Finite Group Databasebeta

← #14425732

Subgroups — #14425732

Order 1920.

1600 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4144C211yesmaximal, characteristic, complemented
2C5×#5804C211yesmaximal, characteristic, complemented
3#11755426C212yesmaximal, complemented
4C2×#91118C211yesmaximal, characteristic, complemented
5C2×#90995C212yesmaximal, complemented
6#11749181C214yesmaximal, complemented
7#11749175C214yesmaximal, complemented
8#10615106—31nomaximal, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C22×C5×#67C2211yescharacteristic, complemented
11C22×C3×C22⋊D5—21no—
12C22×C3×C22⋊D5C2212yes—
13C3×#4144—22no—
14C3×#4144—24no—
15C3×#4144C2214yes—
16C2×C3×#711—22no—
17C2×C3×D4×D5—22no—
18C2×C3×#709—21no—
19C2×#6719C2212yescomplemented
20C5×#953—21no—
21C2×C5×#217C2211yescharacteristic, complemented
22C5×#939—22no—
23C5×#937C2212yescomplemented
24C5×#935C2212yescomplemented
25C5×#929—21no—
26#91571—24no—
27#91568C2214yescomplemented
28#91563—22no—
29#91539—22no—
30#91536C2214yescomplemented
31#91118C2214yescomplemented
32#90995C2218yescomplemented
33#84110—51nomaximal, Hall
34C2×#4144D311yescharacteristic, complemented
35C5×#534—31no—
36#66138—32no—
37C2×#3966—31no—
38#65342—34no—
39#65321—34no—
40C2×#3938—32no—
41C24×C3×C5D411yescharacteristic, complemented
42C24×C3×C5C2311yescharacteristic
43C23×C5×D3C2311yescharacteristic, complemented
44C23×C3×D5—21no—
45C23×C3×D5D412yescomplemented
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4—22no—
48C2×C3×C5×D4D412yescomplemented
49C2×C3×C5×D4C2312yes—
50C22×C3×C4×C5D411yescharacteristic, complemented
51C22×Dic15—22no—
52C2×C5×#67—21no—
53C2×C5×#67—21no—
54C2×C5×#67—22no—
55C2×C5×#67—22no—
56C2×C5×#67—22no—
57C22×C5×Dic3—21no—
58C22×C5×Dic3C2311yescharacteristic, complemented
59C2×C3×C22⋊D5—22no—
60C2×C3×C22⋊D5—24no—
61C2×C3×C22⋊D5—22no—
62C2×C3×C22⋊D5—22no—
63C2×C3×C22⋊D5—22no—
64C2×C3×C22⋊D5—22no—
65C2×C3×C22⋊D5—24no—
66C2×C3×C22⋊D5D414yescomplemented
67C2×C3×C22⋊D5C2314yes—
68C22×C3×Dic5—21no—
69C22×C3×Dic5D412yescomplemented
70C3×#711—28no—
71C3×#711—28no—
72C3×D4×D5—28no—
73C3×D4×D5—28no—
74C3×#709—24no—
75C3×#709—24no—
76C2×C3×D20—21no—
77C2×C3×C4×D5—22no—
78C2×C3×Dic10—21no—
79C2×#1201—24no—
80#6719—22no—
81#6719C2314yescomplemented
82C5×#217—21no—
83C5×#217—22no—
84C5×#217C2312yescomplemented
85C5×(C2×D3)⋊C4—22no—
86C5×Dic3⋊C4—22no—
87#6669—24no—
88(C2×D3)⋊Dic5—24no—
89#6665—24no—
90C2×C3×H(5,𝔽2)—51no—
91#5804—52no—
92#5804D511yescharacteristic, complemented
93#5244—54no—
94#5243—54no—
95C2×#834—51no—
96C2×#834—52no—
97C22×C5×D4—31no—
98C22×C5×D4C2×D311yescharacteristic
99C22×C22⋊D5—21no—
100C22×C22⋊D5C2×D312yes—
101#4144—22no—
102#4144—24no—
103#4144C2×D314yes—
104C2×#711—22no—
105C2×D4×D5—22no—
106C2×#709—21no—
107C5×#123—32no—
108C5×#123—61no—
109C5×#121—62no—
110C5×#120—32no—
111C5×#120—61no—
112C2×C5×C22⋊C4—31no—
113#4099—34no—
114C2×C22⋊Dic5—32no—
115#4085—64no—
116#4083—62no—
117#4081—62no—
118#4075—34no—
119#3966—34no—
120#3938—38no—
121#3026—151noSylow
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—81no—
125C23×C3×C5—81no—
126C23×C3×C5C2×D412yes—
127C23×C3×C5C2×D412yes—
128C23×C3×C5C2×D412yes—
129C23×C3×C5C2×D411yescharacteristic
130C23×C3×C5C2×D411yescharacteristic
131C23×C3×C5C2411yescharacteristic
132C22×C5×D3—22no—
133C22×C5×D3—21no—
134C22×C5×D3—81no—
135C22×C3×D5—22no—
136C22×C3×D5—24no—
137C22×C3×D5—22no—
138C22×C3×D5—82no—
139C22×C3×D5—82no—
140C22×C3×D5C2×D414yes—
141C3×C5×D4—24no—
142C3×C5×D4—24no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5C2×D412yes—
146C2×Dic15—24no—
147C2×Dic15—82no—
148C5×#67—41no—
149C5×#67—42no—
150C5×#67—41no—
151C5×#67—82no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—21no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—81no—
156C3×C22⋊D5—28no—
157C3×C22⋊D5—24no—
158C3×C22⋊D5—24no—
159C3×C22⋊D5—24no—
160C3×C22⋊D5—24no—
161C2×C3×Dic5—24no—
162C2×C3×Dic5—22no—
163C2×C3×Dic5—22no—
164C2×C3×Dic5C2×D414yes—
165C3×D20—24no—
166C3×C4×D5—28no—
167C3×Dic10—24no—
168#1201—84no—
169C3×H(5,𝔽2)—54no—
170C3×H(5,𝔽2)—104no—
171C3×H(5,𝔽2)—102no—
172C2×C3×#41—102no—
173C2×C3×#41—101no—
174C22×C3×D4—52no—
175C22×C3×D4—102no—
176C22×C3×D4—101no—
177C22×C3×D4C2×D511yescharacteristic
178C22×#67C2×D511yescharacteristic, complemented
179#953—21no—
180#953—102no—
181C2×#217—52no—
182C2×#217C2×D511yescharacteristic, complemented
183#939—22no—
184#939—104no—
185#937—54no—
186#937C2×D512yescomplemented
187#935—54no—
188#935C2×D512yescomplemented
189#929—21no—
190#929—102no—
191#834—54no—
192#834—58no—
193C24×C5—31no—
194C24×C5#6711yescharacteristic, complemented
195C24×C5C22×D311yescharacteristic
196C23×D5—21no—
197C23×D5#6712yescomplemented
198C2×C5×D4—22no—
199C2×C5×D4—22no—
200C2×C5×D4—61no—
201C2×C5×D4—61no—
202C2×C5×D4—62no—
203C2×C5×D4—62no—
204C2×C5×D4—62no—
205C2×C5×D4#6712yescomplemented
206C2×C5×D4C22×D312yes—
207C22×C4×C5—31no—
208C22×C4×C5—61no—
209C22×C4×C5#6711yescharacteristic, complemented
210C2×C22⋊D5—22no—
211C2×C22⋊D5—24no—
212C2×C22⋊D5—22no—
213C2×C22⋊D5—22no—
214C2×C22⋊D5—22no—
215C2×C22⋊D5—22no—
216C2×C22⋊D5—24no—
217C2×C22⋊D5—64no—
218C2×C22⋊D5#6714yescomplemented
219C2×C22⋊D5C22×D314yes—
220C22×Dic5—21no—
221C22×Dic5—62no—
222C22×Dic5#6712yescomplemented
223#711—28no—
224#711—28no—
225D4×D5—28no—
226D4×D5—28no—
227#709—24no—
228#709—24no—
229C2×D20—21no—
230C2×C4×D5—22no—
231C2×Dic10—21no—
232C5×C4⋊C4—62no—
233C5×C22⋊C4—32no—
234C5×C22⋊C4—61no—
235C5×C22⋊C4—62no—
236C5×C22⋊C4—62no—
237C22⋊Dic5—34no—
238C22⋊Dic5—62no—
239C22⋊Dic5—64no—
240#685—64no—
241#683—64no—
242C2×H(5,𝔽2)—51no—
243#534—31no—
244#534—152no—
245#458—154no—
246B(4,𝔽2)—154no—
247C2×#99—152no—
248C2×#99—151no—
249C22×C3×C5—22no—
250C22×C3×C5—41no—
251C22×C3×C5—42no—
252C22×C3×C5—42no—
253C22×C3×C5—41no—
254C22×C3×C5—41no—
255C22×C3×C5—42no—
256C22×C3×C5—42no—
257C22×C3×C5—81no—
258C22×C3×C5#12011yescharacteristic
259C22×C3×C5#12012yes—
260C22×C3×C5#12011yescharacteristic
261C22×C3×C5C22×D412yes—
262C22×C3×C5C22×D411yescharacteristic
263C2×C5×D3—44no—
264C2×C5×D3—42no—
265C2×C5×D3—41no—
266C2×C3×D5—24no—
267C2×C3×D5—44no—
268C2×C3×D5—44no—
269C2×C3×D5—82no—
270C2×C3×D5—84no—
271C3×C4×C5—22no—
272Dic15—44no—
273C3×Dic5—24no—
274C5×Dic3—42no—
275C5×Dic3—41no—
276C24×C3—52no—
277C24×C3—101no—
278C24×C3C22⋊D511yescharacteristic, complemented
279C24×C3C22×D511yescharacteristic
280C23×D3C22×D511yescharacteristic, complemented
281C3×#41—104no—
282C3×#41—108no—
283C3×#41—108no—
284C3×#41—104no—
285C2×C3×Q8—101no—
286C2×C3×D4—22no—
287C2×C3×D4—22no—
288C2×C3×D4—54no—
289C2×C3×D4—54no—
290C2×C3×D4—101no—
291C2×C3×D4—108no—
292C2×C3×D4—108no—
293C2×C3×D4—102no—
294C2×C3×D4—104no—
295C2×C3×D4—102no—
296C2×C3×D4—102no—
297C2×C3×D4—102no—
298C2×C3×D4—102no—
299C2×C3×D4—104no—
300C2×C3×D4C22⋊D512yescomplemented
301C2×C3×D4C22×D512yes—
302C22×C3×C4—52no—
303C22×C3×C4—102no—
304C22×C3×C4—101no—
305C22×C3×C4C22⋊D511yescharacteristic, complemented
306C2×#67—22no—
307C2×#67—22no—
308C2×#67—21no—
309C2×#67—21no—
310C2×#67—22no—
311C2×#67—104no—
312C22×Dic3—21no—
313C22×Dic3—102no—
314C22×Dic3C22×D511yescharacteristic, complemented
315#217—21no—
316#217—22no—
317#217—54no—
318#217—104no—
319#217—102no—
320#217C22×D512yescomplemented
321(C2×D3)⋊C4—22no—
322(C2×D3)⋊C4—104no—
323Dic3⋊C4—22no—
324Dic3⋊C4—104no—
325C23×C5—21no—
326C23×C5—21no—
327C23×C5—62no—
328C23×C5—61no—
329C23×C5—81no—
330C23×C5—81no—
331C23×C5—241no—
332C23×C5C2×#6712yes—
333C23×C5C2×#6711yescharacteristic
334C23×C5C2×#6711yescharacteristic
335C23×C5C2×#6712yes—
336C23×C5C2×#6712yes—
337C23×C5C23×D311yescharacteristic
338C22×D5—22no—
339C22×D5—24no—
340C22×D5—22no—
341C22×D5—82no—
342C22×D5—82no—
343C22×D5C2×#6714yes—
344C5×D4—24no—
345C5×D4—24no—
346C5×D4—121no—
347C5×D4—122no—
348C5×D4—121no—
349C5×D4—242no—
350C2×C4×C5—21no—
351C2×C4×C5—21no—
352C2×C4×C5—62no—
353C2×C4×C5—62no—
354C2×C4×C5—61no—
355C2×C4×C5—241no—
356C2×C4×C5C2×#6712yes—
357C22⋊D5—28no—
358C22⋊D5—24no—
359C22⋊D5—24no—
360C22⋊D5—24no—
361C22⋊D5—24no—
362C22⋊D5—244no—
363C2×Dic5—24no—
364C2×Dic5—22no—
365C2×Dic5—22no—
366C2×Dic5—64no—
367C2×Dic5—242no—
368C2×Dic5C2×#6714yes—
369D20—24no—
370C4×D5—28no—
371Dic10—24no—
372H(5,𝔽2)—54no—
373H(5,𝔽2)—104no—
374H(5,𝔽2)—102no—
375C2×#41—102no—
376C2×#41—101no—
377C22×D4—31no—
378C22×D4—52no—
379C22×D4—102no—
380C22×D4—101no—
381C22×D4D3×D511yescharacteristic
382#123—32no—
383#123—61no—
384#123—154no—
385#123—302no—
386#121—62no—
387#121—304no—
388#120—32no—
389#120—61no—
390#120—154no—
391#120—302no—
392C2×C22⋊C4—31no—
393C2×C22⋊C4—152no—
394#99—158no—
395#99—154no—
396C2×C3×C5—21no—
397C2×C3×C5—22no—
398C2×C3×C5—21no—
399C2×C3×C5—22no—
400C2×C3×C5—21no—
401C2×C3×C5—42no—
402C2×C3×C5C2×#12011yescharacteristic
403C3×D5—44no—
404C5×D3—81no—
405C23×C3—21no—
406C23×C3—21no—
407C23×C3—54no—
408C23×C3—81no—
409C23×C3—81no—
410C23×C3—104no—
411C23×C3—102no—
412C23×C3—102no—
413C23×C3—402no—
414C23×C3—402no—
415C23×C3D4×D512yes—
416C23×C3D4×D512yes—
417C23×C3C2×C22⋊D512yes—
418C23×C3C2×C22⋊D511yescharacteristic
419C23×C3C2×C22⋊D511yescharacteristic
420C23×C3C23×D511yescharacteristic
421C22×D3—22no—
422C22×D3—21no—
423C22×D3—81no—
424C3×Q8—104no—
425C3×D4—24no—
426C3×D4—24no—
427C3×D4—104no—
428C3×D4—108no—
429C3×D4—104no—
430C3×D4—104no—
431C3×D4—104no—
432C3×D4—104no—
433C2×C3×C4—21no—
434C2×C3×C4—21no—
435C2×C3×C4—54no—
436C2×C3×C4—108no—
437C2×C3×C4—104no—
438C2×C3×C4—102no—
439C2×C3×C4—102no—
440C2×C3×C4C2×C22⋊D512yes—
441#67—41no—
442#67—41no—
443#67—42no—
444#67—82no—
445#67—404no—
446C2×Dic3—22no—
447C2×Dic3—21no—
448C2×Dic3—22no—
449C2×Dic3—81no—
450C2×Dic3—104no—
451C2×Dic3—402no—
452C22×C5—22no—
453C22×C5—42no—
454C22×C5—41no—
455C22×C5—42no—
456C22×C5—41no—
457C22×C5—41no—
458C22×C5—42no—
459C22×C5—42no—
460C22×C5—81no—
461C22×C5—124no—
462C22×C5—122no—
463C22×C5—121no—
464C22×C5#95311yescharacteristic
465C22×C5#95312yes—
466C22×C5#95311yescharacteristic
467C22×C5C22×#6711yescharacteristic
468C22×C5C22×#6712yes—
469C2×D5—24no—
470C2×D5—44no—
471C2×D5—44no—
472C2×D5—82no—
473C2×D5—84no—
474C4×C5—22no—
475C4×C5—122no—
476C4×C5—121no—
477Dic5—24no—
478Dic5—124no—
479C24—31no—
480C24—52no—
481C24—101no—
482C24#120111yescharacteristic, complemented
483C24C2×D3×D511yescharacteristic
484#41—104no—
485#41—108no—
486#41—108no—
487#41—104no—
488C2×Q8—101no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—54no—
492C2×D4—54no—
493C2×D4—61no—
494C2×D4—61no—
495C2×D4—62no—
496C2×D4—62no—
497C2×D4—62no—
498C2×D4—101no—
499C2×D4—108no—
500C2×D4—108no—
501C2×D4—102no—
502C2×D4—104no—
503C2×D4—102no—
504C2×D4—102no—
505C2×D4—104no—
506C2×D4—102no—
507C2×D4—102no—
508C2×D4—304no—
509C2×D4#120112yescomplemented
510C2×D4C2×D3×D512yes—
511C22×C4—31no—
512C22×C4—52no—
513C22×C4—61no—
514C22×C4—102no—
515C22×C4—101no—
516C22×C4—302no—
517C22×C4#120111yescharacteristic, complemented
518C4⋊C4—62no—
519C4⋊C4—304no—
520C22⋊C4—32no—
521C22⋊C4—61no—
522C22⋊C4—62no—
523C22⋊C4—62no—
524C22⋊C4—154no—
525C22⋊C4—304no—
526C22⋊C4—304no—
527C22⋊C4—302no—
528C3×C5#302611yescharacteristic, complemented, Hall
529C22×C3—22no—
530C22×C3—41no—
531C22×C3—42no—
532C22×C3—41no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—81no—
538C22×C3—104no—
539C22×C3—204no—
540C22×C3—204no—
541C22×C3—402no—
542C22×C3—404no—
543C22×C3#408311yescharacteristic
544C22×C3#408311yescharacteristic
545C22×C3#408312yes—
546C22×C3C2×D4×D512yes—
547C22×C3C22×C22⋊D511yescharacteristic
548C2×D3—42no—
549C2×D3—44no—
550C2×D3—41no—
551C3×C4—22no—
552C3×C4—104no—
553Dic3—42no—
554Dic3—41no—
555Dic3—204no—
556C2×C5—22no—
557C2×C5—21no—
558C2×C5—21no—
559C2×C5—22no—
560C2×C5—21no—
561C2×C5—42no—
562C2×C5—241no—
563C2×C5C2×#95311yescharacteristic
564D5—44no—
565C23—21no—
566C23—21no—
567C23—54no—
568C23—61no—
569C23—62no—
570C23—81no—
571C23—81no—
572C23—104no—
573C23—102no—
574C23—102no—
575C23—241no—
576C23—402no—
577C23—402no—
578C23C2×#120112yes—
579C23C2×#120111yescharacteristic
580C23C2×#120111yescharacteristic
581C23D5×#6712yes—
582C23D5×#6712yes—
583C23C22×D3×D511yescharacteristic
584Q8—104no—
585D4—24no—
586D4—24no—
587D4—104no—
588D4—108no—
589D4—104no—
590D4—104no—
591D4—104no—
592D4—104no—
593D4—121no—
594D4—121no—
595D4—122no—
596D4—242no—
597D4—1204no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—54no—
601C2×C4—62no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—108no—
605C2×C4—104no—
606C2×C4—102no—
607C2×C4—102no—
608C2×C4—241no—
609C2×C4—304no—
610C2×C4—1202no—
611C2×C4C2×#120112yes—
612C2×C3—21no—
613C2×C3—22no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—22no—
617C2×C3—42no—
618C2×C3—204no—
619C2×C3C2×#408311yescharacteristic
620D3—81no—
621C5#8411011yescharacteristic, complemented, Sylow
622C22—22no—
623C22—41no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—104no—
632C22—122no—
633C22—124no—
634C22—121no—
635C22—204no—
636C22—204no—
637C22—402no—
638C22—404no—
639C22#9156311yescharacteristic
640C22#9156311yescharacteristic
641C22#9156312yes—
642C22C22×#120111yescharacteristic
643C22C2×D5×#6712yes—
644C4—22no—
645C4—104no—
646C4—122no—
647C4—121no—
648C4—604no—
649C3#1061510611yescharacteristic, complemented, Sylow
650C2—21no—
651C2—22no—
652C2—21no—
653C2—21no—
654C2—22no—
655C2—42no—
656C2—204no—
657C2—241no—
658C2C2×#9156311yescharacteristic
subgroupG/H
centerC2C2×#91563
commutator, cosocleC23×C3×C5C24
FrattiniC23C22×D3×D5
FittingC22×C3×C5×D4C22
socleC2×C3×C5C2×#120
hypercenterC22×D4D3×D5
hypocenterC3×C5#3026
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4144C2semidirect
2. C5×#5804C2semidirect
3. #11755426C2semidirect
4. C2×#91118C2semidirect
5. C2×#90995C2semidirect
6. #11749181C2semidirect
7. #11749175C2semidirect
10. C22×C5×#67C22semidirect
19. C2×#6719C22semidirect
21. C2×C5×#217C22semidirect
23. C5×#937C22semidirect
24. C5×#935C22semidirect
27. #91568C22semidirect
30. #91536C22semidirect
31. #91118C22semidirect
32. #90995C22semidirect
34. C2×#4144D3semidirect
41. C24×C3×C5D4semidirect
43. C23×C5×D3C23semidirect
45. C23×C3×D5D4semidirect
48. C2×C3×C5×D4D4semidirect
50. C22×C3×C4×C5D4semidirect
58. C22×C5×Dic3C23semidirect
66. C2×C3×C22⋊D5D4semidirect
69. C22×C3×Dic5D4semidirect
81. #6719C23semidirect
84. C5×#217C23semidirect
92. #5804D5semidirect
178. C22×#67C2×D5semidirect
182. C2×#217C2×D5semidirect
186. #937C2×D5semidirect
188. #935C2×D5semidirect
194. C24×C5#67semidirect
197. C23×D5#67semidirect
205. C2×C5×D4#67semidirect
209. C22×C4×C5#67semidirect
218. C2×C22⋊D5#67semidirect
222. C22×Dic5#67semidirect
278. C24×C3C22⋊D5semidirect
280. C23×D3C22×D5semidirect
300. C2×C3×D4C22⋊D5semidirect
305. C22×C3×C4C22⋊D5semidirect
314. C22×Dic3C22×D5semidirect
320. #217C22×D5semidirect
482. C24#1201semidirect
509. C2×D4#1201semidirect
517. C22×C4#1201semidirect
528. C3×C5#3026semidirect
621. C5#84110semidirect
649. C3#10615106semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41442
C5×#58042
#117554262
C2×#911182
C2×#909952
#117491812
#117491752
#106151063
#841105