Finite Group Databasebeta

← #11753180

Subgroups — #11753180

Order 960.

500 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91771C211yesmaximal, characteristic, complemented
2C5×#878C211yesmaximal, characteristic, complemented
3C3×#4027C211yesmaximal, characteristic, complemented
4#91572C211yesmaximal, characteristic, complemented
5#91569C211yesmaximal, characteristic, complemented
6#91552C211yesmaximal, characteristic, complemented
7C2×#6667C211yesmaximal, characteristic, complemented
8#91459C211yesmaximal, characteristic, complemented
9#91455C211yesmaximal, characteristic, complemented
10#91450C211yesmaximal, characteristic, complemented
11#91433C211yesmaximal, characteristic, complemented
12#91387C211yesmaximal, characteristic, complemented
13#91375C211yesmaximal, characteristic, complemented
14#91348C211yesmaximal, characteristic, complemented
15#91337C211yesmaximal, characteristic, complemented
16#65978—31nomaximal, Hall
17C23×D15C2211yescharacteristic
18C2×#1220C2211yescharacteristic
19C2×D60C2211yescharacteristic
20C2×D60C2211yescharacteristic
21C22×C5×Dic3C2211yescharacteristic
22C2×C5×Dic6C2211yescharacteristic, complemented
23C2×C3×C22⋊D5C2211yescharacteristic
24C22×C3×Dic5C2211yescharacteristic
25C2×C3×C4×D5C2211yescharacteristic, complemented
26C2×#1202C2211yescharacteristic
27C2×#1202C2211yescharacteristic
28C2×#1202C2211yescharacteristic, complemented
29C2×#1200C2211yescharacteristic, complemented
30C3×C5×C22⋊C4C2211yescharacteristic
31#6714C2211yescharacteristic
32#6714C2211yescharacteristic
33C5×#217C2211yescharacteristic
34C5×C4⋊Dic3C2211yescharacteristic, complemented
35C5×Dic3⋊C4C2211yescharacteristic, complemented
36C5×Dic3⋊C4C2211yescharacteristic, complemented
37C3×#685C2211yescharacteristic, complemented
38C3×#683C2211yescharacteristic, complemented
39C3×C4×Dic5C2211yescharacteristic, complemented
40Dic3⋊Dic5C2211yescharacteristic, complemented
41#6669C2211yescharacteristic, complemented
42#6667C2211yescharacteristic
43#6667C2211yescharacteristic
44#6667C2211yescharacteristic
45#6667C2211yescharacteristic
46#6667C2211yescharacteristic
47#6667C2211yescharacteristic
48#6667C2211yescharacteristic, complemented
49#6667C2211yescharacteristic, complemented
50#6665C2211yescharacteristic, complemented
51Dic3×Dic5C2211yescharacteristic, complemented
52#5560—51nomaximal, Hall
53C5×C22⋊Q8—31no—
54#4093—31no—
55#4093—31no—
56#4076—31no—
57C2×#685—31no—
58#4044—31no—
59#4044—31no—
60#4041—31no—
61#4041—31no—
62#4040—31no—
63#4040—31no—
64#4028—31no—
65#4028—31no—
66#4027—31no—
67#4027D311yescharacteristic, complemented
68C23×C3×C5C2311yescharacteristic
69C22×D15—22no—
70C22×D15—41no—
71C22×D15—41no—
72C22×D15C2311yescharacteristic
73C22×D15C2311yescharacteristic
74C22×D15C2311yescharacteristic
75C22×C3×D5C2311yescharacteristic
76C2×C3×C4×C5C2311yescharacteristic
77C2×C3×C4×C5C2311yescharacteristic
78#1220—22no—
79C2×Dic15C2311yescharacteristic
80D60—41no—
81D60—41no—
82C2×C5×Dic3—41no—
83C2×C5×Dic3C2311yescharacteristic
84C2×C5×Dic3C2311yescharacteristic
85C2×C5×Dic3C2311yescharacteristic
86C2×C5×Dic3C2311yescharacteristic
87C5×Dic6—41no—
88C3×C22⋊D5—22no—
89C2×C3×Dic5—41no—
90C2×C3×Dic5C2311yescharacteristic
91C2×C3×Dic5C2311yescharacteristic
92C2×C3×Dic5C2311yescharacteristic
93C3×C4×D5—41no—
94#1202—41no—
95#1202—41no—
96#1202—41no—
97#1200—41no—
98C3×C4×D4—51no—
99#939—51no—
100#937—51no—
101#930—51no—
102#929—51no—
103C4×#67—51no—
104C2×(C2×D3)⋊C4—51no—
105#898—51no—
106#894—51no—
107#885—51no—
108#884—51no—
109#882—51no—
110#878D511yescharacteristic, complemented
111#876—51no—
112#875—51no—
113C23×D5—31no—
114C2×C5×Q8—31no—
115C22×C4×C5—31no—
116C2×C22⋊D5—31no—
117C2×C22⋊D5C2×D311yescharacteristic
118C22×Dic5C2×D311yescharacteristic
119C2×D20—31no—
120C2×D20—31no—
121C2×D20—31no—
122C2×D20—31no—
123C2×D20—31no—
124C2×C4×D5—31no—
125C2×C4×D5C2×D311yescharacteristic, complemented
126C5×C4⋊C4—31no—
127C5×C4⋊C4—31no—
128C5×C4⋊C4—31no—
129C5×C22⋊C4—31no—
130C5×C22⋊C4C2×D311yescharacteristic
131#685—31no—
132#685—31no—
133#685—31no—
134#685—31no—
135#685—31no—
136#685—31no—
137#685—31no—
138#685—31no—
139#685—31no—
140#685—31no—
141#685—31no—
142#685C2×D311yescharacteristic, complemented
143C4⋊Dic5—31no—
144#683—31no—
145#683C2×D311yescharacteristic, complemented
146C4×Dic5—31no—
147C4×Dic5C2×D311yescharacteristic, complemented
148#538—151noSylow
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5#4112yes—
152C22×C3×C5C2411yescharacteristic
153C2×D15—21no—
154C2×D15—21no—
155C2×D15—21no—
156C2×D15—21no—
157C2×D15—21no—
158C2×D15—21no—
159C2×D15—21no—
160C2×D15—21no—
161C2×D15—21no—
162C2×D15—44no—
163C2×C3×D5—21no—
164C2×C3×D5—21no—
165C2×C3×D5—21no—
166C3×C4×C5—21no—
167C3×C4×C5—21no—
168Dic15—21no—
169C3×Dic5—21no—
170C3×Dic5—21no—
171C3×Dic5—21no—
172C5×Dic3—21no—
173C5×Dic3—21no—
174C5×Dic3—21no—
175C5×Dic3—21no—
176C23×D3—51no—
177C2×C3×D4—51no—
178C22×C3×C4—51no—
179C22×C3×C4—51no—
180C2×#67—51no—
181C2×#67—51no—
182C2×#67—51no—
183C2×#67—51no—
184C22×Dic3C2×D511yescharacteristic
185C2×D12—51no—
186C2×D12—51no—
187C2×C4×D3—51no—
188C2×Dic6C2×D511yescharacteristic, complemented
189C3×C4⋊C4—51no—
190C3×C22⋊C4—51no—
191C3×C22⋊C4C2×D511yescharacteristic
192C3×C42—51no—
193#217—51no—
194#217C2×D511yescharacteristic
195(C2×D3)⋊C4—51no—
196(C2×D3)⋊C4—51no—
197(C2×D3)⋊C4—51no—
198(C2×D3)⋊C4—51no—
199(C2×D3)⋊C4—51no—
200(C2×D3)⋊C4—51no—
201(C2×D3)⋊C4—51no—
202(C2×D3)⋊C4—51no—
203(C2×D3)⋊C4—51no—
204(C2×D3)⋊C4—51no—
205C4⋊Dic3C2×D511yescharacteristic, complemented
206Dic3⋊C4—51no—
207Dic3⋊C4—51no—
208Dic3⋊C4C2×D511yescharacteristic, complemented
209Dic3⋊C4C2×D511yescharacteristic, complemented
210C4×Dic3—51no—
211C23×C5C22×D311yescharacteristic
212C22×D5—31no—
213C22×D5—31no—
214C22×D5—31no—
215C22×D5—62no—
216C22×D5—121no—
217C22×D5—121no—
218C22×D5C22×D311yescharacteristic
219C5×Q8—121no—
220C2×C4×C5—31no—
221C2×C4×C5—31no—
222C2×C4×C5—31no—
223C2×C4×C5—31no—
224C2×C4×C5—121no—
225C2×C4×C5C22×D311yescharacteristic
226C2×C4×C5C22×D311yescharacteristic
227C22⋊D5—22no—
228C22⋊D5—62no—
229C2×Dic5—31no—
230C2×Dic5—41no—
231C2×Dic5C22×D311yescharacteristic
232C2×Dic5C22×D311yescharacteristic
233C2×Dic5C22×D311yescharacteristic
234D20—121no—
235D20—121no—
236D20—121no—
237D20—121no—
238D20—121no—
239C4×D5—41no—
240C4×D5—121no—
241#126—151no—
242#126—151no—
243#124—151no—
244#124—151no—
245#123—151no—
246#123—151no—
247C22⋊Q8—31no—
248#121—151no—
249#121—151no—
250#121—151no—
251#120—151no—
252#120—151no—
253C4×D4—51no—
254C4×D4—151no—
255C2×C22⋊C4—151no—
256C2×C3×C5—22no—
257C2×C3×C5C2×#4111yescharacteristic
258C2×C3×C5H(5,𝔽2)11yescharacteristic
259C2×C3×C5H(5,𝔽2)11yescharacteristic
260D15—41no—
261D15—41no—
262D15—41no—
263C3×D5—41no—
264C23×C3—51no—
265C23×C3C22×D511yescharacteristic
266C22×D3—51no—
267C22×D3—51no—
268C22×D3—51no—
269C22×D3—102no—
270C22×D3—201no—
271C22×D3—201no—
272C3×D4—102no—
273C2×C3×C4—51no—
274C2×C3×C4—51no—
275C2×C3×C4—51no—
276C2×C3×C4—201no—
277C2×C3×C4—201no—
278C2×C3×C4C22×D511yescharacteristic
279C2×C3×C4C22×D511yescharacteristic
280#67—102no—
281#67—201no—
282#67—201no—
283#67—201no—
284C2×Dic3—41no—
285C2×Dic3—51no—
286C2×Dic3C22×D511yescharacteristic
287C2×Dic3C22×D511yescharacteristic
288C2×Dic3C22×D511yescharacteristic
289C2×Dic3C22×D511yescharacteristic
290D12—201no—
291D12—201no—
292C4×D3—201no—
293Dic6—41no—
294C22×C5—21no—
295C22×C5—21no—
296C22×C5#23512yes—
297C22×C5C23×D311yescharacteristic
298C2×D5—21no—
299C2×D5—21no—
300C2×D5—21no—
301C2×D5—61no—
302C2×D5—61no—
303C2×D5—61no—
304C2×D5—61no—
305C2×D5—61no—
306C2×D5—61no—
307C2×D5—61no—
308C2×D5—61no—
309C2×D5—61no—
310C2×D5—124no—
311C4×C5—21no—
312C4×C5—21no—
313C4×C5—61no—
314C4×C5—61no—
315C4×C5—61no—
316C4×C5—61no—
317Dic5—21no—
318Dic5—21no—
319Dic5—21no—
320Dic5—61no—
321C24—151no—
322C2×Q8—31no—
323C2×D4—51no—
324C2×D4—151no—
325C2×D4—151no—
326C2×D4—151no—
327C2×D4—151no—
328C2×D4—151no—
329C2×D4—151no—
330C22×C4—31no—
331C22×C4—51no—
332C22×C4—51no—
333C22×C4—151no—
334C4⋊C4—31no—
335C4⋊C4—31no—
336C4⋊C4—31no—
337C4⋊C4—51no—
338C4⋊C4—151no—
339C4⋊C4—151no—
340C22⋊C4—31no—
341C22⋊C4—51no—
342C22⋊C4—151no—
343C22⋊C4—151no—
344C22⋊C4—151no—
345C22⋊C4—151no—
346C22⋊C4—151no—
347C22⋊C4—151no—
348C22⋊C4—151no—
349C22⋊C4—151no—
350C22⋊C4—151no—
351C22⋊C4—151no—
352C22⋊C4—151no—
353C22⋊C4D3×D511yescharacteristic
354C42—51no—
355C42—151no—
356C3×C5#53811yescharacteristic, complemented, Hall
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—101no—
360C22×C3—101no—
361C22×C3—101no—
362C22×C3#71312yes—
363C22×C3C23×D511yescharacteristic
364C2×D3—101no—
365C2×D3—101no—
366C2×D3—101no—
367C2×D3—101no—
368C2×D3—101no—
369C2×D3—101no—
370C2×D3—101no—
371C2×D3—101no—
372C2×D3—101no—
373C2×D3—204no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—101no—
377C3×C4—101no—
378C3×C4—101no—
379Dic3—21no—
380Dic3—21no—
381Dic3—21no—
382Dic3—21no—
383Dic3—101no—
384C2×C5—22no—
385C2×C5C2×#23511yescharacteristic
386C2×C5#100411yescharacteristic
387C2×C5#100911yescharacteristic
388D5—41no—
389D5—121no—
390D5—121no—
391D5—121no—
392C23—51no—
393C23—151no—
394C23—151no—
395C23—151no—
396C23—302no—
397C23—601no—
398C23—601no—
399C23C2×D3×D511yescharacteristic
400Q8—121no—
401D4—102no—
402D4—302no—
403D4—601no—
404D4—601no—
405D4—601no—
406D4—601no—
407D4—601no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—51no—
413C2×C4—51no—
414C2×C4—51no—
415C2×C4—121no—
416C2×C4—151no—
417C2×C4—201no—
418C2×C4—201no—
419C2×C4—601no—
420C2×C4C2×D3×D511yescharacteristic
421C2×C4C2×D3×D511yescharacteristic
422C2×C3—22no—
423C2×C3—201no—
424C2×C3#414411yescharacteristic
425C2×C3C2×#71311yescharacteristic
426C2×C3#414911yescharacteristic
427D3—201no—
428D3—201no—
429D3—201no—
430C5#556011yescharacteristic, complemented, Sylow
431C22—21no—
432C22—21no—
433C22—101no—
434C22—101no—
435C22—101no—
436C22—301no—
437C22—301no—
438C22—301no—
439C22—301no—
440C22—301no—
441C22—301no—
442C22—301no—
443C22—301no—
444C22—301no—
445C22—604no—
446C22#677312yes—
447C22C22×D3×D511yescharacteristic
448C4—21no—
449C4—21no—
450C4—61no—
451C4—61no—
452C4—61no—
453C4—61no—
454C4—101no—
455C4—101no—
456C4—101no—
457C4—301no—
458C3#6597811yescharacteristic, complemented, Sylow
459C2—22no—
460C2—201no—
461C2—601no—
462C2—601no—
463C2—601no—
464C2C2×#677311yescharacteristic
465C2#9202611yescharacteristic
466C2#9202711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#538
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. #91771C2semidirect
2. C5×#878C2semidirect
3. C3×#4027C2semidirect
4. #91572C2semidirect
5. #91569C2semidirect
6. #91552C2semidirect
7. C2×#6667C2semidirect
8. #91459C2semidirect
9. #91455C2semidirect
10. #91450C2semidirect
11. #91433C2semidirect
12. #91387C2semidirect
13. #91375C2semidirect
14. #91348C2semidirect
15. #91337C2semidirect
22. C2×C5×Dic6C22semidirect
25. C2×C3×C4×D5C22semidirect
28. C2×#1202C22semidirect
29. C2×#1200C22semidirect
34. C5×C4⋊Dic3C22semidirect
35. C5×Dic3⋊C4C22semidirect
36. C5×Dic3⋊C4C22semidirect
37. C3×#685C22semidirect
38. C3×#683C22semidirect
39. C3×C4×Dic5C22semidirect
40. Dic3⋊Dic5C22semidirect
41. #6669C22semidirect
48. #6667C22semidirect
49. #6667C22semidirect
50. #6665C22semidirect
51. Dic3×Dic5C22semidirect
67. #4027D3semidirect
110. #878D5semidirect
125. C2×C4×D5C2×D3semidirect
142. #685C2×D3semidirect
145. #683C2×D3semidirect
147. C4×Dic5C2×D3semidirect
188. C2×Dic6C2×D5semidirect
205. C4⋊Dic3C2×D5semidirect
208. Dic3⋊C4C2×D5semidirect
209. Dic3⋊C4C2×D5semidirect
356. C3×C5#538semidirect
430. C5#5560semidirect
458. C3#65978semidirect

Maximal subgroups

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

subgroupindex
#917712
C5×#8782
C3×#40272
#915722
#915692
#915522
C2×#66672
#914592
#914552
#914502
#914332
#913872
#913752
#913482
#913372
#659783
#55605