Finite Group Databasebeta

← #13106396

Subgroups — #13106396

Order 1600.

798 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#711C211yesmaximal, characteristic, complemented
2C2×#87886C211yesmaximal, characteristic, complemented
3#11717257C211yesmaximal, characteristic, complemented
4C5×#4085C211yesmaximal, characteristic, complemented
5#11717101C211yesmaximal, characteristic, complemented
6#11717096C211yesmaximal, characteristic, complemented
7Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
8#11717089C211yesmaximal, characteristic, complemented
9C2×#87779C211yesmaximal, characteristic, complemented
10#11717079C211yesmaximal, characteristic, complemented
11#11717070C211yesmaximal, characteristic, complemented
12D5×#685C211yesmaximal, characteristic, complemented
13#11717025C211yesmaximal, characteristic, complemented
14#11717008C211yesmaximal, characteristic, complemented
15#11716968C211yesmaximal, characteristic, complemented
16C23×C5⋊D5C2211yescharacteristic
17C22×D52C2211yescharacteristic
18C2×C52×D4C2211yescharacteristic, complemented
19C2×C22⋊(C5⋊D5)C2211yescharacteristic
20C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C5×C22⋊D5C2211yescharacteristic
23C2×C5×C22⋊D5C2211yescharacteristic, complemented
24C2×C5×C22⋊D5C2211yescharacteristic, complemented
25C22×C5×Dic5C2211yescharacteristic, complemented
26C22×C5×Dic5C2211yescharacteristic, complemented
27C22×C5×Dic5C2211yescharacteristic, complemented
28C5×#711—24no—
29C2×C4×C5×D5C2211yescharacteristic, complemented
30C2×C5×Dic10C2211yescharacteristic, complemented
31#87886—24no—
32C2×#6038C2211yescharacteristic
33C2×#6038C2211yescharacteristic
34C2×#6036C2211yescharacteristic
35C2×D5×Dic5C2211yescharacteristic, complemented
36C22⋊#1039C2211yescharacteristic, complemented
37#87808C2211yescharacteristic
38#87808C2211yescharacteristic, complemented
39C5×C22⋊Dic5C2211yescharacteristic, complemented
40C5×#685C2211yescharacteristic, complemented
41C5×#683C2211yescharacteristic, complemented
42#87781C2211yescharacteristic, complemented
43#87781C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87779C2211yescharacteristic
46#87779C2211yescharacteristic
47#87779C2211yescharacteristic, complemented
48#87779C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#66163—51nomaximal
54#65944—51nomaximal
55C23×C52C2311yescharacteristic
56C23×C52C2311yescharacteristic
57C22×C5⋊D5—22no—
58C22×C5⋊D5—22no—
59C22×C5⋊D5—41no—
60C22×C5⋊D5C2311yescharacteristic
61C22×C5⋊D5C2311yescharacteristic
62C22×C5×D5C2311yescharacteristic
63C22×C5×D5C2311yescharacteristic
64C2×D52—22no—
65C2×D52—22no—
66C2×D52—41no—
67C52×D4—22no—
68C2×C4×C52C2311yescharacteristic
69C22⋊(C5⋊D5)—22no—
70C22⋊(C5⋊D5)—41no—
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C5×C22⋊D5—22no—
74C5×C22⋊D5—22no—
75C5×C22⋊D5—41no—
76C5×C22⋊D5D414yescomplemented
77C2×C5×Dic5—22no—
78C2×C5×Dic5—22no—
79C2×C5×Dic5—41no—
80C2×C5×Dic5C2311yescharacteristic
81C2×C5×Dic5C2311yescharacteristic
82C2×C5×Dic5C2311yescharacteristic
83C2×C5×Dic5C2311yescharacteristic
84C2×C5×Dic5C2311yescharacteristic
85C2×C5×Dic5C2311yescharacteristic
86C4×C5×D5—22no—
87C5×Dic10—22no—
88#6038—22no—
89#6038—22no—
90#6036—22no—
91D5×Dic5—41no—
92C2×C5×#41—51no—
93C2×#711D511yescharacteristic, complemented
94C2×D4×D5—51no—
95C2×D4×D5—51no—
96C5×#121—51no—
97#4093—51no—
98#4092—51no—
99#4085—51no—
100#4085—51no—
101#4085D511yescharacteristic, complemented
102#4083—51no—
103#4083—51no—
104#4083—51no—
105#4083—51no—
106#4083—102no—
107#4081—51no—
108D4×Dic5—51no—
109#4076—51no—
110#4075—51no—
111#4075—51no—
112C4×C22⋊D5—51no—
113C4×C22⋊D5—51no—
114C2×#685—51no—
115C2×#685—51no—
116C2×#685—51no—
117#4042—51no—
118#4040—51no—
119#4031—51no—
120#4030—51no—
121#4027—51no—
122D5×C22⋊C4—51no—
123C22×C52—21no—
124C22×C52—21no—
125C22×C52—21no—
126C22×C52—21no—
127C22×C52—21no—
128C22×C52C2×D412yes—
129C22×C52C2411yescharacteristic
130C2×C5⋊D5—21no—
131C2×C5⋊D5—21no—
132C2×C5⋊D5—21no—
133C2×C5⋊D5—21no—
134C2×C5⋊D5—21no—
135C2×C5⋊D5—44no—
136C2×C5⋊D5#4112yescomplemented
137C2×C5×D5—21no—
138C2×C5×D5—21no—
139C2×C5×D5—21no—
140C2×C5×D5—21no—
141C2×C5×D5—21no—
142C2×C5×D5C2×D412yes—
143D52—44no—
144C4×C52—21no—
145#1039—21no—
146#1039—21no—
147C5×Dic5—21no—
148C5×Dic5—21no—
149C5×Dic5—21no—
150C5×Dic5—21no—
151C5×Dic5—21no—
152C5×Dic5C2×D412yes—
153C23×D5—51no—
154C23×D5—51no—
155C23×D5—51no—
156C23×D5—51no—
157C23×D5—102no—
158C5×#41—104no—
159C2×C5×Q8—51no—
160C2×C5×D4—22no—
161C2×C5×D4—51no—
162C2×C5×D4—51no—
163C2×C5×D4—51no—
164C2×C5×D4—51no—
165C2×C5×D4C2×D511yescharacteristic, complemented
166C2×C5×D4C2×D511yescharacteristic, complemented
167C22×C4×C5—51no—
168C22×C4×C5—51no—
169C22×C4×C5—51no—
170C22×C4×C5—51no—
171C2×C22⋊D5—51no—
172C2×C22⋊D5—51no—
173C2×C22⋊D5—51no—
174C2×C22⋊D5—51no—
175C2×C22⋊D5—51no—
176C2×C22⋊D5—51no—
177C2×C22⋊D5—102no—
178C2×C22⋊D5—102no—
179C2×C22⋊D5C2×D511yescharacteristic
180C2×C22⋊D5C2×D511yescharacteristic
181C2×C22⋊D5C2×D511yescharacteristic, complemented
182C2×C22⋊D5C2×D511yescharacteristic, complemented
183C22×Dic5—51no—
184C22×Dic5C2×D511yescharacteristic, complemented
185C22×Dic5C2×D511yescharacteristic, complemented
186C22×Dic5C2×D511yescharacteristic, complemented
187#711—24no—
188D4×D5—104no—
189D4×D5—104no—
190C2×D20—51no—
191C2×D20—51no—
192C2×C4×D5—51no—
193C2×C4×D5—51no—
194C2×C4×D5—51no—
195C2×C4×D5C2×D511yescharacteristic, complemented
196C2×Dic10C2×D511yescharacteristic, complemented
197C5×C4⋊C4—51no—
198C5×C22⋊C4—51no—
199C5×C22⋊C4—51no—
200C22⋊Dic5—51no—
201C22⋊Dic5—51no—
202C22⋊Dic5—51no—
203C22⋊Dic5—51no—
204C22⋊Dic5—51no—
205C22⋊Dic5—102no—
206C22⋊Dic5C2×D511yescharacteristic, complemented
207#685—51no—
208#685—51no—
209#685—51no—
210#685—51no—
211#685—51no—
212#685—51no—
213#685—51no—
214#685—51no—
215#685—51no—
216#685—51no—
217#685—51no—
218#685—51no—
219#685—51no—
220#685—51no—
221#685—51no—
222#685—102no—
223#685—102no—
224#685C2×D511yescharacteristic, complemented
225C4⋊Dic5—51no—
226#683—51no—
227#683—51no—
228#683—51no—
229#683—51no—
230#683—51no—
231#683C2×D511yescharacteristic, complemented
232C4×Dic5—51no—
233C4×Dic5—51no—
234#546—251noSylow
235C2×C52—22no—
236C2×C52—41no—
237C2×C52C22×D411yescharacteristic
238C2×C52C2×#4111yescharacteristic
239C2×C52H(5,𝔽2)11yescharacteristic
240C5⋊D5—22no—
241C5⋊D5—41no—
242C5×D5—22no—
243C5×D5—41no—
244C23×C5—22no—
245C23×C5—22no—
246C23×C5—51no—
247C23×C5—51no—
248C23×C5C22×D511yescharacteristic
249C23×C5C22×D511yescharacteristic
250C23×C5C22×D511yescharacteristic
251C23×C5C22×D511yescharacteristic
252C22×D5—51no—
253C22×D5—51no—
254C22×D5—51no—
255C22×D5—51no—
256C22×D5—102no—
257C22×D5—102no—
258C22×D5—102no—
259C22×D5—102no—
260C22×D5—102no—
261C22×D5—102no—
262C22×D5—102no—
263C22×D5—102no—
264C22×D5—102no—
265C22×D5—102no—
266C22×D5—204no—
267C22×D5—204no—
268C22×D5—201no—
269C22×D5—201no—
270C22×D5—201no—
271C22×D5—201no—
272C22×D5—204no—
273C22×D5C22×D511yescharacteristic
274C22×D5C22×D511yescharacteristic
275C5×Q8—102no—
276C5×D4—22no—
277C5×D4—22no—
278C5×D4—44no—
279C5×D4—54no—
280C5×D4—102no—
281C5×D4—102no—
282C5×D4—201no—
283C2×C4×C5—22no—
284C2×C4×C5—51no—
285C2×C4×C5—51no—
286C2×C4×C5—51no—
287C2×C4×C5—51no—
288C2×C4×C5—51no—
289C2×C4×C5—51no—
290C2×C4×C5—102no—
291C2×C4×C5—102no—
292C2×C4×C5—102no—
293C2×C4×C5—201no—
294C2×C4×C5C22×D511yescharacteristic
295C2×C4×C5C22×D511yescharacteristic
296C22⋊D5—22no—
297C22⋊D5—22no—
298C22⋊D5—41no—
299C22⋊D5—102no—
300C22⋊D5—102no—
301C22⋊D5—102no—
302C22⋊D5—102no—
303C22⋊D5—204no—
304C22⋊D5—201no—
305C22⋊D5—201no—
306C22⋊D5—204no—
307C22⋊D5C22⋊D514yescomplemented
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—41no—
311C2×Dic5—51no—
312C2×Dic5—51no—
313C2×Dic5—51no—
314C2×Dic5—51no—
315C2×Dic5—102no—
316C2×Dic5—102no—
317C2×Dic5—201no—
318C2×Dic5C22×D511yescharacteristic
319C2×Dic5C22×D511yescharacteristic
320C2×Dic5C22×D511yescharacteristic
321C2×Dic5C22×D511yescharacteristic
322C2×Dic5C22×D511yescharacteristic
323C2×Dic5C22×D511yescharacteristic
324D20—102no—
325D20—102no—
326C4×D5—22no—
327C4×D5—102no—
328C4×D5—102no—
329C4×D5—201no—
330Dic10—22no—
331C2×#41—51no—
332C22×D4—251no—
333#124—251no—
334#123—251no—
335#123—251no—
336C22⋊Q8—251no—
337#121—51no—
338#121—251no—
339#121—251no—
340#120—251no—
341#120—251no—
342C4×D4—251no—
343C4×D4—251no—
344C2×C22⋊C4—251no—
345C2×C22⋊C4—251no—
346C52#54611yescharacteristic, complemented, Sylow
347C22×C5—21no—
348C22×C5—21no—
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—21no—
356C22×C5—21no—
357C22×C5—24no—
358C22×C5—22no—
359C22×C5—42no—
360C22×C5—42no—
361C22×C5—42no—
362C22×C5—42no—
363C22×C5—42no—
364C22×C5—52no—
365C22×C5—101no—
366C22×C5—101no—
367C22×C5—101no—
368C22×C5—101no—
369C22×C5—101no—
370C22×C5D4×D512yes—
371C22×C5C2×C22⋊D512yes—
372C22×C5C23×D511yescharacteristic
373C22×C5C23×D511yescharacteristic
374C2×D5—21no—
375C2×D5—21no—
376C2×D5—21no—
377C2×D5—21no—
378C2×D5—21no—
379C2×D5—52no—
380C2×D5—52no—
381C2×D5—104no—
382C2×D5—101no—
383C2×D5—101no—
384C2×D5—101no—
385C2×D5—101no—
386C2×D5—104no—
387C2×D5—101no—
388C2×D5—101no—
389C2×D5—101no—
390C2×D5—101no—
391C2×D5—101no—
392C2×D5—101no—
393C2×D5—202no—
394C2×D5—202no—
395C2×D5—202no—
396C2×D5—202no—
397C2×D5—204no—
398C2×D5—204no—
399C2×D5—204no—
400C2×D5—204no—
401C2×D5—2016no—
402C2×D5C2×C22⋊D512yes—
403C4×C5—21no—
404C4×C5—21no—
405C4×C5—42no—
406C4×C5—52no—
407C4×C5—101no—
408C4×C5—101no—
409C4×C5—101no—
410C4×C5—101no—
411C4×C5—101no—
412Dic5—21no—
413Dic5—21no—
414Dic5—21no—
415Dic5—21no—
416Dic5—21no—
417Dic5—101no—
418Dic5—101no—
419Dic5—101no—
420Dic5—101no—
421Dic5—202no—
422Dic5—202no—
423Dic5C2×C22⋊D512yes—
424C24—251no—
425C24—251no—
426#41—104no—
427C2×Q8—51no—
428C2×D4—51no—
429C2×D4—51no—
430C2×D4—51no—
431C2×D4—51no—
432C2×D4—251no—
433C2×D4—251no—
434C2×D4—251no—
435C2×D4—251no—
436C2×D4—504no—
437C2×D4D5211yescharacteristic, complemented
438C22×C4—51no—
439C22×C4—51no—
440C22×C4—51no—
441C22×C4—51no—
442C22×C4—251no—
443C22×C4—251no—
444C4⋊C4—51no—
445C4⋊C4—251no—
446C4⋊C4—251no—
447C4⋊C4—251no—
448C22⋊C4—51no—
449C22⋊C4—51no—
450C22⋊C4—251no—
451C22⋊C4—251no—
452C22⋊C4—251no—
453C22⋊C4—251no—
454C22⋊C4—251no—
455C22⋊C4—251no—
456C22⋊C4—251no—
457C22⋊C4—251no—
458C22⋊C4—251no—
459C22⋊C4—251no—
460C42—251no—
461C2×C5—22no—
462C2×C5—22no—
463C2×C5—22no—
464C2×C5—22no—
465C2×C5—22no—
466C2×C5—44no—
467C2×C5—41no—
468C2×C5—41no—
469C2×C5—82no—
470C2×C5—102no—
471C2×C5—201no—
472C2×C5C2×D4×D511yescharacteristic
473C2×C5#414411yescharacteristic
474C2×C5D5×#4111yescharacteristic
475C2×C5D5×#4111yescharacteristic
476C2×C5#414911yescharacteristic
477C2×C5C22×C22⋊D511yescharacteristic
478D5—22no—
479D5—41no—
480D5—102no—
481D5—102no—
482D5—204no—
483D5—204no—
484D5—201no—
485D5—201no—
486C23—51no—
487C23—51no—
488C23—251no—
489C23—251no—
490C23—502no—
491C23—502no—
492C23—502no—
493C23—502no—
494C23—1001no—
495C23—1001no—
496C23C2×D5211yescharacteristic
497C23C2×D5211yescharacteristic
498Q8—102no—
499D4—22no—
500D4—54no—
501D4—102no—
502D4—102no—
503D4—201no—
504D4—502no—
505D4—502no—
506D4—502no—
507D4—1001no—
508C2×C4—51no—
509C2×C4—51no—
510C2×C4—51no—
511C2×C4—51no—
512C2×C4—51no—
513C2×C4—51no—
514C2×C4—102no—
515C2×C4—102no—
516C2×C4—102no—
517C2×C4—201no—
518C2×C4—251no—
519C2×C4—251no—
520C2×C4—502no—
521C2×C4—1001no—
522C2×C4C2×D5211yescharacteristic
523C5—22no—
524C5#6594411yescharacteristic, complemented
525C5#6616311yescharacteristic, complemented
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—52no—
532C22—101no—
533C22—101no—
534C22—101no—
535C22—101no—
536C22—101no—
537C22—252no—
538C22—501no—
539C22—501no—
540C22—501no—
541C22—501no—
542C22—501no—
543C22—1004no—
544C22—1004no—
545C22D5×C22⋊D512yes—
546C22C22×D5211yescharacteristic
547C4—21no—
548C4—52no—
549C4—101no—
550C4—101no—
551C4—101no—
552C4—101no—
553C4—101no—
554C4—501no—
555C4—501no—
556C2—22no—
557C2—41no—
558C2—102no—
559C2—201no—
560C2—502no—
561C2—1001no—
562C2#1171747711yescharacteristic
563C2#1171747911yescharacteristic
564C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#711C2semidirect
2. C2×#87886C2semidirect
3. #11717257C2semidirect
4. C5×#4085C2semidirect
5. #11717101C2semidirect
6. #11717096C2semidirect
7. Dic5×C22⋊D5C2semidirect
8. #11717089C2semidirect
9. C2×#87779C2semidirect
10. #11717079C2semidirect
11. #11717070C2semidirect
12. D5×#685C2semidirect
13. #11717025C2semidirect
14. #11717008C2semidirect
15. #11716968C2semidirect
18. C2×C52×D4C22semidirect
20. C2×C22⋊(C5⋊D5)C22semidirect
23. C2×C5×C22⋊D5C22semidirect
24. C2×C5×C22⋊D5C22semidirect
25. C22×C5×Dic5C22semidirect
26. C22×C5×Dic5C22semidirect
27. C22×C5×Dic5C22semidirect
29. C2×C4×C5×D5C22semidirect
30. C2×C5×Dic10C22semidirect
35. C2×D5×Dic5C22semidirect
36. C22⋊#1039C22semidirect
38. #87808C22semidirect
39. C5×C22⋊Dic5C22semidirect
40. C5×#685C22semidirect
41. C5×#683C22semidirect
42. #87781C22semidirect
43. #87781C22semidirect
44. #87780C22semidirect
47. #87779C22semidirect
48. #87779C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
76. C5×C22⋊D5D4semidirect
93. C2×#711D5semidirect
101. #4085D5semidirect
136. C2×C5⋊D5#41semidirect
165. C2×C5×D4C2×D5semidirect
166. C2×C5×D4C2×D5semidirect
181. C2×C22⋊D5C2×D5semidirect
182. C2×C22⋊D5C2×D5semidirect
184. C22×Dic5C2×D5semidirect
185. C22×Dic5C2×D5semidirect
186. C22×Dic5C2×D5semidirect
195. C2×C4×D5C2×D5semidirect
196. C2×Dic10C2×D5semidirect
206. C22⋊Dic5C2×D5semidirect
224. #685C2×D5semidirect
231. #683C2×D5semidirect
307. C22⋊D5C22⋊D5semidirect
346. C52#546semidirect
437. C2×D4D52semidirect
524. C5#65944semidirect
525. C5#66163semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#7112
C2×#878862
#117172572
C5×#40852
#117171012
#117170962
Dic5×C22⋊D52
#117170892
C2×#877792
#117170792
#117170702
D5×#6852
#117170252
#117170082
#117169682
#661635
#659445