Finite Group Databasebeta

← #14417262

Subgroups — #14417262

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
1C5×#5516C211yesmaximal, characteristic, complemented
2C3×#65857C211yesmaximal, characteristic
3C2×#91307C211yesmaximal, characteristic
4#11751098C211yesmaximal, characteristic, complemented
5#11748470C211yesmaximal, characteristic, complemented
6C2×#90972C211yesmaximal, characteristic, complemented
7#11748364C211yesmaximal, characteristic
8#11747089C212yesmaximal, complemented
9C4×#6661C212yesmaximal, complemented
10#11747048C212yesmaximal, complemented
11C4×#6655C212yesmaximal, complemented
12#10614278—31nomaximal, Hall
13C22×C5×Dic6C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C22×C3⋊#168C2211yescharacteristic
18C5×#926C2211yescharacteristic
19C5×#887—21no—
20C5×#887—21no—
21C5×#877—22no—
22C4×C5×Dic6C2212yes—
23C4×C5×Dic6C2212yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C3×#4038—21no—
26C3×D5×C4⋊C4—21no—
27C3×#4027—22no—
28C3×C4×D20C2212yes—
29C3×C4×Dic10C2212yes—
30C2×#6661C2212yes—
31#91307C418yescomplemented
32C2×#6655C2212yes—
33#91087C2212yes—
34C4×C3⋊#168C2212yes—
35#90989C2212yes—
36#90986C2212yes—
37#90972C2212yes—
38#90972C2212yes—
39#90970C2211yescharacteristic
40#90970C2211yescharacteristic
41#90964C2211yescharacteristic
42#90964C2211yescharacteristic
43#82571—51nomaximal, Hall
44C5×#519—31no—
45C2×#4088—31no—
46#65857D311yescharacteristic
47C2×#3967—31no—
48#65319—32no—
49C4×#689—32no—
50#65315—32no—
51C4×#688—32no—
52#65291—31no—
53#65286—31no—
54#65284—31no—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C5×Dic3—21no—
57C2×C5×Dic6—21no—
58C2×C5×Dic6—22no—
59C2×C5×Dic6C2×C414yescomplemented
60C2×C5×Dic6C2312yes—
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#6710C2×C414yescomplemented
74C2×C3⋊#168—22no—
75C2×C3⋊#168C2311yescharacteristic
76C2×C3⋊#168C2311yescharacteristic
77C2×C3⋊#168C2312yes—
78C5×#217—21no—
79C5×C4⋊Dic3C2311yescharacteristic
80C5×C4⋊Dic3C2311yescharacteristic
81C5×Dic3⋊C4—22no—
82C5×Dic3⋊C4—22no—
83C4×C5×Dic3—22no—
84C4×C5×Dic3—22no—
85C4×C5×Dic3—21no—
86C3×#685—22no—
87C3×C4⋊Dic5—21no—
88C3×C4⋊Dic5C2312yes—
89C3×#683—22no—
90C3×C4×Dic5—22no—
91#6661C2×C418yescomplemented
92#6655C2×C418yescomplemented
93C3×#520—51no—
94C2×#951—51no—
95#5516D511yescharacteristic, complemented
96#5233—51no—
97#5030—52no—
98C4×#216—52no—
99#5018—52no—
100C4×#214—52no—
101#4997—51no—
102C2×#810—51no—
103#4964—51no—
104C22×C5×Q8—31no—
105C2×#709C2×D311yescharacteristic
106C4×C5×Q8—32no—
107C4×C5×Q8—32no—
108C4×C5×Q8—61no—
109C4×C5×Q8—61no—
110C5×#117—31no—
111C5×#117—62no—
112C5×#117C2×D311yescharacteristic
113C2×#689—32no—
114#4088—38no—
115C2×#688—32no—
116C2×C4⋊Dic5C2×D311yescharacteristic
117C2×#681—31no—
118C22×#168—31no—
119#4038—21no—
120D5×C4⋊C4—21no—
121#4027—22no—
122C4×D20C2×D312yes—
123C4×Dic10C2×D312yes—
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×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×C5×Dic3—24no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—21no—
147C5×Dic6—24no—
148C5×Dic6—22no—
149C5×Dic6C22×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—
160C3⋊#168—21no—
161C3⋊#168—21no—
162C3⋊#168C22×C414yes—
163C2×C3×#41—51no—
164C22×Dic6C2×D511yescharacteristic
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#926C2×D511yescharacteristic
176C2×#208—51no—
177C22×C3⋊C8—51no—
178#887—21no—
179#887—21no—
180#877—22no—
181C4×Dic6C2×D512yes—
182C4×Dic6C2×D512yes—
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#803—52no—
192C4×C3⋊C8—52no—
193C2×C5×Q8—34no—
194C2×C5×Q8—32no—
195C2×C5×Q8—62no—
196C2×C5×Q8—61no—
197C22×C4×C5—61no—
198C22×C4×C5C22×D311yescharacteristic
199C2×C22⋊D5—21no—
200C22×Dic5—21no—
201#709—22no—
202#709C4×D314yes—
203C2×D20C22×D311yescharacteristic
204C2×C4×D5—21no—
205C2×C4×D5—22no—
206C2×Dic10C22×D311yescharacteristic
207C5×C4⋊C4—31no—
208C5×C4⋊C4—31no—
209C5×C4⋊C4—62no—
210C5×C4⋊C4—62no—
211C5×C4⋊C4#6711yescharacteristic
212C5×C4⋊C4#6711yescharacteristic
213C5×C22⋊C4—61no—
214C5×C22⋊C4#6712yes—
215C42×C5—61no—
216C42×C5—62no—
217C42×C5—62no—
218C42×C5C22×D312yes—
219#689—38no—
220#688—38no—
221#685—22no—
222C4⋊Dic5—21no—
223C4⋊Dic5C22×D312yes—
224#683—22no—
225C4×Dic5—22no—
226#681—34no—
227C2×#168—32no—
228C2×#168—31no—
229C2×#168—31no—
230C2×#168—62no—
231C2×#137—151no—
232#520—51no—
233#519—31no—
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×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
254C3×Dic5—22no—
255C3×Dic5—42no—
256C5×Dic3—24no—
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—101no—
265C22×C3×C4—101no—
266C22×C3×C4—102no—
267C22×C3×C4C22×D511yescharacteristic
268C22×Dic3—21no—
269C2×Dic6—22no—
270C2×Dic6—21no—
271C2×Dic6C4×D514yescomplemented
272C2×Dic6C22×D512yes—
273C3×C4⋊C4—52no—
274C3×C4⋊C4—102no—
275C3×C4⋊C4—101no—
276C3×C4⋊C4C22⋊D511yescharacteristic
277C3×C4⋊C4C22⋊D511yescharacteristic
278C3×C22⋊C4—102no—
279C3×C22⋊C4C22⋊D512yes—
280C3×C42—102no—
281C3×C42C22×D512yes—
282#217—21no—
283#216—58no—
284#214—58no—
285C4⋊Dic3C22×D511yescharacteristic
286C4⋊Dic3C22×D511yescharacteristic
287Dic3⋊C4—22no—
288Dic3⋊C4—22no—
289C4×Dic3—22no—
290C4×Dic3—22no—
291C4×Dic3—21no—
292#208—54no—
293C2×C3⋊C8—52no—
294C2×C3⋊C8—51no—
295C2×C3⋊C8—51no—
296C2×C3⋊C8—102no—
297C23×C5C2×#6711yescharacteristic
298C22×D5—21no—
299C5×Q8—34no—
300C5×Q8—64no—
301C5×Q8—62no—
302C2×C4×C5—64no—
303C2×C4×C5—62no—
304C2×C4×C5—61no—
305C2×C4×C5—61no—
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—24no—
313C2×Dic5—21no—
314C2×Dic5—22no—
315C2×Dic5—81no—
316D20—21no—
317D20C2×C4×D312yes—
318C4×D5—24no—
319C4×D5—82no—
320Dic10—21no—
321Dic10C2×C4×D312yes—
322#168—34no—
323#168—61no—
324#168—61no—
325C2×#41—51no—
326C22×Q8—31no—
327#137—158no—
328C2×Q16—152no—
329C2×SD16—152no—
330C2×M16—151no—
331C22×C8—151no—
332C4×Q8—32no—
333C4×Q8—32no—
334C4×Q8—52no—
335C4×Q8—61no—
336C4×Q8—61no—
337C4×D4—52no—
338C4×D4—102no—
339#117—31no—
340#117—62no—
341#117—101no—
342#117D3×D511yescharacteristic
343C2×C4⋊C4—51no—
344C2×C4⋊C4—101no—
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—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×C3C2×C22⋊D511yescharacteristic
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×C4#70912yes—
374C2×C3×C4C2×C22⋊D512yes—
375C2×C3×C4C2×C22⋊D512yes—
376C2×C3×C4C2×C22⋊D511yescharacteristic
377C2×C3×C4C23×D511yescharacteristic
378C2×Dic3—22no—
379C2×Dic3—24no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382Dic6—22no—
383Dic6—24no—
384Dic6C2×C4×D514yes—
385C3⋊C8—54no—
386C3⋊C8—101no—
387C3⋊C8—101no—
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×C5—64no—
398C4×C5—121no—
399C4×C5—121no—
400C4×C5C4×#6712yes—
401C4×C5C22×C4×D311yescharacteristic
402C4×C5C2×#23511yescharacteristic
403Dic5—22no—
404Dic5—42no—
405#41—54no—
406#41—102no—
407C2×Q8—34no—
408C2×Q8—32no—
409C2×Q8—51no—
410C2×Q8—62no—
411C2×Q8—61no—
412C2×D4—51no—
413C2×D4—101no—
414C22×C4—61no—
415C22×C4—101no—
416C22×C4—101no—
417C22×C4—102no—
418C22×C4C2×D3×D511yescharacteristic
419Q16—158no—
420SD16—158no—
421M16—154no—
422C2×C8—152no—
423C2×C8—151no—
424C2×C8—151no—
425C2×C8—302no—
426C4⋊C4—31no—
427C4⋊C4—31no—
428C4⋊C4—52no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—101no—
432C4⋊C4—102no—
433C4⋊C4#120111yescharacteristic
434C4⋊C4#120111yescharacteristic
435C22⋊C4—61no—
436C22⋊C4—102no—
437C22⋊C4#120112yes—
438C42—61no—
439C42—62no—
440C42—62no—
441C42—102no—
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×C22⋊D512yes—
450C22×C3C22×C22⋊D511yescharacteristic
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—102no—
454C3×C4—202no—
455C3×C4C4×C22⋊D512yes—
456C3×C4C22×C4×D511yescharacteristic
457C3×C4C2×#70911yescharacteristic
458Dic3—24no—
459Dic3—41no—
460Dic3—41no—
461C2×C5—22no—
462C2×C5C2×C4×#6711yescharacteristic
463C2×C5#583611yescharacteristic
464C2×C5#583811yescharacteristic
465D5—42no—
466C23—101no—
467C23C2×#120111yescharacteristic
468Q8—34no—
469Q8—52no—
470Q8—64no—
471Q8—62no—
472Q8—101no—
473D4—52no—
474D4—104no—
475D4—101no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—101no—
481C2×C4—104no—
482C2×C4—102no—
483C2×C4—402no—
484C2×C4—401no—
485C2×C4#677112yes—
486C2×C4C4×D3×D512yes—
487C2×C4C2×#120112yes—
488C2×C4C2×#120112yes—
489C2×C4C2×#120111yescharacteristic
490C2×C4C22×D3×D511yescharacteristic
491C8—154no—
492C8—301no—
493C8—301no—
494C2×C3—22no—
495C2×C3—202no—
496C2×C3C2×C4×C22⋊D511yescharacteristic
497C2×C3#6617111yescharacteristic
498C2×C3#6617211yescharacteristic
499C5#8257111yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—102no—
503C22—201no—
504C22—201no—
505C22C4×#120112yes—
506C22C22×#120111yescharacteristic
507C4—22no—
508C4—22no—
509C4—64no—
510C4—102no—
511C4—121no—
512C4—121no—
513C4—202no—
514C4C4×#120112yes—
515C4C2×#677111yescharacteristic
516C4C2×C4×D3×D511yescharacteristic
517C3#1061427811yescharacteristic, complemented, Sylow
518C2—22no—
519C2—202no—
520C2C2×C4×#120111yescharacteristic
521C2#1175530211yescharacteristic
522C2#1175530311yescharacteristic
subgroupG/H
centerC22C22×#1201
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. C5×#5516C2semidirect
4. #11751098C2semidirect
5. #11748470C2semidirect
6. C2×#90972C2semidirect
8. #11747089C2semidirect
9. C4×#6661C2semidirect
10. #11747048C2semidirect
11. C4×#6655C2semidirect
31. #91307C4semidirect
59. C2×C5×Dic6C2×C4semidirect
73. #6710C2×C4semidirect
91. #6661C2×C4semidirect
92. #6655C2×C4semidirect
95. #5516D5semidirect
271. C2×Dic6C4×D5semidirect
443. C3×C5#2947semidirect
499. C5#82571semidirect
517. C3#10614278semidirect

Maximal subgroups

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

subgroupindex
C5×#55162
C3×#658572
C2×#913072
#117510982
#117484702
C2×#909722
#117483642
#117470892
C4×#66612
#117470482
C4×#66552
#106142783
#825715