Finite Group Databasebeta

← #13105088

Subgroups — #13105088

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
1C2×C4×D52C211yesmaximal, characteristic, complemented
2C2×#87873C211yesmaximal, characteristic, complemented
3C4×C4⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
4C4×C5×D20C211yesmaximal, characteristic, complemented
5C5×#4018C211yesmaximal, characteristic, complemented
6#11717041C211yesmaximal, characteristic, complemented
7#11717040C211yesmaximal, characteristic, complemented
8#11717028C211yesmaximal, characteristic, complemented
9C4×#6038C211yesmaximal, characteristic, complemented
10C4×#6038C211yesmaximal, characteristic, complemented
11#11717011C211yesmaximal, characteristic, complemented
12#11717009C211yesmaximal, characteristic, complemented
13#11717007C211yesmaximal, characteristic, complemented
14#11717005C211yesmaximal, characteristic, complemented
15#11717003C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C4⋊(C5⋊D5)C2211yescharacteristic
18C2×C4×C5⋊D5C2211yescharacteristic
19C2×C4×C5⋊D5C2211yescharacteristic, complemented
20C2×C5×D20C2211yescharacteristic
21C2×C4×C5×D5C2211yescharacteristic
22C2×C4×C5×D5C2211yescharacteristic
23C2×C4×C5×D5C2211yescharacteristic, complemented
24C2×#6039C2211yescharacteristic, complemented
25C2×#6038C2211yescharacteristic
26C2×#6038C2211yescharacteristic
27C2×#6038C2211yescharacteristic
28C2×#6038C2211yescharacteristic, complemented
29C2×#6037C2211yescharacteristic
30C2×#6036C2211yescharacteristic
31C2×D5×Dic5C2211yescharacteristic
32C2×D5×Dic5C2211yescharacteristic
33C4×D52—24no—
34#87873—24no—
35C42×C52C2211yescharacteristic, complemented
36#87808C2211yescharacteristic
37#87808C2211yescharacteristic, complemented
38C4⋊#1039C2211yescharacteristic, complemented
39C5×#685C2211yescharacteristic
40C5×#685C2211yescharacteristic
41C5×#685C2211yescharacteristic
42C5×#685C2211yescharacteristic, complemented
43C5×C4⋊Dic5C2211yescharacteristic, complemented
44C5×#683C2211yescharacteristic, complemented
45C5×#683C2211yescharacteristic, complemented
46C4×C5×Dic5C2211yescharacteristic, complemented
47#87780C2211yescharacteristic, complemented
48#87780C2211yescharacteristic, complemented
49#87779C2211yescharacteristic
50#87779C2211yescharacteristic, complemented
51#87778C2211yescharacteristic
52#87778C2211yescharacteristic, complemented
53#65881—51nomaximal
54#65858—51nomaximal
55C22×C5⋊D5C2311yescharacteristic
56C22×C5⋊D5C2311yescharacteristic
57C22×C5×D5C2311yescharacteristic
58C22×C5×D5C2311yescharacteristic
59C22×C5×D5C2311yescharacteristic
60C2×D52—22no—
61C2×D52—22no—
62C2×D52—22no—
63C2×C4×C52C2311yescharacteristic
64C2×C4×C52C2311yescharacteristic
65C2×C4×C52C2311yescharacteristic
66C2×#1039C2311yescharacteristic
67C2×#1039C2311yescharacteristic
68C4⋊(C5⋊D5)—22no—
69C4×C5⋊D5—22no—
70C4×C5⋊D5—22no—
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C5×D20—22no—
77C4×C5×D5—22no—
78C4×C5×D5—22no—
79C4×C5×D5D414yescomplemented
80#6039—22no—
81#6038—22no—
82#6038—22no—
83#6038—22no—
84#6038—22no—
85#6037—22no—
86#6036—22no—
87D5×Dic5—22no—
88D5×Dic5—22no—
89C2×#709—51no—
90C2×#709—51no—
91C22×C4×D5—51no—
92C22×C4×D5—51no—
93C4×C5×D4—51no—
94C5×#117—51no—
95#4085—51no—
96#4083—51no—
97C4×C22⋊D5—51no—
98C4×C22⋊D5—51no—
99#4043—51no—
100#4043—51no—
101#4042—51no—
102#4041—51no—
103#4041—51no—
104#4040—51no—
105#4032—51no—
106#4032—51no—
107#4030—51no—
108#4029—51no—
109#4028—51no—
110#4028—51no—
111#4028—51no—
112#4024—51no—
113C4×D20—51no—
114C4×D20—51no—
115C4×D20—51no—
116C4×D20—51no—
117C4×D20—102no—
118C4×D20D511yescharacteristic, complemented
119#4018D511yescharacteristic, complemented
120C22×C52C2411yescharacteristic
121C2×C5⋊D5—21no—
122C2×C5⋊D5—21no—
123C2×C5⋊D5—21no—
124C2×C5⋊D5—21no—
125C2×C5⋊D5—21no—
126C2×C5⋊D5#4112yescomplemented
127C2×C5×D5—21no—
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×D5C2×D412yes—
135C2×C5×D5#4112yescomplemented
136D52—28no—
137C4×C52—21no—
138C4×C52—21no—
139C4×C52C2×D412yes—
140#1039—21no—
141#1039—21no—
142C5×Dic5—21no—
143C5×Dic5—21no—
144C5×Dic5—21no—
145C5×Dic5—21no—
146C5×Dic5C2×D412yes—
147C23×D5—51no—
148C23×D5—51no—
149C2×C5×D4—51no—
150C22×C4×C5—51no—
151C22×C4×C5—51no—
152C22×C4×C5—51no—
153C2×C22⋊D5—51no—
154C2×C22⋊D5—51no—
155C2×C22⋊D5—51no—
156C2×C22⋊D5—51no—
157C2×C22⋊D5—51no—
158C2×C22⋊D5—51no—
159C22×Dic5—51no—
160C22×Dic5—51no—
161#709—104no—
162#709—104no—
163C2×D20—51no—
164C2×D20—51no—
165C2×D20—51no—
166C2×D20—51no—
167C2×D20—51no—
168C2×D20—51no—
169C2×D20—102no—
170C2×D20C2×D511yescharacteristic
171C2×C4×D5—51no—
172C2×C4×D5—51no—
173C2×C4×D5—51no—
174C2×C4×D5—51no—
175C2×C4×D5—51no—
176C2×C4×D5—51no—
177C2×C4×D5—51no—
178C2×C4×D5—51no—
179C2×C4×D5—102no—
180C2×C4×D5—102no—
181C2×C4×D5—104no—
182C2×C4×D5—104no—
183C2×C4×D5C2×D511yescharacteristic
184C2×C4×D5C2×D511yescharacteristic
185C2×C4×D5C2×D511yescharacteristic, complemented
186C2×Dic10—51no—
187C2×Dic10—51no—
188C5×C4⋊C4—51no—
189C5×C4⋊C4—51no—
190C5×C4⋊C4—51no—
191C5×C22⋊C4—51no—
192C5×C22⋊C4—51no—
193C5×C22⋊C4—51no—
194C5×C22⋊C4—51no—
195C42×C5—22no—
196C42×C5—51no—
197C42×C5C2×D511yescharacteristic, complemented
198C42×C5C2×D511yescharacteristic, complemented
199C22⋊Dic5—51no—
200C22⋊Dic5—51no—
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—51no—
211#685—102no—
212#685—102no—
213#685C2×D511yescharacteristic
214#685C2×D511yescharacteristic
215#685C2×D511yescharacteristic
216#685C2×D511yescharacteristic, complemented
217C4⋊Dic5—51no—
218C4⋊Dic5—51no—
219C4⋊Dic5—51no—
220C4⋊Dic5—51no—
221C4⋊Dic5—102no—
222C4⋊Dic5C2×D511yescharacteristic, complemented
223#683—51no—
224#683—51no—
225#683C2×D511yescharacteristic, complemented
226#683C2×D511yescharacteristic, complemented
227C4×Dic5C2×D511yescharacteristic, complemented
228#525—251noSylow
229C2×C52C22×D411yescharacteristic
230C2×C52C2×#4111yescharacteristic
231C2×C52C2×#4111yescharacteristic
232C5⋊D5—22no—
233C5⋊D5—41no—
234C5×D5—22no—
235C5×D5—22no—
236C5×D5—41no—
237C23×C5—51no—
238C23×C5—51no—
239C23×C5—51no—
240C22×D5—51no—
241C22×D5—51no—
242C22×D5—51no—
243C22×D5—51no—
244C22×D5—102no—
245C22×D5—102no—
246C22×D5—102no—
247C22×D5—102no—
248C22×D5—102no—
249C22×D5—102no—
250C22×D5—102no—
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—51no—
260C2×C4×C5—54no—
261C2×C4×C5—51no—
262C2×C4×C5—51no—
263C2×C4×C5—51no—
264C2×C4×C5—51no—
265C2×C4×C5—102no—
266C2×C4×C5—102no—
267C2×C4×C5C22×D511yescharacteristic
268C2×C4×C5C22×D511yescharacteristic
269C2×C4×C5C22×D511yescharacteristic
270C2×C4×C5C22×D511yescharacteristic
271C2×C4×C5C22×D511yescharacteristic
272C2×C4×C5C22×D511yescharacteristic
273C22⋊D5—102no—
274C22⋊D5—102no—
275C22⋊D5—102no—
276C22⋊D5—102no—
277C22⋊D5—102no—
278C22⋊D5—102no—
279C2×Dic5—51no—
280C2×Dic5—51no—
281C2×Dic5—51no—
282C2×Dic5—51no—
283C2×Dic5—102no—
284C2×Dic5—102no—
285C2×Dic5—102no—
286C2×Dic5—102no—
287C2×Dic5C22×D511yescharacteristic
288C2×Dic5C22×D511yescharacteristic
289C2×Dic5C22×D511yescharacteristic
290C2×Dic5C22×D511yescharacteristic
291C2×Dic5C22×D511yescharacteristic
292D20—22no—
293D20—102no—
294D20—102no—
295D20—108no—
296D20—102no—
297D20—102no—
298D20—102no—
299D20—102no—
300C4×D5—22no—
301C4×D5—22no—
302C4×D5—102no—
303C4×D5—102no—
304C4×D5—102no—
305C4×D5—102no—
306C4×D5—102no—
307C4×D5—102no—
308C4×D5—102no—
309C4×D5—102no—
310C4×D5—204no—
311C4×D5—204no—
312C4×D5D2014yescomplemented
313Dic10—102no—
314Dic10—102no—
315C2×#41—251no—
316C23×C4—251no—
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—51no—
330C52#52511yescharacteristic, complemented, Sylow
331C22×C5—22no—
332C22×C5—52no—
333C22×C5—52no—
334C22×C5—101no—
335C22×C5—101no—
336C22×C5—101no—
337C22×C5—101no—
338C22×C5—101no—
339C22×C5—101no—
340C22×C5—101no—
341C22×C5C23×D511yescharacteristic
342C22×C5C23×D511yescharacteristic
343C2×D5—21no—
344C2×D5—21no—
345C2×D5—21no—
346C2×D5—21no—
347C2×D5—21no—
348C2×D5—21no—
349C2×D5—21no—
350C2×D5—52no—
351C2×D5—52no—
352C2×D5—104no—
353C2×D5—108no—
354C2×D5—108no—
355C2×D5—101no—
356C2×D5—101no—
357C2×D5—101no—
358C2×D5—101no—
359C2×D5—104no—
360C2×D5—101no—
361C2×D5—101no—
362C2×D5—101no—
363C2×D5—101no—
364C2×D5—101no—
365C2×D5—101no—
366C2×D5—202no—
367C2×D5—202no—
368C2×D5—202no—
369C2×D5—202no—
370C2×D5C2×D2012yes—
371C2×D5#70912yescomplemented
372C4×C5—24no—
373C4×C5—21no—
374C4×C5—21no—
375C4×C5—21no—
376C4×C5—21no—
377C4×C5—24no—
378C4×C5—42no—
379C4×C5—52no—
380C4×C5—101no—
381C4×C5—101no—
382C4×C5—101no—
383C4×C5—101no—
384C4×C5C2×D2012yes—
385C4×C5D4×D512yes—
386Dic5—21no—
387Dic5—21no—
388Dic5—21no—
389Dic5—21no—
390Dic5—101no—
391Dic5—101no—
392Dic5—101no—
393Dic5—101no—
394Dic5—202no—
395Dic5—202no—
396Dic5C2×D2012yes—
397C24—251no—
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—251no—
411C22×C4—251no—
412C22×C4—251no—
413C22×C4—251no—
414C22×C4—251no—
415C22×C4—504no—
416C4⋊C4—51no—
417C4⋊C4—51no—
418C4⋊C4—51no—
419C4⋊C4—251no—
420C4⋊C4—251no—
421C4⋊C4—251no—
422C22⋊C4—51no—
423C22⋊C4—51no—
424C22⋊C4—51no—
425C22⋊C4—51no—
426C22⋊C4—251no—
427C22⋊C4—251no—
428C22⋊C4—251no—
429C22⋊C4—251no—
430C22⋊C4—251no—
431C22⋊C4—251no—
432C42—51no—
433C42D5211yescharacteristic, complemented
434C2×C5—22no—
435C2×C5—22no—
436C2×C5—22no—
437C2×C5—102no—
438C2×C5—102no—
439C2×C5—201no—
440C2×C5C22×D2011yescharacteristic
441C2×C5C2×#70911yescharacteristic
442C2×C5C2×D4×D511yescharacteristic
443C2×C5D5×#4111yescharacteristic
444C2×C5D5×#4111yescharacteristic
445C2×C5D5×#4111yescharacteristic
446D5—22no—
447D5—22no—
448D5—41no—
449D5—102no—
450D5—102no—
451D5—204no—
452D5—204no—
453D5—201no—
454D5—201no—
455C23—51no—
456C23—51no—
457C23—51no—
458C23—251no—
459C23—251no—
460C23—502no—
461C23—502no—
462C23—502no—
463Q8—502no—
464D4—102no—
465D4—502no—
466D4—502no—
467D4—502no—
468D4—502no—
469D4—502no—
470D4—502no—
471C2×C4—51no—
472C2×C4—54no—
473C2×C4—51no—
474C2×C4—51no—
475C2×C4—51no—
476C2×C4—51no—
477C2×C4—102no—
478C2×C4—102no—
479C2×C4—251no—
480C2×C4—251no—
481C2×C4—502no—
482C2×C4—502no—
483C2×C4—502no—
484C2×C4—502no—
485C2×C4—502no—
486C2×C4C2×D5211yescharacteristic
487C2×C4C2×D5211yescharacteristic
488C2×C4C2×D5211yescharacteristic
489C5—22no—
490C5#6585811yescharacteristic, complemented
491C5#6588111yescharacteristic, complemented
492C22—52no—
493C22—52no—
494C22—101no—
495C22—101no—
496C22—101no—
497C22—101no—
498C22—101no—
499C22—101no—
500C22—101no—
501C22—252no—
502C22—508no—
503C22—501no—
504C22—501no—
505C22—501no—
506C22—501no—
507C22—501no—
508C22C22×D5211yescharacteristic
509C4—21no—
510C4—21no—
511C4—52no—
512C4—101no—
513C4—101no—
514C4—101no—
515C4—101no—
516C4—501no—
517C4—501no—
518C4D5×D2012yes—
519C2—102no—
520C2—102no—
521C2—201no—
522C2—502no—
523C2—1001no—
524C2C2×D5×D2011yescharacteristic
525C2D5×#70911yescharacteristic
526C2#1171747111yescharacteristic
subgroupG/H
centerC2×C4C2×D52
commutator, socle, cosocleC22×C52C24
FrattiniC22C22×D52
FittingC42×C52C22
hypercenterC42D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×D52C2semidirect
2. C2×#87873C2semidirect
3. C4×C4⋊(C5⋊D5)C2semidirect
4. C4×C5×D20C2semidirect
5. C5×#4018C2semidirect
6. #11717041C2semidirect
7. #11717040C2semidirect
8. #11717028C2semidirect
9. C4×#6038C2semidirect
10. C4×#6038C2semidirect
11. #11717011C2semidirect
12. #11717009C2semidirect
13. #11717007C2semidirect
14. #11717005C2semidirect
15. #11717003C2semidirect
19. C2×C4×C5⋊D5C22semidirect
23. C2×C4×C5×D5C22semidirect
24. C2×#6039C22semidirect
28. C2×#6038C22semidirect
35. C42×C52C22semidirect
37. #87808C22semidirect
38. C4⋊#1039C22semidirect
42. C5×#685C22semidirect
43. C5×C4⋊Dic5C22semidirect
44. C5×#683C22semidirect
45. C5×#683C22semidirect
46. C4×C5×Dic5C22semidirect
47. #87780C22semidirect
48. #87780C22semidirect
50. #87779C22semidirect
52. #87778C22semidirect
79. C4×C5×D5D4semidirect
118. C4×D20D5semidirect
119. #4018D5semidirect
126. C2×C5⋊D5#41semidirect
135. C2×C5×D5#41semidirect
185. C2×C4×D5C2×D5semidirect
197. C42×C5C2×D5semidirect
198. C42×C5C2×D5semidirect
216. #685C2×D5semidirect
222. C4⋊Dic5C2×D5semidirect
225. #683C2×D5semidirect
226. #683C2×D5semidirect
227. C4×Dic5C2×D5semidirect
312. C4×D5D20semidirect
330. C52#525semidirect
371. C2×D5#709semidirect
433. C42D52semidirect
490. C5#65858semidirect
491. C5#65881semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D522
C2×#878732
C4×C4⋊(C5⋊D5)2
C4×C5×D202
C5×#40182
#117170412
#117170402
#117170282
C4×#60382
#117170112
#117170092
#117170072
#117170052
#117170032
#658815
#658585