Finite Group Databasebeta

← #13106578

Subgroups — #13106578

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
1C22×C5×C22⋊D5C211yesmaximal, characteristic, complemented
2C2×#87879C211yesmaximal, characteristic, complemented
3#11717273C211yesmaximal, characteristic, complemented
4C2×C5×C22⋊Dic5C211yesmaximal, characteristic, complemented
5#11717096C211yesmaximal, characteristic, complemented
6#11717095C211yesmaximal, characteristic, complemented
7Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
8#11717089C211yesmaximal, characteristic, complemented
9#11717087C211yesmaximal, characteristic, complemented
10D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
11#11717082C211yesmaximal, characteristic, complemented
12#11717079C211yesmaximal, characteristic, complemented
13#11717077C211yesmaximal, characteristic, complemented
14#11717076C211yesmaximal, characteristic, complemented
15#11717070C211yesmaximal, characteristic, complemented
16C24×C52C2211yescharacteristic
17C23×C5×D5C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
20C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
21C2×C5×C22⋊D5—24no—
22C2×C5×C22⋊D5C2211yescharacteristic
23C2×C5×C22⋊D5C2211yescharacteristic
24C2×C5×C22⋊D5C2211yescharacteristic
25C2×C5×C22⋊D5C2211yescharacteristic
26C22×C5×Dic5C2211yescharacteristic
27C22×C5×Dic5C2211yescharacteristic, complemented
28C22×C5×Dic5C2211yescharacteristic, complemented
29C2×#6039C2211yescharacteristic, complemented
30C2×#6038C2211yescharacteristic, complemented
31C2×#6038C2211yescharacteristic, complemented
32C2×#6036C2211yescharacteristic, complemented
33#87879—24no—
34C2×D5×Dic5C2211yescharacteristic, complemented
35C2×D5×Dic5C2211yescharacteristic, complemented
36C22⋊#1039C2211yescharacteristic
37C22⋊#1039C2211yescharacteristic, complemented
38C22⋊#1039C2211yescharacteristic, complemented
39C5×C22⋊Dic5C2211yescharacteristic
40C5×C22⋊Dic5C2211yescharacteristic
41C5×C22⋊Dic5C2211yescharacteristic, complemented
42C5×C22⋊Dic5C2211yescharacteristic, complemented
43#87781C2211yescharacteristic, complemented
44#87781C2211yescharacteristic, complemented
45#87780C2211yescharacteristic, complemented
46#87780C2211yescharacteristic, complemented
47#87779C2211yescharacteristic, complemented
48#87778C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#66140—51nomaximal
54#65824—51nomaximal
55C23×C52—22no—
56C23×C52—22no—
57C23×C52—41no—
58C23×C52C2311yescharacteristic
59C23×C52C2311yescharacteristic
60C23×C52C2311yescharacteristic
61C22×C5⋊D5C2311yescharacteristic
62C22×C5×D5—22no—
63C22×C5×D5—22no—
64C22×C5×D5—41no—
65C22×C5×D5C2311yescharacteristic
66C22×C5×D5C2311yescharacteristic
67C22⋊(C5⋊D5)—22no—
68C22⋊(C5⋊D5)—22no—
69C22⋊(C5⋊D5)—41no—
70C2×#1039C2311yescharacteristic
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C5×C22⋊D5—22no—
74C5×C22⋊D5—22no—
75C5×C22⋊D5—22no—
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
86#6039—22no—
87#6038—22no—
88#6038—41no—
89#6036—22no—
90D5×Dic5—22no—
91D5×Dic5—41no—
92C22×C5×D4—51no—
93C22×C22⋊D5D511yescharacteristic, complemented
94C2×#711—51no—
95C2×#709—51no—
96C2×C5×C22⋊C4—51no—
97#4099—51no—
98#4099—51no—
99#4099—51no—
100#4099—102no—
101C2×C22⋊Dic5—51no—
102C2×C22⋊Dic5D511yescharacteristic, complemented
103#4085—51no—
104#4085—51no—
105#4084—51no—
106#4082—51no—
107#4081—51no—
108#4081—51no—
109D4×Dic5—51no—
110#4076—51no—
111#4075—51no—
112C4×C22⋊D5—51no—
113C4×C22⋊D5—51no—
114#4070—51no—
115#4031—51no—
116#4030—51no—
117#4030—51no—
118#4029—51no—
119#4028—51no—
120#4027—51no—
121D5×C22⋊C4—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—21no—
140C2×C5×D5—21no—
141C2×C5×D5—21no—
142C2×C5×D5—44no—
143C2×C5×D5C2×D412yes—
144#1039—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—
153C24×C5—22no—
154C24×C5—51no—
155C24×C5C2×D511yescharacteristic
156C24×C5C2×D511yescharacteristic
157C23×D5C2×D511yescharacteristic
158C2×C5×D4—51no—
159C2×C5×D4—51no—
160C2×C5×D4—51no—
161C2×C5×D4—51no—
162C2×C5×D4—104no—
163C22×C4×C5—51no—
164C22×C4×C5—51no—
165C22×C4×C5—51no—
166C2×C22⋊D5—24no—
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—51no—
175C2×C22⋊D5—102no—
176C2×C22⋊D5—102no—
177C2×C22⋊D5—102no—
178C2×C22⋊D5C2×D511yescharacteristic
179C2×C22⋊D5C2×D511yescharacteristic
180C2×C22⋊D5C2×D511yescharacteristic
181C2×C22⋊D5C2×D511yescharacteristic
182C22×Dic5—51no—
183C22×Dic5—51no—
184C22×Dic5C2×D511yescharacteristic
185C22×Dic5C2×D511yescharacteristic, complemented
186C22×Dic5C2×D511yescharacteristic, complemented
187#711—104no—
188#709—104no—
189C2×D20—51no—
190C2×D20—51no—
191C2×C4×D5—51no—
192C2×C4×D5—51no—
193C2×C4×D5—51no—
194C2×C4×D5—51no—
195C2×Dic10—51no—
196C2×Dic10—51no—
197C5×C22⋊C4—51no—
198C5×C22⋊C4—51no—
199C5×C22⋊C4—51no—
200C5×C22⋊C4—51no—
201C22⋊Dic5—51no—
202C22⋊Dic5—51no—
203C22⋊Dic5—51no—
204C22⋊Dic5—51no—
205C22⋊Dic5—51no—
206C22⋊Dic5—51no—
207C22⋊Dic5—51no—
208C22⋊Dic5—51no—
209C22⋊Dic5—51no—
210C22⋊Dic5—51no—
211C22⋊Dic5—102no—
212C22⋊Dic5—102no—
213C22⋊Dic5—102no—
214C22⋊Dic5C2×D511yescharacteristic
215C22⋊Dic5C2×D511yescharacteristic
216C22⋊Dic5C2×D511yescharacteristic, complemented
217C22⋊Dic5C2×D511yescharacteristic, complemented
218#685—51no—
219#685—51no—
220#685—51no—
221#685—51no—
222#685—51no—
223#685—51no—
224C4⋊Dic5—51no—
225C4⋊Dic5—51no—
226#683—51no—
227#683—51no—
228#683—51no—
229#683—51no—
230#683—51no—
231#683—51no—
232C4×Dic5—51no—
233C4×Dic5—51no—
234#546—251noSylow
235C2×C52—22no—
236C2×C52—22no—
237C2×C52—41no—
238C2×C52C22×D411yescharacteristic
239C2×C52C2×#4111yescharacteristic
240C2×C52H(5,𝔽2)11yescharacteristic
241C5⋊D5—41no—
242C5×D5—22no—
243C5×D5—41no—
244C23×C5—22no—
245C23×C5—22no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5—44no—
252C23×C5—44no—
253C23×C5—41no—
254C23×C5—41no—
255C23×C5—44no—
256C23×C5—51no—
257C23×C5—51no—
258C23×C5—102no—
259C23×C5—102no—
260C23×C5—201no—
261C23×C5C22×D511yescharacteristic
262C23×C5C22×D511yescharacteristic
263C23×C5C22×D511yescharacteristic
264C23×C5C22×D511yescharacteristic
265C23×C5C22×D511yescharacteristic
266C23×C5C22×D511yescharacteristic
267C22×D5—22no—
268C22×D5—22no—
269C22×D5—41no—
270C22×D5—51no—
271C22×D5—51no—
272C22×D5—102no—
273C22×D5C22×D511yescharacteristic
274C22×D5C22×D511yescharacteristic
275C5×D4—54no—
276C5×D4—102no—
277C5×D4—102no—
278C5×D4—102no—
279C2×C4×C5—51no—
280C2×C4×C5—51no—
281C2×C4×C5—51no—
282C2×C4×C5—51no—
283C2×C4×C5—51no—
284C2×C4×C5—51no—
285C2×C4×C5—102no—
286C2×C4×C5—102no—
287C2×C4×C5—201no—
288C22⋊D5—22no—
289C22⋊D5—22no—
290C22⋊D5—22no—
291C22⋊D5—102no—
292C22⋊D5—102no—
293C22⋊D5—102no—
294C22⋊D5—102no—
295C22⋊D5—102no—
296C22⋊D5—204no—
297C22⋊D5—204no—
298C22⋊D5—201no—
299C22⋊D5—201no—
300C22⋊D5—201no—
301C22⋊D5—204no—
302C22⋊D5C22⋊D514yescomplemented
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—41no—
306C2×Dic5—51no—
307C2×Dic5—51no—
308C2×Dic5—51no—
309C2×Dic5—51no—
310C2×Dic5—51no—
311C2×Dic5—51no—
312C2×Dic5—102no—
313C2×Dic5—102no—
314C2×Dic5—102no—
315C2×Dic5—102no—
316C2×Dic5—201no—
317C2×Dic5C22×D511yescharacteristic
318C2×Dic5C22×D511yescharacteristic
319C2×Dic5C22×D511yescharacteristic
320C2×Dic5C22×D511yescharacteristic
321C2×Dic5C22×D511yescharacteristic
322C2×Dic5C22×D511yescharacteristic
323D20—102no—
324D20—201no—
325C4×D5—102no—
326C4×D5—102no—
327C4×D5—102no—
328C4×D5—201no—
329Dic10—102no—
330Dic10—102no—
331C2×#41—251no—
332C22×D4—51no—
333#124—251no—
334#123—251no—
335#123—251no—
336C22⋊Q8—251no—
337#121—251no—
338#121—251no—
339#121—251no—
340#120—251no—
341#120—251no—
342C4×D4—251no—
343C4×D4—251no—
344C2×C22⋊C4—51no—
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—24no—
356C22×C5—21no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5—21no—
360C22×C5—21no—
361C22×C5—21no—
362C22×C5—24no—
363C22×C5—24no—
364C22×C5—22no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5—42no—
369C22×C5—42no—
370C22×C5—42no—
371C22×C5—44no—
372C22×C5—44no—
373C22×C5—416no—
374C22×C5—52no—
375C22×C5—101no—
376C22×C5—101no—
377C22×C5—101no—
378C22×C5—101no—
379C22×C5—101no—
380C22×C5—204no—
381C22×C5#70912yescomplemented
382C22×C5D4×D512yes—
383C22×C5#71112yescomplemented
384C22×C5C2×C22⋊D512yes—
385C22×C5C23×D511yescharacteristic
386C22×C5C23×D511yescharacteristic
387C2×D5—21no—
388C2×D5—21no—
389C2×D5—21no—
390C2×D5—21no—
391C2×D5—21no—
392C2×D5—44no—
393C2×D5—101no—
394C2×D5—101no—
395C2×D5—101no—
396C2×D5—101no—
397C2×D5—101no—
398C2×D5—101no—
399C2×D5—202no—
400C2×D5—202no—
401C2×D5—202no—
402C2×D5C2×C22⋊D512yes—
403C4×C5—52no—
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—101no—
415Dic5—101no—
416Dic5—101no—
417Dic5—101no—
418Dic5—101no—
419Dic5—101no—
420Dic5—202no—
421Dic5—202no—
422Dic5—202no—
423Dic5C2×C22⋊D512yes—
424C24—51no—
425C24D5211yescharacteristic
426#41—504no—
427C2×Q8—251no—
428C2×D4—51no—
429C2×D4—51no—
430C2×D4—51no—
431C2×D4—51no—
432C2×D4—104no—
433C2×D4—251no—
434C2×D4—251no—
435C2×D4—251no—
436C2×D4—251no—
437C2×D4—251no—
438C22×C4—51no—
439C22×C4—51no—
440C22×C4—51no—
441C22×C4—251no—
442C22×C4—251no—
443C22×C4—251no—
444C4⋊C4—251no—
445C4⋊C4—251no—
446C4⋊C4—251no—
447C4⋊C4—251no—
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—
459C22⋊C4—251no—
460C42—251no—
461C2×C5—22no—
462C2×C5—22no—
463C2×C5—22no—
464C2×C5—22no—
465C2×C5—22no—
466C2×C5—22no—
467C2×C5—22no—
468C2×C5—44no—
469C2×C5—44no—
470C2×C5—44no—
471C2×C5—41no—
472C2×C5—41no—
473C2×C5—102no—
474C2×C5—201no—
475C2×C5C2×#70911yescharacteristic
476C2×C5C2×D4×D511yescharacteristic
477C2×C5C2×#71111yescharacteristic
478C2×C5#414411yescharacteristic
479C2×C5#414411yescharacteristic
480C2×C5C22×C22⋊D511yescharacteristic
481D5—22no—
482D5—41no—
483D5—201no—
484D5—201no—
485D5—402no—
486C23—22no—
487C23—22no—
488C23—41no—
489C23—51no—
490C23—51no—
491C23—102no—
492C23—102no—
493C23—201no—
494C23—251no—
495C23C2×D5211yescharacteristic
496C23C2×D5211yescharacteristic
497C23C2×D5211yescharacteristic
498Q8—502no—
499D4—54no—
500D4—102no—
501D4—102no—
502D4—102no—
503D4—502no—
504D4—502no—
505D4—502no—
506D4—1001no—
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—201no—
517C2×C4—251no—
518C2×C4—251no—
519C2×C4—251no—
520C2×C4—502no—
521C2×C4—502no—
522C2×C4—1001no—
523C5—22no—
524C5#6582411yescharacteristic, complemented
525C5#6614011yescharacteristic, complemented
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22—52no—
535C22—101no—
536C22—101no—
537C22—101no—
538C22—101no—
539C22—101no—
540C22—204no—
541C22—501no—
542C22—501no—
543C22—501no—
544C22#8787912yescomplemented
545C22D5×C22⋊D512yes—
546C22C22×D5211yescharacteristic
547C4—52no—
548C4—101no—
549C4—101no—
550C4—101no—
551C4—101no—
552C4—101no—
553C4—501no—
554C4—501no—
555C4—501no—
556C2—22no—
557C2—22no—
558C2—41no—
559C2—102no—
560C2—201no—
561C2—1001no—
562C2C2×#8787911yescharacteristic
563C2C2×D5×C22⋊D511yescharacteristic
564C2#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×C5×C22⋊D5C2semidirect
2. C2×#87879C2semidirect
3. #11717273C2semidirect
4. C2×C5×C22⋊Dic5C2semidirect
5. #11717096C2semidirect
6. #11717095C2semidirect
7. Dic5×C22⋊D5C2semidirect
8. #11717089C2semidirect
9. #11717087C2semidirect
10. D5×C22⋊Dic5C2semidirect
11. #11717082C2semidirect
12. #11717079C2semidirect
13. #11717077C2semidirect
14. #11717076C2semidirect
15. #11717070C2semidirect
19. C2×C22⋊(C5⋊D5)C22semidirect
20. C2×C22⋊(C5⋊D5)C22semidirect
27. C22×C5×Dic5C22semidirect
28. C22×C5×Dic5C22semidirect
29. C2×#6039C22semidirect
30. C2×#6038C22semidirect
31. C2×#6038C22semidirect
32. C2×#6036C22semidirect
34. C2×D5×Dic5C22semidirect
35. C2×D5×Dic5C22semidirect
37. C22⋊#1039C22semidirect
38. C22⋊#1039C22semidirect
41. C5×C22⋊Dic5C22semidirect
42. C5×C22⋊Dic5C22semidirect
43. #87781C22semidirect
44. #87781C22semidirect
45. #87780C22semidirect
46. #87780C22semidirect
47. #87779C22semidirect
48. #87778C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
76. C5×C22⋊D5D4semidirect
93. C22×C22⋊D5D5semidirect
102. C2×C22⋊Dic5D5semidirect
132. C22×C52#41semidirect
185. C22×Dic5C2×D5semidirect
186. C22×Dic5C2×D5semidirect
216. C22⋊Dic5C2×D5semidirect
217. C22⋊Dic5C2×D5semidirect
302. C22⋊D5C22⋊D5semidirect
346. C52#546semidirect
381. C22×C5#709semidirect
383. C22×C5#711semidirect
524. C5#65824semidirect
525. C5#66140semidirect
544. C22#87879semidirect

Maximal subgroups

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

subgroupindex
C22×C5×C22⋊D52
C2×#878792
#117172732
C2×C5×C22⋊Dic52
#117170962
#117170952
Dic5×C22⋊D52
#117170892
#117170872
D5×C22⋊Dic52
#117170822
#117170792
#117170772
#117170762
#117170702
#661405
#658245