Finite Group Databasebeta

← #13106579

Subgroups — #13106579

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
1C22×C5×C22⋊D5C211yesmaximal, characteristic, complemented
2C2×#87880C211yesmaximal, characteristic, complemented
3C2×C22⋊#1039C211yesmaximal, characteristic, complemented
4C5×#4099C211yesmaximal, characteristic, complemented
5#11717098C211yesmaximal, characteristic, complemented
6#11717094C211yesmaximal, characteristic, complemented
7#11717091C211yesmaximal, characteristic, complemented
8Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
9#11717088C211yesmaximal, characteristic, complemented
10#11717088C211yesmaximal, characteristic, complemented
11#11717087C211yesmaximal, characteristic, complemented
12D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
13#11717082C211yesmaximal, characteristic, complemented
14#11717080C211yesmaximal, characteristic, complemented
15#11717077C211yesmaximal, characteristic, complemented
16C24×C52C2211yescharacteristic
17C23×C5×D5C2211yescharacteristic
18C22×#1039C2211yescharacteristic, complemented
19C22×#1039C2211yescharacteristic, complemented
20C2×C5×C22⋊D5—24no—
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C5×C22⋊D5C2211yescharacteristic
23C2×C5×C22⋊D5C2211yescharacteristic
24C2×C5×C22⋊D5C2211yescharacteristic
25C2×C5×C22⋊D5C2211yescharacteristic
26C2×C5×C22⋊D5C2211yescharacteristic, complemented
27C2×C5×C22⋊D5C2211yescharacteristic, complemented
28C22×C5×Dic5C2211yescharacteristic
29C2×#6039C2211yescharacteristic, complemented
30C2×#6037C2211yescharacteristic, complemented
31C2×#6037C2211yescharacteristic, complemented
32#87880—24no—
33C2×D5×Dic5C2211yescharacteristic, complemented
34C2×D5×Dic5C2211yescharacteristic, complemented
35C2×D5×Dic5C2211yescharacteristic, complemented
36C22⋊#1039C2211yescharacteristic
37C22⋊#1039C2211yescharacteristic
38C22⋊#1039C2211yescharacteristic, complemented
39C22⋊#1039C2211yescharacteristic, complemented
40C5×C22⋊Dic5C2211yescharacteristic
41C5×C22⋊Dic5C2211yescharacteristic, complemented
42C5×C22⋊Dic5C2211yescharacteristic, complemented
43#87781C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87780C2211yescharacteristic, complemented
46#87780C2211yescharacteristic, complemented
47#87778C2211yescharacteristic, complemented
48#87778C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#66140—51nomaximal
54#65925—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
67C2×#1039—22no—
68C2×#1039—41no—
69C2×#1039C2311yescharacteristic
70C2×#1039C2311yescharacteristic
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C5×C22⋊D5—22no—
74C5×C22⋊D5—22no—
75C5×C22⋊D5—22no—
76C5×C22⋊D5—22no—
77C5×C22⋊D5—22no—
78C5×C22⋊D5—41no—
79C5×C22⋊D5D414yescomplemented
80C2×C5×Dic5—22no—
81C2×C5×Dic5C2311yescharacteristic
82C2×C5×Dic5C2311yescharacteristic
83C2×C5×Dic5C2311yescharacteristic
84C2×C5×Dic5C2311yescharacteristic
85C2×C5×Dic5C2311yescharacteristic
86#6039—22no—
87#6037—22no—
88#6037—41no—
89D5×Dic5—22no—
90D5×Dic5—22no—
91D5×Dic5—41no—
92C22×C5×D4—51no—
93C22×C22⋊D5D511yescharacteristic, complemented
94C2×#711—51no—
95C2×#711—51no—
96C5×#120—51no—
97#4099—51no—
98#4099D511yescharacteristic, complemented
99C2×C22⋊Dic5—51no—
100C2×C22⋊Dic5—51no—
101C2×C22⋊Dic5—51no—
102C2×C22⋊Dic5—102no—
103#4085—51no—
104#4085—51no—
105#4085—51no—
106#4085—51no—
107#4084—51no—
108#4084—51no—
109#4083—51no—
110#4082—51no—
111#4081—51no—
112#4081—51no—
113D4×Dic5—51no—
114D4×Dic5—51no—
115C4×C22⋊D5—51no—
116#4070—51no—
117#4032—51no—
118#4031—51no—
119#4029—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—
147#1039—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—51no—
163C2×C5×D4—51no—
164C2×C5×D4—51no—
165C2×C5×D4—104no—
166C22×C4×C5—51no—
167C2×C22⋊D5—24no—
168C2×C22⋊D5—51no—
169C2×C22⋊D5—51no—
170C2×C22⋊D5—51no—
171C2×C22⋊D5—51no—
172C2×C22⋊D5C2×D511yescharacteristic
173C2×C22⋊D5C2×D511yescharacteristic
174C2×C22⋊D5C2×D511yescharacteristic
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—51no—
185C22×Dic5—51no—
186C22×Dic5—102no—
187C22×Dic5—102no—
188C22×Dic5C2×D511yescharacteristic
189#711—104no—
190#711—104no—
191C2×C4×D5—51no—
192C2×C4×D5—51no—
193C2×C4×D5—51no—
194C2×Dic10—51no—
195C2×Dic10—51no—
196C5×C22⋊C4—51no—
197C5×C22⋊C4—51no—
198C5×C22⋊C4—51no—
199C22⋊Dic5—51no—
200C22⋊Dic5—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—51no—
212C22⋊Dic5—102no—
213C22⋊Dic5—102no—
214C22⋊Dic5—102no—
215C22⋊Dic5—102no—
216C22⋊Dic5C2×D511yescharacteristic
217C22⋊Dic5C2×D511yescharacteristic, complemented
218C22⋊Dic5C2×D511yescharacteristic, complemented
219#685—51no—
220#685—51no—
221#685—51no—
222#685—51no—
223#685—51no—
224C4⋊Dic5—51no—
225C4⋊Dic5—51no—
226C4⋊Dic5—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—22no—
242C5×D5—41no—
243C5×D5—41no—
244C23×C5—28no—
245C23×C5—22no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5—22no—
252C23×C5—44no—
253C23×C5—41no—
254C23×C5—41no—
255C23×C5—44no—
256C23×C5—51no—
257C23×C5—51no—
258C23×C5—51no—
259C23×C5—102no—
260C23×C5—102no—
261C23×C5—201no—
262C23×C5C22×D511yescharacteristic
263C23×C5C22×D511yescharacteristic
264C23×C5C22×D511yescharacteristic
265C23×C5C22×D511yescharacteristic
266C23×C5C22×D511yescharacteristic
267C23×C5C22×D511yescharacteristic
268C22×D5—22no—
269C22×D5—22no—
270C22×D5—41no—
271C22×D5C22×D511yescharacteristic
272C22×D5C22×D511yescharacteristic
273C22×D5C22×D511yescharacteristic
274C5×D4—54no—
275C5×D4—102no—
276C5×D4—102no—
277C5×D4—102no—
278C5×D4—102no—
279C5×D4—102no—
280C5×D4—201no—
281C2×C4×C5—51no—
282C2×C4×C5—51no—
283C2×C4×C5—51no—
284C2×C4×C5—51no—
285C2×C4×C5—51no—
286C2×C4×C5—102no—
287C22⋊D5—22no—
288C22⋊D5—22no—
289C22⋊D5—22no—
290C22⋊D5—22no—
291C22⋊D5—22no—
292C22⋊D5—41no—
293C22⋊D5—102no—
294C22⋊D5—102no—
295C22⋊D5—201no—
296C22⋊D5—201no—
297C22⋊D5C22⋊D514yescomplemented
298C2×Dic5—22no—
299C2×Dic5—51no—
300C2×Dic5—51no—
301C2×Dic5—51no—
302C2×Dic5—51no—
303C2×Dic5—51no—
304C2×Dic5—51no—
305C2×Dic5—51no—
306C2×Dic5—51no—
307C2×Dic5—102no—
308C2×Dic5—102no—
309C2×Dic5—102no—
310C2×Dic5—102no—
311C2×Dic5—102no—
312C2×Dic5—102no—
313C2×Dic5—102no—
314C2×Dic5—102no—
315C2×Dic5—204no—
316C2×Dic5—201no—
317C2×Dic5—201no—
318C2×Dic5—201no—
319C2×Dic5—204no—
320C2×Dic5C22×D511yescharacteristic
321C2×Dic5C22×D511yescharacteristic
322C2×Dic5C22×D511yescharacteristic
323C2×Dic5C22×D511yescharacteristic
324C2×Dic5C22×D511yescharacteristic
325C4×D5—102no—
326C4×D5—102no—
327C4×D5—201no—
328Dic10—102no—
329Dic10—102no—
330C2×#41—251no—
331C22×D4—51no—
332#124—251no—
333#123—251no—
334#123—251no—
335C22⋊Q8—251no—
336#121—251no—
337#121—251no—
338#121—251no—
339#120—51no—
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—21no—
353C22×C5—21no—
354C22×C5—21no—
355C22×C5—21no—
356C22×C5—24no—
357C22×C5—21no—
358C22×C5—21no—
359C22×C5—21no—
360C22×C5—21no—
361C22×C5—21no—
362C22×C5—21no—
363C22×C5—24no—
364C22×C5—24no—
365C22×C5—22no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5—42no—
369C22×C5—42no—
370C22×C5—44no—
371C22×C5—44no—
372C22×C5—416no—
373C22×C5—52no—
374C22×C5—101no—
375C22×C5—101no—
376C22×C5—101no—
377C22×C5—101no—
378C22×C5—101no—
379C22×C5—101no—
380C22×C5—101no—
381C22×C5—101no—
382C22×C5—204no—
383C22×C5D4×D512yes—
384C22×C5#71112yescomplemented
385C22×C5#71112yescomplemented
386C22×C5C2×C22⋊D512yes—
387C22×C5C23×D511yescharacteristic
388C22×C5C23×D511yescharacteristic
389C2×D5—21no—
390C2×D5—21no—
391C2×D5—21no—
392C2×D5—21no—
393C2×D5—21no—
394C2×D5—21no—
395C2×D5—21no—
396C2×D5—21no—
397C2×D5—44no—
398C2×D5C2×C22⋊D512yes—
399C4×C5—52no—
400C4×C5—101no—
401C4×C5—101no—
402C4×C5—101no—
403C4×C5—101no—
404Dic5—21no—
405Dic5—21no—
406Dic5—21no—
407Dic5—21no—
408Dic5—101no—
409Dic5—101no—
410Dic5—101no—
411Dic5—101no—
412Dic5—101no—
413Dic5—101no—
414Dic5—101no—
415Dic5—101no—
416Dic5—202no—
417Dic5—202no—
418Dic5—202no—
419Dic5—202no—
420Dic5C2×C22⋊D512yes—
421C24—51no—
422C24D5211yescharacteristic
423#41—504no—
424C2×Q8—251no—
425C2×D4—51no—
426C2×D4—51no—
427C2×D4—51no—
428C2×D4—51no—
429C2×D4—51no—
430C2×D4—51no—
431C2×D4—51no—
432C2×D4—104no—
433C2×D4—251no—
434C2×D4—251no—
435C22×C4—51no—
436C22×C4—251no—
437C22×C4—251no—
438C22×C4—251no—
439C22×C4—251no—
440C22×C4—251no—
441C4⋊C4—251no—
442C4⋊C4—251no—
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—22no—
465C2×C5—22no—
466C2×C5—44no—
467C2×C5—44no—
468C2×C5—41no—
469C2×C5—41no—
470C2×C5—102no—
471C2×C5—201no—
472C2×C5—201no—
473C2×C5C2×D4×D511yescharacteristic
474C2×C5C2×#71111yescharacteristic
475C2×C5C2×#71111yescharacteristic
476C2×C5#414411yescharacteristic
477C2×C5#414411yescharacteristic
478C2×C5C22×C22⋊D511yescharacteristic
479D5—22no—
480D5—41no—
481D5—41no—
482C23—22no—
483C23—22no—
484C23—41no—
485C23—51no—
486C23—51no—
487C23—51no—
488C23—102no—
489C23—102no—
490C23—201no—
491C23C2×D5211yescharacteristic
492C23C2×D5211yescharacteristic
493C23C2×D5211yescharacteristic
494Q8—502no—
495D4—54no—
496D4—102no—
497D4—102no—
498D4—102no—
499D4—102no—
500D4—102no—
501D4—201no—
502D4—502no—
503D4—1001no—
504C2×C4—51no—
505C2×C4—51no—
506C2×C4—51no—
507C2×C4—51no—
508C2×C4—51no—
509C2×C4—102no—
510C2×C4—251no—
511C2×C4—251no—
512C2×C4—251no—
513C2×C4—251no—
514C2×C4—502no—
515C2×C4—502no—
516C2×C4—502no—
517C2×C4—1001no—
518C2×C4—1001no—
519C5—22no—
520C5#6592511yescharacteristic, complemented
521C5#6614011yescharacteristic, complemented
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—44no—
530C22—52no—
531C22—101no—
532C22—101no—
533C22—101no—
534C22—101no—
535C22—101no—
536C22—101no—
537C22—101no—
538C22—101no—
539C22—204no—
540C22#8788012yescomplemented
541C22D5×C22⋊D512yes—
542C22C22×D5211yescharacteristic
543C4—52no—
544C4—101no—
545C4—101no—
546C4—101no—
547C4—101no—
548C4—501no—
549C4—501no—
550C4—501no—
551C4—501no—
552C2—22no—
553C2—22no—
554C2—41no—
555C2—102no—
556C2—201no—
557C2—201no—
558C2C2×#8788011yescharacteristic
559C2C2×D5×C22⋊D511yescharacteristic
560C2#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×#87880C2semidirect
3. C2×C22⋊#1039C2semidirect
4. C5×#4099C2semidirect
5. #11717098C2semidirect
6. #11717094C2semidirect
7. #11717091C2semidirect
8. Dic5×C22⋊D5C2semidirect
9. #11717088C2semidirect
10. #11717088C2semidirect
11. #11717087C2semidirect
12. D5×C22⋊Dic5C2semidirect
13. #11717082C2semidirect
14. #11717080C2semidirect
15. #11717077C2semidirect
18. C22×#1039C22semidirect
19. C22×#1039C22semidirect
26. C2×C5×C22⋊D5C22semidirect
27. C2×C5×C22⋊D5C22semidirect
29. C2×#6039C22semidirect
30. C2×#6037C22semidirect
31. C2×#6037C22semidirect
33. C2×D5×Dic5C22semidirect
34. C2×D5×Dic5C22semidirect
35. C2×D5×Dic5C22semidirect
38. C22⋊#1039C22semidirect
39. C22⋊#1039C22semidirect
41. C5×C22⋊Dic5C22semidirect
42. C5×C22⋊Dic5C22semidirect
43. #87781C22semidirect
44. #87780C22semidirect
45. #87780C22semidirect
46. #87780C22semidirect
47. #87778C22semidirect
48. #87778C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
79. C5×C22⋊D5D4semidirect
93. C22×C22⋊D5D5semidirect
98. #4099D5semidirect
132. C22×C52#41semidirect
177. C2×C22⋊D5C2×D5semidirect
178. C2×C22⋊D5C2×D5semidirect
217. C22⋊Dic5C2×D5semidirect
218. C22⋊Dic5C2×D5semidirect
297. C22⋊D5C22⋊D5semidirect
345. C52#546semidirect
384. C22×C5#711semidirect
385. C22×C5#711semidirect
520. C5#65925semidirect
521. C5#66140semidirect
540. C22#87880semidirect

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×#878802
C2×C22⋊#10392
C5×#40992
#117170982
#117170942
#117170912
Dic5×C22⋊D52
#117170882
#117170872
D5×C22⋊Dic52
#117170822
#117170802
#117170772
#661405
#659255