Finite Group Databasebeta

← #14326272

Subgroups — #14326272

Order 1920.

1584 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6714C211yesmaximal, characteristic
2C2×C3×#4072C211yesmaximal, characteristic, complemented
3C2×#91561C211yesmaximal, characteristic
4C5×#5211C211yesmaximal, characteristic, complemented
5#11749516C211yesmaximal, characteristic, complemented
6#11749502C211yesmaximal, characteristic
7#11749487C211yesmaximal, characteristic, complemented
8#11748898C212yesmaximal, complemented
9#11748875C212yesmaximal, complemented
10#11748829C212yesmaximal, complemented
11#11748678C212yesmaximal, complemented
12#10605853—31nomaximal, Hall
13C24×D15C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C22×#1200C2212yes—
17C2×#6714C418yescomplemented
18C2×#6714C2211yescharacteristic
19C2×#6714C2211yescharacteristic
20C2×#6714C2211yescharacteristic
21C2×#6714C2211yescharacteristic
22C2×C5×#217C2211yescharacteristic
23C2×C5×#217C2211yescharacteristic
24C2×C3×C22⋊Dic5C2211yescharacteristic
25C2×C3×C22⋊Dic5C2211yescharacteristic
26C3×#4072—24no—
27C2×C3×C4⋊Dic5C2212yescomplemented
28C2×C3×C4×Dic5C2212yescomplemented
29#91561—24no—
30C2×#6669C2212yescomplemented
31C2×#6667C2212yes—
32C2×#6667C2212yes—
33C2×#6667C2212yes—
34C2×Dic3×Dic5C2212yescomplemented
35C5×C3⋊#95C2212yescomplemented
36C5×C3⋊#95C2212yescomplemented
37#90994C2212yescomplemented
38#90994C2212yescomplemented
39#82271—51nomaximal, Hall
40C22×#685—31no—
41C2×#4072D311yescharacteristic, complemented
42C2×D5×C22⋊C4—31no—
43C5×#381—31no—
44#65244—31no—
45#65231—31no—
46C22⋊C4×Dic5—31no—
47#64959—32no—
48#64958—32no—
49#64957—32no—
50#64955—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15—22no—
53C23×D15—22no—
54C23×D15—24no—
55C23×D15—42no—
56C23×D15C2×C414yes—
57C23×D15C2311yescharacteristic
58C23×D15C2311yescharacteristic
59C22×C3×C4×C5—22no—
60C22×C3×C4×C5—22no—
61C22×C3×C4×C5C2×C414yescomplemented
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×Dic15—42no—
65C22×Dic15C2×C414yescomplemented
66C22×Dic15C2311yescharacteristic
67C22×Dic15C2311yescharacteristic
68C22×C5×Dic3C2312yes—
69C22×C5×Dic3C2312yes—
70C22×C3×Dic5C2312yes—
71C22×C3×Dic5C2312yes—
72C2×#1200—24no—
73C2×#1200—24no—
74C2×#1200—42no—
75#6714C2×C414yes—
76#6714C2×C414yes—
77#6714C2×C414yescomplemented
78#6714C2×C414yescomplemented
79C5×#217—22no—
80C5×#217—22no—
81C3×C22⋊Dic5—22no—
82C3×C22⋊Dic5D414yes—
83C3×C4⋊Dic5—24no—
84C3×C4×Dic5—24no—
85#6669—42no—
86#6667—24no—
87#6667—24no—
88#6667—42no—
89Dic3×Dic5—42no—
90C2×C3×#117—51no—
91C22×(C2×D3)⋊C4—51no—
92C2×D3×C22⋊C4—51no—
93#5211D511yescharacteristic, complemented
94#4990—52no—
95#4989—52no—
96#4955—51no—
97#4942—51no—
98Dic3×C22⋊C4—51no—
99#4940—52no—
100C4×(C2×D3)⋊C4—52no—
101C24×D5—31no—
102C23×C4×C5C2×D311yescharacteristic
103C23×Dic5—31no—
104C22×C4×D5—32no—
105C2×C5×C22⋊C4—31no—
106C2×C5×C22⋊C4—31no—
107C2×C22⋊Dic5C2×D311yescharacteristic
108C2×C22⋊Dic5C2×D311yescharacteristic
109C2×#685—38no—
110C2×#685—31no—
111C2×#685—31no—
112C2×#685—31no—
113C2×#685—31no—
114C2×#685—32no—
115C2×#685—32no—
116C2×#685—32no—
117#4072—24no—
118C2×C4⋊Dic5C2×D312yescomplemented
119C2×#683—32no—
120C2×C4×Dic5—32no—
121C2×C4×Dic5C2×D312yescomplemented
122D5×C22⋊C4—64no—
123C5×#95—32no—
124C5×#95—32no—
125#3963—32no—
126#3963—32no—
127#2342—151noSylow
128C23×C3×C5—22no—
129C23×C3×C5—22no—
130C23×C3×C5C22×C412yes—
131C23×C3×C5C2×D412yes—
132C23×C3×C5#4112yes—
133C23×C3×C5C2411yescharacteristic
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—22no—
137C22×D15—28no—
138C22×D15—21no—
139C22×D15—22no—
140C22×D15—21no—
141C22×D15—22no—
142C22×D15—28no—
143C22×D15—44no—
144C22×D15—48no—
145C22×D15—48no—
146C22×D15C22×C412yes—
147C22×D15C22×C412yes—
148C22×D15#4114yes—
149C2×C3×C4×C5—28no—
150C2×C3×C4×C5—21no—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5C22×C412yes—
155C2×C3×C4×C5C22×C412yes—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15—44no—
159C2×Dic15C22×C412yes—
160C2×Dic15C22×C412yes—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—22no—
171C2×C3×Dic5—22no—
172C2×C3×Dic5C2×D414yes—
173#1200—48no—
174C24×D3—51no—
175C23×C3×C4C2×D511yescharacteristic
176C23×Dic3—51no—
177C22×C4×D3—52no—
178C3×#117—104no—
179C2×C3×C4⋊C4—52no—
180C2×C3×C22⋊C4—51no—
181C2×C3×C22⋊C4—51no—
182C2×C3×C42—52no—
183C2×#217C2×D511yescharacteristic
184C2×#217C2×D511yescharacteristic
185C2×(C2×D3)⋊C4—58no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—51no—
188C2×(C2×D3)⋊C4—51no—
189C2×(C2×D3)⋊C4—51no—
190C2×(C2×D3)⋊C4—52no—
191C2×(C2×D3)⋊C4—52no—
192C2×(C2×D3)⋊C4—52no—
193C2×Dic3⋊C4—52no—
194C2×C4×Dic3—52no—
195D3×C22⋊C4—104no—
196C3⋊#95—52no—
197C3⋊#95—52no—
198C3⋊#95C2×D512yescomplemented
199C3⋊#95C2×D512yescomplemented
200C24×C5C22×D311yescharacteristic
201C23×D5—31no—
202C23×D5—31no—
203C23×D5—34no—
204C23×D5—64no—
205C23×D5—62no—
206C23×D5—62no—
207C23×D5—122no—
208C22×C4×C5—22no—
209C22×C4×C5—22no—
210C22×C4×C5—32no—
211C22×C4×C5—32no—
212C22×C4×C5C4×D314yescomplemented
213C22×C4×C5C22×D311yescharacteristic
214C22×C4×C5C22×D311yescharacteristic
215C22×Dic5—31no—
216C22×Dic5—31no—
217C22×Dic5—34no—
218C22×Dic5—122no—
219C22×Dic5C22×D312yes—
220C22×Dic5C22×D312yes—
221C2×C4×D5—64no—
222C2×C4×D5—64no—
223C2×C4×D5—122no—
224C5×C22⋊C4—62no—
225C5×C22⋊C4—62no—
226C22⋊Dic5—22no—
227C22⋊Dic5#6714yes—
228#685—34no—
229#685—34no—
230#685—34no—
231#685—34no—
232#685—64no—
233#685—64no—
234#685—122no—
235C4⋊Dic5—24no—
236#683—122no—
237C4×Dic5—24no—
238C4×Dic5—122no—
239C2×#117—51no—
240C22×C22⋊C4—151no—
241C22×C22⋊C4—151no—
242#388—152no—
243#388—152no—
244#385—152no—
245#381—31no—
246#380—151no—
247#379—151no—
248C4×C22⋊C4—152no—
249C4×C22⋊C4—151no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—24no—
253C22×C3×C5—24no—
254C22×C3×C5C4×D414yes—
255C22×C3×C5C23×C411yescharacteristic
256C22×C3×C5C22×D411yescharacteristic
257C22×C3×C5C2×#4111yescharacteristic
258C22×C3×C5C2×#4112yes—
259C22×C3×C5H(5,𝔽2)12yes—
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—24no—
263C2×D15—24no—
264C2×D15—22no—
265C2×D15—22no—
266C2×D15—22no—
267C2×D15—416no—
268C2×D15—42no—
269C2×D15—42no—
270C2×D15#11714yescomplemented
271C3×C4×C5—22no—
272C3×C4×C5—22no—
273Dic15—22no—
274Dic15—22no—
275C3×Dic5—24no—
276C3×Dic5—42no—
277C5×Dic3—42no—
278C5×Dic3—42no—
279C24×C3C22×D511yescharacteristic
280C23×D3—51no—
281C23×D3—51no—
282C23×D3—54no—
283C23×D3—104no—
284C23×D3—102no—
285C23×D3—102no—
286C23×D3—202no—
287C22×C3×C4—22no—
288C22×C3×C4—22no—
289C22×C3×C4—52no—
290C22×C3×C4—52no—
291C22×C3×C4C4×D514yescomplemented
292C22×C3×C4C22×D511yescharacteristic
293C22×C3×C4C22×D511yescharacteristic
294C22×Dic3—51no—
295C22×Dic3—51no—
296C22×Dic3—54no—
297C22×Dic3—202no—
298C22×Dic3C22×D512yes—
299C22×Dic3C22×D512yes—
300C2×C4×D3—104no—
301C2×C4×D3—104no—
302C2×C4×D3—202no—
303C3×C4⋊C4—104no—
304C3×C22⋊C4—54no—
305C3×C22⋊C4—102no—
306C3×C42—104no—
307#217—22no—
308#217—22no—
309(C2×D3)⋊C4—54no—
310(C2×D3)⋊C4—54no—
311(C2×D3)⋊C4—54no—
312(C2×D3)⋊C4—54no—
313(C2×D3)⋊C4—104no—
314(C2×D3)⋊C4—104no—
315(C2×D3)⋊C4—202no—
316Dic3⋊C4—202no—
317C4×Dic3—202no—
318C23×C5—22no—
319C23×C5—22no—
320C23×C5C2×C4×D312yes—
321C23×C5#23512yes—
322C23×C5C2×#6712yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—32no—
325C22×D5—34no—
326C22×D5—32no—
327C22×D5—68no—
328C22×D5—61no—
329C22×D5—62no—
330C22×D5—61no—
331C22×D5—68no—
332C22×D5—61no—
333C22×D5—62no—
334C22×D5—62no—
335C22×D5—61no—
336C22×D5—124no—
337C22×D5—128no—
338C22×D5—128no—
339C2×C4×C5—28no—
340C2×C4×C5—21no—
341C2×C4×C5—21no—
342C2×C4×C5—21no—
343C2×C4×C5—21no—
344C2×C4×C5—62no—
345C2×C4×C5—62no—
346C2×C4×C5—62no—
347C2×C4×C5—62no—
348C2×C4×C5—62no—
349C2×C4×C5—62no—
350C2×C4×C5C2×C4×D312yes—
351C2×C4×C5C2×C4×D312yes—
352C2×Dic5—22no—
353C2×Dic5—22no—
354C2×Dic5—22no—
355C2×Dic5—22no—
356C2×Dic5—22no—
357C2×Dic5—32no—
358C2×Dic5—32no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5—124no—
362C2×Dic5C2×#6714yes—
363C4×D5—128no—
364C25—151no—
365C23×C4—152no—
366C23×C4—151no—
367C23×C4D3×D511yescharacteristic
368#117—104no—
369C2×C4⋊C4—52no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—31no—
372C2×C22⋊C4—31no—
373C2×C22⋊C4—51no—
374C2×C22⋊C4—51no—
375C2×C22⋊C4—158no—
376C2×C22⋊C4—151no—
377C2×C22⋊C4—151no—
378C2×C22⋊C4—151no—
379C2×C22⋊C4—151no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4—152no—
382C2×C22⋊C4—152no—
383C2×C22⋊C4—304no—
384C2×C42—52no—
385C2×C42—152no—
386#95—32no—
387#95—32no—
388#95—152no—
389#95—152no—
390C2×C3×C5—24no—
391C2×C3×C5C2×#11711yescharacteristic
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5#51811yescharacteristic
394C2×C3×C5#54612yes—
395C2×C3×C5#55112yes—
396D15—24no—
397D15—42no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3C2×C4×D512yes—
401C23×C3D4×D512yes—
402C23×C3#71112yes—
403C23×C3C23×D511yescharacteristic
404C22×D3—52no—
405C22×D3—54no—
406C22×D3—52no—
407C22×D3—108no—
408C22×D3—101no—
409C22×D3—101no—
410C22×D3—102no—
411C22×D3—101no—
412C22×D3—101no—
413C22×D3—102no—
414C22×D3—102no—
415C22×D3—108no—
416C22×D3—204no—
417C22×D3—208no—
418C22×D3—208no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—28no—
424C2×C3×C4—54no—
425C2×C3×C4—102no—
426C2×C3×C4—102no—
427C2×C3×C4—102no—
428C2×C3×C4—102no—
429C2×C3×C4—102no—
430C2×C3×C4C2×C4×D512yes—
431C2×C3×C4C2×C4×D512yes—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—22no—
438C2×Dic3—52no—
439C2×Dic3—52no—
440C2×Dic3—102no—
441C2×Dic3—102no—
442C2×Dic3—204no—
443C4×D3—208no—
444C22×C5—22no—
445C22×C5—22no—
446C22×C5—24no—
447C22×C5—24no—
448C22×C5C4×#6714yes—
449C22×C5C22×C4×D311yescharacteristic
450C22×C5C2×#23511yescharacteristic
451C22×C5D3×#4112yes—
452C22×C5#100912yes—
453C22×C5C22×#6711yescharacteristic
454C2×D5—34no—
455C2×D5—62no—
456C2×D5—64no—
457C2×D5—62no—
458C2×D5—64no—
459C2×D5—62no—
460C2×D5—62no—
461C2×D5—62no—
462C2×D5—1216no—
463C2×D5—122no—
464C2×D5—122no—
465C4×C5—22no—
466C4×C5—22no—
467C4×C5—122no—
468C4×C5—122no—
469Dic5—24no—
470Dic5—42no—
471Dic5—62no—
472Dic5—62no—
473C24—151no—
474C24—151no—
475C24—154no—
476C24—304no—
477C24—302no—
478C24—302no—
479C24—602no—
480C24C2×D3×D511yescharacteristic
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—32no—
484C22×C4—32no—
485C22×C4—52no—
486C22×C4—52no—
487C22×C4—151no—
488C22×C4—151no—
489C22×C4—154no—
490C22×C4—304no—
491C22×C4—304no—
492C22×C4—602no—
493C22×C4—602no—
494C22×C4#120014yescomplemented
495C22×C4C2×D3×D511yescharacteristic
496C22×C4C2×D3×D511yescharacteristic
497C4⋊C4—104no—
498C4⋊C4—602no—
499C22⋊C4—54no—
500C22⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—102no—
503C22⋊C4—154no—
504C22⋊C4—154no—
505C22⋊C4—154no—
506C22⋊C4—154no—
507C22⋊C4—304no—
508C22⋊C4—304no—
509C22⋊C4—602no—
510C42—104no—
511C42—602no—
512C3×C5#234211yescharacteristic, complemented, Hall
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—24no—
516C22×C3—24no—
517C22×C3#402714yes—
518C22×C3C22×C4×D511yescharacteristic
519C22×C3C2×D4×D511yescharacteristic
520C22×C3C2×#71111yescharacteristic
521C22×C3D5×#4112yes—
522C22×C3#414912yes—
523C2×D3—54no—
524C2×D3—102no—
525C2×D3—102no—
526C2×D3—104no—
527C2×D3—104no—
528C2×D3—102no—
529C2×D3—102no—
530C2×D3—102no—
531C2×D3—2016no—
532C2×D3—202no—
533C2×D3—202no—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—104no—
537C3×C4—202no—
538Dic3—42no—
539Dic3—42no—
540Dic3—102no—
541Dic3—102no—
542C2×C5—24no—
543C2×C5D3×#11711yescharacteristic
544C2×C5#552111yescharacteristic
545C2×C5#552712yes—
546C2×C5C2×C4×#6711yescharacteristic
547C2×C5#582912yes—
548D5—64no—
549D5—122no—
550C23—22no—
551C23—22no—
552C23—154no—
553C23—152no—
554C23—152no—
555C23—308no—
556C23—302no—
557C23—301no—
558C23—301no—
559C23—302no—
560C23—302no—
561C23—301no—
562C23—301no—
563C23—308no—
564C23—604no—
565C23—608no—
566C23—608no—
567C23#678212yes—
568C23C2×#120012yes—
569C23D5×#6712yes—
570C23C22×D3×D511yescharacteristic
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—54no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—102no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—152no—
589C2×C4—152no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—608no—
593C2×C4—604no—
594C2×C4C2×#120012yes—
595C2×C4C2×#120012yes—
596C2×C3—24no—
597C2×C3C2×#402711yescharacteristic
598C2×C3D5×#11711yescharacteristic
599C2×C3#6585511yescharacteristic
600C2×C3#6598812yes—
601C2×C3#6599212yes—
602D3—104no—
603D3—202no—
604C5#8227111yescharacteristic, complemented, Sylow
605C22—22no—
606C22—24no—
607C22—22no—
608C22—24no—
609C22—154no—
610C22—302no—
611C22—302no—
612C22—304no—
613C22—304no—
614C22—302no—
615C22—302no—
616C22—302no—
617C22—6016no—
618C22—602no—
619C22—602no—
620C22#9155214yes—
621C22#9201612yes—
622C22#9201912yes—
623C22C2×#678211yescharacteristic
624C22C22×#120011yescharacteristic
625C22C2×D5×#6711yescharacteristic
626C4—22no—
627C4—22no—
628C4—104no—
629C4—122no—
630C4—122no—
631C4—202no—
632C4—302no—
633C4—302no—
634C3#1060585311yescharacteristic, complemented, Sylow
635C2—24no—
636C2—304no—
637C2—602no—
638C2#1175446111yescharacteristic
639C2#1175446212yes—
640C2#1175451511yescharacteristic
641C2#1175463912yes—
642C2C2×#9155211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC24×D15C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2342
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4072C2semidirect
4. C5×#5211C2semidirect
5. #11749516C2semidirect
7. #11749487C2semidirect
8. #11748898C2semidirect
9. #11748875C2semidirect
10. #11748829C2semidirect
11. #11748678C2semidirect
17. C2×#6714C4semidirect
27. C2×C3×C4⋊Dic5C22semidirect
28. C2×C3×C4×Dic5C22semidirect
30. C2×#6669C22semidirect
34. C2×Dic3×Dic5C22semidirect
35. C5×C3⋊#95C22semidirect
36. C5×C3⋊#95C22semidirect
37. #90994C22semidirect
38. #90994C22semidirect
41. C2×#4072D3semidirect
61. C22×C3×C4×C5C2×C4semidirect
65. C22×Dic15C2×C4semidirect
77. #6714C2×C4semidirect
78. #6714C2×C4semidirect
93. #5211D5semidirect
118. C2×C4⋊Dic5C2×D3semidirect
121. C2×C4×Dic5C2×D3semidirect
198. C3⋊#95C2×D5semidirect
199. C3⋊#95C2×D5semidirect
212. C22×C4×C5C4×D3semidirect
270. C2×D15#117semidirect
291. C22×C3×C4C4×D5semidirect
494. C22×C4#1200semidirect
512. C3×C5#2342semidirect
604. C5#82271semidirect
634. C3#10605853semidirect

Maximal subgroups

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

subgroupindex
C22×#67142
C2×C3×#40722
C2×#915612
C5×#52112
#117495162
#117495022
#117494872
#117488982
#117488752
#117488292
#117486782
#106058533
#822715