Finite Group Databasebeta

← #13106067

Subgroups — #13106067

Order 1600.

754 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C5×D5C211yesmaximal, characteristic, complemented
2C2×#87873C211yesmaximal, characteristic, complemented
3C4×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
4C4×C5×C22⋊D5C211yesmaximal, characteristic, complemented
5#11717096C211yesmaximal, characteristic, complemented
6#11717095C211yesmaximal, characteristic, complemented
7#11717094C211yesmaximal, characteristic, complemented
8#11717082C211yesmaximal, characteristic, complemented
9#11717024C211yesmaximal, characteristic, complemented
10#11717023C211yesmaximal, characteristic, complemented
11C4×#6038C211yesmaximal, characteristic, complemented
12C4×#6037C211yesmaximal, characteristic, complemented
13#11716973C211yesmaximal, characteristic, complemented
14#11716969C211yesmaximal, characteristic, complemented
15#11716968C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C22×C4×C52C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C2×C4×C5⋊D5C2211yescharacteristic, complemented
20C2×C5×C22⋊D5C2211yescharacteristic
21C22×C5×Dic5C2211yescharacteristic
22C2×C4×C5×D5—24no—
23C2×C4×C5×D5C2211yescharacteristic
24C2×C4×C5×D5C2211yescharacteristic
25C2×C4×C5×D5C2211yescharacteristic
26C2×C4×C5×D5C2211yescharacteristic
27C2×C4×C5×D5C2211yescharacteristic, complemented
28C2×#6039C2211yescharacteristic, complemented
29C2×#6038C2211yescharacteristic
30C2×#6038C2211yescharacteristic
31C2×#6038C2211yescharacteristic, complemented
32C2×#6037C2211yescharacteristic
33C2×#6037C2211yescharacteristic
34#87873—24no—
35C22⋊#1039C2211yescharacteristic
36#87808C2211yescharacteristic, complemented
37#87806C2211yescharacteristic, complemented
38C4×#1039C2211yescharacteristic, complemented
39C5×C22⋊Dic5C2211yescharacteristic
40C5×#685C2211yescharacteristic, complemented
41C5×#683C2211yescharacteristic, complemented
42C4×C5×Dic5C2211yescharacteristic, complemented
43#87781C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87780C2211yescharacteristic, complemented
46#87780C2211yescharacteristic, complemented
47#87779C2211yescharacteristic, complemented
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50#87778C2211yescharacteristic
51#87778C2211yescharacteristic
52#87778C2211yescharacteristic, complemented
53#66125—51nomaximal
54#65881—51nomaximal
55C23×C52C2311yescharacteristic
56C22×C5⋊D5C2311yescharacteristic
57C22×C5×D5—22no—
58C22×C5×D5—22no—
59C22×C5×D5—22no—
60C22×C5×D5C2311yescharacteristic
61C22×C5×D5C2311yescharacteristic
62C22×C5×D5C2311yescharacteristic
63C2×C4×C52—22no—
64C2×C4×C52C2311yescharacteristic
65C2×C4×C52C2311yescharacteristic
66C22⋊(C5⋊D5)—22no—
67C2×#1039C2311yescharacteristic
68C2×#1039C2311yescharacteristic
69C2×#1039C2311yescharacteristic
70C4×C5⋊D5—22no—
71C5×C22⋊D5—22no—
72C2×C5×Dic5—22no—
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78C4×C5×D5—22no—
79C4×C5×D5—22no—
80C4×C5×D5—22no—
81C4×C5×D5—22no—
82C4×C5×D5D414yescomplemented
83#6039—22no—
84#6038—22no—
85#6038—22no—
86#6038—22no—
87#6037—22no—
88#6037—22no—
89C23×C4×C5—51no—
90C2×#709—51no—
91C2×#709—51no—
92C22×C4×D5D511yescharacteristic, complemented
93C4×C5×D4—51no—
94#4099—51no—
95#4099—51no—
96#4085—51no—
97#4083—51no—
98#4076—51no—
99#4076—51no—
100#4075—51no—
101#4075—51no—
102C4×C22⋊D5—51no—
103C4×C22⋊D5—51no—
104C4×C22⋊D5—51no—
105C4×C22⋊D5—51no—
106C4×C22⋊D5—51no—
107C4×C22⋊D5—102no—
108C4×C22⋊D5D511yescharacteristic, complemented
109#4072—51no—
110#4070—51no—
111#4070—51no—
112#4042—51no—
113#4040—51no—
114#4030—51no—
115#4029—51no—
116#4028—51no—
117#4024—51no—
118C4×D20—51no—
119#4018—51no—
120C22×C52—21no—
121C22×C52—21no—
122C22×C52#4112yescomplemented
123C22×C52C2411yescharacteristic
124C2×C5⋊D5—21no—
125C2×C5⋊D5—21no—
126C2×C5⋊D5—21no—
127C2×C5×D5—28no—
128C2×C5×D5—21no—
129C2×C5×D5—21no—
130C2×C5×D5—21no—
131C2×C5×D5—21no—
132C2×C5×D5—21no—
133C2×C5×D5—21no—
134C2×C5×D5—21no—
135C2×C5×D5C2×D412yes—
136C2×C5×D5#4112yescomplemented
137C4×C52—21no—
138C4×C52C2×D412yes—
139#1039—21no—
140#1039—21no—
141#1039—21no—
142C5×Dic5—21no—
143C5×Dic5—21no—
144C5×Dic5—21no—
145C5×Dic5—21no—
146C5×Dic5C2×D412yes—
147C24×C5—51no—
148C23×D5C2×D511yescharacteristic
149C2×C5×D4—51no—
150C22×C4×C5—22no—
151C22×C4×C5—51no—
152C22×C4×C5—51no—
153C22×C4×C5—51no—
154C22×C4×C5—51no—
155C22×C4×C5—51no—
156C22×C4×C5—51no—
157C22×C4×C5—104no—
158C22×C4×C5C2×D511yescharacteristic
159C22×C4×C5C2×D511yescharacteristic
160C2×C22⋊D5—51no—
161C2×C22⋊D5—51no—
162C2×C22⋊D5—51no—
163C2×C22⋊D5—51no—
164C2×C22⋊D5—51no—
165C2×C22⋊D5—51no—
166C2×C22⋊D5—51no—
167C2×C22⋊D5—51no—
168C2×C22⋊D5—51no—
169C2×C22⋊D5—102no—
170C2×C22⋊D5C2×D511yescharacteristic
171C22×Dic5C2×D511yescharacteristic
172#709—104no—
173#709—104no—
174C2×D20—51no—
175C2×D20—51no—
176C2×D20—51no—
177C2×C4×D5—24no—
178C2×C4×D5—51no—
179C2×C4×D5—51no—
180C2×C4×D5—102no—
181C2×C4×D5C2×D511yescharacteristic
182C2×C4×D5C2×D511yescharacteristic
183C2×C4×D5C2×D511yescharacteristic
184C2×C4×D5C2×D511yescharacteristic
185C2×C4×D5C2×D511yescharacteristic, complemented
186C2×Dic10—51no—
187C2×Dic10—51no—
188C5×C4⋊C4—51no—
189C5×C22⋊C4—51no—
190C5×C22⋊C4—51no—
191C42×C5—51no—
192C22⋊Dic5—51no—
193C22⋊Dic5—51no—
194C22⋊Dic5—51no—
195C22⋊Dic5—51no—
196C22⋊Dic5—51no—
197C22⋊Dic5—51no—
198C22⋊Dic5—51no—
199C22⋊Dic5—102no—
200C22⋊Dic5C2×D511yescharacteristic
201#685—51no—
202#685—51no—
203#685—51no—
204#685—51no—
205#685—51no—
206#685—51no—
207#685—51no—
208#685—51no—
209#685—51no—
210#685—102no—
211#685C2×D511yescharacteristic, complemented
212C4⋊Dic5—51no—
213C4⋊Dic5—51no—
214C4⋊Dic5—51no—
215#683—51no—
216#683—51no—
217#683—51no—
218#683—51no—
219#683—51no—
220#683—51no—
221#683—51no—
222#683—102no—
223#683C2×D511yescharacteristic, complemented
224C4×Dic5—51no—
225C4×Dic5—51no—
226C4×Dic5—102no—
227C4×Dic5C2×D511yescharacteristic, complemented
228#525—251noSylow
229C2×C52—22no—
230C2×C52C22×D411yescharacteristic
231C2×C52C2×#4111yescharacteristic
232C2×C52C2×#4111yescharacteristic
233C5⋊D5—41no—
234C5×D5—22no—
235C5×D5—22no—
236C5×D5—41no—
237C23×C5—22no—
238C23×C5—51no—
239C23×C5—51no—
240C23×C5—51no—
241C23×C5—102no—
242C23×C5—102no—
243C23×C5—102no—
244C23×C5C22×D511yescharacteristic
245C23×C5C22×D511yescharacteristic
246C22×D5—22no—
247C22×D5—22no—
248C22×D5—22no—
249C22×D5—51no—
250C22×D5—51no—
251C22×D5—102no—
252C22×D5C22×D511yescharacteristic
253C22×D5C22×D511yescharacteristic
254C22×D5C22×D511yescharacteristic
255C5×D4—102no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—44no—
261C2×C4×C5—51no—
262C2×C4×C5—54no—
263C2×C4×C5—51no—
264C2×C4×C5—51no—
265C2×C4×C5—51no—
266C2×C4×C5—51no—
267C2×C4×C5—102no—
268C2×C4×C5—102no—
269C2×C4×C5—102no—
270C2×C4×C5—102no—
271C2×C4×C5—102no—
272C2×C4×C5C22×D511yescharacteristic
273C2×C4×C5C22×D511yescharacteristic
274C2×C4×C5C22×D511yescharacteristic
275C2×C4×C5C22×D511yescharacteristic
276C22⋊D5—22no—
277C22⋊D5—102no—
278C22⋊D5—102no—
279C22⋊D5—102no—
280C22⋊D5—102no—
281C22⋊D5—108no—
282C22⋊D5—102no—
283C22⋊D5—102no—
284C22⋊D5—102no—
285C22⋊D5—102no—
286C22⋊D5—102no—
287C2×Dic5—22no—
288C2×Dic5—51no—
289C2×Dic5—51no—
290C2×Dic5—51no—
291C2×Dic5—51no—
292C2×Dic5—51no—
293C2×Dic5—51no—
294C2×Dic5—102no—
295C2×Dic5—102no—
296C2×Dic5—102no—
297C2×Dic5C22×D511yescharacteristic
298C2×Dic5C22×D511yescharacteristic
299C2×Dic5C22×D511yescharacteristic
300C2×Dic5C22×D511yescharacteristic
301C2×Dic5C22×D511yescharacteristic
302D20—102no—
303D20—102no—
304D20—102no—
305C4×D5—22no—
306C4×D5—22no—
307C4×D5—22no—
308C4×D5—22no—
309C4×D5—102no—
310C4×D5—102no—
311C4×D5—204no—
312C4×D5C22⋊D514yescomplemented
313Dic10—102no—
314Dic10—102no—
315C2×#41—251no—
316C23×C4—51no—
317#123—251no—
318#123—251no—
319C22⋊Q8—251no—
320C22⋊Q8—251no—
321#121—251no—
322#121—251no—
323#120—251no—
324#120—251no—
325C4×D4—51no—
326C4×D4—251no—
327C4×D4—251no—
328C4×D4—251no—
329#117—251no—
330C52#52511yescharacteristic, complemented, Sylow
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—21no—
334C22×C5—21no—
335C22×C5—24no—
336C22×C5—22no—
337C22×C5—42no—
338C22×C5—42no—
339C22×C5—52no—
340C22×C5—52no—
341C22×C5—108no—
342C22×C5—101no—
343C22×C5—101no—
344C22×C5—101no—
345C22×C5—101no—
346C22×C5—101no—
347C22×C5—101no—
348C22×C5—101no—
349C22×C5#70912yescomplemented
350C22×C5#70912yescomplemented
351C22×C5C23×D511yescharacteristic
352C22×C5C23×D511yescharacteristic
353C2×D5—28no—
354C2×D5—21no—
355C2×D5—21no—
356C2×D5—21no—
357C2×D5—21no—
358C2×D5—21no—
359C2×D5—21no—
360C2×D5—21no—
361C2×D5—104no—
362C2×D5—101no—
363C2×D5—101no—
364C2×D5—101no—
365C2×D5—101no—
366C2×D5—101no—
367C2×D5—101no—
368C2×D5—202no—
369C2×D5—202no—
370C2×D5#70912yescomplemented
371C2×D5C2×C22⋊D512yes—
372C4×C5—21no—
373C4×C5—21no—
374C4×C5—24no—
375C4×C5—42no—
376C4×C5—52no—
377C4×C5—101no—
378C4×C5—101no—
379C4×C5—101no—
380C4×C5—101no—
381C4×C5D4×D512yes—
382C4×C5C2×C22⋊D512yes—
383Dic5—21no—
384Dic5—21no—
385Dic5—21no—
386Dic5—21no—
387Dic5—104no—
388Dic5—101no—
389Dic5—101no—
390Dic5—101no—
391Dic5—101no—
392Dic5—101no—
393Dic5—101no—
394Dic5—202no—
395Dic5—202no—
396Dic5C2×C22⋊D512yes—
397C24—51no—
398#41—504no—
399C2×Q8—251no—
400C2×D4—51no—
401C2×D4—251no—
402C2×D4—251no—
403C2×D4—251no—
404C2×D4—251no—
405C2×D4—251no—
406C2×D4—251no—
407C22×C4—51no—
408C22×C4—51no—
409C22×C4—51no—
410C22×C4—51no—
411C22×C4—51no—
412C22×C4—51no—
413C22×C4—104no—
414C22×C4—251no—
415C22×C4D5211yescharacteristic
416C4⋊C4—51no—
417C4⋊C4—251no—
418C4⋊C4—251no—
419C4⋊C4—251no—
420C4⋊C4—251no—
421C4⋊C4—251no—
422C22⋊C4—51no—
423C22⋊C4—51no—
424C22⋊C4—251no—
425C22⋊C4—251no—
426C22⋊C4—251no—
427C22⋊C4—251no—
428C22⋊C4—251no—
429C22⋊C4—251no—
430C22⋊C4—251no—
431C22⋊C4—251no—
432C42—51no—
433C42—251no—
434C2×C5—22no—
435C2×C5—22no—
436C2×C5—22no—
437C2×C5—22no—
438C2×C5—22no—
439C2×C5—44no—
440C2×C5—102no—
441C2×C5—102no—
442C2×C5—201no—
443C2×C5C2×#70911yescharacteristic
444C2×C5C2×#70911yescharacteristic
445C2×C5C2×#70911yescharacteristic
446C2×C5C2×D4×D511yescharacteristic
447C2×C5D5×#4111yescharacteristic
448C2×C5C22×C22⋊D511yescharacteristic
449D5—22no—
450D5—22no—
451D5—41no—
452D5—204no—
453D5—201no—
454D5—201no—
455C23—51no—
456C23—51no—
457C23—51no—
458C23—102no—
459C23—102no—
460C23—102no—
461C23—251no—
462C23C2×D5211yescharacteristic
463Q8—502no—
464D4—102no—
465D4—502no—
466D4—502no—
467D4—502no—
468D4—502no—
469D4—502no—
470D4—502no—
471C2×C4—22no—
472C2×C4—51no—
473C2×C4—54no—
474C2×C4—51no—
475C2×C4—51no—
476C2×C4—51no—
477C2×C4—51no—
478C2×C4—102no—
479C2×C4—102no—
480C2×C4—102no—
481C2×C4—102no—
482C2×C4—102no—
483C2×C4—251no—
484C2×C4—251no—
485C2×C4—251no—
486C2×C4—502no—
487C2×C4C2×D5211yescharacteristic
488C2×C4C2×D5211yescharacteristic
489C5—22no—
490C5#6588111yescharacteristic, complemented
491C5#6612511yescharacteristic, complemented
492C22—21no—
493C22—21no—
494C22—52no—
495C22—52no—
496C22—108no—
497C22—101no—
498C22—101no—
499C22—101no—
500C22—101no—
501C22—101no—
502C22—101no—
503C22—101no—
504C22—501no—
505C22—501no—
506C22—501no—
507C22#8787312yescomplemented
508C22C22×D5211yescharacteristic
509C4—21no—
510C4—52no—
511C4—101no—
512C4—101no—
513C4—101no—
514C4—101no—
515C4—501no—
516C4—501no—
517C4—501no—
518C4D5×C22⋊D512yes—
519C2—22no—
520C2—102no—
521C2—102no—
522C2—201no—
523C2—1001no—
524C2C2×#8787311yescharacteristic
525C2D5×#70911yescharacteristic
526C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
centerC2×C4C2×D52
commutator, socle, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC22×C4×C52C22
hypercenterC22×C4D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×C5×D5C2semidirect
2. C2×#87873C2semidirect
3. C4×C22⋊(C5⋊D5)C2semidirect
4. C4×C5×C22⋊D5C2semidirect
5. #11717096C2semidirect
6. #11717095C2semidirect
7. #11717094C2semidirect
8. #11717082C2semidirect
9. #11717024C2semidirect
10. #11717023C2semidirect
11. C4×#6038C2semidirect
12. C4×#6037C2semidirect
13. #11716973C2semidirect
14. #11716969C2semidirect
15. #11716968C2semidirect
19. C2×C4×C5⋊D5C22semidirect
27. C2×C4×C5×D5C22semidirect
28. C2×#6039C22semidirect
31. C2×#6038C22semidirect
36. #87808C22semidirect
37. #87806C22semidirect
38. C4×#1039C22semidirect
40. C5×#685C22semidirect
41. C5×#683C22semidirect
42. C4×C5×Dic5C22semidirect
43. #87781C22semidirect
44. #87780C22semidirect
45. #87780C22semidirect
46. #87780C22semidirect
47. #87779C22semidirect
52. #87778C22semidirect
82. C4×C5×D5D4semidirect
92. C22×C4×D5D5semidirect
108. C4×C22⋊D5D5semidirect
122. C22×C52#41semidirect
136. C2×C5×D5#41semidirect
185. C2×C4×D5C2×D5semidirect
211. #685C2×D5semidirect
223. #683C2×D5semidirect
227. C4×Dic5C2×D5semidirect
312. C4×D5C22⋊D5semidirect
330. C52#525semidirect
349. C22×C5#709semidirect
350. C22×C5#709semidirect
370. C2×D5#709semidirect
490. C5#65881semidirect
491. C5#66125semidirect
507. C22#87873semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5×D52
C2×#878732
C4×C22⋊(C5⋊D5)2
C4×C5×C22⋊D52
#117170962
#117170952
#117170942
#117170822
#117170242
#117170232
C4×#60382
C4×#60372
#117169732
#117169692
#117169682
#661255
#658815