Finite Group Databasebeta

← #11753241

Subgroups — #11753241

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6779C211yesmaximal, characteristic, complemented
2C2×D3×D20C211yesmaximal, characteristic, complemented
3#91773C211yesmaximal, characteristic, complemented
4C5×#67⋊C4C211yesmaximal, characteristic, complemented
5C3×#4030C211yesmaximal, characteristic, complemented
6#91577C211yesmaximal, characteristic, complemented
7#91577C211yesmaximal, characteristic, complemented
8#91569C211yesmaximal, characteristic, complemented
9#91473C211yesmaximal, characteristic, complemented
10D5×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
11C4×#1202C211yesmaximal, characteristic, complemented
12#91433C211yesmaximal, characteristic, complemented
13#91407C211yesmaximal, characteristic, complemented
14#91353C211yesmaximal, characteristic, complemented
15#91352C211yesmaximal, characteristic, complemented
16#65884—31nomaximal, Hall
17C22×D3×D5C2211yescharacteristic
18C22×D3×D5C2211yescharacteristic
19C2×#1220C2211yescharacteristic, complemented
20C2×#1220C2211yescharacteristic, complemented
21C2×D60C2211yescharacteristic
22C2×C4×D15C2211yescharacteristic, complemented
23C2×C5×#67C2211yescharacteristic
24C22×C5×Dic3C2211yescharacteristic, complemented
25C2×C4×C5×D3C2211yescharacteristic
26C2×C3×C22⋊D5C2211yescharacteristic, complemented
27C2×C3×C22⋊D5C2211yescharacteristic, complemented
28C2×C3×D20C2211yescharacteristic
29C2×C3×C4×D5C2211yescharacteristic, complemented
30C2×#1204C2211yescharacteristic, complemented
31C2×#1202C2211yescharacteristic
32C2×#1202C2211yescharacteristic
33C2×#1200C2211yescharacteristic, complemented
34#6779—24no—
35C2×D5×Dic3C2211yescharacteristic, complemented
36D3×D20—24no—
37C3×C5×C22⋊C4C2211yescharacteristic, complemented
38#6714C2211yescharacteristic, complemented
39#6712C2211yescharacteristic, complemented
40C5×(C2×D3)⋊C4C2211yescharacteristic
41C5×Dic3⋊C4C2211yescharacteristic, complemented
42C4×C5×Dic3C2211yescharacteristic, complemented
43C3×#685C2211yescharacteristic, complemented
44C3×#683C2211yescharacteristic, complemented
45Dic3⋊Dic5C2211yescharacteristic, complemented
46#6667C2211yescharacteristic, complemented
47#6667C2211yescharacteristic, complemented
48#6667C2211yescharacteristic, complemented
49(C2×D3)⋊Dic5C2211yescharacteristic
50(C2×D3)⋊Dic5C2211yescharacteristic
51#6665C2211yescharacteristic, complemented
52#6665C2211yescharacteristic, complemented
53#6665C2211yescharacteristic, complemented
54#5610—51nomaximal, Hall
55C2×#709—31no—
56C22×D20—31no—
57C4×C5×D4—31no—
58#4083—31no—
59#4083—31no—
60#4076—31no—
61#4070—31no—
62#4040—31no—
63#4040—31no—
64#4030—31no—
65#4030D311yescharacteristic, complemented
66D5×C22⋊C4—31no—
67D5×C22⋊C4—31no—
68#4021—31no—
69C4×D20—31no—
70C23×C3×C5C2311yescharacteristic
71C22×D15C2311yescharacteristic
72C22×D15C2311yescharacteristic
73C22×C5×D3C2311yescharacteristic
74C22×C3×D5C2311yescharacteristic
75C22×C3×D5C2311yescharacteristic
76C2×D3×D5—22no—
77C2×D3×D5—22no—
78C2×D3×D5—22no—
79C2×D3×D5—22no—
80C2×D3×D5—41no—
81C2×D3×D5—41no—
82C2×C3×C4×C5C2311yescharacteristic
83C2×C3×C4×C5C2311yescharacteristic
84#1220—22no—
85#1220—41no—
86C2×Dic15C2311yescharacteristic
87C2×Dic15C2311yescharacteristic
88D60—22no—
89C4×D15—41no—
90C5×#67—22no—
91C2×C5×Dic3—22no—
92C2×C5×Dic3C2311yescharacteristic
93C2×C5×Dic3C2311yescharacteristic
94C2×C5×Dic3C2311yescharacteristic
95C4×C5×D3—22no—
96C3×C22⋊D5—22no—
97C3×C22⋊D5—41no—
98C2×C3×Dic5C2311yescharacteristic
99C2×C3×Dic5C2311yescharacteristic
100C3×D20—22no—
101C3×C4×D5—41no—
102#1204—22no—
103#1202—22no—
104#1202D414yescomplemented
105#1200—22no—
106D5×Dic3—22no—
107C2×#237—51no—
108C2×D3×D4—51no—
109C3×#121—51no—
110#939—51no—
111#937—51no—
112#937—51no—
113#935—51no—
114C4×#67—51no—
115C2×(C2×D3)⋊C4—51no—
116#896—51no—
117#894—51no—
118#885—51no—
119#884—51no—
120#67⋊C4D511yescharacteristic, complemented
121D3×C22⋊C4—51no—
122C23×D5—31no—
123C23×D5—31no—
124C2×C5×D4—31no—
125C22×C4×C5—31no—
126C22×C4×C5—31no—
127C2×C22⋊D5—31no—
128C2×C22⋊D5—31no—
129C2×C22⋊D5C2×D311yescharacteristic, complemented
130C2×C22⋊D5C2×D311yescharacteristic, complemented
131#709—64no—
132C2×D20—31no—
133C2×D20—31no—
134C2×D20—31no—
135C2×D20—64no—
136C2×D20C2×D311yescharacteristic
137C2×C4×D5—31no—
138C2×C4×D5—31no—
139C2×C4×D5—31no—
140C2×C4×D5C2×D311yescharacteristic, complemented
141C2×Dic10—31no—
142C5×C4⋊C4—31no—
143C5×C22⋊C4—31no—
144C5×C22⋊C4C2×D311yescharacteristic, complemented
145C42×C5—31no—
146C22⋊Dic5—31no—
147C22⋊Dic5—31no—
148#685—31no—
149#685—31no—
150#685—31no—
151#685—31no—
152#685—31no—
153#685—31no—
154#685—31no—
155#685C2×D311yescharacteristic, complemented
156C4⋊Dic5—31no—
157#683—31no—
158#683C2×D311yescharacteristic, complemented
159#546—151noSylow
160C22×C3×C5—21no—
161C22×C3×C5—21no—
162C22×C3×C5—21no—
163C22×C3×C5C2411yescharacteristic
164C2×D15—21no—
165C2×D15—21no—
166C2×D15—21no—
167C2×D15—21no—
168C2×D15—21no—
169C2×D15C2×D412yes—
170C2×C5×D3—21no—
171C2×C5×D3—21no—
172C2×C5×D3#4112yescomplemented
173C2×C3×D5—21no—
174C2×C3×D5—21no—
175C2×C3×D5—21no—
176C2×C3×D5—21no—
177C2×C3×D5—21no—
178C2×C3×D5C2×D412yes—
179D3×D5—44no—
180D3×D5—44no—
181C3×C4×C5—21no—
182C3×C4×C5—21no—
183Dic15—21no—
184Dic15—21no—
185C3×Dic5—21no—
186C3×Dic5—21no—
187C5×Dic3—21no—
188C5×Dic3—21no—
189C5×Dic3C2×D412yes—
190C23×D3—51no—
191C23×D3—51no—
192C2×C3×D4—51no—
193C2×C3×D4—51no—
194C2×C3×D4—51no—
195C22×C3×C4—51no—
196C2×#67—51no—
197C2×#67—51no—
198C2×#67—51no—
199C2×#67—51no—
200C2×#67C2×D511yescharacteristic
201C22×Dic3—51no—
202C22×Dic3C2×D511yescharacteristic, complemented
203#237—104no—
204D3×D4—104no—
205C2×D12—51no—
206C2×C4×D3—51no—
207C2×C4×D3—51no—
208C2×C4×D3C2×D511yescharacteristic
209C2×Dic6—51no—
210C3×C4⋊C4—51no—
211C3×C22⋊C4—51no—
212C3×C22⋊C4C2×D511yescharacteristic, complemented
213#217—51no—
214#217—51no—
215#217—51no—
216(C2×D3)⋊C4—51no—
217(C2×D3)⋊C4—51no—
218(C2×D3)⋊C4—51no—
219(C2×D3)⋊C4—51no—
220(C2×D3)⋊C4—51no—
221(C2×D3)⋊C4—51no—
222(C2×D3)⋊C4C2×D511yescharacteristic
223Dic3⋊C4—51no—
224Dic3⋊C4—51no—
225Dic3⋊C4C2×D511yescharacteristic, complemented
226C4×Dic3C2×D511yescharacteristic, complemented
227C23×C5—31no—
228C23×C5C22×D311yescharacteristic
229C22×D5—31no—
230C22×D5—31no—
231C22×D5—62no—
232C22×D5—62no—
233C22×D5—62no—
234C22×D5—62no—
235C22×D5—121no—
236C22×D5—121no—
237C22×D5C22×D311yescharacteristic
238C22×D5C22×D311yescharacteristic
239C5×D4—62no—
240C2×C4×C5—31no—
241C2×C4×C5—31no—
242C2×C4×C5—31no—
243C2×C4×C5—62no—
244C2×C4×C5—62no—
245C2×C4×C5C22×D311yescharacteristic
246C2×C4×C5C22×D311yescharacteristic
247C22⋊D5—22no—
248C22⋊D5—41no—
249C22⋊D5—62no—
250C22⋊D5—121no—
251C2×Dic5—31no—
252C2×Dic5—31no—
253C2×Dic5C22×D311yescharacteristic
254C2×Dic5C22×D311yescharacteristic
255D20—22no—
256D20—34no—
257D20—62no—
258D20—62no—
259C4×D5—41no—
260C4×D5—62no—
261C4×D5—62no—
262C4×D5—121no—
263Dic10—62no—
264C2×#41—151no—
265C22×D4—151no—
266#124—151no—
267#123—151no—
268#123—151no—
269C22⋊Q8—151no—
270#121—51no—
271#121—151no—
272#121—151no—
273#120—151no—
274#120—151no—
275C4×D4—31no—
276C4×D4—151no—
277C2×C22⋊C4—151no—
278C2×C22⋊C4—151no—
279C2×C3×C5—41no—
280C2×C3×C5C22×D411yescharacteristic
281C2×C3×C5C2×#4111yescharacteristic
282C2×C3×C5H(5,𝔽2)11yescharacteristic
283D15—22no—
284D15—41no—
285C3×D5—22no—
286C3×D5—41no—
287C5×D3—22no—
288C23×C3—51no—
289C23×C3—51no—
290C23×C3C22×D511yescharacteristic
291C22×D3—51no—
292C22×D3—51no—
293C22×D3—102no—
294C22×D3—102no—
295C22×D3—102no—
296C22×D3—102no—
297C22×D3—201no—
298C22×D3—201no—
299C22×D3C22×D511yescharacteristic
300C3×D4—102no—
301C3×D4—102no—
302C3×D4—201no—
303C2×C3×C4—51no—
304C2×C3×C4—51no—
305C2×C3×C4—201no—
306C2×C3×C4C22×D511yescharacteristic
307C2×C3×C4C22×D511yescharacteristic
308#67—22no—
309#67—54no—
310#67—102no—
311#67—102no—
312#67—201no—
313C2×Dic3—22no—
314C2×Dic3—51no—
315C2×Dic3—51no—
316C2×Dic3—102no—
317C2×Dic3C22×D511yescharacteristic
318C2×Dic3C22×D511yescharacteristic
319C2×Dic3C22×D511yescharacteristic
320D12—102no—
321C4×D3—22no—
322C4×D3—102no—
323C4×D3—201no—
324Dic6—102no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—21no—
328C22×C5—32no—
329C22×C5—61no—
330C22×C5—61no—
331C22×C5C23×D311yescharacteristic
332C2×D5—21no—
333C2×D5—21no—
334C2×D5—21no—
335C2×D5—21no—
336C2×D5—21no—
337C2×D5—32no—
338C2×D5—61no—
339C2×D5—61no—
340C2×D5—61no—
341C2×D5—61no—
342C2×D5—61no—
343C2×D5—124no—
344C2×D5—124no—
345C2×D5D3×D412yes—
346C4×C5—21no—
347C4×C5—21no—
348C4×C5—32no—
349C4×C5—61no—
350C4×C5—61no—
351Dic5—21no—
352Dic5—21no—
353Dic5—61no—
354Dic5—61no—
355C24—151no—
356C24—151no—
357#41—304no—
358C2×Q8—151no—
359C2×D4—31no—
360C2×D4—51no—
361C2×D4—51no—
362C2×D4—51no—
363C2×D4—151no—
364C2×D4—151no—
365C2×D4—151no—
366C2×D4—151no—
367C2×D4—151no—
368C2×D4—304no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—51no—
372C22×C4—151no—
373C22×C4—151no—
374C22×C4—151no—
375C4⋊C4—31no—
376C4⋊C4—51no—
377C4⋊C4—151no—
378C4⋊C4—151no—
379C22⋊C4—31no—
380C22⋊C4—51no—
381C22⋊C4—151no—
382C22⋊C4—151no—
383C22⋊C4—151no—
384C22⋊C4—151no—
385C22⋊C4—151no—
386C22⋊C4—151no—
387C22⋊C4—151no—
388C22⋊C4—151no—
389C22⋊C4—151no—
390C22⋊C4D3×D511yescharacteristic, complemented
391C42—31no—
392C3×C5#54611yescharacteristic, complemented, Hall
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—52no—
397C22×C3—101no—
398C22×C3—101no—
399C22×C3—101no—
400C22×C3—101no—
401C22×C3—101no—
402C22×C3C23×D511yescharacteristic
403C2×D3—21no—
404C2×D3—21no—
405C2×D3—52no—
406C2×D3—101no—
407C2×D3—101no—
408C2×D3—101no—
409C2×D3—101no—
410C2×D3—101no—
411C2×D3—204no—
412C2×D3—204no—
413C2×D3#70912yescomplemented
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—101no—
417C3×C4—101no—
418Dic3—21no—
419Dic3—21no—
420Dic3—101no—
421Dic3—101no—
422Dic3D4×D512yes—
423C2×C5—41no—
424C2×C5—62no—
425C2×C5C2×D3×D411yescharacteristic
426C2×C5#100411yescharacteristic
427C2×C5D3×#4111yescharacteristic
428D5—22no—
429D5—41no—
430D5—62no—
431D5—121no—
432C23—31no—
433C23—51no—
434C23—51no—
435C23—151no—
436C23—151no—
437C23—302no—
438C23—302no—
439C23—302no—
440C23—302no—
441C23—601no—
442C23—601no—
443C23C2×D3×D511yescharacteristic
444Q8—302no—
445D4—62no—
446D4—102no—
447D4—102no—
448D4—154no—
449D4—201no—
450D4—302no—
451D4—302no—
452D4—302no—
453D4—601no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—51no—
458C2×C4—51no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—151no—
462C2×C4—151no—
463C2×C4—201no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—601no—
467C2×C4C2×D3×D511yescharacteristic
468C2×C4C2×D3×D511yescharacteristic
469C2×C3—41no—
470C2×C3—102no—
471C2×C3—201no—
472C2×C3C2×#70911yescharacteristic
473C2×C3C2×D4×D511yescharacteristic
474C2×C3#414911yescharacteristic
475D3—22no—
476D3—102no—
477D3—201no—
478C5#561011yescharacteristic, complemented, Sylow
479C22—21no—
480C22—21no—
481C22—21no—
482C22—32no—
483C22—52no—
484C22—61no—
485C22—61no—
486C22—101no—
487C22—101no—
488C22—101no—
489C22—101no—
490C22—101no—
491C22—152no—
492C22—301no—
493C22—301no—
494C22—301no—
495C22—301no—
496C22—301no—
497C22—604no—
498C22—604no—
499C22C22×D3×D511yescharacteristic
500C4—21no—
501C4—21no—
502C4—32no—
503C4—61no—
504C4—61no—
505C4—101no—
506C4—101no—
507C4—301no—
508C4—301no—
509C3#6588411yescharacteristic, complemented, Sylow
510C2—41no—
511C2—62no—
512C2—102no—
513C2—201no—
514C2—302no—
515C2—601no—
516C2D3×#70911yescharacteristic
517C2D3×D4×D511yescharacteristic
518C2#9202611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC3×C5×C22⋊C4C22
hypercenterC22⋊C4D3×D5
hypocenterC3×C5#546
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. C2×#6779C2semidirect
2. C2×D3×D20C2semidirect
3. #91773C2semidirect
4. C5×#67⋊C4C2semidirect
5. C3×#4030C2semidirect
6. #91577C2semidirect
7. #91577C2semidirect
8. #91569C2semidirect
9. #91473C2semidirect
10. D5×(C2×D3)⋊C4C2semidirect
11. C4×#1202C2semidirect
12. #91433C2semidirect
13. #91407C2semidirect
14. #91353C2semidirect
15. #91352C2semidirect
19. C2×#1220C22semidirect
20. C2×#1220C22semidirect
22. C2×C4×D15C22semidirect
24. C22×C5×Dic3C22semidirect
26. C2×C3×C22⋊D5C22semidirect
27. C2×C3×C22⋊D5C22semidirect
29. C2×C3×C4×D5C22semidirect
30. C2×#1204C22semidirect
33. C2×#1200C22semidirect
35. C2×D5×Dic3C22semidirect
37. C3×C5×C22⋊C4C22semidirect
38. #6714C22semidirect
39. #6712C22semidirect
41. C5×Dic3⋊C4C22semidirect
42. C4×C5×Dic3C22semidirect
43. C3×#685C22semidirect
44. C3×#683C22semidirect
45. Dic3⋊Dic5C22semidirect
46. #6667C22semidirect
47. #6667C22semidirect
48. #6667C22semidirect
51. #6665C22semidirect
52. #6665C22semidirect
53. #6665C22semidirect
65. #4030D3semidirect
104. #1202D4semidirect
120. #67⋊C4D5semidirect
129. C2×C22⋊D5C2×D3semidirect
130. C2×C22⋊D5C2×D3semidirect
140. C2×C4×D5C2×D3semidirect
144. C5×C22⋊C4C2×D3semidirect
155. #685C2×D3semidirect
158. #683C2×D3semidirect
172. C2×C5×D3#41semidirect
202. C22×Dic3C2×D5semidirect
212. C3×C22⋊C4C2×D5semidirect
225. Dic3⋊C4C2×D5semidirect
226. C4×Dic3C2×D5semidirect
390. C22⋊C4D3×D5semidirect
392. C3×C5#546semidirect
413. C2×D3#709semidirect
478. C5#5610semidirect
509. C3#65884semidirect

Maximal subgroups

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

subgroupindex
C2×#67792
C2×D3×D202
#917732
C5×#67⋊C42
C3×#40302
#915772
#915692
#914732
D5×(C2×D3)⋊C42
C4×#12022
#914332
#914072
#913532
#913522
#658843
#56105