Finite Group Databasebeta

← #14417207

Subgroups — #14417207

Order 1920.

940 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91807C211yesmaximal, characteristic
2#11755149C211yesmaximal, characteristic
3#11755124C211yesmaximal, characteristic, complemented
4C5×#5184C211yesmaximal, characteristic, complemented
5C3×Q16×Dic5C211yesmaximal, characteristic, complemented
6#11749257C211yesmaximal, characteristic, complemented
7#11749250C211yesmaximal, characteristic, complemented
8#11749232C211yesmaximal, characteristic, complemented
9#11749225C211yesmaximal, characteristic, complemented
10#11749215C211yesmaximal, characteristic
11#11749203C211yesmaximal, characteristic, complemented
12#11748053C211yesmaximal, characteristic, complemented
13#11747800C211yesmaximal, characteristic, complemented
14#11747760C211yesmaximal, characteristic, complemented
15#11747640C211yesmaximal, characteristic, complemented
16#10611561—31nomaximal, Hall
17C2×#6821C2211yescharacteristic
18C2×Q8×D15C2211yescharacteristic
19C2×C3×C5×Q16C2211yescharacteristic
20C2×#6718C2211yescharacteristic
21C2×#6717C2211yescharacteristic
22#91807C418yescomplemented
23C2×#6706C2211yescharacteristic
24C2×#6705C2211yescharacteristic
25C5×Q8×Dic3C2211yescharacteristic
26C5×Q8×Dic3C2211yescharacteristic
27C3×Q8×Dic5C2211yescharacteristic
28C3×Q8×Dic5C2211yescharacteristic
29#91437—21no—
30#91437C2211yescharacteristic
31#91428—21no—
32#91428C2211yescharacteristic
33#91418—21no—
34#91418C2211yescharacteristic
35#91386—21no—
36#91386—21no—
37#91386C2211yescharacteristic
38#91347—21no—
39#91347C2211yescharacteristic
40#91340—21no—
41#91340C2211yescharacteristic
42#91333—21no—
43#91333C2211yescharacteristic
44C2×#6648C2211yescharacteristic
45C5×#835C2211yescharacteristic
46C5×#835C2211yescharacteristic
47C5×#817C2211yescharacteristic
48C5×#815C2211yescharacteristic
49C3×#3967C2211yescharacteristic
50C3×#3967C2211yescharacteristic
51C3×#3950C2211yescharacteristic
52C3×C8×Dic5C2211yescharacteristic
53#90989C2211yescharacteristic
54#90986C2211yescharacteristic
55#90975C2211yescharacteristic
56#90974C2211yescharacteristic
57#90969C2211yescharacteristic
58#90968C2211yescharacteristic
59#90952C2211yescharacteristic
60Dic3×#168C2211yescharacteristic
61#79719—51nomaximal, Hall
62C2×#4064—31no—
63#65908—31no—
64#65906—31no—
65C5×#441—31no—
66Q16×Dic5D311yescharacteristic, complemented
67C4×#689—31no—
68C4×#688—31no—
69#65150—31no—
70#65141—31no—
71#65106—31no—
72#65093—31no—
73D5×#103—31no—
74#65078—31no—
75#64999—31no—
76#64997—31no—
77C2×C3×C5×Q8C2311yescharacteristic
78C2×C3×C5×Q8C2311yescharacteristic
79#6821—22no—
80#6821C2×C414yes—
81Q8×D15—22no—
82Q8×D15C2×C414yescomplemented
83C2×D60—21no—
84C2×D60C2311yescharacteristic
85C2×C4×D15—21no—
86C2×C4×D15—21no—
87C2×C4×D15C2311yescharacteristic
88C2×Dic30—21no—
89C2×Dic30C2311yescharacteristic
90C2×#1200—21no—
91C2×#1200—21no—
92C2×#1200—21no—
93C3×C5×Q16C2×C414yescomplemented
94C2×C3×C5×C8C2311yescharacteristic
95#6718C2×C414yescomplemented
96#6717C2×C414yescomplemented
97C2×C3⋊#168C2311yescharacteristic
98#6706C2×C414yescomplemented
99#6705C2×C414yescomplemented
100C5×C4⋊Dic3—21no—
101C5×C4⋊Dic3—21no—
102C5×C4⋊Dic3C2311yescharacteristic
103C5×C4⋊Dic3C2311yescharacteristic
104C4×C5×Dic3—21no—
105C4×C5×Dic3—21no—
106C4×C5×Dic3C2311yescharacteristic
107C2×C5×C3⋊C8C2311yescharacteristic
108C3×C4⋊Dic5—21no—
109C3×C4⋊Dic5—21no—
110C3×C4⋊Dic5C2311yescharacteristic
111C3×C4⋊Dic5C2311yescharacteristic
112C3×C4×Dic5—21no—
113C3×C4×Dic5—21no—
114C3×C4×Dic5C2311yescharacteristic
115C2×C3×#168C2311yescharacteristic
116#6669—21no—
117#6669—21no—
118#6669—21no—
119#6669D411yescharacteristic
120#6669D411yescharacteristic
121#6667—21no—
122#6667—21no—
123#6667—21no—
124#6667D411yescharacteristic
125#6667D411yescharacteristic
126Dic3×Dic5—21no—
127Dic3×Dic5—21no—
128Dic3×Dic5—21no—
129#6648—22no—
130C2×#918—51no—
131#5574—51no—
132#5572—51no—
133C3×C4×Q16—51no—
134#5184D511yescharacteristic, complemented
135#5030—51no—
136#5026—51no—
137#4889—51no—
138#4881—51no—
139#4804—51no—
140D3×#103—51no—
141#4790—51no—
142#4788—51no—
143C4×#204—51no—
144#4689—51no—
145C2×#713—31no—
146C2×Q8×D5—31no—
147C2×C5×Q16C2×D311yescharacteristic
148C4×C5×Q8—31no—
149C4×C5×Q8—31no—
150Q8×Dic5C2×D311yescharacteristic
151Q8×Dic5C2×D311yescharacteristic
152C2×#689—31no—
153C2×#688—31no—
154#4064—38no—
155C2×#677—31no—
156C2×#676—31no—
157C2×C8×D5—31no—
158#4039—31no—
159#4039—61no—
160#4038—31no—
161#4038—61no—
162D5×C4⋊C4—31no—
163D5×C4⋊C4—61no—
164#4033—31no—
165#4033—61no—
166C4×D20—31no—
167C4×D20—61no—
168#4018—31no—
169#4018—61no—
170#4018—61no—
171C4×Dic10—31no—
172C4×Dic10—61no—
173C5×#106—31no—
174C5×#103—31no—
175C5×#103—31no—
176C5×#97—31no—
177#3967C2×D311yescharacteristic
178#3967C2×D311yescharacteristic
179#3953—31no—
180#3950C2×D311yescharacteristic
181#3948—31no—
182#3947—31no—
183C8×Dic5C2×D311yescharacteristic
184#3942—31no—
185#3941—31no—
186#3940—31no—
187#3939—31no—
188C4×#168—31no—
189#2947—151noSylow
190C22×D15—21no—
191C22×D15C2×D411yescharacteristic
192C3×C5×Q8—21no—
193C3×C5×Q8—21no—
194C3×C5×Q8C22×C412yes—
195C3×C5×Q8C22×C412yes—
196C2×C3×C4×C5—21no—
197C2×C3×C4×C5—21no—
198C2×C3×C4×C5C2411yescharacteristic
199C2×Dic15—21no—
200C2×Dic15C2×D411yescharacteristic
201D60—24no—
202D60—21no—
203D60C22×C412yes—
204C4×D15—24no—
205C4×D15—24no—
206C4×D15C22×C412yes—
207C4×D15#4112yes—
208Dic30—21no—
209Dic30—24no—
210Dic30C22×C412yes—
211C2×C5×Dic3—21no—
212C2×C5×Dic3—21no—
213C2×C5×Dic3C2×D411yescharacteristic
214C2×C5×Dic3C2×D411yescharacteristic
215C2×C3×Dic5—21no—
216C2×C3×Dic5—21no—
217C2×C3×Dic5C2×D411yescharacteristic
218C2×C3×Dic5C2×D411yescharacteristic
219#1200—81no—
220#1200—81no—
221#1200—81no—
222C3×C5×C8C22×C412yes—
223C3⋊#168C22×C412yes—
224C3×#168—21no—
225C5×C3⋊C8—21no—
226C2×#239—51no—
227C2×D3×Q8—51no—
228C2×C3×Q16C2×D511yescharacteristic
229C3×C4×Q8—51no—
230C3×C4×Q8—51no—
231Q8×Dic3C2×D511yescharacteristic
232Q8×Dic3C2×D511yescharacteristic
233C2×#216—51no—
234C2×#215—51no—
235#918—58no—
236C2×#204—51no—
237C2×ΓL(1,𝔽25)—51no—
238C2×D3×C8—51no—
239#893—51no—
240#893—101no—
241#892—51no—
242#892—101no—
243D3×C4⋊C4—51no—
244D3×C4⋊C4—101no—
245#887—51no—
246#887—101no—
247C4×D12—51no—
248C4×D12—101no—
249#872—51no—
250#872—101no—
251#872—101no—
252C4×Dic6—51no—
253C4×Dic6—101no—
254C3×#107—51no—
255C3×#103—51no—
256C3×#103—51no—
257C3×C4×C8—51no—
258#835C2×D511yescharacteristic
259#835C2×D511yescharacteristic
260#821—51no—
261#817C2×D511yescharacteristic
262#816—51no—
263#815C2×D511yescharacteristic
264C8×Dic3—51no—
265#810—51no—
266#809—51no—
267#808—51no—
268#807—51no—
269#803—51no—
270C2×C5×Q8C22×D311yescharacteristic
271C2×C5×Q8C22×D311yescharacteristic
272#713—34no—
273#713—62no—
274Q8×D5—34no—
275Q8×D5—62no—
276C2×D20—31no—
277C2×D20—61no—
278C2×C4×D5—31no—
279C2×C4×D5—61no—
280C2×C4×D5—61no—
281C2×C4×D5—61no—
282C2×C4×D5—61no—
283C2×C4×D5—61no—
284C2×Dic10—31no—
285C2×Dic10—61no—
286C5×Q16C4×D314yescomplemented
287C2×C5×C8—31no—
288C2×C5×C8C22×D311yescharacteristic
289C5×C4⋊C4—31no—
290C5×C4⋊C4—31no—
291C5×C4⋊C4—61no—
292C5×C4⋊C4—61no—
293C42×C5—31no—
294C42×C5—61no—
295C42×C5—61no—
296#689—34no—
297#688—34no—
298#685—31no—
299#685—31no—
300#685—61no—
301#685—61no—
302#685—61no—
303C4⋊Dic5—21no—
304C4⋊Dic5—21no—
305C4⋊Dic5C22×D311yescharacteristic
306C4⋊Dic5C22×D311yescharacteristic
307#683—31no—
308#683—31no—
309#683—61no—
310#683—61no—
311#683—61no—
312C4×Dic5—21no—
313C4×Dic5—21no—
314C4×Dic5—61no—
315C4×Dic5—61no—
316C4×Dic5—61no—
317C4×Dic5C22×D311yescharacteristic
318C2×#168—31no—
319C2×#168C22×D311yescharacteristic
320#677—34no—
321#676—34no—
322C8×D5—62no—
323C2×#137—151no—
324#520—151no—
325#519—151no—
326#441—31no—
327#441—151no—
328#440—151no—
329#440—151no—
330C4×Q16—51no—
331C4×Q16—151no—
332C4×SD16—151no—
333C4×SD16—151no—
334#428—151no—
335#416—151no—
336C2×#103—151no—
337#405—151no—
338C22×C3×C5C22×D411yescharacteristic
339C2×D15—21no—
340C2×D15—21no—
341C2×D15—22no—
342C2×D15—41no—
343C2×D15—41no—
344C2×D15C4×D412yes—
345C3×C4×C5—22no—
346C3×C4×C5—22no—
347C3×C4×C5C23×C411yescharacteristic
348C3×C4×C5C2×#4111yescharacteristic
349Dic15—22no—
350Dic15C4×D412yes—
351C3×Dic5—21no—
352C3×Dic5—21no—
353C3×Dic5—41no—
354C3×Dic5—41no—
355C5×Dic3—21no—
356C5×Dic3—21no—
357C5×Dic3—41no—
358C5×Dic3—41no—
359C2×C3×Q8C22×D511yescharacteristic
360C2×C3×Q8C22×D511yescharacteristic
361#239—54no—
362#239—102no—
363D3×Q8—54no—
364D3×Q8—102no—
365C2×D12—51no—
366C2×D12—101no—
367C2×C4×D3—51no—
368C2×C4×D3—101no—
369C2×C4×D3—101no—
370C2×C4×D3—101no—
371C2×C4×D3—101no—
372C2×C4×D3—101no—
373C2×Dic6—51no—
374C2×Dic6—101no—
375C3×Q16C4×D514yescomplemented
376C2×C3×C8—51no—
377C2×C3×C8C22×D511yescharacteristic
378C3×C4⋊C4—51no—
379C3×C4⋊C4—51no—
380C3×C4⋊C4—101no—
381C3×C4⋊C4—101no—
382C3×C42—51no—
383C3×C42—101no—
384C3×C42—101no—
385#216—54no—
386#215—54no—
387(C2×D3)⋊C4—51no—
388(C2×D3)⋊C4—51no—
389(C2×D3)⋊C4—101no—
390(C2×D3)⋊C4—101no—
391(C2×D3)⋊C4—101no—
392C4⋊Dic3—21no—
393C4⋊Dic3—21no—
394C4⋊Dic3C22×D511yescharacteristic
395C4⋊Dic3C22×D511yescharacteristic
396Dic3⋊C4—51no—
397Dic3⋊C4—51no—
398Dic3⋊C4—101no—
399Dic3⋊C4—101no—
400Dic3⋊C4—101no—
401C4×Dic3—21no—
402C4×Dic3—21no—
403C4×Dic3—101no—
404C4×Dic3—101no—
405C4×Dic3—101no—
406C4×Dic3C22×D511yescharacteristic
407C2×C3⋊C8—51no—
408C2×C3⋊C8C22×D511yescharacteristic
409#204—54no—
410ΓL(1,𝔽25)—54no—
411D3×C8—102no—
412C22×D5—31no—
413C22×D5—61no—
414C5×Q8—21no—
415C5×Q8—21no—
416C5×Q8C2×C4×D312yes—
417C5×Q8C2×C4×D312yes—
418C2×C4×C5—21no—
419C2×C4×C5—21no—
420C2×C4×C5—31no—
421C2×C4×C5—31no—
422C2×C4×C5—61no—
423C2×C4×C5—61no—
424C2×C4×C5C23×D311yescharacteristic
425C2×Dic5—21no—
426C2×Dic5—21no—
427C2×Dic5—31no—
428C2×Dic5—61no—
429C2×Dic5D3×D411yescharacteristic
430C2×Dic5D3×D411yescharacteristic
431D20—32no—
432D20—64no—
433D20—61no—
434C4×D5—32no—
435C4×D5—32no—
436C4×D5—64no—
437C4×D5—64no—
438C4×D5—241no—
439C4×D5—241no—
440C4×D5—241no—
441Dic10—32no—
442Dic10—64no—
443Dic10—61no—
444C5×C8—61no—
445C5×C8C2×C4×D312yes—
446#168—21no—
447#168—32no—
448C2×#41—151no—
449C22×Q8—151no—
450#137—158no—
451C2×Q16—151no—
452C2×Q16D3×D511yescharacteristic
453C2×SD16—151no—
454C2×SD16—151no—
455C2×M16—151no—
456C22×C8—151no—
457C4×Q8—31no—
458C4×Q8—31no—
459C4×Q8—51no—
460C4×Q8—51no—
461C4×Q8—151no—
462C4×Q8—151no—
463C4×Q8—301no—
464C4×Q8—301no—
465C4×D4—151no—
466C4×D4—151no—
467C4×D4—301no—
468C4×D4—301no—
469#117—151no—
470#117—151no—
471#117—301no—
472#117—301no—
473#117—301no—
474C2×C4⋊C4—151no—
475C2×C4⋊C4—301no—
476#107—51no—
477#107—151no—
478#106—31no—
479#106—151no—
480#103—31no—
481#103—31no—
482#103—51no—
483#103—51no—
484#103—151no—
485#103—151no—
486#102—151no—
487#102—151no—
488#97—31no—
489#97—151no—
490C4×C8—51no—
491C4×C8—151no—
492C2×C3×C5C2×C4×D411yescharacteristic
493C2×C3×C5#57711yescharacteristic
494C2×C3×C5#57811yescharacteristic
495D15—22no—
496D15—42no—
497C22×D3—51no—
498C22×D3—101no—
499C3×Q8—21no—
500C3×Q8—21no—
501C3×Q8C2×C4×D512yes—
502C3×Q8C2×C4×D512yes—
503C2×C3×C4—21no—
504C2×C3×C4—21no—
505C2×C3×C4—51no—
506C2×C3×C4—51no—
507C2×C3×C4—101no—
508C2×C3×C4—101no—
509C2×C3×C4C23×D511yescharacteristic
510C2×Dic3—21no—
511C2×Dic3—21no—
512C2×Dic3—51no—
513C2×Dic3—101no—
514C2×Dic3D4×D511yescharacteristic
515C2×Dic3D4×D511yescharacteristic
516D12—52no—
517D12—104no—
518D12—101no—
519C4×D3—52no—
520C4×D3—52no—
521C4×D3—104no—
522C4×D3—104no—
523C4×D3—401no—
524C4×D3—401no—
525C4×D3—401no—
526Dic6—52no—
527Dic6—104no—
528Dic6—101no—
529C3×C8—101no—
530C3×C8C2×C4×D512yes—
531C3⋊C8—21no—
532C3⋊C8—52no—
533C22×C5C2×D3×D411yescharacteristic
534C2×D5—32no—
535C2×D5—61no—
536C2×D5—61no—
537C2×D5—62no—
538C2×D5—121no—
539C2×D5—121no—
540C4×C5—22no—
541C4×C5—22no—
542C4×C5—61no—
543C4×C5—61no—
544C4×C5—121no—
545C4×C5—121no—
546C4×C5C22×C4×D311yescharacteristic
547C4×C5D3×#4111yescharacteristic
548Dic5—21no—
549Dic5—21no—
550Dic5—32no—
551Dic5—41no—
552Dic5—41no—
553Dic5—62no—
554#41—154no—
555#41—302no—
556C2×Q8—154no—
557C2×Q8—151no—
558C2×Q8—301no—
559C2×Q8—302no—
560C2×Q8C2×D3×D511yescharacteristic
561C2×Q8C2×D3×D511yescharacteristic
562C2×D4—151no—
563C2×D4—301no—
564C22×C4—151no—
565C22×C4—301no—
566C22×C4—301no—
567C22×C4—301no—
568C22×C4—301no—
569C22×C4—301no—
570Q16—154no—
571Q16#120014yescomplemented
572SD16—154no—
573SD16—154no—
574M16—154no—
575C2×C8—31no—
576C2×C8—51no—
577C2×C8—151no—
578C2×C8—302no—
579C2×C8C2×D3×D511yescharacteristic
580C4⋊C4—31no—
581C4⋊C4—31no—
582C4⋊C4—51no—
583C4⋊C4—51no—
584C4⋊C4—61no—
585C4⋊C4—61no—
586C4⋊C4—101no—
587C4⋊C4—101no—
588C4⋊C4—151no—
589C4⋊C4—151no—
590C4⋊C4—301no—
591C4⋊C4—301no—
592C4⋊C4—301no—
593C22⋊C4—151no—
594C22⋊C4—151no—
595C22⋊C4—301no—
596C22⋊C4—301no—
597C22⋊C4—301no—
598C42—31no—
599C42—51no—
600C42—61no—
601C42—61no—
602C42—101no—
603C42—101no—
604C42—301no—
605C42—301no—
606C42—301no—
607C3×C5#294711yescharacteristic, complemented, Hall
608C22×C3C2×D4×D511yescharacteristic
609C2×D3—52no—
610C2×D3—101no—
611C2×D3—101no—
612C2×D3—102no—
613C2×D3—201no—
614C2×D3—201no—
615C3×C4—22no—
616C3×C4—22no—
617C3×C4—101no—
618C3×C4—101no—
619C3×C4—201no—
620C3×C4—201no—
621C3×C4C22×C4×D511yescharacteristic
622C3×C4D5×#4111yescharacteristic
623Dic3—21no—
624Dic3—21no—
625Dic3—41no—
626Dic3—41no—
627Dic3—52no—
628Dic3—102no—
629C2×C5C4×D3×D411yescharacteristic
630C2×C5#576611yescharacteristic
631C2×C5#577011yescharacteristic
632D5—62no—
633D5—122no—
634C23—151no—
635C23—301no—
636Q8—21no—
637Q8—21no—
638Q8—152no—
639Q8—304no—
640Q8—301no—
641Q8C2×#120012yes—
642Q8C2×#120012yes—
643D4—152no—
644D4—304no—
645D4—301no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—31no—
649C2×C4—31no—
650C2×C4—51no—
651C2×C4—51no—
652C2×C4—61no—
653C2×C4—61no—
654C2×C4—101no—
655C2×C4—101no—
656C2×C4—151no—
657C2×C4—152no—
658C2×C4—152no—
659C2×C4—304no—
660C2×C4—304no—
661C2×C4—301no—
662C2×C4—1201no—
663C2×C4—1201no—
664C2×C4—1201no—
665C2×C4C22×D3×D511yescharacteristic
666C8—61no—
667C8—101no—
668C8—152no—
669C8C2×#120012yes—
670C2×C3C4×D4×D511yescharacteristic
671C2×C3#6611211yescharacteristic
672C2×C3#6611311yescharacteristic
673D3—102no—
674D3—202no—
675C5#7971911yescharacteristic, complemented, Sylow
676C22—152no—
677C22—301no—
678C22—301no—
679C22—302no—
680C22—601no—
681C22—601no—
682C22D3×D4×D511yescharacteristic
683C4—22no—
684C4—22no—
685C4—61no—
686C4—61no—
687C4—101no—
688C4—101no—
689C4—121no—
690C4—121no—
691C4—152no—
692C4—201no—
693C4—201no—
694C4—302no—
695C4#9202411yescharacteristic
696C4C22×#120011yescharacteristic
697C3#1061156111yescharacteristic, complemented, Sylow
698C2—302no—
699C2—602no—
700C2#1175410711yescharacteristic
701C2#1175411411yescharacteristic
702C2D4×#120011yescharacteristic
subgroupG/H
centerC22D3×D4×D5
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×Q16C22
socleC22×C3×C5C22×D4
prime radicalC2×#6821C22
cosocleC2×C3×C4×C5C24
hypercenterC2×Q16D3×D5
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
3. #11755124C2semidirect
4. C5×#5184C2semidirect
5. C3×Q16×Dic5C2semidirect
6. #11749257C2semidirect
7. #11749250C2semidirect
8. #11749232C2semidirect
9. #11749225C2semidirect
11. #11749203C2semidirect
12. #11748053C2semidirect
13. #11747800C2semidirect
14. #11747760C2semidirect
15. #11747640C2semidirect
22. #91807C4semidirect
66. Q16×Dic5D3semidirect
82. Q8×D15C2×C4semidirect
93. C3×C5×Q16C2×C4semidirect
95. #6718C2×C4semidirect
96. #6717C2×C4semidirect
98. #6706C2×C4semidirect
99. #6705C2×C4semidirect
134. #5184D5semidirect
286. C5×Q16C4×D3semidirect
375. C3×Q16C4×D5semidirect
571. Q16#1200semidirect
607. C3×C5#2947semidirect
675. C5#79719semidirect
697. C3#10611561semidirect

Maximal subgroups

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

subgroupindex
C2×#918072
#117551492
#117551242
C5×#51842
C3×Q16×Dic52
#117492572
#117492502
#117492322
#117492252
#117492152
#117492032
#117480532
#117478002
#117477602
#117476402
#106115613
#797195