Finite Group Databasebeta

← #13106398

Subgroups — #13106398

Order 1600.

832 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#87902C211yesmaximal, characteristic, complemented
2C2×D5×C22⋊D5C211yesmaximal, characteristic, complemented
3C5×#4085C211yesmaximal, characteristic, complemented
4C5×#4083C211yesmaximal, characteristic, complemented
5#11717101C211yesmaximal, characteristic, complemented
6#11717098C211yesmaximal, characteristic, complemented
7#11717091C211yesmaximal, characteristic, complemented
8#11717080C211yesmaximal, characteristic, complemented
9C2×#87778C211yesmaximal, characteristic, complemented
10#11717076C211yesmaximal, characteristic, complemented
11#11717070C211yesmaximal, characteristic, complemented
12#11717039C211yesmaximal, characteristic, complemented
13#11717031C211yesmaximal, characteristic, complemented
14#11717015C211yesmaximal, characteristic, complemented
15#11716971C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C22×D52C2211yescharacteristic
18C2×C52×D4C2211yescharacteristic, complemented
19C2×C22⋊(C5⋊D5)C2211yescharacteristic
20C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
21C22×#1039C2211yescharacteristic, complemented
22C22×#1039C2211yescharacteristic, complemented
23#87902—24no—
24C2×C4×C5⋊D5C2211yescharacteristic, complemented
25C2×#6045C2211yescharacteristic, complemented
26C2×C5×C22⋊D5C2211yescharacteristic
27C2×C5×C22⋊D5C2211yescharacteristic
28C2×C5×C22⋊D5C2211yescharacteristic, complemented
29C2×C5×C22⋊D5C2211yescharacteristic, complemented
30C22×C5×Dic5C2211yescharacteristic, complemented
31D5×C22⋊D5—24no—
32C2×#6038C2211yescharacteristic
33C2×#6037C2211yescharacteristic
34C2×#6036C2211yescharacteristic, complemented
35C2×D5×Dic5C2211yescharacteristic
36C5×C22⋊Dic5C2211yescharacteristic, complemented
37C5×C22⋊Dic5C2211yescharacteristic, complemented
38C5×#685C2211yescharacteristic
39C5×#685C2211yescharacteristic, complemented
40C5×#685C2211yescharacteristic, complemented
41C5×#683C2211yescharacteristic, complemented
42#87781C2211yescharacteristic, complemented
43#87780C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87779C2211yescharacteristic, complemented
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#65944—51nomaximal
54#65925—51nomaximal
55C23×C52C2311yescharacteristic
56C23×C52C2311yescharacteristic
57C22×C5⋊D5C2311yescharacteristic
58C22×C5×D5—22no—
59C22×C5×D5—22no—
60C22×C5×D5—41no—
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)D414yescomplemented
71C2×#1039—22no—
72C2×#1039—22no—
73C2×#1039C2311yescharacteristic
74C2×#1039C2311yescharacteristic
75C2×#1039C2311yescharacteristic
76C4×C5⋊D5—22no—
77#6045—22no—
78C5×C22⋊D5—22no—
79C5×C22⋊D5—22no—
80C5×C22⋊D5—41no—
81C5×C22⋊D5—41no—
82C2×C5×Dic5—41no—
83C2×C5×Dic5C2311yescharacteristic
84C2×C5×Dic5C2311yescharacteristic
85C2×C5×Dic5C2311yescharacteristic
86C2×C5×Dic5C2311yescharacteristic
87C2×C5×Dic5C2311yescharacteristic
88#6038—22no—
89#6037—22no—
90#6036—41no—
91D5×Dic5—22no—
92C22×C22⋊D5—51no—
93C2×#711—51no—
94C2×#711—51no—
95C2×#711—102no—
96C2×D4×D5—51no—
97C5×#121—51no—
98C5×#120—51no—
99C2×C22⋊Dic5—51no—
100#4085—51no—
101#4085—51no—
102#4085D511yescharacteristic, complemented
103#4084—51no—
104#4083—51no—
105#4083—51no—
106#4083D511yescharacteristic, complemented
107#4081—51no—
108D4×Dic5—51no—
109C4×C22⋊D5—51no—
110C2×#685—51no—
111#4042—51no—
112#4040—51no—
113#4032—51no—
114#4031—51no—
115#4031—51no—
116#4030—51no—
117#4029—51no—
118#4027—51no—
119#4027—51no—
120D5×C22⋊C4—51no—
121D5×C22⋊C4—51no—
122#4024—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⋊D5C2×D412yes—
133C2×C5×D5—21no—
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—44no—
142C2×C5×D5#4112yescomplemented
143D52—44no—
144C4×C52—21no—
145#1039—21no—
146#1039—21no—
147#1039C2×D412yes—
148C5×Dic5—21no—
149C5×Dic5—21no—
150C5×Dic5—21no—
151C5×Dic5—21no—
152C5×Dic5—21no—
153C24×C5—51no—
154C23×D5—51no—
155C23×D5—51no—
156C23×D5C2×D511yescharacteristic
157C2×C5×D4—22no—
158C2×C5×D4—51no—
159C2×C5×D4—51no—
160C2×C5×D4—51no—
161C2×C5×D4—51no—
162C2×C5×D4C2×D511yescharacteristic, complemented
163C2×C5×D4C2×D511yescharacteristic, complemented
164C22×C4×C5—51no—
165C2×C22⋊D5—51no—
166C2×C22⋊D5—51no—
167C2×C22⋊D5—51no—
168C2×C22⋊D5—51no—
169C2×C22⋊D5—51no—
170C2×C22⋊D5—51no—
171C2×C22⋊D5—51no—
172C2×C22⋊D5—102no—
173C2×C22⋊D5—102no—
174C2×C22⋊D5—104no—
175C2×C22⋊D5C2×D511yescharacteristic
176C2×C22⋊D5C2×D511yescharacteristic
177C2×C22⋊D5C2×D511yescharacteristic, complemented
178C2×C22⋊D5C2×D511yescharacteristic, complemented
179C22×Dic5—51no—
180C22×Dic5—51no—
181C22×Dic5—51no—
182C22×Dic5—51no—
183C22×Dic5—51no—
184C22×Dic5—102no—
185C22×Dic5—102no—
186C22×Dic5C2×D511yescharacteristic, complemented
187#711—104no—
188#711—104no—
189#711—1016no—
190D4×D5—104no—
191C2×D20—51no—
192C2×C4×D5—51no—
193C2×C4×D5—51no—
194C2×C4×D5—51no—
195C2×C4×D5—51no—
196C2×C4×D5—51no—
197C2×C4×D5—102no—
198C2×Dic10—51no—
199C2×Dic10—51no—
200C2×Dic10—102no—
201C5×C4⋊C4—51no—
202C5×C22⋊C4—51no—
203C5×C22⋊C4—51no—
204C5×C22⋊C4—51no—
205C5×C22⋊C4—51no—
206C5×C22⋊C4—51no—
207C22⋊Dic5—51no—
208C22⋊Dic5—51no—
209C22⋊Dic5—51no—
210C22⋊Dic5—51no—
211C22⋊Dic5—51no—
212C22⋊Dic5—51no—
213C22⋊Dic5C2×D511yescharacteristic, complemented
214C22⋊Dic5C2×D511yescharacteristic, complemented
215#685—51no—
216#685—51no—
217#685—51no—
218#685—51no—
219#685—51no—
220#685—51no—
221#685—51no—
222#685—51no—
223#685C2×D511yescharacteristic
224#685C2×D511yescharacteristic, complemented
225#685C2×D511yescharacteristic, complemented
226C4⋊Dic5—51no—
227C4⋊Dic5—51no—
228#683—51no—
229#683—51no—
230#683—51no—
231#683—51no—
232#683C2×D511yescharacteristic, complemented
233C4×Dic5—51no—
234C4×Dic5—51no—
235#546—251noSylow
236C2×C52—22no—
237C2×C52—41no—
238C2×C52C22×D411yescharacteristic
239C2×C52C2×#4111yescharacteristic
240C2×C52H(5,𝔽2)11yescharacteristic
241C5⋊D5—22no—
242C5×D5—22no—
243C5×D5—41no—
244C5×D5—41no—
245C23×C5—22no—
246C23×C5—22no—
247C23×C5—51no—
248C23×C5—51no—
249C23×C5—51no—
250C23×C5—102no—
251C23×C5—102no—
252C23×C5—201no—
253C23×C5C22×D511yescharacteristic
254C23×C5C22×D511yescharacteristic
255C23×C5C22×D511yescharacteristic
256C23×C5C22×D511yescharacteristic
257C22×D5—22no—
258C22×D5—22no—
259C22×D5—41no—
260C22×D5—51no—
261C22×D5—51no—
262C22×D5—102no—
263C22×D5—102no—
264C22×D5—102no—
265C22×D5—102no—
266C22×D5—102no—
267C22×D5—201no—
268C22×D5—201no—
269C22×D5C22×D511yescharacteristic
270C22×D5C22×D511yescharacteristic
271C22×D5C22×D511yescharacteristic
272C5×D4—28no—
273C5×D4—22no—
274C5×D4—22no—
275C5×D4—102no—
276C5×D4—102no—
277C5×D4—201no—
278C5×D4—201no—
279C2×C4×C5—22no—
280C2×C4×C5—51no—
281C2×C4×C5—51no—
282C2×C4×C5—51no—
283C2×C4×C5—51no—
284C2×C4×C5—51no—
285C2×C4×C5—201no—
286C2×C4×C5C22×D511yescharacteristic
287C2×C4×C5C22×D511yescharacteristic
288C22⋊D5—22no—
289C22⋊D5—22no—
290C22⋊D5—41no—
291C22⋊D5—41no—
292C22⋊D5—54no—
293C22⋊D5—54no—
294C22⋊D5—108no—
295C22⋊D5—102no—
296C22⋊D5—102no—
297C22⋊D5—102no—
298C22⋊D5—102no—
299C22⋊D5—102no—
300C22⋊D5—108no—
301C2×Dic5—41no—
302C2×Dic5—51no—
303C2×Dic5—51no—
304C2×Dic5—51no—
305C2×Dic5—51no—
306C2×Dic5—51no—
307C2×Dic5—51no—
308C2×Dic5—102no—
309C2×Dic5—102no—
310C2×Dic5—102no—
311C2×Dic5—102no—
312C2×Dic5—102no—
313C2×Dic5—102no—
314C2×Dic5—108no—
315C2×Dic5—108no—
316C2×Dic5—102no—
317C2×Dic5—102no—
318C2×Dic5C22×D511yescharacteristic
319C2×Dic5C22×D511yescharacteristic
320C2×Dic5C22×D511yescharacteristic
321C2×Dic5C22×D511yescharacteristic
322C2×Dic5C22×D511yescharacteristic
323D20—102no—
324C4×D5—102no—
325C4×D5—102no—
326C4×D5—102no—
327C4×D5—108no—
328C4×D5—201no—
329C4×D5—201no—
330Dic10—102no—
331Dic10—102no—
332Dic10—108no—
333C2×#41—251no—
334C22×D4—251no—
335#124—251no—
336#123—251no—
337#123—251no—
338C22⋊Q8—251no—
339#121—51no—
340#121—251no—
341#121—251no—
342#120—51no—
343#120—251no—
344C4×D4—251no—
345C4×D4—251no—
346C2×C22⋊C4—251no—
347C2×C22⋊C4—251no—
348C52#54611yescharacteristic, complemented, Sylow
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—24no—
354C22×C5—21no—
355C22×C5—21no—
356C22×C5—21no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5—21no—
360C22×C5—24no—
361C22×C5—22no—
362C22×C5—42no—
363C22×C5—42no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5—52no—
367C22×C5—101no—
368C22×C5—101no—
369C22×C5—101no—
370C22×C5—101no—
371C22×C5—101no—
372C22×C5—101no—
373C22×C5—101no—
374C22×C5—101no—
375C22×C5—204no—
376C22×C5D4×D512yes—
377C22×C5D4×D512yes—
378C22×C5C23×D511yescharacteristic
379C22×C5C23×D511yescharacteristic
380C2×D5—21no—
381C2×D5—21no—
382C2×D5—21no—
383C2×D5—21no—
384C2×D5—21no—
385C2×D5—21no—
386C2×D5—21no—
387C2×D5—21no—
388C2×D5—44no—
389C2×D5—52no—
390C2×D5—52no—
391C2×D5—104no—
392C2×D5—101no—
393C2×D5—101no—
394C2×D5—101no—
395C2×D5—101no—
396C2×D5—202no—
397C2×D5—202no—
398C2×D5—204no—
399C2×D5—204no—
400C2×D5#71112yescomplemented
401C4×C5—24no—
402C4×C5—21no—
403C4×C5—21no—
404C4×C5—101no—
405C4×C5—101no—
406C4×C5—101no—
407C4×C5—101no—
408C4×C5—101no—
409Dic5—21no—
410Dic5—21no—
411Dic5—21no—
412Dic5—21no—
413Dic5—21no—
414Dic5—52no—
415Dic5—52no—
416Dic5—104no—
417Dic5—104no—
418Dic5—104no—
419Dic5—101no—
420Dic5—101no—
421Dic5—101no—
422Dic5—101no—
423C24—51no—
424C24—251no—
425#41—504no—
426C2×Q8—251no—
427C2×D4—51no—
428C2×D4—51no—
429C2×D4—51no—
430C2×D4—51no—
431C2×D4—251no—
432C2×D4—251no—
433C2×D4—251no—
434C2×D4—251no—
435C2×D4—504no—
436C2×D4D5211yescharacteristic, complemented
437C22×C4—51no—
438C22×C4—251no—
439C22×C4—251no—
440C22×C4—251no—
441C22×C4—251no—
442C22×C4—251no—
443C4⋊C4—51no—
444C4⋊C4—251no—
445C4⋊C4—251no—
446C4⋊C4—251no—
447C22⋊C4—51no—
448C22⋊C4—51no—
449C22⋊C4—51no—
450C22⋊C4—51no—
451C22⋊C4—51no—
452C22⋊C4—251no—
453C22⋊C4—251no—
454C22⋊C4—251no—
455C22⋊C4—251no—
456C22⋊C4—251no—
457C22⋊C4—251no—
458C22⋊C4—251no—
459C42—251no—
460C2×C5—22no—
461C2×C5—22no—
462C2×C5—22no—
463C2×C5—22no—
464C2×C5—22no—
465C2×C5—44no—
466C2×C5—44no—
467C2×C5—41no—
468C2×C5—41no—
469C2×C5—102no—
470C2×C5—201no—
471C2×C5—201no—
472C2×C5C2×D4×D511yescharacteristic
473C2×C5C2×D4×D511yescharacteristic
474C2×C5C2×#71111yescharacteristic
475C2×C5#414411yescharacteristic
476C2×C5#414411yescharacteristic
477C2×C5D5×#4111yescharacteristic
478D5—22no—
479D5—41no—
480D5—41no—
481D5—102no—
482D5—102no—
483D5—204no—
484C23—51no—
485C23—51no—
486C23—51no—
487C23—102no—
488C23—102no—
489C23—201no—
490C23—251no—
491C23—502no—
492C23—502no—
493C23—1001no—
494C23C2×D5211yescharacteristic
495C23C2×D5211yescharacteristic
496Q8—502no—
497D4—22no—
498D4—102no—
499D4—102no—
500D4—201no—
501D4—201no—
502D4—254no—
503D4—502no—
504D4—502no—
505D4—502no—
506C2×C4—51no—
507C2×C4—51no—
508C2×C4—51no—
509C2×C4—51no—
510C2×C4—51no—
511C2×C4—201no—
512C2×C4—251no—
513C2×C4—251no—
514C2×C4—251no—
515C2×C4—502no—
516C2×C4—502no—
517C2×C4—502no—
518C2×C4—502no—
519C2×C4—1001no—
520C2×C4C2×D5211yescharacteristic
521C5—22no—
522C5#6592511yescharacteristic, complemented
523C5#6594411yescharacteristic, complemented
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—52no—
530C22—101no—
531C22—101no—
532C22—101no—
533C22—101no—
534C22—101no—
535C22—101no—
536C22—101no—
537C22—101no—
538C22—204no—
539C22—252no—
540C22—501no—
541C22—501no—
542C22—1004no—
543C22#8788612yes—
544C22C22×D5211yescharacteristic
545C4—21no—
546C4—101no—
547C4—101no—
548C4—101no—
549C4—101no—
550C4—101no—
551C4—252no—
552C4—501no—
553C4—501no—
554C2—22no—
555C2—41no—
556C2—102no—
557C2—201no—
558C2—201no—
559C2—502no—
560C2D5×#71111yescharacteristic
561C2#1171747811yescharacteristic
562C2C2×#8788611yescharacteristic
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×#87902C2semidirect
2. C2×D5×C22⋊D5C2semidirect
3. C5×#4085C2semidirect
4. C5×#4083C2semidirect
5. #11717101C2semidirect
6. #11717098C2semidirect
7. #11717091C2semidirect
8. #11717080C2semidirect
9. C2×#87778C2semidirect
10. #11717076C2semidirect
11. #11717070C2semidirect
12. #11717039C2semidirect
13. #11717031C2semidirect
14. #11717015C2semidirect
15. #11716971C2semidirect
18. C2×C52×D4C22semidirect
20. C2×C22⋊(C5⋊D5)C22semidirect
21. C22×#1039C22semidirect
22. C22×#1039C22semidirect
24. C2×C4×C5⋊D5C22semidirect
25. C2×#6045C22semidirect
28. C2×C5×C22⋊D5C22semidirect
29. C2×C5×C22⋊D5C22semidirect
30. C22×C5×Dic5C22semidirect
34. C2×#6036C22semidirect
36. C5×C22⋊Dic5C22semidirect
37. C5×C22⋊Dic5C22semidirect
39. C5×#685C22semidirect
40. C5×#685C22semidirect
41. C5×#683C22semidirect
42. #87781C22semidirect
43. #87780C22semidirect
44. #87780C22semidirect
45. #87779C22semidirect
48. #87778C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
70. C22⋊(C5⋊D5)D4semidirect
102. #4085D5semidirect
106. #4083D5semidirect
142. C2×C5×D5#41semidirect
162. C2×C5×D4C2×D5semidirect
163. C2×C5×D4C2×D5semidirect
177. C2×C22⋊D5C2×D5semidirect
178. C2×C22⋊D5C2×D5semidirect
186. C22×Dic5C2×D5semidirect
213. C22⋊Dic5C2×D5semidirect
214. C22⋊Dic5C2×D5semidirect
224. #685C2×D5semidirect
225. #685C2×D5semidirect
232. #683C2×D5semidirect
348. C52#546semidirect
400. C2×D5#711semidirect
436. C2×D4D52semidirect
522. C5#65925semidirect
523. C5#65944semidirect

Maximal subgroups

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

subgroupindex
C2×#879022
C2×D5×C22⋊D52
C5×#40852
C5×#40832
#117171012
#117170982
#117170912
#117170802
C2×#877782
#117170762
#117170702
#117170392
#117170312
#117170152
#117169712
#659445
#659255