Finite Group Databasebeta

← #14424800

Subgroups — #14424800

Order 1920.

1424 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92101C211yesmaximal, characteristic, complemented
2C2×C3×C5×#135C211yesmaximal, characteristic, complemented
3#11756592C211yesmaximal, characteristic, complemented
4C2×#91840C211yesmaximal, characteristic, complemented
5#11751333C211yesmaximal, characteristic, complemented
6#11751287C212yesmaximal, complemented
7#11751274C212yesmaximal, complemented
8#11751271C212yesmaximal, complemented
9#11751259C212yesmaximal, complemented
10#11751211C211yesmaximal, characteristic, complemented
11C2×#91100C211yesmaximal, characteristic, complemented
12#10615644—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C2×C3×C5×#41C2211yescharacteristic
15#92101—24no—
16#92101—24no—
17C2×Q8×D15C2212yes—
18C2×#6819—22no—
19C2×#6819C2212yes—
20C2×#6817—21no—
21C2×#6817C2211yescharacteristic
22C22×Dic30—21no—
23C22×Dic30C2211yescharacteristic
24C3×C5×#135—24no—
25C2×C3×C5×Q16C2211yescharacteristic
26C2×C3×C5×SD16C2211yescharacteristic
27C2×C3×C5×SD16C2211yescharacteristic
28C2×C3×C5×D8C2211yescharacteristic
29C22×C3×C5×C8C2211yescharacteristic
30#91840—24no—
31#91835C2212yes—
32C2×#6718C2212yes—
33#91828—22no—
34#91827C2212yes—
35C2×#6716C2212yes—
36#91819—21no—
37C2×C4⋊Dic15C2211yescharacteristic
38#91813—21no—
39C2×#6710C2211yescharacteristic
40#91119C2212yes—
41#91116C2212yes—
42#91104C2212yes—
43#91104C2212yes—
44#91100C2212yes—
45#91100C2212yes—
46#83913—51nomaximal, Hall
47C2×#4150—31no—
48C2×C5×#135D311yescharacteristic, complemented
49#66167—31no—
50C2×#4097—31no—
51#65521—31no—
52#65475—32no—
53#65462—32no—
54#65459—32no—
55#65447—32no—
56#65399—31no—
57C2×#3948—31no—
58C3×C5×#41—22no—
59C3×C5×#41—22no—
60C3×C5×#41—22no—
61C3×C5×#41D414yescomplemented
62C2×C3×C5×Q8C2311yescharacteristic
63C2×C3×C5×Q8C2311yescharacteristic
64C2×C3×C5×D4—21no—
65C2×C3×C5×D4—21no—
66C2×C3×C5×D4C2311yescharacteristic
67C2×C3×C5×D4C2311yescharacteristic
68C22×C3×C4×C5—21no—
69C22×C3×C4×C5—21no—
70C22×C3×C4×C5C2311yescharacteristic
71C2×#1220—22no—
72C2×#1220—22no—
73C2×#1220D412yes—
74C22×Dic15—21no—
75C22×Dic15—22no—
76C22×Dic15—21no—
77Q8×D15—24no—
78Q8×D15—24no—
79#6819—216no—
80#6819—24no—
81#6819—24no—
82#6817—28no—
83#6817—22no—
84#6817—22no—
85C2×D60D411yescharacteristic
86C2×C4×D15—22no—
87C2×C4×D15C2312yes—
88C2×Dic30—28no—
89C2×Dic30—21no—
90C2×Dic30—22no—
91C2×Dic30—22no—
92C2×Dic30D411yescharacteristic
93C2×Dic30D414yescomplemented
94C2×Dic30C2312yes—
95C3×C5×Q16—22no—
96C3×C5×SD16—22no—
97C3×C5×SD16—22no—
98C3×C5×D8—22no—
99C2×C3×C5×C8—22no—
100C2×C3×C5×C8C2311yescharacteristic
101C2×C3×C5×C8C2311yescharacteristic
102#6719—22no—
103#6718—24no—
104#6716—24no—
105#6714—22no—
106C4⋊Dic15—21no—
107C4⋊Dic15C2312yes—
108#6712—22no—
109#6710—22no—
110C2×C3⋊#168C2312yes—
111C2×#1010—51no—
112C2×C3×#135D511yescharacteristic, complemented
113#5833—51no—
114C2×#951—51no—
115#5232—51no—
116#5186—52no—
117#5173—52no—
118#5170—52no—
119#5158—52no—
120#5110—51no—
121C2×#816—51no—
122C2×C5×#41C2×D311yescharacteristic
123C2×C5×#41C2×D311yescharacteristic
124#4150—64no—
125#4150—64no—
126C2×Q8×D5—32no—
127C2×#711—32no—
128C2×#711—62no—
129C2×#709—31no—
130C2×#709—61no—
131C22×Dic10—31no—
132C22×Dic10—61no—
133C5×#135—24no—
134C2×C5×Q16C2×D311yescharacteristic
135C2×C5×SD16C2×D311yescharacteristic
136C2×C5×SD16C2×D311yescharacteristic
137C2×C5×D8C2×D311yescharacteristic
138C22×C5×C8C2×D311yescharacteristic
139#4097—64no—
140#4092—32no—
141C2×#689—32no—
142#4085—62no—
143#4084—32no—
144C2×#687—32no—
145#4076—61no—
146C2×C4⋊Dic5—31no—
147#4070—61no—
148C2×#681—31no—
149#3967—32no—
150#3964—32no—
151#3952—32no—
152#3952—32no—
153#3948—32no—
154#3948—32no—
155#3013—151noSylow
156C23×C3×C5—21no—
157C23×C3×C5—21no—
158C23×C3×C5C2×D411yescharacteristic
159C22×D15C2×D412yes—
160C3×C5×Q8—21no—
161C3×C5×Q8—21no—
162C3×C5×Q8—21no—
163C3×C5×Q8C2×D412yes—
164C3×C5×D4—24no—
165C3×C5×D4—21no—
166C3×C5×D4—21no—
167C3×C5×D4—21no—
168C3×C5×D4—42no—
169C3×C5×D4C2×D412yes—
170C2×C3×C4×C5—21no—
171C2×C3×C4×C5—24no—
172C2×C3×C4×C5—21no—
173C2×C3×C4×C5—42no—
174C2×C3×C4×C5C2×D411yescharacteristic
175C2×C3×C4×C5C2×D412yes—
176C2×C3×C4×C5C2×D412yes—
177C2×C3×C4×C5C2411yescharacteristic
178#1220—28no—
179#1220—24no—
180#1220—82no—
181C2×Dic15—28no—
182C2×Dic15—24no—
183C2×Dic15—22no—
184C2×Dic15—22no—
185C2×Dic15—81no—
186C2×Dic15C2×D412yes—
187D60—22no—
188C4×D15—28no—
189C4×D15—22no—
190C4×D15—22no—
191Dic30—28no—
192Dic30—22no—
193Dic30—24no—
194Dic30—22no—
195Dic30C2×D414yes—
196C3×C5×C8—21no—
197C3×C5×C8—21no—
198C3⋊#168—22no—
199C2×C3×#41C2×D511yescharacteristic
200C2×C3×#41C2×D511yescharacteristic
201#1010—104no—
202#1010—104no—
203C2×D3×Q8—52no—
204C2×#237—52no—
205C2×#237—102no—
206C2×#235—51no—
207C2×#235—101no—
208C22×Dic6—51no—
209C22×Dic6—101no—
210C3×#135—24no—
211C2×C3×Q16C2×D511yescharacteristic
212C2×C3×SD16C2×D511yescharacteristic
213C2×C3×SD16C2×D511yescharacteristic
214C2×C3×D8C2×D511yescharacteristic
215C22×C3×C8C2×D511yescharacteristic
216#951—104no—
217#946—52no—
218C2×#216—52no—
219#939—102no—
220#938—52no—
221C2×#214—52no—
222#930—101no—
223C2×C4⋊Dic3—51no—
224#924—101no—
225C2×#208—51no—
226#835—52no—
227#832—52no—
228(C2×D3)⋊C8—52no—
229(C2×D3)⋊C8—52no—
230#816—52no—
231#816—52no—
232C5×#41—22no—
233C5×#41—22no—
234C5×#41—22no—
235C5×#41#6714yescomplemented
236C2×C5×Q8C22×D311yescharacteristic
237C2×C5×Q8C22×D311yescharacteristic
238C2×C5×D4—21no—
239C2×C5×D4—21no—
240C2×C5×D4C22×D311yescharacteristic
241C2×C5×D4C22×D311yescharacteristic
242C22×C4×C5—21no—
243C22×C4×C5—21no—
244C22×C4×C5C22×D311yescharacteristic
245C2×C22⋊D5—32no—
246C2×C22⋊D5—62no—
247C2×C22⋊D5—62no—
248C22×Dic5—62no—
249C22×Dic5—61no—
250C22×Dic5—61no—
251Q8×D5—64no—
252Q8×D5—64no—
253#711—64no—
254#711—616no—
255#711—64no—
256#709—68no—
257#709—62no—
258#709—62no—
259C2×D20—31no—
260C2×C4×D5—32no—
261C2×C4×D5—62no—
262C2×Dic10—34no—
263C2×Dic10—32no—
264C2×Dic10—31no—
265C2×Dic10—62no—
266C2×Dic10—68no—
267C2×Dic10—61no—
268C2×Dic10—62no—
269C5×Q16—22no—
270C5×SD16—22no—
271C5×SD16—22no—
272C5×D8—22no—
273C2×C5×C8—22no—
274C2×C5×C8C22×D311yescharacteristic
275C2×C5×C8C22×D311yescharacteristic
276C22⋊Dic5—62no—
277#689—64no—
278#687—64no—
279#685—62no—
280C4⋊Dic5—32no—
281C4⋊Dic5—61no—
282#683—62no—
283#681—62no—
284C2×#168—32no—
285C2×2−1+4—151no—
286C2×#137—151no—
287C2×#135D1511yescharacteristic, complemented
288#537—151no—
289#452—152no—
290#451—152no—
291#449—152no—
292#448—152no—
293#417—151no—
294C2×#103—151no—
295#408—151no—
296C22×C3×C5—21no—
297C22×C3×C5—21no—
298C22×C3×C5—22no—
299C22×C3×C5—41no—
300C22×C3×C5—41no—
301C22×C3×C5—41no—
302C22×C3×C5—41no—
303C22×C3×C5—41no—
304C22×C3×C5#12012yes—
305C22×C3×C5C22×D411yescharacteristic
306C2×D15—22no—
307C2×D15—22no—
308C2×D15—22no—
309C3×C4×C5—22no—
310C3×C4×C5—41no—
311C3×C4×C5#12012yes—
312C3×C4×C5C22×D411yescharacteristic
313C3×C4×C5C22×D411yescharacteristic
314Dic15—22no—
315Dic15—24no—
316Dic15—42no—
317C3×#41—22no—
318C3×#41—22no—
319C3×#41—22no—
320C3×#41C22⋊D514yescomplemented
321C2×C3×Q8C22×D511yescharacteristic
322C2×C3×Q8C22×D511yescharacteristic
323C2×C3×D4—21no—
324C2×C3×D4—21no—
325C2×C3×D4C22×D511yescharacteristic
326C2×C3×D4C22×D511yescharacteristic
327C22×C3×C4—21no—
328C22×C3×C4—21no—
329C22×C3×C4C22×D511yescharacteristic
330C2×#67—52no—
331C2×#67—102no—
332C2×#67—102no—
333C22×Dic3—102no—
334C22×Dic3—101no—
335C22×Dic3—101no—
336D3×Q8—104no—
337D3×Q8—104no—
338#237—104no—
339#237—1016no—
340#237—104no—
341#235—108no—
342#235—102no—
343#235—102no—
344C2×D12—51no—
345C2×C4×D3—52no—
346C2×C4×D3—102no—
347C2×Dic6—54no—
348C2×Dic6—52no—
349C2×Dic6—51no—
350C2×Dic6—102no—
351C2×Dic6—108no—
352C2×Dic6—101no—
353C2×Dic6—102no—
354C3×Q16—22no—
355C3×SD16—22no—
356C3×SD16—22no—
357C3×D8—22no—
358C2×C3×C8—22no—
359C2×C3×C8C22×D511yescharacteristic
360C2×C3×C8C22×D511yescharacteristic
361#217—102no—
362#216—104no—
363#214—104no—
364(C2×D3)⋊C4—102no—
365C4⋊Dic3—52no—
366C4⋊Dic3—101no—
367Dic3⋊C4—102no—
368#208—102no—
369C2×C3⋊C8—52no—
370C23×C5—21no—
371C23×C5—21no—
372C23×C5D3×D411yescharacteristic
373C22×D5—32no—
374C5×Q8—21no—
375C5×Q8—21no—
376C5×Q8—21no—
377C5×Q8C2×#6712yes—
378C5×D4—24no—
379C5×D4—21no—
380C5×D4—21no—
381C5×D4—21no—
382C5×D4—42no—
383C5×D4C2×#6712yes—
384C2×C4×C5—21no—
385C2×C4×C5—24no—
386C2×C4×C5—21no—
387C2×C4×C5—42no—
388C2×C4×C5D3×D411yescharacteristic
389C2×C4×C5D3×D412yes—
390C2×C4×C5C2×#6712yes—
391C2×C4×C5C23×D311yescharacteristic
392C22⋊D5—68no—
393C22⋊D5—64no—
394C22⋊D5—242no—
395C2×Dic5—32no—
396C2×Dic5—68no—
397C2×Dic5—64no—
398C2×Dic5—62no—
399C2×Dic5—62no—
400C2×Dic5—241no—
401D20—62no—
402C4×D5—68no—
403C4×D5—62no—
404C4×D5—62no—
405Dic10—34no—
406Dic10—68no—
407Dic10—64no—
408Dic10—62no—
409Dic10—62no—
410C5×C8—21no—
411C5×C8—21no—
412#168—62no—
4132−1+4—304no—
4142−1+4—304no—
415C2×#41—151no—
416C2×#41—152no—
417C2×#41—301no—
418C2×#41—302no—
419C2×#41C2×D1511yescharacteristic
420C2×#41C2×D1511yescharacteristic
421C22×Q8—152no—
422C22×Q8—151no—
423C22×Q8—301no—
424#137—304no—
425#135—24no—
426C2×Q16—152no—
427C2×Q16C2×D1511yescharacteristic
428C2×SD16—152no—
429C2×SD16C2×D1511yescharacteristic
430C2×SD16C2×D1511yescharacteristic
431C2×D8C2×D1511yescharacteristic
432C2×M16—151no—
433C22×C8C2×D1511yescharacteristic
434C22⋊Q8—152no—
435C22⋊Q8—301no—
436#121—152no—
437#121—301no—
438#121—302no—
439C2×C4⋊C4—151no—
440#103—152no—
441#103—152no—
442#103—152no—
443#102—152no—
444C22⋊C8—152no—
445C22⋊C8—152no—
446C2×C3×C5—22no—
447C2×C3×C5—42no—
448C2×C3×C5—81no—
449C2×C3×C5C2×#12011yescharacteristic
450C2×C3×C5#57711yescharacteristic
451C2×C3×C5#57811yescharacteristic
452D15—42no—
453C23×C3—21no—
454C23×C3—21no—
455C23×C3D4×D511yescharacteristic
456C22×D3—52no—
457C3×Q8—21no—
458C3×Q8—21no—
459C3×Q8—21no—
460C3×Q8C2×C22⋊D512yes—
461C3×D4—24no—
462C3×D4—21no—
463C3×D4—21no—
464C3×D4—21no—
465C3×D4—42no—
466C3×D4C2×C22⋊D512yes—
467C2×C3×C4—21no—
468C2×C3×C4—24no—
469C2×C3×C4—21no—
470C2×C3×C4—42no—
471C2×C3×C4D4×D512yes—
472C2×C3×C4D4×D511yescharacteristic
473C2×C3×C4C2×C22⋊D512yes—
474C2×C3×C4C23×D511yescharacteristic
475#67—108no—
476#67—104no—
477#67—402no—
478C2×Dic3—52no—
479C2×Dic3—108no—
480C2×Dic3—102no—
481C2×Dic3—104no—
482C2×Dic3—102no—
483C2×Dic3—401no—
484D12—102no—
485C4×D3—108no—
486C4×D3—102no—
487C4×D3—102no—
488Dic6—54no—
489Dic6—108no—
490Dic6—102no—
491Dic6—104no—
492Dic6—102no—
493C3×C8—21no—
494C3×C8—21no—
495C3⋊C8—102no—
496C22×C5—21no—
497C22×C5—21no—
498C22×C5—22no—
499C22×C5—41no—
500C22×C5—41no—
501C22×C5—41no—
502C22×C5—41no—
503C22×C5—41no—
504C22×C5#93712yes—
505C22×C5C2×D3×D411yescharacteristic
506C2×D5—62no—
507C2×D5—62no—
508C2×D5—62no—
509C4×C5—22no—
510C4×C5—41no—
511C4×C5#93712yes—
512C4×C5C2×D3×D411yescharacteristic
513C4×C5C22×#6711yescharacteristic
514Dic5—64no—
515Dic5—62no—
516Dic5—122no—
517#41—22no—
518#41—22no—
519#41—22no—
520#41—308no—
521#41—304no—
522#41—3016no—
523#41—304no—
524#41—302no—
525#41—302no—
526#41#122014yescomplemented
527C2×Q8—151no—
528C2×Q8—154no—
529C2×Q8—152no—
530C2×Q8—302no—
531C2×Q8—308no—
532C2×Q8—304no—
533C2×Q8—304no—
534C2×Q8—301no—
535C2×Q8—302no—
536C2×Q8C22×D1511yescharacteristic
537C2×Q8C22×D1511yescharacteristic
538C2×D4—21no—
539C2×D4—21no—
540C2×D4—152no—
541C2×D4—151no—
542C2×D4—302no—
543C2×D4—302no—
544C2×D4C22×D1511yescharacteristic
545C2×D4C22×D1511yescharacteristic
546C22×C4—21no—
547C22×C4—21no—
548C22×C4—152no—
549C22×C4—302no—
550C22×C4—302no—
551C22×C4—301no—
552C22×C4—301no—
553C22×C4C22×D1511yescharacteristic
554Q16—22no—
555Q16—304no—
556SD16—22no—
557SD16—22no—
558SD16—304no—
559D8—22no—
560M16—302no—
561C2×C8—22no—
562C2×C8—152no—
563C2×C8C22×D1511yescharacteristic
564C2×C8C22×D1511yescharacteristic
565C4⋊C4—152no—
566C4⋊C4—302no—
567C4⋊C4—301no—
568C22⋊C4—302no—
569C22⋊C4—302no—
570C3×C5#301311yescharacteristic, complemented, Hall
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—22no—
574C22×C3—41no—
575C22×C3—41no—
576C22×C3—41no—
577C22×C3—41no—
578C22×C3—41no—
579C22×C3#408312yes—
580C22×C3C2×D4×D511yescharacteristic
581C2×D3—102no—
582C2×D3—102no—
583C2×D3—102no—
584C3×C4—22no—
585C3×C4—41no—
586C3×C4#408312yes—
587C3×C4C2×D4×D511yescharacteristic
588C3×C4C22×C22⋊D511yescharacteristic
589Dic3—104no—
590Dic3—102no—
591Dic3—202no—
592C2×C5—22no—
593C2×C5—42no—
594C2×C5—81no—
595C2×C5#577011yescharacteristic
596C2×C5#577111yescharacteristic
597C2×C5C2×#93711yescharacteristic
598D5—122no—
599C23—21no—
600C23—21no—
601C23—152no—
602C23D4×D1511yescharacteristic
603Q8—21no—
604Q8—21no—
605Q8—21no—
606Q8—154no—
607Q8—308no—
608Q8—302no—
609Q8—304no—
610Q8—302no—
611Q8C2×#122012yes—
612D4—24no—
613D4—21no—
614D4—21no—
615D4—21no—
616D4—42no—
617D4—308no—
618D4—304no—
619D4—302no—
620D4—1202no—
621D4C2×#122012yes—
622C2×C4—21no—
623C2×C4—24no—
624C2×C4—21no—
625C2×C4—42no—
626C2×C4—152no—
627C2×C4—308no—
628C2×C4—308no—
629C2×C4—304no—
630C2×C4—302no—
631C2×C4—302no—
632C2×C4—302no—
633C2×C4—302no—
634C2×C4—1201no—
635C2×C4D4×D1511yescharacteristic
636C2×C4D4×D1512yes—
637C2×C4C2×#122012yes—
638C2×C4C23×D1511yescharacteristic
639C8—21no—
640C8—21no—
641C8—302no—
642C2×C3—22no—
643C2×C3—42no—
644C2×C3—81no—
645C2×C3#6610411yescharacteristic
646C2×C3#6610511yescharacteristic
647C2×C3C2×#408311yescharacteristic
648D3—202no—
649C5#8391311yescharacteristic, complemented, Sylow
650C22—21no—
651C22—21no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—302no—
659C22—302no—
660C22—302no—
661C22#9182612yes—
662C22C2×D4×D1511yescharacteristic
663C4—22no—
664C4—41no—
665C4—302no—
666C4—304no—
667C4—602no—
668C4#9182612yes—
669C4C2×D4×D1511yescharacteristic
670C4C22×#122011yescharacteristic
671C3#1061564411yescharacteristic, complemented, Sylow
672C2—22no—
673C2—42no—
674C2—81no—
675C2—602no—
676C2#1175652911yescharacteristic
677C2#1175653011yescharacteristic
678C2C2×#9182611yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×#135C2
socleC22×C3×C5C22×D4
hypercenterC2×#135D15
hypocenterC3×C5#3013
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#92101C2semidirect
2. C2×C3×C5×#135C2semidirect
3. #11756592C2semidirect
4. C2×#91840C2semidirect
5. #11751333C2semidirect
6. #11751287C2semidirect
7. #11751274C2semidirect
8. #11751271C2semidirect
9. #11751259C2semidirect
10. #11751211C2semidirect
11. C2×#91100C2semidirect
48. C2×C5×#135D3semidirect
61. C3×C5×#41D4semidirect
93. C2×Dic30D4semidirect
112. C2×C3×#135D5semidirect
235. C5×#41#67semidirect
287. C2×#135D15semidirect
320. C3×#41C22⋊D5semidirect
526. #41#1220semidirect
570. C3×C5#3013semidirect
649. C5#83913semidirect
671. C3#10615644semidirect

Maximal subgroups

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

subgroupindex
C2×#921012
C2×C3×C5×#1352
#117565922
C2×#918402
#117513332
#117512872
#117512742
#117512712
#117512592
#117512112
C2×#911002
#106156443
#839135