Finite Group Databasebeta

← #14424812

Subgroups — #14424812

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×C3×C5×2−1+4C211yesmaximal, characteristic, complemented
2#11756592C211yesmaximal, characteristic, complemented
3C2×#91839C211yesmaximal, characteristic, complemented
4C2×#91831C211yesmaximal, characteristic, complemented
5#11751340C212yesmaximal, complemented
6#11751339C212yesmaximal, complemented
7#11751335C211yesmaximal, characteristic, complemented
8#11751333C211yesmaximal, characteristic, complemented
9#11751329C212yesmaximal, complemented
10#11751328C212yesmaximal, complemented
11C2×#91119C211yesmaximal, characteristic, complemented
12#10615845—31nomaximal, Hall
13C3×C5×2−1+4—24no—
14C3×C5×2−1+4—24no—
15C2×C3×C5×#41—21no—
16C2×C3×C5×#41—22no—
17C2×C3×C5×#41C2212yes—
18C2×C3×C5×#41C2211yescharacteristic
19C2×C3×C5×#41C2211yescharacteristic
20C22×C3×C5×Q8—21no—
21C22×C3×C5×Q8C2212yes—
22C22×C3×C5×Q8C2211yescharacteristic
23C2×#6817C2211yescharacteristic
24#91839—24no—
25#91835—21no—
26C2×#6718C2212yes—
27C2×#6718C2211yescharacteristic
28#91831—24no—
29C2×#6717C2212yes—
30C2×#6717C2211yescharacteristic
31#91828—22no—
32#91827—21no—
33C2×#6716C2211yescharacteristic
34C2×#6715C2211yescharacteristic
35#91819C2212yes—
36C2×C4⋊Dic15C2211yescharacteristic
37#91813C2212yes—
38C2×#6710C2211yescharacteristic
39C22×C3⋊#168C2211yescharacteristic
40#91119C2212yes—
41#91119C2212yes—
42#91119C2212yes—
43#91116C2212yes—
44#91114C2212yes—
45#91114C2212yes—
46#84114—51nomaximal, Hall
47C2×C5×2−1+4D311yescharacteristic, complemented
48#66167—31no—
49C2×#4096—31no—
50C2×#4088—31no—
51#65528—32no—
52#65527—32no—
53#65523—31no—
54#65521—31no—
55#65517—32no—
56#65516—32no—
57C2×#3967—31no—
58C3×C5×#41—216no—
59C3×C5×#41—28no—
60C3×C5×#41—24no—
61C3×C5×#41—24no—
62C3×C5×#41—22no—
63C3×C5×#41—22no—
64C3×C5×#41—22no—
65C3×C5×#41D414yescomplemented
66C2×C3×C5×Q8—28no—
67C2×C3×C5×Q8—24no—
68C2×C3×C5×Q8—24no—
69C2×C3×C5×Q8—22no—
70C2×C3×C5×Q8—21no—
71C2×C3×C5×Q8—22no—
72C2×C3×C5×Q8D411yescharacteristic
73C2×C3×C5×Q8D414yescomplemented
74C2×C3×C5×Q8C2311yescharacteristic
75C2×C3×C5×Q8C2312yes—
76C2×C3×C5×D4—22no—
77C2×C3×C5×D4—21no—
78C2×C3×C5×D4D411yescharacteristic
79C2×C3×C5×D4D412yes—
80C2×C3×C5×D4C2311yescharacteristic
81C22×C3×C4×C5—22no—
82C22×C3×C4×C5—21no—
83C22×C3×C4×C5—22no—
84C22×C3×C4×C5—21no—
85C22×C3×C4×C5C2312yes—
86C22×C3×C4×C5C2311yescharacteristic
87C2×#1220—21no—
88C2×#1220—22no—
89C22×Dic15—21no—
90#6817—22no—
91#6817—22no—
92C2×D60C2311yescharacteristic
93C2×C4×D15—21no—
94C2×Dic30C2311yescharacteristic
95#6719—22no—
96#6718—24no—
97#6718—22no—
98#6717—24no—
99#6717—22no—
100#6716—22no—
101#6715—22no—
102#6714—22no—
103C4⋊Dic15—21no—
104C4⋊Dic15C2312yes—
105#6712—22no—
106#6710—22no—
107C2×C3⋊#168—22no—
108C2×C3⋊#168C2311yescharacteristic
109C2×C3⋊#168C2311yescharacteristic
110C2×C3⋊#168C2312yes—
111C2×C3×2−1+4D511yescharacteristic, complemented
112#5833—51no—
113C2×#950—51no—
114C2×#942—51no—
115#5239—52no—
116#5238—52no—
117#5234—51no—
118#5232—51no—
119#5228—52no—
120#5227—52no—
121C2×#835—51no—
122C5×2−1+4—24no—
123C5×2−1+4—24no—
124C2×C5×#41—21no—
125C2×C5×#41—22no—
126C2×C5×#41C2×D312yes—
127C2×C5×#41C2×D311yescharacteristic
128C2×C5×#41C2×D311yescharacteristic
129C22×C5×Q8—21no—
130C22×C5×Q8C2×D312yes—
131C22×C5×Q8C2×D311yescharacteristic
132C2×#709—31no—
133#4096—64no—
134#4092—61no—
135C2×#689—32no—
136C2×#689—31no—
137#4088—64no—
138C2×#688—32no—
139C2×#688—31no—
140#4085—62no—
141#4084—61no—
142C2×#687—31no—
143C2×#686—31no—
144#4076—32no—
145C2×C4⋊Dic5—31no—
146#4070—32no—
147C2×#681—31no—
148C22×#168—31no—
149#3967—32no—
150#3967—32no—
151#3967—32no—
152#3964—32no—
153C22⋊#168—32no—
154C22⋊#168—32no—
155#3013—151noSylow
156C23×C3×C5—21no—
157C23×C3×C5C2×D412yes—
158C23×C3×C5C2×D411yescharacteristic
159C22×D15—21no—
160C3×C5×Q8—28no—
161C3×C5×Q8—22no—
162C3×C5×Q8—24no—
163C3×C5×Q8—21no—
164C3×C5×Q8—22no—
165C3×C5×Q8C2×D412yes—
166C3×C5×Q8C2×D414yes—
167C3×C5×D4—28no—
168C3×C5×D4—24no—
169C3×C5×D4—24no—
170C3×C5×D4—22no—
171C3×C5×D4—21no—
172C3×C5×D4C2×D412yes—
173C2×C3×C4×C5—28no—
174C2×C3×C4×C5—28no—
175C2×C3×C4×C5—24no—
176C2×C3×C4×C5—22no—
177C2×C3×C4×C5—21no—
178C2×C3×C4×C5—24no—
179C2×C3×C4×C5—22no—
180C2×C3×C4×C5—22no—
181C2×C3×C4×C5C2×D412yes—
182C2×C3×C4×C5C2×D411yescharacteristic
183C2×C3×C4×C5C2×D412yes—
184C2×C3×C4×C5C2×D412yes—
185C2×C3×C4×C5C2411yescharacteristic
186#1220—42no—
187#1220—82no—
188C2×Dic15—21no—
189C2×Dic15—22no—
190C2×Dic15—81no—
191D60—21no—
192D60—21no—
193C4×D15—42no—
194Dic30—21no—
195Dic30—21no—
196C3⋊#168—21no—
197C3⋊#168—21no—
198C3⋊#168—22no—
199C3×2−1+4—24no—
200C3×2−1+4—24no—
201C2×C3×#41—21no—
202C2×C3×#41—22no—
203C2×C3×#41C2×D512yes—
204C2×C3×#41C2×D511yescharacteristic
205C2×C3×#41C2×D511yescharacteristic
206C22×C3×Q8—21no—
207C22×C3×Q8C2×D512yes—
208C22×C3×Q8C2×D511yescharacteristic
209C2×#235—51no—
210#950—104no—
211#946—101no—
212C2×#216—51no—
213C2×#216—52no—
214#942—104no—
215C2×#215—51no—
216C2×#215—52no—
217#939—102no—
218#938—101no—
219C2×#214—51no—
220C2×#213—51no—
221#930—52no—
222C2×C4⋊Dic3—51no—
223#924—52no—
224C2×#208—51no—
225C22×C3⋊C8—51no—
226#835—52no—
227#835—52no—
228#835—52no—
229#832—52no—
230#830—52no—
231#830—52no—
232C5×#41—216no—
233C5×#41—28no—
234C5×#41—24no—
235C5×#41—24no—
236C5×#41—22no—
237C5×#41—22no—
238C5×#41—22no—
239C5×#41#6714yescomplemented
240C2×C5×Q8—28no—
241C2×C5×Q8—24no—
242C2×C5×Q8—24no—
243C2×C5×Q8—22no—
244C2×C5×Q8—21no—
245C2×C5×Q8—22no—
246C2×C5×Q8#6711yescharacteristic
247C2×C5×Q8#6714yescomplemented
248C2×C5×Q8C22×D311yescharacteristic
249C2×C5×Q8C22×D312yes—
250C2×C5×D4—22no—
251C2×C5×D4—21no—
252C2×C5×D4#6711yescharacteristic
253C2×C5×D4#6712yes—
254C2×C5×D4C22×D311yescharacteristic
255C22×C4×C5—22no—
256C22×C4×C5—21no—
257C22×C4×C5—22no—
258C22×C4×C5—21no—
259C22×C4×C5C22×D312yes—
260C22×C4×C5C22×D311yescharacteristic
261C2×C22⋊D5—61no—
262C2×C22⋊D5—62no—
263C22×Dic5—61no—
264#709—62no—
265#709—62no—
266C2×D20—31no—
267C2×C4×D5—61no—
268C2×Dic10—31no—
269C22⋊Dic5—62no—
270#689—62no—
271#689—64no—
272#688—62no—
273#688—64no—
274#687—62no—
275#686—62no—
276#685—62no—
277C4⋊Dic5—32no—
278C4⋊Dic5—61no—
279#683—62no—
280#681—62no—
281C2×#168—32no—
282C2×#168—31no—
283C2×#168—31no—
284C2×#168—62no—
285C2×2−1+4D1511yescharacteristic, complemented
286C2×#137—151no—
287C2×#135—151no—
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—22no—
300C22×C3×C5—22no—
301C22×C3×C5—22no—
302C22×C3×C5—41no—
303C22×C3×C5—41no—
304C22×C3×C5#12012yes—
305C22×C3×C5C22×D411yescharacteristic
306C2×D15—41no—
307C2×D15—41no—
308C2×D15—41no—
309C3×C4×C5—24no—
310C3×C4×C5—22no—
311C3×C4×C5—22no—
312C3×C4×C5#12012yes—
313C3×C4×C5C22×D411yescharacteristic
314C3×C4×C5C22×D411yescharacteristic
315Dic15—41no—
316Dic15—42no—
317C3×#41—216no—
318C3×#41—28no—
319C3×#41—24no—
320C3×#41—24no—
321C3×#41—22no—
322C3×#41—22no—
323C3×#41—22no—
324C3×#41C22⋊D514yescomplemented
325C2×C3×Q8—28no—
326C2×C3×Q8—24no—
327C2×C3×Q8—24no—
328C2×C3×Q8—22no—
329C2×C3×Q8—21no—
330C2×C3×Q8—22no—
331C2×C3×Q8C22⋊D511yescharacteristic
332C2×C3×Q8C22⋊D514yescomplemented
333C2×C3×Q8C22×D511yescharacteristic
334C2×C3×Q8C22×D512yes—
335C2×C3×D4—22no—
336C2×C3×D4—21no—
337C2×C3×D4C22⋊D511yescharacteristic
338C2×C3×D4C22⋊D512yes—
339C2×C3×D4C22×D511yescharacteristic
340C22×C3×C4—22no—
341C22×C3×C4—21no—
342C22×C3×C4—22no—
343C22×C3×C4—21no—
344C22×C3×C4C22×D512yes—
345C22×C3×C4C22×D511yescharacteristic
346C2×#67—101no—
347C2×#67—102no—
348C22×Dic3—101no—
349#235—102no—
350#235—102no—
351C2×D12—51no—
352C2×C4×D3—101no—
353C2×Dic6—51no—
354#217—102no—
355#216—104no—
356#216—102no—
357#215—104no—
358#215—102no—
359#214—102no—
360#213—102no—
361(C2×D3)⋊C4—102no—
362C4⋊Dic3—52no—
363C4⋊Dic3—101no—
364Dic3⋊C4—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—28no—
375C5×Q8—22no—
376C5×Q8—24no—
377C5×Q8—21no—
378C5×Q8—22no—
379C5×Q8C2×#6712yes—
380C5×Q8C2×#6714yes—
381C5×D4—28no—
382C5×D4—24no—
383C5×D4—24no—
384C5×D4—22no—
385C5×D4—21no—
386C5×D4C2×#6712yes—
387C2×C4×C5—28no—
388C2×C4×C5—28no—
389C2×C4×C5—24no—
390C2×C4×C5—22no—
391C2×C4×C5—21no—
392C2×C4×C5—24no—
393C2×C4×C5—22no—
394C2×C4×C5—22no—
395C2×C4×C5C2×#6712yes—
396C2×C4×C5C2×#6711yescharacteristic
397C2×C4×C5C2×#6712yes—
398C2×C4×C5C2×#6712yes—
399C2×C4×C5C23×D311yescharacteristic
400C22⋊D5—122no—
401C22⋊D5—242no—
402C2×Dic5—61no—
403C2×Dic5—62no—
404C2×Dic5—241no—
405D20—61no—
406D20—61no—
407C4×D5—122no—
408Dic10—61no—
409Dic10—61no—
410#168—61no—
411#168—61no—
412#168—62no—
4132−1+4—24no—
4142−1+4—24no—
415C2×#41—21no—
416C2×#41—22no—
417C2×#41—151no—
418C2×#41C2×D1512yes—
419C2×#41C2×D1511yescharacteristic
420C2×#41C2×D1511yescharacteristic
421C22×Q8—21no—
422C22×Q8C2×D1512yes—
423C22×Q8C2×D1511yescharacteristic
424#137—304no—
425#135—304no—
426C2×Q16—151no—
427C2×Q16—152no—
428C2×SD16—151no—
429C2×SD16—152no—
430C2×SD16—151no—
431C2×D8—151no—
432C2×M16—151no—
433C22×C8—151no—
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—42no—
449C2×C3×C5C2×#12011yescharacteristic
450C2×C3×C5#57711yescharacteristic
451C2×C3×C5#57811yescharacteristic
452D15—81no—
453C23×C3—21no—
454C23×C3C2×C22⋊D512yes—
455C23×C3C2×C22⋊D511yescharacteristic
456C22×D3—101no—
457C3×Q8—28no—
458C3×Q8—22no—
459C3×Q8—24no—
460C3×Q8—21no—
461C3×Q8—22no—
462C3×Q8C2×C22⋊D512yes—
463C3×Q8C2×C22⋊D514yes—
464C3×D4—28no—
465C3×D4—24no—
466C3×D4—24no—
467C3×D4—22no—
468C3×D4—21no—
469C3×D4C2×C22⋊D512yes—
470C2×C3×C4—28no—
471C2×C3×C4—28no—
472C2×C3×C4—24no—
473C2×C3×C4—22no—
474C2×C3×C4—21no—
475C2×C3×C4—24no—
476C2×C3×C4—22no—
477C2×C3×C4—22no—
478C2×C3×C4C2×C22⋊D512yes—
479C2×C3×C4C2×C22⋊D511yescharacteristic
480C2×C3×C4C2×C22⋊D512yes—
481C2×C3×C4C2×C22⋊D512yes—
482C2×C3×C4C23×D511yescharacteristic
483#67—202no—
484#67—402no—
485C2×Dic3—101no—
486C2×Dic3—102no—
487C2×Dic3—401no—
488D12—101no—
489D12—101no—
490C4×D3—202no—
491Dic6—101no—
492Dic6—101no—
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—22no—
502C22×C5—41no—
503C22×C5—41no—
504C22×C5#95312yes—
505C22×C5C22×#6711yescharacteristic
506C2×D5—121no—
507C2×D5—121no—
508C2×D5—121no—
509C4×C5—24no—
510C4×C5—22no—
511C4×C5—22no—
512C4×C5#95312yes—
513C4×C5C22×#6711yescharacteristic
514C4×C5C22×#6711yescharacteristic
515Dic5—122no—
516Dic5—121no—
517#41—216no—
518#41—28no—
519#41—24no—
520#41—24no—
521#41—22no—
522#41—22no—
523#41—22no—
524#41—302no—
525#41—302no—
526#41#122014yescomplemented
527C2×Q8—28no—
528C2×Q8—24no—
529C2×Q8—24no—
530C2×Q8—22no—
531C2×Q8—21no—
532C2×Q8—22no—
533C2×Q8—151no—
534C2×Q8#122011yescharacteristic
535C2×Q8#122014yescomplemented
536C2×Q8C22×D1511yescharacteristic
537C2×Q8C22×D1512yes—
538C2×D4—22no—
539C2×D4—21no—
540C2×D4—151no—
541C2×D4—301no—
542C2×D4—302no—
543C2×D4#122012yes—
544C2×D4#122011yescharacteristic
545C2×D4C22×D1511yescharacteristic
546C22×C4—22no—
547C22×C4—21no—
548C22×C4—22no—
549C22×C4—21no—
550C22×C4—301no—
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—152no—
566C4⋊C4—301no—
567C4⋊C4—302no—
568C22⋊C4—302no—
569C22⋊C4—302no—
570C3×C5#301311yescharacteristic, complemented, Hall
571C22×C3—21no—
572C22×C3—21no—
573C22×C3—22no—
574C22×C3—22no—
575C22×C3—22no—
576C22×C3—22no—
577C22×C3—41no—
578C22×C3—41no—
579C22×C3#409912yes—
580C22×C3C22×C22⋊D511yescharacteristic
581C2×D3—201no—
582C2×D3—201no—
583C2×D3—201no—
584C3×C4—24no—
585C3×C4—22no—
586C3×C4—22no—
587C3×C4#409912yes—
588C3×C4C22×C22⋊D511yescharacteristic
589C3×C4C22×C22⋊D511yescharacteristic
590Dic3—202no—
591Dic3—201no—
592C2×C5—22no—
593C2×C5—42no—
594C2×C5—42no—
595C2×C5#583711yescharacteristic
596C2×C5#583811yescharacteristic
597C2×C5C2×#95311yescharacteristic
598D5—241no—
599C23—21no—
600C23—301no—
601C23C2×#122012yes—
602C23C2×#122011yescharacteristic
603Q8—28no—
604Q8—22no—
605Q8—24no—
606Q8—21no—
607Q8—22no—
608Q8—301no—
609Q8—301no—
610Q8C2×#122012yes—
611Q8C2×#122014yes—
612D4—28no—
613D4—24no—
614D4—24no—
615D4—22no—
616D4—21no—
617D4—301no—
618D4—301no—
619D4—602no—
620D4—1202no—
621D4C2×#122012yes—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—21no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—302no—
631C2×C4—301no—
632C2×C4—602no—
633C2×C4—1201no—
634C2×C4C2×#122012yes—
635C2×C4C2×#122011yescharacteristic
636C2×C4C2×#122012yes—
637C2×C4C2×#122012yes—
638C2×C4C23×D1511yescharacteristic
639C8—301no—
640C8—301no—
641C8—302no—
642C2×C3—22no—
643C2×C3—42no—
644C2×C3—42no—
645C2×C3#6617111yescharacteristic
646C2×C3#6617211yescharacteristic
647C2×C3C2×#409911yescharacteristic
648D3—401no—
649C5#8411411yescharacteristic, complemented, Sylow
650C22—21no—
651C22—21no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—601no—
659C22—601no—
660C22—601no—
661C22#9184212yes—
662C22C22×#122011yescharacteristic
663C4—24no—
664C4—22no—
665C4—22no—
666C4—602no—
667C4—601no—
668C4#9184212yes—
669C4C22×#122011yescharacteristic
670C4C22×#122011yescharacteristic
671C3#1061584511yescharacteristic, complemented, Sylow
672C2—22no—
673C2—42no—
674C2—42no—
675C2—1201no—
676C2#1175659611yescharacteristic
677C2#1175659711yescharacteristic
678C2C2×#9184211yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×2−1+4C2
socleC22×C3×C5C22×D4
hypercenterC2×2−1+4D15
hypocenterC3×C5#3013
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×2−1+4C2semidirect
2. #11756592C2semidirect
3. C2×#91839C2semidirect
4. C2×#91831C2semidirect
5. #11751340C2semidirect
6. #11751339C2semidirect
7. #11751335C2semidirect
8. #11751333C2semidirect
9. #11751329C2semidirect
10. #11751328C2semidirect
11. C2×#91119C2semidirect
47. C2×C5×2−1+4D3semidirect
65. C3×C5×#41D4semidirect
73. C2×C3×C5×Q8D4semidirect
111. C2×C3×2−1+4D5semidirect
239. C5×#41#67semidirect
247. C2×C5×Q8#67semidirect
285. C2×2−1+4D15semidirect
324. C3×#41C22⋊D5semidirect
332. C2×C3×Q8C22⋊D5semidirect
526. #41#1220semidirect
535. C2×Q8#1220semidirect
570. C3×C5#3013semidirect
649. C5#84114semidirect
671. C3#10615845semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×2−1+42
#117565922
C2×#918392
C2×#918312
#117513402
#117513392
#117513352
#117513332
#117513292
#117513282
C2×#911192
#106158453
#841145