Finite Group Databasebeta

← #14327680

Subgroups — #14327680

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#117C211yesmaximal, characteristic, complemented
2C2×C4×#1220C211yesmaximal, characteristic
3C2×#91781C211yesmaximal, characteristic, complemented
4#11751313C211yesmaximal, characteristic, complemented
5#11751088C211yesmaximal, characteristic, complemented
6#11751076C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
8#11751058C212yesmaximal
9#11751045C212yesmaximal, complemented
10#11751041C212yesmaximal, complemented
11C4×#6712C212yesmaximal, complemented
12#10613972—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×C4×D15C2211yescharacteristic
16C3×C5×#117—24no—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2212yes—
20C2×C3×C42×C5C2212yes—
21C2×#6719C2212yes—
22C2×#6719C2211yescharacteristic
23C4×#1220C418yescomplemented
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×#6714C2211yescharacteristic
27C2×C4⋊Dic15C2211yescharacteristic
28C2×C4⋊Dic15C2211yescharacteristic
29C2×#6712C2212yes—
30C2×#6712C2212yes—
31C2×#6712C2211yescharacteristic
32C2×C4×Dic15C2212yes—
33C2×C4×Dic15C2212yes—
34C2×C4×Dic15C2211yescharacteristic
35#91781—24no—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#82241—51nomaximal, Hall
40C2×C5×#117D311yescharacteristic, complemented
41C2×C4×C22⋊D5—31no—
42C2×#4038—31no—
43#65501—31no—
44#65276—31no—
45#65265—31no—
46C4⋊C4×Dic5—31no—
47#65246—32no—
48#65233—32no—
49#65229—32no—
50C4×#683—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—22no—
61C2×#1220—22no—
62C2×#1220C2×C414yes—
63C22×Dic15C2312yes—
64C22×Dic15C2312yes—
65C22×Dic15C2311yescharacteristic
66C22×Dic15C2311yescharacteristic
67C22×Dic15C2311yescharacteristic
68C2×C4×D15—22no—
69C2×C4×D15—22no—
70C2×C4×D15C2×C414yes—
71C3×C5×C4⋊C4—22no—
72C3×C5×C4⋊C4D414yescomplemented
73C3×C5×C22⋊C4—24no—
74C3×C42×C5—24no—
75#6719—42no—
76#6719C2×C414yes—
77#6714—24no—
78#6714—42no—
79#6714C2×C414yes—
80C4⋊Dic15—22no—
81C4⋊Dic15—22no—
82#6712—24no—
83#6712—42no—
84#6712C2×C414yes—
85C4×Dic15—24no—
86C4×Dic15—24no—
87C4×Dic15C2×C414yes—
88C2×C3×#117D511yescharacteristic, complemented
89C2×C4×#67—51no—
90C2×#892—51no—
91#5212—51no—
92#4987—51no—
93#4975—51no—
94Dic3×C4⋊C4—51no—
95#4957—52no—
96#4944—52no—
97#4940—52no—
98C4×Dic3⋊C4—52no—
99C23×C4×C5C2×D311yescharacteristic
100C22×C22⋊D5—31no—
101C22×C4×D5—31no—
102C5×#117—24no—
103C2×C5×C4⋊C4C2×D311yescharacteristic
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C5×C22⋊C4C2×D312yes—
106C2×C42×C5C2×D312yes—
107C2×C22⋊Dic5—31no—
108C2×C22⋊Dic5—32no—
109C4×C22⋊D5—38no—
110C2×#685—31no—
111C2×#685—32no—
112C2×#685—32no—
113C2×C4⋊Dic5—31no—
114C2×C4⋊Dic5—31no—
115C2×#683—31no—
116C2×#683—32no—
117C2×#683—32no—
118C2×C4×Dic5—31no—
119C2×C4×Dic5—32no—
120C2×C4×Dic5—32no—
121#4038—64no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125#2354—151noSylow
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5—21no—
131C23×C3×C5—21no—
132C23×C3×C5C22×C412yes—
133C23×C3×C5C2411yescharacteristic
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—21no—
137C22×D15—21no—
138C22×D15—21no—
139C22×D15—21no—
140C22×D15C22×C412yes—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—28no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5C22×C412yes—
150C2×C3×C4×C5C22×C412yes—
151C2×C3×C4×C5C2×D414yes—
152C2×C3×C4×C5C2×D412yes—
153C2×C3×C4×C5#4112yes—
154#1220—28no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—21no—
158C2×Dic15—21no—
159C2×Dic15—22no—
160C2×Dic15—22no—
161C2×Dic15—22no—
162C2×Dic15—22no—
163C2×Dic15—22no—
164C2×Dic15—22no—
165C2×Dic15C22×C412yes—
166C2×Dic15C22×C412yes—
167C2×Dic15C22×C412yes—
168C2×Dic15#4112yes—
169C2×Dic15#4112yes—
170C4×D15—28no—
171C23×C3×C4C2×D511yescharacteristic
172C22×#67—51no—
173C22×C4×D3—51no—
174C3×#117—24no—
175C2×C3×C4⋊C4C2×D511yescharacteristic
176C2×C3×C4⋊C4C2×D511yescharacteristic
177C2×C3×C22⋊C4C2×D512yes—
178C2×C3×C42C2×D512yes—
179C2×#217—51no—
180C2×#217—52no—
181C4×#67—58no—
182C2×(C2×D3)⋊C4—51no—
183C2×(C2×D3)⋊C4—52no—
184C2×(C2×D3)⋊C4—52no—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3—51no—
187C2×Dic3⋊C4—51no—
188C2×Dic3⋊C4—52no—
189C2×Dic3⋊C4—52no—
190C2×C4×Dic3—51no—
191C2×C4×Dic3—52no—
192C2×C4×Dic3—52no—
193#892—104no—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C24×C5C22×D311yescharacteristic
198C23×D5—31no—
199C22×C4×C5—22no—
200C22×C4×C5—22no—
201C22×C4×C5C4×D314yes—
202C22×C4×C5C22×D312yes—
203C22×C4×C5C22×D311yescharacteristic
204C22×C4×C5C22×D312yes—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—34no—
207C2×C22⋊D5—62no—
208C2×C22⋊D5—62no—
209C22×Dic5—31no—
210C22×Dic5—31no—
211C22×Dic5—31no—
212C22×Dic5—32no—
213C22×Dic5—32no—
214C2×C4×D5—34no—
215C2×C4×D5—62no—
216C2×C4×D5—62no—
217C5×C4⋊C4—22no—
218C5×C4⋊C4#6714yescomplemented
219C5×C22⋊C4—24no—
220C42×C5—24no—
221C22⋊Dic5—34no—
222C22⋊Dic5—122no—
223#685—34no—
224#685—64no—
225#685—122no—
226C4⋊Dic5—62no—
227C4⋊Dic5—62no—
228#683—34no—
229#683—64no—
230#683—122no—
231C4×Dic5—34no—
232C4×Dic5—64no—
233C4×Dic5—64no—
234C2×C4×D4—151no—
235C2×#117—151no—
236C2×#117D1511yescharacteristic, complemented
237#390—152no—
238#388—152no—
239#388—152no—
240#384—151no—
241#380—151no—
242#380—151no—
243C4×C4⋊C4—151no—
244C4×C4⋊C4—152no—
245C22×C3×C5—22no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5—41no—
249C22×C3×C5—41no—
250C22×C3×C5—41no—
251C22×C3×C5—41no—
252C22×C3×C5C23×C411yescharacteristic
253C22×C3×C5C22×D411yescharacteristic
254C22×C3×C5C2×#4111yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C22×C3×C5C2×#4111yescharacteristic
257C22×C3×C5H(5,𝔽2)11yescharacteristic
258C22×C3×C52−1+411yescharacteristic
259C2×D15—22no—
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—41no—
263C2×D15—41no—
264C2×D15—41no—
265C2×D15—41no—
266C3×C4×C5—22no—
267C3×C4×C5—24no—
268C3×C4×C5—42no—
269C3×C4×C5C4×D414yes—
270Dic15—22no—
271Dic15—22no—
272Dic15—42no—
273Dic15—42no—
274Dic15#11714yescomplemented
275C24×C3C22×D511yescharacteristic
276C23×D3—51no—
277C22×C3×C4—22no—
278C22×C3×C4—22no—
279C22×C3×C4C4×D514yes—
280C22×C3×C4C22×D512yes—
281C22×C3×C4C22×D511yescharacteristic
282C22×C3×C4C22×D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C2×#67—54no—
285C2×#67—102no—
286C2×#67—102no—
287C22×Dic3—51no—
288C22×Dic3—51no—
289C22×Dic3—51no—
290C22×Dic3—52no—
291C22×Dic3—52no—
292C2×C4×D3—54no—
293C2×C4×D3—102no—
294C2×C4×D3—102no—
295C3×C4⋊C4—22no—
296C3×C4⋊C4C22⋊D514yescomplemented
297C3×C22⋊C4—24no—
298C3×C42—24no—
299#217—54no—
300#217—202no—
301(C2×D3)⋊C4—54no—
302(C2×D3)⋊C4—104no—
303(C2×D3)⋊C4—202no—
304C4⋊Dic3—102no—
305C4⋊Dic3—102no—
306Dic3⋊C4—54no—
307Dic3⋊C4—104no—
308Dic3⋊C4—202no—
309C4×Dic3—54no—
310C4×Dic3—104no—
311C4×Dic3—104no—
312C23×C5—21no—
313C23×C5—21no—
314C23×C5—21no—
315C23×C5—21no—
316C23×C5—21no—
317C23×C5—21no—
318C23×C5C2×C4×D312yes—
319C23×C5C23×D311yescharacteristic
320C22×D5—32no—
321C22×D5—61no—
322C22×D5—61no—
323C22×D5—61no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C2×C4×C5—21no—
328C2×C4×C5—22no—
329C2×C4×C5—22no—
330C2×C4×C5—21no—
331C2×C4×C5—28no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5—22no—
335C2×C4×C5C2×C4×D312yes—
336C2×C4×C5C2×C4×D312yes—
337C2×C4×C5#23512yes—
338C2×C4×C5C2×#6714yes—
339C2×C4×C5C2×#6712yes—
340C22⋊D5—68no—
341C2×Dic5—32no—
342C2×Dic5—32no—
343C2×Dic5—32no—
344C2×Dic5—32no—
345C2×Dic5—32no—
346C2×Dic5—61no—
347C2×Dic5—61no—
348C2×Dic5—61no—
349C2×Dic5—61no—
350C2×Dic5—62no—
351C2×Dic5—62no—
352C2×Dic5—62no—
353C2×Dic5—62no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C4×D5—68no—
357C22×D4—151no—
358C23×C4—151no—
359C23×C4C2×D1511yescharacteristic
360C4×D4—158no—
361#117—24no—
362#117—304no—
363C2×C4⋊C4—151no—
364C2×C4⋊C4—151no—
365C2×C4⋊C4—151no—
366C2×C4⋊C4—152no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4C2×D1511yescharacteristic
369C2×C4⋊C4C2×D1511yescharacteristic
370C2×C22⋊C4—151no—
371C2×C22⋊C4—151no—
372C2×C22⋊C4—152no—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4C2×D1512yes—
376C2×C42—151no—
377C2×C42—152no—
378C2×C42—152no—
379C2×C42C2×D1512yes—
380#95—152no—
381#95—152no—
382#95—152no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#11711yescharacteristic
385C2×C3×C5C2×C4×D411yescharacteristic
386C2×C3×C5#52011yescharacteristic
387C2×C3×C5#54711yescharacteristic
388C2×C3×C5#55011yescharacteristic
389C2×C3×C5#55511yescharacteristic
390C2×C3×C5#55611yescharacteristic
391D15—42no—
392C23×C3—21no—
393C23×C3—21no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3—21no—
397C23×C3—21no—
398C23×C3C2×C4×D512yes—
399C23×C3C23×D511yescharacteristic
400C22×D3—52no—
401C22×D3—101no—
402C22×D3—101no—
403C22×D3—101no—
404C22×D3—101no—
405C22×D3—101no—
406C22×D3—101no—
407C2×C3×C4—21no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—28no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4C2×C4×D512yes—
416C2×C3×C4C2×C4×D512yes—
417C2×C3×C4#70912yes—
418C2×C3×C4C2×C22⋊D514yes—
419C2×C3×C4C2×C22⋊D512yes—
420#67—108no—
421C2×Dic3—52no—
422C2×Dic3—52no—
423C2×Dic3—52no—
424C2×Dic3—52no—
425C2×Dic3—52no—
426C2×Dic3—101no—
427C2×Dic3—101no—
428C2×Dic3—101no—
429C2×Dic3—101no—
430C2×Dic3—102no—
431C2×Dic3—102no—
432C2×Dic3—102no—
433C2×Dic3—102no—
434C2×Dic3—102no—
435C2×Dic3—102no—
436C4×D3—108no—
437C22×C5—22no—
438C22×C5—22no—
439C22×C5—22no—
440C22×C5—41no—
441C22×C5—41no—
442C22×C5—41no—
443C22×C5—41no—
444C22×C5C22×C4×D311yescharacteristic
445C22×C5C2×#23511yescharacteristic
446C22×C5D3×#4111yescharacteristic
447C22×C5D3×#4111yescharacteristic
448C22×C5#100911yescharacteristic
449C22×C5#101011yescharacteristic
450C22×C5C22×#6711yescharacteristic
451C2×D5—62no—
452C2×D5—62no—
453C2×D5—62no—
454C2×D5—121no—
455C2×D5—121no—
456C2×D5—121no—
457C2×D5—121no—
458C4×C5—22no—
459C4×C5—24no—
460C4×C5—42no—
461C4×C5C4×#6714yes—
462Dic5—34no—
463Dic5—62no—
464Dic5—62no—
465Dic5—122no—
466Dic5—122no—
467C24—151no—
468C24C22×D1511yescharacteristic
469C2×D4—154no—
470C2×D4—302no—
471C2×D4—302no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—151no—
475C22×C4—151no—
476C22×C4—151no—
477C22×C4—154no—
478C22×C4—152no—
479C22×C4—152no—
480C22×C4—302no—
481C22×C4—302no—
482C22×C4C4×D1514yes—
483C22×C4C22×D1512yes—
484C22×C4C22×D1511yescharacteristic
485C22×C4C22×D1512yes—
486C22×C4C22×D1511yescharacteristic
487C4⋊C4—22no—
488C4⋊C4—154no—
489C4⋊C4—304no—
490C4⋊C4—302no—
491C4⋊C4—302no—
492C4⋊C4—602no—
493C4⋊C4#122014yescomplemented
494C22⋊C4—24no—
495C22⋊C4—154no—
496C22⋊C4—154no—
497C22⋊C4—304no—
498C22⋊C4—602no—
499C22⋊C4—602no—
500C42—24no—
501C42—154no—
502C42—304no—
503C42—304no—
504C3×C5#235411yescharacteristic, complemented, Hall
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—41no—
511C22×C3—41no—
512C22×C3C22×C4×D511yescharacteristic
513C22×C3C2×#70911yescharacteristic
514C22×C3D5×#4111yescharacteristic
515C22×C3D5×#4111yescharacteristic
516C22×C3#414911yescharacteristic
517C22×C3#415011yescharacteristic
518C22×C3C22×C22⋊D511yescharacteristic
519C2×D3—102no—
520C2×D3—102no—
521C2×D3—102no—
522C2×D3—201no—
523C2×D3—201no—
524C2×D3—201no—
525C2×D3—201no—
526C3×C4—22no—
527C3×C4—24no—
528C3×C4—42no—
529C3×C4C4×C22⋊D514yes—
530Dic3—54no—
531Dic3—102no—
532Dic3—102no—
533Dic3—202no—
534Dic3—202no—
535C2×C5—42no—
536C2×C5D3×#11711yescharacteristic
537C2×C5#552311yescharacteristic
538C2×C5#553211yescharacteristic
539C2×C5#553311yescharacteristic
540C2×C5C2×C4×#6711yescharacteristic
541C2×C5#582411yescharacteristic
542C2×C5#583411yescharacteristic
543D5—122no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—152no—
551C23—301no—
552C23—301no—
553C23—301no—
554C23—301no—
555C23—301no—
556C23—301no—
557C23C2×C4×D1512yes—
558C23C23×D1511yescharacteristic
559D4—308no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—152no—
569C2×C4—152no—
570C2×C4—152no—
571C2×C4—152no—
572C2×C4—152no—
573C2×C4—301no—
574C2×C4—301no—
575C2×C4—301no—
576C2×C4—301no—
577C2×C4—308no—
578C2×C4—302no—
579C2×C4—302no—
580C2×C4—302no—
581C2×C4—302no—
582C2×C4—302no—
583C2×C4—302no—
584C2×C4C2×C4×D1512yes—
585C2×C4C2×C4×D1512yes—
586C2×C4#681712yes—
587C2×C4C2×#122014yes—
588C2×C4C2×#122012yes—
589C2×C3—42no—
590C2×C3D5×#11711yescharacteristic
591C2×C3#6585711yescharacteristic
592C2×C3#6586611yescharacteristic
593C2×C3#6586711yescharacteristic
594C2×C3C2×C4×C22⋊D511yescharacteristic
595C2×C3#6615811yescharacteristic
596C2×C3#6616811yescharacteristic
597D3—202no—
598C5#8224111yescharacteristic, complemented, Sylow
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—302no—
607C22—302no—
608C22—302no—
609C22—601no—
610C22—601no—
611C22—601no—
612C22—601no—
613C22C22×C4×D1511yescharacteristic
614C22C2×#681711yescharacteristic
615C22#41×D1511yescharacteristic
616C22#41×D1511yescharacteristic
617C22#9210011yescharacteristic
618C22#9210111yescharacteristic
619C22C22×#122011yescharacteristic
620C4—22no—
621C4—24no—
622C4—42no—
623C4—154no—
624C4—302no—
625C4—302no—
626C4—602no—
627C4—602no—
628C4C4×#122014yes—
629C3#1061397211yescharacteristic, complemented, Sylow
630C2—42no—
631C2—602no—
632C2D15×#11711yescharacteristic
633C2#1175628211yescharacteristic
634C2#1175629111yescharacteristic
635C2#1175629211yescharacteristic
636C2C2×C4×#122011yescharacteristic
637C2#1175658311yescharacteristic
638C2#1175659311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×#117C2
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC2×#117D15
hypocenterC3×C5#2354
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#117C2semidirect
3. C2×#91781C2semidirect
4. #11751313C2semidirect
5. #11751088C2semidirect
6. #11751076C2semidirect
7. C4⋊C4×Dic15C2semidirect
9. #11751045C2semidirect
10. #11751041C2semidirect
11. C4×#6712C2semidirect
23. C4×#1220C4semidirect
40. C2×C5×#117D3semidirect
72. C3×C5×C4⋊C4D4semidirect
88. C2×C3×#117D5semidirect
218. C5×C4⋊C4#67semidirect
236. C2×#117D15semidirect
274. Dic15#117semidirect
296. C3×C4⋊C4C22⋊D5semidirect
493. C4⋊C4#1220semidirect
504. C3×C5#2354semidirect
598. C5#82241semidirect
629. C3#10613972semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1172
C2×C4×#12202
C2×#917812
#117513132
#117510882
#117510762
C4⋊C4×Dic152
#117510582
#117510452
#117510412
C4×#67122
#106139723
#822415