Finite Group Databasebeta

← #14331220

Subgroups — #14331220

Order 1920.

1296 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4086C211yesmaximal, characteristic
2#11751321C211yesmaximal, characteristic, complemented
3C5×#5220C211yesmaximal, characteristic, complemented
4#11749518C212yesmaximal, complemented
5#11749518C212yesmaximal, complemented
6#11749500C212yesmaximal, complemented
7#11749495C212yesmaximal, complemented
8#11748896C212yesmaximal, complemented
9#11748612C212yesmaximal, complemented
10#10606416—31nomaximal, Hall
11C22×C3×C5×D4C2211yescharacteristic
12C23×Dic15C2211yescharacteristic
13C23×C5×Dic3C2211yescharacteristic
14C22×C3×C22⋊D5C2212yes—
15C22×C3×C22⋊D5C2212yes—
16C22×C3×D20C2211yescharacteristic
17C22×D5×Dic3C2212yes—
18C2×#6719C2212yes—
19C2×#6719C2211yescharacteristic
20C2×C5×#217C2212yes—
21C2×C5×#217C2211yescharacteristic
22C3×#4086C418yescomplemented
23C2×C3×C4×Dic5C2211yescharacteristic
24C2×#6665C2212yes—
25C2×#6665C2212yes—
26C2×#6665C2212yes—
27#91115C2212yes—
28C5×C3⋊#95C2212yes—
29#90994C2212yes—
30#90994C2212yes—
31#90994C2212yes—
32#90994C2212yes—
33#83174—51nomaximal, Hall
34C2×#4086D311yescharacteristic
35C5×#386—31no—
36#65509—31no—
37#65502—32no—
38#65247—32no—
39#65245—32no—
40#65245—32no—
41#64954—32no—
42#64940—32no—
43C24×C3×C5C2311yescharacteristic
44C24×C3×C5C2311yescharacteristic
45C23×C3×D5C2312yes—
46C2×C3×C5×D4—41no—
47C2×C3×C5×D4—41no—
48C2×C3×C5×D4C2×C414yescomplemented
49C22×C3×C4×C5C2311yescharacteristic
50C22×Dic15—41no—
51C22×Dic15—42no—
52C22×Dic15C2312yes—
53C22×Dic15C2312yes—
54C22×C5×Dic3—41no—
55C22×C5×Dic3—42no—
56C22×C5×Dic3C2312yes—
57C22×C5×Dic3C2312yes—
58C2×C3×C22⋊D5—42no—
59C2×C3×C22⋊D5—42no—
60C2×C3×C22⋊D5—42no—
61C2×C3×C22⋊D5—42no—
62C2×C3×C22⋊D5C2×C418yescomplemented
63C2×C3×C22⋊D5C2×C418yescomplemented
64C22×C3×Dic5C2312yes—
65C2×C3×D20—41no—
66C2×C3×D20—41no—
67C2×C3×D20C2×C414yescomplemented
68C2×D5×Dic3—42no—
69C2×D5×Dic3—42no—
70C2×D5×Dic3—42no—
71#6719—41no—
72#6719—42no—
73C5×#217—41no—
74C5×#217—42no—
75C3×C4×Dic5C2×C414yescomplemented
76#6665—42no—
77#6665—42no—
78#6665—42no—
79C2×C3×#127—51no—
80#5220—51no—
81#5220—52no—
82#5220—52no—
83#5220—52no—
84#5220—52no—
85#5220—52no—
86#5220D511yescharacteristic, complemented
87#4937—52no—
88C22×C5×D4C2×D311yescharacteristic
89C23×C4×C5—31no—
90C22×C22⋊D5C2×D312yes—
91C22×C22⋊D5C2×D312yes—
92C23×Dic5—31no—
93C22×D20C2×D311yescharacteristic
94C22×C4×D5—32no—
95C2×C5×C22⋊C4—31no—
96C2×C5×C22⋊C4—32no—
97C2×C22⋊Dic5—31no—
98C2×C22⋊Dic5—32no—
99#4086Dic318yescomplemented
100C2×#685—32no—
101C2×#685—32no—
102C2×#685—32no—
103C2×C4×Dic5C2×D311yescharacteristic
104C5×#95—32no—
105#3963—32no—
106#3963—32no—
107#3963—32no—
108#3963—32no—
109#3963—32no—
110#2380—151noSylow
111C23×C3×C5—22no—
112C23×C3×C5—21no—
113C23×C3×C5—22no—
114C23×C3×C5—21no—
115C23×C3×C5—22no—
116C23×C3×C5—22no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5C22×C412yes—
120C23×C3×C5C22×C412yes—
121C23×C3×C5C2411yescharacteristic
122C22×C3×D5—22no—
123C22×C3×D5—22no—
124C22×C3×D5—22no—
125C22×C3×D5—22no—
126C22×C3×D5—22no—
127C22×C3×D5—22no—
128C22×C3×D5C22×C414yes—
129C3×C5×D4—44no—
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5C22×C412yes—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—22no—
136C2×Dic15—22no—
137C2×Dic15—22no—
138C2×Dic15—22no—
139C2×Dic15—44no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—44no—
147C3×C22⋊D5—48no—
148C3×C22⋊D5—48no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5C22×C414yes—
152C3×D20—44no—
153D5×Dic3—48no—
154C22×C3×D4—52no—
155C22×C3×D4—52no—
156C22×C3×D4—51no—
157C22×C3×D4C2×D511yescharacteristic
158C23×Dic3—51no—
159C23×Dic3—52no—
160C23×Dic3C2×D511yescharacteristic
161C3×#127—58no—
162C2×C3×C42—51no—
163C2×#217—52no—
164C2×#217—51no—
165C2×#217—52no—
166C2×#217—52no—
167C2×#217—52no—
168C2×#217C2×D512yes—
169C2×#217C2×D511yescharacteristic
170C3⋊#95—52no—
171C3⋊#95—52no—
172C3⋊#95—52no—
173C3⋊#95—52no—
174C3⋊#95—52no—
175C3⋊#95C2×D512yes—
176C24×C5C22×D311yescharacteristic
177C24×C5C22×D311yescharacteristic
178C23×D5C22×D312yes—
179C2×C5×D4—41no—
180C2×C5×D4—41no—
181C2×C5×D4C2×Dic314yescomplemented
182C22×C4×C5—32no—
183C22×C4×C5—32no—
184C22×C4×C5—121no—
185C22×C4×C5—122no—
186C22×C4×C5C22×D311yescharacteristic
187C2×C22⋊D5—42no—
188C2×C22⋊D5—42no—
189C2×C22⋊D5—42no—
190C2×C22⋊D5—42no—
191C2×C22⋊D5C2×Dic318yescomplemented
192C2×C22⋊D5C2×Dic318yescomplemented
193C22×Dic5—32no—
194C22×Dic5—32no—
195C22×Dic5—121no—
196C22×Dic5—122no—
197C22×Dic5C22×D312yes—
198C2×D20—41no—
199C2×D20—41no—
200C2×D20C2×Dic314yescomplemented
201C2×C4×D5—122no—
202C2×C4×D5—122no—
203C2×C4×D5—122no—
204C5×C22⋊C4—121no—
205C5×C22⋊C4—122no—
206C22⋊Dic5—121no—
207C22⋊Dic5—122no—
208#685—122no—
209#685—122no—
210#685—122no—
211C4×Dic5C2×Dic314yescomplemented
212C2×#127—51no—
213#386—31no—
214#386—152no—
215#386—152no—
216#386—152no—
217#386—152no—
218#386—152no—
219#386—151no—
220#383—152no—
221C22×C3×C5—24no—
222C22×C3×C5—24no—
223C22×C3×C5—22no—
224C22×C3×C5—22no—
225C22×C3×C5—41no—
226C22×C3×C5—41no—
227C22×C3×C5—41no—
228C22×C3×C5—42no—
229C22×C3×C5—42no—
230C22×C3×C5—41no—
231C22×C3×C5C23×C411yescharacteristic
232C22×C3×C5H(5,𝔽2)12yes—
233C22×C3×C5H(5,𝔽2)11yescharacteristic
234C22×C3×C5H(5,𝔽2)11yescharacteristic
235C22×C3×C5H(5,𝔽2)12yes—
236C2×C3×D5—24no—
237C2×C3×D5—24no—
238C2×C3×D5—24no—
239C2×C3×D5—42no—
240C2×C3×D5—42no—
241C2×C3×D5—42no—
242C2×C3×D5—42no—
243C3×C4×C5—22no—
244Dic15—42no—
245Dic15—42no—
246C3×Dic5—24no—
247C5×Dic3—42no—
248C5×Dic3—42no—
249C24×C3—52no—
250C24×C3C22×D511yescharacteristic
251C24×C3C22×D511yescharacteristic
252C2×C3×D4—41no—
253C2×C3×D4—41no—
254C2×C3×D4—58no—
255C2×C3×D4—58no—
256C2×C3×D4—54no—
257C2×C3×D4—201no—
258C2×C3×D4—201no—
259C2×C3×D4—202no—
260C2×C3×D4—202no—
261C2×C3×D4—202no—
262C2×C3×D4—202no—
263C2×C3×D4C4×D514yescomplemented
264C22×C3×C4—52no—
265C22×C3×C4C22×D511yescharacteristic
266C22×Dic3—42no—
267C22×Dic3—41no—
268C22×Dic3—52no—
269C22×Dic3—52no—
270C22×Dic3—202no—
271C22×Dic3—202no—
272C22×Dic3—202no—
273C22×Dic3—201no—
274C22×Dic3—202no—
275C22×Dic3C22×D512yes—
276C22×Dic3C22×D512yes—
277C3×C42—54no—
278#217—42no—
279#217—41no—
280#217—202no—
281#217—202no—
282#217—202no—
283#217—201no—
284#217—202no—
285C23×C5—22no—
286C23×C5—21no—
287C23×C5—22no—
288C23×C5—21no—
289C23×C5—22no—
290C23×C5—21no—
291C23×C5—22no—
292C23×C5—21no—
293C23×C5C22×Dic312yes—
294C23×C5C22×Dic312yes—
295C23×C5C23×D311yescharacteristic
296C22×D5—22no—
297C22×D5—22no—
298C22×D5—22no—
299C22×D5—22no—
300C22×D5—22no—
301C22×D5—22no—
302C22×D5C22×Dic314yes—
303C5×D4—44no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—62no—
307C2×C4×C5—62no—
308C2×C4×C5—62no—
309C2×C4×C5—62no—
310C2×C4×C5—62no—
311C2×C4×C5—62no—
312C2×C4×C5—124no—
313C2×C4×C5C22×Dic312yes—
314C22⋊D5—48no—
315C22⋊D5—48no—
316C2×Dic5—22no—
317C2×Dic5—22no—
318C2×Dic5—62no—
319C2×Dic5—62no—
320C2×Dic5—62no—
321C2×Dic5—62no—
322C2×Dic5—62no—
323C2×Dic5—62no—
324C2×Dic5—124no—
325C2×Dic5C22×Dic314yes—
326D20—44no—
327C4×D5—128no—
328C22×D4—52no—
329C22×D4—52no—
330C22×D4—51no—
331C22×D4D3×D511yescharacteristic
332C23×C4—31no—
333C23×C4—152no—
334C23×C4—151no—
335#127—58no—
336C2×C22⋊C4—31no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—152no—
339C2×C22⋊C4—152no—
340C2×C22⋊C4—152no—
341C2×C22⋊C4—151no—
342C2×C22⋊C4—152no—
343C2×C42—51no—
344#95—32no—
345#95—152no—
346#95—152no—
347#95—152no—
348#95—152no—
349#95—152no—
350C2×C3×C5—42no—
351C2×C3×C5—42no—
352C2×C3×C5#51811yescharacteristic
353C2×C3×C5#51812yes—
354C2×C3×C5#56011yescharacteristic
355C2×C3×C5#56012yes—
356C2×C3×C5#56011yescharacteristic
357C3×D5—44no—
358C23×C3—21no—
359C23×C3—22no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—22no—
363C23×C3—21no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—54no—
367C23×C3—102no—
368C23×C3—102no—
369C23×C3—102no—
370C23×C3—102no—
371C23×C3—102no—
372C23×C3—102no—
373C23×C3C2×C4×D512yes—
374C23×C3C2×C4×D512yes—
375C23×C3C23×D511yescharacteristic
376C3×D4—44no—
377C3×D4—204no—
378C3×D4—208no—
379C3×D4—208no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—54no—
383C2×C3×C4—102no—
384C2×C3×C4—102no—
385C2×C3×C4C2×C4×D512yes—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—44no—
393C2×Dic3—102no—
394C2×Dic3—102no—
395C2×Dic3—102no—
396C2×Dic3—102no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3—208no—
400C2×Dic3—204no—
401C22×C5—24no—
402C22×C5—24no—
403C22×C5—22no—
404C22×C5—22no—
405C22×C5—41no—
406C22×C5—41no—
407C22×C5—42no—
408C22×C5—42no—
409C22×C5—41no—
410C22×C5—41no—
411C22×C5#100412yes—
412C22×C5#100412yes—
413C22×C5#100411yescharacteristic
414C22×C5#100411yescharacteristic
415C22×C5C23×Dic311yescharacteristic
416C2×D5—24no—
417C2×D5—24no—
418C2×D5—24no—
419C2×D5—42no—
420C2×D5—42no—
421C2×D5—42no—
422C2×D5—42no—
423C4×C5—22no—
424C4×C5—122no—
425C4×C5—122no—
426Dic5—24no—
427Dic5—122no—
428Dic5—122no—
429C24—52no—
430C24C2×D3×D511yescharacteristic
431C24C2×D3×D511yescharacteristic
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—58no—
435C2×D4—58no—
436C2×D4—54no—
437C2×D4—201no—
438C2×D4—201no—
439C2×D4—202no—
440C2×D4—202no—
441C2×D4—202no—
442C2×D4—202no—
443C2×D4D5×Dic314yescomplemented
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—52no—
447C22×C4—122no—
448C22×C4—121no—
449C22×C4—152no—
450C22×C4—152no—
451C22×C4—602no—
452C22×C4—602no—
453C22×C4—602no—
454C22×C4—602no—
455C22×C4—601no—
456C22×C4C2×D3×D511yescharacteristic
457C22⋊C4—122no—
458C22⋊C4—121no—
459C22⋊C4—602no—
460C22⋊C4—602no—
461C22⋊C4—602no—
462C22⋊C4—602no—
463C22⋊C4—601no—
464C42—54no—
465C3×C5#238011yescharacteristic, complemented, Hall
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—41no—
471C22×C3—41no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—104no—
477C22×C3—104no—
478C22×C3—104no—
479C22×C3—202no—
480C22×C3—202no—
481C22×C3—202no—
482C22×C3—202no—
483C22×C3C22×C4×D511yescharacteristic
484C22×C3#414412yes—
485C22×C3#414411yescharacteristic
486C22×C3#414411yescharacteristic
487C22×C3#414912yes—
488C3×C4—22no—
489C3×C4—104no—
490Dic3—42no—
491Dic3—42no—
492Dic3—202no—
493Dic3—202no—
494C2×C5—42no—
495C2×C5—42no—
496C2×C5#572012yes—
497C2×C5#572011yescharacteristic
498C2×C5#572011yescharacteristic
499C2×C5#579811yescharacteristic
500C2×C5#579812yes—
501D5—44no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—54no—
511C23—102no—
512C23—102no—
513C23—102no—
514C23—102no—
515C23—102no—
516C23—102no—
517C23C2×D5×Dic312yes—
518C23C2×D5×Dic312yes—
519C23C22×D3×D511yescharacteristic
520D4—44no—
521D4—204no—
522D4—208no—
523D4—208no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—54no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—102no—
534C2×C4—102no—
535C2×C4—124no—
536C2×C4—302no—
537C2×C4—302no—
538C2×C4—302no—
539C2×C4—302no—
540C2×C4—302no—
541C2×C4—302no—
542C2×C4—608no—
543C2×C4—604no—
544C2×C4C2×D5×Dic312yes—
545C2×C3—42no—
546C2×C3—42no—
547C2×C3—204no—
548C2×C3#6582011yescharacteristic
549C2×C3#6587912yes—
550C2×C3#6592911yescharacteristic
551C2×C3#6595412yes—
552C2×C3#6600011yescharacteristic
553C5#8317411yescharacteristic, complemented, Sylow
554C22—22no—
555C22—22no—
556C22—24no—
557C22—24no—
558C22—42no—
559C22—42no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—104no—
565C22—104no—
566C22—104no—
567C22—202no—
568C22—202no—
569C22—202no—
570C22—202no—
571C22#9202512yes—
572C22#9202612yes—
573C22C22×D5×Dic311yescharacteristic
574C22#9204911yescharacteristic
575C22#9204911yescharacteristic
576C4—22no—
577C4—104no—
578C4—122no—
579C4—122no—
580C4—602no—
581C4—602no—
582C3#1060641611yescharacteristic, complemented, Sylow
583C2—42no—
584C2—42no—
585C2—204no—
586C2#1175491012yes—
587C2#1175499111yescharacteristic
588C2#1175499811yescharacteristic
589C2#1175508012yes—
590C2#1175540611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC22×C3×C5×D4C22
socle, cosocleC23×C3×C5C24
prime radicalC2×C3×#4086C2
hypercenterC22×D4D3×D5
hypocenterC3×C5#2380
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #11751321C2semidirect
3. C5×#5220C2semidirect
4. #11749518C2semidirect
5. #11749518C2semidirect
6. #11749500C2semidirect
7. #11749495C2semidirect
8. #11748896C2semidirect
9. #11748612C2semidirect
22. C3×#4086C4semidirect
48. C2×C3×C5×D4C2×C4semidirect
62. C2×C3×C22⋊D5C2×C4semidirect
63. C2×C3×C22⋊D5C2×C4semidirect
67. C2×C3×D20C2×C4semidirect
75. C3×C4×Dic5C2×C4semidirect
86. #5220D5semidirect
99. #4086Dic3semidirect
181. C2×C5×D4C2×Dic3semidirect
191. C2×C22⋊D5C2×Dic3semidirect
192. C2×C22⋊D5C2×Dic3semidirect
200. C2×D20C2×Dic3semidirect
211. C4×Dic5C2×Dic3semidirect
263. C2×C3×D4C4×D5semidirect
443. C2×D4D5×Dic3semidirect
465. C3×C5#2380semidirect
553. C5#83174semidirect
582. C3#10606416semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#40862
#117513212
C5×#52202
#117495182
#117495002
#117494952
#117488962
#117486122
#106064163
#831745