Finite Group Databasebeta

← #14424377

Subgroups — #14424377

Order 1920.

1504 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4149C211yesmaximal, characteristic
2#11756584C211yesmaximal, characteristic, complemented
3C2×#91313C211yesmaximal, characteristic, complemented
4C22×#6657C211yesmaximal, characteristic, complemented
5C5×#5234C211yesmaximal, characteristic, complemented
6#11748531C214yesmaximal, complemented
7#11748530C214yesmaximal, complemented
8#11748367C211yesmaximal, characteristic, complemented
9#11748365C211yesmaximal, characteristic, complemented
10#10610402—31nomaximal, Hall
11C2×C3×C5×#41C2211yescharacteristic
12C22×Dic30C2211yescharacteristic, complemented
13C3×#4149—24no—
14C3×#4149—24no—
15C2×C3×#713—21no—
16C2×C3×D4×D5—22no—
17C2×C3×D4×D5—21no—
18C2×C3×#709C2212yes—
19C2×C3×#709C2211yescharacteristic
20C22×C3×D20C2212yes—
21C22×C3×D20C2211yescharacteristic
22#91833—21no—
23#91825—21no—
24#91824—22no—
25#91815C2211yescharacteristic, complemented
26#91813C2214yescomplemented
27C2×C5×#208C2211yescharacteristic, complemented
28C22×C5×C3⋊C8C2211yescharacteristic, complemented
29C2×#6662C2212yescomplemented
30#91313—24no—
31C2×#6657—24no—
32C2×#6657C2214yescomplemented
33C2×#6657C2212yescomplemented
34C5×#830C2214yescomplemented
35#90971C2212yescomplemented
36#90965C2212yescomplemented
37#90965C2214yescomplemented
38#84105—51nomaximal, Hall
39C2×#4149D311yescharacteristic
40#66159—31no—
41C2×#4055—31no—
42C22×#677—31no—
43C5×#408—31no—
44#65420—31no—
45#65402—31no—
46#65022—34no—
47#65020—34no—
48C23×C3×D5—22no—
49C23×C3×D5—21no—
50C3×C5×#41—22no—
51C3×C5×#41—22no—
52C2×C3×C5×Q8D411yescharacteristic
53C2×C3×C5×D4D412yes—
54C2×C3×C5×D4D411yescharacteristic
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15—21no—
58C2×Dic30—22no—
59C2×Dic30—22no—
60C2×Dic30—21no—
61C2×Dic30C2312yes—
62C2×C3×C22⋊D5—22no—
63C2×C3×C22⋊D5—21no—
64C3×#713—28no—
65C3×D4×D5—216no—
66C3×D4×D5—28no—
67C3×#709—22no—
68C3×#709—24no—
69C3×#709—24no—
70C3×#709D414yes—
71C2×C3×D20—22no—
72C2×C3×D20—21no—
73C2×C3×D20—22no—
74C2×C3×D20—24no—
75C2×C3×D20—24no—
76C2×C3×D20D414yes—
77C2×C3×D20C2311yescharacteristic
78C2×C3×D20C2312yes—
79C2×C3×C4×D5—22no—
80C2×C3×C4×D5—21no—
81C2×C3×Dic10C2311yescharacteristic
82#6719—24no—
83#6719—21no—
84C4⋊Dic15C2312yes—
85#6712—24no—
86C4×Dic15—21no—
87C5×#208—22no—
88C2×C5×C3⋊C8—22no—
89C2×C5×C3⋊C8C2312yes—
90C2×C5×C3⋊C8C2312yes—
91#6662—24no—
92#6657—28no—
93#6657—24no—
94C2×C3×H(5,𝔽2)—51no—
95#5825—51no—
96C2×#951—51no—
97C22×#214—51no—
98#5239—54no—
99#5234D511yescharacteristic, complemented
100#5233—51no—
101#5218—54no—
102#5215—51no—
103C2×C5×#41C2×D311yescharacteristic
104#4149—24no—
105#4149—24no—
106C2×#713—21no—
107C2×D4×D5—22no—
108C2×D4×D5—21no—
109C2×#709C2×D312yes—
110C2×#709C2×D311yescharacteristic
111C22×D20C2×D312yes—
112C22×D20C2×D311yescharacteristic
113C22×Dic10—31no—
114C2×C5×M16—31no—
115C22×C5×C8—31no—
116#4090—61no—
117#4082—61no—
118#4081—62no—
119#4072—31no—
120#4070—34no—
121#4055—64no—
122C2×Dic20—32no—
123C2×#677—34no—
124C2×#677—32no—
125C2×#677—64no—
126C5×C22⋊C8—34no—
127#3953—32no—
128#3953—34no—
129#3948—32no—
130#3009—151noSylow
131C23×C3×C5C2×D412yes—
132C23×C3×C5C2×D411yescharacteristic
133C22×C3×D5—22no—
134C22×C3×D5—21no—
135C22×C3×D5—28no—
136C22×C3×D5—24no—
137C22×C3×D5—82no—
138C22×C3×D5—84no—
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
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—42no—
152Dic30—22no—
153Dic30—22no—
154Dic30—42no—
155C3×C22⋊D5—28no—
156C3×C22⋊D5—24no—
157C2×C3×Dic5—21no—
158C3×D20—28no—
159C3×D20—24no—
160C3×D20—21no—
161C3×D20—22no—
162C3×D20C2×D412yes—
163C3×D20C2×D414yes—
164C3×C4×D5—28no—
165C3×C4×D5—24no—
166C3×Dic10—21no—
167C3×Dic10C2×D412yes—
168C5×C3⋊C8—22no—
169C5×C3⋊C8—22no—
170C3×H(5,𝔽2)—104no—
171C3×H(5,𝔽2)—104no—
172C2×C3×#41—51no—
173C2×C3×#41—52no—
174C2×C3×#41—101no—
175C2×C3×#41C2×D511yescharacteristic
176C22×C3×D4—51no—
177C22×C3×D4—52no—
178C22×C3×D4—102no—
179C22×C3×D4—101no—
180C22×Dic6—51no—
181#951—104no—
182#944—101no—
183C2×#216—52no—
184#936—101no—
185#935—102no—
186C2×#214—52no—
187C2×#214—54no—
188C2×#214—104no—
189#926—51no—
190#924—54no—
191C2×#208C2×D511yescharacteristic, complemented
192C22×C3⋊C8C2×D511yescharacteristic, complemented
193#835—52no—
194#832—54no—
195#832—52no—
196#830C2×D514yescomplemented
197C23×D5—22no—
198C23×D5—21no—
199C5×#41—22no—
200C5×#41—22no—
201C2×C5×Q8#6711yescharacteristic
202C2×C5×D4#6712yes—
203C2×C5×D4#6711yescharacteristic
204C22×C4×C5C22×D312yes—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—22no—
207C2×C22⋊D5—21no—
208C22×Dic5—61no—
209#713—28no—
210D4×D5—216no—
211D4×D5—28no—
212#709—22no—
213#709—24no—
214#709—24no—
215#709#6714yes—
216C2×D20—22no—
217C2×D20—21no—
218C2×D20—22no—
219C2×D20—24no—
220C2×D20—24no—
221C2×D20#6714yes—
222C2×D20C22×D311yescharacteristic
223C2×D20C22×D312yes—
224C2×C4×D5—22no—
225C2×C4×D5—21no—
226C2×Dic10—32no—
227C2×Dic10—62no—
228C2×Dic10—62no—
229C2×Dic10—61no—
230C2×Dic10C22×D311yescharacteristic
231C5×M16—62no—
232C2×C5×C8—32no—
233C2×C5×C8—32no—
234C2×C5×C8—62no—
235C22⋊Dic5—64no—
236C22⋊Dic5—61no—
237C4⋊Dic5—32no—
238#683—64no—
239C4×Dic5—61no—
240Dic20—64no—
241#677—68no—
242#677—64no—
243C2×H(5,𝔽2)—51no—
244C2×#137—151no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—151no—
251#408—31no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—24no—
255C22×C3×C5#12012yes—
256C22×C3×C5C22×D411yescharacteristic
257C2×C3×D5—24no—
258C2×C3×D5—22no—
259C2×C3×D5—44no—
260C2×C3×D5—42no—
261C2×C3×D5—84no—
262C2×C3×D5—88no—
263C3×C4×C5—22no—
264C3×C4×C5#12012yes—
265C3×C4×C5C22×D411yescharacteristic
266C3×C4×C5C22×D411yescharacteristic
267Dic15—42no—
268Dic15—42no—
269C3×Dic5—22no—
270C24×C3—102no—
271C24×C3—101no—
272C3×#41—22no—
273C3×#41—22no—
274C3×#41—54no—
275C3×#41—104no—
276C3×#41—104no—
277C3×#41—108no—
278C3×#41—102no—
279C2×C3×Q8—51no—
280C2×C3×Q8D2011yescharacteristic
281C2×C3×D4—54no—
282C2×C3×D4—52no—
283C2×C3×D4—51no—
284C2×C3×D4—104no—
285C2×C3×D4—104no—
286C2×C3×D4—1016no—
287C2×C3×D4—108no—
288C2×C3×D4—102no—
289C2×C3×D4—101no—
290C2×C3×D4—102no—
291C2×C3×D4—101no—
292C2×C3×D4—102no—
293C2×C3×D4D2012yes—
294C2×C3×D4D2011yescharacteristic
295C22×C3×C4—102no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D511yescharacteristic
299C22×Dic3—101no—
300C2×Dic6—52no—
301C2×Dic6—102no—
302C2×Dic6—102no—
303C2×Dic6—101no—
304#217—104no—
305#217—101no—
306#216—104no—
307#214—108no—
308#214—104no—
309C4⋊Dic3—52no—
310Dic3⋊C4—104no—
311C4×Dic3—101no—
312#208—22no—
313C2×C3⋊C8—22no—
314C2×C3⋊C8C22×D512yes—
315C2×C3⋊C8C22×D512yes—
316C23×C5C2×#6712yes—
317C23×C5C2×#6711yescharacteristic
318C22×D5—22no—
319C22×D5—21no—
320C22×D5—28no—
321C22×D5—24no—
322C22×D5—82no—
323C22×D5—84no—
324C5×Q8—22no—
325C5×D4—24no—
326C5×D4—22no—
327C2×C4×C5—22no—
328C2×C4×C5—22no—
329C2×C4×C5C2×#6712yes—
330C2×C4×C5C2×#6711yescharacteristic
331C2×C4×C5C2×#6712yes—
332C2×C4×C5C2×#6712yes—
333C2×C4×C5C23×D311yescharacteristic
334C22⋊D5—28no—
335C22⋊D5—24no—
336C2×Dic5—21no—
337C2×Dic5—62no—
338C2×Dic5—62no—
339C2×Dic5—122no—
340D20—28no—
341D20—24no—
342D20—21no—
343D20—22no—
344D20C2×#6712yes—
345D20C2×#6714yes—
346C4×D5—28no—
347C4×D5—24no—
348Dic10—21no—
349Dic10—62no—
350Dic10—62no—
351Dic10—122no—
352Dic10C2×#6712yes—
353C5×C8—62no—
354C5×C8—62no—
355H(5,𝔽2)—104no—
356H(5,𝔽2)—104no—
357C2×#41—51no—
358C2×#41—52no—
359C2×#41—101no—
360C2×#41D3×D511yescharacteristic
361C22×Q8—151no—
362C22×D4—51no—
363C22×D4—52no—
364C22×D4—102no—
365C22×D4—101no—
366#137—304no—
367C2×Q16—152no—
368C2×SD16—154no—
369C2×SD16—152no—
370C2×SD16—304no—
371C2×M16—31no—
372C22×C8—31no—
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—151no—
378#103—152no—
379#102—152no—
380#102—154no—
381C22⋊C8—34no—
382C2×C3×C5—22no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386C3×D5—44no—
387C3×D5—42no—
388C23×C3—108no—
389C23×C3—102no—
390C23×C3—101no—
391C23×C3—104no—
392C23×C3—402no—
393C23×C3—404no—
394C23×C3C2×D2012yes—
395C23×C3C2×D2011yescharacteristic
396C3×Q8—22no—
397C3×Q8—52no—
398C3×Q8—101no—
399C3×D4—24no—
400C3×D4—22no—
401C3×D4—54no—
402C3×D4—52no—
403C3×D4—108no—
404C3×D4—108no—
405C3×D4—104no—
406C3×D4—101no—
407C3×D4—102no—
408C3×D4—104no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—108no—
412C2×C3×C4—101no—
413C2×C3×C4—104no—
414C2×C3×C4C2×D2012yes—
415C2×C3×C4C2×D2011yescharacteristic
416C2×C3×C4D4×D512yes—
417C2×C3×C4D4×D512yes—
418C2×C3×C4C23×D511yescharacteristic
419C2×Dic3—102no—
420C2×Dic3—102no—
421C2×Dic3—202no—
422Dic6—102no—
423Dic6—102no—
424Dic6—202no—
425C3⋊C8—22no—
426C3⋊C8—22no—
427C22×C5—22no—
428C22×C5—24no—
429C22×C5—22no—
430C22×C5#95312yes—
431C22×C5C22×#6711yescharacteristic
432C2×D5—24no—
433C2×D5—22no—
434C2×D5—44no—
435C2×D5—42no—
436C2×D5—84no—
437C2×D5—88no—
438C4×C5—22no—
439C4×C5#95312yes—
440C4×C5C22×#6711yescharacteristic
441C4×C5C22×#6711yescharacteristic
442Dic5—22no—
443Dic5—122no—
444Dic5—122no—
445C24—102no—
446C24—101no—
447#41—22no—
448#41—22no—
449#41—54no—
450#41—104no—
451#41—104no—
452#41—108no—
453#41—102no—
454C2×Q8—51no—
455C2×Q8—152no—
456C2×Q8—301no—
457C2×Q8—302no—
458C2×Q8—302no—
459C2×Q8#120211yescharacteristic
460C2×D4—54no—
461C2×D4—52no—
462C2×D4—51no—
463C2×D4—104no—
464C2×D4—104no—
465C2×D4—1016no—
466C2×D4—108no—
467C2×D4—102no—
468C2×D4—101no—
469C2×D4—102no—
470C2×D4—101no—
471C2×D4—102no—
472C2×D4#120212yes—
473C2×D4#120211yescharacteristic
474C22×C4—102no—
475C22×C4—101no—
476C22×C4—301no—
477C22×C4C2×D3×D512yes—
478C22×C4C2×D3×D511yescharacteristic
479Q16—304no—
480SD16—308no—
481SD16—304no—
482M16—62no—
483C2×C8—32no—
484C2×C8—32no—
485C2×C8—62no—
486C4⋊C4—152no—
487C4⋊C4—304no—
488C22⋊C4—301no—
489C22⋊C4—304no—
490C42—301no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—24no—
495C22×C3—104no—
496C22×C3—102no—
497C22×C3—204no—
498C22×C3—202no—
499C22×C3—404no—
500C22×C3—408no—
501C22×C3#402812yes—
502C22×C3C22×D2011yescharacteristic
503C3×C4—22no—
504C3×C4—102no—
505C3×C4#402812yes—
506C3×C4C2×D4×D511yescharacteristic
507C3×C4C2×D4×D511yescharacteristic
508Dic3—202no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—42no—
512C2×C5#583612yes—
513C2×C5C2×#95311yescharacteristic
514D5—44no—
515D5—42no—
516C23—108no—
517C23—102no—
518C23—101no—
519C23—104no—
520C23—402no—
521C23—404no—
522C23C2×#120212yes—
523C23C2×#120211yescharacteristic
524Q8—22no—
525Q8—52no—
526Q8—101no—
527Q8—302no—
528Q8—302no—
529Q8—602no—
530D4—24no—
531D4—22no—
532D4—54no—
533D4—52no—
534D4—108no—
535D4—108no—
536D4—104no—
537D4—101no—
538D4—102no—
539D4—104no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—108no—
543C2×C4—101no—
544C2×C4—104no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—602no—
548C2×C4C2×#120212yes—
549C2×C4C2×#120211yescharacteristic
550C2×C4D5×#6712yes—
551C2×C4D5×#6712yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—62no—
554C8—62no—
555C2×C3—22no—
556C2×C3—42no—
557C2×C3—204no—
558C2×C3—202no—
559C2×C3C2×#402811yescharacteristic
560C2×C3#6608512yes—
561C5#8410511yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—22no—
565C22—104no—
566C22—102no—
567C22—204no—
568C22—202no—
569C22—404no—
570C22—408no—
571C22#9156412yes—
572C22C22×#120211yescharacteristic
573C4—22no—
574C4—102no—
575C4—602no—
576C4—602no—
577C4#9156412yes—
578C4C2×D5×#6711yescharacteristic
579C4C2×D5×#6711yescharacteristic
580C3#1061040211yescharacteristic, complemented, Sylow
581C2—22no—
582C2—42no—
583C2—204no—
584C2—202no—
585C2#1175530912yes—
586C2C2×#9156411yescharacteristic
subgroupG/H
centerC22C22×#1202
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C5×#41C22
socleC22×C3×C5C22×D4
prime radicalC2×C3×#4149C2
hypercenterC2×#41D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #11756584C2semidirect
3. C2×#91313C2semidirect
4. C22×#6657C2semidirect
5. C5×#5234C2semidirect
6. #11748531C2semidirect
7. #11748530C2semidirect
8. #11748367C2semidirect
9. #11748365C2semidirect
12. C22×Dic30C22semidirect
25. #91815C22semidirect
26. #91813C22semidirect
27. C2×C5×#208C22semidirect
28. C22×C5×C3⋊C8C22semidirect
29. C2×#6662C22semidirect
32. C2×#6657C22semidirect
33. C2×#6657C22semidirect
34. C5×#830C22semidirect
35. #90971C22semidirect
36. #90965C22semidirect
37. #90965C22semidirect
99. #5234D5semidirect
191. C2×#208C2×D5semidirect
192. C22×C3⋊C8C2×D5semidirect
196. #830C2×D5semidirect
491. C3×C5#3009semidirect
561. C5#84105semidirect
580. C3#10610402semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41492
#117565842
C2×#913132
C22×#66572
C5×#52342
#117485312
#117485302
#117483672
#117483652
#106104023
#841055