Finite Group Databasebeta

← #14417176

Subgroups — #14417176

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
1C3×C5×#519C211yesmaximal, characteristic, complemented
2C2×#91831C211yesmaximal, characteristic
3#11756282C211yesmaximal, characteristic
4C2×#91119C211yesmaximal, characteristic, complemented
5#11751131C212yesmaximal, complemented
6C4×#6718C212yesmaximal, complemented
7#11751127C212yesmaximal, complemented
8C4×#6717C212yesmaximal, complemented
9#11751103C211yesmaximal, characteristic
10#11751098C211yesmaximal, characteristic, complemented
11#11751096C211yesmaximal, characteristic, complemented
12#10614278—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C2×#6817C2211yescharacteristic
15C3×C4×C5×Q8—21no—
16C3×C4×C5×Q8—21no—
17C3×C4×C5×Q8C2212yes—
18C3×C4×C5×Q8C2212yes—
19C3×C5×#117—22no—
20C3×C5×#117C2211yescharacteristic
21C3×C5×#117C2211yescharacteristic
22C2×#6718C2212yes—
23#91831C418yescomplemented
24C2×#6717C2212yes—
25C2×C4⋊Dic15C2211yescharacteristic
26C2×#6710C2211yescharacteristic
27C22×C3⋊#168C2211yescharacteristic
28#91781—21no—
29C4⋊C4×D15—21no—
30#91770—22no—
31C4×D60C2212yes—
32C4×Dic30C2212yes—
33#91119C2212yes—
34#91119C2212yes—
35#91094C2211yescharacteristic
36#91094C2211yescharacteristic
37#91093C2211yescharacteristic
38#91093C2211yescharacteristic
39#91092C2212yes—
40#91091C2212yes—
41#91087C2212yes—
42C4×C3⋊#168C2212yes—
43#82547—51nomaximal, Hall
44C5×#519D311yescharacteristic, complemented
45C2×#4088—31no—
46#65857—31no—
47C2×#3967—31no—
48#65319—32no—
49C4×#689—32no—
50#65315—32no—
51C4×#688—32no—
52#65291—31no—
53#65286—31no—
54#65284—31no—
55C2×C3×C5×Q8—21no—
56C2×C3×C5×Q8—22no—
57C2×C3×C5×Q8C2×C414yescomplemented
58C2×C3×C5×Q8C2312yes—
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5C2311yescharacteristic
61C2×#1220—21no—
62C22×Dic15—21no—
63#6817—22no—
64#6817C2×C414yes—
65C2×D60C2311yescharacteristic
66C2×C4×D15—21no—
67C2×C4×D15—22no—
68C2×Dic30C2311yescharacteristic
69C3×C5×C4⋊C4—22no—
70C3×C5×C4⋊C4—22no—
71C3×C5×C4⋊C4D411yescharacteristic
72C3×C5×C4⋊C4D411yescharacteristic
73C3×C5×C4⋊C4C2311yescharacteristic
74C3×C5×C4⋊C4C2311yescharacteristic
75C3×C5×C22⋊C4—21no—
76C3×C5×C22⋊C4D412yes—
77C3×C42×C5—22no—
78C3×C42×C5—22no—
79C3×C42×C5—21no—
80C3×C42×C5C2312yes—
81#6718C2×C418yescomplemented
82#6717C2×C418yescomplemented
83#6714—22no—
84C4⋊Dic15—21no—
85C4⋊Dic15C2312yes—
86#6712—22no—
87C4×Dic15—22no—
88#6710C2×C414yescomplemented
89C2×C3⋊#168—22no—
90C2×C3⋊#168C2311yescharacteristic
91C2×C3⋊#168C2311yescharacteristic
92C2×C3⋊#168C2312yes—
93C3×#519D511yescharacteristic, complemented
94C2×#942—51no—
95#5523—51no—
96C2×#835—51no—
97#5030—52no—
98C4×#216—52no—
99#5026—52no—
100C4×#215—52no—
101#5002—51no—
102#4997—51no—
103#4995—51no—
104C22×C5×Q8C2×D311yescharacteristic
105C2×#709—31no—
106C4×C5×Q8—21no—
107C4×C5×Q8—21no—
108C4×C5×Q8C2×D312yes—
109C4×C5×Q8C2×D312yes—
110C5×#117—22no—
111C5×#117C2×D311yescharacteristic
112C5×#117C2×D311yescharacteristic
113C2×#689—32no—
114#4088—38no—
115C2×#688—32no—
116C2×C4⋊Dic5—31no—
117C2×#681—31no—
118C22×#168—31no—
119#4038—61no—
120D5×C4⋊C4—61no—
121#4027—62no—
122C4×D20—32no—
123C4×Dic10—32no—
124#3967—32no—
125#3967—32no—
126#3942—31no—
127#3942—31no—
128#3941—31no—
129#3941—31no—
130#3940—32no—
131#3939—32no—
132#3935—32no—
133C4×#168—32no—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—21no—
137C3×C5×Q8—24no—
138C3×C5×Q8—22no—
139C3×C5×Q8C22×C414yes—
140C2×C3×C4×C5—24no—
141C2×C3×C4×C5—22no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5#4112yes—
149C2×C3×C4×C5C2411yescharacteristic
150#1220—24no—
151C2×Dic15—21no—
152C2×Dic15—22no—
153C2×Dic15—81no—
154D60—21no—
155D60C22×C412yes—
156C4×D15—24no—
157C4×D15—82no—
158Dic30—21no—
159Dic30C22×C412yes—
160C3⋊#168—21no—
161C3⋊#168—21no—
162C3⋊#168C22×C414yes—
163C22×C3×Q8C2×D511yescharacteristic
164C2×#235—51no—
165C3×C4×Q8—21no—
166C3×C4×Q8—21no—
167C3×C4×Q8C2×D512yes—
168C3×C4×Q8C2×D512yes—
169C3×#117—22no—
170C3×#117C2×D511yescharacteristic
171C3×#117C2×D511yescharacteristic
172C2×#216—52no—
173#942—58no—
174C2×#215—52no—
175C2×C4⋊Dic3—51no—
176C2×#208—51no—
177C22×C3⋊C8—51no—
178#892—101no—
179D3×C4⋊C4—101no—
180#67⋊C4—102no—
181C4×D12—52no—
182C4×Dic6—52no—
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—
193C2×C5×Q8—22no—
194C2×C5×Q8—21no—
195C2×C5×Q8C4×D314yescomplemented
196C2×C5×Q8C22×D312yes—
197C22×C4×C5—21no—
198C22×C4×C5C22×D311yescharacteristic
199C2×C22⋊D5—61no—
200C22×Dic5—61no—
201#709—34no—
202#709—62no—
203C2×D20—31no—
204C2×C4×D5—61no—
205C2×C4×D5—62no—
206C2×Dic10—31no—
207C5×C4⋊C4—22no—
208C5×C4⋊C4—22no—
209C5×C4⋊C4#6711yescharacteristic
210C5×C4⋊C4#6711yescharacteristic
211C5×C4⋊C4C22×D311yescharacteristic
212C5×C4⋊C4C22×D311yescharacteristic
213C5×C22⋊C4—21no—
214C5×C22⋊C4#6712yes—
215C42×C5—22no—
216C42×C5—22no—
217C42×C5—21no—
218C42×C5C22×D312yes—
219#689—38no—
220#688—38no—
221#685—62no—
222C4⋊Dic5—32no—
223C4⋊Dic5—61no—
224#683—62no—
225C4×Dic5—62no—
226#681—34no—
227C2×#168—32no—
228C2×#168—31no—
229C2×#168—31no—
230C2×#168—62no—
231C2×#137—151no—
232#520—151no—
233#519D1511yescharacteristic, complemented
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×D15—22no—
247C2×D15—41no—
248C2×D15—41no—
249C3×C4×C5—24no—
250C3×C4×C5—22no—
251C3×C4×C5—22no—
252C3×C4×C5—41no—
253C3×C4×C5—41no—
254C3×C4×C5C4×D412yes—
255C3×C4×C5C23×C411yescharacteristic
256C3×C4×C5C2×#4111yescharacteristic
257Dic15—22no—
258Dic15—42no—
259C2×C3×Q8—22no—
260C2×C3×Q8—21no—
261C2×C3×Q8C4×D514yescomplemented
262C2×C3×Q8C22×D512yes—
263C22×C3×C4—21no—
264C22×C3×C4C22×D511yescharacteristic
265C2×#67—101no—
266C22×Dic3—101no—
267#235—54no—
268#235—102no—
269C2×D12—51no—
270C2×C4×D3—101no—
271C2×C4×D3—102no—
272C2×Dic6—51no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4C22⋊D511yescharacteristic
276C3×C4⋊C4C22⋊D511yescharacteristic
277C3×C4⋊C4C22×D511yescharacteristic
278C3×C4⋊C4C22×D511yescharacteristic
279C3×C22⋊C4—21no—
280C3×C22⋊C4C22⋊D512yes—
281C3×C42—22no—
282C3×C42—22no—
283C3×C42—21no—
284C3×C42C22×D512yes—
285#216—58no—
286#215—58no—
287(C2×D3)⋊C4—102no—
288C4⋊Dic3—52no—
289C4⋊Dic3—101no—
290Dic3⋊C4—102no—
291C4×Dic3—102no—
292#208—54no—
293C2×C3⋊C8—52no—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—51no—
296C2×C3⋊C8—102no—
297C23×C5C2×#6711yescharacteristic
298C22×D5—61no—
299C5×Q8—24no—
300C5×Q8—22no—
301C5×Q8C2×C4×D314yes—
302C2×C4×C5—24no—
303C2×C4×C5—22no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5C2×C4×D312yes—
307C2×C4×C5#23512yes—
308C2×C4×C5C2×#6712yes—
309C2×C4×C5C2×#6712yes—
310C2×C4×C5C2×#6711yescharacteristic
311C2×C4×C5C23×D311yescharacteristic
312C22⋊D5—64no—
313C2×Dic5—61no—
314C2×Dic5—62no—
315C2×Dic5—241no—
316D20—32no—
317D20—61no—
318C4×D5—64no—
319C4×D5—242no—
320Dic10—32no—
321Dic10—61no—
322#168—34no—
323#168—61no—
324#168—61no—
325C2×#41—151no—
326C22×Q8C2×D1511yescharacteristic
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—151no—
331C22×C8—151no—
332C4×Q8—21no—
333C4×Q8—21no—
334C4×Q8—152no—
335C4×Q8C2×D1512yes—
336C4×Q8C2×D1512yes—
337C4×D4—152no—
338C4×D4—302no—
339#117—22no—
340#117—301no—
341#117C2×D1511yescharacteristic
342#117C2×D1511yescharacteristic
343C2×C4⋊C4—151no—
344C2×C4⋊C4—301no—
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
359D15—42no—
360C23×C3C2×C22⋊D511yescharacteristic
361C22×D3—101no—
362C3×Q8—22no—
363C3×Q8—24no—
364C3×Q8C2×C4×D514yes—
365C2×C3×C4—22no—
366C2×C3×C4—24no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
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—104no—
376C2×Dic3—101no—
377C2×Dic3—102no—
378C2×Dic3—401no—
379D12—52no—
380D12—101no—
381C4×D3—104no—
382C4×D3—402no—
383Dic6—52no—
384Dic6—101no—
385C3⋊C8—54no—
386C3⋊C8—101no—
387C3⋊C8—101no—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5C4×#6712yes—
391C22×C5C22×#6711yescharacteristic
392C2×D5—62no—
393C2×D5—121no—
394C2×D5—121no—
395C4×C5—24no—
396C4×C5—22no—
397C4×C5—22no—
398C4×C5—41no—
399C4×C5—41no—
400C4×C5C4×#6712yes—
401C4×C5C22×C4×D311yescharacteristic
402C4×C5C2×#23511yescharacteristic
403Dic5—62no—
404Dic5—122no—
405#41—154no—
406#41—302no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—151no—
410C2×Q8C4×D1514yescomplemented
411C2×Q8C22×D1512yes—
412C2×D4—151no—
413C2×D4—301no—
414C22×C4—21no—
415C22×C4—301no—
416C22×C4—301no—
417C22×C4—302no—
418C22×C4C22×D1511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—154no—
422C2×C8—152no—
423C2×C8—151no—
424C2×C8—151no—
425C2×C8—302no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—152no—
429C4⋊C4—301no—
430C4⋊C4—302no—
431C4⋊C4#122011yescharacteristic
432C4⋊C4#122011yescharacteristic
433C4⋊C4C22×D1511yescharacteristic
434C4⋊C4C22×D1511yescharacteristic
435C22⋊C4—21no—
436C22⋊C4—302no—
437C22⋊C4#122012yes—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—302no—
442C42C22×D1512yes—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3C4×C22⋊D512yes—
447C22×C3C22×C22⋊D511yescharacteristic
448C2×D3—102no—
449C2×D3—201no—
450C2×D3—201no—
451C3×C4—24no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4C4×C22⋊D512yes—
457C3×C4C22×C4×D511yescharacteristic
458C3×C4C2×#70911yescharacteristic
459Dic3—102no—
460Dic3—202no—
461C2×C5—22no—
462C2×C5C2×C4×#6711yescharacteristic
463C2×C5#583711yescharacteristic
464C2×C5#583811yescharacteristic
465D5—122no—
466C23—301no—
467C23C2×#122011yescharacteristic
468Q8—24no—
469Q8—22no—
470Q8—152no—
471Q8—301no—
472Q8C2×C4×D1514yes—
473D4—152no—
474D4—304no—
475D4—301no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—301no—
481C2×C4—304no—
482C2×C4—302no—
483C2×C4—1202no—
484C2×C4—1201no—
485C2×C4C2×C4×D1512yes—
486C2×C4#681712yes—
487C2×C4C2×#122012yes—
488C2×C4C2×#122012yes—
489C2×C4C2×#122011yescharacteristic
490C2×C4C23×D1511yescharacteristic
491C8—154no—
492C8—301no—
493C8—301no—
494C2×C3—22no—
495C2×C3C2×C4×C22⋊D511yescharacteristic
496C2×C3#6617111yescharacteristic
497C2×C3#6617211yescharacteristic
498D3—202no—
499C5#8254711yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—302no—
503C22—601no—
504C22—601no—
505C22C4×#122012yes—
506C22C22×#122011yescharacteristic
507C4—24no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—302no—
513C4—602no—
514C4C4×#122012yes—
515C4C22×C4×D1511yescharacteristic
516C4C2×#681711yescharacteristic
517C3#1061427811yescharacteristic, complemented, Sylow
518C2—22no—
519C2—602no—
520C2C2×C4×#122011yescharacteristic
521C2#1175659611yescharacteristic
522C2#1175659711yescharacteristic
subgroupG/H
centerC22C22×#1220
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#519C2
socleC22×C3×C5C22×D4
prime radicalC2×#6817C22
cosocleC2×C3×C4×C5C24
hypercenter#519D15
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
1. C3×C5×#519C2semidirect
4. C2×#91119C2semidirect
5. #11751131C2semidirect
6. C4×#6718C2semidirect
7. #11751127C2semidirect
8. C4×#6717C2semidirect
10. #11751098C2semidirect
11. #11751096C2semidirect
23. #91831C4semidirect
44. C5×#519D3semidirect
57. C2×C3×C5×Q8C2×C4semidirect
81. #6718C2×C4semidirect
82. #6717C2×C4semidirect
88. #6710C2×C4semidirect
93. C3×#519D5semidirect
195. C2×C5×Q8C4×D3semidirect
233. #519D15semidirect
261. C2×C3×Q8C4×D5semidirect
410. C2×Q8C4×D15semidirect
443. C3×C5#2947semidirect
499. C5#82547semidirect
517. C3#10614278semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5192
C2×#918312
#117562822
C2×#911192
#117511312
C4×#67182
#117511272
C4×#67172
#117511032
#117510982
#117510962
#106142783
#825475