31no—
155C4×C5×Q8—33no—
156C4×C5×Q8—33no—
157C4×C5×Q8C2×D311yescharacteristic
158C5×#117—33no—
159C5×#117—33no—
160#4093—33no—
161#4093—33no—
162#4092—33no—
163#4092—33no—
164Q8×Dic5—31no—
165Q8×Dic5C2×D311yescharacteristic
166#4076—33no—
167#4076—33no—
168C4×C22⋊D5—33no—
169C4×C22⋊D5—33no—
170#4044—33no—
171#4042—33no—
172#4041—33no—
173#4041—33no—
174#4040—36no—
175#4040—33no—
176#4040—36no—
177#4039—33no—
178#4039C2×D313yes—
179#4038—33no—
180D5×C4⋊C4C2×D313yes—
181#4036—33no—
182#4035—33no—
183#4034—33no—
184#4031—33no—
185#4030—36no—
186#4030—33no—
187#4029—33no—
188#4025—31no—
189#4022—36no—
190#4021—36no—
191#4020—33no—
192C4×D20—33no—
193C4×D20—36no—
194C4×D20—33no—
195C4×D20C2×D313yes—
196#4018—36no—
197#4018C2×D313yes—
198C42×D5—33no—
199#4016—33no—
200C4×Dic10—33no—
201#3524—151noSylow
202C22×D15C2413yes—
203C22×C5×D3C2411yescharacteristic
204C22×C3×D5C2413yes—
205C3×C5×Q8#4114yes—
206C2×C3×C4×C5C2411yescharacteristic
207C2×C3×C4×C5C2413yes—
208C2×C3×C4×C5C2413yes—
209C2×Dic15C2411yescharacteristic
210C2×Dic15C2411yescharacteristic
211C2×Dic15C2413yes—
212D60—26no—
213C4×D15—26no—
214C4×D15—43no—
215C4×D15—43no—
216C2×C5×Dic3C2413yes—
217C2×C5×Dic3C2413yes—
218C2×C5×Dic3C2411yescharacteristic
219C4×C5×D3—26no—
220C5×Dic6—26no—
221C2×C3×Dic5C2411yescharacteristic
222C2×C3×Dic5C2411yescharacteristic
223C2×C3×Dic5C2413yes—
224C3×D20—26no—
225C3×C4×D5—26no—
226C3×C4×D5—46no—
227#1204—43no—
228#1203—43no—
229#1202—43no—
230#1202—46no—
231#1201—43no—
232C2×C3×#41—51no—
233C2×#239—51no—
234C2×D3×Q8C2×D511yescharacteristic
235C3×C4×Q8—51no—
236C3×C4×Q8C2×D511yescharacteristic
237C3×C4×D4—53no—
238C3×C4×D4—53no—
239C3×#117—53no—
240C2×C3×C4⋊C4—53no—
241Q8×Dic3—51no—
242Q8×Dic3C2×D511yescharacteristic
243#940—53no—
244#938—53no—
245#938—53no—
246#936—53no—
247#930—56no—
248#930—53no—
249#929—56no—
250C4×#67—53no—
251C4×#67—53no—
252C4×#67—56no—
253#926—56no—
254#924—53no—
255#898—51no—
256#898—56no—
257#897—53no—
258#896—53no—
259#895—53no—
260#894—53no—
261#893—53no—
262#892—53no—
263#892C2×D513yes—
264#890—53no—
265#890—53no—
266#889—53no—
267#887C2×D513yes—
268#885—53no—
269#885—56no—
270#884—53no—
271#883—53no—
272#883—56no—
273#879—53no—
274#875—53no—
275C4×D12—53no—
276C4×D12—53no—
277#872C2×D513yes—
278C42×D3—53no—
279#869—53no—
280C4×Dic6—53no—
281C4×Dic6C2×D513yes—
282C2×C5×Q8—33no—
283C2×C5×Q8—64no—
284C2×C5×Q8C22×D311yescharacteristic
285C22×C4×C5—33no—
286C2×C22⋊D5—33no—
287C2×C22⋊D5—33no—
288#713—24no—
289#713—64no—
290C2×D20—33no—
291C2×D20—33no—
292C2×D20—36no—
293C2×D20C22×D313yes—
294C2×C4×D5—33no—
295C2×C4×D5—33no—
296C2×C4×D5—33no—
297C2×C4×D5C22×D313yes—
298C2×C4×D5C22×D316yes—
299C2×Dic10—33no—
300C5×C4⋊C4—33no—
301C5×C4⋊C4—33no—
302C5×C4⋊C4—33no—
303C5×C4⋊C4C22×D313yes—
304C5×C22⋊C4—31no—
305C5×C22⋊C4—33no—
306C42×C5—33no—
307C42×C5—36no—
308C42×C5C22×D313yes—
309C22⋊Dic5—31no—
310C22⋊Dic5—31no—
311#685—33no—
312#685—36no—
313#685—33no—
314#685—36no—
315#685—33no—
316#685—36no—
317#685—33no—
318#685C22×D313yes—
319#685C22×D313yes—
320C4⋊Dic5—33no—
321C4⋊Dic5—33no—
322C4⋊Dic5—31no—
323C4⋊Dic5C22×D313yes—
324#683—31no—
325#683—36no—
326#683—33no—
327#683—33no—
328#683—33no—
329#683C22×D311yescharacteristic
330#683C22×D313yes—
331C4×Dic5—33no—
332C4×Dic5—31no—
333C4×Dic5—33no—
334C4×Dic5C22×D313yes—
335#559—153no—
336#557—151no—
337#555—153no—
338#553—153no—
339#552—153no—
340#550—151no—
341#548—151no—
342#548—151no—
343#542—153no—
344#540—153no—
345#529—153no—
346#529—153no—
347#520—51no—
348#519—31no—
349C4×#41—151no—
350C22×C3×C5C2511yescharacteristic
351C2×D15—23no—
352C2×D15—23no—
353C2×D15—23no—
354C2×C5×D3—21no—
355C2×C5×D3—21no—
356C2×C5×D3—21no—
357C2×C3×D5—23no—
358C2×C3×D5—23no—
359C2×C3×D5—23no—
360C3×C4×C5—23no—
361C3×C4×C5—21no—
362C3×C4×C5C2×#4116yes—
363Dic15—21no—
364Dic15—23no—
365Dic15—21no—
366C3×Dic5—21no—
367C3×Dic5—23no—
368C3×Dic5—21no—
369C5×Dic3—23no—
370C5×Dic3—21no—
371C5×Dic3—23no—
372C3×#41—104no—
373C2×C3×Q8C22×D511yescharacteristic
374C2×C3×D4—53no—
375C22×C3×C4—53no—
376C22×C3×C4—56no—
377C2×#67—53no—
378C2×#67—53no—
379C2×#67—56no—
380#239—104no—
381D3×Q8—24no—
382C2×D12—53no—
383C2×D12—53no—
384C2×C4×D3—53no—
385C2×C4×D3—53no—
386C2×C4×D3—53no—
387C2×C4×D3C22×D513yes—
388C2×Dic6—53no—
389C2×Dic6C22×D513yes—
390C3×C4⋊C4—53no—
391C3×C4⋊C4—51no—
392C3×C4⋊C4—53no—
393C3×C4⋊C4C22×D513yes—
394C3×C22⋊C4—53no—
395C3×C22⋊C4—53no—
396C3×C42—53no—
397C3×C42C22×D513yes—
398#217—53no—
399#217—53no—
400#217—56no—
401(C2×D3)⋊C4—51no—
402(C2×D3)⋊C4—51no—
403(C2×D3)⋊C4—56no—
404(C2×D3)⋊C4—53no—
405(C2×D3)⋊C4—56no—
406(C2×D3)⋊C4—53no—
407(C2×D3)⋊C4C22×D511yescharacteristic
408(C2×D3)⋊C4C22×D513yes—
409C4⋊Dic3—53no—
410C4⋊Dic3—51no—
411C4⋊Dic3—56no—
412C4⋊Dic3—53no—
413C4⋊Dic3C22×D513yes—
414Dic3⋊C4—53no—
415Dic3⋊C4—53no—
416Dic3⋊C4—53no—
417Dic3⋊C4—51no—
418Dic3⋊C4C22×D513yes—
419Dic3⋊C4C22×D513yes—
420C4×Dic3—53no—
421C4×Dic3—51no—
422C4×Dic3—53no—
423C4×Dic3C22×D513yes—
424C4×Dic3C22×D516yes—
425C23×C5—31no—
426C22×D5—33no—
427C22×D5C23×D313yes—
428C5×Q8—66no—
429C5×Q8#23514yes—
430C2×C4×C5—31no—
431C2×C4×C5—33no—
432C2×C4×C5—33no—
433C2×C4×C5—66no—
434C2×C4×C5C23×D313yes—
435C2×C4×C5C23×D311yescharacteristic
436C2×C4×C5C23×D313yes—
437C22⋊D5—123no—
438C22⋊D5—123no—
439C2×Dic5—31no—
440C2×Dic5—31no—
441C2×Dic5—33no—
442C2×Dic5C23×D311yescharacteristic
443C2×Dic5C23×D313yes—
444C2×Dic5C23×D311yescharacteristic
445D20—26no—
446D20—66no—
447D20—123no—
448D20—126no—
449C4×D5—26no—
450C4×D5—46no—
451C4×D5—66no—
452C4×D5—123no—
453C4×D5—123no—
454Dic10—123no—
455C2×#41—51no—
456C2×#41—151no—
457C22×Q8—31no—
458C4⋊Q8—153no—
459#127—153no—
460#126—156no—
461#126—151no—
462#126—153no—
463#125—153no—
464#125—153no—
465#125—153no—
466#124—156no—
467#124—153no—
468#124—153no—
469#124—153no—
470#123—153no—
471#123—156no—
472#123—153no—
473#123—156no—
474C22⋊Q8—153no—
475C22⋊Q8—153no—
476C22⋊Q8—153no—
477#121—153no—
478#121—156no—
479#121—153no—
480#121—153no—
481#121—153no—
482#121—153no—
483C4×Q8—31no—
484C4×Q8—33no—
485C4×Q8—33no—
486C4×Q8—51no—
487C4×Q8—153no—
488C4×Q8—151no—
489C4×Q8D3×D511yescharacteristic
490C4×D4—53no—
491C4×D4—53no—
492C4×D4—153no—
493C4×D4—156no—
494C4×D4—153no—
495C4×D4—153no—
496C4×D4—153no—
497C4×D4—153no—
498#117—33no—
499#117—33no—
500#117—53no—
501#117—156no—
502#117—153no—
503C2×C4⋊C4—53no—
504C2×C42—153no—
505C2×C3×C5C22×#4111yescharacteristic
506C2×C3×C5#58511yescharacteristic
507C2×C3×C5#58511yescharacteristic
508D15—43no—
509C3×D5—43no—
510C5×D3—41no—
511C23×C3—53no—
512C22×D3—53no—
513C22×D3C23×D511yescharacteristic
514C3×Q8#70914yes—
515C3×D4—106no—
516C2×C3×C4—51no—
517C2×C3×C4—53no—
518C2×C3×C4—51no—
519C2×C3×C4—106no—
520C2×C3×C4—206no—
521C2×C3×C4C23×D513yes—
522C2×C3×C4C23×D511yescharacteristic
523C2×C3×C4C23×D513yes—
524#67—206no—
525#67—203no—
526#67—203no—
527C2×Dic3—51no—
528C2×Dic3—51no—
529C2×Dic3—53no—
530C2×Dic3C23×D511yescharacteristic
531C2×Dic3C23×D513yes—
532C2×Dic3C23×D513yes—
533D12—106no—
534D12—203no—
535C4×D3—26no—
536C4×D3—106no—
537C4×D3—203no—
538C4×D3—203no—
539Dic6—26no—
540Dic6—203no—
541C22×C5—61no—
542C22×C5—61no—
543C22×C5—61no—
544C22×C5C24×D311yescharacteristic
545C2×D5—23no—
546C2×D5—23no—
547C2×D5—23no—
548C2×D5—63no—
549C2×D5—63no—
550C2×D5—63no—
551C4×C5—23no—
552C4×C5—21no—
553C4×C5—63no—
554C4×C5—61no—
555C4×C5—63no—
556C4×C5C2×#23516yes—
557Dic5—21no—
558Dic5—23no—
559Dic5—21no—
560Dic5—61no—
561Dic5—61no—
562Dic5—63no—
563#41—104no—
564#41—304no—
565C2×Q8—33no—
566C2×Q8—64no—
567C2×Q8—153no—
568C2×Q8C2×D3×D511yescharacteristic
569C2×D4—53no—
570C2×D4—156no—
571C2×D4—153no—
572C2×D4—153no—
573C2×D4—153no—
574C2×D4—153no—
575C22×C4—33no—
576C22×C4—53no—
577C22×C4—56no—
578C22×C4—153no—
579C22×C4—153no—
580C22×C4—153no—
581C4⋊C4—33no—
582C4⋊C4—33no—
583C4⋊C4—33no—
584C4⋊C4—53no—
585C4⋊C4—53no—
586C4⋊C4—51no—
587C4⋊C4—153no—
588C4⋊C4—156no—
589C4⋊C4—151no—
590C4⋊C4—153no—
591C4⋊C4—153no—
592C4⋊C4—151no—
593C4⋊C4—153no—
594C4⋊C4—153no—
595C4⋊C4C2×D3×D513yes—
596C22⋊C4—31no—
597C22⋊C4—33no—
598C22⋊C4—53no—
599C22⋊C4—53no—
600C22⋊C4—153no—
601C22⋊C4—156no—
602C22⋊C4—153no—
603C22⋊C4—153no—
604C22⋊C4—156no—
605C22⋊C4—153no—
606C22⋊C4—151no—
607C22⋊C4—151no—
608C22⋊C4—156no—
609C42—36no—
610C42—33no—
611C42—53no—
612C42—151no—
613C42—153no—
614C42—153no—
615C42C2×D3×D513yes—
616C3×C5#352411yescharacteristic, complemented, Hall
617C22×C3—103no—
618C22×C3—103no—
619C22×C3—103no—
620C22×C3C24×D511yescharacteristic
621C2×D3—21no—
622C2×D3—21no—
623C2×D3—21no—
624C2×D3—103no—
625C2×D3—103no—
626C2×D3—103no—
627C3×C4—21no—
628C3×C4—23no—
629C3×C4—101no—
630C3×C4—101no—
631C3×C4—103no—
632C3×C4C2×#70916yes—
633Dic3—23no—
634Dic3—23no—
635Dic3—21no—
636Dic3—101no—
637Dic3—103no—
638Dic3—101no—
639C2×C5—121no—
640C2×C5C22×#23511yescharacteristic
641C2×C5#596611yescharacteristic
642C2×C5#596811yescharacteristic
643D5—43no—
644D5—123no—
645C23—31no—
646C23—53no—
647C23—153no—
648Q8—66no—
649Q8—603no—
650Q8#677114yes—
651D4—106no—
652D4—306no—
653D4—603no—
654D4—606no—
655D4—603no—
656D4—603no—
657C2×C4—33no—
658C2×C4—31no—
659C2×C4—33no—
660C2×C4—51no—
661C2×C4—53no—
662C2×C4—51no—
663C2×C4—66no—
664C2×C4—106no—
665C2×C4—151no—
666C2×C4—151no—
667C2×C4—153no—
668C2×C4—206no—
669C2×C4—306no—
670C2×C4—603no—
671C2×C4—603no—
672C2×C4C22×D3×D513yes—
673C2×C4C22×D3×D511yescharacteristic
674C2×C4C22×D3×D513yes—
675C2×C3—203no—
676C2×C3C22×#70911yescharacteristic
677C2×C3#6628711yescharacteristic
678C2×C3#6628711yescharacteristic
679D3—41no—
680D3—203no—
681C5#8589811yescharacteristic, complemented, Sylow
682C22—61no—
683C22—61no—
684C22—61no—
685C22—103no—
686C22—103no—
687C22—103no—
688C22—303no—
689C22—303no—
690C22—303no—
691C22C23×D3×D511yescharacteristic
692C4—23no—
693C4—21no—
694C4—61no—
695C4—63no—
696C4—63no—
697C4—103no—
698C4—101no—
699C4—101no—
700C4—301no—
701C4—301no—
702C4—303no—
703C4C2×#677116yes—
704C3#1061778011yescharacteristic, complemented, Sylow
705C2—121no—
706C2—203no—
707C2—603no—
708C2C22×#677111yescharacteristic
709C2#1175705011yescharacteristic
710C2#1175706211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C4×C5×Q8C22
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3524
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×#6821C2semidirect
2. C5×#5572C2semidirect
3. C3×#65908C2semidirect
4. #11755267C2semidirect
5. #11755251C2semidirect
6. #11755247C2semidirect
7. #11755131C2semidirect
8. #11753781C2semidirect
9. #11753740C2semidirect
10. #11753736C2semidirect
11. #11753347C2semidirect
12. #11752709C2semidirect
13. #11752703C2semidirect
14. #11752504C2semidirect
15. #11752501C2semidirect
76. #65908D3semidirect
141. #5572D5semidirect
616. C3×C5#3524semidirect
681. C5#85898semidirect
704. C3#10617780semidirect

Maximal subgroups

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

subgroupindex
C4×#68212
C5×#55722
C3×#659082
#117552672
#117552512
#117552472
#117551312
#117537812
#117537402
#117537362
#117533472
#117527092
#117527032
#117525042
#117525012
#106177803
#858985