Finite Group Databasebeta

← #11754677

Subgroups — #11754677

Order 960.

496 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#91818C211yesmaximal, characteristic, complemented
2C4×C5×#67C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊D5C211yesmaximal, characteristic, complemented
4#91577C211yesmaximal, characteristic
5#91555C211yesmaximal, characteristic, complemented
6#91548C211yesmaximal, characteristic, complemented
7#91526C211yesmaximal, characteristic, complemented
8D3×#685C211yesmaximal, characteristic, complemented
9D5×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#91398C211yesmaximal, characteristic, complemented
11#91393C211yesmaximal, characteristic, complemented
12#91387C211yesmaximal, characteristic, complemented
13#91380C211yesmaximal, characteristic, complemented
14#91352C211yesmaximal, characteristic, complemented
15#91349C211yesmaximal, characteristic, complemented
16#65890—31nomaximal, Hall
17C22×D3×D5C2211yescharacteristic
18C22×C3×C4×C5C2211yescharacteristic, complemented
19C2×#1220C2211yescharacteristic
20C2×C5×#67C2211yescharacteristic
21C2×C4×C5×D3C2211yescharacteristic
22C2×C3×C22⋊D5C2211yescharacteristic
23C2×C3×C4×D5C2211yescharacteristic
24C2×#1204C2211yescharacteristic, complemented
25C2×D3×Dic5C2211yescharacteristic
26C2×D5×Dic3C2211yescharacteristic
27#6719C2211yescharacteristic, complemented
28#6714C2211yescharacteristic
29#6714C2211yescharacteristic
30#6712C2211yescharacteristic, complemented
31#6712C2211yescharacteristic, complemented
32C5×#217C2211yescharacteristic, complemented
33C5×(C2×D3)⋊C4C2211yescharacteristic
34C5×Dic3⋊C4C2211yescharacteristic, complemented
35C4×C5×Dic3C2211yescharacteristic, complemented
36C3×C22⋊Dic5C2211yescharacteristic, complemented
37C3×#685C2211yescharacteristic
38C3×#683C2211yescharacteristic, complemented
39C3×C4×Dic5C2211yescharacteristic, complemented
40Dic3⋊Dic5C2211yescharacteristic, complemented
41#6669C2211yescharacteristic, complemented
42#6669C2211yescharacteristic, complemented
43#6668C2211yescharacteristic, complemented
44#6667C2211yescharacteristic
45#6667C2211yescharacteristic
46#6667C2211yescharacteristic
47(C2×D3)⋊Dic5C2211yescharacteristic
48(C2×D3)⋊Dic5C2211yescharacteristic
49#6665C2211yescharacteristic
50#6665C2211yescharacteristic
51Dic3×Dic5C2211yescharacteristic, complemented
52#5556—51nomaximal, Hall
53C4×C5×D4—31no—
54#4083—31no—
55#4081—31no—
56#4075—31no—
57C4×C22⋊D5D311yescharacteristic, complemented
58C2×#685—31no—
59#4070—31no—
60#4044—31no—
61#4042—31no—
62#4031—31no—
63#4029—31no—
64D5×C22⋊C4—31no—
65#4023—31no—
66#4022—31no—
67#4018—31no—
68C23×C3×C5C2311yescharacteristic
69C22×D15C2311yescharacteristic
70C22×C5×D3C2311yescharacteristic
71C22×C3×D5C2311yescharacteristic
72C2×D3×D5—22no—
73C2×D3×D5—22no—
74C2×D3×D5—41no—
75C2×C3×C4×C5—41no—
76C2×C3×C4×C5C2311yescharacteristic
77C2×C3×C4×C5C2311yescharacteristic
78#1220—41no—
79C2×Dic15C2311yescharacteristic
80C2×Dic15C2311yescharacteristic
81C2×Dic15C2311yescharacteristic
82C5×#67—22no—
83C2×C5×Dic3C2311yescharacteristic
84C2×C5×Dic3C2311yescharacteristic
85C2×C5×Dic3C2311yescharacteristic
86C4×C5×D3—41no—
87C3×C22⋊D5—22no—
88C2×C3×Dic5C2311yescharacteristic
89C2×C3×Dic5C2311yescharacteristic
90C2×C3×Dic5C2311yescharacteristic
91C3×C4×D5—41no—
92#1204—41no—
93D3×Dic5—22no—
94D5×Dic3—22no—
95C3×C4×D4—51no—
96#937—51no—
97#935—51no—
98#929—51no—
99C4×#67D511yescharacteristic, complemented
100C2×(C2×D3)⋊C4—51no—
101#924—51no—
102#898—51no—
103#896—51no—
104#885—51no—
105#883—51no—
106D3×C22⋊C4—51no—
107#877—51no—
108#876—51no—
109#872—51no—
110C23×D5—31no—
111C2×C5×D4—31no—
112C22×C4×C5—31no—
113C22×C4×C5C2×D311yescharacteristic, complemented
114C2×C22⋊D5—31no—
115C2×C22⋊D5C2×D311yescharacteristic
116C22×Dic5—31no—
117C2×C4×D5—31no—
118C2×C4×D5C2×D311yescharacteristic
119C2×Dic10—31no—
120C5×C4⋊C4—31no—
121C5×C22⋊C4—31no—
122C5×C22⋊C4—31no—
123C42×C5—31no—
124C22⋊Dic5—31no—
125C22⋊Dic5—31no—
126C22⋊Dic5—31no—
127C22⋊Dic5C2×D311yescharacteristic, complemented
128#685—31no—
129#685—31no—
130#685—31no—
131#685—31no—
132#685—31no—
133#685—31no—
134#685—31no—
135#685C2×D311yescharacteristic
136C4⋊Dic5—31no—
137#683—31no—
138#683—31no—
139#683—31no—
140#683—31no—
141#683—31no—
142#683C2×D311yescharacteristic, complemented
143C4×Dic5—31no—
144C4×Dic5C2×D311yescharacteristic, complemented
145#551—151noSylow
146C22×C3×C5—21no—
147C22×C3×C5—21no—
148C22×C3×C5—21no—
149C22×C3×C5C2411yescharacteristic
150C2×D15—21no—
151C2×D15—21no—
152C2×D15—21no—
153C2×C5×D3—21no—
154C2×C5×D3—21no—
155C2×C5×D3#4112yes—
156C2×C3×D5—21no—
157C2×C3×D5—21no—
158C2×C3×D5#4112yes—
159D3×D5—44no—
160C3×C4×C5—21no—
161C3×C4×C5—21no—
162Dic15—21no—
163Dic15—21no—
164Dic15—21no—
165C3×Dic5—21no—
166C3×Dic5—21no—
167C3×Dic5—21no—
168C5×Dic3—21no—
169C5×Dic3—21no—
170C5×Dic3—21no—
171C23×D3—51no—
172C2×C3×D4—51no—
173C22×C3×C4—51no—
174C22×C3×C4C2×D511yescharacteristic, complemented
175C2×#67—51no—
176C2×#67C2×D511yescharacteristic
177C22×Dic3—51no—
178C2×C4×D3—51no—
179C2×C4×D3C2×D511yescharacteristic
180C2×Dic6—51no—
181C3×C4⋊C4—51no—
182C3×C22⋊C4—51no—
183C3×C22⋊C4—51no—
184C3×C42—51no—
185#217—51no—
186#217—51no—
187#217—51no—
188#217C2×D511yescharacteristic, complemented
189(C2×D3)⋊C4—51no—
190(C2×D3)⋊C4—51no—
191(C2×D3)⋊C4—51no—
192(C2×D3)⋊C4—51no—
193(C2×D3)⋊C4—51no—
194(C2×D3)⋊C4—51no—
195(C2×D3)⋊C4—51no—
196(C2×D3)⋊C4C2×D511yescharacteristic
197C4⋊Dic3—51no—
198Dic3⋊C4—51no—
199Dic3⋊C4—51no—
200Dic3⋊C4—51no—
201Dic3⋊C4—51no—
202Dic3⋊C4—51no—
203Dic3⋊C4C2×D511yescharacteristic, complemented
204C4×Dic3—51no—
205C4×Dic3C2×D511yescharacteristic, complemented
206C23×C5—31no—
207C23×C5C22×D311yescharacteristic
208C22×D5—31no—
209C22×D5—62no—
210C22×D5—62no—
211C22×D5—121no—
212C22×D5C22×D311yescharacteristic
213C5×D4—62no—
214C2×C4×C5—31no—
215C2×C4×C5—31no—
216C2×C4×C5—31no—
217C2×C4×C5—41no—
218C2×C4×C5—121no—
219C2×C4×C5C22×D311yescharacteristic
220C2×C4×C5C22×D311yescharacteristic
221C22⋊D5—22no—
222C22⋊D5—121no—
223C2×Dic5—31no—
224C2×Dic5—31no—
225C2×Dic5—31no—
226C2×Dic5—62no—
227C2×Dic5C22×D311yescharacteristic
228C2×Dic5C22×D311yescharacteristic
229C2×Dic5C22×D311yescharacteristic
230C4×D5—41no—
231C4×D5—62no—
232Dic10—121no—
233#126—151no—
234#126—151no—
235#124—151no—
236#123—151no—
237#123—151no—
238#123—151no—
239C22⋊Q8—151no—
240C22⋊Q8—151no—
241#120—151no—
242C4×D4—31no—
243C4×D4—51no—
244#117—151no—
245#117—151no—
246C2×C22⋊C4—151no—
247C2×C22⋊C4—151no—
248C2×C3×C5—41no—
249C2×C3×C5C2×#4111yescharacteristic
250C2×C3×C5C2×#4111yescharacteristic
251C2×C3×C5H(5,𝔽2)11yescharacteristic
252D15—41no—
253C3×D5—22no—
254C5×D3—22no—
255C23×C3—51no—
256C23×C3C22×D511yescharacteristic
257C22×D3—51no—
258C22×D3—102no—
259C22×D3—102no—
260C22×D3—201no—
261C22×D3C22×D511yescharacteristic
262C3×D4—102no—
263C2×C3×C4—41no—
264C2×C3×C4—51no—
265C2×C3×C4—51no—
266C2×C3×C4—51no—
267C2×C3×C4—201no—
268C2×C3×C4C22×D511yescharacteristic
269C2×C3×C4C22×D511yescharacteristic
270#67—22no—
271#67—201no—
272C2×Dic3—51no—
273C2×Dic3—51no—
274C2×Dic3—51no—
275C2×Dic3—102no—
276C2×Dic3C22×D511yescharacteristic
277C2×Dic3C22×D511yescharacteristic
278C2×Dic3C22×D511yescharacteristic
279C4×D3—41no—
280C4×D3—102no—
281Dic6—201no—
282C22×C5—21no—
283C22×C5—21no—
284C22×C5—21no—
285C22×C5—32no—
286C22×C5—61no—
287C22×C5—61no—
288C22×C5C23×D311yescharacteristic
289C2×D5—21no—
290C2×D5—21no—
291C2×D5—61no—
292C2×D5—61no—
293C2×D5—61no—
294C2×D5—124no—
295C2×D5#23512yes—
296C4×C5—21no—
297C4×C5—21no—
298C4×C5—61no—
299C4×C5—61no—
300C4×C5—61no—
301Dic5—21no—
302Dic5—21no—
303Dic5—21no—
304Dic5—61no—
305Dic5—61no—
306Dic5—61no—
307C24—151no—
308C2×Q8—151no—
309C2×D4—31no—
310C2×D4—51no—
311C2×D4—151no—
312C22×C4—31no—
313C22×C4—51no—
314C22×C4—151no—
315C22×C4—151no—
316C22×C4D3×D511yescharacteristic, complemented
317C4⋊C4—31no—
318C4⋊C4—51no—
319C4⋊C4—151no—
320C4⋊C4—151no—
321C4⋊C4—151no—
322C4⋊C4—151no—
323C4⋊C4—151no—
324C4⋊C4—151no—
325C22⋊C4—31no—
326C22⋊C4—31no—
327C22⋊C4—51no—
328C22⋊C4—51no—
329C22⋊C4—151no—
330C22⋊C4—151no—
331C22⋊C4—151no—
332C22⋊C4—151no—
333C22⋊C4—151no—
334C22⋊C4—151no—
335C22⋊C4—151no—
336C22⋊C4—151no—
337C22⋊C4—151no—
338C22⋊C4—151no—
339C42—31no—
340C42—51no—
341C42—151no—
342C3×C5#55111yescharacteristic, complemented, Hall
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—52no—
347C22×C3—101no—
348C22×C3—101no—
349C22×C3C23×D511yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—101no—
353C2×D3—101no—
354C2×D3—101no—
355C2×D3—204no—
356C2×D3#70912yes—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—101no—
360C3×C4—101no—
361C3×C4—101no—
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—101no—
366Dic3—101no—
367Dic3—101no—
368C2×C5—41no—
369C2×C5—62no—
370C2×C5C2×#23511yescharacteristic
371C2×C5#100411yescharacteristic
372C2×C5D3×#4111yescharacteristic
373D5—22no—
374D5—121no—
375C23—31no—
376C23—51no—
377C23—151no—
378C23—302no—
379C23—302no—
380C23—601no—
381C23C2×D3×D511yescharacteristic
382Q8—601no—
383D4—62no—
384D4—102no—
385D4—601no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—41no—
390C2×C4—51no—
391C2×C4—51no—
392C2×C4—51no—
393C2×C4—121no—
394C2×C4—151no—
395C2×C4—151no—
396C2×C4—151no—
397C2×C4—201no—
398C2×C4—302no—
399C2×C4—302no—
400C2×C4C2×D3×D511yescharacteristic
401C2×C4C2×D3×D511yescharacteristic
402C2×C3—41no—
403C2×C3—102no—
404C2×C3C2×#70911yescharacteristic
405C2×C3#414411yescharacteristic
406C2×C3D5×#4111yescharacteristic
407D3—22no—
408D3—201no—
409C5#555611yescharacteristic, complemented, Sylow
410C22—21no—
411C22—21no—
412C22—21no—
413C22—32no—
414C22—52no—
415C22—61no—
416C22—61no—
417C22—101no—
418C22—101no—
419C22—301no—
420C22—301no—
421C22—301no—
422C22—604no—
423C22C22×D3×D511yescharacteristic
424C4—21no—
425C4—21no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—101no—
430C4—101no—
431C4—101no—
432C4—301no—
433C4—301no—
434C4—301no—
435C3#6589011yescharacteristic, complemented, Sylow
436C2—41no—
437C2—62no—
438C2—102no—
439C2—601no—
440C2D5×#23511yescharacteristic
441C2D3×#70911yescharacteristic
442C2#9204911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
prime radical#91577C2
hypercenterC22×C4D3×D5
hypocenterC3×C5#551
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. #91818C2semidirect
2. C4×C5×#67C2semidirect
3. C3×C4×C22⋊D5C2semidirect
5. #91555C2semidirect
6. #91548C2semidirect
7. #91526C2semidirect
8. D3×#685C2semidirect
9. D5×(C2×D3)⋊C4C2semidirect
10. #91398C2semidirect
11. #91393C2semidirect
12. #91387C2semidirect
13. #91380C2semidirect
14. #91352C2semidirect
15. #91349C2semidirect
18. C22×C3×C4×C5C22semidirect
24. C2×#1204C22semidirect
27. #6719C22semidirect
30. #6712C22semidirect
31. #6712C22semidirect
32. C5×#217C22semidirect
34. C5×Dic3⋊C4C22semidirect
35. C4×C5×Dic3C22semidirect
36. C3×C22⋊Dic5C22semidirect
38. C3×#683C22semidirect
39. C3×C4×Dic5C22semidirect
40. Dic3⋊Dic5C22semidirect
41. #6669C22semidirect
42. #6669C22semidirect
43. #6668C22semidirect
51. Dic3×Dic5C22semidirect
57. C4×C22⋊D5D3semidirect
99. C4×#67D5semidirect
113. C22×C4×C5C2×D3semidirect
127. C22⋊Dic5C2×D3semidirect
142. #683C2×D3semidirect
144. C4×Dic5C2×D3semidirect
174. C22×C3×C4C2×D5semidirect
188. #217C2×D5semidirect
203. Dic3⋊C4C2×D5semidirect
205. C4×Dic3C2×D5semidirect
316. C22×C4D3×D5semidirect
342. C3×C5#551semidirect
409. C5#5556semidirect
435. C3#65890semidirect

Maximal subgroups

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

subgroupindex
#918182
C4×C5×#672
C3×C4×C22⋊D52
#915772
#915552
#915482
#915262
D3×#6852
D5×(C2×D3)⋊C42
#913982
#913932
#913872
#913802
#913522
#913492
#658903
#55565