Finite Group Databasebeta

← #11757034

Subgroups — #11757034

Order 960.

1620 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#92095C211yesmaximal, characteristic, complemented
2C5×#1004C211yesmaximal, characteristic, complemented
3C2×C3×#711C211yesmaximal, characteristic, complemented
4#92049C214yesmaximal, complemented
5C2×#6782C214yesmaximal, complemented
6#92027C214yesmaximal, complemented
7#92024C214yesmaximal, complemented
8D3×#711C214yesmaximal, complemented
9#92020C214yesmaximal, complemented
10D5×#235C212yesmaximal, complemented
11#92004C212yesmaximal, complemented
12#66285—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic, complemented
14C2×#1220C2214yescomplemented
15#6819C2214yescomplemented
16D4×D15C2214yescomplemented
17#6817C2212yescomplemented
18C2×C5×#67C2214yescomplemented
19C5×#237C2214yescomplemented
20C5×D3×D4C2214yescomplemented
21C5×#235C2212yescomplemented
22C2×C3×C22⋊D5C2212yescomplemented
23C22×C3×Dic5C2212yescomplemented
24C3×#711C2214yescomplemented
25C3×#711C2214yescomplemented
26C2×C3×C4×D5C2211yescharacteristic, complemented
27C2×C3×Dic10C2211yescharacteristic, complemented
28#6790C2218yescomplemented
29D3×C22⋊D5C2218yescomplemented
30D5×#67C2214yescomplemented
31C2×#1204C2214yescomplemented
32C2×#1203C2214yescomplemented
33C2×#1200C2214yescomplemented
34#6782C2218yescomplemented
35#6782C2214yescomplemented
36#6782C2218yescomplemented
37#6782C2218yescomplemented
38C2×D3×Dic5C2214yescomplemented
39#6780C2218yescomplemented
40#6779C2218yescomplemented
41D5×D12C2212yescomplemented
42C4×D3×D5C2214yescomplemented
43#6773C2212yescomplemented
44#6772C2212yescomplemented
45#6771C2214yescomplemented
46D15⋊Q8C2214yescomplemented
47#6769C2214yescomplemented
48#6768C2214yescomplemented
49D3×Dic10C2214yescomplemented
50D5×Dic6C2212yescomplemented
51#5964—51nomaximal, Hall
52C5×H(5,𝔽2)—31no—
53#4150—34no—
54#4150—32no—
55D5×#41—34no—
56D5×#41—32no—
57#4144—31no—
58#4144—34no—
59#4144—34no—
60C2×#711—34no—
61C2×#711—34no—
62C2×#711D311yescharacteristic, complemented
63C23×C3×C5C2312yes—
64C22×D15C2314yes—
65C22×C5×D3C2314yes—
66C22×C3×D5C2311yescharacteristic
67C2×D3×D5C2314yes—
68C3×C5×D4C2314yes—
69C2×C3×C4×C5C2311yescharacteristic
70#1220C2318yes—
71#1220C2314yes—
72C2×Dic15C2314yes—
73D60C2312yes—
74C4×D15C2314yes—
75Dic30C2312yes—
76C5×#67C2318yes—
77C5×#67C2314yes—
78C2×C5×Dic3C2314yes—
79C5×D12C2312yes—
80C4×C5×D3C2314yes—
81C5×Dic6C2312yes—
82C3×C22⋊D5C2318yes—
83C2×C3×Dic5C2314yes—
84C2×C3×Dic5C2312yes—
85C2×C3×Dic5C2311yescharacteristic
86C2×C3×Dic5C2314yes—
87C3×C4×D5C2312yes—
88C3×C4×D5C2312yes—
89C3×Dic10C2314yes—
90#1204C2314yes—
91#1204C2318yes—
92#1203C2318yes—
93#1203C2314yes—
94#1202C2314yes—
95#1201C2314yes—
96#1200C2318yes—
97#1200C2314yes—
98D3×Dic5C2318yes—
99D3×Dic5C2314yes—
100D5×Dic3C2314yes—
101C2×C3×#41—51no—
102#1010—54no—
103#1009—54no—
104D3×#41—54no—
105D3×#41—54no—
106#1007—52no—
107#1004—51no—
108#1004—54no—
109#1004D511yescharacteristic, complemented
110C2×#235—54no—
111C2×#235—52no—
112C5×#41—34no—
113C5×#41—32no—
114C2×C5×D4—34no—
115C2×C5×D4—34no—
116C2×C5×D4C2×D311yescharacteristic, complemented
117C2×C22⋊D5—34no—
118C2×C22⋊D5—38no—
119C2×C22⋊D5—34no—
120C2×C22⋊D5C2×D312yescomplemented
121C22×Dic5—34no—
122C22×Dic5C2×D312yescomplemented
123#713—32no—
124Q8×D5—32no—
125Q8×D5—34no—
126#711—34no—
127#711—38no—
128#711—38no—
129#711—34no—
130#711—38no—
131#711—38no—
132#711—32no—
133#711—34no—
134#711C2×D314yescomplemented
135#711C2×D314yescomplemented
136D4×D5—34no—
137D4×D5—34no—
138D4×D5—38no—
139D4×D5—32no—
140#709—32no—
141#709—38no—
142#709—34no—
143#709—34no—
144C2×C4×D5—34no—
145C2×C4×D5—34no—
146C2×C4×D5C2×D311yescharacteristic, complemented
147C2×Dic10—34no—
148C2×Dic10—34no—
149C2×Dic10C2×D311yescharacteristic, complemented
150#585—151noSylow
151C22×C3×C5—42no—
152C22×C3×C5C2411yescharacteristic
153C22×C3×C5C2414yes—
154C2×D15—44no—
155C2×D15C2414yes—
156C2×C5×D3—44no—
157C2×C5×D3C2414yes—
158C2×C3×D5—41no—
159C2×C3×D5C2412yes—
160D3×D5—44no—
161C3×C4×C5C2412yes—
162Dic15C2414yes—
163C3×Dic5C2411yescharacteristic
164C3×Dic5C2414yes—
165C3×Dic5C2411yescharacteristic
166C5×Dic3C2414yes—
167C3×#41—54no—
168C3×#41—54no—
169C2×C3×Q8—51no—
170C2×C3×D4—52no—
171C2×C3×D4C2×D511yescharacteristic, complemented
172C22×C3×C4—52no—
173C22×C3×C4—51no—
174C2×#67—54no—
175C2×#67—54no—
176C2×#67C2×D514yescomplemented
177#239—54no—
178#239—54no—
179D3×Q8—54no—
180D3×Q8—54no—
181#237—54no—
182#237—58no—
183#237—58no—
184#237C2×D514yescomplemented
185D3×D4—58no—
186D3×D4—54no—
187D3×D4—58no—
188D3×D4C2×D514yescomplemented
189#235—52no—
190#235—54no—
191#235—58no—
192#235—58no—
193#235—58no—
194#235—52no—
195#235—54no—
196#235—52no—
197#235C2×D512yescomplemented
198C2×D12—54no—
199C2×D12—52no—
200C2×C4×D3—54no—
201C2×C4×D3—54no—
202C2×C4×D3—54no—
203C2×Dic6—54no—
204C2×Dic6—52no—
205C23×C5—34no—
206C23×C5C22×D312yes—
207C22×D5—34no—
208C22×D5—34no—
209C22×D5C22×D311yescharacteristic
210C5×Q8—32no—
211C5×D4—38no—
212C5×D4—34no—
213C5×D4—32no—
214C5×D4C22×D314yes—
215C2×C4×C5—34no—
216C2×C4×C5—34no—
217C2×C4×C5C22×D311yescharacteristic
218C22⋊D5—38no—
219C22⋊D5—34no—
220C22⋊D5—34no—
221C22⋊D5—34no—
222C22⋊D5—38no—
223C22⋊D5C22×D318yes—
224C2×Dic5—34no—
225C2×Dic5—38no—
226C2×Dic5—34no—
227C2×Dic5C22×D314yes—
228C2×Dic5C22×D312yes—
229C2×Dic5C22×D314yes—
230C2×Dic5C22×D311yescharacteristic
231D20—34no—
232D20—32no—
233C4×D5—34no—
234C4×D5—34no—
235C4×D5—34no—
236C4×D5—38no—
237C4×D5C22×D312yes—
238C4×D5C22×D312yes—
239Dic10—32no—
240Dic10—34no—
241Dic10—38no—
242Dic10C22×D314yes—
2432−1+4—152no—
2442−1+4—154no—
245H(5,𝔽2)—31no—
246H(5,𝔽2)—154no—
247H(5,𝔽2)—154no—
248H(5,𝔽2)—151no—
249C2×#41—51no—
250C2×#41—154no—
251C2×#41—152no—
252C2×#41—154no—
253C2×#41—154no—
254C2×C3×C5—21no—
255C2×C3×C5—24no—
256C2×C3×C5C2511yescharacteristic
257D15—24no—
258C3×D5—22no—
259C5×D3—24no—
260C23×C3—51no—
261C23×C3C22×D512yes—
262C22×D3—54no—
263C22×D3—54no—
264C22×D3C22×D514yes—
265C3×Q8—54no—
266C3×D4—58no—
267C3×D4C22×D514yes—
268C2×C3×C4—54no—
269C2×C3×C4—54no—
270C2×C3×C4—51no—
271C2×C3×C4—52no—
272C2×C3×C4—52no—
273C2×C3×C4—52no—
274C2×C3×C4C22×D511yescharacteristic
275#67—54no—
276#67—58no—
277#67—54no—
278#67—54no—
279#67C22×D514yes—
280#67C22×D518yes—
281C2×Dic3—54no—
282C2×Dic3—54no—
283C2×Dic3C22×D514yes—
284D12—54no—
285D12—58no—
286D12—52no—
287D12C22×D512yes—
288C4×D3—54no—
289C4×D3—58no—
290C4×D3—54no—
291C4×D3—54no—
292C4×D3—58no—
293C4×D3C22×D514yes—
294Dic6—52no—
295Dic6—54no—
296Dic6—58no—
297Dic6C22×D512yes—
298C22×C5—34no—
299C22×C5—42no—
300C22×C5—124no—
301C22×C5C23×D314yes—
302C22×C5C23×D311yescharacteristic
303C2×D5—34no—
304C2×D5—41no—
305C2×D5—124no—
306C2×D5—124no—
307C2×D5C23×D312yes—
308C4×C5—34no—
309C4×C5C23×D312yes—
310Dic5—34no—
311Dic5C23×D311yescharacteristic
312Dic5C23×D314yes—
313Dic5C23×D311yescharacteristic
314#41—34no—
315#41—32no—
316#41—54no—
317#41—54no—
318#41—154no—
319#41—158no—
320#41—154no—
321#41—154no—
322#41—158no—
323#41—152no—
324#41—158no—
325#41—158no—
326#41—152no—
327#41—158no—
328#41—154no—
329#41—154no—
330#41—152no—
331C2×Q8—51no—
332C2×Q8—154no—
333C2×Q8—154no—
334C2×Q8—152no—
335C2×Q8—154no—
336C2×D4—34no—
337C2×D4—34no—
338C2×D4—52no—
339C2×D4—152no—
340C2×D4—154no—
341C2×D4—158no—
342C2×D4—154no—
343C2×D4—158no—
344C2×D4—154no—
345C2×D4—154no—
346C2×D4D3×D511yescharacteristic, complemented
347C22×C4—52no—
348C22×C4—51no—
349C22×C4—154no—
350C22×C4—154no—
351C22×C4—154no—
352C3×C5#58511yescharacteristic, complemented, Hall
353C22×C3—42no—
354C22×C3—52no—
355C22×C3—201no—
356C22×C3C23×D514yes—
357C22×C3C23×D511yescharacteristic
358C2×D3—44no—
359C2×D3—54no—
360C2×D3—204no—
361C2×D3—204no—
362C2×D3C23×D514yes—
363C3×C4—51no—
364C3×C4—54no—
365C3×C4—51no—
366C3×C4C23×D512yes—
367Dic3—54no—
368Dic3C23×D514yes—
369C2×C5—21no—
370C2×C5—24no—
371C2×C5—64no—
372C2×C5C24×D311yescharacteristic
373D5—22no—
374D5—64no—
375C23—34no—
376C23—51no—
377C23—154no—
378C23—154no—
379C23C2×D3×D512yes—
380Q8—32no—
381Q8—54no—
382Q8—154no—
383Q8—158no—
384Q8—152no—
385D4—38no—
386D4—34no—
387D4—32no—
388D4—58no—
389D4—154no—
390D4—154no—
391D4—158no—
392D4—154no—
393D4—158no—
394D4—154no—
395D4—152no—
396D4C2×D3×D514yes—
397C2×C4—34no—
398C2×C4—34no—
399C2×C4—51no—
400C2×C4—54no—
401C2×C4—52no—
402C2×C4—54no—
403C2×C4—52no—
404C2×C4—52no—
405C2×C4—154no—
406C2×C4—158no—
407C2×C4—154no—
408C2×C4—154no—
409C2×C4—158no—
410C2×C4—154no—
411C2×C4—154no—
412C2×C4C2×D3×D511yescharacteristic
413C2×C3—24no—
414C2×C3—21no—
415C2×C3—102no—
416C2×C3C24×D511yescharacteristic
417D3—24no—
418D3—104no—
419C5#596411yescharacteristic, complemented, Sylow
420C22—34no—
421C22—42no—
422C22—52no—
423C22—124no—
424C22—154no—
425C22—201no—
426C22—604no—
427C22—604no—
428C22C22×D3×D514yes—
429C22C22×D3×D511yescharacteristic
430C4—34no—
431C4—51no—
432C4—51no—
433C4—54no—
434C4—154no—
435C4C22×D3×D512yes—
436C3#6628511yescharacteristic, complemented, Sylow
437C2—21no—
438C2—24no—
439C2—64no—
440C2—102no—
441C2—304no—
442C2C23×D3×D511yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×D5
commutator, socle, cosocleC2×C3×C5C25
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#585
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #92095C2semidirect
2. C5×#1004C2semidirect
3. C2×C3×#711C2semidirect
4. #92049C2semidirect
5. C2×#6782C2semidirect
6. #92027C2semidirect
7. #92024C2semidirect
8. D3×#711C2semidirect
9. #92020C2semidirect
10. D5×#235C2semidirect
11. #92004C2semidirect
13. C2×C3×C5×D4C22semidirect
14. C2×#1220C22semidirect
15. #6819C22semidirect
16. D4×D15C22semidirect
17. #6817C22semidirect
18. C2×C5×#67C22semidirect
19. C5×#237C22semidirect
20. C5×D3×D4C22semidirect
21. C5×#235C22semidirect
22. C2×C3×C22⋊D5C22semidirect
23. C22×C3×Dic5C22semidirect
24. C3×#711C22semidirect
25. C3×#711C22semidirect
26. C2×C3×C4×D5C22semidirect
27. C2×C3×Dic10C22semidirect
28. #6790C22semidirect
29. D3×C22⋊D5C22semidirect
30. D5×#67C22semidirect
31. C2×#1204C22semidirect
32. C2×#1203C22semidirect
33. C2×#1200C22semidirect
34. #6782C22semidirect
35. #6782C22semidirect
36. #6782C22semidirect
37. #6782C22semidirect
38. C2×D3×Dic5C22semidirect
39. #6780C22semidirect
40. #6779C22semidirect
41. D5×D12C22semidirect
42. C4×D3×D5C22semidirect
43. #6773C22semidirect
44. #6772C22semidirect
45. #6771C22semidirect
46. D15⋊Q8C22semidirect
47. #6769C22semidirect
48. #6768C22semidirect
49. D3×Dic10C22semidirect
50. D5×Dic6C22semidirect
62. C2×#711D3semidirect
109. #1004D5semidirect
116. C2×C5×D4C2×D3semidirect
120. C2×C22⋊D5C2×D3semidirect
122. C22×Dic5C2×D3semidirect
134. #711C2×D3semidirect
135. #711C2×D3semidirect
146. C2×C4×D5C2×D3semidirect
149. C2×Dic10C2×D3semidirect
171. C2×C3×D4C2×D5semidirect
176. C2×#67C2×D5semidirect
184. #237C2×D5semidirect
188. D3×D4C2×D5semidirect
197. #235C2×D5semidirect
346. C2×D4D3×D5semidirect
352. C3×C5#585semidirect
419. C5#5964semidirect
436. C3#66285semidirect

Maximal subgroups

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

subgroupindex
#920952
C5×#10042
C2×C3×#7112
#920492
C2×#67822
#920272
#920242
D3×#7112
#920202
D5×#2352
#920042
#662853
#59645