Finite Group Databasebeta

← #14323609

Subgroups — #14323609

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×#6712C211yesmaximal, characteristic, complemented
2C2×C3×C4×C22⋊D5C211yesmaximal, characteristic
3C2×#91537C211yesmaximal, characteristic
4C4×C5×#217C211yesmaximal, characteristic, complemented
5#11749517C211yesmaximal, characteristic, complemented
6#11749500C211yesmaximal, characteristic
7#11749473C211yesmaximal, characteristic, complemented
8#11748820C212yesmaximal, complemented
9#11748814C212yesmaximal, complemented
10#11748706C212yesmaximal, complemented
11#11748704C212yesmaximal, complemented
12#10605177—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C22×C3×C22⋊D5C2211yescharacteristic
17C22×C3×C4×D5C2211yescharacteristic
18C2×#6712—24no—
19C2×#6712C2212yes—
20C2×#6712C2212yes—
21C2×C5×#217C2211yescharacteristic
22C2×C5×#217C2211yescharacteristic
23C2×C4×C5×Dic3C2212yes—
24C2×C3×C22⋊Dic5C2211yescharacteristic
25C3×C4×C22⋊D5—24no—
26C2×C3×#685C2211yescharacteristic
27C2×C3×#683C2211yescharacteristic
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×Dic3⋊Dic5C2212yes—
30C2×Dic3⋊Dic5C2212yes—
31#91537C418yescomplemented
32C2×#6665C2212yes—
33C2×#6665C2212yes—
34C5×C3⋊#95C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82326—51nomaximal, Hall
40C2×C4×C22⋊D5D311yescharacteristic
41C22×#683—31no—
42C2×#4032—31no—
43C4×C5×C22⋊C4—31no—
44#65247—31no—
45#65239—31no—
46#65232—31no—
47#65229—32no—
48#65217—32no—
49#64957—32no—
50#64942—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C3×D5C2311yescharacteristic
53C22×C3×C4×C5—42no—
54C22×C3×C4×C5D414yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15—22no—
58C22×Dic15—22no—
59C22×Dic15—22no—
60C22×Dic15—42no—
61C22×Dic15C2×C414yescomplemented
62C22×Dic15C2312yes—
63C22×Dic15C2312yes—
64C22×C5×Dic3C2312yes—
65C22×C5×Dic3C2312yes—
66C2×C3×C22⋊D5—22no—
67C2×C3×C22⋊D5—22no—
68C2×C3×C22⋊D5C2×C414yes—
69C22×C3×Dic5C2311yescharacteristic
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2311yescharacteristic
72C2×C3×C4×D5—22no—
73C2×C3×C4×D5—42no—
74#6712—24no—
75#6712—24no—
76C5×#217—22no—
77C5×#217C2×C414yescomplemented
78C4×C5×Dic3—42no—
79C3×C22⋊Dic5—22no—
80C3×#685—22no—
81C3×#683—22no—
82C3×C4×Dic5—22no—
83Dic3⋊Dic5—42no—
84Dic3⋊Dic5C2×C418yescomplemented
85#6665—42no—
86#6665C2×C418yescomplemented
87C2×C3×C4×D4—51no—
88C2×#935—51no—
89C22×Dic3⋊C4—51no—
90#5220—51no—
91#5210—52no—
92C4×#217D511yescharacteristic, complemented
93#4976—52no—
94#4946—51no—
95#4945—52no—
96#4943—51no—
97#4933—52no—
98C23×C4×C5C2×D311yescharacteristic
99C22×C22⋊D5C2×D311yescharacteristic
100C23×Dic5—31no—
101C23×Dic5—31no—
102C22×C4×D5C2×D311yescharacteristic
103C2×C5×C22⋊C4—31no—
104C2×C5×C22⋊C4—31no—
105C2×C42×C5—32no—
106C2×C22⋊Dic5C2×D311yescharacteristic
107C4×C22⋊D5—24no—
108C2×#685—32no—
109C2×#685—32no—
110C2×#685C2×D311yescharacteristic
111C2×C4⋊Dic5—32no—
112C2×C4⋊Dic5—32no—
113C2×#683—32no—
114C2×#683—32no—
115C2×#683—64no—
116C2×#683C2×D311yescharacteristic
117C2×C4×Dic5C2×D311yescharacteristic
118#4032—38no—
119C5×#95—32no—
120#3963—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—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—44no—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D412yes—
141C2×Dic15—28no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—22no—
147C2×Dic15—44no—
148C2×Dic15C22×C414yes—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3C22×C414yes—
155C3×C22⋊D5—28no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—21no—
160C2×C3×Dic5—22no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5C22×C412yes—
163C3×C4×D5—44no—
164C22×C3×D4—51no—
165C23×C3×C4—51no—
166C23×C3×C4C2×D511yescharacteristic
167C23×Dic3—51no—
168C23×Dic3—51no—
169C3×C4×D4—104no—
170C2×C3×C4⋊C4—51no—
171C2×C3×C22⋊C4—51no—
172C2×C3×C22⋊C4—51no—
173C2×C3×C42—51no—
174C2×#217—52no—
175C2×#217—52no—
176C2×#217C2×D511yescharacteristic
177C2×#217C2×D511yescharacteristic
178#935—58no—
179C2×Dic3⋊C4—52no—
180C2×Dic3⋊C4—52no—
181C2×Dic3⋊C4—52no—
182C2×Dic3⋊C4—52no—
183C2×Dic3⋊C4—104no—
184C2×C4×Dic3C2×D512yes—
185C3⋊#95—52no—
186C3⋊#95—52no—
187C3⋊#95—52no—
188C3⋊#95—52no—
189C3⋊#95C2×D512yes—
190C24×C5C22×D311yescharacteristic
191C23×D5C22×D311yescharacteristic
192C22×C4×C5—32no—
193C22×C4×C5—32no—
194C22×C4×C5—42no—
195C22×C4×C5#6714yes—
196C22×C4×C5C22×D311yescharacteristic
197C22×C4×C5C22×D311yescharacteristic
198C2×C22⋊D5—22no—
199C2×C22⋊D5—22no—
200C2×C22⋊D5C4×D314yes—
201C22×Dic5—34no—
202C22×Dic5—32no—
203C22×Dic5—32no—
204C22×Dic5—62no—
205C22×Dic5—62no—
206C22×Dic5—62no—
207C22×Dic5—122no—
208C22×Dic5C22×D311yescharacteristic
209C22×Dic5C22×D311yescharacteristic
210C22×Dic5C22×D311yescharacteristic
211C2×C4×D5—22no—
212C2×C4×D5—42no—
213C5×C22⋊C4—34no—
214C5×C22⋊C4—62no—
215C42×C5—122no—
216C22⋊Dic5—22no—
217#685—22no—
218#685—38no—
219#685—122no—
220C4⋊Dic5—38no—
221C4⋊Dic5—122no—
222#683—22no—
223#683—64no—
224#683—64no—
225C4×Dic5—22no—
226C2×#123—151no—
227C2×C4×D4—51no—
228C22×C4⋊C4—151no—
229#389—152no—
230#388—152no—
231#387—152no—
232#386—151no—
233#385—152no—
234#385—151no—
235#381—151no—
236C4×C22⋊C4—31no—
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—22no—
253C3×C4×C5—22no—
254Dic15—24no—
255Dic15—42no—
256C3×Dic5—22no—
257C3×Dic5—41no—
258C3×Dic5—41no—
259C5×Dic3—24no—
260C5×Dic3—42no—
261C24×C3—51no—
262C24×C3C22×D511yescharacteristic
263C2×C3×D4—54no—
264C2×C3×D4—102no—
265C2×C3×D4—102no—
266C22×C3×C4—42no—
267C22×C3×C4—51no—
268C22×C3×C4—51no—
269C22×C3×C4—51no—
270C22×C3×C4—102no—
271C22×C3×C4—202no—
272C22×C3×C4D2014yes—
273C22×C3×C4C22×D511yescharacteristic
274C22×C3×C4C22×D511yescharacteristic
275C22×Dic3—54no—
276C22×Dic3—52no—
277C22×Dic3—52no—
278C22×Dic3—102no—
279C22×Dic3—102no—
280C22×Dic3—102no—
281C22×Dic3—202no—
282C22×Dic3C22×D512yes—
283C22×Dic3C22×D512yes—
284C3×C4⋊C4—102no—
285C3×C22⋊C4—102no—
286C3×C22⋊C4—102no—
287C3×C42—102no—
288#217—22no—
289#217—58no—
290#217—202no—
291#217C4×D514yescomplemented
292Dic3⋊C4—58no—
293Dic3⋊C4—104no—
294Dic3⋊C4—104no—
295Dic3⋊C4—202no—
296C4×Dic3—42no—
297C23×C5—22no—
298C23×C5—22no—
299C23×C5C2×C4×D312yes—
300C23×C5#23512yes—
301C23×C5C2×#6712yes—
302C23×C5C23×D311yescharacteristic
303C22×D5—21no—
304C22×D5—22no—
305C22×D5—22no—
306C22×D5—21no—
307C22×D5C2×C4×D312yes—
308C2×C4×C5—22no—
309C2×C4×C5—22no—
310C2×C4×C5—34no—
311C2×C4×C5—44no—
312C2×C4×C5—62no—
313C2×C4×C5—62no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5—62no—
317C2×C4×C5C2×#6712yes—
318C2×C4×C5C2×#6712yes—
319C22⋊D5—28no—
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—21no—
325C2×Dic5—21no—
326C2×Dic5—34no—
327C2×Dic5—68no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C2×Dic5—62no—
332C2×Dic5—62no—
333C2×Dic5—124no—
334C2×Dic5C2×C4×D312yes—
335C4×D5—44no—
336C22×D4—51no—
337C23×C4—51no—
338C23×C4—151no—
339C23×C4—151no—
340C23×C4D3×D511yescharacteristic
341#123—158no—
342C4×D4—104no—
343C2×C4⋊C4—51no—
344C2×C4⋊C4—152no—
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—51no—
353C2×C22⋊C4—152no—
354C2×C22⋊C4—152no—
355C2×C42—32no—
356C2×C42—51no—
357#95—32no—
358#95—152no—
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×C3C2×D2012yes—
378C23×C3#70912yes—
379C23×C3C23×D511yescharacteristic
380C3×D4—108no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—44no—
384C2×C3×C4—52no—
385C2×C3×C4—101no—
386C2×C3×C4—101no—
387C2×C3×C4—101no—
388C2×C3×C4—101no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—204no—
392C2×C3×C4C2×D2012yes—
393C2×C3×C4C2×D2012yes—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—54no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—108no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C2×Dic3—102no—
406C2×Dic3—204no—
407C2×Dic3C2×C4×D514yes—
408C22×C5—24no—
409C22×C5—22no—
410C22×C5—24no—
411C22×C5—22no—
412C22×C5C4×#6714yes—
413C22×C5C22×C4×D311yescharacteristic
414C22×C5C2×#23511yescharacteristic
415C22×C5#100412yes—
416C22×C5#101012yes—
417C22×C5C22×#6711yescharacteristic
418C2×D5—22no—
419C2×D5—24no—
420C2×D5—41no—
421C2×D5—42no—
422C2×D5—41no—
423C4×C5—22no—
424C4×C5—22no—
425C4×C5—64no—
426C4×C5—122no—
427Dic5—22no—
428Dic5—41no—
429Dic5—41no—
430Dic5—64no—
431Dic5—122no—
432C24—51no—
433C24C2×D3×D511yescharacteristic
434C2×D4—54no—
435C2×D4—102no—
436C2×D4—102no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—42no—
440C22×C4—51no—
441C22×C4—51no—
442C22×C4—51no—
443C22×C4—102no—
444C22×C4—154no—
445C22×C4—152no—
446C22×C4—152no—
447C22×C4—202no—
448C22×C4—302no—
449C22×C4—302no—
450C22×C4—302no—
451C22×C4—602no—
452C22×C4#120214yes—
453C22×C4C2×D3×D511yescharacteristic
454C22×C4C2×D3×D511yescharacteristic
455C4⋊C4—102no—
456C4⋊C4—158no—
457C4⋊C4—304no—
458C4⋊C4—304no—
459C4⋊C4—602no—
460C22⋊C4—34no—
461C22⋊C4—62no—
462C22⋊C4—102no—
463C22⋊C4—102no—
464C22⋊C4—158no—
465C22⋊C4—602no—
466C42—102no—
467C42—122no—
468C3×C5#232211yescharacteristic, complemented, Hall
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—102no—
474C22×C3—104no—
475C22×C3—201no—
476C22×C3—201no—
477C22×C3—202no—
478C22×C3C4×D2014yes—
479C22×C3C22×C4×D511yescharacteristic
480C22×C3C22×D2011yescharacteristic
481C22×C3C2×#70911yescharacteristic
482C22×C3#414412yes—
483C22×C3#415012yes—
484C3×C4—22no—
485C3×C4—22no—
486C3×C4—102no—
487C3×C4—201no—
488C3×C4—201no—
489Dic3—24no—
490Dic3—42no—
491Dic3—104no—
492Dic3—202no—
493C2×C5—24no—
494C2×C5#554612yes—
495C2×C5#556412yes—
496C2×C5C2×C4×#6711yescharacteristic
497C2×C5#580411yescharacteristic
498C2×C5#582511yescharacteristic
499D5—42no—
500C23—22no—
501C23—22no—
502C23—52no—
503C23—102no—
504C23—101no—
505C23—101no—
506C23—102no—
507C23#677112yes—
508C23C4×D3×D512yes—
509C23C2×#120212yes—
510C23C22×D3×D511yescharacteristic
511D4—108no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—34no—
515C2×C4—44no—
516C2×C4—52no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—102no—
525C2×C4—101no—
526C2×C4—101no—
527C2×C4—102no—
528C2×C4—154no—
529C2×C4—204no—
530C2×C4—308no—
531C2×C4—302no—
532C2×C4—302no—
533C2×C4—302no—
534C2×C4—302no—
535C2×C4—302no—
536C2×C4—604no—
537C2×C4C2×#120212yes—
538C2×C4C2×#120212yes—
539C2×C3—24no—
540C2×C3—202no—
541C2×C3C2×C4×D2011yescharacteristic
542C2×C3#6582711yescharacteristic
543C2×C3#6586011yescharacteristic
544C2×C3#6588012yes—
545C2×C3#6589812yes—
546C5#8232611yescharacteristic, complemented, Sylow
547C22—22no—
548C22—24no—
549C22—24no—
550C22—22no—
551C22—102no—
552C22—104no—
553C22—201no—
554C22—202no—
555C22—201no—
556C22C4×#120214yes—
557C22#9200112yes—
558C22C2×#677111yescharacteristic
559C22C2×C4×D3×D511yescharacteristic
560C22C22×#120211yescharacteristic
561C22#9204912yes—
562C4—22no—
563C4—22no—
564C4—64no—
565C4—102no—
566C4—122no—
567C4—201no—
568C4—201no—
569C4—304no—
570C4—602no—
571C3#1060517711yescharacteristic, complemented, Sylow
572C2—24no—
573C2—202no—
574C2#1175425211yescharacteristic
575C2#1175428612yes—
576C2C2×C4×#120211yescharacteristic
577C2#1175459612yes—
578C2#1175542711yescharacteristic
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×#6712C2semidirect
4. C4×C5×#217C2semidirect
5. #11749517C2semidirect
7. #11749473C2semidirect
8. #11748820C2semidirect
9. #11748814C2semidirect
10. #11748706C2semidirect
11. #11748704C2semidirect
31. #91537C4semidirect
61. C22×Dic15C2×C4semidirect
77. C5×#217C2×C4semidirect
84. Dic3⋊Dic5C2×C4semidirect
86. #6665C2×C4semidirect
92. C4×#217D5semidirect
291. #217C4×D5semidirect
468. C3×C5#2322semidirect
546. C5#82326semidirect
571. C3#10605177semidirect

Maximal subgroups

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

subgroupindex
C22×#67122
C2×C3×C4×C22⋊D52
C2×#915372
C4×C5×#2172
#117495172
#117495002
#117494732
#117488202
#117488142
#117487062
#117487042
#106051773
#823265