Finite Group Databasebeta

← #14417265

Subgroups — #14417265

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
1#11756282C211yesmaximal, characteristic
2C3×#65850C211yesmaximal, characteristic, complemented
3C2×#91315C211yesmaximal, characteristic
4C5×#4997C211yesmaximal, characteristic, complemented
5#11748455C211yesmaximal, characteristic, complemented
6#11748411C211yesmaximal, characteristic
7C2×#90971C211yesmaximal, characteristic, complemented
8#11747091C212yesmaximal, complemented
9C4×#6662C212yesmaximal, complemented
10#11747054C212yesmaximal, complemented
11C4×#6658C212yesmaximal, complemented
12#10610302—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17#91781—21no—
18C4⋊C4×D15—21no—
19#91770—22no—
20C4×D60C2212yes—
21C4×Dic30C2212yes—
22C2×C5×#208C2211yescharacteristic
23C22×C5×C3⋊C8C2211yescharacteristic
24C3×#4072C2211yescharacteristic
25C3×#4033—21no—
26C3×#4033—21no—
27C3×#4023—22no—
28C3×C4×Dic10C2212yes—
29C3×C4×Dic10C2212yes—
30C2×#6662C2212yes—
31#91315C418yescomplemented
32C2×#6658C2212yes—
33C5×#803C2212yes—
34C4×C5×C3⋊C8C2212yes—
35#90990C2212yes—
36#90987C2212yes—
37#90974C2211yescharacteristic
38#90974C2211yescharacteristic
39#90971C2212yes—
40#90971C2212yes—
41#90968C2211yescharacteristic
42#90968C2211yescharacteristic
43#82547—51nomaximal, Hall
44C2×#4055—31no—
45#65857—31no—
46#65850D311yescharacteristic, complemented
47C5×#405—31no—
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—
58#6817—22no—
59#6817C2×C414yes—
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×C4×D15—22no—
63C2×Dic30C2311yescharacteristic
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#6714—22no—
74C4⋊Dic15—21no—
75C4⋊Dic15C2312yes—
76#6712—22no—
77C4×Dic15—22no—
78C5×#208C2×C414yescomplemented
79C2×C5×C3⋊C8—22no—
80C2×C5×C3⋊C8C2311yescharacteristic
81C2×C5×C3⋊C8C2311yescharacteristic
82C2×C5×C3⋊C8C2312yes—
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#6662C2×C418yescomplemented
92#6658C2×C418yescomplemented
93C3×#519—51no—
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#4997D511yescharacteristic, complemented
103#4995—51no—
104C2×#709—31no—
105C22×Dic10C2×D311yescharacteristic
106C2×C5×M16—31no—
107C22×C5×C8—31no—
108C5×#117C2×D311yescharacteristic
109#4072C2×D311yescharacteristic
110C2×C4⋊Dic5—31no—
111#4055—38no—
112C2×Dic20—32no—
113C2×#677—32no—
114#4038—61no—
115D5×C4⋊C4—61no—
116#4033—21no—
117#4033—21no—
118#4027—62no—
119#4023—22no—
120C4×D20—32no—
121C4×Dic10—32no—
122C4×Dic10C2×D312yes—
123C4×Dic10C2×D312yes—
124C5×#97—32no—
125C4×C5×C8—32no—
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—21no—
145C2×Dic15—22no—
146C2×Dic15—81no—
147D60—21no—
148D60C22×C412yes—
149C4×D15—24no—
150C4×D15—82no—
151Dic30—21no—
152Dic30C22×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—
160C5×C3⋊C8—21no—
161C5×C3⋊C8—21no—
162C5×C3⋊C8C22×C414yes—
163C22×C3×Q8—51no—
164C2×#235—51no—
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⋊Dic3—51no—
176C2×#208C2×D511yescharacteristic
177C22×C3⋊C8C2×D511yescharacteristic
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#803C2×D512yes—
192C4×C3⋊C8C2×D512yes—
193C22×C4×C5C22×D311yescharacteristic
194C2×C22⋊D5—61no—
195C22×Dic5—21no—
196C22×Dic5—61no—
197#709—34no—
198#709—62no—
199C2×D20—31no—
200C2×C4×D5—61no—
201C2×C4×D5—62no—
202C2×Dic10—22no—
203C2×Dic10—21no—
204C2×Dic10—31no—
205C2×Dic10C4×D314yescomplemented
206C2×Dic10C22×D312yes—
207C5×M16—34no—
208C2×C5×C8—32no—
209C2×C5×C8—31no—
210C2×C5×C8—31no—
211C2×C5×C8—62no—
212C5×C4⋊C4#6711yescharacteristic
213C5×C4⋊C4#6711yescharacteristic
214C5×C22⋊C4#6712yes—
215C42×C5C22×D312yes—
216C22⋊Dic5—21no—
217#685—62no—
218C4⋊Dic5—32no—
219C4⋊Dic5—61no—
220C4⋊Dic5C22×D311yescharacteristic
221C4⋊Dic5C22×D311yescharacteristic
222#683—22no—
223#683—22no—
224#683—62no—
225C4×Dic5—22no—
226C4×Dic5—22no—
227C4×Dic5—21no—
228C4×Dic5—62no—
229Dic20—38no—
230#677—38no—
231C2×#137—151no—
232#520—151no—
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—31no—
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—22no—
255Dic15—42no—
256C3×Dic5—24no—
257C3×Dic5—41no—
258C3×Dic5—41no—
259C2×C3×Q8—54no—
260C2×C3×Q8—52no—
261C2×C3×Q8—101no—
262C2×C3×Q8—102no—
263C22×C3×C4—101no—
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—51no—
274C3×C4⋊C4—51no—
275C3×C4⋊C4—102no—
276C3×C4⋊C4—102no—
277C3×C4⋊C4D2011yescharacteristic
278C3×C4⋊C4D2011yescharacteristic
279C3×C22⋊C4—101no—
280C3×C22⋊C4D2012yes—
281C3×C42—101no—
282C3×C42—102no—
283C3×C42—102no—
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#208C4×D514yescomplemented
293C2×C3⋊C8—22no—
294C2×C3⋊C8C22×D511yescharacteristic
295C2×C3⋊C8C22×D511yescharacteristic
296C2×C3⋊C8C22×D512yes—
297C23×C5C2×#6711yescharacteristic
298C22×D5—61no—
299C2×C4×C5C2×C4×D312yes—
300C2×C4×C5#23512yes—
301C2×C4×C5C2×#6712yes—
302C2×C4×C5C2×#6712yes—
303C2×C4×C5C2×#6711yescharacteristic
304C2×C4×C5C23×D311yescharacteristic
305C22⋊D5—64no—
306C2×Dic5—24no—
307C2×Dic5—22no—
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—61no—
311C2×Dic5—62no—
312C2×Dic5—241no—
313D20—32no—
314D20—61no—
315C4×D5—64no—
316C4×D5—242no—
317Dic10—24no—
318Dic10—22no—
319Dic10—32no—
320Dic10—61no—
321Dic10C2×C4×D314yes—
322C5×C8—34no—
323C5×C8—61no—
324C5×C8—61no—
325C2×#41—151no—
326C22×Q8—51no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—31no—
331C22×C8—31no—
332C4×Q8—52no—
333C4×Q8—52no—
334C4×Q8—101no—
335C4×Q8—101no—
336C4×Q8—152no—
337C4×D4—152no—
338C4×D4—302no—
339#117—51no—
340#117—102no—
341#117—301no—
342#117D3×D511yescharacteristic
343C2×C4⋊C4—151no—
344C2×C4⋊C4—301no—
345#107—152no—
346#106—152no—
347#103—151no—
348#103—151no—
349#103—152no—
350#103—152no—
351#102—151no—
352#102—151no—
353#97—32no—
354C4×C8—32no—
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—
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×C4C2×D2012yes—
371C2×C3×C4C2×D2012yes—
372C2×C3×C4C2×D2011yescharacteristic
373C2×C3×C4#70912yes—
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—21no—
386C3⋊C8—21no—
387C3⋊C8C2×C4×D514yes—
388C22×C5—21no—
389C22×C5—21no—
390C22×C5C4×#6712yes—
391C22×C5C22×#6711yescharacteristic
392C2×D5—62no—
393C2×D5—121no—
394C2×D5—121no—
395C4×C5—22no—
396C4×C5—22no—
397C4×C5C4×#6712yes—
398C4×C5C22×C4×D311yescharacteristic
399C4×C5C2×#23511yescharacteristic
400Dic5—24no—
401Dic5—41no—
402Dic5—41no—
403Dic5—62no—
404Dic5—122no—
405#41—154no—
406#41—302no—
407C2×Q8—54no—
408C2×Q8—52no—
409C2×Q8—102no—
410C2×Q8—101no—
411C2×Q8—151no—
412C2×D4—151no—
413C2×D4—301no—
414C22×C4—101no—
415C22×C4—301no—
416C22×C4—301no—
417C22×C4—302no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—34no—
422C2×C8—32no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—62no—
426C4⋊C4—51no—
427C4⋊C4—51no—
428C4⋊C4—102no—
429C4⋊C4—102no—
430C4⋊C4—152no—
431C4⋊C4—301no—
432C4⋊C4—302no—
433C4⋊C4#120211yescharacteristic
434C4⋊C4#120211yescharacteristic
435C22⋊C4—101no—
436C22⋊C4—302no—
437C22⋊C4#120212yes—
438C42—101no—
439C42—102no—
440C42—102no—
441C42—302no—
442C42C2×D3×D512yes—
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×C4—104no—
454C3×C4—201no—
455C3×C4—201no—
456C3×C4C4×D2012yes—
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×#120211yescharacteristic
468Q8—54no—
469Q8—104no—
470Q8—102no—
471Q8—152no—
472Q8—301no—
473D4—152no—
474D4—304no—
475D4—301no—
476C2×C4—104no—
477C2×C4—102no—
478C2×C4—101no—
479C2×C4—101no—
480C2×C4—301no—
481C2×C4—304no—
482C2×C4—302no—
483C2×C4—1202no—
484C2×C4—1201no—
485C2×C4#677112yes—
486C2×C4C4×D3×D512yes—
487C2×C4C2×#120212yes—
488C2×C4C2×#120212yes—
489C2×C4C2×#120211yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—34no—
492C8—61no—
493C8—61no—
494C2×C3—22no—
495C2×C3C2×C4×D2011yescharacteristic
496C2×C3#6608511yescharacteristic
497C2×C3#6608711yescharacteristic
498D3—202no—
499C5#8254711yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—302no—
503C22—601no—
504C22—601no—
505C22C4×#120212yes—
506C22C22×#120211yescharacteristic
507C4—22no—
508C4—22no—
509C4—104no—
510C4—201no—
511C4—201no—
512C4—302no—
513C4—602no—
514C4C4×#120212yes—
515C4C2×#677111yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061030211yescharacteristic, complemented, Sylow
518C2—22no—
519C2—602no—
520C2C2×C4×#120211yescharacteristic
521C2#1175531011yescharacteristic
522C2#1175531211yescharacteristic
subgroupG/H
centerC22C22×#1202
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#6817C22
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. C5×#4997C2semidirect
5. #11748455C2semidirect
7. C2×#90971C2semidirect
8. #11747091C2semidirect
9. C4×#6662C2semidirect
10. #11747054C2semidirect
11. C4×#6658C2semidirect
31. #91315C4semidirect
46. #65850D3semidirect
67. C2×C3×Dic10C2×C4semidirect
78. C5×#208C2×C4semidirect
91. #6662C2×C4semidirect
92. #6658C2×C4semidirect
102. #4997D5semidirect
205. C2×Dic10C4×D3semidirect
292. #208C4×D5semidirect
443. C3×C5#2947semidirect
499. C5#82547semidirect
517. C3#10610302semidirect

Maximal subgroups

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

subgroupindex
#117562822
C3×#658502
C2×#913152
C5×#49972
#117484552
#117484112
C2×#909712
#117470912
C4×#66622
#117470542
C4×#66582
#106103023
#825475