Finite Group Databasebeta

← #14417264

Subgroups — #14417264

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#11756275C211yesmaximal, characteristic, complemented
2C3×#65857C211yesmaximal, characteristic
3C2×#91313C211yesmaximal, characteristic
4C5×#4997C211yesmaximal, characteristic, complemented
5#11748453C211yesmaximal, characteristic, complemented
6C2×#90974C211yesmaximal, characteristic, complemented
7#11748365C211yesmaximal, characteristic
8#11747091C212yesmaximal, complemented
9C4×#6662C212yesmaximal, complemented
10#11747052C212yesmaximal, complemented
11C4×#6657C212yesmaximal, complemented
12#10610302—31nomaximal, Hall
13C22×Dic30C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16#91815C2211yescharacteristic
17#91776—21no—
18#91776—21no—
19#91766—22no—
20C4×Dic30C2212yes—
21C4×Dic30C2212yes—
22C2×C5×#208C2211yescharacteristic
23C22×C5×C3⋊C8C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C3×#4038—21no—
26C3×D5×C4⋊C4—21no—
27C3×#4027—22no—
28C3×C4×D20C2212yes—
29C3×C4×Dic10C2212yes—
30C2×#6662C2212yes—
31#91313C418yescomplemented
32C2×#6657C2212yes—
33C5×#803C2212yes—
34C4×C5×C3⋊C8C2212yes—
35#90990C2212yes—
36#90987C2212yes—
37#90974C2212yes—
38#90974C2212yes—
39#90971C2211yescharacteristic
40#90971C2211yescharacteristic
41#90965C2211yescharacteristic
42#90965C2211yescharacteristic
43#82571—51nomaximal, Hall
44C2×#4055—31no—
45#65857D311yescharacteristic
46#65850—31no—
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
56C22×Dic15—21no—
57C2×Dic30—21no—
58C2×Dic30—22no—
59C2×Dic30C2×C414yescomplemented
60C2×Dic30C2312yes—
61C2×C3×C22⋊D5—21no—
62C22×C3×Dic5—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⋊C4D411yescharacteristic
70C3×C5×C4⋊C4D411yescharacteristic
71C3×C5×C22⋊C4D412yes—
72C3×C42×C5C2312yes—
73#6719—21no—
74C4⋊Dic15C2311yescharacteristic
75C4⋊Dic15C2311yescharacteristic
76#6712—22no—
77#6712—22no—
78C4×Dic15—22no—
79C4×Dic15—22no—
80C4×Dic15—21no—
81C5×#208C2×C414yescomplemented
82C2×C5×C3⋊C8—22no—
83C2×C5×C3⋊C8C2311yescharacteristic
84C2×C5×C3⋊C8C2311yescharacteristic
85C2×C5×C3⋊C8C2312yes—
86C3×#685—22no—
87C3×C4⋊Dic5—21no—
88C3×C4⋊Dic5C2312yes—
89C3×#683—22no—
90C3×C4×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#4997D511yescharacteristic, complemented
102C2×#810—51no—
103#4964—51no—
104C2×#709C2×D311yescharacteristic
105C22×Dic10—31no—
106C2×C5×M16—31no—
107C22×C5×C8—31no—
108C5×#117C2×D311yescharacteristic
109#4072—31no—
110C2×C4⋊Dic5C2×D311yescharacteristic
111#4055—38no—
112C2×Dic20—32no—
113C2×#677—32no—
114#4038—21no—
115D5×C4⋊C4—21no—
116#4033—61no—
117#4033—61no—
118#4027—22no—
119#4023—62no—
120C4×D20C2×D312yes—
121C4×Dic10—32no—
122C4×Dic10—32no—
123C4×Dic10C2×D312yes—
124C5×#97—32no—
125C4×C5×C8—32no—
126#3953—31no—
127#3953—31no—
128#3950—32no—
129#3949—32no—
130#3948—31no—
131#3948—31no—
132#3948—32no—
133#3948—32no—
134#2947—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C3×D5—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
143C2×Dic15—24no—
144C2×Dic15—22no—
145C2×Dic15—21no—
146C2×Dic15—21no—
147Dic30—24no—
148Dic30—22no—
149Dic30C22×C414yes—
150C3×C22⋊D5—24no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—81no—
154C3×D20—21no—
155C3×D20C22×C412yes—
156C3×C4×D5—24no—
157C3×C4×D5—82no—
158C3×Dic10—21no—
159C3×Dic10C22×C412yes—
160C5×C3⋊C8—21no—
161C5×C3⋊C8—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—101no—
169C3×#117C2×D511yescharacteristic
170C2×C3×C4⋊C4—51no—
171C2×C3×C4⋊C4—101no—
172#951—58no—
173C2×#216—52no—
174C2×#214—52no—
175#926—51no—
176C2×#208C2×D511yescharacteristic
177C22×C3⋊C8C2×D511yescharacteristic
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#808—52no—
190#807—52no—
191#803C2×D512yes—
192C4×C3⋊C8C2×D512yes—
193C22×C4×C5C22×D311yescharacteristic
194C2×C22⋊D5—21no—
195C22×Dic5—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—
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—61no—
217#685—22no—
218C4⋊Dic5—21no—
219C4⋊Dic5—31no—
220C4⋊Dic5—31no—
221C4⋊Dic5C22×D312yes—
222#683—22no—
223#683—62no—
224#683—62no—
225C4×Dic5—22no—
226C4×Dic5—62no—
227C4×Dic5—62no—
228C4×Dic5—61no—
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—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×C3×D5—22no—
247C2×C3×D5—41no—
248C2×C3×D5—41no—
249C3×C4×C5—22no—
250C3×C4×C5—22no—
251C3×C4×C5C4×D412yes—
252C3×C4×C5C23×C411yescharacteristic
253C3×C4×C5C2×#4111yescharacteristic
254Dic15—24no—
255Dic15—41no—
256Dic15—41no—
257C3×Dic5—22no—
258C3×Dic5—42no—
259C3×#41—54no—
260C3×#41—102no—
261C2×C3×Q8—51no—
262C2×C3×D4—51no—
263C2×C3×D4—101no—
264C22×C3×C4—101no—
265C22×C3×C4—101no—
266C22×C3×C4—102no—
267C22×C3×C4C22×D511yescharacteristic
268C22×Dic3—101no—
269C2×Dic6—54no—
270C2×Dic6—52no—
271C2×Dic6—101no—
272C2×Dic6—102no—
273C3×C4⋊C4—52no—
274C3×C4⋊C4—102no—
275C3×C4⋊C4—101no—
276C3×C4⋊C4D2011yescharacteristic
277C3×C4⋊C4D2011yescharacteristic
278C3×C22⋊C4—102no—
279C3×C22⋊C4D2012yes—
280C3×C42—102no—
281C3×C42C22×D512yes—
282#217—101no—
283#216—58no—
284#214—58no—
285C4⋊Dic3—51no—
286C4⋊Dic3—51no—
287Dic3⋊C4—102no—
288Dic3⋊C4—102no—
289C4×Dic3—102no—
290C4×Dic3—102no—
291C4×Dic3—101no—
292#208C4×D514yescomplemented
293C2×C3⋊C8—22no—
294C2×C3⋊C8C22×D511yescharacteristic
295C2×C3⋊C8C22×D511yescharacteristic
296C2×C3⋊C8C22×D512yes—
297C23×C5C2×#6711yescharacteristic
298C22×D5—21no—
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—24no—
306C2×Dic5—21no—
307C2×Dic5—22no—
308C2×Dic5—64no—
309C2×Dic5—62no—
310C2×Dic5—61no—
311C2×Dic5—61no—
312C2×Dic5—81no—
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—
323C5×C8—61no—
324C5×C8—61no—
325C2×#41—51no—
326C22×Q8—151no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—31no—
331C22×C8—31no—
332C4×Q8—52no—
333C4×Q8—152no—
334C4×Q8—152no—
335C4×Q8—301no—
336C4×Q8—301no—
337C4×D4—52no—
338C4×D4—102no—
339#117—101no—
340#117—151no—
341#117—302no—
342#117D3×D511yescharacteristic
343C2×C4⋊C4—51no—
344C2×C4⋊C4—101no—
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—32no—
354C4×C8—32no—
355C2×C3×C5—22no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#57711yescharacteristic
358C2×C3×C5#57811yescharacteristic
359C3×D5—42no—
360C23×C3—101no—
361C23×C3C2×D2011yescharacteristic
362C3×Q8—52no—
363C3×Q8—101no—
364C3×D4—52no—
365C3×D4—104no—
366C3×D4—101no—
367C2×C3×C4—101no—
368C2×C3×C4—104no—
369C2×C3×C4—102no—
370C2×C3×C4—402no—
371C2×C3×C4—401no—
372C2×C3×C4C2×C4×D512yes—
373C2×C3×C4C2×D2012yes—
374C2×C3×C4C2×D2012yes—
375C2×C3×C4C2×D2011yescharacteristic
376C2×C3×C4#70912yes—
377C2×C3×C4C23×D511yescharacteristic
378C2×Dic3—102no—
379C2×Dic3—104no—
380C2×Dic3—101no—
381C2×Dic3—101no—
382Dic6—54no—
383Dic6—102no—
384Dic6—104no—
385C3⋊C8—21no—
386C3⋊C8—21no—
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—22no—
396C4×C5—22no—
397C4×C5C4×#6712yes—
398C4×C5C22×C4×D311yescharacteristic
399C4×C5C2×#23511yescharacteristic
400Dic5—22no—
401Dic5—42no—
402Dic5—64no—
403Dic5—121no—
404Dic5—121no—
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—101no—
415C22×C4—101no—
416C22×C4—102no—
417C22×C4—301no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—34no—
422C2×C8—32no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—62no—
426C4⋊C4—52no—
427C4⋊C4—101no—
428C4⋊C4—102no—
429C4⋊C4—151no—
430C4⋊C4—151no—
431C4⋊C4—302no—
432C4⋊C4—302no—
433C4⋊C4#120211yescharacteristic
434C4⋊C4#120211yescharacteristic
435C22⋊C4—102no—
436C22⋊C4—301no—
437C22⋊C4#120212yes—
438C42—102no—
439C42—301no—
440C42—302no—
441C42—302no—
442C42C2×D3×D512yes—
443C3×C5#294711yescharacteristic, complemented, Hall
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—102no—
447C22×C3—201no—
448C22×C3—201no—
449C22×C3C4×D2012yes—
450C22×C3C22×D2011yescharacteristic
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—102no—
454C3×C4—202no—
455C3×C4C4×D2012yes—
456C3×C4C22×C4×D511yescharacteristic
457C3×C4C2×#70911yescharacteristic
458Dic3—104no—
459Dic3—201no—
460Dic3—201no—
461C2×C5—22no—
462C2×C5C2×C4×#6711yescharacteristic
463C2×C5#583611yescharacteristic
464C2×C5#583811yescharacteristic
465D5—42no—
466C23—101no—
467C23C2×#120211yescharacteristic
468Q8—52no—
469Q8—101no—
470Q8—154no—
471Q8—304no—
472Q8—302no—
473D4—52no—
474D4—104no—
475D4—101no—
476C2×C4—101no—
477C2×C4—104no—
478C2×C4—102no—
479C2×C4—304no—
480C2×C4—302no—
481C2×C4—301no—
482C2×C4—301no—
483C2×C4—402no—
484C2×C4—401no—
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×C3—202no—
496C2×C3C2×C4×D2011yescharacteristic
497C2×C3#6608511yescharacteristic
498C2×C3#6608711yescharacteristic
499C5#8257111yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—102no—
503C22—201no—
504C22—201no—
505C22C4×#120212yes—
506C22C22×#120211yescharacteristic
507C4—22no—
508C4—22no—
509C4—102no—
510C4—202no—
511C4—304no—
512C4—601no—
513C4—601no—
514C4C4×#120212yes—
515C4C2×#677111yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061030211yescharacteristic, complemented, Sylow
518C2—22no—
519C2—202no—
520C2C2×C4×#120211yescharacteristic
521C2#1175530911yescharacteristic
522C2#1175531211yescharacteristic
subgroupG/H
centerC22C22×#1202
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#709C22
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
1. #11756275C2semidirect
4. C5×#4997C2semidirect
5. #11748453C2semidirect
6. C2×#90974C2semidirect
8. #11747091C2semidirect
9. C4×#6662C2semidirect
10. #11747052C2semidirect
11. C4×#6657C2semidirect
31. #91313C4semidirect
59. C2×Dic30C2×C4semidirect
81. C5×#208C2×C4semidirect
91. #6662C2×C4semidirect
92. #6657C2×C4semidirect
101. #4997D5semidirect
292. #208C4×D5semidirect
443. C3×C5#2947semidirect
499. C5#82571semidirect
517. C3#10610302semidirect

Maximal subgroups

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

subgroupindex
#117562752
C3×#658572
C2×#913132
C5×#49972
#117484532
C2×#909742
#117483652
#117470912
C4×#66622
#117470522
C4×#66572
#106103023
#825715