Finite Group Databasebeta

← #14424803

Subgroups — #14424803

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
2C3×C5×#537C211yesmaximal, characteristic, complemented
3C2×#91840C211yesmaximal, characteristic, complemented
4C2×#91839C211yesmaximal, characteristic, complemented
5#11751335C211yesmaximal, characteristic, complemented
6#11751151C212yesmaximal, complemented
7#11751148C212yesmaximal, complemented
8#11751141C212yesmaximal, complemented
9#11751138C212yesmaximal, complemented
10C2×#91094C211yesmaximal, characteristic, complemented
11#11751067C211yesmaximal, characteristic, complemented
12#10615158—31nomaximal, Hall
13C2×C3×C5×#41C2211yescharacteristic
14C2×C3×C5×#41C2211yescharacteristic
15#92101—24no—
16#92101—24no—
17C2×Q8×D15—21no—
18C2×#6819—22no—
19C2×#6819—21no—
20C2×#6817C2212yes—
21C2×#6817C2211yescharacteristic
22C22×Dic30C2212yes—
23C22×Dic30C2211yescharacteristic
24C3×C5×C22⋊Q8—21no—
25C3×C5×C22⋊Q8C2212yes—
26C3×C5×#121—21no—
27C3×C5×#121—22no—
28C3×C5×#121C2212yes—
29C2×C3×C5×C4⋊C4C2211yescharacteristic
30#91840—24no—
31#91839—24no—
32C2×#6718C2211yescharacteristic
33C2×#6718C2212yes—
34C2×#6717C2211yescharacteristic
35C2×#6716C2211yescharacteristic
36C2×#6716C2212yes—
37C2×#6715C2211yescharacteristic
38C2×#6710C2211yescharacteristic
39C22×C3⋊#168C2211yescharacteristic
40#91114C2212yes—
41#91114C2212yes—
42#91094C2212yes—
43#91094C2212yes—
44#91094C2212yes—
45#91093C2212yes—
46#83427—51nomaximal, Hall
47C2×#4150—31no—
48C5×#537D311yescharacteristic, complemented
49C2×#4097—31no—
50C2×#4096—31no—
51#65523—31no—
52#65339—32no—
53#65336—32no—
54#65329—32no—
55#65326—32no—
56C2×#3942—31no—
57#65255—31no—
58C3×C5×#41—22no—
59C3×C5×#41—22no—
60C3×C5×#41—22no—
61C3×C5×#41—22no—
62C2×C3×C5×Q8D411yescharacteristic
63C2×C3×C5×Q8C2311yescharacteristic
64C2×C3×C5×D4—21no—
65C2×C3×C5×D4—22no—
66C2×C3×C5×D4D412yes—
67C2×C3×C5×D4D411yescharacteristic
68C2×C3×C5×D4C2311yescharacteristic
69C22×C3×C4×C5—21no—
70C22×C3×C4×C5—21no—
71C22×C3×C4×C5C2312yes—
72C22×C3×C4×C5C2311yescharacteristic
73C2×#1220—22no—
74C2×#1220—21no—
75C22×Dic15—22no—
76C22×Dic15—21no—
77Q8×D15—28no—
78#6819—216no—
79#6819—28no—
80#6817—22no—
81#6817—24no—
82#6817—24no—
83#6817D414yescomplemented
84C2×D60C2311yescharacteristic
85C2×C4×D15—22no—
86C2×C4×D15—21no—
87C2×Dic30—22no—
88C2×Dic30—21no—
89C2×Dic30—22no—
90C2×Dic30—24no—
91C2×Dic30—24no—
92C2×Dic30D414yescomplemented
93C2×Dic30C2311yescharacteristic
94C2×Dic30C2312yes—
95C3×C5×C4⋊C4—21no—
96C3×C5×C4⋊C4—22no—
97C3×C5×C4⋊C4C2312yes—
98C3×C5×C22⋊C4—22no—
99C3×C5×C22⋊C4—22no—
100#6718—22no—
101#6718—24no—
102#6717—22no—
103#6716—22no—
104#6716—24no—
105#6715—22no—
106#6710—22no—
107C2×C3⋊#168—22no—
108C2×C3⋊#168C2311yescharacteristic
109C2×C3⋊#168C2311yescharacteristic
110C2×C3⋊#168C2312yes—
111C2×#1010—51no—
112C3×#537D511yescharacteristic, complemented
113C2×#951—51no—
114C2×#950—51no—
115#5234—51no—
116#5050—52no—
117#5047—52no—
118#5040—52no—
119#5037—52no—
120C2×#810—51no—
121#4966—51no—
122C2×C5×#41C2×D311yescharacteristic
123C2×C5×#41C2×D311yescharacteristic
124#4150—64no—
125#4150—64no—
126C2×Q8×D5—61no—
127C2×#711—62no—
128C2×#711—61no—
129C2×#709—31no—
130C2×#709—32no—
131C22×Dic10—31no—
132C22×Dic10—32no—
133C5×C22⋊Q8—21no—
134C5×C22⋊Q8C2×D312yes—
135C5×#121—21no—
136C5×#121—22no—
137C5×#121C2×D312yes—
138C2×C5×C4⋊C4C2×D311yescharacteristic
139#4097—64no—
140#4096—64no—
141C2×#689—32no—
142C2×#689—31no—
143C2×#688—31no—
144C2×#687—32no—
145C2×#687—31no—
146C2×#686—31no—
147C2×#681—31no—
148C22×#168—31no—
149C22⋊#168—32no—
150C22⋊#168—32no—
151#3942—32no—
152#3942—32no—
153#3942—32no—
154#3941—32no—
155#3013—151noSylow
156C23×C3×C5—21no—
157C23×C3×C5C2×D412yes—
158C23×C3×C5C2×D411yescharacteristic
159C22×D15—21no—
160C3×C5×Q8—22no—
161C3×C5×Q8—21no—
162C3×C5×Q8—21no—
163C3×C5×D4—24no—
164C3×C5×D4—22no—
165C3×C5×D4—21no—
166C3×C5×D4—21no—
167C3×C5×D4—42no—
168C3×C5×D4—82no—
169C2×C3×C4×C5—22no—
170C2×C3×C4×C5—22no—
171C2×C3×C4×C5—22no—
172C2×C3×C4×C5—21no—
173C2×C3×C4×C5—42no—
174C2×C3×C4×C5—81no—
175C2×C3×C4×C5C2×D412yes—
176C2×C3×C4×C5C2×D411yescharacteristic
177C2×C3×C4×C5C2×D412yes—
178C2×C3×C4×C5C2×D412yes—
179C2×C3×C4×C5C2411yescharacteristic
180#1220—28no—
181#1220—24no—
182C2×Dic15—28no—
183C2×Dic15—24no—
184C2×Dic15—21no—
185C2×Dic15—22no—
186D60—21no—
187D60C2×D412yes—
188C4×D15—28no—
189C4×D15—24no—
190Dic30—28no—
191Dic30—24no—
192Dic30—22no—
193Dic30—21no—
194Dic30C2×D412yes—
195Dic30C2×D414yes—
196C3⋊#168—21no—
197C3⋊#168—21no—
198C3⋊#168—22no—
199C2×C3×#41C2×D511yescharacteristic
200C2×C3×#41C2×D511yescharacteristic
201#1010—104no—
202#1010—104no—
203C2×D3×Q8—101no—
204C2×#237—102no—
205C2×#237—101no—
206C2×#235—51no—
207C2×#235—52no—
208C22×Dic6—51no—
209C22×Dic6—52no—
210C3×C22⋊Q8—21no—
211C3×C22⋊Q8C2×D512yes—
212C3×#121—21no—
213C3×#121—22no—
214C3×#121C2×D512yes—
215C2×C3×C4⋊C4C2×D511yescharacteristic
216#951—104no—
217#950—104no—
218C2×#216—52no—
219C2×#216—51no—
220C2×#215—51no—
221C2×#214—52no—
222C2×#214—51no—
223C2×#213—51no—
224C2×#208—51no—
225C22×C3⋊C8—51no—
226#830—52no—
227#830—52no—
228#810—52no—
229#810—52no—
230#810—52no—
231#809—52no—
232C5×#41—22no—
233C5×#41—22no—
234C5×#41—22no—
235C5×#41—22no—
236C2×C5×Q8#6711yescharacteristic
237C2×C5×Q8C22×D311yescharacteristic
238C2×C5×D4—21no—
239C2×C5×D4—22no—
240C2×C5×D4#6712yes—
241C2×C5×D4#6711yescharacteristic
242C2×C5×D4C22×D311yescharacteristic
243C22×C4×C5—21no—
244C22×C4×C5—21no—
245C22×C4×C5C22×D312yes—
246C22×C4×C5C22×D311yescharacteristic
247C2×C22⋊D5—62no—
248C2×C22⋊D5—61no—
249C22×Dic5—62no—
250C22×Dic5—61no—
251Q8×D5—68no—
252#711—616no—
253#711—68no—
254#709—34no—
255#709—64no—
256#709—64no—
257#709—62no—
258C2×D20—31no—
259C2×C4×D5—62no—
260C2×C4×D5—61no—
261C2×Dic10—34no—
262C2×Dic10—32no—
263C2×Dic10—31no—
264C2×Dic10—64no—
265C2×Dic10—64no—
266C2×Dic10—62no—
267C2×Dic10—62no—
268C2×Dic10—61no—
269C5×C4⋊C4—21no—
270C5×C4⋊C4—22no—
271C5×C4⋊C4C22×D312yes—
272C5×C22⋊C4—22no—
273C5×C22⋊C4—22no—
274#689—62no—
275#689—64no—
276#688—62no—
277#687—62no—
278#687—64no—
279#686—62no—
280#681—62no—
281C2×#168—32no—
282C2×#168—31no—
283C2×#168—31no—
284C2×#168—62no—
285C2×2−1+4—151no—
286C2×#137—151no—
287C2×#135—151no—
288#537D1511yescharacteristic, complemented
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—22no—
300C22×C3×C5—22no—
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—41no—
308C2×D15—41no—
309C3×C4×C5—22no—
310C3×C4×C5—41no—
311C3×C4×C5—42no—
312C3×C4×C5#12012yes—
313C3×C4×C5C22×D411yescharacteristic
314C3×C4×C5C22×D411yescharacteristic
315Dic15—22no—
316Dic15—24no—
317C3×#41—22no—
318C3×#41—22no—
319C3×#41—22no—
320C3×#41—22no—
321C2×C3×Q8C22⋊D511yescharacteristic
322C2×C3×Q8C22×D511yescharacteristic
323C2×C3×D4—21no—
324C2×C3×D4—22no—
325C2×C3×D4C22⋊D512yes—
326C2×C3×D4C22⋊D511yescharacteristic
327C2×C3×D4C22×D511yescharacteristic
328C22×C3×C4—21no—
329C22×C3×C4—21no—
330C22×C3×C4C22×D512yes—
331C22×C3×C4C22×D511yescharacteristic
332C2×#67—102no—
333C2×#67—101no—
334C22×Dic3—102no—
335C22×Dic3—101no—
336D3×Q8—108no—
337#237—1016no—
338#237—108no—
339#235—54no—
340#235—104no—
341#235—104no—
342#235—102no—
343C2×D12—51no—
344C2×C4×D3—102no—
345C2×C4×D3—101no—
346C2×Dic6—54no—
347C2×Dic6—52no—
348C2×Dic6—51no—
349C2×Dic6—104no—
350C2×Dic6—104no—
351C2×Dic6—102no—
352C2×Dic6—102no—
353C2×Dic6—101no—
354C3×C4⋊C4—21no—
355C3×C4⋊C4—22no—
356C3×C4⋊C4C22×D512yes—
357C3×C22⋊C4—22no—
358C3×C22⋊C4—22no—
359#216—102no—
360#216—104no—
361#215—102no—
362#214—102no—
363#214—104no—
364#213—102no—
365#208—102no—
366C2×C3⋊C8—52no—
367C2×C3⋊C8—51no—
368C2×C3⋊C8—51no—
369C2×C3⋊C8—102no—
370C23×C5—21no—
371C23×C5C2×#6712yes—
372C23×C5C2×#6711yescharacteristic
373C22×D5—61no—
374C5×Q8—22no—
375C5×Q8—21no—
376C5×Q8—21no—
377C5×D4—24no—
378C5×D4—22no—
379C5×D4—21no—
380C5×D4—21no—
381C5×D4—42no—
382C5×D4—82no—
383C2×C4×C5—22no—
384C2×C4×C5—22no—
385C2×C4×C5—22no—
386C2×C4×C5—21no—
387C2×C4×C5—42no—
388C2×C4×C5—81no—
389C2×C4×C5D3×D412yes—
390C2×C4×C5D3×D412yes—
391C2×C4×C5C2×#6712yes—
392C2×C4×C5C2×#6711yescharacteristic
393C2×C4×C5C23×D311yescharacteristic
394C22⋊D5—68no—
395C22⋊D5—64no—
396C2×Dic5—68no—
397C2×Dic5—64no—
398C2×Dic5—62no—
399C2×Dic5—61no—
400D20—32no—
401D20—61no—
402C4×D5—68no—
403C4×D5—64no—
404Dic10—34no—
405Dic10—32no—
406Dic10—68no—
407Dic10—64no—
408Dic10—61no—
409Dic10—62no—
410#168—61no—
411#168—61no—
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—151no—
422C22×Q8—152no—
423C22×Q8—301no—
424#137—304no—
425#135—304no—
426C2×Q16—151no—
427C2×Q16—152no—
428C2×SD16—151no—
429C2×SD16—151no—
430C2×SD16—152no—
431C2×D8—151no—
432C2×M16—151no—
433C22×C8—151no—
434C22⋊Q8—21no—
435C22⋊Q8C2×D1512yes—
436#121—21no—
437#121—22no—
438#121C2×D1512yes—
439C2×C4⋊C4C2×D1511yescharacteristic
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×C3C2×C22⋊D512yes—
455C23×C3C2×C22⋊D511yescharacteristic
456C22×D3—101no—
457C3×Q8—22no—
458C3×Q8—21no—
459C3×Q8—21no—
460C3×D4—24no—
461C3×D4—21no—
462C3×D4—21no—
463C3×D4—22no—
464C3×D4—42no—
465C3×D4—82no—
466C2×C3×C4—22no—
467C2×C3×C4—22no—
468C2×C3×C4—22no—
469C2×C3×C4—21no—
470C2×C3×C4—42no—
471C2×C3×C4—81no—
472C2×C3×C4D4×D512yes—
473C2×C3×C4D4×D512yes—
474C2×C3×C4C2×C22⋊D512yes—
475C2×C3×C4C2×C22⋊D511yescharacteristic
476C2×C3×C4C23×D511yescharacteristic
477#67—108no—
478#67—104no—
479C2×Dic3—108no—
480C2×Dic3—102no—
481C2×Dic3—101no—
482C2×Dic3—104no—
483D12—52no—
484D12—101no—
485C4×D3—108no—
486C4×D3—104no—
487Dic6—54no—
488Dic6—52no—
489Dic6—108no—
490Dic6—101no—
491Dic6—102no—
492Dic6—104no—
493C3⋊C8—101no—
494C3⋊C8—101no—
495C3⋊C8—102no—
496C22×C5—21no—
497C22×C5—21no—
498C22×C5—22no—
499C22×C5—22no—
500C22×C5—22no—
501C22×C5—41no—
502C22×C5—41no—
503C22×C5—41no—
504C22×C5#93712yes—
505C22×C5C22×#6711yescharacteristic
506C2×D5—62no—
507C2×D5—121no—
508C2×D5—121no—
509C4×C5—22no—
510C4×C5—41no—
511C4×C5—42no—
512C4×C5#93712yes—
513C4×C5C2×D3×D411yescharacteristic
514C4×C5C2×D3×D411yescharacteristic
515Dic5—64no—
516Dic5—62no—
517#41—22no—
518#41—22no—
519#41—22no—
520#41—22no—
521#41—154no—
522#41—304no—
523#41—304no—
524#41—3016no—
525#41—308no—
526#41—302no—
527C2×Q8—154no—
528C2×Q8—152no—
529C2×Q8—151no—
530C2×Q8—304no—
531C2×Q8—304no—
532C2×Q8—308no—
533C2×Q8—302no—
534C2×Q8—302no—
535C2×Q8—301no—
536C2×Q8#122011yescharacteristic
537C2×Q8C22×D1511yescharacteristic
538C2×D4—21no—
539C2×D4—22no—
540C2×D4—151no—
541C2×D4—302no—
542C2×D4—301no—
543C2×D4#122012yes—
544C2×D4#122011yescharacteristic
545C2×D4C22×D1511yescharacteristic
546C22×C4—21no—
547C22×C4—21no—
548C22×C4—302no—
549C22×C4—301no—
550C22×C4—302no—
551C22×C4—301no—
552C22×C4C22×D1512yes—
553C22×C4C22×D1511yescharacteristic
554Q16—302no—
555Q16—304no—
556SD16—302no—
557SD16—304no—
558SD16—302no—
559D8—302no—
560M16—302no—
561C2×C8—152no—
562C2×C8—151no—
563C2×C8—151no—
564C2×C8—302no—
565C4⋊C4—21no—
566C4⋊C4—22no—
567C4⋊C4C22×D1512yes—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C3×C5#301311yescharacteristic, complemented, Hall
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—22no—
574C22×C3—22no—
575C22×C3—22no—
576C22×C3—41no—
577C22×C3—41no—
578C22×C3—41no—
579C22×C3#408312yes—
580C22×C3C22×C22⋊D511yescharacteristic
581C2×D3—102no—
582C2×D3—201no—
583C2×D3—201no—
584C3×C4—22no—
585C3×C4—41no—
586C3×C4—42no—
587C3×C4#408312yes—
588C3×C4C2×D4×D511yescharacteristic
589C3×C4C2×D4×D511yescharacteristic
590Dic3—104no—
591Dic3—102no—
592C2×C5—22no—
593C2×C5—42no—
594C2×C5—81no—
595C2×C5C2×#93711yescharacteristic
596C2×C5#583611yescharacteristic
597C2×C5#583811yescharacteristic
598D5—122no—
599C23—21no—
600C23—301no—
601C23C2×#122012yes—
602C23C2×#122011yescharacteristic
603Q8—22no—
604Q8—21no—
605Q8—21no—
606Q8—154no—
607Q8—152no—
608Q8—308no—
609Q8—304no—
610Q8—302no—
611Q8—301no—
612D4—24no—
613D4—22no—
614D4—21no—
615D4—21no—
616D4—42no—
617D4—82no—
618D4—152no—
619D4—308no—
620D4—304no—
621D4—301no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—42no—
627C2×C4—81no—
628C2×C4—308no—
629C2×C4—308no—
630C2×C4—304no—
631C2×C4—302no—
632C2×C4—301no—
633C2×C4—304no—
634C2×C4D4×D1512yes—
635C2×C4D4×D1512yes—
636C2×C4C2×#122012yes—
637C2×C4C2×#122011yescharacteristic
638C2×C4C23×D1511yescharacteristic
639C8—301no—
640C8—301no—
641C8—302no—
642C2×C3—22no—
643C2×C3—42no—
644C2×C3—81no—
645C2×C3C2×#408311yescharacteristic
646C2×C3#6617011yescharacteristic
647C2×C3#6617211yescharacteristic
648D3—202no—
649C5#8342711yescharacteristic, complemented, Sylow
650C22—21no—
651C22—21no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—302no—
659C22—601no—
660C22—601no—
661C22#9182612yes—
662C22C22×#122011yescharacteristic
663C4—22no—
664C4—41no—
665C4—42no—
666C4—304no—
667C4—302no—
668C4#9182612yes—
669C4C2×D4×D1511yescharacteristic
670C4C2×D4×D1511yescharacteristic
671C3#1061515811yescharacteristic, complemented, Sylow
672C2—22no—
673C2—42no—
674C2—81no—
675C2—602no—
676C2C2×#9182611yescharacteristic
677C2#1175659511yescharacteristic
678C2#1175659711yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#537C2
socleC22×C3×C5C22×D4
hypercenter#537D15
hypocenterC3×C5#3013
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#92101C2semidirect
2. C3×C5×#537C2semidirect
3. C2×#91840C2semidirect
4. C2×#91839C2semidirect
5. #11751335C2semidirect
6. #11751151C2semidirect
7. #11751148C2semidirect
8. #11751141C2semidirect
9. #11751138C2semidirect
10. C2×#91094C2semidirect
11. #11751067C2semidirect
48. C5×#537D3semidirect
83. #6817D4semidirect
92. C2×Dic30D4semidirect
112. C3×#537D5semidirect
288. #537D15semidirect
570. C3×C5#3013semidirect
649. C5#83427semidirect
671. C3#10615158semidirect

Maximal subgroups

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

subgroupindex
C2×#921012
C3×C5×#5372
C2×#918402
C2×#918392
#117513352
#117511512
#117511482
#117511412
#117511382
C2×#910942
#117510672
#106151583
#834275