Finite Group Databasebeta

← #14425260

Subgroups — #14425260

Order 1920.

1448 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66283C211yesmaximal, characteristic, complemented
2C5×#5832C211yesmaximal, characteristic, complemented
3#11754138C212yesmaximal, complemented
4#11751338C211yesmaximal, characteristic, complemented
5#11749390C212yesmaximal, complemented
6#11749383C212yesmaximal, complemented
7#11749144C212yesmaximal, complemented
8#11749137C212yesmaximal, complemented
9C2×#90977C212yesmaximal, complemented
10#10615053—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C2×C5×#235C2211yescharacteristic, complemented
13C3×#4150—22no—
14C3×#4149—22no—
15C3×D5×#41—24no—
16C3×#4147C2212yes—
17C3×#4144—22no—
18C3×#4144C2212yes—
19C2×C3×#709—22no—
20C2×C3×#709C2212yes—
21C2×#6710C2212yescomplemented
22C5×#947C2211yescharacteristic, complemented
23C5×#944C2211yescharacteristic, complemented
24C5×#940C2211yescharacteristic, complemented
25C5×#939—22no—
26C5×#936C2211yescharacteristic, complemented
27C4×C5×#67—22no—
28C2×C4×C5×Dic3C2211yescharacteristic, complemented
29#91309—22no—
30#91307C2214yescomplemented
31#91299C2214yescomplemented
32#91298—22no—
33#91297—24no—
34#91120C2212yescomplemented
35#91117C2212yescomplemented
36#90977C2214yescomplemented
37#90977C2214yescomplemented
38#84121—51nomaximal, Hall
39#66283D311yescharacteristic, complemented
40C5×#532—31no—
41#66156—32no—
42#65526—31no—
43#65384—32no—
44#65366—32no—
45#65351—32no—
46#65350—32no—
47C2×#3937—32no—
48C3×C5×#41—22no—
49C3×C5×#41—22no—
50C2×C3×C5×Q8C2311yescharacteristic
51C2×C3×C5×D4D412yes—
52C2×C3×C5×D4C2311yescharacteristic
53C22×C3×C4×C5D412yes—
54C22×C3×C4×C5C2311yescharacteristic
55C2×C5×#67—22no—
56C2×C5×#67—21no—
57C22×C5×Dic3—21no—
58C5×#235—22no—
59C5×#235—22no—
60C2×C5×D12C2311yescharacteristic
61C2×C4×C5×D3—21no—
62C2×C5×Dic6C2311yescharacteristic
63C2×C3×C22⋊D5—24no—
64C2×C3×C22⋊D5—22no—
65C3×#713—24no—
66C3×Q8×D5—24no—
67C3×#711—28no—
68C3×#711—24no—
69C3×D4×D5—28no—
70C3×D4×D5—24no—
71C3×#709—24no—
72C3×#709—24no—
73C3×#709—22no—
74C3×#709—24no—
75C3×#709—22no—
76C3×#709D414yescomplemented
77C3×#709C2314yes—
78C2×C3×D20—22no—
79C2×C3×D20D412yescomplemented
80C2×C3×C4×D5—24no—
81C2×C3×C4×D5—22no—
82C2×C3×Dic10—22no—
83C2×C3×Dic10D412yescomplemented
84#6710—22no—
85#6710C2314yes—
86C2×C3⋊#168—22no—
87C5×#217—22no—
88C5×(C2×D3)⋊C4—22no—
89C5×Dic3⋊C4—22no—
90C4×C5×Dic3—21no—
91C4×C5×Dic3C2311yescharacteristic
92C4×C5×Dic3C2311yescharacteristic
93#6661—24no—
94#6656—24no—
95#6655—24no—
96#6652—24no—
97C3×#585—51no—
98#5832D511yescharacteristic, complemented
99#5822—52no—
100#5246—52no—
101#5245—52no—
102#5242—52no—
103#5241—52no—
104#5237—51no—
105C2×#837—52no—
106C2×C5×#41—31no—
107C2×C5×#41C2×D311yescharacteristic
108#4150—22no—
109#4149—22no—
110D5×#41—24no—
111#4147C2×D312yes—
112#4144—22no—
113#4144C2×D312yes—
114C2×#709—22no—
115C2×#709C2×D312yes—
116C5×C4⋊Q8—31no—
117C5×#127—31no—
118C5×#124—31no—
119C5×#124—31no—
120C5×#121—62no—
121C4×C5×D4—62no—
122C2×C42×C5—31no—
123#4097—62no—
124#4096—64no—
125#4095—62no—
126#4088—34no—
127#4078—34no—
128C2×#681—32no—
129#3968—32no—
130#3965—32no—
131#3937—34no—
132#3937—34no—
133#3018—151noSylow
134C23×C3×C5C2×D412yes—
135C23×C3×C5C2×D411yescharacteristic
136C22×C5×D3—21no—
137C22×C3×D5—24no—
138C22×C3×D5—22no—
139C3×C5×Q8—22no—
140C3×C5×D4—24no—
141C3×C5×D4—22no—
142C2×C3×C4×C5—22no—
143C2×C3×C4×C5—22no—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5C2411yescharacteristic
149C5×#67—44no—
150C5×#67—42no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3—21no—
154C2×C5×Dic3—42no—
155C5×D12—22no—
156C4×C5×D3—41no—
157C4×C5×D3—41no—
158C5×Dic6—22no—
159C3×C22⋊D5—28no—
160C3×C22⋊D5—24no—
161C3×C22⋊D5—24no—
162C2×C3×Dic5—24no—
163C2×C3×Dic5—22no—
164C3×D20—24no—
165C3×D20—22no—
166C3×D20C2×D414yes—
167C3×C4×D5—28no—
168C3×C4×D5—24no—
169C3×C4×D5—24no—
170C3×Dic10—24no—
171C3×Dic10—22no—
172C3×Dic10C2×D414yes—
173C3⋊#168—24no—
174C3×2−1+4—52no—
175C3×2−1+4—102no—
176C3×H(5,𝔽2)—52no—
177C3×H(5,𝔽2)—102no—
178C3×H(5,𝔽2)—102no—
179C2×C3×#41—52no—
180C2×C3×#41—104no—
181C2×C3×#41—102no—
182C2×C3×#41C2×D511yescharacteristic
183C2×#235C2×D511yescharacteristic, complemented
184#951—54no—
185#950—104no—
186#949—54no—
187#947C2×D511yescharacteristic, complemented
188#944C2×D511yescharacteristic, complemented
189#942—102no—
190#940C2×D511yescharacteristic, complemented
191#939—22no—
192#936C2×D511yescharacteristic, complemented
193#932—102no—
194C4×#67—22no—
195C2×C4×Dic3C2×D511yescharacteristic, complemented
196C2×#208—52no—
197#837—54no—
198#837—54no—
199#836—52no—
200#833—52no—
201C5×#41—22no—
202C5×#41—22no—
203C5×#41—62no—
204C5×#41—62no—
205C2×C5×Q8—31no—
206C2×C5×Q8C22×D311yescharacteristic
207C2×C5×D4—31no—
208C2×C5×D4—62no—
209C2×C5×D4—61no—
210C2×C5×D4#6712yes—
211C2×C5×D4C22×D311yescharacteristic
212C22×C4×C5—61no—
213C22×C4×C5—61no—
214C22×C4×C5#6712yes—
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—24no—
217C2×C22⋊D5—22no—
218#713—24no—
219Q8×D5—24no—
220#711—28no—
221#711—24no—
222D4×D5—28no—
223D4×D5—24no—
224#709—24no—
225#709—24no—
226#709—22no—
227#709—24no—
228#709—22no—
229#709#6714yescomplemented
230#709C22×D314yes—
231C2×D20—22no—
232C2×D20#6712yescomplemented
233C2×C4×D5—24no—
234C2×C4×D5—22no—
235C2×Dic10—22no—
236C2×Dic10#6712yescomplemented
237C5×C4⋊C4—62no—
238C5×C22⋊C4—62no—
239C5×C22⋊C4—62no—
240C42×C5—31no—
241C42×C5—31no—
242C42×C5—61no—
243#689—64no—
244#688—64no—
245#687—64no—
246#686—64no—
247#681—34no—
248#681—62no—
249C2×#168—62no—
250#585—51no—
251#575—152no—
252#532—31no—
253#456—152no—
254#455—152no—
255#454—152no—
256#453—152no—
257C2×#104—152no—
258#413—151no—
259C22×C3×C5—22no—
260C22×C3×C5—41no—
261C22×C3×C5—42no—
262C22×C3×C5#12012yes—
263C22×C3×C5C22×D411yescharacteristic
264C2×C5×D3—42no—
265C2×C5×D3—41no—
266C2×C3×D5—24no—
267C2×C3×D5—82no—
268C2×C3×D5—84no—
269C3×C4×C5—22no—
270C3×C4×C5#12011yescharacteristic
271C3×C4×C5#12011yescharacteristic
272C3×C4×C5C22×D412yes—
273C3×Dic5—24no—
274C5×Dic3—22no—
275C5×Dic3—41no—
276C5×Dic3—41no—
277C3×#41—22no—
278C3×#41—22no—
279C3×#41—54no—
280C3×#41—54no—
281C3×#41—104no—
282C3×#41—104no—
283C3×#41—104no—
284C3×#41—108no—
285C3×#41—104no—
286C3×#41—102no—
287C3×#41—102no—
288C3×#41—104no—
289C2×C3×Q8—52no—
290C2×C3×Q8—102no—
291C2×C3×Q8—104no—
292C2×C3×Q8C22×D511yescharacteristic
293C2×C3×D4—52no—
294C2×C3×D4—102no—
295C2×C3×D4—104no—
296C2×C3×D4—108no—
297C2×C3×D4—102no—
298C2×C3×D4—104no—
299C2×C3×D4C22⋊D512yes—
300C2×C3×D4C22×D511yescharacteristic
301C22×C3×C4—104no—
302C22×C3×C4—102no—
303C22×C3×C4C22⋊D512yes—
304C22×C3×C4C22×D511yescharacteristic
305C2×#67—22no—
306C2×#67—21no—
307C22×Dic3—21no—
308#235—22no—
309#235—22no—
310C2×D12C22×D511yescharacteristic
311C2×C4×D3—21no—
312C2×Dic6C22×D511yescharacteristic
313#217—22no—
314#216—104no—
315#215—104no—
316#214—104no—
317#213—104no—
318(C2×D3)⋊C4—22no—
319Dic3⋊C4—22no—
320C4×Dic3—21no—
321C4×Dic3C22×D511yescharacteristic
322C4×Dic3C22×D511yescharacteristic
323#208—54no—
324#208—102no—
325C2×C3⋊C8—102no—
326C23×C5—61no—
327C23×C5C2×#6712yes—
328C23×C5C2×#6711yescharacteristic
329C22×D5—24no—
330C22×D5—22no—
331C5×Q8—22no—
332C5×Q8—62no—
333C5×D4—24no—
334C5×D4—22no—
335C5×D4—62no—
336C5×D4—124no—
337C5×D4—122no—
338C2×C4×C5—22no—
339C2×C4×C5—22no—
340C2×C4×C5—61no—
341C2×C4×C5—61no—
342C2×C4×C5—61no—
343C2×C4×C5—122no—
344C2×C4×C5—121no—
345C2×C4×C5—121no—
346C2×C4×C5C2×#6712yes—
347C2×C4×C5C2×#6711yescharacteristic
348C2×C4×C5C2×#6712yes—
349C2×C4×C5C2×#6712yes—
350C2×C4×C5C23×D311yescharacteristic
351C22⋊D5—28no—
352C22⋊D5—24no—
353C22⋊D5—24no—
354C2×Dic5—24no—
355C2×Dic5—22no—
356D20—24no—
357D20—22no—
358D20C2×#6714yes—
359C4×D5—28no—
360C4×D5—24no—
361C4×D5—24no—
362Dic10—24no—
363Dic10—22no—
364Dic10C2×#6714yes—
365#168—64no—
3662−1+4—52no—
3672−1+4—102no—
368H(5,𝔽2)—52no—
369H(5,𝔽2)—102no—
370H(5,𝔽2)—102no—
371C2×#41—31no—
372C2×#41—52no—
373C2×#41—104no—
374C2×#41—102no—
375C2×#41D3×D511yescharacteristic
376#137—154no—
377#137—302no—
378AGL(1,ℤ/8)—154no—
379AGL(1,ℤ/8)—302no—
380#135—304no—
381C2×M16—152no—
382C4⋊Q8—31no—
383#127—31no—
384#124—31no—
385#124—31no—
386#121—62no—
387C4×D4—62no—
388C2×C42—31no—
389#104—154no—
390#104—154no—
391#101—152no—
392#100—152no—
393C2×C3×C5—21no—
394C2×C3×C5—22no—
395C2×C3×C5—42no—
396C2×C3×C5C2×#12011yescharacteristic
397C3×D5—44no—
398C5×D3—81no—
399C23×C3—104no—
400C23×C3—102no—
401C23×C3C2×C22⋊D512yes—
402C23×C3C2×C22⋊D511yescharacteristic
403C22×D3—21no—
404C3×Q8—22no—
405C3×Q8—54no—
406C3×Q8—104no—
407C3×Q8—102no—
408C3×D4—24no—
409C3×D4—22no—
410C3×D4—54no—
411C3×D4—104no—
412C3×D4—108no—
413C3×D4—104no—
414C3×D4—104no—
415C3×D4—102no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—108no—
419C2×C3×C4—104no—
420C2×C3×C4—102no—
421C2×C3×C4—104no—
422C2×C3×C4—104no—
423C2×C3×C4D4×D512yes—
424C2×C3×C4D4×D512yes—
425C2×C3×C4C2×C22⋊D512yes—
426C2×C3×C4C2×C22⋊D511yescharacteristic
427C2×C3×C4C23×D511yescharacteristic
428#67—44no—
429#67—42no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—42no—
434D12—22no—
435C4×D3—41no—
436C4×D3—41no—
437Dic6—22no—
438C3⋊C8—104no—
439C22×C5—22no—
440C22×C5—41no—
441C22×C5—42no—
442C22×C5—122no—
443C22×C5—121no—
444C22×C5#95312yes—
445C22×C5C22×#6711yescharacteristic
446C2×D5—24no—
447C2×D5—82no—
448C2×D5—84no—
449C4×C5—22no—
450C4×C5—62no—
451C4×C5—121no—
452C4×C5—121no—
453C4×C5#95311yescharacteristic
454C4×C5#95311yescharacteristic
455C4×C5C22×#6712yes—
456Dic5—24no—
457#41—22no—
458#41—22no—
459#41—54no—
460#41—54no—
461#41—62no—
462#41—62no—
463#41—104no—
464#41—104no—
465#41—104no—
466#41—108no—
467#41—104no—
468#41—102no—
469#41—104no—
470#41—102no—
471C2×Q8—31no—
472C2×Q8—52no—
473C2×Q8—102no—
474C2×Q8—104no—
475C2×Q8C2×D3×D511yescharacteristic
476C2×D4—31no—
477C2×D4—52no—
478C2×D4—62no—
479C2×D4—61no—
480C2×D4—102no—
481C2×D4—104no—
482C2×D4—108no—
483C2×D4—102no—
484C2×D4—104no—
485C2×D4#120112yes—
486C2×D4C2×D3×D511yescharacteristic
487C22×C4—61no—
488C22×C4—61no—
489C22×C4—104no—
490C22×C4—102no—
491C22×C4#120112yes—
492C22×C4C2×D3×D511yescharacteristic
493Q16—304no—
494SD16—304no—
495SD16—304no—
496D8—304no—
497M16—154no—
498M16—302no—
499C2×C8—302no—
500C4⋊C4—62no—
501C22⋊C4—62no—
502C22⋊C4—62no—
503C42—31no—
504C42—31no—
505C42—61no—
506C3×C5#301811yescharacteristic, complemented, Hall
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—42no—
510C22×C3—104no—
511C22×C3—402no—
512C22×C3—404no—
513C22×C3#408312yes—
514C22×C3C22×C22⋊D511yescharacteristic
515C2×D3—42no—
516C2×D3—41no—
517C3×C4—22no—
518C3×C4—104no—
519C3×C4#408311yescharacteristic
520C3×C4#408311yescharacteristic
521C3×C4C2×D4×D512yes—
522Dic3—22no—
523Dic3—41no—
524Dic3—41no—
525C2×C5—21no—
526C2×C5—22no—
527C2×C5—42no—
528C2×C5—241no—
529C2×C5C2×#95311yescharacteristic
530D5—44no—
531C23—61no—
532C23—104no—
533C23—102no—
534C23C2×#120112yes—
535C23C2×#120111yescharacteristic
536Q8—22no—
537Q8—54no—
538Q8—62no—
539Q8—104no—
540Q8—102no—
541D4—24no—
542D4—22no—
543D4—54no—
544D4—62no—
545D4—104no—
546D4—108no—
547D4—104no—
548D4—104no—
549D4—102no—
550D4—124no—
551D4—122no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—108no—
558C2×C4—104no—
559C2×C4—102no—
560C2×C4—104no—
561C2×C4—104no—
562C2×C4—122no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4C2×#120112yes—
566C2×C4C2×#120111yescharacteristic
567C2×C4D5×#6712yes—
568C2×C4D5×#6712yes—
569C2×C4C22×D3×D511yescharacteristic
570C8—304no—
571C2×C3—21no—
572C2×C3—22no—
573C2×C3—42no—
574C2×C3—204no—
575C2×C3C2×#408311yescharacteristic
576D3—81no—
577C5#8412111yescharacteristic, complemented, Sylow
578C22—22no—
579C22—41no—
580C22—42no—
581C22—104no—
582C22—122no—
583C22—121no—
584C22—402no—
585C22—404no—
586C22#9156312yes—
587C22C22×#120111yescharacteristic
588C4—22no—
589C4—62no—
590C4—104no—
591C4—121no—
592C4—121no—
593C4#9156311yescharacteristic
594C4#9156311yescharacteristic
595C4C2×D5×#6712yes—
596C3#1061505311yescharacteristic, complemented, Sylow
597C2—21no—
598C2—22no—
599C2—42no—
600C2—204no—
601C2—241no—
602C2C2×#9156311yescharacteristic
subgroupG/H
centerC4#91563
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC2×C3×C5C2×#120
hypercenterC2×#41D3×D5
hypocenterC3×C5#3018
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C3×#66283C2semidirect
2. C5×#5832C2semidirect
3. #11754138C2semidirect
4. #11751338C2semidirect
5. #11749390C2semidirect
6. #11749383C2semidirect
7. #11749144C2semidirect
8. #11749137C2semidirect
9. C2×#90977C2semidirect
12. C2×C5×#235C22semidirect
21. C2×#6710C22semidirect
22. C5×#947C22semidirect
23. C5×#944C22semidirect
24. C5×#940C22semidirect
26. C5×#936C22semidirect
28. C2×C4×C5×Dic3C22semidirect
30. #91307C22semidirect
31. #91299C22semidirect
34. #91120C22semidirect
35. #91117C22semidirect
36. #90977C22semidirect
37. #90977C22semidirect
39. #66283D3semidirect
76. C3×#709D4semidirect
79. C2×C3×D20D4semidirect
83. C2×C3×Dic10D4semidirect
98. #5832D5semidirect
183. C2×#235C2×D5semidirect
187. #947C2×D5semidirect
188. #944C2×D5semidirect
190. #940C2×D5semidirect
192. #936C2×D5semidirect
195. C2×C4×Dic3C2×D5semidirect
229. #709#67semidirect
232. C2×D20#67semidirect
236. C2×Dic10#67semidirect
506. C3×C5#3018semidirect
577. C5#84121semidirect
596. C3#10615053semidirect

Maximal subgroups

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

subgroupindex
C3×#662832
C5×#58322
#117541382
#117513382
#117493902
#117493832
#117491442
#117491372
C2×#909772
#106150533
#841215