Finite Group Databasebeta

← #11753847

Subgroups — #11753847

Order 960.

692 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6777C211yesmaximal, characteristic, complemented
2C2×D5×D12C211yesmaximal, characteristic, complemented
3#91783C211yesmaximal, characteristic, complemented
4C5×#895C211yesmaximal, characteristic, complemented
5C3×#4040C211yesmaximal, characteristic, complemented
6#91476C211yesmaximal, characteristic, complemented
7#91475C211yesmaximal, characteristic, complemented
8#91475C211yesmaximal, characteristic, complemented
9#91474C211yesmaximal, characteristic, complemented
10D3×#685C211yesmaximal, characteristic, complemented
11#91454C211yesmaximal, characteristic, complemented
12#91453C211yesmaximal, characteristic, complemented
13#91447C211yesmaximal, characteristic, complemented
14#91416C211yesmaximal, characteristic, complemented
15#91355C211yesmaximal, characteristic, complemented
16#65939—31nomaximal, Hall
17C22×D3×D5C2211yescharacteristic
18C22×D3×D5C2211yescharacteristic, complemented
19C22×D3×D5C2211yescharacteristic, complemented
20C2×D60C2211yescharacteristic
21C2×C4×D15C2211yescharacteristic, complemented
22C2×C5×D12C2211yescharacteristic, complemented
23C2×C5×D12C2211yescharacteristic, complemented
24C2×C5×D12C2211yescharacteristic, complemented
25C2×C4×C5×D3C2211yescharacteristic, complemented
26C2×C3×D20C2211yescharacteristic
27C2×C3×C4×D5C2211yescharacteristic, complemented
28C2×#1203C2211yescharacteristic
29C2×#1203C2211yescharacteristic, complemented
30C2×#1203C2211yescharacteristic, complemented
31C2×#1202C2211yescharacteristic, complemented
32C2×#1201C2211yescharacteristic
33C2×#1201C2211yescharacteristic, complemented
34C2×#1201C2211yescharacteristic, complemented
35C2×D3×Dic5C2211yescharacteristic, complemented
36#6777—24no—
37D5×D12—24no—
38C3×C5×C4⋊C4C2211yescharacteristic, complemented
39#6714C2211yescharacteristic, complemented
40#6714C2211yescharacteristic, complemented
41#6714C2211yescharacteristic, complemented
42#6712C2211yescharacteristic, complemented
43C5×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C5×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C3×#685C2211yescharacteristic, complemented
46C3×#685C2211yescharacteristic, complemented
47C3×#685C2211yescharacteristic, complemented
48C3×#683C2211yescharacteristic, complemented
49Dic3⋊Dic5C2211yescharacteristic, complemented
50#6667C2211yescharacteristic, complemented
51(C2×D3)⋊Dic5C2211yescharacteristic, complemented
52(C2×D3)⋊Dic5C2211yescharacteristic, complemented
53#6665C2211yescharacteristic, complemented
54#5653—51nomaximal, Hall
55C2×D4×D5—31no—
56C2×D4×D5—31no—
57C5×#121—31no—
58#4085—31no—
59#4085—31no—
60#4083—31no—
61#4083—31no—
62#4076—31no—
63C2×#685—31no—
64#4040—31no—
65#4040D311yescharacteristic, complemented
66#4029—31no—
67#4029—31no—
68#4028—31no—
69#4028—31no—
70C22×D15C2311yescharacteristic
71C22×D15C2311yescharacteristic
72C22×C5×D3C2311yescharacteristic
73C22×C5×D3C2311yescharacteristic
74C22×C5×D3C2311yescharacteristic
75C22×C3×D5C2311yescharacteristic
76C22×C3×D5C2311yescharacteristic
77C2×D3×D5—22no—
78C2×D3×D5—22no—
79C2×D3×D5—41no—
80C2×D3×D5—41no—
81C2×D3×D5—41no—
82C2×D3×D5—41no—
83C2×D3×D5—41no—
84C2×D3×D5—41no—
85C2×D3×D5D414yescomplemented
86C2×C3×C4×C5C2311yescharacteristic
87C2×C3×C4×C5C2311yescharacteristic
88C2×C3×C4×C5C2311yescharacteristic
89C2×Dic15C2311yescharacteristic
90C2×Dic15C2311yescharacteristic
91D60—22no—
92C4×D15—22no—
93C2×C5×Dic3C2311yescharacteristic
94C5×D12—22no—
95C5×D12—22no—
96C5×D12—41no—
97C4×C5×D3—41no—
98C2×C3×Dic5C2311yescharacteristic
99C2×C3×Dic5C2311yescharacteristic
100C3×D20—22no—
101C3×C4×D5—22no—
102#1203—22no—
103#1203—22no—
104#1203—41no—
105#1202—41no—
106#1201—22no—
107#1201—22no—
108#1201—41no—
109D3×Dic5—41no—
110C2×D3×D4—51no—
111C22×D12—51no—
112C3×#123—51no—
113#937—51no—
114#929—51no—
115#895—51no—
116#895D511yescharacteristic, complemented
117#894—51no—
118#894—51no—
119#884—51no—
120#884—51no—
121#882—51no—
122#882—51no—
123#882—51no—
124D3×C22⋊C4—51no—
125C23×D5—31no—
126C23×D5—31no—
127C23×D5—31no—
128C2×C5×D4—31no—
129C2×C5×D4—31no—
130C2×C5×D4—31no—
131C22×C4×C5—31no—
132C2×C22⋊D5—31no—
133C2×C22⋊D5—31no—
134C2×C22⋊D5—31no—
135C2×C22⋊D5—31no—
136C2×C22⋊D5—31no—
137C2×C22⋊D5—31no—
138C22×Dic5—31no—
139D4×D5—64no—
140D4×D5—64no—
141C2×D20—31no—
142C2×D20—31no—
143C2×D20C2×D311yescharacteristic
144C2×C4×D5—31no—
145C2×C4×D5C2×D311yescharacteristic, complemented
146C5×C4⋊C4C2×D311yescharacteristic, complemented
147C5×C22⋊C4—31no—
148C5×C22⋊C4—31no—
149C22⋊Dic5—31no—
150C22⋊Dic5—31no—
151#685—31no—
152#685—31no—
153#685—31no—
154#685—31no—
155#685—31no—
156#685C2×D311yescharacteristic, complemented
157#685C2×D311yescharacteristic, complemented
158#685C2×D311yescharacteristic, complemented
159C4⋊Dic5—31no—
160#683—31no—
161#683C2×D311yescharacteristic, complemented
162#534—151noSylow
163C22×C3×C5C2411yescharacteristic
164C2×D15—21no—
165C2×D15—21no—
166C2×D15—21no—
167C2×D15—21no—
168C2×D15—21no—
169C2×D15C2×D412yes—
170C2×C5×D3—21no—
171C2×C5×D3—21no—
172C2×C5×D3—21no—
173C2×C5×D3—21no—
174C2×C5×D3—21no—
175C2×C5×D3—21no—
176C2×C5×D3—21no—
177C2×C5×D3—21no—
178C2×C5×D3C2×D412yes—
179C2×C3×D5—21no—
180C2×C3×D5—21no—
181C2×C3×D5—21no—
182C2×C3×D5—21no—
183C2×C3×D5—21no—
184C2×C3×D5C2×D412yes—
185D3×D5—44no—
186D3×D5—44no—
187D3×D5—44no—
188C3×C4×C5—21no—
189C3×C4×C5—21no—
190C3×C4×C5—21no—
191Dic15—21no—
192Dic15—21no—
193C3×Dic5—21no—
194C3×Dic5—21no—
195C5×Dic3—21no—
196C23×D3—51no—
197C23×D3—51no—
198C23×D3—51no—
199C2×C3×D4—51no—
200C22×C3×C4—51no—
201C2×#67—51no—
202C2×#67—51no—
203C2×#67—51no—
204C2×#67—51no—
205D3×D4—104no—
206C2×D12—51no—
207C2×D12—51no—
208C2×D12—51no—
209C2×D12—51no—
210C2×D12—104no—
211C2×D12C2×D511yescharacteristic, complemented
212C2×D12C2×D511yescharacteristic, complemented
213C2×D12C2×D511yescharacteristic, complemented
214C2×C4×D3—51no—
215C2×C4×D3—51no—
216C2×C4×D3C2×D511yescharacteristic, complemented
217C3×C4⋊C4—51no—
218C3×C4⋊C4C2×D511yescharacteristic, complemented
219C3×C22⋊C4—51no—
220C3×C22⋊C4—51no—
221C3×C22⋊C4—51no—
222#217—51no—
223(C2×D3)⋊C4—51no—
224(C2×D3)⋊C4—51no—
225(C2×D3)⋊C4—51no—
226(C2×D3)⋊C4—51no—
227(C2×D3)⋊C4—51no—
228(C2×D3)⋊C4—51no—
229(C2×D3)⋊C4C2×D511yescharacteristic, complemented
230(C2×D3)⋊C4C2×D511yescharacteristic, complemented
231Dic3⋊C4—51no—
232Dic3⋊C4—51no—
233C23×C5—31no—
234C23×C5—31no—
235C23×C5—31no—
236C22×D5—31no—
237C22×D5—34no—
238C22×D5—31no—
239C22×D5—62no—
240C22×D5—62no—
241C22×D5—121no—
242C22×D5—121no—
243C22×D5—121no—
244C22×D5—121no—
245C22×D5—121no—
246C22×D5—121no—
247C22×D5C22×D311yescharacteristic
248C22×D5C22×D311yescharacteristic
249C5×D4—62no—
250C5×D4—62no—
251C5×D4—121no—
252C2×C4×C5—31no—
253C2×C4×C5—121no—
254C2×C4×C5C22×D311yescharacteristic
255C2×C4×C5C22×D311yescharacteristic
256C2×C4×C5C22×D311yescharacteristic
257C22⋊D5—62no—
258C22⋊D5—62no—
259C22⋊D5—62no—
260C22⋊D5—62no—
261C22⋊D5—121no—
262C22⋊D5—121no—
263C2×Dic5—31no—
264C2×Dic5—31no—
265C2×Dic5—121no—
266C2×Dic5C22×D311yescharacteristic
267C2×Dic5C22×D311yescharacteristic
268D20—22no—
269D20—62no—
270D20—121no—
271C4×D5—22no—
272C4×D5—62no—
273C22×D4—151no—
274C22×D4—151no—
275#123—51no—
276#123—151no—
277#123—151no—
278#123—151no—
279#121—31no—
280#121—151no—
281#121—151no—
282#121—151no—
283#120—151no—
284#120—151no—
285#120—151no—
286#120—151no—
287C2×C22⋊C4—151no—
288C2×C3×C5C22×D411yescharacteristic
289C2×C3×C5H(5,𝔽2)11yescharacteristic
290C2×C3×C5H(5,𝔽2)11yescharacteristic
291D15—22no—
292D15—41no—
293C3×D5—22no—
294C3×D5—41no—
295C5×D3—22no—
296C5×D3—41no—
297C5×D3—41no—
298C23×C3—51no—
299C23×C3—51no—
300C22×D3—51no—
301C22×D3—54no—
302C22×D3—51no—
303C22×D3—102no—
304C22×D3—102no—
305C22×D3—201no—
306C22×D3—201no—
307C22×D3—201no—
308C22×D3—201no—
309C22×D3—201no—
310C22×D3—201no—
311C22×D3C22×D511yescharacteristic
312C22×D3C22×D511yescharacteristic
313C22×D3C22×D511yescharacteristic
314C3×D4—102no—
315C2×C3×C4—51no—
316C2×C3×C4—51no—
317C2×C3×C4—102no—
318C2×C3×C4C22×D511yescharacteristic
319C2×C3×C4C22×D511yescharacteristic
320C2×C3×C4C22×D511yescharacteristic
321#67—102no—
322#67—102no—
323#67—201no—
324#67—201no—
325C2×Dic3—51no—
326C2×Dic3—51no—
327C2×Dic3C22×D511yescharacteristic
328D12—22no—
329D12—22no—
330D12—41no—
331D12—102no—
332D12—102no—
333D12—102no—
334D12—201no—
335C4×D3—41no—
336C4×D3—102no—
337C4×D3—201no—
338C22×C5—32no—
339C22×C5—61no—
340C22×C5—61no—
341C22×C5—61no—
342C22×C5—61no—
343C22×C5—61no—
344C22×C5—61no—
345C22×C5—61no—
346C22×C5—61no—
347C22×C5C23×D311yescharacteristic
348C2×D5—21no—
349C2×D5—21no—
350C2×D5—21no—
351C2×D5—21no—
352C2×D5—21no—
353C2×D5—32no—
354C2×D5—61no—
355C2×D5—61no—
356C2×D5—61no—
357C2×D5—61no—
358C2×D5—61no—
359C2×D5—124no—
360C2×D5—124no—
361C2×D5—124no—
362C2×D5D3×D412yes—
363C4×C5—21no—
364C4×C5—21no—
365C4×C5—21no—
366C4×C5—61no—
367Dic5—21no—
368Dic5—21no—
369Dic5—61no—
370Dic5—61no—
371C24—151no—
372C24—151no—
373C24—151no—
374C2×D4—31no—
375C2×D4—31no—
376C2×D4—31no—
377C2×D4—51no—
378C2×D4—151no—
379C2×D4—151no—
380C2×D4—151no—
381C2×D4—151no—
382C2×D4—151no—
383C2×D4—151no—
384C2×D4—151no—
385C2×D4—151no—
386C2×D4—304no—
387C2×D4—304no—
388C22×C4—31no—
389C22×C4—51no—
390C22×C4—151no—
391C22×C4—151no—
392C4⋊C4—51no—
393C4⋊C4—151no—
394C4⋊C4—151no—
395C4⋊C4D3×D511yescharacteristic, complemented
396C22⋊C4—31no—
397C22⋊C4—31no—
398C22⋊C4—51no—
399C22⋊C4—51no—
400C22⋊C4—51no—
401C22⋊C4—151no—
402C22⋊C4—151no—
403C22⋊C4—151no—
404C22⋊C4—151no—
405C22⋊C4—151no—
406C22⋊C4—151no—
407C22⋊C4—151no—
408C3×C5#53411yescharacteristic, complemented, Hall
409C22×C3—52no—
410C22×C3—101no—
411C22×C3—101no—
412C22×C3—101no—
413C22×C3—101no—
414C22×C3—101no—
415C22×C3C23×D511yescharacteristic
416C2×D3—21no—
417C2×D3—21no—
418C2×D3—21no—
419C2×D3—21no—
420C2×D3—21no—
421C2×D3—21no—
422C2×D3—21no—
423C2×D3—21no—
424C2×D3—52no—
425C2×D3—101no—
426C2×D3—101no—
427C2×D3—101no—
428C2×D3—101no—
429C2×D3—101no—
430C2×D3—204no—
431C2×D3—204no—
432C2×D3—204no—
433C2×D3D4×D512yes—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—101no—
438C3×C4—101no—
439Dic3—21no—
440Dic3—101no—
441Dic3—101no—
442C2×C5—62no—
443C2×C5—121no—
444C2×C5—121no—
445C2×C5C2×D3×D411yescharacteristic
446C2×C5#100911yescharacteristic
447C2×C5#100911yescharacteristic
448D5—22no—
449D5—41no—
450D5—62no—
451D5—121no—
452C23—31no—
453C23—31no—
454C23—31no—
455C23—51no—
456C23—51no—
457C23—154no—
458C23—151no—
459C23—151no—
460C23—302no—
461C23—302no—
462C23—601no—
463C23—601no—
464C23—601no—
465C23—601no—
466C23—601no—
467C23—601no—
468D4—62no—
469D4—62no—
470D4—102no—
471D4—121no—
472D4—302no—
473D4—302no—
474D4—302no—
475D4—302no—
476D4—302no—
477D4—601no—
478D4—601no—
479D4—601no—
480C2×C4—31no—
481C2×C4—51no—
482C2×C4—51no—
483C2×C4—102no—
484C2×C4—121no—
485C2×C4—151no—
486C2×C4—151no—
487C2×C4—302no—
488C2×C4—601no—
489C2×C4C2×D3×D511yescharacteristic
490C2×C4C2×D3×D511yescharacteristic
491C2×C4C2×D3×D511yescharacteristic
492C2×C3—102no—
493C2×C3—201no—
494C2×C3C2×D4×D511yescharacteristic
495C2×C3#414411yescharacteristic
496C2×C3#414911yescharacteristic
497D3—22no—
498D3—41no—
499D3—41no—
500D3—102no—
501D3—201no—
502C5#565311yescharacteristic, complemented, Sylow
503C22—32no—
504C22—52no—
505C22—61no—
506C22—61no—
507C22—61no—
508C22—61no—
509C22—61no—
510C22—61no—
511C22—61no—
512C22—61no—
513C22—101no—
514C22—101no—
515C22—101no—
516C22—101no—
517C22—101no—
518C22—152no—
519C22—301no—
520C22—301no—
521C22—301no—
522C22—301no—
523C22—301no—
524C22—604no—
525C22—604no—
526C22—604no—
527C22C22×D3×D511yescharacteristic
528C4—21no—
529C4—21no—
530C4—21no—
531C4—61no—
532C4—101no—
533C4—101no—
534C4—301no—
535C4—301no—
536C3#6593911yescharacteristic, complemented, Sylow
537C2—62no—
538C2—102no—
539C2—121no—
540C2—121no—
541C2—201no—
542C2—302no—
543C2—601no—
544C2#9201811yescharacteristic
545C2D3×D4×D511yescharacteristic
546C2#9203511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C4⋊C4C22
hypercenterC4⋊C4D3×D5
hypocenterC3×C5#534
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#6777C2semidirect
2. C2×D5×D12C2semidirect
3. #91783C2semidirect
4. C5×#895C2semidirect
5. C3×#4040C2semidirect
6. #91476C2semidirect
7. #91475C2semidirect
8. #91475C2semidirect
9. #91474C2semidirect
10. D3×#685C2semidirect
11. #91454C2semidirect
12. #91453C2semidirect
13. #91447C2semidirect
14. #91416C2semidirect
15. #91355C2semidirect
18. C22×D3×D5C22semidirect
19. C22×D3×D5C22semidirect
21. C2×C4×D15C22semidirect
22. C2×C5×D12C22semidirect
23. C2×C5×D12C22semidirect
24. C2×C5×D12C22semidirect
25. C2×C4×C5×D3C22semidirect
27. C2×C3×C4×D5C22semidirect
29. C2×#1203C22semidirect
30. C2×#1203C22semidirect
31. C2×#1202C22semidirect
33. C2×#1201C22semidirect
34. C2×#1201C22semidirect
35. C2×D3×Dic5C22semidirect
38. C3×C5×C4⋊C4C22semidirect
39. #6714C22semidirect
40. #6714C22semidirect
41. #6714C22semidirect
42. #6712C22semidirect
43. C5×(C2×D3)⋊C4C22semidirect
44. C5×(C2×D3)⋊C4C22semidirect
45. C3×#685C22semidirect
46. C3×#685C22semidirect
47. C3×#685C22semidirect
48. C3×#683C22semidirect
49. Dic3⋊Dic5C22semidirect
50. #6667C22semidirect
51. (C2×D3)⋊Dic5C22semidirect
52. (C2×D3)⋊Dic5C22semidirect
53. #6665C22semidirect
65. #4040D3semidirect
85. C2×D3×D5D4semidirect
116. #895D5semidirect
145. C2×C4×D5C2×D3semidirect
146. C5×C4⋊C4C2×D3semidirect
156. #685C2×D3semidirect
157. #685C2×D3semidirect
158. #685C2×D3semidirect
161. #683C2×D3semidirect
211. C2×D12C2×D5semidirect
212. C2×D12C2×D5semidirect
213. C2×D12C2×D5semidirect
216. C2×C4×D3C2×D5semidirect
218. C3×C4⋊C4C2×D5semidirect
229. (C2×D3)⋊C4C2×D5semidirect
230. (C2×D3)⋊C4C2×D5semidirect
395. C4⋊C4D3×D5semidirect
408. C3×C5#534semidirect
502. C5#5653semidirect
536. C3#65939semidirect

Maximal subgroups

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

subgroupindex
C2×#67772
C2×D5×D122
#917832
C5×#8952
C3×#40402
#914762
#914752
#914742
D3×#6852
#914542
#914532
#914472
#914162
#913552
#659393
#56535