Finite Group Databasebeta

← #13106159

Subgroups — #13106159

Order 1600.

800 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#709C211yesmaximal, characteristic, complemented
2C2×#87886C211yesmaximal, characteristic, complemented
3C2×#87808C211yesmaximal, characteristic, complemented
4C5×#4075C211yesmaximal, characteristic, complemented
5#11717099C211yesmaximal, characteristic, complemented
6#11717098C211yesmaximal, characteristic, complemented
7#11717091C211yesmaximal, characteristic, complemented
8#11717040C211yesmaximal, characteristic, complemented
9D5×#685C211yesmaximal, characteristic, complemented
10#11717025C211yesmaximal, characteristic, complemented
11#11717024C211yesmaximal, characteristic, complemented
12#11717022C211yesmaximal, characteristic, complemented
13#11717008C211yesmaximal, characteristic, complemented
14#11717005C211yesmaximal, characteristic, complemented
15#11716968C211yesmaximal, characteristic, complemented
16C23×C5⋊D5C2211yescharacteristic
17C22×D52C2211yescharacteristic
18C22×C4×C52C2211yescharacteristic, complemented
19C22×#1039C2211yescharacteristic, complemented
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C5×C22⋊D5C2211yescharacteristic, complemented
23C5×#709—24no—
24C2×C5×D20C2211yescharacteristic, complemented
25C2×C4×C5×D5C2211yescharacteristic, complemented
26C2×C4×C5×D5C2211yescharacteristic, complemented
27C2×C5×Dic10C2211yescharacteristic, complemented
28#87886—24no—
29C2×#6038C2211yescharacteristic
30C2×#6038C2211yescharacteristic
31C2×#6038C2211yescharacteristic
32C2×#6038C2211yescharacteristic, complemented
33C2×#6037C2211yescharacteristic, complemented
34C2×#6036C2211yescharacteristic
35C2×D5×Dic5C2211yescharacteristic, complemented
36#87808C2211yescharacteristic
37#87808C2211yescharacteristic
38#87808C2211yescharacteristic, complemented
39#87808C2211yescharacteristic, complemented
40C5×C22⋊Dic5C2211yescharacteristic, complemented
41C5×#685C2211yescharacteristic, complemented
42C5×#685C2211yescharacteristic, complemented
43C5×#683C2211yescharacteristic, complemented
44C5×#683C2211yescharacteristic, complemented
45#87781C2211yescharacteristic, complemented
46#87780C2211yescharacteristic, complemented
47#87779C2211yescharacteristic
48#87779C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#66163—51nomaximal
54#65988—51nomaximal
55C23×C52C2311yescharacteristic
56C22×C5⋊D5—22no—
57C22×C5⋊D5—22no—
58C22×C5⋊D5—41no—
59C22×C5⋊D5C2311yescharacteristic
60C22×C5⋊D5C2311yescharacteristic
61C22×C5×D5C2311yescharacteristic
62C22×C5×D5C2311yescharacteristic
63C22×C5×D5C2311yescharacteristic
64C2×D52—22no—
65C2×D52—22no—
66C2×D52—41no—
67C2×C4×C52—22no—
68C2×C4×C52C2311yescharacteristic
69C2×C4×C52C2311yescharacteristic
70C2×#1039—41no—
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C5×C22⋊D5—22no—
74C5×C22⋊D5—22no—
75C5×C22⋊D5D414yescomplemented
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78C2×C5×Dic5C2311yescharacteristic
79C2×C5×Dic5C2311yescharacteristic
80C2×C5×Dic5C2311yescharacteristic
81C5×D20—22no—
82C4×C5×D5—22no—
83C4×C5×D5—22no—
84C5×Dic10—22no—
85#6038—22no—
86#6038—22no—
87#6038—22no—
88#6038—41no—
89#6037—41no—
90#6036—22no—
91D5×Dic5—41no—
92C2×C5×#41—51no—
93C2×D4×D5—51no—
94C2×D4×D5—51no—
95C2×#709D511yescharacteristic, complemented
96C5×#123—51no—
97#4093—51no—
98#4092—51no—
99#4085—51no—
100#4085—51no—
101#4085—51no—
102#4083—51no—
103#4083—51no—
104D4×Dic5—51no—
105#4076—51no—
106#4075—51no—
107#4075D511yescharacteristic, complemented
108C4×C22⋊D5—51no—
109C2×#685—51no—
110C2×#685—51no—
111C2×#685—51no—
112C2×#685—102no—
113#4042—51no—
114#4041—51no—
115#4040—51no—
116#4039—51no—
117#4031—51no—
118#4030—51no—
119#4028—51no—
120#4028—51no—
121#4027—51no—
122D5×C22⋊C4—51no—
123C22×C52—21no—
124C22×C52—21no—
125C22×C52C2×D412yes—
126C22×C52C2411yescharacteristic
127C2×C5⋊D5—21no—
128C2×C5⋊D5—21no—
129C2×C5⋊D5—21no—
130C2×C5⋊D5—21no—
131C2×C5⋊D5—21no—
132C2×C5⋊D5—44no—
133C2×C5⋊D5#4112yescomplemented
134C2×C5×D5—21no—
135C2×C5×D5—21no—
136C2×C5×D5—21no—
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—
145C4×C52—21no—
146#1039—21no—
147#1039—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—51no—
161C2×C5×D4—51no—
162C2×C5×D4—51no—
163C2×C5×D4—51no—
164C22×C4×C5—22no—
165C22×C4×C5—51no—
166C22×C4×C5—51no—
167C22×C4×C5C2×D511yescharacteristic, complemented
168C22×C4×C5C2×D511yescharacteristic, complemented
169C2×C22⋊D5—51no—
170C2×C22⋊D5—51no—
171C2×C22⋊D5—51no—
172C2×C22⋊D5—51no—
173C2×C22⋊D5—51no—
174C2×C22⋊D5—51no—
175C2×C22⋊D5C2×D511yescharacteristic
176C2×C22⋊D5C2×D511yescharacteristic
177C2×C22⋊D5C2×D511yescharacteristic, complemented
178C22×Dic5—51no—
179C22×Dic5—51no—
180C22×Dic5—51no—
181C22×Dic5—102no—
182D4×D5—104no—
183D4×D5—104no—
184#709—24no—
185C2×D20—51no—
186C2×D20—51no—
187C2×D20—51no—
188C2×D20—51no—
189C2×D20C2×D511yescharacteristic, complemented
190C2×C4×D5—51no—
191C2×C4×D5—51no—
192C2×C4×D5—51no—
193C2×C4×D5C2×D511yescharacteristic, complemented
194C2×C4×D5C2×D511yescharacteristic, complemented
195C2×Dic10C2×D511yescharacteristic, complemented
196C5×C4⋊C4—51no—
197C5×C4⋊C4—51no—
198C5×C22⋊C4—51no—
199C5×C22⋊C4—51no—
200C5×C22⋊C4—51no—
201C22⋊Dic5—51no—
202C22⋊Dic5—51no—
203C22⋊Dic5—51no—
204C22⋊Dic5C2×D511yescharacteristic, complemented
205#685—51no—
206#685—51no—
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—102no—
221#685—102no—
222#685—102no—
223#685—102no—
224#685C2×D511yescharacteristic, complemented
225#685C2×D511yescharacteristic, complemented
226C4⋊Dic5—51no—
227#683—51no—
228#683—51no—
229#683—51no—
230#683C2×D511yescharacteristic, complemented
231#683C2×D511yescharacteristic, complemented
232C4×Dic5—51no—
233C4×Dic5—51no—
234#546—251noSylow
235C2×C52—22no—
236C2×C52C22×D411yescharacteristic
237C2×C52C2×#4111yescharacteristic
238C2×C52H(5,𝔽2)11yescharacteristic
239C5⋊D5—22no—
240C5⋊D5—41no—
241C5×D5—22no—
242C5×D5—41no—
243C5×D5—41no—
244C23×C5—22no—
245C23×C5—51no—
246C23×C5—51no—
247C23×C5—51no—
248C23×C5C22×D511yescharacteristic
249C23×C5C22×D511yescharacteristic
250C22×D5—51no—
251C22×D5—51no—
252C22×D5—51no—
253C22×D5—51no—
254C22×D5—102no—
255C22×D5—102no—
256C22×D5—102no—
257C22×D5—102no—
258C22×D5—102no—
259C22×D5—102no—
260C22×D5—102no—
261C22×D5—108no—
262C22×D5—102no—
263C22×D5—102no—
264C22×D5—102no—
265C22×D5—204no—
266C22×D5—201no—
267C22×D5—201no—
268C22×D5—201no—
269C22×D5—201no—
270C22×D5—204no—
271C22×D5C22×D511yescharacteristic
272C22×D5C22×D511yescharacteristic
273C22×D5C22×D511yescharacteristic
274C5×Q8—102no—
275C5×D4—54no—
276C5×D4—102no—
277C5×D4—102no—
278C5×D4—102no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—22no—
282C2×C4×C5—22no—
283C2×C4×C5—44no—
284C2×C4×C5—51no—
285C2×C4×C5—51no—
286C2×C4×C5—51no—
287C2×C4×C5—51no—
288C2×C4×C5—51no—
289C2×C4×C5—102no—
290C2×C4×C5—102no—
291C2×C4×C5C22×D511yescharacteristic
292C2×C4×C5C22×D511yescharacteristic
293C2×C4×C5C22×D511yescharacteristic
294C2×C4×C5C22×D511yescharacteristic
295C22⋊D5—22no—
296C22⋊D5—22no—
297C22⋊D5—102no—
298C22⋊D5—102no—
299C22⋊D5—102no—
300C22⋊D5—201no—
301C22⋊D5—201no—
302C22⋊D5—201no—
303C22⋊D5C22⋊D514yescomplemented
304C2×Dic5—51no—
305C2×Dic5—51no—
306C2×Dic5—51no—
307C2×Dic5—51no—
308C2×Dic5—102no—
309C2×Dic5—102no—
310C2×Dic5—201no—
311C2×Dic5—201no—
312C2×Dic5—201no—
313C2×Dic5—204no—
314C2×Dic5C22×D511yescharacteristic
315C2×Dic5C22×D511yescharacteristic
316C2×Dic5C22×D511yescharacteristic
317C2×Dic5C22×D511yescharacteristic
318C2×Dic5C22×D511yescharacteristic
319D20—22no—
320D20—102no—
321D20—102no—
322D20—102no—
323D20—201no—
324C4×D5—22no—
325C4×D5—22no—
326C4×D5—102no—
327C4×D5—102no—
328C4×D5—201no—
329Dic10—22no—
330C2×#41—51no—
331C22×D4—251no—
332#124—251no—
333#123—51no—
334#123—251no—
335C22⋊Q8—251no—
336#121—251no—
337#121—251no—
338#121—251no—
339#120—251no—
340#120—251no—
341C4×D4—251no—
342C4×D4—251no—
343C2×C22⋊C4—251no—
344C2×C22⋊C4—251no—
345C52#54611yescharacteristic, complemented, Sylow
346C22×C5—24no—
347C22×C5—24no—
348C22×C5—21no—
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—21no—
352C22×C5—24no—
353C22×C5—22no—
354C22×C5—52no—
355C22×C5—101no—
356C22×C5—101no—
357C22×C5—101no—
358C22×C5—101no—
359C22×C5—101no—
360C22×C5—101no—
361C22×C5—101no—
362C22×C5—101no—
363C22×C5D4×D512yes—
364C22×C5C2×C22⋊D512yes—
365C22×C5C23×D511yescharacteristic
366C22×C5C23×D511yescharacteristic
367C2×D5—21no—
368C2×D5—21no—
369C2×D5—21no—
370C2×D5—21no—
371C2×D5—21no—
372C2×D5—21no—
373C2×D5—21no—
374C2×D5—21no—
375C2×D5—52no—
376C2×D5—52no—
377C2×D5—104no—
378C2×D5—101no—
379C2×D5—101no—
380C2×D5—101no—
381C2×D5—101no—
382C2×D5—104no—
383C2×D5—101no—
384C2×D5—101no—
385C2×D5—101no—
386C2×D5—101no—
387C2×D5—101no—
388C2×D5—101no—
389C2×D5—202no—
390C2×D5—202no—
391C2×D5—202no—
392C2×D5—202no—
393C2×D5—204no—
394C2×D5—204no—
395C2×D5—204no—
396C2×D5—204no—
397C2×D5—2016no—
398C2×D5C2×C22⋊D512yes—
399C4×C5—21no—
400C4×C5—21no—
401C4×C5—21no—
402C4×C5—21no—
403C4×C5—42no—
404C4×C5—42no—
405C4×C5—52no—
406C4×C5—101no—
407C4×C5—101no—
408C4×C5—101no—
409C4×C5—101no—
410Dic5—21no—
411Dic5—21no—
412Dic5—21no—
413Dic5—21no—
414Dic5—101no—
415Dic5—101no—
416Dic5—101no—
417Dic5—101no—
418Dic5—202no—
419Dic5—202no—
420Dic5C2×C22⋊D512yes—
421C24—251no—
422C24—251no—
423#41—104no—
424C2×Q8—51no—
425C2×D4—51no—
426C2×D4—51no—
427C2×D4—51no—
428C2×D4—51no—
429C2×D4—251no—
430C2×D4—251no—
431C2×D4—251no—
432C2×D4—251no—
433C2×D4—251no—
434C2×D4—504no—
435C22×C4—51no—
436C22×C4—51no—
437C22×C4—251no—
438C22×C4—251no—
439C22×C4—251no—
440C22×C4D5211yescharacteristic, complemented
441C4⋊C4—51no—
442C4⋊C4—51no—
443C4⋊C4—251no—
444C4⋊C4—251no—
445C22⋊C4—51no—
446C22⋊C4—51no—
447C22⋊C4—51no—
448C22⋊C4—251no—
449C22⋊C4—251no—
450C22⋊C4—251no—
451C22⋊C4—251no—
452C22⋊C4—251no—
453C22⋊C4—251no—
454C22⋊C4—251no—
455C22⋊C4—251no—
456C22⋊C4—251no—
457C42—251no—
458C2×C5—28no—
459C2×C5—22no—
460C2×C5—22no—
461C2×C5—22no—
462C2×C5—22no—
463C2×C5—22no—
464C2×C5—102no—
465C2×C5—201no—
466C2×C5—201no—
467C2×C5C2×D4×D511yescharacteristic
468C2×C5D5×#4111yescharacteristic
469C2×C5D5×#4111yescharacteristic
470C2×C5#414911yescharacteristic
471C2×C5#414911yescharacteristic
472C2×C5C22×C22⋊D511yescharacteristic
473D5—22no—
474D5—41no—
475D5—41no—
476D5—102no—
477D5—102no—
478D5—204no—
479D5—204no—
480D5—201no—
481D5—201no—
482C23—51no—
483C23—51no—
484C23—51no—
485C23—251no—
486C23—251no—
487C23—502no—
488C23—502no—
489C23—502no—
490C23—502no—
491C23—1001no—
492C23—1001no—
493C23C2×D5211yescharacteristic
494Q8—102no—
495D4—54no—
496D4—102no—
497D4—102no—
498D4—102no—
499D4—502no—
500D4—502no—
501D4—502no—
502D4—1001no—
503D4—1001no—
504C2×C4—22no—
505C2×C4—51no—
506C2×C4—51no—
507C2×C4—51no—
508C2×C4—51no—
509C2×C4—51no—
510C2×C4—102no—
511C2×C4—102no—
512C2×C4—251no—
513C2×C4—251no—
514C2×C4—502no—
515C2×C4—1001no—
516C2×C4—1001no—
517C2×C4C2×D5211yescharacteristic
518C2×C4C2×D5211yescharacteristic
519C5—22no—
520C5#6598811yescharacteristic, complemented
521C5#6616311yescharacteristic, complemented
522C22—21no—
523C22—21no—
524C22—52no—
525C22—101no—
526C22—101no—
527C22—101no—
528C22—101no—
529C22—101no—
530C22—101no—
531C22—101no—
532C22—101no—
533C22—252no—
534C22—501no—
535C22—501no—
536C22—501no—
537C22—501no—
538C22—501no—
539C22—1004no—
540C22—1004no—
541C22D5×C22⋊D512yes—
542C22C22×D5211yescharacteristic
543C4—21no—
544C4—21no—
545C4—52no—
546C4—101no—
547C4—101no—
548C4—101no—
549C4—101no—
550C4—501no—
551C4—501no—
552C2—22no—
553C2—102no—
554C2—201no—
555C2—201no—
556C2—502no—
557C2—1001no—
558C2#1171747111yescharacteristic
559C2#1171747311yescharacteristic
560C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
hypercenterC22×C4D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×#709C2semidirect
2. C2×#87886C2semidirect
3. C2×#87808C2semidirect
4. C5×#4075C2semidirect
5. #11717099C2semidirect
6. #11717098C2semidirect
7. #11717091C2semidirect
8. #11717040C2semidirect
9. D5×#685C2semidirect
10. #11717025C2semidirect
11. #11717024C2semidirect
12. #11717022C2semidirect
13. #11717008C2semidirect
14. #11717005C2semidirect
15. #11716968C2semidirect
18. C22×C4×C52C22semidirect
19. C22×#1039C22semidirect
22. C2×C5×C22⋊D5C22semidirect
24. C2×C5×D20C22semidirect
25. C2×C4×C5×D5C22semidirect
26. C2×C4×C5×D5C22semidirect
27. C2×C5×Dic10C22semidirect
32. C2×#6038C22semidirect
33. C2×#6037C22semidirect
35. C2×D5×Dic5C22semidirect
38. #87808C22semidirect
39. #87808C22semidirect
40. C5×C22⋊Dic5C22semidirect
41. C5×#685C22semidirect
42. C5×#685C22semidirect
43. C5×#683C22semidirect
44. C5×#683C22semidirect
45. #87781C22semidirect
46. #87780C22semidirect
48. #87779C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
75. C5×C22⋊D5D4semidirect
95. C2×#709D5semidirect
107. #4075D5semidirect
133. C2×C5⋊D5#41semidirect
167. C22×C4×C5C2×D5semidirect
168. C22×C4×C5C2×D5semidirect
177. C2×C22⋊D5C2×D5semidirect
189. C2×D20C2×D5semidirect
193. C2×C4×D5C2×D5semidirect
194. C2×C4×D5C2×D5semidirect
195. C2×Dic10C2×D5semidirect
204. C22⋊Dic5C2×D5semidirect
224. #685C2×D5semidirect
225. #685C2×D5semidirect
230. #683C2×D5semidirect
231. #683C2×D5semidirect
303. C22⋊D5C22⋊D5semidirect
345. C52#546semidirect
440. C22×C4D52semidirect
520. C5#65988semidirect
521. C5#66163semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#7092
C2×#878862
C2×#878082
C5×#40752
#117170992
#117170982
#117170912
#117170402
D5×#6852
#117170252
#117170242
#117170222
#117170082
#117170052
#117169682
#661635
#659885