Finite Group Databasebeta

← #14417260

Subgroups — #14417260

Order 1920.

940 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5523C211yesmaximal, characteristic
2C3×#65850C211yesmaximal, characteristic, complemented
3C2×#91309C211yesmaximal, characteristic
4#11751098C211yesmaximal, characteristic, complemented
5#11748472C211yesmaximal, characteristic, complemented
6C2×#90970C211yesmaximal, characteristic, complemented
7#11748382C211yesmaximal, characteristic
8#11747089C212yesmaximal, complemented
9C4×#6661C212yesmaximal, complemented
10#11747050C212yesmaximal, complemented
11C4×#6656C212yesmaximal, complemented
12#10614302—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C22×C3⋊#168C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C5×#892—21no—
20C5×D3×C4⋊C4—21no—
21C5×#67⋊C4—22no—
22C4×C5×D12C2212yes—
23C4×C5×Dic6C2212yes—
24C3×#4072C2211yescharacteristic
25C3×#4033—21no—
26C3×#4033—21no—
27C3×#4023—22no—
28C3×C4×Dic10C2212yes—
29C3×C4×Dic10C2212yes—
30C2×#6661C2212yes—
31#91309C418yescomplemented
32C2×#6656C2212yes—
33#91087C2212yes—
34C4×C3⋊#168C2212yes—
35#90989C2212yes—
36#90986C2212yes—
37#90972C2211yescharacteristic
38#90972C2211yescharacteristic
39#90970C2212yes—
40#90970C2212yes—
41#90966C2211yescharacteristic
42#90966C2211yescharacteristic
43#82547—51nomaximal, Hall
44C5×#520—31no—
45C2×#4097—31no—
46#65850D311yescharacteristic, complemented
47#65522—31no—
48#65319—32no—
49C4×#689—32no—
50#65307—32no—
51C4×#687—32no—
52#65286—31no—
53C2×#3942—31no—
54#65253—31no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×C5×#67—21no—
57C22×C5×Dic3—21no—
58C5×#235—22no—
59C5×#235C2×C414yes—
60C2×C5×D12C2311yescharacteristic
61C2×C4×C5×D3—21no—
62C2×C4×C5×D3—22no—
63C2×C5×Dic6C2311yescharacteristic
64C22×C3×Dic5—21no—
65C2×C3×Dic10—21no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10C2×C414yescomplemented
68C2×C3×Dic10C2312yes—
69C3×C5×C4⋊C4D411yescharacteristic
70C3×C5×C4⋊C4D411yescharacteristic
71C3×C5×C22⋊C4D412yes—
72C3×C42×C5C2312yes—
73#6710C2×C414yescomplemented
74C2×C3⋊#168—22no—
75C2×C3⋊#168C2311yescharacteristic
76C2×C3⋊#168C2311yescharacteristic
77C2×C3⋊#168C2312yes—
78C5×(C2×D3)⋊C4—22no—
79C5×C4⋊Dic3—21no—
80C5×C4⋊Dic3C2312yes—
81C5×Dic3⋊C4—22no—
82C4×C5×Dic3—22no—
83C3×C22⋊Dic5—21no—
84C3×C4⋊Dic5C2311yescharacteristic
85C3×C4⋊Dic5C2311yescharacteristic
86C3×#683—22no—
87C3×#683—22no—
88C3×C4×Dic5—22no—
89C3×C4×Dic5—22no—
90C3×C4×Dic5—21no—
91#6661C2×C418yescomplemented
92#6656C2×C418yescomplemented
93C3×#519—51no—
94C2×#942—51no—
95#5523D511yescharacteristic
96C2×#835—51no—
97#5030—52no—
98C4×#216—52no—
99#5026—52no—
100C4×#215—52no—
101#5002—51no—
102#4997—51no—
103#4995—51no—
104C2×C5×#41—31no—
105C22×Dic10C2×D311yescharacteristic
106C4×C5×Q8—32no—
107C4×C5×D4—32no—
108C4×C5×D4—62no—
109C5×#117—61no—
110C5×#117C2×D311yescharacteristic
111C2×C5×C4⋊C4—31no—
112C2×C5×C4⋊C4—61no—
113#4097—38no—
114C2×#689—32no—
115C2×#687—32no—
116#4072C2×D311yescharacteristic
117C2×#681—31no—
118C22×#168—31no—
119#4033—21no—
120#4033—21no—
121#4023—22no—
122C4×Dic10C2×D312yes—
123C4×Dic10C2×D312yes—
124#3967—31no—
125#3967—31no—
126#3964—31no—
127#3964—31no—
128#3942—32no—
129#3942—32no—
130#3940—32no—
131#3939—32no—
132#3935—32no—
133C4×#168—32no—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C5×D3—21no—
137C2×C3×C4×C5C22×C412yes—
138C2×C3×C4×C5C2×D412yes—
139C2×C3×C4×C5C2×D412yes—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5#4112yes—
142C2×C3×C4×C5C2411yescharacteristic
143C5×#67—24no—
144C2×C5×Dic3—21no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—81no—
147C5×D12—21no—
148C5×D12C22×C412yes—
149C4×C5×D3—24no—
150C4×C5×D3—82no—
151C5×Dic6—21no—
152C5×Dic6C22×C412yes—
153C2×C3×Dic5—24no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C3×Dic10—24no—
158C3×Dic10—22no—
159C3×Dic10C22×C414yes—
160C3⋊#168—21no—
161C3⋊#168—21no—
162C3⋊#168C22×C414yes—
163C22×C3×Q8—51no—
164C2×#235C2×D511yescharacteristic
165C3×C4×Q8—52no—
166C3×C4×Q8—52no—
167C3×C4×Q8—101no—
168C3×C4×Q8—101no—
169C3×#117—51no—
170C3×#117—102no—
171C3×#117C2×D511yescharacteristic
172C2×#216—52no—
173#942—58no—
174C2×#215—52no—
175C2×C4⋊Dic3C2×D511yescharacteristic
176C2×#208—51no—
177C22×C3⋊C8—51no—
178#892—21no—
179D3×C4⋊C4—21no—
180#67⋊C4—22no—
181C4×D12C2×D512yes—
182C4×Dic6C2×D512yes—
183#835—52no—
184#835—52no—
185#810—51no—
186#810—51no—
187#809—51no—
188#809—51no—
189#808—52no—
190#807—52no—
191#803—52no—
192C4×C3⋊C8—52no—
193C5×#41—34no—
194C5×#41—62no—
195C2×C5×Q8—31no—
196C2×C5×D4—31no—
197C2×C5×D4—61no—
198C22×C4×C5—61no—
199C22×C4×C5—61no—
200C22×C4×C5—62no—
201C22×C4×C5C22×D311yescharacteristic
202C22×Dic5—21no—
203C2×Dic10—22no—
204C2×Dic10—21no—
205C2×Dic10C4×D314yescomplemented
206C2×Dic10C22×D312yes—
207C5×C4⋊C4—32no—
208C5×C4⋊C4—61no—
209C5×C4⋊C4—62no—
210C5×C4⋊C4#6711yescharacteristic
211C5×C4⋊C4#6711yescharacteristic
212C5×C22⋊C4—62no—
213C5×C22⋊C4#6712yes—
214C42×C5—62no—
215C42×C5C22×D312yes—
216C22⋊Dic5—21no—
217#689—38no—
218#687—38no—
219C4⋊Dic5C22×D311yescharacteristic
220C4⋊Dic5C22×D311yescharacteristic
221#683—22no—
222#683—22no—
223C4×Dic5—22no—
224C4×Dic5—22no—
225C4×Dic5—21no—
226#681—34no—
227C2×#168—32no—
228C2×#168—31no—
229C2×#168—31no—
230C2×#168—62no—
231C2×#137—151no—
232#520—31no—
233#519—51no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—151no—
239#416—151no—
240C2×#103—151no—
241#405—151no—
242C22×C3×C5—21no—
243C22×C3×C5—21no—
244C22×C3×C5C4×D412yes—
245C22×C3×C5C22×D411yescharacteristic
246C2×C5×D3—22no—
247C2×C5×D3—41no—
248C2×C5×D3—41no—
249C3×C4×C5—22no—
250C3×C4×C5—22no—
251C3×C4×C5C4×D412yes—
252C3×C4×C5C23×C411yescharacteristic
253C3×C4×C5C2×#4111yescharacteristic
254C3×Dic5—24no—
255C3×Dic5—41no—
256C3×Dic5—41no—
257C5×Dic3—22no—
258C5×Dic3—42no—
259C2×C3×Q8—54no—
260C2×C3×Q8—52no—
261C2×C3×Q8—102no—
262C2×C3×Q8—101no—
263C22×C3×C4—101no—
264C22×C3×C4C22×D511yescharacteristic
265C2×#67—21no—
266C22×Dic3—21no—
267#235—22no—
268#235C4×D514yes—
269C2×D12C22×D511yescharacteristic
270C2×C4×D3—21no—
271C2×C4×D3—22no—
272C2×Dic6C22×D511yescharacteristic
273C3×C4⋊C4—51no—
274C3×C4⋊C4—51no—
275C3×C4⋊C4—102no—
276C3×C4⋊C4—102no—
277C3×C4⋊C4C22⋊D511yescharacteristic
278C3×C4⋊C4C22⋊D511yescharacteristic
279C3×C22⋊C4—101no—
280C3×C22⋊C4C22⋊D512yes—
281C3×C42—101no—
282C3×C42—102no—
283C3×C42—102no—
284C3×C42C22×D512yes—
285#216—58no—
286#215—58no—
287(C2×D3)⋊C4—22no—
288C4⋊Dic3—21no—
289C4⋊Dic3C22×D512yes—
290Dic3⋊C4—22no—
291C4×Dic3—22no—
292#208—54no—
293C2×C3⋊C8—52no—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—51no—
296C2×C3⋊C8—102no—
297C23×C5—61no—
298C23×C5C2×#6711yescharacteristic
299C5×Q8—32no—
300C5×Q8—61no—
301C5×D4—32no—
302C5×D4—64no—
303C5×D4—61no—
304C2×C4×C5—61no—
305C2×C4×C5—64no—
306C2×C4×C5—62no—
307C2×C4×C5—242no—
308C2×C4×C5—241no—
309C2×C4×C5C2×C4×D312yes—
310C2×C4×C5#23512yes—
311C2×C4×C5C2×#6712yes—
312C2×C4×C5C2×#6712yes—
313C2×C4×C5C2×#6711yescharacteristic
314C2×C4×C5C23×D311yescharacteristic
315C2×Dic5—24no—
316C2×Dic5—22no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319Dic10—24no—
320Dic10—22no—
321Dic10C2×C4×D314yes—
322#168—34no—
323#168—61no—
324#168—61no—
325C2×#41—31no—
326C22×Q8—51no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—151no—
331C22×C8—151no—
332C4×Q8—32no—
333C4×Q8—52no—
334C4×Q8—52no—
335C4×Q8—101no—
336C4×Q8—101no—
337C4×D4—32no—
338C4×D4—62no—
339#117—51no—
340#117—61no—
341#117—102no—
342#117D3×D511yescharacteristic
343C2×C4⋊C4—31no—
344C2×C4⋊C4—61no—
345#107—152no—
346#106—152no—
347#103—152no—
348#103—152no—
349#103—151no—
350#103—151no—
351#102—151no—
352#102—151no—
353#97—152no—
354C4×C8—152no—
355C2×C3×C5—22no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#57711yescharacteristic
358C2×C3×C5#57811yescharacteristic
359C5×D3—42no—
360C23×C3C2×C22⋊D511yescharacteristic
361C22×D3—21no—
362C3×Q8—54no—
363C3×Q8—102no—
364C3×Q8—104no—
365C2×C3×C4—102no—
366C2×C3×C4—104no—
367C2×C3×C4—101no—
368C2×C3×C4—101no—
369C2×C3×C4C2×C4×D512yes—
370C2×C3×C4#70912yes—
371C2×C3×C4C2×C22⋊D512yes—
372C2×C3×C4C2×C22⋊D512yes—
373C2×C3×C4C2×C22⋊D511yescharacteristic
374C2×C3×C4C23×D511yescharacteristic
375#67—24no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—81no—
379D12—21no—
380D12C2×C4×D512yes—
381C4×D3—24no—
382C4×D3—82no—
383Dic6—21no—
384Dic6C2×C4×D512yes—
385C3⋊C8—54no—
386C3⋊C8—101no—
387C3⋊C8—101no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5—62no—
391C22×C5—121no—
392C22×C5—121no—
393C22×C5C4×#6712yes—
394C22×C5C22×#6711yescharacteristic
395C4×C5—22no—
396C4×C5—22no—
397C4×C5—62no—
398C4×C5—122no—
399C4×C5C4×#6712yes—
400C4×C5C22×C4×D311yescharacteristic
401C4×C5C2×#23511yescharacteristic
402Dic5—24no—
403Dic5—41no—
404Dic5—41no—
405#41—34no—
406#41—62no—
407C2×Q8—31no—
408C2×Q8—54no—
409C2×Q8—52no—
410C2×Q8—102no—
411C2×Q8—101no—
412C2×D4—31no—
413C2×D4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—62no—
417C22×C4—101no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—154no—
422C2×C8—152no—
423C2×C8—151no—
424C2×C8—151no—
425C2×C8—302no—
426C4⋊C4—32no—
427C4⋊C4—51no—
428C4⋊C4—51no—
429C4⋊C4—61no—
430C4⋊C4—62no—
431C4⋊C4—102no—
432C4⋊C4—102no—
433C4⋊C4#120111yescharacteristic
434C4⋊C4#120111yescharacteristic
435C22⋊C4—62no—
436C22⋊C4—101no—
437C22⋊C4#120112yes—
438C42—62no—
439C42—101no—
440C42—102no—
441C42—102no—
442C42C2×D3×D512yes—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3C4×C22⋊D512yes—
447C22×C3C22×C22⋊D511yescharacteristic
448C2×D3—22no—
449C2×D3—41no—
450C2×D3—41no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—104no—
454C3×C4—201no—
455C3×C4—201no—
456C3×C4C4×C22⋊D512yes—
457C3×C4C22×C4×D511yescharacteristic
458C3×C4C2×#70911yescharacteristic
459Dic3—22no—
460Dic3—42no—
461C2×C5—22no—
462C2×C5—122no—
463C2×C5C2×C4×#6711yescharacteristic
464C2×C5#583711yescharacteristic
465C2×C5#583811yescharacteristic
466C23—61no—
467C23C2×#120111yescharacteristic
468Q8—32no—
469Q8—54no—
470Q8—61no—
471Q8—104no—
472Q8—102no—
473D4—32no—
474D4—64no—
475D4—61no—
476C2×C4—61no—
477C2×C4—64no—
478C2×C4—62no—
479C2×C4—104no—
480C2×C4—102no—
481C2×C4—101no—
482C2×C4—101no—
483C2×C4—242no—
484C2×C4—241no—
485C2×C4#677112yes—
486C2×C4C4×D3×D512yes—
487C2×C4C2×#120112yes—
488C2×C4C2×#120112yes—
489C2×C4C2×#120111yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—154no—
492C8—301no—
493C8—301no—
494C2×C3—22no—
495C2×C3C2×C4×C22⋊D511yescharacteristic
496C2×C3#6617011yescharacteristic
497C2×C3#6617211yescharacteristic
498D3—42no—
499C5#8254711yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—62no—
503C22—121no—
504C22—121no—
505C22C4×#120112yes—
506C22C22×#120111yescharacteristic
507C4—22no—
508C4—22no—
509C4—62no—
510C4—104no—
511C4—122no—
512C4—201no—
513C4—201no—
514C4C4×#120112yes—
515C4C2×#677111yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061430211yescharacteristic, complemented, Sylow
518C2—22no—
519C2—122no—
520C2C2×C4×#120111yescharacteristic
521C2#1175530211yescharacteristic
522C2#1175530411yescharacteristic
subgroupG/H
centerC22C22×#1201
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×C5×#235C22
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C3×#65850C2semidirect
4. #11751098C2semidirect
5. #11748472C2semidirect
6. C2×#90970C2semidirect
8. #11747089C2semidirect
9. C4×#6661C2semidirect
10. #11747050C2semidirect
11. C4×#6656C2semidirect
31. #91309C4semidirect
46. #65850D3semidirect
67. C2×C3×Dic10C2×C4semidirect
73. #6710C2×C4semidirect
91. #6661C2×C4semidirect
92. #6656C2×C4semidirect
205. C2×Dic10C4×D3semidirect
443. C3×C5#2947semidirect
499. C5#82547semidirect
517. C3#10614302semidirect

Maximal subgroups

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

subgroupindex
C5×#55232
C3×#658502
C2×#913092
#117510982
#117484722
C2×#909702
#117483822
#117470892
C4×#66612
#117470502
C4×#66562
#106143023
#825475