Finite Group Databasebeta

← #14412221

Subgroups — #14412221

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92004C211yesmaximal, characteristic
2#11756504C211yesmaximal, characteristic, complemented
3#11754130C211yesmaximal, characteristic, complemented
4C5×#5132C211yesmaximal, characteristic, complemented
5C2×C3×#3956C211yesmaximal, characteristic, complemented
6C2×#90961C211yesmaximal, characteristic, complemented
7#11748355C211yesmaximal, characteristic, complemented
8#11748239C214yesmaximal, complemented
9#11748227C214yesmaximal, complemented
10#10610446—31nomaximal, Hall
11C2×#6817C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C2×#6782C414yes—
15C2×D15⋊Q8C2212yes—
16C2×#6769C412yes—
17#92004—22no—
18#92004—22no—
19#92004—22no—
20#92004—22no—
21C2×#6768C2212yes—
22C2×D3×Dic10C412yes—
23C2×C3×C5×M16C2211yescharacteristic
24C2×#6710C2211yescharacteristic
25#91796—22no—
26M16×D15C2214yes—
27#91790—22no—
28C2×C5×#208C2211yescharacteristic
29C2×C3×#681C2211yescharacteristic
30#91292C2214yes—
31#91291—22no—
32#91290—22no—
33C5×#824C2212yes—
34C5×#823C2212yes—
35C3×#3956C2214yes—
36#90962C2212yes—
37#90961C2214yes—
38#90959C2212yes—
39#78706—51nomaximal, Hall
40C2×#4150—31no—
41#66079—31no—
42#66079—31no—
43C5×#413—31no—
44#65526—31no—
45C2×#3956—31no—
46C2×#3956D311yescharacteristic, complemented
47D5×#101—34no—
48#65034—34no—
49C22×C3×C4×C5C2311yescharacteristic
50C2×#1220C2×C412yes—
51#6817D412yes—
52#6817D412yes—
53#6817D412yes—
54#6817D412yes—
55C2×D60C2×C411yescharacteristic
56C2×C4×D15C2312yes—
57C2×Dic30C2×C411yescharacteristic
58C2×C5×#67C2×C412yes—
59C5×#235—21no—
60C5×#235—21no—
61C5×#235—21no—
62C5×#235—21no—
63C2×C5×D12C2311yescharacteristic
64C2×C4×C5×D3C2×C412yes—
65C2×C5×Dic6C2311yescharacteristic
66C22×C3×Dic5C2×C412yes—
67C2×C3×Dic10—21no—
68C2×C3×Dic10—21no—
69C2×C3×Dic10—21no—
70C2×C3×Dic10—21no—
71C2×C3×Dic10C2×C412yes—
72C2×C3×Dic10C2312yes—
73C2×#1204C2×C414yes—
74C2×#1203C2×C414yes—
75C2×#1200C2×C414yes—
76#6782—28no—
77#6782—28no—
78C2×D3×Dic5C2×C414yes—
79D15⋊Q8—24no—
80D15⋊Q8—24no—
81#6769—24no—
82#6769—24no—
83#6768—24no—
84#6768—24no—
85D3×Dic10—24no—
86D3×Dic10—24no—
87C3×C5×M16—21no—
88C3×C5×M16C2312yes—
89C2×C3×C5×C8—21no—
90#6710—21no—
91#6710C2312yes—
92C2×C3⋊#168—21no—
93#6705—24no—
94C8×D15—24no—
95C5×#208—21no—
96C5×#208C2312yes—
97C2×C5×C3⋊C8—21no—
98C3×#681—21no—
99C3×#681C2312yes—
100C2×C3×#168—21no—
101#6651—24no—
102#6648—24no—
103C2×#1007—51no—
104#5745—51no—
105#5745—51no—
106C2×C3×#101—51no—
107C2×#836—51no—
108#5132—51no—
109#5132D511yescharacteristic, complemented
110#4751—54no—
111D3×#101—54no—
112C2×C5×#41—31no—
113#4150—62no—
114#4150—62no—
115#4150—62no—
116#4150—62no—
117C2×Q8×D5—32no—
118C2×#711—34no—
119C2×#711—32no—
120C2×#709—31no—
121C2×#709—32no—
122C22×Dic10—32no—
123C22×Dic10C2×D311yescharacteristic
124C2×C5×M16—31no—
125C2×C5×M16C2×D311yescharacteristic
126C2×#681—31no—
127C2×#681C2×D311yescharacteristic
128#4053—62no—
129#4053—62no—
130D5×M16—34no—
131D5×M16—34no—
132#4047—62no—
133#4047—62no—
134C5×#101—32no—
135C5×#100—32no—
136#3968—32no—
137#3965—32no—
138#3956—34no—
139#3956C2×D314yes—
140#2889—151noSylow
141C23×C3×C5C22×C411yescharacteristic
142C22×D15C22×C412yes—
143C22×C5×D3C22×C412yes—
144C2×C3×C4×C5C22×C411yescharacteristic
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D411yescharacteristic
147C2×C3×C4×C5C2×D411yescharacteristic
148C2×C3×C4×C5C2×D411yescharacteristic
149C2×C3×C4×C5C2411yescharacteristic
150#1220C22⋊C414yes—
151#1220C22⋊C414yes—
152C2×Dic15C22×C412yes—
153D60C22⋊C412yes—
154D60C22⋊C412yes—
155C4×D15C2×D414yes—
156C4×D15C2×D414yes—
157Dic30C22⋊C412yes—
158Dic30C22⋊C412yes—
159C5×#67—22no—
160C5×#67—22no—
161C2×C5×Dic3C22×C412yes—
162C5×D12—21no—
163C5×D12—21no—
164C4×C5×D3—22no—
165C4×C5×D3—22no—
166C5×Dic6—21no—
167C5×Dic6—21no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5C22×C414yes—
171C3×Dic10—22no—
172C3×Dic10—22no—
173C3×Dic10—22no—
174C3×Dic10—22no—
175#1204—28no—
176#1203—28no—
177#1200—28no—
178D3×Dic5—28no—
179C3×C5×C8—22no—
180C3⋊#168—22no—
181C3×#168—22no—
182C5×C3⋊C8—22no—
183C22×C3×Q8—51no—
184#1007—102no—
185#1007—102no—
186#1007—102no—
187#1007—102no—
188C2×#239—52no—
189C2×#239—52no—
190C2×D3×Q8—52no—
191C2×D3×Q8—52no—
192C2×#235—51no—
193C2×#235—54no—
194C2×#235C2×D511yescharacteristic
195C2×C3×M16—51no—
196C2×C3×M16C2×D511yescharacteristic
197C2×#208—51no—
198C2×#208C2×D511yescharacteristic
199#907—102no—
200#907—102no—
201D3×M16—54no—
202D3×M16—54no—
203#901—102no—
204#901—102no—
205C3×#101—54no—
206#836—54no—
207#824—52no—
208#824C2×D512yes—
209#823—52no—
210#823C2×D512yes—
211C5×#41—61no—
212C5×#41—61no—
213C5×#41—61no—
214C5×#41—61no—
215C2×C5×Q8—31no—
216C2×C5×D4—32no—
217C2×C5×D4—31no—
218C22×C4×C5—32no—
219C22×C4×C5C22×D311yescharacteristic
220C2×C22⋊D5—32no—
221C2×C22⋊D5—34no—
222C22×Dic5—34no—
223C22×Dic5C4×D312yes—
224Q8×D5—64no—
225Q8×D5—64no—
226#711—64no—
227#711—64no—
228#711—68no—
229#711—68no—
230#709—32no—
231#709—32no—
232#709—32no—
233#709—32no—
234#709—64no—
235#709—64no—
236C2×D20—31no—
237C2×C4×D5—32no—
238C2×C4×D5—34no—
239C2×Dic10—21no—
240C2×Dic10—21no—
241C2×Dic10—21no—
242C2×Dic10—21no—
243C2×Dic10—31no—
244C2×Dic10—34no—
245C2×Dic10—64no—
246C2×Dic10—64no—
247C2×Dic10C4×D312yes—
248C2×Dic10C22×D312yes—
249C5×M16—21no—
250C5×M16—32no—
251C5×M16—61no—
252C5×M16C22×D312yes—
253C2×C5×C8—21no—
254C2×C5×C8—61no—
255#681—21no—
256#681—32no—
257#681—61no—
258#681C22×D312yes—
259C2×#168—21no—
260C2×#168—61no—
261#676—64no—
262#676—64no—
263C8×D5—64no—
264C8×D5—64no—
265C2×2−1+4—151no—
266#568—151no—
267#568—151no—
268#413—31no—
269#413—154no—
270#413—151no—
271C2×#101—51no—
272C2×#101—154no—
273C2×#101—151no—
274C22×C3×C5—41no—
275C22×C3×C5C2×C22⋊C411yescharacteristic
276C22×C3×C5C2×C22⋊C411yescharacteristic
277C22×C3×C5C23×C411yescharacteristic
278C2×D15—42no—
279C2×D15C2×C22⋊C414yes—
280C2×C5×D3—22no—
281C2×C5×D3—42no—
282C3×C4×C5C2×C22⋊C411yescharacteristic
283C3×C4×C5C2×C22⋊C411yescharacteristic
284C3×C4×C5C22×D411yescharacteristic
285C3×C4×C5C22×D411yescharacteristic
286Dic15C2×C22⋊C414yes—
287C3×Dic5—24no—
288C5×Dic3—22no—
289C2×C3×Q8—52no—
290C2×C3×Q8—52no—
291C2×C3×Q8—101no—
292C2×C3×Q8—101no—
293C2×C3×Q8—101no—
294C2×C3×Q8—101no—
295C22×C3×C4—52no—
296C22×C3×C4C22×D511yescharacteristic
297C2×#67—52no—
298C2×#67C4×D512yes—
299#239—104no—
300#239—104no—
301#239—104no—
302#239—104no—
303D3×Q8—104no—
304D3×Q8—104no—
305D3×Q8—104no—
306D3×Q8—104no—
307#235—21no—
308#235—21no—
309#235—21no—
310#235—21no—
311#235—52no—
312#235—52no—
313#235—52no—
314#235—52no—
315#235—108no—
316#235—108no—
317C2×D12—51no—
318C2×D12—54no—
319C2×D12C22×D511yescharacteristic
320C2×C4×D3—52no—
321C2×C4×D3—54no—
322C2×C4×D3—54no—
323C2×C4×D3C4×D512yes—
324C2×Dic6—51no—
325C2×Dic6—54no—
326C2×Dic6C22×D511yescharacteristic
327C3×M16—21no—
328C3×M16—52no—
329C3×M16—101no—
330C3×M16C22×D512yes—
331C2×C3×C8—21no—
332C2×C3×C8—101no—
333#208—21no—
334#208—52no—
335#208—101no—
336#208C22×D512yes—
337C2×C3⋊C8—21no—
338C2×C3⋊C8—101no—
339ΓL(1,𝔽25)—104no—
340ΓL(1,𝔽25)—104no—
341D3×C8—104no—
342D3×C8—104no—
343C23×C5—32no—
344C23×C5C2×C4×D311yescharacteristic
345C22×D5—32no—
346C5×Q8—61no—
347C5×Q8—61no—
348C5×D4—62no—
349C5×D4—62no—
350C5×D4—61no—
351C5×D4—61no—
352C2×C4×C5—32no—
353C2×C4×C5—62no—
354C2×C4×C5—62no—
355C2×C4×C5C2×C4×D311yescharacteristic
356C2×C4×C5D3×D411yescharacteristic
357C2×C4×C5D3×D411yescharacteristic
358C2×C4×C5D3×D411yescharacteristic
359C2×C4×C5D3×D411yescharacteristic
360C2×C4×C5C23×D311yescharacteristic
361C22⋊D5—34no—
362C22⋊D5—34no—
363C22⋊D5—68no—
364C2×Dic5—22no—
365C2×Dic5—22no—
366C2×Dic5—32no—
367C2×Dic5—68no—
368C2×Dic5C2×C4×D314yes—
369D20—32no—
370D20—32no—
371C4×D5—34no—
372C4×D5—34no—
373C4×D5—68no—
374Dic10—22no—
375Dic10—22no—
376Dic10—22no—
377Dic10—22no—
378Dic10—32no—
379Dic10—32no—
380Dic10—68no—
381C5×C8—22no—
382C5×C8—62no—
383#168—22no—
384#168—62no—
3852−1+4—302no—
3862−1+4—302no—
3872−1+4—302no—
3882−1+4—302no—
389C2×#41—31no—
390C2×#41—152no—
391C2×#41—154no—
392C2×#41—152no—
393C2×#41—151no—
394C22×Q8—51no—
395C22×Q8—152no—
396C22×Q8—152no—
397#131—302no—
398#131—302no—
399#131—302no—
400#131—302no—
401C2×M16—31no—
402C2×M16—51no—
403C2×M16—154no—
404C2×M16—151no—
405C2×M16—154no—
406C2×M16D3×D511yescharacteristic
407#101—32no—
408#101—54no—
409#101—154no—
410#101—152no—
411#100—32no—
412#100—152no—
413C2×C3×C5—21no—
414C2×C3×C5—21no—
415C2×C3×C5—21no—
416C2×C3×C5C22×C22⋊C411yescharacteristic
417D15—24no—
418C5×D3—42no—
419C23×C3C2×C4×D511yescharacteristic
420C22×D3—52no—
421C22×D3C2×C4×D512yes—
422C3×Q8—102no—
423C3×Q8—102no—
424C3×Q8—102no—
425C3×Q8—102no—
426C2×C3×C4—54no—
427C2×C3×C4—102no—
428C2×C3×C4—102no—
429C2×C3×C4C2×C4×D511yescharacteristic
430C2×C3×C4D4×D511yescharacteristic
431C2×C3×C4D4×D511yescharacteristic
432C2×C3×C4D4×D511yescharacteristic
433C2×C3×C4D4×D511yescharacteristic
434C2×C3×C4C23×D511yescharacteristic
435#67—22no—
436#67—22no—
437#67—54no—
438#67—54no—
439C2×Dic3—52no—
440C2×Dic3C2×C4×D512yes—
441D12—21no—
442D12—21no—
443D12—52no—
444D12—52no—
445D12—108no—
446C4×D3—22no—
447C4×D3—22no—
448C4×D3—54no—
449C4×D3—54no—
450C4×D3—108no—
451C4×D3—108no—
452Dic6—21no—
453Dic6—21no—
454Dic6—52no—
455Dic6—52no—
456Dic6—108no—
457C3×C8—22no—
458C3×C8—102no—
459C3⋊C8—22no—
460C3⋊C8—102no—
461C22×C5—41no—
462C22×C5—62no—
463C22×C5—122no—
464C22×C5D3×C22⋊C411yescharacteristic
465C22×C5D3×C22⋊C411yescharacteristic
466C22×C5C22×C4×D311yescharacteristic
467C2×D5—34no—
468C2×D5—122no—
469C4×C5—62no—
470C4×C5D3×C22⋊C411yescharacteristic
471C4×C5D3×C22⋊C411yescharacteristic
472C4×C5C2×D3×D411yescharacteristic
473C4×C5C2×D3×D411yescharacteristic
474Dic5—24no—
475Dic5—34no—
476#41—61no—
477#41—61no—
478#41—61no—
479#41—61no—
480#41—152no—
481#41—152no—
482#41—152no—
483#41—152no—
484#41—304no—
485#41—304no—
486#41—308no—
487#41—308no—
488#41—304no—
489#41—304no—
490C2×Q8—31no—
491C2×Q8—52no—
492C2×Q8—52no—
493C2×Q8—101no—
494C2×Q8—101no—
495C2×Q8—101no—
496C2×Q8—101no—
497C2×Q8—154no—
498C2×Q8—151no—
499C2×Q8—304no—
500C2×Q8—304no—
501C2×Q8—304no—
502C2×Q8—304no—
503C2×D4—32no—
504C2×D4—31no—
505C2×D4—154no—
506C2×D4—152no—
507C2×D4—151no—
508C22×C4—32no—
509C22×C4—52no—
510C22×C4—154no—
511C22×C4—154no—
512C22×C4—152no—
513C22×C4C2×D3×D511yescharacteristic
514M16—21no—
515M16—32no—
516M16—52no—
517M16—61no—
518M16—101no—
519M16—152no—
520M16—304no—
521M16—304no—
522M16—301no—
523M16C2×D3×D512yes—
524C2×C8—21no—
525C2×C8—61no—
526C2×C8—101no—
527C2×C8—304no—
528C2×C8—304no—
529C2×C8—301no—
530C3×C5#288911yescharacteristic, complemented, Hall
531C22×C3—41no—
532C22×C3D5×C22⋊C411yescharacteristic
533C22×C3D5×C22⋊C411yescharacteristic
534C22×C3C22×C4×D511yescharacteristic
535C2×D3—22no—
536C2×D3—42no—
537C2×D3—54no—
538C2×D3—202no—
539C3×C4—104no—
540C3×C4D5×C22⋊C411yescharacteristic
541C3×C4D5×C22⋊C411yescharacteristic
542C3×C4C2×D4×D511yescharacteristic
543C3×C4C2×D4×D511yescharacteristic
544Dic3—22no—
545Dic3—54no—
546C2×C5—21no—
547C2×C5—21no—
548C2×C5—21no—
549C2×C5—122no—
550C2×C5C2×D3×C22⋊C411yescharacteristic
551D5—64no—
552C23—32no—
553C23—152no—
554C23C4×D3×D511yescharacteristic
555Q8—61no—
556Q8—61no—
557Q8—102no—
558Q8—102no—
559Q8—102no—
560Q8—102no—
561Q8—152no—
562Q8—152no—
563Q8—308no—
564D4—62no—
565D4—62no—
566D4—61no—
567D4—61no—
568D4—154no—
569D4—154no—
570D4—152no—
571D4—152no—
572D4—308no—
573C2×C4—32no—
574C2×C4—54no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—102no—
578C2×C4—102no—
579C2×C4—152no—
580C2×C4—154no—
581C2×C4—154no—
582C2×C4—308no—
583C2×C4—308no—
584C2×C4C4×D3×D511yescharacteristic
585C2×C4#677711yescharacteristic
586C2×C4#677711yescharacteristic
587C2×C4#679011yescharacteristic
588C2×C4#679011yescharacteristic
589C2×C4C22×D3×D511yescharacteristic
590C8—22no—
591C8—62no—
592C8—102no—
593C8—302no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3C2×D5×C22⋊C411yescharacteristic
598D3—42no—
599D3—104no—
600C5#7870611yescharacteristic, complemented, Sylow
601C22—41no—
602C22—62no—
603C22—122no—
604C22—154no—
605C22—602no—
606C22#9147311yescharacteristic
607C22#9147311yescharacteristic
608C22C2×C4×D3×D511yescharacteristic
609C4—62no—
610C4—104no—
611C4—154no—
612C4#9147311yescharacteristic
613C4#9147311yescharacteristic
614C4C2×#677711yescharacteristic
615C4C2×#679011yescharacteristic
616C3#1061044611yescharacteristic, complemented, Sylow
617C2—21no—
618C2—21no—
619C2—21no—
620C2—122no—
621C2—304no—
622C2C2×#9147311yescharacteristic
subgroupG/H
centerC2C2×#91473
commutatorC22×C3×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5C22×C22⋊C4
prime radicalC2×#92004C2
cosocleC2×C3×C4×C5C24
hypercenterC2×M16D3×D5
hypocenterC3×C5#2889
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #11756504C2semidirect
3. #11754130C2semidirect
4. C5×#5132C2semidirect
5. C2×C3×#3956C2semidirect
6. C2×#90961C2semidirect
7. #11748355C2semidirect
8. #11748239C2semidirect
9. #11748227C2semidirect
46. C2×#3956D3semidirect
109. #5132D5semidirect
530. C3×C5#2889semidirect
600. C5#78706semidirect
616. C3#10610446semidirect

Maximal subgroups

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

subgroupindex
C2×#920042
#117565042
#117541302
C5×#51322
C2×C3×#39562
C2×#909612
#117483552
#117482392
#117482272
#106104463
#787065