Finite Group Databasebeta

← #14416893

Subgroups — #14416893

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65856C211yesmaximal, characteristic
2C2×#91333C211yesmaximal, characteristic, complemented
3C2×#91313C211yesmaximal, characteristic
4#11751105C211yesmaximal, characteristic, complemented
5C5×#4995C211yesmaximal, characteristic, complemented
6#11748368C211yesmaximal, characteristic
7Dic5×#208C211yesmaximal, characteristic, complemented
8#11747472C212yesmaximal, complemented
9#11747270C212yesmaximal, complemented
10#11747269C212yesmaximal, complemented
11#11747268C212yesmaximal, complemented
12#10610931—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C2×C5×C4⋊Dic3C2211yescharacteristic
18C2×C5×#208C2211yescharacteristic
19C2×C3×C4×Dic5C2211yescharacteristic
20C3×#4039C2211yescharacteristic
21C3×#4039C2211yescharacteristic
22C3×#4033C2211yescharacteristic
23C3×#4033C2211yescharacteristic
24C3×#4027—21no—
25C3×#4027—21no—
26C3×#4018—21no—
27C3×C42×D5—21no—
28C2×#6669—21no—
29C2×Dic3×Dic5—21no—
30#91333—21no—
31#91333—21no—
32#91333C2212yes—
33#91333C2212yes—
34C2×#6662C2212yes—
35#91313C418yescomplemented
36C2×#6657C2212yes—
37#91094C2212yes—
38#91094C2212yes—
39C5×#808C2212yes—
40C5×#807C2212yes—
41#90970C2212yes—
42#90964C2212yes—
43#90952C2212yes—
44Dic5×C3⋊C8C2212yes—
45C4×#951—51nomaximal, Hall
46C2×#4055—31no—
47#65856D311yescharacteristic
48C2×#4033—31no—
49C5×#428—31no—
50M16×Dic5—31no—
51#65293—31no—
52#65255—31no—
53#65178—32no—
54#65162—32no—
55#65142—32no—
56#65141—32no—
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15—21no—
59C2×Dic30—21no—
60C2×Dic30—22no—
61C2×Dic30C2×C414yescomplemented
62C2×Dic30C2312yes—
63C22×C5×Dic3—21no—
64C2×C3×C22⋊D5—21no—
65C22×C3×Dic5D411yescharacteristic
66C22×C3×Dic5D411yescharacteristic
67C3×#709—22no—
68C3×#709C2×C414yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×C4×D5—21no—
72C2×C3×C4×D5—21no—
73C2×C3×Dic10C2311yescharacteristic
74C3×C5×C4⋊C4C2311yescharacteristic
75C3×C5×C4⋊C4C2311yescharacteristic
76C3×C5×C22⋊C4—21no—
77C3×C42×C5—21no—
78#6710—22no—
79C2×C3⋊#168C2312yes—
80C5×C4⋊Dic3—21no—
81C5×C4⋊Dic3C2312yes—
82C5×#208C2×C414yescomplemented
83C2×C5×C3⋊C8C2312yes—
84C3×#685—21no—
85C3×#685—21no—
86C3×#683—21no—
87C3×#683—21no—
88C3×C4×Dic5—21no—
89C3×C4×Dic5—21no—
90C3×C4×Dic5D411yescharacteristic
91C3×C4×Dic5D411yescharacteristic
92C3×C4×Dic5C2311yescharacteristic
93C3×C4×Dic5C2311yescharacteristic
94#6669—22no—
95#6669—22no—
96Dic3×Dic5—22no—
97Dic3×Dic5—22no—
98#6662C2×C418yescomplemented
99#6657C2×C418yescomplemented
100C3×C4×#41—51no—
101C2×#951—51no—
102C2×C4×Dic6—51no—
103#5232—51no—
104#5030—52no—
105C4×#216—52no—
106#5018—52no—
107C4×#214—52no—
108#5004—51no—
109#4995D511yescharacteristic, complemented
110C4×#208—51no—
111C2×#709C2×D311yescharacteristic
112C22×Dic10—31no—
113C2×C5×M16—31no—
114C5×#117C2×D311yescharacteristic
115C2×C5×C4⋊C4—31no—
116C2×#683—61no—
117C2×C4×Dic5—61no—
118C2×C4×Dic5C2×D311yescharacteristic
119C2×#681—31no—
120#4055—38no—
121C2×Dic20—32no—
122C2×#677—32no—
123#4039C2×D311yescharacteristic
124#4039C2×D311yescharacteristic
125#4033—32no—
126#4033—32no—
127#4033—61no—
128#4033—61no—
129#4033C2×D311yescharacteristic
130#4033C2×D311yescharacteristic
131#4027—21no—
132#4027—21no—
133#4018—21no—
134C42×D5—21no—
135C5×#107—32no—
136C5×#106—32no—
137#3947—32no—
138C8×Dic5—32no—
139#3942—32no—
140#3942—32no—
141#3942—32no—
142#3941—32no—
143C4×#137—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×C3×D5—21no—
146C2×C3×C4×C5—21no—
147C2×C3×C4×C5—21no—
148C2×C3×C4×C5C22×C412yes—
149C2×C3×C4×C5C2×D411yescharacteristic
150C2×C3×C4×C5#4112yes—
151C2×C3×C4×C5C2411yescharacteristic
152C2×Dic15—24no—
153C2×Dic15—22no—
154Dic30—24no—
155Dic30—22no—
156Dic30C22×C414yes—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—42no—
159C3×C22⋊D5—24no—
160C2×C3×Dic5—21no—
161C2×C3×Dic5—22no—
162C2×C3×Dic5—41no—
163C2×C3×Dic5C2×D411yescharacteristic
164C2×C3×Dic5C2×D411yescharacteristic
165C2×C3×Dic5C2×D411yescharacteristic
166C2×C3×Dic5C2×D411yescharacteristic
167C3×D20—21no—
168C3×D20C22×C412yes—
169C3×C4×D5—24no—
170C3×C4×D5—42no—
171C3×C4×D5—42no—
172C3×Dic10—21no—
173C3×Dic10C22×C412yes—
174C3⋊#168—22no—
175C5×C3⋊C8C22×C414yes—
176C2×C3×#41—51no—
177C22×Dic6—51no—
178C3×C4×Q8—51no—
179C3×C4×Q8—51no—
180C3×C4×D4—51no—
181C3×C4×D4—51no—
182C3×C4×D4—101no—
183C3×C4×D4—101no—
184C3×#117—101no—
185C3×#117C2×D511yescharacteristic
186C2×C3×C42—51no—
187C2×C3×C42—101no—
188#951—58no—
189C2×#216—52no—
190C2×#214—52no—
191C2×C4⋊Dic3C2×D511yescharacteristic
192C2×Dic3⋊C4—101no—
193C2×C4×Dic3—101no—
194C2×#208—51no—
195C2×#208C2×D511yescharacteristic
196C4×Dic6—52no—
197C4×Dic6—52no—
198C4×Dic6—101no—
199C4×Dic6—101no—
200#835—52no—
201#832—52no—
202#810—52no—
203#810—52no—
204#808C2×D512yes—
205#807C2×D512yes—
206#803—52no—
207C4×C3⋊C8—52no—
208C22×C4×C5—61no—
209C22×C4×C5C22×D311yescharacteristic
210C2×C22⋊D5—21no—
211C22×Dic5—61no—
212C22×Dic5#6711yescharacteristic
213C22×Dic5#6711yescharacteristic
214#709—22no—
215#709C4×D314yes—
216C2×D20C22×D311yescharacteristic
217C2×C4×D5—21no—
218C2×C4×D5—21no—
219C2×C4×D5—21no—
220C2×Dic10—34no—
221C2×Dic10—32no—
222C2×Dic10—62no—
223C2×Dic10—61no—
224C2×Dic10C22×D311yescharacteristic
225C5×M16—34no—
226C2×C5×C8—32no—
227C5×C4⋊C4—32no—
228C5×C4⋊C4—61no—
229C5×C4⋊C4C22×D311yescharacteristic
230C5×C4⋊C4C22×D311yescharacteristic
231C5×C22⋊C4—21no—
232C42×C5—21no—
233#685—21no—
234#685—21no—
235#683—21no—
236#683—21no—
237#683—62no—
238#683—62no—
239C4×Dic5—21no—
240C4×Dic5—21no—
241C4×Dic5—62no—
242C4×Dic5—62no—
243C4×Dic5#6711yescharacteristic
244C4×Dic5#6711yescharacteristic
245C4×Dic5C22×D311yescharacteristic
246C4×Dic5C22×D311yescharacteristic
247#681—62no—
248C2×#168—32no—
249Dic20—38no—
250#677—38no—
251C2×#137—151no—
252C4×#41—51no—
253C2×C4×Q8—151no—
254#441—152no—
255#440—152no—
256C4×Q16—152no—
257C4×SD16—152no—
258#428—31no—
259#419—151no—
260#417—151no—
261C4×M16—151no—
262C22×C3×C5—21no—
263C22×C3×C5—21no—
264C22×C3×C5C4×D412yes—
265C22×C3×C5C22×D411yescharacteristic
266C2×C3×D5—22no—
267C2×C3×D5—41no—
268C2×C3×D5—41no—
269C3×C4×C5—41no—
270C3×C4×C5—41no—
271C3×C4×C5C4×D412yes—
272C3×C4×C5C23×C411yescharacteristic
273C3×C4×C5C2×#4111yescharacteristic
274Dic15—24no—
275C3×Dic5—22no—
276C3×Dic5—21no—
277C3×Dic5—21no—
278C3×Dic5—21no—
279C3×Dic5#13712yescomplemented
280C5×Dic3—42no—
281C3×#41—54no—
282C3×#41—102no—
283C2×C3×Q8—51no—
284C2×C3×D4—51no—
285C2×C3×D4—101no—
286C22×C3×C4—51no—
287C22×C3×C4—51no—
288C22×C3×C4—101no—
289C22×C3×C4—101no—
290C22×C3×C4—101no—
291C22×C3×C4C22×D511yescharacteristic
292C22×Dic3—21no—
293C22×Dic3—101no—
294C2×Dic6—54no—
295C2×Dic6—52no—
296C2×Dic6—101no—
297C2×Dic6—102no—
298C3×C4⋊C4—101no—
299C3×C4⋊C4—101no—
300C3×C4⋊C4C22×D511yescharacteristic
301C3×C4⋊C4C22×D511yescharacteristic
302C3×C22⋊C4—21no—
303C3×C22⋊C4—101no—
304C3×C22⋊C4—101no—
305C3×C42—21no—
306C3×C42—51no—
307C3×C42—51no—
308C3×C42—51no—
309C3×C42—51no—
310C3×C42—101no—
311C3×C42—101no—
312#216—58no—
313#214—58no—
314C4⋊Dic3—21no—
315C4⋊Dic3C22×D512yes—
316Dic3⋊C4—102no—
317Dic3⋊C4—102no—
318C4×Dic3—102no—
319C4×Dic3—102no—
320#208—102no—
321#208C4×D514yescomplemented
322C2×C3⋊C8—52no—
323C2×C3⋊C8C22×D512yes—
324C23×C5C2×#6711yescharacteristic
325C22×D5—21no—
326C2×C4×C5—21no—
327C2×C4×C5—21no—
328C2×C4×C5—62no—
329C2×C4×C5—122no—
330C2×C4×C5C2×C4×D312yes—
331C2×C4×C5#23512yes—
332C2×C4×C5C2×#6711yescharacteristic
333C2×C4×C5C23×D311yescharacteristic
334C22⋊D5—24no—
335C2×Dic5—21no—
336C2×Dic5—22no—
337C2×Dic5—41no—
338C2×Dic5—64no—
339C2×Dic5—62no—
340C2×Dic5C2×#6711yescharacteristic
341C2×Dic5C2×#6711yescharacteristic
342C2×Dic5C2×#6711yescharacteristic
343C2×Dic5C2×#6711yescharacteristic
344D20—21no—
345D20C2×C4×D312yes—
346C4×D5—24no—
347C4×D5—42no—
348C4×D5—42no—
349Dic10—21no—
350Dic10—34no—
351Dic10—64no—
352Dic10—62no—
353Dic10C2×C4×D312yes—
354C5×C8—34no—
355#168—62no—
356C2×#41—51no—
357C22×Q8—151no—
358#137—158no—
359C2×Q16—152no—
360C2×SD16—152no—
361C2×M16—31no—
362C2×M16—151no—
363C4×Q8—51no—
364C4×Q8—51no—
365C4×Q8—152no—
366C4×Q8—152no—
367C4×Q8—301no—
368C4×Q8—301no—
369C4×D4—51no—
370C4×D4—51no—
371C4×D4—101no—
372C4×D4—101no—
373#117—101no—
374#117D3×D511yescharacteristic
375C2×C4⋊C4—31no—
376C2×C4⋊C4—301no—
377C2×C42—51no—
378C2×C42—101no—
379C2×C42—301no—
380#107—32no—
381#106—32no—
382#103—152no—
383#103—152no—
384#103—152no—
385#102—152no—
386#97—152no—
387C4×C8—152no—
388C2×C3×C5—22no—
389C2×C3×C5C2×C4×D411yescharacteristic
390C2×C3×C5C2×#13711yescharacteristic
391C2×C3×C5#57511yescharacteristic
392C3×D5—42no—
393C23×C3—101no—
394C23×C3D4×D511yescharacteristic
395C3×Q8—52no—
396C3×Q8—101no—
397C3×D4—52no—
398C3×D4—104no—
399C3×D4—101no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—51no—
403C2×C3×C4—51no—
404C2×C3×C4—51no—
405C2×C3×C4—51no—
406C2×C3×C4—102no—
407C2×C3×C4—101no—
408C2×C3×C4—104no—
409C2×C3×C4—202no—
410C2×C3×C4—202no—
411C2×C3×C4—201no—
412C2×C3×C4C2×C4×D512yes—
413C2×C3×C4D4×D511yescharacteristic
414C2×C3×C4#71312yes—
415C2×C3×C4C23×D511yescharacteristic
416C2×Dic3—22no—
417C2×Dic3—42no—
418C2×Dic3—102no—
419C2×Dic3—104no—
420Dic6—54no—
421Dic6—102no—
422Dic6—104no—
423C3⋊C8—102no—
424C3⋊C8C2×C4×D514yes—
425C22×C5—21no—
426C22×C5—21no—
427C22×C5C4×#6712yes—
428C22×C5C22×#6711yescharacteristic
429C2×D5—22no—
430C2×D5—41no—
431C2×D5—41no—
432C4×C5—41no—
433C4×C5—41no—
434C4×C5—122no—
435C4×C5C4×#6712yes—
436C4×C5C22×C4×D311yescharacteristic
437C4×C5C2×#23511yescharacteristic
438Dic5—22no—
439Dic5—21no—
440Dic5—21no—
441Dic5—21no—
442Dic5—64no—
443Dic5#95112yescomplemented
444#41—54no—
445#41—102no—
446C2×Q8—51no—
447C2×Q8—154no—
448C2×Q8—152no—
449C2×Q8—302no—
450C2×Q8—301no—
451C2×D4—51no—
452C2×D4—101no—
453C22×C4—51no—
454C22×C4—51no—
455C22×C4—61no—
456C22×C4—101no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—301no—
460C22×C4C2×D3×D511yescharacteristic
461Q16—158no—
462SD16—158no—
463M16—34no—
464M16—302no—
465C2×C8—32no—
466C2×C8—152no—
467C4⋊C4—32no—
468C4⋊C4—61no—
469C4⋊C4—101no—
470C4⋊C4—101no—
471C4⋊C4—302no—
472C4⋊C4—302no—
473C4⋊C4C2×D3×D511yescharacteristic
474C4⋊C4C2×D3×D511yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—101no—
477C22⋊C4—101no—
478C42—21no—
479C42—51no—
480C42—51no—
481C42—51no—
482C42—51no—
483C42—101no—
484C42—101no—
485C42—302no—
486C42—302no—
487C3×C5C4×#13711yescharacteristic, complemented, Hall
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—102no—
491C22×C3—201no—
492C22×C3—201no—
493C22×C3#403912yes—
494C22×C3C2×D4×D511yescharacteristic
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—52no—
498C3×C4—102no—
499C3×C4—101no—
500C3×C4—101no—
501C3×C4—101no—
502C3×C4#403912yes—
503C3×C4C22×C4×D511yescharacteristic
504C3×C4C2×#71311yescharacteristic
505Dic3—42no—
506Dic3—104no—
507C2×C5—22no—
508C2×C5C2×C4×#6711yescharacteristic
509C2×C5#582211yescharacteristic
510C2×C5C2×#95111yescharacteristic
511D5—42no—
512C23—101no—
513C23D5×#6711yescharacteristic
514Q8—52no—
515Q8—101no—
516Q8—154no—
517Q8—304no—
518Q8—302no—
519D4—52no—
520D4—104no—
521D4—101no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—51no—
525C2×C4—51no—
526C2×C4—51no—
527C2×C4—51no—
528C2×C4—62no—
529C2×C4—102no—
530C2×C4—101no—
531C2×C4—104no—
532C2×C4—122no—
533C2×C4—202no—
534C2×C4—202no—
535C2×C4—201no—
536C2×C4—304no—
537C2×C4—302no—
538C2×C4#677312yes—
539C2×C4C4×D3×D512yes—
540C2×C4D5×#6711yescharacteristic
541C2×C4C22×D3×D511yescharacteristic
542C8—34no—
543C8—302no—
544C2×C3—22no—
545C2×C3—202no—
546C2×C3C2×#403911yescharacteristic
547C2×C3#6610711yescharacteristic
548C2×C3D5×#13711yescharacteristic
549C5C4×#95111yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—102no—
553C22—201no—
554C22—201no—
555C22#9144412yes—
556C22C2×D5×#6711yescharacteristic
557C4—41no—
558C4—41no—
559C4—52no—
560C4—102no—
561C4—101no—
562C4—101no—
563C4—101no—
564C4—122no—
565C4—304no—
566C4#9144412yes—
567C4C2×#677311yescharacteristic
568C4C2×C4×D3×D511yescharacteristic
569C3#1061093111yescharacteristic, complemented, Sylow
570C2—22no—
571C2—202no—
572C2C2×#9144411yescharacteristic
573C2D5×#95111yescharacteristic
574C2#1175534311yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5C4×#137
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C2×#91333C2semidirect
4. #11751105C2semidirect
5. C5×#4995C2semidirect
7. Dic5×#208C2semidirect
8. #11747472C2semidirect
9. #11747270C2semidirect
10. #11747269C2semidirect
11. #11747268C2semidirect
35. #91313C4semidirect
61. C2×Dic30C2×C4semidirect
82. C5×#208C2×C4semidirect
98. #6662C2×C4semidirect
99. #6657C2×C4semidirect
109. #4995D5semidirect
279. C3×Dic5#137semidirect
321. #208C4×D5semidirect
443. Dic5#951semidirect
487. C3×C5C4×#137semidirect
549. C5C4×#951semidirect
569. C3#10610931semidirect

Maximal subgroups

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

subgroupindex
C3×#658562
C2×#913332
C2×#913132
#117511052
C5×#49952
#117483682
Dic5×#2082
#117474722
#117472702
#117472692
#117472682
#106109313
C4×#9515