Finite Group Databasebeta

← #14424381

Subgroups — #14424381

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×#92019C211yesmaximal, characteristic
2C22×#6706C211yesmaximal, characteristic, complemented
3#11754271C211yesmaximal, characteristic, complemented
4C2×#91313C211yesmaximal, characteristic, complemented
5C5×#5110C211yesmaximal, characteristic, complemented
6C3×#65402C211yesmaximal, characteristic, complemented
7#11748392C211yesmaximal, characteristic, complemented
8#11748166C214yesmaximal, complemented
9#11748120C214yesmaximal, complemented
10#10610402—31nomaximal, Hall
11C22×D60C2211yescharacteristic
12C22×Dic30C2211yescharacteristic, complemented
13C2×C5×#235C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C2×#6790—22no—
16#92019—24no—
17#92019—24no—
18C2×D3×D20C2212yes—
19C2×D5×D12—21no—
20C2×#6773—21no—
21C2×#6769C2212yes—
22C22×C3×C5×C8C2211yescharacteristic, complemented
23C2×#6706—24no—
24C2×#6706C2214yescomplemented
25C2×C5×#208C2211yescharacteristic, complemented
26C3×#4072C2211yescharacteristic, complemented
27#91543—22no—
28#91376—21no—
29#91373—21no—
30#91365C2214yescomplemented
31C2×#6662C2212yescomplemented
32#91313—24no—
33C2×#6657C2212yescomplemented
34C5×(C2×D3)⋊C8C2214yescomplemented
35C3×#3953C2212yescomplemented
36C3×#3948C2212yescomplemented
37#90968C2214yescomplemented
38#78835—51nomaximal, Hall
39C2×#4149—31no—
40#66159—31no—
41C2×#4055—31no—
42C22×#677—31no—
43C5×#408—31no—
44#65420—31no—
45#65402D311yescharacteristic, complemented
46#65022—34no—
47#65020—34no—
48C23×D15—21no—
49C22×D3×D5—22no—
50C22×C3×C4×C5C2311yescharacteristic
51C22×Dic15—21no—
52C2×D60—21no—
53C2×D60—22no—
54C2×D60D414yes—
55C2×D60C2312yes—
56C2×Dic30—22no—
57C2×Dic30—22no—
58C2×Dic30—21no—
59C2×Dic30C2312yes—
60C2×C5×#67D412yes—
61C5×#235—22no—
62C5×#235—22no—
63C2×C5×D12D411yescharacteristic
64C2×C4×C5×D3C2312yes—
65C2×C5×Dic6D411yescharacteristic
66C2×C3×C22⋊D5—21no—
67C3×#709—22no—
68C3×#709D414yes—
69C2×C3×D20C2311yescharacteristic
70C2×C3×C4×D5—21no—
71C2×C3×Dic10C2311yescharacteristic
72#6790—216no—
73C2×#1203—22no—
74C2×#1202—22no—
75C2×#1200—22no—
76D3×D20—24no—
77D3×D20—24no—
78D5×D12—28no—
79#6773—28no—
80#6769—24no—
81#6769—24no—
82C2×C3×C5×C8—22no—
83C2×C3×C5×C8C2312yes—
84#6706—28no—
85C5×#208—22no—
86C2×C5×C3⋊C8C2312yes—
87C3×C22⋊Dic5—21no—
88C3×C4⋊Dic5C2312yes—
89C3×C4×Dic5—21no—
90#6668—24no—
91(C2×D3)⋊Dic5—24no—
92#6662—24no—
93#6657—24no—
94C2×#1009—51no—
95C2×#951—51no—
96C22×#204—51no—
97#5526—51no—
98C3×#416—51no—
99#5110D511yescharacteristic, complemented
100#5001—51no—
101#4808—54no—
102#4776—54no—
103C2×C5×#41—31no—
104#4149—64no—
105#4149—64no—
106C2×#713—61no—
107C2×D4×D5—61no—
108C2×D4×D5—62no—
109C2×#709—32no—
110C2×#709C2×D311yescharacteristic
111C22×D20—31no—
112C22×D20—32no—
113C22×Dic10—31no—
114C2×C5×M16—31no—
115C22×C5×C8C2×D311yescharacteristic, complemented
116#4090—61no—
117#4082—61no—
118#4081—62no—
119#4072C2×D311yescharacteristic, complemented
120#4070—34no—
121#4055—64no—
122C2×Dic20—32no—
123C2×#677—32no—
124C2×#677—34no—
125C2×#677—64no—
126C5×C22⋊C8—34no—
127#3953—34no—
128#3953C2×D312yescomplemented
129#3948C2×D312yescomplemented
130#3009—151noSylow
131C23×C3×C5C2×D411yescharacteristic
132C22×D15—22no—
133C22×D15—24no—
134C22×D15—82no—
135C22×C5×D3C2×D412yes—
136C22×C3×D5—21no—
137C2×D3×D5—28no—
138C2×D3×D5—84no—
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2411yescharacteristic
143C2×Dic15—22no—
144C2×Dic15—42no—
145D60—24no—
146D60—22no—
147D60C2×D414yes—
148Dic30—22no—
149Dic30—22no—
150Dic30—42no—
151C5×#67—24no—
152C2×C5×Dic3C2×D412yes—
153C5×D12—22no—
154C4×C5×D3—22no—
155C4×C5×D3—22no—
156C5×Dic6—22no—
157C3×C22⋊D5—24no—
158C2×C3×Dic5—21no—
159C2×C3×Dic5—22no—
160C3×D20—21no—
161C3×D20C2×D412yes—
162C3×C4×D5—24no—
163C3×Dic10—21no—
164C3×Dic10C2×D412yes—
165#1203—28no—
166#1202—28no—
167#1200—28no—
168C3×C5×C8—22no—
169C5×C3⋊C8—22no—
170C2×C3×#41—51no—
171#1009—104no—
172#1009—104no—
173C2×#239—52no—
174C2×D3×D4—52no—
175C2×D3×D4—102no—
176C2×#235—101no—
177C2×#235C2×D511yescharacteristic
178C22×D12—51no—
179C22×D12—101no—
180C22×Dic6—51no—
181C22×C3×C8C2×D511yescharacteristic, complemented
182C3×#117—51no—
183#951—104no—
184C2×#216—52no—
185C2×#214—52no—
186C2×#208C2×D511yescharacteristic, complemented
187C2×#204—54no—
188C2×#204—104no—
189#897—54no—
190#886—102no—
191#875—101no—
192#869—101no—
193C3×#103—52no—
194C3×#102—52no—
195(C2×D3)⋊C8C2×D514yescomplemented
196#809—54no—
197C23×D5—62no—
198C23×D5—61no—
199C5×#41—62no—
200C5×#41—62no—
201C2×C5×Q8—31no—
202C2×C5×D4—32no—
203C2×C5×D4—31no—
204C22×C4×C5—32no—
205C22×C4×C5C22×D311yescharacteristic
206C2×C22⋊D5—21no—
207C2×C22⋊D5—62no—
208C22×Dic5—61no—
209#713—68no—
210D4×D5—68no—
211D4×D5—616no—
212#709—22no—
213#709—64no—
214#709—64no—
215#709D1214yes—
216C2×D20—34no—
217C2×D20—32no—
218C2×D20—64no—
219C2×D20—62no—
220C2×D20—62no—
221C2×D20—61no—
222C2×D20—64no—
223C2×D20C22×D311yescharacteristic
224C2×C4×D5—21no—
225C2×C4×D5—62no—
226C2×Dic10—32no—
227C2×Dic10—62no—
228C2×Dic10—62no—
229C2×Dic10—61no—
230C2×Dic10C22×D311yescharacteristic
231C5×M16—62no—
232C2×C5×C8—22no—
233C2×C5×C8—32no—
234C2×C5×C8C22×D312yes—
235C22⋊Dic5—21no—
236C22⋊Dic5—64no—
237C4⋊Dic5C22×D312yes—
238#683—64no—
239C4×Dic5—21no—
240Dic20—64no—
241#677—68no—
242#677—64no—
243C2×H(5,𝔽2)—151no—
244C2×#137—151no—
245C22×SD16—151no—
246#536—151no—
247#452—154no—
248#450—154no—
249#418—151no—
250#416—51no—
251#408—31no—
252C22×C3×C5—22no—
253C22×C3×C5#12012yes—
254C22×C3×C5C22×D411yescharacteristic
255C2×D15—24no—
256C2×D15—44no—
257C2×D15—84no—
258C2×C5×D3—22no—
259C2×C5×D3—24no—
260C2×C3×D5—22no—
261C2×C3×D5—42no—
262D3×D5—88no—
263C3×C4×C5#12012yes—
264C3×C4×C5C22×D411yescharacteristic
265C3×C4×C5C22×D411yescharacteristic
266Dic15—42no—
267C3×Dic5—22no—
268C3×Dic5—42no—
269C5×Dic3—22no—
270C23×D3—101no—
271C23×D3—102no—
272C3×#41—54no—
273C3×#41—102no—
274C2×C3×Q8—51no—
275C2×C3×D4—51no—
276C2×C3×D4—101no—
277C22×C3×C4—101no—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—102no—
280C2×#67D2012yes—
281C22×Dic3—101no—
282#239—104no—
283#239—104no—
284D3×D4—104no—
285D3×D4—104no—
286D3×D4—1016no—
287#235—22no—
288#235—22no—
289#235—108no—
290C2×D12—54no—
291C2×D12—52no—
292C2×D12—108no—
293C2×D12—102no—
294C2×D12—101no—
295C2×D12—102no—
296C2×D12D2011yescharacteristic
297C2×C4×D3—102no—
298C2×C4×D3C22×D512yes—
299C2×Dic6—52no—
300C2×Dic6—102no—
301C2×Dic6—102no—
302C2×Dic6—101no—
303C2×Dic6D2011yescharacteristic
304C2×C3×C8—22no—
305C2×C3×C8C22×D512yes—
306C3×C4⋊C4—52no—
307C3×C22⋊C4—101no—
308C3×C42—101no—
309#216—104no—
310#214—104no—
311(C2×D3)⋊C4—104no—
312C4⋊Dic3—104no—
313#208—22no—
314C2×C3⋊C8C22×D512yes—
315#204—108no—
316C23×C5—32no—
317C23×C5D3×D411yescharacteristic
318C22×D5—21no—
319C22×D5—68no—
320C22×D5—62no—
321C22×D5—64no—
322C22×D5—244no—
323C22×D5—242no—
324C5×Q8—62no—
325C5×D4—64no—
326C5×D4—62no—
327C2×C4×C5—32no—
328C2×C4×C5—62no—
329C2×C4×C5—62no—
330C2×C4×C5C2×D1212yes—
331C2×C4×C5D3×D411yescharacteristic
332C2×C4×C5D3×D412yes—
333C2×C4×C5C23×D311yescharacteristic
334C22⋊D5—24no—
335C22⋊D5—68no—
336C2×Dic5—21no—
337C2×Dic5—22no—
338C2×Dic5—62no—
339C2×Dic5—122no—
340D20—21no—
341D20—34no—
342D20—68no—
343D20—64no—
344D20—62no—
345D20C2×D1212yes—
346C4×D5—24no—
347C4×D5—68no—
348Dic10—21no—
349Dic10—62no—
350Dic10—62no—
351Dic10—122no—
352Dic10C2×D1212yes—
353C5×C8—22no—
354C5×C8—62no—
355H(5,𝔽2)—304no—
356H(5,𝔽2)—304no—
357C2×#41—31no—
358C2×#41—51no—
359C2×#41—152no—
360C2×#41—301no—
361C22×Q8—151no—
362C22×D4—152no—
363C22×D4—151no—
364C22×D4—301no—
365C22×D4—302no—
366#137—304no—
367C2×Q16—152no—
368C2×SD16—152no—
369C2×SD16—154no—
370C2×SD16—304no—
371C2×M16—31no—
372C22×C8D3×D511yescharacteristic, complemented
373C4⋊Q8—301no—
374#124—301no—
375#123—302no—
376C22⋊Q8—154no—
377#117—51no—
378#103—52no—
379#102—52no—
380#102—154no—
381C22⋊C8—34no—
382C2×C3×C5—22no—
383C2×C3×C5C2×#12011yescharacteristic
384C2×C3×C5#57712yes—
385D15—44no—
386C3×D5—42no—
387C5×D3—42no—
388C23×C3—101no—
389C23×C3C2×D2011yescharacteristic
390C22×D3—108no—
391C22×D3—102no—
392C22×D3—104no—
393C22×D3—404no—
394C22×D3—402no—
395C22×D3C2×D2012yes—
396C3×Q8—52no—
397C3×Q8—101no—
398C3×D4—52no—
399C3×D4—104no—
400C3×D4—101no—
401C2×C3×C4—101no—
402C2×C3×C4—104no—
403C2×C3×C4—102no—
404C2×C3×C4C2×D2011yescharacteristic
405C2×C3×C4D4×D512yes—
406C2×C3×C4D4×D512yes—
407C2×C3×C4C23×D511yescharacteristic
408#67—24no—
409#67—108no—
410C2×Dic3—102no—
411C2×Dic3—202no—
412C2×Dic3C2×D2012yes—
413D12—22no—
414D12—54no—
415D12—108no—
416D12—102no—
417D12—104no—
418C4×D3—22no—
419C4×D3—22no—
420C4×D3—108no—
421Dic6—22no—
422Dic6—102no—
423Dic6—102no—
424Dic6—202no—
425C3×C8—22no—
426C3⋊C8—22no—
427C22×C5—22no—
428C22×C5—62no—
429C22×C5—64no—
430C22×C5#88212yes—
431C22×C5C2×D3×D411yescharacteristic
432C2×D5—22no—
433C2×D5—42no—
434C2×D5—64no—
435C2×D5—124no—
436C2×D5—248no—
437C2×D5—244no—
438C4×C5—62no—
439C4×C5#88212yes—
440C4×C5C22×D1211yescharacteristic
441C4×C5C2×D3×D411yescharacteristic
442Dic5—22no—
443Dic5—42no—
444Dic5—122no—
445C24—302no—
446C24—301no—
447#41—54no—
448#41—62no—
449#41—62no—
450#41—102no—
451#41—304no—
452#41—304no—
453#41—308no—
454C2×Q8—31no—
455C2×Q8—51no—
456C2×Q8—152no—
457C2×Q8—301no—
458C2×Q8—302no—
459C2×Q8—302no—
460C2×D4—32no—
461C2×D4—31no—
462C2×D4—51no—
463C2×D4—101no—
464C2×D4—154no—
465C2×D4—152no—
466C2×D4—302no—
467C2×D4—304no—
468C2×D4—304no—
469C2×D4—308no—
470C2×D4—302no—
471C2×D4—3016no—
472C2×D4—302no—
473C2×D4—301no—
474C22×C4—32no—
475C22×C4—101no—
476C22×C4—302no—
477C22×C4—301no—
478C22×C4C2×D3×D511yescharacteristic
479Q16—304no—
480SD16—308no—
481SD16—304no—
482M16—62no—
483C2×C8—22no—
484C2×C8—32no—
485C2×C8C2×D3×D512yes—
486C4⋊C4—52no—
487C4⋊C4—304no—
488C22⋊C4—101no—
489C22⋊C4—304no—
490C42—101no—
491C3×C5#300911yescharacteristic, complemented, Hall
492C22×C3—22no—
493C22×C3—102no—
494C22×C3—202no—
495C22×C3#402812yes—
496C22×C3C22×D2011yescharacteristic
497C2×D3—22no—
498C2×D3—24no—
499C2×D3—104no—
500C2×D3—204no—
501C2×D3—408no—
502C2×D3—404no—
503C3×C4—102no—
504C3×C4—202no—
505C3×C4#402812yes—
506C3×C4C2×D4×D511yescharacteristic
507C3×C4C2×D4×D511yescharacteristic
508Dic3—22no—
509Dic3—202no—
510C2×C5—22no—
511C2×C5—122no—
512C2×C5C2×#88211yescharacteristic
513C2×C5#577012yes—
514D5—42no—
515D5—124no—
516C23—32no—
517C23—101no—
518C23—308no—
519C23—302no—
520C23—304no—
521C23—1202no—
522C23—1204no—
523C23D3×D2011yescharacteristic
524Q8—52no—
525Q8—62no—
526Q8—101no—
527Q8—302no—
528Q8—302no—
529Q8—602no—
530D4—52no—
531D4—64no—
532D4—62no—
533D4—104no—
534D4—101no—
535D4—154no—
536D4—308no—
537D4—308no—
538D4—304no—
539D4—302no—
540C2×C4—32no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—101no—
544C2×C4—104no—
545C2×C4—102no—
546C2×C4—308no—
547C2×C4—302no—
548C2×C4—602no—
549C2×C4D5×D1212yes—
550C2×C4D3×D2011yescharacteristic
551C2×C4#679012yes—
552C2×C4C22×D3×D511yescharacteristic
553C8—22no—
554C8—62no—
555C2×C3—22no—
556C2×C3—202no—
557C2×C3C2×#402811yescharacteristic
558C2×C3#6608512yes—
559D3—42no—
560D3—204no—
561C5#7883511yescharacteristic, complemented, Sylow
562C22—22no—
563C22—62no—
564C22—64no—
565C22—102no—
566C22—202no—
567C22—304no—
568C22—604no—
569C22—1204no—
570C22—1208no—
571C22#9147612yes—
572C22C2×D3×D2011yescharacteristic
573C4—62no—
574C4—102no—
575C4—202no—
576C4—602no—
577C4#9147612yes—
578C4C2×D5×D1211yescharacteristic
579C4C2×#679011yescharacteristic
580C3#1061040211yescharacteristic, complemented, Sylow
581C2—22no—
582C2—122no—
583C2—202no—
584C2—604no—
585C2#1175397212yes—
586C2C2×#9147611yescharacteristic
subgroupG/H
centerC22C2×D3×D20
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5C22×D4
prime radicalC2×#92019C2
hypercenterC22×C8D3×D5
hypocenterC3×C5#3009
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×#6706C2semidirect
3. #11754271C2semidirect
4. C2×#91313C2semidirect
5. C5×#5110C2semidirect
6. C3×#65402C2semidirect
7. #11748392C2semidirect
8. #11748166C2semidirect
9. #11748120C2semidirect
12. C22×Dic30C22semidirect
22. C22×C3×C5×C8C22semidirect
24. C2×#6706C22semidirect
25. C2×C5×#208C22semidirect
26. C3×#4072C22semidirect
30. #91365C22semidirect
31. C2×#6662C22semidirect
33. C2×#6657C22semidirect
34. C5×(C2×D3)⋊C8C22semidirect
35. C3×#3953C22semidirect
36. C3×#3948C22semidirect
37. #90968C22semidirect
45. #65402D3semidirect
99. #5110D5semidirect
115. C22×C5×C8C2×D3semidirect
119. #4072C2×D3semidirect
128. #3953C2×D3semidirect
129. #3948C2×D3semidirect
181. C22×C3×C8C2×D5semidirect
186. C2×#208C2×D5semidirect
195. (C2×D3)⋊C8C2×D5semidirect
372. C22×C8D3×D5semidirect
491. C3×C5#3009semidirect
561. C5#78835semidirect
580. C3#10610402semidirect

Maximal subgroups

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

subgroupindex
C2×#920192
C22×#67062
#117542712
C2×#913132
C5×#51102
C3×#654022
#117483922
#117481662
#117481202
#106104023
#788355