Finite Group Databasebeta

← #14323459

Subgroups — #14323459

Order 1920.

1616 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6714C211yesmaximal, characteristic, complemented
2C2×#91572C211yesmaximal, characteristic
3C2×#91559C211yesmaximal, characteristic, complemented
4C5×#5210C211yesmaximal, characteristic, complemented
5C3×#65502C211yesmaximal, characteristic, complemented
6#11749502C211yesmaximal, characteristic, complemented
7#11749487C211yesmaximal, characteristic, complemented
8#11748913C212yesmaximal, complemented
9#11748872C212yesmaximal, complemented
10#11748671C212yesmaximal, complemented
11#11748669C212yesmaximal, complemented
12#10606323—31nomaximal, Hall
13C24×D15C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C22×C3×C22⋊D5C2211yescharacteristic
17C22×#1202C2212yes—
18C22×D5×Dic3C2211yescharacteristic, complemented
19C2×#6714—24no—
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×C5×#217C2211yescharacteristic
23C2×C5×#217C2211yescharacteristic
24C2×C5×Dic3⋊C4C2212yescomplemented
25C2×C3×C22⋊Dic5C2211yescharacteristic
26C2×C3×#685C2212yescomplemented
27#91572C418yescomplemented
28#91559—24no—
29C2×#6668C2211yescharacteristic, complemented
30C2×#6667C2212yes—
31C2×#6667C2212yes—
32C2×#6667C2212yes—
33C2×#6665C2211yescharacteristic, complemented
34C2×Dic3×Dic5C2211yescharacteristic, complemented
35C2×Dic3×Dic5C2212yescomplemented
36C5×C3⋊#95C2212yescomplemented
37C3×#3963C2212yescomplemented
38#90994C2212yescomplemented
39#74875—51nomaximal, Hall
40C22×#685—31no—
41C2×#4028—31no—
42C2×#4027—31no—
43C5×#385—31no—
44#65502D311yescharacteristic, complemented
45#65279—32no—
46#65277—32no—
47#65244—31no—
48C22⋊C4×Dic5—31no—
49#64958—32no—
50#64954—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15—22no—
53C23×D15—22no—
54C23×D15—42no—
55C23×D15—42no—
56C23×D15C2×C414yescomplemented
57C23×D15C2312yes—
58C23×C3×D5C2311yescharacteristic
59C22×C3×C4×C5—22no—
60C22×C3×C4×C5—42no—
61C22×C3×C4×C5C2312yes—
62C22×Dic15—42no—
63C22×Dic15D414yescomplemented
64C22×Dic15C2311yescharacteristic
65C22×Dic15C2311yescharacteristic
66C22×C5×Dic3C2311yescharacteristic
67C22×C5×Dic3C2311yescharacteristic
68C22×C5×Dic3C2312yes—
69C2×C3×C22⋊D5—22no—
70C2×C3×C22⋊D5—22no—
71C2×C3×C22⋊D5C2×C414yes—
72C22×C3×Dic5C2312yes—
73C22×C3×Dic5C2311yescharacteristic
74C2×#1202—42no—
75C2×#1202—42no—
76C2×#1202C2×C418yescomplemented
77C2×D5×Dic3—22no—
78C2×D5×Dic3—42no—
79#6714—24no—
80#6714—24no—
81C5×#217—22no—
82C5×#217C2×C414yescomplemented
83C5×Dic3⋊C4—42no—
84C3×C22⋊Dic5—22no—
85C3×#685—42no—
86#6668—22no—
87#6667—42no—
88#6667—42no—
89#6667C2×C418yescomplemented
90#6665—22no—
91Dic3×Dic5—22no—
92Dic3×Dic5—42no—
93C2×#937—51no—
94C2×D4×Dic3—51no—
95C22×(C2×D3)⋊C4—51no—
96C3×#386—51no—
97#5210D511yescharacteristic, complemented
98#4958—52no—
99#4957—52no—
100#4955—51no—
101Dic3×C22⋊C4—51no—
102#4670—52no—
103#4669—52no—
104C24×D5—31no—
105C23×C4×C5C2×D311yescharacteristic
106C22×C22⋊D5C2×D311yescharacteristic
107C23×Dic5—31no—
108C22×D20—32no—
109C22×C4×D5—31no—
110C2×C5×C4⋊C4—32no—
111C2×C5×C22⋊C4—31no—
112C2×C5×C22⋊C4—31no—
113C2×C22⋊Dic5C2×D311yescharacteristic
114C2×#685—32no—
115C2×#685—31no—
116C2×#685—32no—
117C2×#685—32no—
118C2×#685—32no—
119C2×#685—32no—
120C2×#685—64no—
121C2×#685C2×D312yescomplemented
122C2×#683—31no—
123C2×C4×Dic5—31no—
124C2×C4×Dic5—32no—
125#4028—38no—
126#4027—64no—
127C5×#95—32no—
128#3963—32no—
129#3963C2×D312yescomplemented
130#2321—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5C22×C412yes—
134C23×C3×C5C2×D412yes—
135C23×C3×C5#4112yes—
136C23×C3×C5C2411yescharacteristic
137C22×D15—28no—
138C22×D15—22no—
139C22×D15—22no—
140C22×D15—22no—
141C22×D15—22no—
142C22×D15—22no—
143C22×D15—22no—
144C22×D15—48no—
145C22×D15—44no—
146C22×D15—44no—
147C22×D15—48no—
148C22×D15C22×C414yes—
149C22×C3×D5—21no—
150C22×C3×D5—21no—
151C22×C3×D5—22no—
152C22×C3×D5—22no—
153C22×C3×D5C22×C412yes—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5—44no—
158C2×Dic15—22no—
159C2×Dic15—22no—
160C2×Dic15—44no—
161C2×Dic15C2×D412yes—
162C2×Dic15C2×D412yes—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—21no—
166C2×C5×Dic3—21no—
167C2×C5×Dic3—22no—
168C2×C5×Dic3—22no—
169C2×C5×Dic3C22×C414yes—
170C3×C22⋊D5—28no—
171C2×C3×Dic5—21no—
172C2×C3×Dic5—21no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5—22no—
175C2×C3×Dic5—22no—
176C2×C3×Dic5C22×C412yes—
177#1202—48no—
178D5×Dic3—44no—
179C24×D3—51no—
180C22×C3×D4—51no—
181C23×C3×C4C2×D511yescharacteristic
182C22×#67—52no—
183C23×Dic3—51no—
184C23×Dic3—51no—
185C2×C3×C22⋊C4—51no—
186C2×C3×C22⋊C4—52no—
187C2×#217—51no—
188C2×#217C2×D511yescharacteristic
189C2×#217C2×D511yescharacteristic
190#937—58no—
191D4×Dic3—104no—
192C2×(C2×D3)⋊C4—52no—
193C2×(C2×D3)⋊C4—52no—
194C2×(C2×D3)⋊C4—52no—
195C2×(C2×D3)⋊C4—52no—
196C2×(C2×D3)⋊C4—52no—
197C2×(C2×D3)⋊C4—104no—
198C2×C4⋊Dic3—51no—
199C2×Dic3⋊C4C2×D512yescomplemented
200C2×C4×Dic3—51no—
201C2×C4×Dic3—52no—
202C3×#95—52no—
203C3⋊#95—52no—
204C3⋊#95C2×D512yescomplemented
205C24×C5C22×D311yescharacteristic
206C23×D5—32no—
207C23×D5—34no—
208C23×D5—62no—
209C23×D5—62no—
210C23×D5—122no—
211C23×D5—122no—
212C23×D5C22×D311yescharacteristic
213C22×C4×C5—22no—
214C22×C4×C5—32no—
215C22×C4×C5—31no—
216C22×C4×C5—31no—
217C22×C4×C5—42no—
218C22×C4×C5C22×D312yes—
219C2×C22⋊D5—22no—
220C2×C22⋊D5—22no—
221C2×C22⋊D5C4×D314yes—
222C22×Dic5—31no—
223C22×Dic5—31no—
224C22×Dic5—34no—
225C22×Dic5—122no—
226C22×Dic5C22×D312yes—
227C22×Dic5C22×D311yescharacteristic
228C2×D20—38no—
229C2×D20—122no—
230C2×D20—122no—
231C2×C4×D5—62no—
232C2×C4×D5—122no—
233C5×C4⋊C4—122no—
234C5×C22⋊C4—34no—
235C5×C22⋊C4—62no—
236C22⋊Dic5—22no—
237#685—38no—
238#685—42no—
239#685—64no—
240#685—64no—
241#685—62no—
242#685—122no—
243#685—122no—
244#683—62no—
245C4×Dic5—62no—
246C4×Dic5—122no—
247C2×#120—151no—
248C2×C4×D4—151no—
249C22×C22⋊C4—151no—
250#390—152no—
251#388—152no—
252#388—152no—
253#386—51no—
254#386—152no—
255#385—31no—
256#379—151no—
257C4×C22⋊C4—151no—
258C22×C3×C5—24no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—24no—
262C22×C3×C5C4×D414yes—
263C22×C3×C5C23×C411yescharacteristic
264C22×C3×C5C22×D411yescharacteristic
265C22×C3×C5C2×#4111yescharacteristic
266C22×C3×C5H(5,𝔽2)12yes—
267C22×C3×C5H(5,𝔽2)12yes—
268C2×D15—24no—
269C2×D15—24no—
270C2×D15—24no—
271C2×D15—416no—
272C2×D15—42no—
273C2×D15—42no—
274C2×D15—42no—
275C2×D15—42no—
276C2×C3×D5—22no—
277C2×C3×D5—24no—
278C2×C3×D5—41no—
279C2×C3×D5—41no—
280C2×C3×D5—42no—
281C3×C4×C5—42no—
282Dic15—22no—
283Dic15—22no—
284C3×Dic5—22no—
285C3×Dic5—42no—
286C5×Dic3—24no—
287C5×Dic3—41no—
288C5×Dic3—41no—
289C24×C3—51no—
290C24×C3C22×D511yescharacteristic
291C23×D3—52no—
292C23×D3—54no—
293C23×D3—102no—
294C23×D3—102no—
295C23×D3—202no—
296C23×D3—202no—
297C2×C3×D4—54no—
298C2×C3×D4—102no—
299C2×C3×D4—102no—
300C22×C3×C4—22no—
301C22×C3×C4—42no—
302C22×C3×C4—51no—
303C22×C3×C4—52no—
304C22×C3×C4C22×D512yes—
305C2×#67—58no—
306C2×#67—202no—
307C2×#67—202no—
308C22×Dic3—51no—
309C22×Dic3—51no—
310C22×Dic3—54no—
311C22×Dic3—102no—
312C22×Dic3—202no—
313C22×Dic3—202no—
314C22×Dic3C22×D511yescharacteristic
315C22×Dic3C22×D511yescharacteristic
316C22×Dic3C22×D512yes—
317C3×C22⋊C4—102no—
318C3×C22⋊C4—202no—
319#217—22no—
320#217—102no—
321#217C4×D514yescomplemented
322(C2×D3)⋊C4—58no—
323(C2×D3)⋊C4—104no—
324(C2×D3)⋊C4—104no—
325(C2×D3)⋊C4—202no—
326(C2×D3)⋊C4—202no—
327C4⋊Dic3—102no—
328Dic3⋊C4—42no—
329C4×Dic3—102no—
330C4×Dic3—202no—
331C23×C5—22no—
332C23×C5—22no—
333C23×C5C2×C4×D312yes—
334C23×C5D3×D412yes—
335C23×C5#23712yes—
336C23×C5C23×D311yescharacteristic
337C22×D5—21no—
338C22×D5—21no—
339C22×D5—22no—
340C22×D5—22no—
341C22×D5—34no—
342C22×D5—68no—
343C22×D5—62no—
344C22×D5—62no—
345C22×D5—62no—
346C22×D5—62no—
347C22×D5—62no—
348C22×D5—62no—
349C22×D5—128no—
350C22×D5—124no—
351C22×D5—124no—
352C22×D5—128no—
353C22×D5C2×C4×D312yes—
354C2×C4×C5—22no—
355C2×C4×C5—22no—
356C2×C4×C5—22no—
357C2×C4×C5—34no—
358C2×C4×C5—44no—
359C2×C4×C5—62no—
360C2×C4×C5—62no—
361C2×C4×C5—61no—
362C2×C4×C5—61no—
363C2×C4×C5—62no—
364C2×C4×C5—62no—
365C22⋊D5—28no—
366C2×Dic5—21no—
367C2×Dic5—21no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—22no—
371C2×Dic5—32no—
372C2×Dic5—32no—
373C2×Dic5—62no—
374C2×Dic5—62no—
375C2×Dic5—124no—
376C2×Dic5C2×C4×D312yes—
377D20—128no—
378C4×D5—124no—
379C25—151no—
380C22×D4—51no—
381C22×D4—152no—
382C23×C4—151no—
383C23×C4—151no—
384C23×C4D3×D511yescharacteristic
385#120—158no—
386C4×D4—304no—
387C2×C4⋊C4—32no—
388C2×C4⋊C4—151no—
389C2×C22⋊C4—31no—
390C2×C22⋊C4—31no—
391C2×C22⋊C4—51no—
392C2×C22⋊C4—52no—
393C2×C22⋊C4—152no—
394C2×C22⋊C4—151no—
395C2×C22⋊C4—152no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—304no—
400C2×C42—151no—
401C2×C42—152no—
402#95—32no—
403#95—52no—
404#95—152no—
405C2×C3×C5—24no—
406C2×C3×C5C2×C4×D411yescharacteristic
407C2×C3×C5#51812yes—
408C2×C3×C5#53411yescharacteristic
409C2×C3×C5#53511yescharacteristic
410C2×C3×C5#53812yes—
411D15—44no—
412C3×D5—42no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3—52no—
416C23×C3—101no—
417C23×C3—102no—
418C23×C3—101no—
419C23×C3—102no—
420C23×C3C2×C4×D512yes—
421C23×C3D4×D512yes—
422C23×C3#71312yes—
423C23×C3C23×D511yescharacteristic
424C22×D3—54no—
425C22×D3—102no—
426C22×D3—102no—
427C22×D3—102no—
428C22×D3—102no—
429C22×D3—102no—
430C22×D3—102no—
431C22×D3—108no—
432C22×D3—208no—
433C22×D3—204no—
434C22×D3—208no—
435C22×D3—204no—
436C3×D4—108no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—44no—
441C2×C3×C4—52no—
442C2×C3×C4—101no—
443C2×C3×C4—101no—
444C2×C3×C4—102no—
445C2×C3×C4—102no—
446C2×C3×C4—102no—
447#67—208no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—21no—
451C2×Dic3—21no—
452C2×Dic3—22no—
453C2×Dic3—22no—
454C2×Dic3—52no—
455C2×Dic3—52no—
456C2×Dic3—102no—
457C2×Dic3—102no—
458C2×Dic3—204no—
459C2×Dic3—204no—
460C2×Dic3C2×C4×D514yes—
461C22×C5—22no—
462C22×C5—24no—
463C22×C5—22no—
464C22×C5—24no—
465C22×C5#67⋊C414yes—
466C22×C5C22×C4×D311yescharacteristic
467C22×C5C2×D3×D411yescharacteristic
468C22×C5C2×#23711yescharacteristic
469C22×C5#100412yes—
470C22×C5#100912yes—
471C2×D5—22no—
472C2×D5—24no—
473C2×D5—41no—
474C2×D5—41no—
475C2×D5—42no—
476C2×D5—64no—
477C2×D5—64no—
478C2×D5—64no—
479C2×D5—1216no—
480C2×D5—122no—
481C2×D5—122no—
482C2×D5—122no—
483C2×D5—122no—
484C4×C5—42no—
485C4×C5—64no—
486C4×C5—121no—
487C4×C5—121no—
488Dic5—22no—
489Dic5—42no—
490Dic5—62no—
491Dic5—62no—
492C24—51no—
493C24—152no—
494C24—154no—
495C24—302no—
496C24—302no—
497C24—602no—
498C24—602no—
499C24C2×D3×D511yescharacteristic
500C2×D4—54no—
501C2×D4—102no—
502C2×D4—102no—
503C2×D4—158no—
504C2×D4—602no—
505C2×D4—602no—
506C22×C4—22no—
507C22×C4—32no—
508C22×C4—31no—
509C22×C4—31no—
510C22×C4—42no—
511C22×C4—51no—
512C22×C4—52no—
513C22×C4—151no—
514C22×C4—151no—
515C22×C4—154no—
516C22×C4—302no—
517C22×C4—602no—
518C22×C4—602no—
519C22×C4C2×D3×D512yes—
520C4⋊C4—122no—
521C4⋊C4—302no—
522C22⋊C4—34no—
523C22⋊C4—62no—
524C22⋊C4—102no—
525C22⋊C4—158no—
526C22⋊C4—202no—
527C22⋊C4—304no—
528C22⋊C4—304no—
529C22⋊C4—302no—
530C22⋊C4—602no—
531C22⋊C4—602no—
532C42—302no—
533C42—602no—
534C3×C5#232111yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—22no—
537C22×C3—24no—
538C22×C3—22no—
539C22×C3—102no—
540C22×C3—104no—
541C22×C3—201no—
542C22×C3—202no—
543C22×C3—201no—
544C22×C3#403914yes—
545C22×C3C22×C4×D511yescharacteristic
546C22×C3C2×D4×D511yescharacteristic
547C22×C3#414412yes—
548C22×C3C2×#71311yescharacteristic
549C22×C3#414912yes—
550C2×D3—104no—
551C2×D3—104no—
552C2×D3—104no—
553C2×D3—2016no—
554C2×D3—202no—
555C2×D3—202no—
556C2×D3—202no—
557C2×D3—202no—
558C3×C4—42no—
559C3×C4—102no—
560C3×C4—202no—
561Dic3—24no—
562Dic3—41no—
563Dic3—41no—
564Dic3—102no—
565Dic3—102no—
566C2×C5—24no—
567C2×C5C2×#67⋊C411yescharacteristic
568C2×C5#554512yes—
569C2×C5#559211yescharacteristic
570C2×C5#561712yes—
571C2×C5#565311yescharacteristic
572D5—42no—
573D5—124no—
574C23—22no—
575C23—22no—
576C23—52no—
577C23—101no—
578C23—102no—
579C23—101no—
580C23—102no—
581C23—154no—
582C23—308no—
583C23—302no—
584C23—302no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23—302no—
589C23—608no—
590C23—604no—
591C23—604no—
592C23—608no—
593C23#676612yes—
594C23C4×D3×D512yes—
595C23#679012yes—
596C23C22×D3×D511yescharacteristic
597D4—108no—
598D4—608no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—34no—
603C2×C4—44no—
604C2×C4—52no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—62no—
611C2×C4—101no—
612C2×C4—101no—
613C2×C4—102no—
614C2×C4—102no—
615C2×C4—102no—
616C2×C4—152no—
617C2×C4—152no—
618C2×C4—302no—
619C2×C4—302no—
620C2×C4—604no—
621C2×C4—604no—
622C2×C3—24no—
623C2×C3—202no—
624C2×C3C2×#403911yescharacteristic
625C2×C3#6587912yes—
626C2×C3#6592611yescharacteristic
627C2×C3#6597812yes—
628C2×C3#6598711yescharacteristic
629D3—204no—
630C5#7487511yescharacteristic, complemented, Sylow
631C22—24no—
632C22—22no—
633C22—22no—
634C22—24no—
635C22—102no—
636C22—104no—
637C22—201no—
638C22—201no—
639C22—202no—
640C22—304no—
641C22—304no—
642C22—304no—
643C22—6016no—
644C22—602no—
645C22—602no—
646C22—602no—
647C22—602no—
648C22#9139614yes—
649C22C2×#676611yescharacteristic
650C22C2×C4×D3×D511yescharacteristic
651C22#9201912yes—
652C22C2×#679011yescharacteristic
653C22#9204912yes—
654C4—42no—
655C4—64no—
656C4—102no—
657C4—121no—
658C4—121no—
659C4—202no—
660C4—302no—
661C4—302no—
662C3#1060632311yescharacteristic, complemented, Sylow
663C2—24no—
664C2—202no—
665C2—604no—
666C2C2×#9139611yescharacteristic
667C2#1175447612yes—
668C2#1175462912yes—
669C2#1175466911yescharacteristic
670C2#1175540011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC2×#91572C2
hypercenterC23×C4D3×D5
hypocenterC3×C5#2321
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6714C2semidirect
3. C2×#91559C2semidirect
4. C5×#5210C2semidirect
5. C3×#65502C2semidirect
6. #11749502C2semidirect
7. #11749487C2semidirect
8. #11748913C2semidirect
9. #11748872C2semidirect
10. #11748671C2semidirect
11. #11748669C2semidirect
18. C22×D5×Dic3C22semidirect
24. C2×C5×Dic3⋊C4C22semidirect
26. C2×C3×#685C22semidirect
27. #91572C4semidirect
29. C2×#6668C22semidirect
33. C2×#6665C22semidirect
34. C2×Dic3×Dic5C22semidirect
35. C2×Dic3×Dic5C22semidirect
36. C5×C3⋊#95C22semidirect
37. C3×#3963C22semidirect
38. #90994C22semidirect
44. #65502D3semidirect
56. C23×D15C2×C4semidirect
63. C22×Dic15D4semidirect
76. C2×#1202C2×C4semidirect
82. C5×#217C2×C4semidirect
89. #6667C2×C4semidirect
97. #5210D5semidirect
121. C2×#685C2×D3semidirect
129. #3963C2×D3semidirect
199. C2×Dic3⋊C4C2×D5semidirect
204. C3⋊#95C2×D5semidirect
321. #217C4×D5semidirect
534. C3×C5#2321semidirect
630. C5#74875semidirect
662. C3#10606323semidirect

Maximal subgroups

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

subgroupindex
C22×#67142
C2×#915722
C2×#915592
C5×#52102
C3×#655022
#117495022
#117494872
#117489132
#117488722
#117486712
#117486692
#106063233
#748755