Finite Group Databasebeta

← #14323612

Subgroups — #14323612

Order 1920.

1264 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Dic15C211yesmaximal, characteristic, complemented
2C2×C3×#4075C211yesmaximal, characteristic
3C2×#91559C211yesmaximal, characteristic
4C5×#5211C211yesmaximal, characteristic, complemented
5#11749515C211yesmaximal, characteristic, complemented
6#11749500C211yesmaximal, characteristic
7#11749487C211yesmaximal, characteristic, complemented
8#11748900C212yesmaximal, complemented
9#11748772C212yesmaximal, complemented
10#11748691C212yesmaximal, complemented
11#11748680C212yesmaximal, complemented
12#10605787—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C22×C3×C22⋊D5C2211yescharacteristic
17C22×D5×Dic3C2211yescharacteristic
18C2×C4⋊Dic15—24no—
19C2×C4⋊Dic15C2212yes—
20C2×C4⋊Dic15C2212yes—
21C2×C5×#217C2211yescharacteristic
22C2×C5×#217C2211yescharacteristic
23C2×C3×C22⋊Dic5C2211yescharacteristic
24C3×#4075C418yescomplemented
25C2×C3×#685C2212yes—
26C2×C3×#683C2212yes—
27#91559—24no—
28C2×#6668C2211yescharacteristic
29C2×#6668C2212yes—
30C2×#6665C2211yescharacteristic
31C2×#6665C2212yes—
32C2×Dic3×Dic5C2211yescharacteristic
33C2×Dic3×Dic5C2212yes—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82836—51nomaximal, Hall
40C2×#4075D311yescharacteristic
41C22×C4⋊Dic5—31no—
42C2×#4027—31no—
43C5×#381—31no—
44#65247—31no—
45#65235—31no—
46C22⋊C4×Dic5—31no—
47#64959—32no—
48#64957—32no—
49#64944—32no—
50#64936—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C3×D5C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yescomplemented
56C22×C3×C4×C5C2312yes—
57C22×Dic15—22no—
58C22×Dic15—42no—
59C22×Dic15—42no—
60C22×Dic15D414yes—
61C22×Dic15C2312yes—
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C22×C5×Dic3C2311yescharacteristic
65C22×C5×Dic3C2312yes—
66C22×C5×Dic3C2311yescharacteristic
67C2×C3×C22⋊D5—22no—
68C2×C3×C22⋊D5—22no—
69C2×C3×C22⋊D5C2×C414yes—
70C22×C3×Dic5C2312yes—
71C22×C3×Dic5C2311yescharacteristic
72C2×D5×Dic3—22no—
73C2×D5×Dic3—42no—
74C4⋊Dic15—24no—
75C4⋊Dic15—24no—
76C5×#217—22no—
77C5×#217—22no—
78C3×C22⋊Dic5C2×C414yescomplemented
79C3×#685C2×C418yescomplemented
80C3×#683C2×C418yescomplemented
81#6668—22no—
82#6668—42no—
83#6665—22no—
84#6665—42no—
85Dic3×Dic5—22no—
86Dic3×Dic5—42no—
87C2×C3×#123—51no—
88C2×D4×Dic3—51no—
89C22×C4⋊Dic3—51no—
90#5220—51no—
91#5211D511yescharacteristic, complemented
92#4984—52no—
93#4980—52no—
94#4951—52no—
95#4950—51no—
96#4950—52no—
97Dic3×C22⋊C4—51no—
98C23×C4×C5C2×D311yescharacteristic
99C22×C22⋊D5C2×D311yescharacteristic
100C23×Dic5—31no—
101C23×Dic5—31no—
102C22×C4×D5—31no—
103C2×C5×C22⋊C4—31no—
104C2×C5×C22⋊C4—31no—
105C2×C22⋊Dic5C2×D311yescharacteristic
106#4075Dic318yescomplemented
107C2×#685—31no—
108C2×#685—32no—
109C2×#685C2×D312yes—
110C2×C4⋊Dic5—32no—
111C2×C4⋊Dic5—32no—
112C2×C4⋊Dic5—64no—
113C2×#683—31no—
114C2×#683—32no—
115C2×#683C2×D312yes—
116C2×C4×Dic5—31no—
117C2×C4×Dic5—32no—
118#4027—64no—
119C5×#95—32no—
120C5×#95—32no—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#2322—151noSylow
125C23×C3×C5—22no—
126C23×C3×C5—22no—
127C23×C3×C5C22×C412yes—
128C23×C3×C5C2×D412yes—
129C23×C3×C5#4112yes—
130C23×C3×C5C2411yescharacteristic
131C22×C3×D5—21no—
132C22×C3×D5—21no—
133C22×C3×D5—22no—
134C22×C3×D5—22no—
135C22×C3×D5C22×C412yes—
136C2×C3×C4×C5—28no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C22×C414yes—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—44no—
146C2×Dic15—44no—
147C2×Dic15C2×D412yes—
148C2×Dic15C2×D412yes—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—22no—
156C3×C22⋊D5—28no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—22no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5C22×C412yes—
162C2×C3×Dic5C22×C414yes—
163D5×Dic3—44no—
164C22×C3×D4—51no—
165C23×C3×C4C2×D511yescharacteristic
166C23×Dic3—51no—
167C23×Dic3—51no—
168C23×Dic3—51no—
169C3×#123—58no—
170C2×C3×C4⋊C4—52no—
171C2×C3×C22⋊C4—51no—
172C2×C3×C22⋊C4—52no—
173C2×#217—51no—
174C2×#217—52no—
175C2×#217C2×D511yescharacteristic
176C2×#217C2×D511yescharacteristic
177D4×Dic3—104no—
178C2×C4⋊Dic3—52no—
179C2×C4⋊Dic3—52no—
180C2×C4⋊Dic3—51no—
181C2×C4⋊Dic3—52no—
182C2×C4⋊Dic3—104no—
183C2×C4×Dic3—51no—
184C2×C4×Dic3—52no—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95C2×D512yes—
189C3⋊#95C2×D512yes—
190C24×C5C22×D311yescharacteristic
191C23×D5C22×D311yescharacteristic
192C22×C4×C5—22no—
193C22×C4×C5—22no—
194C22×C4×C5—31no—
195C22×C4×C5—31no—
196C22×C4×C5—32no—
197C22×C4×C5C2×Dic314yescomplemented
198C22×C4×C5C22×D312yes—
199C2×C22⋊D5—22no—
200C2×C22⋊D5—22no—
201C2×C22⋊D5C2×Dic314yes—
202C22×Dic5—31no—
203C22×Dic5—31no—
204C22×Dic5—34no—
205C22×Dic5—32no—
206C22×Dic5—62no—
207C22×Dic5—122no—
208C22×Dic5—122no—
209C22×Dic5C22×D312yes—
210C22×Dic5C22×D311yescharacteristic
211C2×C4×D5—62no—
212C2×C4×D5—122no—
213C5×C22⋊C4—62no—
214C5×C22⋊C4—62no—
215C22⋊Dic5C2×Dic314yescomplemented
216#685—62no—
217#685—122no—
218#685C2×Dic318yescomplemented
219C4⋊Dic5—64no—
220C4⋊Dic5—64no—
221#683—62no—
222#683—122no—
223#683C2×Dic318yescomplemented
224C4×Dic5—62no—
225C4×Dic5—122no—
226C2×#123—51no—
227C2×C4×D4—151no—
228C22×C4⋊C4—151no—
229#389—152no—
230#388—152no—
231#387—152no—
232#386—151no—
233#385—152no—
234#385—151no—
235#381—31no—
236C4×C22⋊C4—151no—
237C22×C3×C5—24no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—24no—
241C22×C3×C5C4×D414yes—
242C22×C3×C5C23×C411yescharacteristic
243C22×C3×C5C22×D411yescharacteristic
244C22×C3×C5C2×#4111yescharacteristic
245C22×C3×C5H(5,𝔽2)12yes—
246C22×C3×C52−1+412yes—
247C2×C3×D5—22no—
248C2×C3×D5—24no—
249C2×C3×D5—41no—
250C2×C3×D5—41no—
251C2×C3×D5—42no—
252C3×C4×C5—24no—
253Dic15—22no—
254Dic15—22no—
255Dic15—42no—
256C3×Dic5—22no—
257C3×Dic5—24no—
258C5×Dic3—41no—
259C5×Dic3—41no—
260C5×Dic3—42no—
261C24×C3—51no—
262C24×C3C22×D511yescharacteristic
263C2×C3×D4—54no—
264C2×C3×D4—102no—
265C2×C3×D4—102no—
266C22×C3×C4—22no—
267C22×C3×C4—22no—
268C22×C3×C4—52no—
269C22×C3×C4—51no—
270C22×C3×C4C4×D514yescomplemented
271C22×C3×C4C22×D512yes—
272C22×Dic3—51no—
273C22×Dic3—51no—
274C22×Dic3—54no—
275C22×Dic3—52no—
276C22×Dic3—102no—
277C22×Dic3—102no—
278C22×Dic3—202no—
279C22×Dic3—202no—
280C22×Dic3—202no—
281C22×Dic3C22×D511yescharacteristic
282C22×Dic3C22×D512yes—
283C22×Dic3C22×D511yescharacteristic
284C3×C4⋊C4—58no—
285C3×C22⋊C4—54no—
286C3×C22⋊C4—58no—
287#217—22no—
288#217—22no—
289#217—102no—
290#217—202no—
291C4⋊Dic3—102no—
292C4⋊Dic3—104no—
293C4⋊Dic3—104no—
294C4⋊Dic3—202no—
295C4×Dic3—102no—
296C4×Dic3—202no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5D3×D412yes—
300C23×C5#23712yes—
301C23×C5C22×Dic312yes—
302C23×C5C23×D311yescharacteristic
303C22×D5—21no—
304C22×D5—21no—
305C22×D5—22no—
306C22×D5—22no—
307C22×D5C22×Dic312yes—
308C2×C4×C5—28no—
309C2×C4×C5—22no—
310C2×C4×C5—22no—
311C2×C4×C5—61no—
312C2×C4×C5—61no—
313C2×C4×C5—62no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5—62no—
317C2×C4×C5—62no—
318C2×C4×C5C22×Dic314yes—
319C22⋊D5—28no—
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—32no—
325C2×Dic5—32no—
326C2×Dic5—62no—
327C2×Dic5—62no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C2×Dic5—124no—
332C2×Dic5—124no—
333C2×Dic5C22×Dic312yes—
334C2×Dic5C22×Dic314yes—
335C4×D5—124no—
336C22×D4—51no—
337C23×C4—151no—
338C23×C4—151no—
339C23×C4—151no—
340C23×C4D3×D511yescharacteristic
341#123—58no—
342C4×D4—304no—
343C2×C4⋊C4—52no—
344C2×C4⋊C4—151no—
345C2×C4⋊C4—152no—
346C2×C4⋊C4—152no—
347C2×C4⋊C4—152no—
348C2×C4⋊C4—304no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—31no—
351C2×C22⋊C4—51no—
352C2×C22⋊C4—52no—
353C2×C22⋊C4—151no—
354C2×C22⋊C4—152no—
355C2×C42—151no—
356C2×C42—152no—
357#95—32no—
358#95—32no—
359#95—152no—
360#95—152no—
361#95—152no—
362C2×C3×C5—24no—
363C2×C3×C5C2×C4×D411yescharacteristic
364C2×C3×C5#52012yes—
365C2×C3×C5#53411yescharacteristic
366C2×C3×C5#53611yescharacteristic
367C2×C3×C5#53912yes—
368C3×D5—42no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—52no—
372C23×C3—101no—
373C23×C3—101no—
374C23×C3—102no—
375C23×C3—102no—
376C23×C3C2×C4×D512yes—
377C23×C3D4×D512yes—
378C23×C3#71112yes—
379C23×C3C23×D511yescharacteristic
380C3×D4—108no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—28no—
384C2×C3×C4—54no—
385C2×C3×C4—52no—
386C2×C3×C4—101no—
387C2×C3×C4—101no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4C2×C4×D514yes—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—52no—
399C2×Dic3—52no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—102no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C2×Dic3—204no—
406C2×Dic3—204no—
407C2×Dic3—204no—
408C22×C5—22no—
409C22×C5—22no—
410C22×C5—24no—
411C22×C5—24no—
412C22×C5D4×Dic314yes—
413C22×C5C2×D3×D411yescharacteristic
414C22×C5C2×#23711yescharacteristic
415C22×C5#100912yes—
416C22×C5#101012yes—
417C22×C5C23×Dic311yescharacteristic
418C2×D5—22no—
419C2×D5—24no—
420C2×D5—41no—
421C2×D5—41no—
422C2×D5—42no—
423C4×C5—24no—
424C4×C5—121no—
425C4×C5—121no—
426C4×C5—122no—
427Dic5—22no—
428Dic5—24no—
429Dic5—62no—
430Dic5—62no—
431Dic5—122no—
432C24—51no—
433C24C2×D3×D511yescharacteristic
434C2×D4—54no—
435C2×D4—102no—
436C2×D4—102no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—31no—
440C22×C4—32no—
441C22×C4—31no—
442C22×C4—52no—
443C22×C4—51no—
444C22×C4—151no—
445C22×C4—151no—
446C22×C4—154no—
447C22×C4—152no—
448C22×C4—302no—
449C22×C4—302no—
450C22×C4—602no—
451C22×C4—602no—
452C22×C4—602no—
453C22×C4D5×Dic314yescomplemented
454C22×C4C2×D3×D512yes—
455C4⋊C4—58no—
456C4⋊C4—304no—
457C4⋊C4—304no—
458C4⋊C4—302no—
459C4⋊C4—602no—
460C22⋊C4—54no—
461C22⋊C4—58no—
462C22⋊C4—62no—
463C22⋊C4—62no—
464C22⋊C4—302no—
465C22⋊C4—602no—
466C42—302no—
467C42—602no—
468C3×C5#232211yescharacteristic, complemented, Hall
469C22×C3—24no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—24no—
473C22×C3—102no—
474C22×C3—104no—
475C22×C3—201no—
476C22×C3—201no—
477C22×C3—202no—
478C22×C3#402714yes—
479C22×C3C22×C4×D511yescharacteristic
480C22×C3C2×D4×D511yescharacteristic
481C22×C3C2×#71111yescharacteristic
482C22×C3#414912yes—
483C22×C3#415012yes—
484C3×C4—24no—
485C3×C4—102no—
486C3×C4—104no—
487Dic3—41no—
488Dic3—41no—
489Dic3—42no—
490Dic3—102no—
491Dic3—102no—
492Dic3—202no—
493C2×C5—24no—
494C2×C5#564811yescharacteristic
495C2×C5#565311yescharacteristic
496C2×C5#566512yes—
497C2×C5C2×D4×Dic311yescharacteristic
498C2×C5#582712yes—
499D5—42no—
500C23—22no—
501C23—22no—
502C23—52no—
503C23—102no—
504C23—102no—
505C23—101no—
506C23—101no—
507C23C2×D5×Dic312yes—
508C23#678012yes—
509C23#679012yes—
510C23C22×D3×D511yescharacteristic
511D4—108no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—54no—
516C2×C4—52no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—101no—
527C2×C4—101no—
528C2×C4—152no—
529C2×C4—152no—
530C2×C4—302no—
531C2×C4—302no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—302no—
535C2×C4—604no—
536C2×C4—604no—
537C2×C4—604no—
538C2×C4C2×D5×Dic314yes—
539C2×C3—24no—
540C2×C3—202no—
541C2×C3C2×#402711yescharacteristic
542C2×C3#6585712yes—
543C2×C3#6598211yescharacteristic
544C2×C3#6598711yescharacteristic
545C2×C3#6599912yes—
546C5#8283611yescharacteristic, complemented, Sylow
547C22—24no—
548C22—24no—
549C22—22no—
550C22—22no—
551C22—104no—
552C22—102no—
553C22—202no—
554C22—201no—
555C22—201no—
556C22#9155914yes—
557C22#9200112yes—
558C22#9201912yes—
559C22C22×D5×Dic311yescharacteristic
560C22C2×#678011yescharacteristic
561C22C2×#679011yescharacteristic
562C4—24no—
563C4—104no—
564C4—102no—
565C4—122no—
566C4—121no—
567C4—121no—
568C4—302no—
569C4—302no—
570C4—602no—
571C3#1060578711yescharacteristic, complemented, Sylow
572C2—24no—
573C2—202no—
574C2#1175437211yescharacteristic
575C2#1175450912yes—
576C2#1175451112yes—
577C2#1175466911yescharacteristic
578C2C2×#9155911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×C3×C22⋊D5C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2322
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×C4⋊Dic15C2semidirect
4. C5×#5211C2semidirect
5. #11749515C2semidirect
7. #11749487C2semidirect
8. #11748900C2semidirect
9. #11748772C2semidirect
10. #11748691C2semidirect
11. #11748680C2semidirect
24. C3×#4075C4semidirect
55. C22×C3×C4×C5C2×C4semidirect
78. C3×C22⋊Dic5C2×C4semidirect
79. C3×#685C2×C4semidirect
80. C3×#683C2×C4semidirect
91. #5211D5semidirect
106. #4075Dic3semidirect
197. C22×C4×C5C2×Dic3semidirect
215. C22⋊Dic5C2×Dic3semidirect
218. #685C2×Dic3semidirect
223. #683C2×Dic3semidirect
270. C22×C3×C4C4×D5semidirect
453. C22×C4D5×Dic3semidirect
468. C3×C5#2322semidirect
546. C5#82836semidirect
571. C3#10605787semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic152
C2×C3×#40752
C2×#915592
C5×#52112
#117495152
#117495002
#117494872
#117489002
#117487722
#117486912
#117486802
#106057873
#828365