Finite Group Databasebeta

← #13106586

Subgroups — #13106586

Order 1600.

904 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C2×#87879C211yesmaximal, characteristic, complemented
3C5×#4099C211yesmaximal, characteristic, complemented
4C2×C5×C22⋊Dic5C211yesmaximal, characteristic, complemented
5#11717100C211yesmaximal, characteristic, complemented
6#11717099C211yesmaximal, characteristic, complemented
7#11717096C211yesmaximal, characteristic, complemented
8#11717093C211yesmaximal, characteristic, complemented
9#11717092C211yesmaximal, characteristic, complemented
10#11717091C211yesmaximal, characteristic, complemented
11#11717089C211yesmaximal, characteristic, complemented
12#11717084C211yesmaximal, characteristic, complemented
13#11717076C211yesmaximal, characteristic, complemented
14#11717071C211yesmaximal, characteristic, complemented
15#11717070C211yesmaximal, characteristic, complemented
16C24×C52C2211yescharacteristic
17C23×C5⋊D5C2211yescharacteristic
18C2×C22⋊(C5⋊D5)—24no—
19C2×C22⋊(C5⋊D5)C2211yescharacteristic
20C2×C22⋊(C5⋊D5)C2211yescharacteristic
21C2×C22⋊(C5⋊D5)C2211yescharacteristic
22C2×C22⋊(C5⋊D5)C2211yescharacteristic
23C22×#1039C2211yescharacteristic
24C2×C5×C22⋊D5C2211yescharacteristic
25C2×C5×C22⋊D5C2211yescharacteristic, complemented
26C2×C5×C22⋊D5C2211yescharacteristic, complemented
27C22×C5×Dic5C2211yescharacteristic, complemented
28C22×C5×Dic5C2211yescharacteristic, complemented
29C2×#6039C2211yescharacteristic, complemented
30C2×#6038C2211yescharacteristic, complemented
31C2×#6038C2211yescharacteristic, complemented
32C2×#6036C2211yescharacteristic, complemented
33C2×#6036C2211yescharacteristic, complemented
34#87879—24no—
35C2×D5×Dic5C2211yescharacteristic, complemented
36C5×C22⋊Dic5C2211yescharacteristic
37C5×C22⋊Dic5C2211yescharacteristic
38C5×C22⋊Dic5C2211yescharacteristic
39C5×C22⋊Dic5C2211yescharacteristic, complemented
40C5×C22⋊Dic5C2211yescharacteristic, complemented
41C5×C22⋊Dic5C2211yescharacteristic, complemented
42C5×C22⋊Dic5C2211yescharacteristic, complemented
43#87781C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87780C2211yescharacteristic, complemented
46#87780C2211yescharacteristic, complemented
47#87779C2211yescharacteristic, complemented
48#87779C2211yescharacteristic, complemented
49#87779C2211yescharacteristic, complemented
50#87779C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#65925—51nomaximal
54#65824—51nomaximal
55C23×C52—22no—
56C23×C52—22no—
57C23×C52—41no—
58C23×C52C2311yescharacteristic
59C23×C52C2311yescharacteristic
60C23×C52C2311yescharacteristic
61C22×C5⋊D5—22no—
62C22×C5⋊D5—22no—
63C22×C5⋊D5—41no—
64C22×C5⋊D5C2311yescharacteristic
65C22×C5⋊D5C2311yescharacteristic
66C22×C5×D5C2311yescharacteristic
67C22⋊(C5⋊D5)—22no—
68C22⋊(C5⋊D5)—22no—
69C22⋊(C5⋊D5)—22no—
70C22⋊(C5⋊D5)D414yescomplemented
71C2×#1039—22no—
72C2×#1039C2311yescharacteristic
73C2×#1039C2311yescharacteristic
74C5×C22⋊D5—22no—
75C5×C22⋊D5—22no—
76C5×C22⋊D5—41no—
77C2×C5×Dic5—22no—
78C2×C5×Dic5—41no—
79C2×C5×Dic5C2311yescharacteristic
80C2×C5×Dic5C2311yescharacteristic
81C2×C5×Dic5C2311yescharacteristic
82C2×C5×Dic5C2311yescharacteristic
83C2×C5×Dic5C2311yescharacteristic
84C2×C5×Dic5C2311yescharacteristic
85C2×C5×Dic5C2311yescharacteristic
86#6039—22no—
87#6038—22no—
88#6038—41no—
89#6036—22no—
90#6036—41no—
91D5×Dic5—22no—
92C22×C22⋊D5—51no—
93C22×C22⋊D5—51no—
94C22×C22⋊D5—102no—
95C2×#711—51no—
96C2×#709—51no—
97C5×#120—51no—
98C2×C5×C22⋊C4—51no—
99#4099D511yescharacteristic, complemented
100C2×C22⋊Dic5D511yescharacteristic, complemented
101#4085—51no—
102#4085—51no—
103#4084—51no—
104#4083—51no—
105D4×Dic5—51no—
106#4076—51no—
107#4075—51no—
108C4×C22⋊D5—51no—
109#4032—51no—
110#4031—51no—
111#4031—51no—
112#4030—51no—
113#4030—51no—
114#4029—51no—
115#4029—51no—
116#4028—51no—
117#4027—51no—
118#4027—51no—
119D5×C22⋊C4—51no—
120D5×C22⋊C4—51no—
121#4024—51no—
122#4024—51no—
123C22×C52—21no—
124C22×C52—21no—
125C22×C52—21no—
126C22×C52—21no—
127C22×C52—21no—
128C22×C52—21no—
129C22×C52—21no—
130C22×C52—44no—
131C22×C52C2×D412yes—
132C22×C52#4112yescomplemented
133C22×C52C2411yescharacteristic
134C2×C5⋊D5—21no—
135C2×C5⋊D5—21no—
136C2×C5⋊D5—21no—
137C2×C5⋊D5—21no—
138C2×C5⋊D5—21no—
139C2×C5⋊D5—44no—
140C2×C5⋊D5C2×D412yes—
141C2×C5×D5—21no—
142C2×C5×D5—21no—
143C2×C5×D5—21no—
144#1039—21no—
145#1039C2×D412yes—
146C5×Dic5—21no—
147C5×Dic5—21no—
148C5×Dic5—21no—
149C5×Dic5—21no—
150C5×Dic5—21no—
151C5×Dic5—21no—
152C5×Dic5—21no—
153C24×C5—22no—
154C24×C5C2×D511yescharacteristic
155C24×C5C2×D511yescharacteristic
156C23×D5—51no—
157C23×D5—51no—
158C23×D5—102no—
159C2×C5×D4—51no—
160C2×C5×D4—51no—
161C2×C5×D4—51no—
162C22×C4×C5—51no—
163C22×C4×C5—51no—
164C2×C22⋊D5—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—51no—
173C2×C22⋊D5—51no—
174C2×C22⋊D5—102no—
175C2×C22⋊D5—102no—
176C2×C22⋊D5—102no—
177C2×C22⋊D5—102no—
178C2×C22⋊D5—104no—
179C2×C22⋊D5—104no—
180C2×C22⋊D5—1016no—
181C2×C22⋊D5C2×D511yescharacteristic
182C2×C22⋊D5C2×D511yescharacteristic, complemented
183C2×C22⋊D5C2×D511yescharacteristic, complemented
184C22×Dic5—51no—
185C22×Dic5—51no—
186C22×Dic5—51no—
187C22×Dic5—102no—
188C22×Dic5C2×D511yescharacteristic, complemented
189C22×Dic5C2×D511yescharacteristic, complemented
190#711—104no—
191#709—104no—
192C2×D20—51no—
193C2×D20—51no—
194C2×C4×D5—51no—
195C2×C4×D5—51no—
196C2×C4×D5—51no—
197C2×C4×D5—51no—
198C2×C4×D5—51no—
199C2×Dic10—51no—
200C2×Dic10—51no—
201C5×C22⋊C4—51no—
202C5×C22⋊C4—51no—
203C5×C22⋊C4—51no—
204C5×C22⋊C4—51no—
205C5×C22⋊C4—51no—
206C5×C22⋊C4—51no—
207C5×C22⋊C4—51no—
208C22⋊Dic5—51no—
209C22⋊Dic5C2×D511yescharacteristic
210C22⋊Dic5C2×D511yescharacteristic
211C22⋊Dic5C2×D511yescharacteristic
212C22⋊Dic5C2×D511yescharacteristic, complemented
213C22⋊Dic5C2×D511yescharacteristic, complemented
214C22⋊Dic5C2×D511yescharacteristic, complemented
215C22⋊Dic5C2×D511yescharacteristic, complemented
216#685—51no—
217#685—51no—
218#685—51no—
219#685—51no—
220#685—51no—
221#685—51no—
222#685—51no—
223#685—51no—
224#685—51no—
225C4⋊Dic5—51no—
226C4⋊Dic5—51no—
227C4⋊Dic5—51no—
228#683—51no—
229#683—51no—
230#683—51no—
231#683—51no—
232#683—51no—
233C4×Dic5—51no—
234C4×Dic5—51no—
235#546—251noSylow
236C2×C52—22no—
237C2×C52—22no—
238C2×C52—41no—
239C2×C52C22×D411yescharacteristic
240C2×C52C2×#4111yescharacteristic
241C2×C52H(5,𝔽2)11yescharacteristic
242C5⋊D5—22no—
243C5⋊D5—41no—
244C5×D5—41no—
245C23×C5—28no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5—22no—
252C23×C5—22no—
253C23×C5—44no—
254C23×C5—41no—
255C23×C5—41no—
256C23×C5—44no—
257C23×C5—51no—
258C23×C5C22×D511yescharacteristic
259C23×C5C22×D511yescharacteristic
260C23×C5C22×D511yescharacteristic
261C23×C5C22×D511yescharacteristic
262C23×C5C22×D511yescharacteristic
263C23×C5C22×D511yescharacteristic
264C22×D5—51no—
265C22×D5—51no—
266C22×D5—51no—
267C22×D5—51no—
268C22×D5—102no—
269C22×D5—102no—
270C22×D5—102no—
271C22×D5—102no—
272C22×D5—108no—
273C22×D5—102no—
274C22×D5—102no—
275C22×D5—204no—
276C22×D5—201no—
277C22×D5—201no—
278C22×D5—204no—
279C22×D5C22×D511yescharacteristic
280C5×D4—102no—
281C5×D4—102no—
282C5×D4—201no—
283C2×C4×C5—51no—
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—201no—
292C22⋊D5—22no—
293C22⋊D5—22no—
294C22⋊D5—41no—
295C22⋊D5—54no—
296C22⋊D5—54no—
297C22⋊D5—108no—
298C22⋊D5—102no—
299C22⋊D5—102no—
300C22⋊D5—102no—
301C22⋊D5—102no—
302C22⋊D5—102no—
303C22⋊D5—108no—
304C22⋊D5—108no—
305C22⋊D5—108no—
306C22⋊D5—102no—
307C22⋊D5—102no—
308C22⋊D5—201no—
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—102no—
318C2×Dic5—102no—
319C2×Dic5—102no—
320C2×Dic5—108no—
321C2×Dic5C22×D511yescharacteristic
322C2×Dic5C22×D511yescharacteristic
323C2×Dic5C22×D511yescharacteristic
324C2×Dic5C22×D511yescharacteristic
325C2×Dic5C22×D511yescharacteristic
326C2×Dic5C22×D511yescharacteristic
327C2×Dic5C22×D511yescharacteristic
328D20—102no—
329D20—201no—
330C4×D5—102no—
331C4×D5—102no—
332C4×D5—102no—
333C4×D5—201no—
334C4×D5—201no—
335Dic10—102no—
336Dic10—102no—
337C2×#41—251no—
338C22×D4—251no—
339#124—251no—
340#123—251no—
341#123—251no—
342C22⋊Q8—251no—
343#121—251no—
344#121—251no—
345#121—251no—
346#120—51no—
347#120—251no—
348C4×D4—251no—
349C4×D4—251no—
350C2×C22⋊C4—51no—
351C2×C22⋊C4—251no—
352C52#54611yescharacteristic, complemented, Sylow
353C22×C5—24no—
354C22×C5—24no—
355C22×C5—21no—
356C22×C5—21no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5—21no—
360C22×C5—21no—
361C22×C5—21no—
362C22×C5—21no—
363C22×C5—24no—
364C22×C5—21no—
365C22×C5—21no—
366C22×C5—21no—
367C22×C5—21no—
368C22×C5—21no—
369C22×C5—21no—
370C22×C5—24no—
371C22×C5—24no—
372C22×C5—22no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—42no—
376C22×C5—42no—
377C22×C5—44no—
378C22×C5—44no—
379C22×C5—416no—
380C22×C5—101no—
381C22×C5—101no—
382C22×C5—101no—
383C22×C5#70912yescomplemented
384C22×C5D4×D512yes—
385C22×C5D4×D512yes—
386C22×C5#71112yescomplemented
387C22×C5C23×D511yescharacteristic
388C22×C5C23×D511yescharacteristic
389C2×D5—21no—
390C2×D5—21no—
391C2×D5—21no—
392C2×D5—52no—
393C2×D5—52no—
394C2×D5—104no—
395C2×D5—101no—
396C2×D5—101no—
397C2×D5—101no—
398C2×D5—101no—
399C2×D5—104no—
400C2×D5—101no—
401C2×D5—101no—
402C2×D5—101no—
403C2×D5—101no—
404C2×D5—101no—
405C2×D5—101no—
406C2×D5—202no—
407C2×D5—202no—
408C2×D5—202no—
409C2×D5—202no—
410C2×D5—204no—
411C2×D5—204no—
412C2×D5—2016no—
413C4×C5—101no—
414C4×C5—101no—
415C4×C5—101no—
416C4×C5—101no—
417C4×C5—101no—
418C4×C5—101no—
419C4×C5—101no—
420Dic5—21no—
421Dic5—21no—
422Dic5—21no—
423Dic5—21no—
424Dic5—21no—
425Dic5—21no—
426Dic5—21no—
427Dic5—52no—
428Dic5—52no—
429Dic5—104no—
430Dic5—104no—
431Dic5—101no—
432Dic5—101no—
433C24—251no—
434C24D5211yescharacteristic
435#41—504no—
436C2×Q8—251no—
437C2×D4—51no—
438C2×D4—51no—
439C2×D4—51no—
440C2×D4—251no—
441C2×D4—251no—
442C2×D4—251no—
443C2×D4—251no—
444C2×D4—251no—
445C2×D4—251no—
446C2×D4—504no—
447C22×C4—51no—
448C22×C4—51no—
449C22×C4—251no—
450C22×C4—251no—
451C22×C4—251no—
452C22×C4—251no—
453C4⋊C4—251no—
454C4⋊C4—251no—
455C4⋊C4—251no—
456C4⋊C4—251no—
457C22⋊C4—51no—
458C22⋊C4—51no—
459C22⋊C4—51no—
460C22⋊C4—51no—
461C22⋊C4—51no—
462C22⋊C4—51no—
463C22⋊C4—51no—
464C22⋊C4—251no—
465C22⋊C4—251no—
466C22⋊C4—251no—
467C22⋊C4—251no—
468C22⋊C4—251no—
469C42—251no—
470C2×C5—28no—
471C2×C5—22no—
472C2×C5—22no—
473C2×C5—22no—
474C2×C5—22no—
475C2×C5—22no—
476C2×C5—22no—
477C2×C5—22no—
478C2×C5—44no—
479C2×C5—44no—
480C2×C5—41no—
481C2×C5—41no—
482C2×C5—201no—
483C2×C5C2×#70911yescharacteristic
484C2×C5C2×D4×D511yescharacteristic
485C2×C5C2×D4×D511yescharacteristic
486C2×C5C2×#71111yescharacteristic
487C2×C5#414411yescharacteristic
488C2×C5#414411yescharacteristic
489D5—41no—
490D5—102no—
491D5—102no—
492D5—204no—
493D5—204no—
494D5—201no—
495D5—201no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—51no—
500C23—251no—
501C23—251no—
502C23—502no—
503C23—502no—
504C23—1001no—
505C23C2×D5211yescharacteristic
506C23C2×D5211yescharacteristic
507C23C2×D5211yescharacteristic
508Q8—502no—
509D4—102no—
510D4—102no—
511D4—201no—
512D4—254no—
513D4—502no—
514D4—502no—
515D4—502no—
516D4—502no—
517D4—1001no—
518C2×C4—51no—
519C2×C4—51no—
520C2×C4—51no—
521C2×C4—51no—
522C2×C4—51no—
523C2×C4—51no—
524C2×C4—51no—
525C2×C4—102no—
526C2×C4—201no—
527C2×C4—251no—
528C2×C4—251no—
529C2×C4—502no—
530C2×C4—502no—
531C2×C4—502no—
532C2×C4—1001no—
533C5—22no—
534C5#6582411yescharacteristic, complemented
535C5#6592511yescharacteristic, complemented
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—44no—
544C22—101no—
545C22—101no—
546C22—101no—
547C22—252no—
548C22—501no—
549C22—501no—
550C22—501no—
551C22—501no—
552C22—501no—
553C22—1004no—
554C22#8787912yescomplemented
555C22#8788612yes—
556C22C22×D5211yescharacteristic
557C4—101no—
558C4—101no—
559C4—101no—
560C4—101no—
561C4—101no—
562C4—101no—
563C4—101no—
564C4—252no—
565C4—501no—
566C2—22no—
567C2—22no—
568C2—41no—
569C2—201no—
570C2—502no—
571C2—1001no—
572C2C2×#8787911yescharacteristic
573C2C2×#8788611yescharacteristic
574C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC24×C52C22
hypercenterC24D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊(C5⋊D5)C2semidirect
2. C2×#87879C2semidirect
3. C5×#4099C2semidirect
4. C2×C5×C22⋊Dic5C2semidirect
5. #11717100C2semidirect
6. #11717099C2semidirect
7. #11717096C2semidirect
8. #11717093C2semidirect
9. #11717092C2semidirect
10. #11717091C2semidirect
11. #11717089C2semidirect
12. #11717084C2semidirect
13. #11717076C2semidirect
14. #11717071C2semidirect
15. #11717070C2semidirect
25. C2×C5×C22⋊D5C22semidirect
26. C2×C5×C22⋊D5C22semidirect
27. C22×C5×Dic5C22semidirect
28. C22×C5×Dic5C22semidirect
29. C2×#6039C22semidirect
30. C2×#6038C22semidirect
31. C2×#6038C22semidirect
32. C2×#6036C22semidirect
33. C2×#6036C22semidirect
35. C2×D5×Dic5C22semidirect
39. C5×C22⋊Dic5C22semidirect
40. C5×C22⋊Dic5C22semidirect
41. C5×C22⋊Dic5C22semidirect
42. C5×C22⋊Dic5C22semidirect
43. #87781C22semidirect
44. #87780C22semidirect
45. #87780C22semidirect
46. #87780C22semidirect
47. #87779C22semidirect
48. #87779C22semidirect
49. #87779C22semidirect
50. #87779C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
70. C22⋊(C5⋊D5)D4semidirect
99. #4099D5semidirect
100. C2×C22⋊Dic5D5semidirect
132. C22×C52#41semidirect
182. C2×C22⋊D5C2×D5semidirect
183. C2×C22⋊D5C2×D5semidirect
188. C22×Dic5C2×D5semidirect
189. C22×Dic5C2×D5semidirect
212. C22⋊Dic5C2×D5semidirect
213. C22⋊Dic5C2×D5semidirect
214. C22⋊Dic5C2×D5semidirect
215. C22⋊Dic5C2×D5semidirect
352. C52#546semidirect
383. C22×C5#709semidirect
386. C22×C5#711semidirect
534. C5#65824semidirect
535. C5#65925semidirect
554. C22#87879semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C5⋊D5)2
C2×#878792
C5×#40992
C2×C5×C22⋊Dic52
#117171002
#117170992
#117170962
#117170932
#117170922
#117170912
#117170892
#117170842
#117170762
#117170712
#117170702
#659255
#658245