Finite Group Databasebeta

← #14417253

Subgroups — #14417253

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×#65838C211yesmaximal, characteristic
2#11754213C211yesmaximal, characteristic, complemented
3C2×#91313C211yesmaximal, characteristic
4C2×#91094C211yesmaximal, characteristic, complemented
5C5×#4964C211yesmaximal, characteristic, complemented
6#11748364C211yesmaximal, characteristic
7#11748320C211yesmaximal, characteristic, complemented
8#11747472C212yesmaximal, complemented
9#11747270C212yesmaximal, complemented
10#11747269C212yesmaximal, complemented
11#11747268C212yesmaximal, complemented
12#10610933—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C22×C3⋊#168C2211yescharacteristic
17C5×#926C2211yescharacteristic
18C2×C5×#208C2211yescharacteristic
19C3×C4×C22⋊D5—22no—
20C3×#4072C2211yescharacteristic
21C3×#4039C2212yes—
22C3×#4038—21no—
23C3×D5×C4⋊C4—21no—
24C3×#4033C2212yes—
25#91532—22no—
26#91340—21no—
27#91340—21no—
28#91333C2212yes—
29#91333C2212yes—
30C2×#6662C2212yes—
31#91313C418yescomplemented
32C2×#6657C2212yes—
33#91094C2212yes—
34#91094C2212yes—
35C5×#808C2212yes—
36C5×#807C2212yes—
37#90970C2211yescharacteristic
38#90970C2211yescharacteristic
39#90964C2211yescharacteristic
40#90964C2211yescharacteristic
41#90952C2212yes—
42Dic5×C3⋊C8C2212yes—
43#82571—51nomaximal, Hall
44C2×#4055—31no—
45#65850—31no—
46#65838D311yescharacteristic
47C5×#428—31no—
48#65411—31no—
49#65291—31no—
50C2×#3942—31no—
51#65178—32no—
52#65162—32no—
53#65142—32no—
54#65141—32no—
55C22×C3×C4×C5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15—21no—
58C2×Dic30—21no—
59C2×Dic30—22no—
60C2×Dic30C2×C414yescomplemented
61C2×Dic30C2312yes—
62C2×C3×C22⋊D5—21no—
63C3×#709—22no—
64C3×#709C2×C414yes—
65C2×C3×D20C2311yescharacteristic
66C2×C3×C4×D5—21no—
67C2×C3×C4×D5—22no—
68C2×C3×Dic10C2311yescharacteristic
69C3×C5×C4⋊C4—21no—
70C3×C5×C4⋊C4C2312yes—
71C2×C3⋊#168—22no—
72C2×C3⋊#168C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C5×#217—21no—
75C5×C4⋊Dic3C2311yescharacteristic
76C5×C4⋊Dic3C2311yescharacteristic
77C4×C5×Dic3—21no—
78C5×#208C2×C414yescomplemented
79C2×C5×C3⋊C8C2312yes—
80C3×C22⋊Dic5D412yes—
81C3×#685—22no—
82C3×C4⋊Dic5D411yescharacteristic
83C3×C4⋊Dic5D411yescharacteristic
84C3×#683—22no—
85C3×C4×Dic5—22no—
86C3×C4×Dic5C2312yes—
87#6669—22no—
88#6669—22no—
89Dic3×Dic5—22no—
90Dic3×Dic5—22no—
91#6662C2×C418yescomplemented
92#6657C2×C418yescomplemented
93C3×#520—51no—
94C2×#951—51no—
95#5516—51no—
96#5233—51no—
97#5030—52no—
98C4×#216—52no—
99#5018—52no—
100C4×#214—52no—
101#4997—51no—
102C2×#810—51no—
103#4964D511yescharacteristic, complemented
104C2×#709C2×D311yescharacteristic
105C22×Dic10—31no—
106C2×C5×M16—31no—
107C5×#117—31no—
108C2×C5×C4⋊C4C2×D311yescharacteristic
109C4×C22⋊D5—22no—
110#4072C2×D311yescharacteristic
111C22×#168—31no—
112#4055—38no—
113C2×Dic20—32no—
114C2×#677—32no—
115#4039C2×D312yes—
116#4038—21no—
117D5×C4⋊C4—21no—
118#4033—32no—
119#4033—32no—
120#4033C2×D312yes—
121#4023—62no—
122C4×Dic10—61no—
123C4×Dic10—61no—
124C5×#107—32no—
125C5×#106—32no—
126#3947—32no—
127C8×Dic5—32no—
128#3942—31no—
129#3942—31no—
130#3942—32no—
131#3942—32no—
132#3941—31no—
133#3941—31no—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C3×D5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—81no—
139C2×C3×C4×C5C22×C412yes—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5#4112yes—
142C2×C3×C4×C5C2411yescharacteristic
143C2×Dic15—24no—
144C2×Dic15—22no—
145Dic30—24no—
146Dic30—22no—
147Dic30C22×C414yes—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C3×C22⋊D5—24no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5C2×D412yes—
153C2×C3×Dic5C2×D412yes—
154C3×D20—21no—
155C3×D20C22×C412yes—
156C3×C4×D5—24no—
157C3×C4×D5—82no—
158C3×Dic10—21no—
159C3×Dic10C22×C412yes—
160C3⋊#168—21no—
161C3⋊#168—21no—
162C5×C3⋊C8C22×C414yes—
163C2×C3×#41—51no—
164C22×Dic6—51no—
165C3×C4×Q8—52no—
166C3×C4×D4—52no—
167C3×C4×D4—102no—
168C3×#117—51no—
169C3×#117—101no—
170C2×C3×C4⋊C4—101no—
171C2×C3×C4⋊C4C2×D511yescharacteristic
172#951—58no—
173C2×#216—52no—
174C2×#214—52no—
175#926C2×D511yescharacteristic
176C2×#208C2×D511yescharacteristic
177C22×C3⋊C8—51no—
178#887—101no—
179#887—101no—
180#877—102no—
181C4×Dic6—52no—
182C4×Dic6—52no—
183#835—51no—
184#835—51no—
185#832—51no—
186#832—51no—
187#810—52no—
188#810—52no—
189#808C2×D512yes—
190#807C2×D512yes—
191#803—52no—
192C4×C3⋊C8—52no—
193C22×C4×C5—21no—
194C22×C4×C5C22×D311yescharacteristic
195C2×C22⋊D5—21no—
196C22×Dic5—61no—
197#709—22no—
198#709C4×D314yes—
199C2×D20C22×D311yescharacteristic
200C2×C4×D5—21no—
201C2×C4×D5—22no—
202C2×Dic10—34no—
203C2×Dic10—32no—
204C2×Dic10—62no—
205C2×Dic10—61no—
206C2×Dic10C22×D311yescharacteristic
207C5×M16—34no—
208C2×C5×C8—32no—
209C5×C4⋊C4—21no—
210C5×C4⋊C4—31no—
211C5×C4⋊C4—31no—
212C5×C4⋊C4C22×D312yes—
213C5×C22⋊C4—61no—
214C42×C5—61no—
215C22⋊Dic5#6712yes—
216#685—22no—
217C4⋊Dic5#6711yescharacteristic
218C4⋊Dic5#6711yescharacteristic
219#683—22no—
220#683—62no—
221#683—62no—
222C4×Dic5—22no—
223C4×Dic5—62no—
224C4×Dic5—62no—
225C4×Dic5C22×D312yes—
226C2×#168—31no—
227C2×#168—31no—
228C2×#168—62no—
229Dic20—38no—
230#677—38no—
231C2×#137—151no—
232#520—51no—
233#519—151no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—31no—
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×C3×D5—22no—
247C2×C3×D5—41no—
248C2×C3×D5—41no—
249C3×C4×C5—42no—
250C3×C4×C5C4×D412yes—
251C3×C4×C5C23×C411yescharacteristic
252C3×C4×C5C2×#4111yescharacteristic
253Dic15—24no—
254C3×Dic5—22no—
255C3×Dic5—22no—
256C3×Dic5—22no—
257C5×Dic3—41no—
258C5×Dic3—41no—
259C3×#41—54no—
260C3×#41—102no—
261C2×C3×Q8—51no—
262C2×C3×D4—51no—
263C2×C3×D4—101no—
264C22×C3×C4—21no—
265C22×C3×C4—102no—
266C22×C3×C4—101no—
267C22×C3×C4C22×D511yescharacteristic
268C22×Dic3—101no—
269C2×Dic6—54no—
270C2×Dic6—52no—
271C2×Dic6—101no—
272C2×Dic6—102no—
273C3×C4⋊C4—21no—
274C3×C4⋊C4—51no—
275C3×C4⋊C4—51no—
276C3×C4⋊C4—102no—
277C3×C4⋊C4C22×D512yes—
278C3×C22⋊C4—52no—
279C3×C22⋊C4—102no—
280C3×C42—52no—
281C3×C42—102no—
282#217—21no—
283#216—58no—
284#214—58no—
285C4⋊Dic3C22×D511yescharacteristic
286C4⋊Dic3C22×D511yescharacteristic
287Dic3⋊C4—102no—
288Dic3⋊C4—102no—
289C4×Dic3—21no—
290C4×Dic3—102no—
291C4×Dic3—102no—
292#208C4×D514yescomplemented
293C2×C3⋊C8—51no—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—102no—
296C2×C3⋊C8C22×D512yes—
297C23×C5C2×#6711yescharacteristic
298C22×D5—21no—
299C2×C4×C5—22no—
300C2×C4×C5—61no—
301C2×C4×C5—61no—
302C2×C4×C5—81no—
303C2×C4×C5C2×C4×D312yes—
304C2×C4×C5#23512yes—
305C2×C4×C5C2×#6711yescharacteristic
306C2×C4×C5C23×D311yescharacteristic
307C22⋊D5—24no—
308C2×Dic5—21no—
309C2×Dic5—64no—
310C2×Dic5—62no—
311C2×Dic5C2×#6712yes—
312C2×Dic5C2×#6712yes—
313D20—21no—
314D20C2×C4×D312yes—
315C4×D5—24no—
316C4×D5—82no—
317Dic10—21no—
318Dic10—34no—
319Dic10—64no—
320Dic10—62no—
321Dic10C2×C4×D312yes—
322C5×C8—34no—
323#168—61no—
324#168—61no—
325C2×#41—51no—
326C22×Q8—151no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—31no—
331C22×C8—151no—
332C4×Q8—52no—
333C4×Q8—152no—
334C4×Q8—152no—
335C4×Q8—301no—
336C4×Q8—301no—
337C4×D4—52no—
338C4×D4—102no—
339#117—31no—
340#117—51no—
341#117—101no—
342#117—302no—
343C2×C4⋊C4—101no—
344C2×C4⋊C4D3×D511yescharacteristic
345#107—32no—
346#106—32no—
347#103—151no—
348#103—151no—
349#103—152no—
350#103—152no—
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
359C3×D5—42no—
360C23×C3—101no—
361C23×C3D4×D511yescharacteristic
362C3×Q8—52no—
363C3×Q8—101no—
364C3×D4—52no—
365C3×D4—104no—
366C3×D4—101no—
367C2×C3×C4—22no—
368C2×C3×C4—52no—
369C2×C3×C4—52no—
370C2×C3×C4—81no—
371C2×C3×C4—101no—
372C2×C3×C4—104no—
373C2×C3×C4—402no—
374C2×C3×C4C2×C4×D512yes—
375C2×C3×C4D4×D511yescharacteristic
376C2×C3×C4#71312yes—
377C2×C3×C4C23×D511yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—102no—
381C2×Dic3—104no—
382Dic6—54no—
383Dic6—102no—
384Dic6—104no—
385C3⋊C8—101no—
386C3⋊C8—101no—
387C3⋊C8C2×C4×D514yes—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5C4×#6712yes—
391C22×C5C22×#6711yescharacteristic
392C2×D5—22no—
393C2×D5—41no—
394C2×D5—41no—
395C4×C5—42no—
396C4×C5—121no—
397C4×C5—121no—
398C4×C5C4×#6712yes—
399C4×C5C22×C4×D311yescharacteristic
400C4×C5C2×#23511yescharacteristic
401Dic5—22no—
402Dic5—22no—
403Dic5—22no—
404Dic5—64no—
405#41—54no—
406#41—102no—
407C2×Q8—51no—
408C2×Q8—154no—
409C2×Q8—152no—
410C2×Q8—302no—
411C2×Q8—301no—
412C2×D4—51no—
413C2×D4—101no—
414C22×C4—21no—
415C22×C4—102no—
416C22×C4—101no—
417C22×C4—301no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—34no—
422C2×C8—32no—
423C2×C8—151no—
424C2×C8—151no—
425C2×C8—302no—
426C4⋊C4—21no—
427C4⋊C4—31no—
428C4⋊C4—31no—
429C4⋊C4—51no—
430C4⋊C4—51no—
431C4⋊C4—102no—
432C4⋊C4—302no—
433C4⋊C4—302no—
434C4⋊C4C2×D3×D512yes—
435C22⋊C4—52no—
436C22⋊C4—61no—
437C22⋊C4—102no—
438C42—52no—
439C42—61no—
440C42—102no—
441C42—302no—
442C42—302no—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—102no—
447C22×C3—201no—
448C22×C3—201no—
449C22×C3#403912yes—
450C22×C3C2×D4×D511yescharacteristic
451C3×C4—42no—
452C3×C4—102no—
453C3×C4—102no—
454C3×C4—102no—
455C3×C4#403912yes—
456C3×C4C22×C4×D511yescharacteristic
457C3×C4C2×#71311yescharacteristic
458Dic3—41no—
459Dic3—41no—
460Dic3—104no—
461C2×C5—22no—
462C2×C5C2×C4×#6711yescharacteristic
463C2×C5#583611yescharacteristic
464C2×C5#583811yescharacteristic
465D5—42no—
466C23—101no—
467C23D5×#6711yescharacteristic
468Q8—52no—
469Q8—101no—
470Q8—154no—
471Q8—304no—
472Q8—302no—
473D4—52no—
474D4—104no—
475D4—101no—
476C2×C4—22no—
477C2×C4—52no—
478C2×C4—52no—
479C2×C4—61no—
480C2×C4—61no—
481C2×C4—81no—
482C2×C4—101no—
483C2×C4—104no—
484C2×C4—304no—
485C2×C4—302no—
486C2×C4—402no—
487C2×C4#677312yes—
488C2×C4C4×D3×D512yes—
489C2×C4D5×#6711yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—34no—
492C8—301no—
493C8—301no—
494C2×C3—22no—
495C2×C3—202no—
496C2×C3C2×#403911yescharacteristic
497C2×C3#6610911yescharacteristic
498C2×C3#6611311yescharacteristic
499C5#8257111yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—102no—
503C22—201no—
504C22—201no—
505C22#9144412yes—
506C22C2×D5×#6711yescharacteristic
507C4—42no—
508C4—102no—
509C4—102no—
510C4—102no—
511C4—121no—
512C4—121no—
513C4—304no—
514C4#9144412yes—
515C4C2×#677311yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061093311yescharacteristic, complemented, Sylow
518C2—22no—
519C2—202no—
520C2C2×#9144411yescharacteristic
521C2#1175470811yescharacteristic
522C2#1175510211yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
cosocleC2×C3×C4×C5C24
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11754213C2semidirect
4. C2×#91094C2semidirect
5. C5×#4964C2semidirect
7. #11748320C2semidirect
8. #11747472C2semidirect
9. #11747270C2semidirect
10. #11747269C2semidirect
11. #11747268C2semidirect
31. #91313C4semidirect
60. C2×Dic30C2×C4semidirect
78. C5×#208C2×C4semidirect
91. #6662C2×C4semidirect
92. #6657C2×C4semidirect
103. #4964D5semidirect
292. #208C4×D5semidirect
443. C3×C5#2947semidirect
499. C5#82571semidirect
517. C3#10610933semidirect

Maximal subgroups

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

subgroupindex
C3×#658382
#117542132
C2×#913132
C2×#910942
C5×#49642
#117483642
#117483202
#117474722
#117472702
#117472692
#117472682
#106109333
#825715