Finite Group Databasebeta

← #11754538

Subgroups — #11754538

Order 960.

660 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
1C22×C4×C5×D3C211yesmaximal, characteristic, complemented
2C2×#6771C211yesmaximal, characteristic, complemented
3C4×#1220C211yesmaximal, characteristic, complemented
4C3×C4×C22⋊D5C211yesmaximal, characteristic, complemented
5#91575C211yesmaximal, characteristic, complemented
6#91570C211yesmaximal, characteristic, complemented
7#91569C211yesmaximal, characteristic, complemented
8#91568C211yesmaximal, characteristic, complemented
9#91454C211yesmaximal, characteristic, complemented
10#91452C211yesmaximal, characteristic, complemented
11C4×#1203C211yesmaximal, characteristic, complemented
12C4×#1201C211yesmaximal, characteristic, complemented
13#91371C211yesmaximal, characteristic, complemented
14#91352C211yesmaximal, characteristic, complemented
15#91351C211yesmaximal, characteristic, complemented
16#66125—31nomaximal, Hall
17C23×C5×D3C2211yescharacteristic
18C22×C3×C4×C5C2211yescharacteristic
19C2×#1220C2211yescharacteristic
20C2×C4×D15C2211yescharacteristic, complemented
21C22×C5×Dic3C2211yescharacteristic
22C2×C4×C5×D3—24no—
23C2×C4×C5×D3C2211yescharacteristic
24C2×C4×C5×D3C2211yescharacteristic
25C2×C4×C5×D3C2211yescharacteristic
26C2×C4×C5×D3C2211yescharacteristic
27C2×C3×C22⋊D5C2211yescharacteristic
28C2×C3×C4×D5C2211yescharacteristic, complemented
29C2×#1204C2211yescharacteristic, complemented
30C2×#1203C2211yescharacteristic
31C2×#1203C2211yescharacteristic
32C2×#1202C2211yescharacteristic, complemented
33C2×#1201C2211yescharacteristic
34C2×#1201C2211yescharacteristic
35#6771—24no—
36#6719C2211yescharacteristic
37#6714C2211yescharacteristic, complemented
38#6712C2211yescharacteristic, complemented
39C4×Dic15C2211yescharacteristic, complemented
40C3×C22⋊Dic5C2211yescharacteristic
41C3×#685C2211yescharacteristic, complemented
42C3×#683C2211yescharacteristic, complemented
43C3×C4×Dic5C2211yescharacteristic, complemented
44Dic3⋊Dic5C2211yescharacteristic, complemented
45Dic3⋊Dic5C2211yescharacteristic, complemented
46#6669C2211yescharacteristic, complemented
47#6668C2211yescharacteristic, complemented
48#6667C2211yescharacteristic, complemented
49(C2×D3)⋊Dic5C2211yescharacteristic
50(C2×D3)⋊Dic5C2211yescharacteristic
51(C2×D3)⋊Dic5C2211yescharacteristic
52(C2×D3)⋊Dic5C2211yescharacteristic
53#6665C2211yescharacteristic, complemented
54#5547—51nomaximal, Hall
55C23×C4×C5—31no—
56C2×#709—31no—
57#4099—31no—
58#4099—31no—
59#4076—31no—
60#4076—31no—
61#4075—31no—
62#4075—31no—
63C4×C22⋊D5—31no—
64C4×C22⋊D5—31no—
65C4×C22⋊D5—31no—
66C4×C22⋊D5D311yescharacteristic, complemented
67#4072—31no—
68#4070—31no—
69#4070—31no—
70C23×C3×C5C2311yescharacteristic
71C22×D15C2311yescharacteristic
72C22×C5×D3—22no—
73C22×C5×D3—22no—
74C22×C5×D3—22no—
75C22×C5×D3C2311yescharacteristic
76C22×C5×D3C2311yescharacteristic
77C22×C3×D5C2311yescharacteristic
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5C2311yescharacteristic
80C2×C3×C4×C5C2311yescharacteristic
81#1220—22no—
82C2×Dic15C2311yescharacteristic
83C2×Dic15C2311yescharacteristic
84C2×Dic15C2311yescharacteristic
85C4×D15—22no—
86C2×C5×Dic3—22no—
87C2×C5×Dic3C2311yescharacteristic
88C2×C5×Dic3C2311yescharacteristic
89C4×C5×D3—22no—
90C4×C5×D3—22no—
91C4×C5×D3—22no—
92C4×C5×D3D414yescomplemented
93C3×C22⋊D5—22no—
94C2×C3×Dic5C2311yescharacteristic
95C2×C3×Dic5C2311yescharacteristic
96C2×C3×Dic5C2311yescharacteristic
97C3×C4×D5—22no—
98#1204—22no—
99#1203—22no—
100#1203—22no—
101#1202—22no—
102#1201—22no—
103#1201—22no—
104C2×#235—51no—
105C22×C4×D3D511yescharacteristic, complemented
106C3×C4×D4—51no—
107#939—51no—
108#937—51no—
109C4×#67—51no—
110C4×#67—51no—
111#896—51no—
112#894—51no—
113#884—51no—
114#883—51no—
115#882—51no—
116#878—51no—
117C4×D12—51no—
118#872—51no—
119C24×C5—31no—
120C22×C4×C5—31no—
121C22×C4×C5—31no—
122C22×C4×C5—31no—
123C22×C4×C5—31no—
124C22×C4×C5—31no—
125C22×C4×C5—64no—
126C22×C4×C5C2×D311yescharacteristic
127C2×C22⋊D5—31no—
128C2×C22⋊D5—31no—
129C2×C22⋊D5—31no—
130C2×C22⋊D5—31no—
131C2×C22⋊D5—31no—
132C2×C22⋊D5C2×D311yescharacteristic
133#709—64no—
134C2×D20—31no—
135C2×C4×D5—31no—
136C2×C4×D5C2×D311yescharacteristic, complemented
137C2×Dic10—31no—
138C22⋊Dic5—31no—
139C22⋊Dic5—31no—
140C22⋊Dic5—31no—
141C22⋊Dic5—31no—
142C22⋊Dic5—31no—
143C22⋊Dic5C2×D311yescharacteristic
144#685—31no—
145#685—31no—
146#685—31no—
147#685C2×D311yescharacteristic, complemented
148C4⋊Dic5—31no—
149C4⋊Dic5—31no—
150#683—31no—
151#683—31no—
152#683—31no—
153#683C2×D311yescharacteristic, complemented
154C4×Dic5—31no—
155C4×Dic5C2×D311yescharacteristic, complemented
156#525—151noSylow
157C22×C3×C5—21no—
158C22×C3×C5—21no—
159C22×C3×C5#4112yescomplemented
160C22×C3×C5C2411yescharacteristic
161C2×D15—21no—
162C2×D15—21no—
163C2×D15—21no—
164C2×C5×D3—28no—
165C2×C5×D3—21no—
166C2×C5×D3—21no—
167C2×C5×D3—21no—
168C2×C5×D3—21no—
169C2×C5×D3C2×D412yes—
170C2×C5×D3#4112yescomplemented
171C2×C3×D5—21no—
172C2×C3×D5—21no—
173C2×C3×D5—21no—
174C3×C4×C5—21no—
175C3×C4×C5C2×D412yes—
176Dic15—21no—
177Dic15—21no—
178Dic15—21no—
179C3×Dic5—21no—
180C3×Dic5—21no—
181C3×Dic5—21no—
182C5×Dic3—21no—
183C5×Dic3C2×D412yes—
184C23×D3C2×D511yescharacteristic
185C2×C3×D4—51no—
186C22×C3×C4—51no—
187C22×C3×C4C2×D511yescharacteristic
188C2×#67—51no—
189C2×#67—51no—
190C2×#67—51no—
191C2×#67—51no—
192C22×Dic3C2×D511yescharacteristic
193#235—104no—
194C2×D12—51no—
195C2×D12—51no—
196C2×C4×D3—24no—
197C2×C4×D3—51no—
198C2×C4×D3C2×D511yescharacteristic
199C2×C4×D3C2×D511yescharacteristic
200C2×C4×D3C2×D511yescharacteristic
201C2×C4×D3C2×D511yescharacteristic
202C2×Dic6—51no—
203C3×C4⋊C4—51no—
204C3×C22⋊C4—51no—
205C3×C22⋊C4—51no—
206C3×C42—51no—
207#217—51no—
208#217—51no—
209(C2×D3)⋊C4—51no—
210(C2×D3)⋊C4—51no—
211(C2×D3)⋊C4—51no—
212(C2×D3)⋊C4—51no—
213(C2×D3)⋊C4—51no—
214(C2×D3)⋊C4—51no—
215C4⋊Dic3—51no—
216Dic3⋊C4—51no—
217Dic3⋊C4—51no—
218Dic3⋊C4—51no—
219Dic3⋊C4—51no—
220C4×Dic3—51no—
221C23×C5—31no—
222C23×C5—31no—
223C23×C5—62no—
224C23×C5—62no—
225C23×C5—62no—
226C23×C5C22×D311yescharacteristic
227C22×D5—31no—
228C22×D5C22×D311yescharacteristic
229C2×C4×C5—22no—
230C2×C4×C5—31no—
231C2×C4×C5—34no—
232C2×C4×C5—31no—
233C2×C4×C5—62no—
234C2×C4×C5—62no—
235C2×C4×C5—62no—
236C2×C4×C5—62no—
237C2×C4×C5C22×D311yescharacteristic
238C2×C4×C5C22×D311yescharacteristic
239C22⋊D5—22no—
240C22⋊D5—62no—
241C22⋊D5—62no—
242C22⋊D5—62no—
243C22⋊D5—62no—
244C22⋊D5—62no—
245C2×Dic5—31no—
246C2×Dic5—31no—
247C2×Dic5—31no—
248C2×Dic5C22×D311yescharacteristic
249C2×Dic5C22×D311yescharacteristic
250C2×Dic5C22×D311yescharacteristic
251D20—62no—
252C4×D5—22no—
253C4×D5—62no—
254Dic10—62no—
255C2×#41—151no—
256C23×C4—31no—
257#123—151no—
258#123—151no—
259C22⋊Q8—151no—
260C22⋊Q8—151no—
261#121—151no—
262#121—151no—
263#120—151no—
264#120—151no—
265C4×D4—51no—
266C4×D4—151no—
267C4×D4—151no—
268C4×D4—151no—
269#117—151no—
270C2×C3×C5—22no—
271C2×C3×C5C22×D411yescharacteristic
272C2×C3×C5C2×#4111yescharacteristic
273C2×C3×C5C2×#4111yescharacteristic
274D15—41no—
275C3×D5—41no—
276C5×D3—22no—
277C5×D3—22no—
278C23×C3—51no—
279C23×C3C22×D511yescharacteristic
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—51no—
284C22×D3C22×D511yescharacteristic
285C22×D3C22×D511yescharacteristic
286C3×D4—102no—
287C2×C3×C4—22no—
288C2×C3×C4—51no—
289C2×C3×C4—51no—
290C2×C3×C4—51no—
291C2×C3×C4—102no—
292C2×C3×C4C22×D511yescharacteristic
293C2×C3×C4C22×D511yescharacteristic
294#67—102no—
295#67—102no—
296#67—102no—
297#67—102no—
298C2×Dic3—22no—
299C2×Dic3—51no—
300C2×Dic3—51no—
301C2×Dic3—51no—
302C2×Dic3C22×D511yescharacteristic
303C2×Dic3C22×D511yescharacteristic
304D12—102no—
305D12—102no—
306C4×D3—22no—
307C4×D3—22no—
308C4×D3—22no—
309C4×D3—102no—
310C4×D3C22⋊D514yescomplemented
311Dic6—102no—
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—32no—
315C22×C5—32no—
316C22×C5—68no—
317C22×C5—61no—
318C22×C5—61no—
319C22×C5—61no—
320C22×C5—61no—
321C22×C5#23512yescomplemented
322C22×C5C23×D311yescharacteristic
323C2×D5—21no—
324C2×D5—21no—
325C2×D5—21no—
326C2×D5—61no—
327C2×D5—61no—
328C2×D5—61no—
329C4×C5—21no—
330C4×C5—32no—
331C4×C5—61no—
332C4×C5D3×D412yes—
333Dic5—21no—
334Dic5—21no—
335Dic5—21no—
336Dic5—61no—
337Dic5—61no—
338Dic5—61no—
339C24—31no—
340#41—304no—
341C2×Q8—151no—
342C2×D4—51no—
343C2×D4—151no—
344C2×D4—151no—
345C2×D4—151no—
346C2×D4—151no—
347C2×D4—151no—
348C2×D4—151no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—51no—
355C22×C4—64no—
356C22×C4—151no—
357C22×C4D3×D511yescharacteristic
358C4⋊C4—51no—
359C4⋊C4—151no—
360C4⋊C4—151no—
361C4⋊C4—151no—
362C4⋊C4—151no—
363C4⋊C4—151no—
364C22⋊C4—51no—
365C22⋊C4—51no—
366C22⋊C4—151no—
367C22⋊C4—151no—
368C22⋊C4—151no—
369C22⋊C4—151no—
370C22⋊C4—151no—
371C22⋊C4—151no—
372C22⋊C4—151no—
373C22⋊C4—151no—
374C42—51no—
375C42—151no—
376C3×C5#52511yescharacteristic, complemented, Hall
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—101no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3#70912yescomplemented
383C22×C3C23×D511yescharacteristic
384C2×D3—28no—
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—101no—
390C2×D3—101no—
391C2×D3—101no—
392C2×D3#70912yescomplemented
393C2×D3C2×C22⋊D512yes—
394C3×C4—21no—
395C3×C4—101no—
396C3×C4—101no—
397C3×C4—101no—
398C3×C4C2×C22⋊D512yes—
399Dic3—21no—
400Dic3—101no—
401Dic3—101no—
402Dic3—101no—
403Dic3C2×C22⋊D512yes—
404C2×C5—22no—
405C2×C5—62no—
406C2×C5—62no—
407C2×C5C2×#23511yescharacteristic
408C2×C5C2×D3×D411yescharacteristic
409C2×C5D3×#4111yescharacteristic
410D5—41no—
411D5—121no—
412C23—31no—
413C23—31no—
414C23—51no—
415C23—62no—
416C23—62no—
417C23—62no—
418C23—151no—
419C23C2×D3×D511yescharacteristic
420Q8—302no—
421D4—102no—
422D4—302no—
423D4—302no—
424D4—302no—
425D4—302no—
426D4—302no—
427D4—302no—
428C2×C4—22no—
429C2×C4—31no—
430C2×C4—34no—
431C2×C4—31no—
432C2×C4—51no—
433C2×C4—51no—
434C2×C4—51no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—102no—
440C2×C4—151no—
441C2×C4—151no—
442C2×C4—151no—
443C2×C4—302no—
444C2×C4C2×D3×D511yescharacteristic
445C2×C4C2×D3×D511yescharacteristic
446C2×C3—22no—
447C2×C3—201no—
448C2×C3C2×#70911yescharacteristic
449C2×C3C2×#70911yescharacteristic
450C2×C3C22×C22⋊D511yescharacteristic
451D3—22no—
452D3—22no—
453D3—201no—
454C5#554711yescharacteristic, complemented, Sylow
455C22—21no—
456C22—21no—
457C22—32no—
458C22—32no—
459C22—68no—
460C22—61no—
461C22—61no—
462C22—61no—
463C22—61no—
464C22—101no—
465C22—101no—
466C22—101no—
467C22—301no—
468C22—301no—
469C22—301no—
470C22#677112yescomplemented
471C22C22×D3×D511yescharacteristic
472C4—21no—
473C4—32no—
474C4—61no—
475C4—101no—
476C4—101no—
477C4—101no—
478C4—301no—
479C4—301no—
480C4—301no—
481C4D3×C22⋊D512yes—
482C3#6612511yescharacteristic, complemented, Sylow
483C2—22no—
484C2—62no—
485C2—62no—
486C2—201no—
487C2—601no—
488C2C2×#677111yescharacteristic
489C2D3×#70911yescharacteristic
490C2C2×D3×C22⋊D511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FrattiniC22C22×D3×D5
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#525
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C5×D3C2semidirect
2. C2×#6771C2semidirect
3. C4×#1220C2semidirect
4. C3×C4×C22⋊D5C2semidirect
5. #91575C2semidirect
6. #91570C2semidirect
7. #91569C2semidirect
8. #91568C2semidirect
9. #91454C2semidirect
10. #91452C2semidirect
11. C4×#1203C2semidirect
12. C4×#1201C2semidirect
13. #91371C2semidirect
14. #91352C2semidirect
15. #91351C2semidirect
20. C2×C4×D15C22semidirect
28. C2×C3×C4×D5C22semidirect
29. C2×#1204C22semidirect
32. C2×#1202C22semidirect
37. #6714C22semidirect
38. #6712C22semidirect
39. C4×Dic15C22semidirect
41. C3×#685C22semidirect
42. C3×#683C22semidirect
43. C3×C4×Dic5C22semidirect
44. Dic3⋊Dic5C22semidirect
45. Dic3⋊Dic5C22semidirect
46. #6669C22semidirect
47. #6668C22semidirect
48. #6667C22semidirect
53. #6665C22semidirect
66. C4×C22⋊D5D3semidirect
92. C4×C5×D3D4semidirect
105. C22×C4×D3D5semidirect
136. C2×C4×D5C2×D3semidirect
147. #685C2×D3semidirect
153. #683C2×D3semidirect
155. C4×Dic5C2×D3semidirect
159. C22×C3×C5#41semidirect
170. C2×C5×D3#41semidirect
310. C4×D3C22⋊D5semidirect
321. C22×C5#235semidirect
376. C3×C5#525semidirect
382. C22×C3#709semidirect
392. C2×D3#709semidirect
454. C5#5547semidirect
470. C22#6771semidirect
482. C3#66125semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5×D32
C2×#67712
C4×#12202
C3×C4×C22⋊D52
#915752
#915702
#915692
#915682
#914542
#914522
C4×#12032
C4×#12012
#913712
#913522
#913512
#661253
#55475