Finite Group Databasebeta

← #11753163

Subgroups — #11753163

Order 960.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91768C211yesmaximal, characteristic, complemented
2C5×#884C211yesmaximal, characteristic, complemented
3C3×#4028C211yesmaximal, characteristic, complemented
4#91565C211yesmaximal, characteristic, complemented
5#91563C211yesmaximal, characteristic, complemented
6#91563C211yesmaximal, characteristic, complemented
7#91549C211yesmaximal, characteristic, complemented
8#91463C211yesmaximal, characteristic, complemented
9#91457C211yesmaximal, characteristic, complemented
10#91456C211yesmaximal, characteristic, complemented
11#91453C211yesmaximal, characteristic, complemented
12#91452C211yesmaximal, characteristic, complemented
13#91408C211yesmaximal, characteristic, complemented
14#91407C211yesmaximal, characteristic, complemented
15#91366C211yesmaximal, characteristic, complemented
16#65954—31nomaximal, Hall
17C23×C3×D5C2211yescharacteristic
18C2×C5×#67C2211yescharacteristic
19C2×C5×#67C2211yescharacteristic
20C2×C5×D12C2211yescharacteristic, complemented
21C2×C4×C5×D3C2211yescharacteristic, complemented
22C2×C3×C22⋊D5C2211yescharacteristic
23C2×C3×D20C2211yescharacteristic
24C2×C3×D20C2211yescharacteristic
25C2×#1201C2211yescharacteristic
26C2×#1201C2211yescharacteristic
27C2×#1201C2211yescharacteristic
28C2×#1201C2211yescharacteristic
29C2×#1201C2211yescharacteristic, complemented
30C2×#1201C2211yescharacteristic, complemented
31C2×D3×Dic5C2211yescharacteristic, complemented
32C2×D5×Dic3C2211yescharacteristic, complemented
33C3×C5×C22⋊C4C2211yescharacteristic
34#6719C2211yescharacteristic
35#6719C2211yescharacteristic
36C4⋊Dic15C2211yescharacteristic, complemented
37#6712C2211yescharacteristic, complemented
38#6712C2211yescharacteristic, complemented
39C4×Dic15C2211yescharacteristic, complemented
40C5×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C5×Dic3⋊C4C2211yescharacteristic, complemented
42C3×#685C2211yescharacteristic
43C3×#685C2211yescharacteristic
44Dic3⋊Dic5C2211yescharacteristic, complemented
45#6668C2211yescharacteristic, complemented
46(C2×D3)⋊Dic5C2211yescharacteristic, complemented
47#6665C2211yescharacteristic
48#6665C2211yescharacteristic
49#6665C2211yescharacteristic
50#6665C2211yescharacteristic
51#6665C2211yescharacteristic, complemented
52#5595—51nomaximal, Hall
53C5×#121—31no—
54#4086—31no—
55#4085—31no—
56#4085—31no—
57#4084—31no—
58#4083—31no—
59#4083—31no—
60#4076—31no—
61#4075—31no—
62#4044—31no—
63#4040—31no—
64#4032—31no—
65#4030—31no—
66#4028D311yescharacteristic, complemented
67#4025—31no—
68C23×C3×C5C2311yescharacteristic
69C22×C5×D3C2311yescharacteristic
70C22×C5×D3C2311yescharacteristic
71C22×C3×D5—41no—
72C22×C3×D5—41no—
73C22×C3×D5—41no—
74C22×C3×D5C2311yescharacteristic
75C22×C3×D5C2311yescharacteristic
76C22×C3×D5C2311yescharacteristic
77C2×C3×C4×C5C2311yescharacteristic
78C2×C3×C4×C5C2311yescharacteristic
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C2×Dic15C2311yescharacteristic
83C5×#67—41no—
84C5×#67—41no—
85C2×C5×Dic3C2311yescharacteristic
86C2×C5×Dic3C2311yescharacteristic
87C5×D12—41no—
88C4×C5×D3—41no—
89C3×C22⋊D5—41no—
90C2×C3×Dic5C2311yescharacteristic
91C3×D20—41no—
92C3×D20—41no—
93#1201—41no—
94#1201—41no—
95#1201—41no—
96#1201—41no—
97#1201—41no—
98#1201—41no—
99D3×Dic5—41no—
100D5×Dic3—41no—
101C3×#120—51no—
102#953—51no—
103#953—51no—
104#940—51no—
105#939—51no—
106#939—51no—
107#938—51no—
108#938—51no—
109#935—51no—
110#884—51no—
111#884D511yescharacteristic, complemented
112#883—51no—
113#883—51no—
114#879—51no—
115#879—51no—
116C23×D5C2×D311yescharacteristic
117C2×C5×D4—31no—
118C2×C5×D4—31no—
119C2×C5×D4—31no—
120C22×C4×C5—31no—
121C2×C22⋊D5—31no—
122C2×C22⋊D5—31no—
123C2×C22⋊D5—31no—
124C2×C22⋊D5—31no—
125C2×C22⋊D5—31no—
126C2×C22⋊D5—31no—
127C2×C22⋊D5C2×D311yescharacteristic
128C22×Dic5—31no—
129C2×D20C2×D311yescharacteristic
130C2×D20C2×D311yescharacteristic
131C2×C4×D5—31no—
132C5×C4⋊C4—31no—
133C5×C22⋊C4—31no—
134C5×C22⋊C4C2×D311yescharacteristic
135C22⋊Dic5—31no—
136C22⋊Dic5—31no—
137C22⋊Dic5—31no—
138#685—31no—
139#685—31no—
140#685—31no—
141#685—31no—
142#685—31no—
143#685C2×D311yescharacteristic
144#685C2×D311yescharacteristic
145C4⋊Dic5—31no—
146C4⋊Dic5—31no—
147#683—31no—
148#683—31no—
149#683—31no—
150C4×Dic5—31no—
151#560—151noSylow
152C22×C3×C5—21no—
153C22×C3×C5—21no—
154C22×C3×C5—21no—
155C22×C3×C5C2411yescharacteristic
156C2×C5×D3—21no—
157C2×C5×D3—21no—
158C2×C5×D3—21no—
159C2×C5×D3—21no—
160C2×C5×D3—21no—
161C2×C5×D3—21no—
162C2×C3×D5—21no—
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C2×C3×D5—21no—
167C2×C3×D5—21no—
168C2×C3×D5—21no—
169C2×C3×D5—21no—
170C2×C3×D5—21no—
171C2×C3×D5—44no—
172C3×C4×C5—21no—
173C3×C4×C5—21no—
174Dic15—21no—
175Dic15—21no—
176Dic15—21no—
177Dic15—21no—
178C3×Dic5—21no—
179C5×Dic3—21no—
180C5×Dic3—21no—
181C24×C3—51no—
182C2×C3×D4—51no—
183C2×C3×D4—51no—
184C2×C3×D4—51no—
185C2×#67—51no—
186C2×#67—51no—
187C2×#67—51no—
188C2×#67—51no—
189C2×#67—51no—
190C2×#67—51no—
191C2×#67C2×D511yescharacteristic
192C2×#67C2×D511yescharacteristic
193C22×Dic3—51no—
194C2×D12C2×D511yescharacteristic, complemented
195C2×C4×D3—51no—
196C2×C4×D3C2×D511yescharacteristic, complemented
197C3×C22⋊C4—51no—
198C3×C22⋊C4—51no—
199C3×C22⋊C4C2×D511yescharacteristic
200#217—51no—
201#217—51no—
202#217—51no—
203#217—51no—
204#217—51no—
205#217—51no—
206#217—51no—
207(C2×D3)⋊C4—51no—
208(C2×D3)⋊C4C2×D511yescharacteristic, complemented
209C4⋊Dic3—51no—
210C4⋊Dic3—51no—
211Dic3⋊C4—51no—
212Dic3⋊C4—51no—
213Dic3⋊C4—51no—
214Dic3⋊C4C2×D511yescharacteristic, complemented
215C4×Dic3—51no—
216C23×C5—31no—
217C23×C5—31no—
218C23×C5C22×D311yescharacteristic
219C22×D5—41no—
220C22×D5—41no—
221C22×D5—41no—
222C22×D5C22×D311yescharacteristic
223C22×D5C22×D311yescharacteristic
224C22×D5C22×D311yescharacteristic
225C5×D4—121no—
226C5×D4—121no—
227C5×D4—121no—
228C2×C4×C5—31no—
229C2×C4×C5—31no—
230C2×C4×C5—121no—
231C2×C4×C5C22×D311yescharacteristic
232C2×C4×C5C22×D311yescharacteristic
233C22⋊D5—41no—
234C22⋊D5—121no—
235C22⋊D5—121no—
236C22⋊D5—121no—
237C22⋊D5—121no—
238C22⋊D5—121no—
239C22⋊D5—121no—
240C2×Dic5—31no—
241C2×Dic5—31no—
242C2×Dic5—31no—
243C2×Dic5—31no—
244C2×Dic5—121no—
245C2×Dic5C22×D311yescharacteristic
246D20—41no—
247D20—41no—
248C4×D5—121no—
249#127—151no—
250#126—151no—
251#126—151no—
252#123—151no—
253#123—151no—
254#123—151no—
255#121—31no—
256#121—151no—
257#121—151no—
258#121—151no—
259#121—151no—
260#121—151no—
261#120—51no—
262#120—151no—
263#120—151no—
264C2×C3×C5—41no—
265C2×C3×C5H(5,𝔽2)11yescharacteristic
266C2×C3×C5H(5,𝔽2)11yescharacteristic
267C2×C3×C5H(5,𝔽2)11yescharacteristic
268C3×D5—41no—
269C3×D5—41no—
270C3×D5—41no—
271C5×D3—41no—
272C5×D3—41no—
273C23×C3—51no—
274C23×C3—51no—
275C23×C3—51no—
276C23×C3—201no—
277C23×C3—201no—
278C23×C3—201no—
279C23×C3C22×D511yescharacteristic
280C22×D3C22×D511yescharacteristic
281C22×D3C22×D511yescharacteristic
282C3×D4—201no—
283C3×D4—201no—
284C3×D4—201no—
285C2×C3×C4—51no—
286C2×C3×C4C22×D511yescharacteristic
287C2×C3×C4C22×D511yescharacteristic
288#67—41no—
289#67—41no—
290#67—201no—
291#67—201no—
292#67—201no—
293#67—201no—
294#67—201no—
295#67—201no—
296C2×Dic3—51no—
297C2×Dic3—51no—
298C2×Dic3—51no—
299C2×Dic3—51no—
300C2×Dic3—201no—
301C2×Dic3C22×D511yescharacteristic
302C2×Dic3C22×D511yescharacteristic
303D12—41no—
304C4×D3—41no—
305C4×D3—201no—
306C22×C5—21no—
307C22×C5—21no—
308C22×C5—21no—
309C22×C5—61no—
310C22×C5—61no—
311C22×C5—61no—
312C22×C5—61no—
313C22×C5—61no—
314C22×C5—61no—
315C22×C5C23×D311yescharacteristic
316C2×D5—21no—
317C2×D5—21no—
318C2×D5—21no—
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—21no—
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—44no—
326C4×C5—21no—
327C4×C5—21no—
328C4×C5—61no—
329C4×C5—61no—
330Dic5—21no—
331Dic5—61no—
332Dic5—61no—
333Dic5—61no—
334Dic5—61no—
335C24—51no—
336C2×D4—31no—
337C2×D4—31no—
338C2×D4—31no—
339C2×D4—51no—
340C2×D4—51no—
341C2×D4—51no—
342C2×D4—151no—
343C2×D4—151no—
344C2×D4—151no—
345C2×D4—151no—
346C2×D4—151no—
347C2×D4—151no—
348C22×C4—31no—
349C22×C4—151no—
350C22×C4—151no—
351C4⋊C4—31no—
352C4⋊C4—151no—
353C4⋊C4—151no—
354C4⋊C4—151no—
355C4⋊C4—151no—
356C4⋊C4—151no—
357C22⋊C4—31no—
358C22⋊C4—51no—
359C22⋊C4—51no—
360C22⋊C4—151no—
361C22⋊C4—151no—
362C22⋊C4—151no—
363C22⋊C4—151no—
364C22⋊C4—151no—
365C22⋊C4—151no—
366C22⋊C4—151no—
367C22⋊C4—151no—
368C22⋊C4D3×D511yescharacteristic
369C42—151no—
370C3×C5#56011yescharacteristic, complemented, Hall
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—101no—
375C22×C3—101no—
376C22×C3—101no—
377C22×C3—101no—
378C22×C3—101no—
379C22×C3—101no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3—101no—
383C22×C3—204no—
384C22×C3C23×D511yescharacteristic
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—21no—
390C2×D3—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—101no—
394Dic3—21no—
395Dic3—21no—
396Dic3—101no—
397Dic3—101no—
398Dic3—101no—
399Dic3—101no—
400C2×C5—41no—
401C2×C5—121no—
402C2×C5—121no—
403C2×C5#100411yescharacteristic
404C2×C5#100411yescharacteristic
405C2×C5#100411yescharacteristic
406D5—41no—
407D5—41no—
408D5—41no—
409C23—31no—
410C23—31no—
411C23—51no—
412C23—51no—
413C23—51no—
414C23—201no—
415C23—201no—
416C23—201no—
417C23C2×D3×D511yescharacteristic
418D4—121no—
419D4—121no—
420D4—121no—
421D4—201no—
422D4—201no—
423D4—201no—
424D4—601no—
425D4—601no—
426D4—601no—
427D4—601no—
428D4—601no—
429D4—601no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—51no—
433C2×C4—121no—
434C2×C4—151no—
435C2×C4—151no—
436C2×C4—151no—
437C2×C4—151no—
438C2×C4—601no—
439C2×C4—601no—
440C2×C4C2×D3×D511yescharacteristic
441C2×C4C2×D3×D511yescharacteristic
442C2×C3—41no—
443C2×C3—201no—
444C2×C3—201no—
445C2×C3—201no—
446C2×C3#414411yescharacteristic
447C2×C3#414411yescharacteristic
448C2×C3#414911yescharacteristic
449D3—41no—
450D3—41no—
451C5#559511yescharacteristic, complemented, Sylow
452C22—21no—
453C22—21no—
454C22—21no—
455C22—61no—
456C22—61no—
457C22—61no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22—101no—
462C22—101no—
463C22—101no—
464C22—101no—
465C22—101no—
466C22—101no—
467C22—101no—
468C22—101no—
469C22—101no—
470C22—204no—
471C22C22×D3×D511yescharacteristic
472C4—21no—
473C4—21no—
474C4—61no—
475C4—61no—
476C4—101no—
477C4—301no—
478C4—301no—
479C4—301no—
480C4—301no—
481C3#6595411yescharacteristic, complemented, Sylow
482C2—41no—
483C2—121no—
484C2—121no—
485C2—201no—
486C2—201no—
487C2—201no—
488C2#9201711yescharacteristic
489C2#9202511yescharacteristic
490C2#9202511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#560
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #91768C2semidirect
2. C5×#884C2semidirect
3. C3×#4028C2semidirect
4. #91565C2semidirect
5. #91563C2semidirect
6. #91563C2semidirect
7. #91549C2semidirect
8. #91463C2semidirect
9. #91457C2semidirect
10. #91456C2semidirect
11. #91453C2semidirect
12. #91452C2semidirect
13. #91408C2semidirect
14. #91407C2semidirect
15. #91366C2semidirect
20. C2×C5×D12C22semidirect
21. C2×C4×C5×D3C22semidirect
29. C2×#1201C22semidirect
30. C2×#1201C22semidirect
31. C2×D3×Dic5C22semidirect
32. C2×D5×Dic3C22semidirect
36. C4⋊Dic15C22semidirect
37. #6712C22semidirect
38. #6712C22semidirect
39. C4×Dic15C22semidirect
40. C5×(C2×D3)⋊C4C22semidirect
41. C5×Dic3⋊C4C22semidirect
44. Dic3⋊Dic5C22semidirect
45. #6668C22semidirect
46. (C2×D3)⋊Dic5C22semidirect
51. #6665C22semidirect
66. #4028D3semidirect
111. #884D5semidirect
194. C2×D12C2×D5semidirect
196. C2×C4×D3C2×D5semidirect
208. (C2×D3)⋊C4C2×D5semidirect
214. Dic3⋊C4C2×D5semidirect
370. C3×C5#560semidirect
451. C5#5595semidirect
481. C3#65954semidirect

Maximal subgroups

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

subgroupindex
#917682
C5×#8842
C3×#40282
#915652
#915632
#915492
#914632
#914572
#914562
#914532
#914522
#914082
#914072
#913662
#659543
#55955