Finite Group Databasebeta

← #14417182

Subgroups — #14417182

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
1C2×#91798C211yesmaximal, characteristic
2#11756282C211yesmaximal, characteristic
3#11756275C211yesmaximal, characteristic, complemented
4C3×C5×#405C211yesmaximal, characteristic, complemented
5#11751222C211yesmaximal, characteristic, complemented
6#11751214C211yesmaximal, characteristic
7C2×#91100C211yesmaximal, characteristic, complemented
8#11750817C212yesmaximal, complemented
9#11750811C212yesmaximal, complemented
10C4×Dic60C212yesmaximal, complemented
11C4×#6706C212yesmaximal, complemented
12#10610302—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×Dic30C2211yescharacteristic
15C2×C3×C5×M16C2211yescharacteristic
16C22×C3×C5×C8C2211yescharacteristic
17C3×C5×#117C2211yescharacteristic
18#91815C2211yescharacteristic
19C2×C4⋊Dic15C2211yescharacteristic
20#91798C418yescomplemented
21C2×Dic60C2212yes—
22C2×#6706C2212yes—
23#91781—21no—
24C4⋊C4×D15—21no—
25#91776—21no—
26#91776—21no—
27#91770—22no—
28#91766—22no—
29C4×D60C2212yes—
30C4×Dic30C2212yes—
31C4×Dic30C2212yes—
32C4×Dic30C2212yes—
33C3×C5×#97C2212yes—
34C3×C4×C5×C8C2212yes—
35#91105C2211yescharacteristic
36#91105C2211yescharacteristic
37#91102C2212yes—
38#91101C2212yes—
39#91100C2212yes—
40#91100C2212yes—
41#91100C2211yescharacteristic
42#91100C2211yescharacteristic
43#78571—51nomaximal, Hall
44C2×#4055—31no—
45#65857—31no—
46#65850—31no—
47C5×#405D311yescharacteristic, complemented
48#65410—31no—
49#65402—31no—
50C2×#3948—31no—
51#65005—32no—
52#64999—32no—
53C4×Dic20—32no—
54C4×#677—32no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×#1220—21no—
57C22×Dic15—21no—
58C22×Dic15—21no—
59#6817—22no—
60#6817C2×C414yes—
61C2×D60C2311yescharacteristic
62C2×C4×D15—21no—
63C2×C4×D15—22no—
64C2×Dic30—21no—
65C2×Dic30—22no—
66C2×Dic30C2×C414yescomplemented
67C2×Dic30C2312yes—
68C2×Dic30C2311yescharacteristic
69C3×C5×M16C2×C414yescomplemented
70C2×C3×C5×C8—22no—
71C2×C3×C5×C8C2311yescharacteristic
72C2×C3×C5×C8C2311yescharacteristic
73C2×C3×C5×C8C2312yes—
74C3×C5×C4⋊C4D411yescharacteristic
75C3×C5×C4⋊C4D411yescharacteristic
76C3×C5×C22⋊C4D412yes—
77C3×C42×C5C2312yes—
78#6719—21no—
79#6714—22no—
80C4⋊Dic15—21no—
81C4⋊Dic15C2311yescharacteristic
82C4⋊Dic15C2311yescharacteristic
83C4⋊Dic15C2312yes—
84#6712—22no—
85#6712—22no—
86#6712—22no—
87C4×Dic15—22no—
88C4×Dic15—22no—
89C4×Dic15—22no—
90C4×Dic15—21no—
91Dic60C2×C418yescomplemented
92#6706C2×C418yescomplemented
93C2×#909—51no—
94#5523—51no—
95#5516—51no—
96C3×#405D511yescharacteristic, complemented
97#5121—51no—
98#5113—51no—
99C2×#816—51no—
100#4716—52no—
101#4710—52no—
102C4×Dic12—52no—
103C4×#204—52no—
104C2×#709—31no—
105C22×Dic10—31no—
106C2×C5×M16C2×D311yescharacteristic
107C22×C5×C8C2×D311yescharacteristic
108C5×#117C2×D311yescharacteristic
109#4072—31no—
110C2×C4⋊Dic5—31no—
111#4055—38no—
112C2×Dic20—32no—
113C2×#677—32no—
114#4038—61no—
115D5×C4⋊C4—61no—
116#4033—61no—
117#4033—61no—
118#4027—62no—
119#4023—62no—
120C4×D20—32no—
121C4×Dic10—32no—
122C4×Dic10—32no—
123C4×Dic10—32no—
124C5×#97C2×D312yes—
125C4×C5×C8C2×D312yes—
126#3953—31no—
127#3953—31no—
128#3950—32no—
129#3949—32no—
130#3948—32no—
131#3948—32no—
132#3948—31no—
133#3948—31no—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×D15—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
143#1220—24no—
144C2×Dic15—24no—
145C2×Dic15—22no—
146C2×Dic15—21no—
147C2×Dic15—21no—
148C2×Dic15—21no—
149C2×Dic15—22no—
150C2×Dic15—81no—
151D60—21no—
152D60C22×C412yes—
153C4×D15—24no—
154C4×D15—82no—
155Dic30—24no—
156Dic30—22no—
157Dic30—21no—
158Dic30C22×C414yes—
159Dic30C22×C412yes—
160C3×C5×C8—21no—
161C3×C5×C8—21no—
162C3×C5×C8C22×C414yes—
163C2×#235—51no—
164C22×Dic6—51no—
165C2×C3×M16C2×D511yescharacteristic
166C22×C3×C8C2×D511yescharacteristic
167C3×#117C2×D511yescharacteristic
168#926—51no—
169C2×C4⋊Dic3—51no—
170#909—58no—
171C2×Dic12—52no—
172C2×#204—52no—
173#892—101no—
174D3×C4⋊C4—101no—
175#887—101no—
176#887—101no—
177#67⋊C4—102no—
178#877—102no—
179C4×D12—52no—
180C4×Dic6—52no—
181C4×Dic6—52no—
182C4×Dic6—52no—
183C3×#97C2×D512yes—
184C3×C4×C8C2×D512yes—
185#821—51no—
186#821—51no—
187#818—52no—
188#817—52no—
189#816—52no—
190#816—51no—
191#816—52no—
192#816—51no—
193C22×C4×C5C22×D311yescharacteristic
194C2×C22⋊D5—61no—
195C22×Dic5—61no—
196C22×Dic5—61no—
197#709—34no—
198#709—62no—
199C2×D20—31no—
200C2×C4×D5—61no—
201C2×C4×D5—62no—
202C2×Dic10—34no—
203C2×Dic10—32no—
204C2×Dic10—31no—
205C2×Dic10—61no—
206C2×Dic10—62no—
207C5×M16C4×D314yescomplemented
208C2×C5×C8—22no—
209C2×C5×C8C22×D311yescharacteristic
210C2×C5×C8C22×D311yescharacteristic
211C2×C5×C8C22×D312yes—
212C5×C4⋊C4D1211yescharacteristic
213C5×C4⋊C4D1211yescharacteristic
214C5×C22⋊C4D1212yes—
215C42×C5C22×D312yes—
216C22⋊Dic5—61no—
217#685—62no—
218C4⋊Dic5—32no—
219C4⋊Dic5—31no—
220C4⋊Dic5—31no—
221C4⋊Dic5—61no—
222#683—62no—
223#683—62no—
224#683—62no—
225C4×Dic5—62no—
226C4×Dic5—62no—
227C4×Dic5—62no—
228C4×Dic5—61no—
229Dic20—38no—
230#677—38no—
231C2×#137—151no—
232#520—151no—
233#519—151no—
234#441—152no—
235#440—152no—
236C4×Q16—152no—
237C4×SD16—152no—
238#428—151no—
239#416—151no—
240C2×#103—151no—
241#405D1511yescharacteristic, complemented
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—22no—
250C3×C4×C5—22no—
251C3×C4×C5C4×D412yes—
252C3×C4×C5C23×C411yescharacteristic
253C3×C4×C5C2×#4111yescharacteristic
254Dic15—24no—
255Dic15—22no—
256Dic15—42no—
257Dic15—41no—
258Dic15—41no—
259C22×C3×C4C22×D511yescharacteristic
260C2×#67—101no—
261C22×Dic3—101no—
262C22×Dic3—101no—
263#235—54no—
264#235—102no—
265C2×D12—51no—
266C2×C4×D3—101no—
267C2×C4×D3—102no—
268C2×Dic6—54no—
269C2×Dic6—52no—
270C2×Dic6—51no—
271C2×Dic6—101no—
272C2×Dic6—102no—
273C3×M16C4×D514yescomplemented
274C2×C3×C8—22no—
275C2×C3×C8C22×D511yescharacteristic
276C2×C3×C8C22×D511yescharacteristic
277C2×C3×C8C22×D512yes—
278C3×C4⋊C4D2011yescharacteristic
279C3×C4⋊C4D2011yescharacteristic
280C3×C22⋊C4D2012yes—
281C3×C42C22×D512yes—
282#217—101no—
283(C2×D3)⋊C4—102no—
284C4⋊Dic3—52no—
285C4⋊Dic3—51no—
286C4⋊Dic3—51no—
287C4⋊Dic3—101no—
288Dic3⋊C4—102no—
289Dic3⋊C4—102no—
290Dic3⋊C4—102no—
291C4×Dic3—102no—
292C4×Dic3—102no—
293C4×Dic3—102no—
294C4×Dic3—101no—
295Dic12—58no—
296#204—58no—
297C23×C5C2×D1211yescharacteristic
298C22×D5—61no—
299C2×C4×C5C2×C4×D312yes—
300C2×C4×C5C2×D1212yes—
301C2×C4×C5C2×D1212yes—
302C2×C4×C5C2×D1211yescharacteristic
303C2×C4×C5#23512yes—
304C2×C4×C5C23×D311yescharacteristic
305C22⋊D5—64no—
306C2×Dic5—64no—
307C2×Dic5—62no—
308C2×Dic5—61no—
309C2×Dic5—61no—
310C2×Dic5—61no—
311C2×Dic5—62no—
312C2×Dic5—241no—
313D20—32no—
314D20—61no—
315C4×D5—64no—
316C4×D5—242no—
317Dic10—34no—
318Dic10—32no—
319Dic10—64no—
320Dic10—62no—
321Dic10—61no—
322C5×C8—21no—
323C5×C8—21no—
324C5×C8C2×C4×D314yes—
325C2×#41—151no—
326C22×Q8—151no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16C2×D1511yescharacteristic
331C22×C8C2×D1511yescharacteristic
332C4×Q8—152no—
333C4×Q8—152no—
334C4×Q8—152no—
335C4×Q8—301no—
336C4×Q8—301no—
337C4×D4—152no—
338C4×D4—302no—
339#117—151no—
340#117—301no—
341#117—302no—
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#97C2×D1512yes—
354C4×C8C2×D1512yes—
355C2×C3×C5—22no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#57711yescharacteristic
358C2×C3×C5#57811yescharacteristic
359D15—42no—
360C23×C3C2×D2011yescharacteristic
361C22×D3—101no—
362C2×C3×C4C2×C4×D512yes—
363C2×C3×C4C2×D2012yes—
364C2×C3×C4C2×D2012yes—
365C2×C3×C4C2×D2011yescharacteristic
366C2×C3×C4#70912yes—
367C2×C3×C4C23×D511yescharacteristic
368#67—104no—
369C2×Dic3—102no—
370C2×Dic3—101no—
371C2×Dic3—104no—
372C2×Dic3—101no—
373C2×Dic3—101no—
374C2×Dic3—102no—
375C2×Dic3—401no—
376D12—52no—
377D12—101no—
378C4×D3—104no—
379C4×D3—402no—
380Dic6—54no—
381Dic6—52no—
382Dic6—102no—
383Dic6—101no—
384Dic6—104no—
385C3×C8—21no—
386C3×C8—21no—
387C3×C8C2×C4×D514yes—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5C4×D1212yes—
391C22×C5C22×D1211yescharacteristic
392C2×D5—62no—
393C2×D5—121no—
394C2×D5—121no—
395C4×C5—22no—
396C4×C5—22no—
397C4×C5C4×D1212yes—
398C4×C5C22×C4×D311yescharacteristic
399C4×C5C2×#23511yescharacteristic
400Dic5—64no—
401Dic5—62no—
402Dic5—122no—
403Dic5—121no—
404Dic5—121no—
405#41—154no—
406#41—302no—
407C2×Q8—154no—
408C2×Q8—152no—
409C2×Q8—151no—
410C2×Q8—301no—
411C2×Q8—302no—
412C2×D4—151no—
413C2×D4—301no—
414C22×C4—301no—
415C22×C4—301no—
416C22×C4—301no—
417C22×C4—302no—
418C22×C4C22×D1511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16C4×D1514yescomplemented
422C2×C8—22no—
423C2×C8C22×D1511yescharacteristic
424C2×C8C22×D1511yescharacteristic
425C2×C8C22×D1512yes—
426C4⋊C4—151no—
427C4⋊C4—151no—
428C4⋊C4—152no—
429C4⋊C4—301no—
430C4⋊C4—302no—
431C4⋊C4—302no—
432C4⋊C4—302no—
433C4⋊C4D6011yescharacteristic
434C4⋊C4D6011yescharacteristic
435C22⋊C4—301no—
436C22⋊C4—302no—
437C22⋊C4D6012yes—
438C42—301no—
439C42—302no—
440C42—302no—
441C42—302no—
442C42C22×D1512yes—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3C4×D2012yes—
447C22×C3C22×D2011yescharacteristic
448C2×D3—102no—
449C2×D3—201no—
450C2×D3—201no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4C4×D2012yes—
454C3×C4C22×C4×D511yescharacteristic
455C3×C4C2×#70911yescharacteristic
456Dic3—104no—
457Dic3—102no—
458Dic3—202no—
459Dic3—201no—
460Dic3—201no—
461C2×C5—22no—
462C2×C5C2×C4×D1211yescharacteristic
463C2×C5#575111yescharacteristic
464C2×C5#575311yescharacteristic
465D5—122no—
466C23—301no—
467C23C2×D6011yescharacteristic
468Q8—154no—
469Q8—152no—
470Q8—304no—
471Q8—302no—
472Q8—301no—
473D4—152no—
474D4—304no—
475D4—301no—
476C2×C4—304no—
477C2×C4—301no—
478C2×C4—302no—
479C2×C4—304no—
480C2×C4—301no—
481C2×C4—301no—
482C2×C4—302no—
483C2×C4—1202no—
484C2×C4—1201no—
485C2×C4C2×C4×D1512yes—
486C2×C4C2×D6012yes—
487C2×C4C2×D6012yes—
488C2×C4C2×D6011yescharacteristic
489C2×C4#681712yes—
490C2×C4C23×D1511yescharacteristic
491C8—21no—
492C8—21no—
493C8C2×C4×D1514yes—
494C2×C3—22no—
495C2×C3C2×C4×D2011yescharacteristic
496C2×C3#6608511yescharacteristic
497C2×C3#6608711yescharacteristic
498D3—202no—
499C5#7857111yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—302no—
503C22—601no—
504C22—601no—
505C22C4×D6012yes—
506C22C22×D6011yescharacteristic
507C4—22no—
508C4—22no—
509C4—302no—
510C4—304no—
511C4—602no—
512C4—601no—
513C4—601no—
514C4C4×D6012yes—
515C4C22×C4×D1511yescharacteristic
516C4C2×#681711yescharacteristic
517C3#1061030211yescharacteristic, complemented, Sylow
518C2—22no—
519C2—602no—
520C2C2×C4×D6011yescharacteristic
521C2#1175651011yescharacteristic
522C2#1175651211yescharacteristic
subgroupG/H
centerC22C22×D60
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C23×D15
FittingC3×C5×#405C2
socleC22×C3×C5C22×D4
prime radicalC2×#6817C22
cosocleC2×C3×C4×C5C24
hypercenter#405D15
hypocenterC3×C5#2947
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
3. #11756275C2semidirect
4. C3×C5×#405C2semidirect
5. #11751222C2semidirect
7. C2×#91100C2semidirect
8. #11750817C2semidirect
9. #11750811C2semidirect
10. C4×Dic60C2semidirect
11. C4×#6706C2semidirect
20. #91798C4semidirect
47. C5×#405D3semidirect
66. C2×Dic30C2×C4semidirect
69. C3×C5×M16C2×C4semidirect
91. Dic60C2×C4semidirect
92. #6706C2×C4semidirect
96. C3×#405D5semidirect
207. C5×M16C4×D3semidirect
241. #405D15semidirect
273. C3×M16C4×D5semidirect
421. M16C4×D15semidirect
443. C3×C5#2947semidirect
499. C5#78571semidirect
517. C3#10610302semidirect

Maximal subgroups

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

subgroupindex
C2×#917982
#117562822
#117562752
C3×C5×#4052
#117512222
#117512142
C2×#911002
#117508172
#117508112
C4×Dic602
C4×#67062
#106103023
#785715