Finite Group Databasebeta

← #13106206

Subgroups — #13106206

Order 1600.

746 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×D5C211yesmaximal, characteristic, complemented
2C2×#87880C211yesmaximal, characteristic, complemented
3#11717255C211yesmaximal, characteristic, complemented
4C5×#4085C211yesmaximal, characteristic, complemented
5#11717094C211yesmaximal, characteristic, complemented
6#11717094C211yesmaximal, characteristic, complemented
7Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
8#11717088C211yesmaximal, characteristic, complemented
9D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
10C2×#87778C211yesmaximal, characteristic, complemented
11#11717077C211yesmaximal, characteristic, complemented
12#11717004C211yesmaximal, characteristic, complemented
13#11717002C211yesmaximal, characteristic, complemented
14#11716997C211yesmaximal, characteristic, complemented
15#11716969C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C23×C5×D5C2211yescharacteristic
18C2×C52×D4C2211yescharacteristic
19C22×#1039C2211yescharacteristic, complemented
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C5×C22⋊D5C2211yescharacteristic, complemented
23C2×C5×C22⋊D5C2211yescharacteristic, complemented
24C22×C5×Dic5C2211yescharacteristic, complemented
25C5×D4×D5—24no—
26C2×C5×D20C2211yescharacteristic
27C2×C4×C5×D5C2211yescharacteristic
28C2×#6039C2211yescharacteristic, complemented
29C2×#6037C2211yescharacteristic
30C2×#6037C2211yescharacteristic, complemented
31C2×#6037C2211yescharacteristic, complemented
32#87880—24no—
33C2×D5×Dic5C2211yescharacteristic, complemented
34C2×D5×Dic5C2211yescharacteristic, complemented
35C2×D5×Dic5C2211yescharacteristic, complemented
36C22⋊#1039C2211yescharacteristic, complemented
37C22⋊#1039C2211yescharacteristic, complemented
38C22⋊#1039C2211yescharacteristic, complemented
39#87806C2211yescharacteristic, complemented
40#87806C2211yescharacteristic, complemented
41C5×C22⋊Dic5C2211yescharacteristic, complemented
42C5×#685C2211yescharacteristic, complemented
43C5×#683C2211yescharacteristic, complemented
44#87780C2211yescharacteristic, complemented
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#66140—51nomaximal
54#65944—51nomaximal
55C23×C52C2311yescharacteristic
56C23×C52C2311yescharacteristic
57C22×C5×D5—22no—
58C22×C5×D5—22no—
59C22×C5×D5—22no—
60C22×C5×D5—22no—
61C22×C5×D5—41no—
62C22×C5×D5—41no—
63C22×C5×D5C2311yescharacteristic
64C22×C5×D5C2311yescharacteristic
65C22×C5×D5C2311yescharacteristic
66C22×C5×D5C2311yescharacteristic
67C52×D4—22no—
68C2×C4×C52C2311yescharacteristic
69C2×#1039—22no—
70C2×#1039C2311yescharacteristic
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C2×#1039C2311yescharacteristic
74C5×C22⋊D5—22no—
75C5×C22⋊D5—22no—
76C5×C22⋊D5—41no—
77C5×C22⋊D5D414yescomplemented
78C2×C5×Dic5—41no—
79C2×C5×Dic5C2311yescharacteristic
80C2×C5×Dic5C2311yescharacteristic
81C2×C5×Dic5C2311yescharacteristic
82C2×C5×Dic5C2311yescharacteristic
83C5×D20—22no—
84C4×C5×D5—22no—
85#6039—22no—
86#6037—22no—
87#6037—22no—
88#6037—41no—
89D5×Dic5—22no—
90D5×Dic5—22no—
91D5×Dic5—41no—
92C22×C5×D4—51no—
93C2×#711—51no—
94C2×#711—51no—
95C2×D4×D5D511yescharacteristic, complemented
96C5×#121—51no—
97#4099—51no—
98#4099—51no—
99C2×C22⋊Dic5—51no—
100C2×C22⋊Dic5—51no—
101#4085—51no—
102#4085—51no—
103#4085D511yescharacteristic, complemented
104#4084—51no—
105#4083—51no—
106#4083—51no—
107#4082—51no—
108#4081—51no—
109#4081—51no—
110#4081—51no—
111#4081—102no—
112D4×Dic5—51no—
113C4×C22⋊D5—51no—
114C4×C22⋊D5—51no—
115C2×#685—51no—
116#4070—51no—
117#4042—51no—
118#4040—51no—
119#4031—51no—
120#4030—51no—
121#4027—51no—
122D5×C22⋊C4—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×D5—21no—
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—44no—
141C2×C5×D5—44no—
142C2×C5×D5C2×D412yes—
143C2×C5×D5#4112yescomplemented
144C4×C52—21no—
145#1039—21no—
146#1039—21no—
147#1039—21no—
148#1039—21no—
149C5×Dic5—21no—
150C5×Dic5—21no—
151C5×Dic5—21no—
152C5×Dic5C2×D412yes—
153C24×C5—51no—
154C24×C5—51no—
155C23×D5C2×D511yescharacteristic
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×D4—51no—
163C2×C5×D4—104no—
164C2×C5×D4C2×D511yescharacteristic
165C2×C5×D4C2×D511yescharacteristic
166C22×C4×C5—51no—
167C22×C4×C5—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⋊D5C2×D511yescharacteristic
175C2×C22⋊D5C2×D511yescharacteristic
176C2×C22⋊D5C2×D511yescharacteristic, complemented
177C2×C22⋊D5C2×D511yescharacteristic, complemented
178C22×Dic5—51no—
179C22×Dic5—51no—
180C22×Dic5—51no—
181C22×Dic5—51no—
182C22×Dic5—51no—
183C22×Dic5—102no—
184C22×Dic5C2×D511yescharacteristic, complemented
185#711—104no—
186#711—104no—
187D4×D5—24no—
188C2×D20C2×D511yescharacteristic
189C2×C4×D5—51no—
190C2×C4×D5—51no—
191C2×C4×D5—51no—
192C2×C4×D5C2×D511yescharacteristic
193C2×Dic10—51no—
194C2×Dic10—51no—
195C5×C4⋊C4—51no—
196C5×C22⋊C4—51no—
197C5×C22⋊C4—51no—
198C22⋊Dic5—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—102no—
212C22⋊Dic5—102no—
213C22⋊Dic5—102no—
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#685C2×D511yescharacteristic, complemented
223C4⋊Dic5—51no—
224#683—51no—
225#683—51no—
226#683—51no—
227#683—51no—
228#683—51no—
229#683—102no—
230#683—102no—
231#683C2×D511yescharacteristic, complemented
232C4×Dic5—51no—
233C4×Dic5—51no—
234#546—251noSylow
235C2×C52—22no—
236C2×C52—41no—
237C2×C52C22×D411yescharacteristic
238C2×C52C2×#4111yescharacteristic
239C2×C52H(5,𝔽2)11yescharacteristic
240C5×D5—22no—
241C5×D5—22no—
242C5×D5—41no—
243C5×D5—41no—
244C23×C5—22no—
245C23×C5—22no—
246C23×C5—51no—
247C23×C5—51no—
248C23×C5—51no—
249C23×C5—51no—
250C23×C5—102no—
251C23×C5—102no—
252C23×C5—102no—
253C23×C5—102no—
254C23×C5—201no—
255C23×C5—201no—
256C23×C5C22×D511yescharacteristic
257C23×C5C22×D511yescharacteristic
258C23×C5C22×D511yescharacteristic
259C23×C5C22×D511yescharacteristic
260C22×D5—22no—
261C22×D5—22no—
262C22×D5—22no—
263C22×D5—22no—
264C22×D5—41no—
265C22×D5—41no—
266C22×D5C22×D511yescharacteristic
267C22×D5C22×D511yescharacteristic
268C22×D5C22×D511yescharacteristic
269C22×D5C22×D511yescharacteristic
270C5×D4—22no—
271C5×D4—22no—
272C5×D4—44no—
273C5×D4—54no—
274C5×D4—102no—
275C5×D4—102no—
276C5×D4—102no—
277C5×D4—201no—
278C2×C4×C5—22no—
279C2×C4×C5—51no—
280C2×C4×C5—51no—
281C2×C4×C5—51no—
282C2×C4×C5—51no—
283C2×C4×C5—102no—
284C2×C4×C5—201no—
285C2×C4×C5C22×D511yescharacteristic
286C2×C4×C5C22×D511yescharacteristic
287C22⋊D5—22no—
288C22⋊D5—22no—
289C22⋊D5—41no—
290C22⋊D5—102no—
291C22⋊D5—102no—
292C22⋊D5—102no—
293C22⋊D5—102no—
294C22⋊D5—201no—
295C22⋊D5—201no—
296C22⋊D5C22⋊D514yescomplemented
297C2×Dic5—41no—
298C2×Dic5—51no—
299C2×Dic5—51no—
300C2×Dic5—51no—
301C2×Dic5—51no—
302C2×Dic5—51no—
303C2×Dic5—51no—
304C2×Dic5—51no—
305C2×Dic5—51no—
306C2×Dic5—102no—
307C2×Dic5—102no—
308C2×Dic5—102no—
309C2×Dic5—102no—
310C2×Dic5—102no—
311C2×Dic5—102no—
312C2×Dic5—102no—
313C2×Dic5—102no—
314C2×Dic5—204no—
315C2×Dic5—201no—
316C2×Dic5C22×D511yescharacteristic
317C2×Dic5C22×D511yescharacteristic
318C2×Dic5C22×D511yescharacteristic
319C2×Dic5C22×D511yescharacteristic
320D20—22no—
321C4×D5—22no—
322C4×D5—102no—
323C4×D5—102no—
324C4×D5—201no—
325Dic10—102no—
326Dic10—102no—
327C2×#41—251no—
328C22×D4—51no—
329#124—251no—
330#123—251no—
331#123—251no—
332C22⋊Q8—251no—
333#121—51no—
334#121—251no—
335#121—251no—
336#120—251no—
337#120—251no—
338C4×D4—251no—
339C4×D4—251no—
340C2×C22⋊C4—251no—
341C2×C22⋊C4—251no—
342C52#54611yescharacteristic, complemented, Sylow
343C22×C5—21no—
344C22×C5—21no—
345C22×C5—21no—
346C22×C5—21no—
347C22×C5—21no—
348C22×C5—21no—
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—24no—
354C22×C5—22no—
355C22×C5—42no—
356C22×C5—42no—
357C22×C5—42no—
358C22×C5—42no—
359C22×C5—42no—
360C22×C5—52no—
361C22×C5—52no—
362C22×C5—101no—
363C22×C5—101no—
364C22×C5—101no—
365C22×C5—101no—
366C22×C5—101no—
367C22×C5—101no—
368C22×C5—101no—
369C22×C5—101no—
370C22×C5—101no—
371C22×C5—101no—
372C22×C5—204no—
373C22×C5—204no—
374C22×C5D4×D512yes—
375C22×C5C2×C22⋊D512yes—
376C22×C5C23×D511yescharacteristic
377C22×C5C23×D511yescharacteristic
378C2×D5—21no—
379C2×D5—21no—
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—44no—
390C2×D5#71112yescomplemented
391C2×D5C2×C22⋊D512yes—
392C4×C5—21no—
393C4×C5—21no—
394C4×C5—42no—
395C4×C5—52no—
396C4×C5—101no—
397C4×C5—101no—
398C4×C5—101no—
399Dic5—21no—
400Dic5—21no—
401Dic5—21no—
402Dic5—101no—
403Dic5—101no—
404Dic5—101no—
405Dic5—101no—
406Dic5—101no—
407Dic5—101no—
408Dic5—101no—
409Dic5—101no—
410Dic5—202no—
411Dic5—202no—
412Dic5—202no—
413Dic5—202no—
414Dic5C2×C22⋊D512yes—
415C24—51no—
416C24—51no—
417#41—504no—
418C2×Q8—251no—
419C2×D4—51no—
420C2×D4—51no—
421C2×D4—51no—
422C2×D4—51no—
423C2×D4—51no—
424C2×D4—104no—
425C2×D4—251no—
426C2×D4—251no—
427C2×D4—251no—
428C2×D4D5211yescharacteristic
429C22×C4—51no—
430C22×C4—51no—
431C22×C4—251no—
432C22×C4—251no—
433C22×C4—251no—
434C22×C4—251no—
435C4⋊C4—51no—
436C4⋊C4—251no—
437C4⋊C4—251no—
438C4⋊C4—251no—
439C22⋊C4—51no—
440C22⋊C4—51no—
441C22⋊C4—251no—
442C22⋊C4—251no—
443C22⋊C4—251no—
444C22⋊C4—251no—
445C22⋊C4—251no—
446C22⋊C4—251no—
447C22⋊C4—251no—
448C22⋊C4—251no—
449C22⋊C4—251no—
450C22⋊C4—251no—
451C42—251no—
452C2×C5—22no—
453C2×C5—22no—
454C2×C5—22no—
455C2×C5—22no—
456C2×C5—22no—
457C2×C5—44no—
458C2×C5—41no—
459C2×C5—41no—
460C2×C5—82no—
461C2×C5—102no—
462C2×C5—102no—
463C2×C5—201no—
464C2×C5—201no—
465C2×C5C2×D4×D511yescharacteristic
466C2×C5C2×#71111yescharacteristic
467C2×C5#414411yescharacteristic
468C2×C5#414411yescharacteristic
469C2×C5D5×#4111yescharacteristic
470C2×C5C22×C22⋊D511yescharacteristic
471D5—22no—
472D5—22no—
473D5—41no—
474D5—41no—
475C23—51no—
476C23—51no—
477C23—51no—
478C23—51no—
479C23—102no—
480C23—102no—
481C23—102no—
482C23—102no—
483C23—201no—
484C23—201no—
485C23C2×D5211yescharacteristic
486C23C2×D5211yescharacteristic
487Q8—502no—
488D4—22no—
489D4—54no—
490D4—102no—
491D4—102no—
492D4—102no—
493D4—201no—
494D4—502no—
495D4—502no—
496D4—1001no—
497C2×C4—51no—
498C2×C4—51no—
499C2×C4—51no—
500C2×C4—51no—
501C2×C4—102no—
502C2×C4—201no—
503C2×C4—251no—
504C2×C4—251no—
505C2×C4—251no—
506C2×C4—251no—
507C2×C4—502no—
508C2×C4—502no—
509C2×C4—502no—
510C2×C4—1001no—
511C2×C4C2×D5211yescharacteristic
512C5—22no—
513C5#6594411yescharacteristic, complemented
514C5#6614011yescharacteristic, complemented
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—52no—
521C22—52no—
522C22—101no—
523C22—101no—
524C22—101no—
525C22—101no—
526C22—101no—
527C22—101no—
528C22—101no—
529C22—101no—
530C22—101no—
531C22—101no—
532C22—204no—
533C22—204no—
534C22D5×C22⋊D512yes—
535C22C22×D5211yescharacteristic
536C4—21no—
537C4—52no—
538C4—101no—
539C4—101no—
540C4—101no—
541C4—501no—
542C4—501no—
543C4—501no—
544C4—501no—
545C2—22no—
546C2—41no—
547C2—102no—
548C2—102no—
549C2—201no—
550C2—201no—
551C2D5×#71111yescharacteristic
552C2#1171747811yescharacteristic
553C2C2×D5×C22⋊D511yescharacteristic
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×C5×D4×D5C2semidirect
2. C2×#87880C2semidirect
3. #11717255C2semidirect
4. C5×#4085C2semidirect
5. #11717094C2semidirect
6. #11717094C2semidirect
7. Dic5×C22⋊D5C2semidirect
8. #11717088C2semidirect
9. D5×C22⋊Dic5C2semidirect
10. C2×#87778C2semidirect
11. #11717077C2semidirect
12. #11717004C2semidirect
13. #11717002C2semidirect
14. #11716997C2semidirect
15. #11716969C2semidirect
19. C22×#1039C22semidirect
22. C2×C5×C22⋊D5C22semidirect
23. C2×C5×C22⋊D5C22semidirect
24. C22×C5×Dic5C22semidirect
28. C2×#6039C22semidirect
30. C2×#6037C22semidirect
31. C2×#6037C22semidirect
33. C2×D5×Dic5C22semidirect
34. C2×D5×Dic5C22semidirect
35. C2×D5×Dic5C22semidirect
36. C22⋊#1039C22semidirect
37. C22⋊#1039C22semidirect
38. C22⋊#1039C22semidirect
39. #87806C22semidirect
40. #87806C22semidirect
41. C5×C22⋊Dic5C22semidirect
42. C5×#685C22semidirect
43. C5×#683C22semidirect
44. #87780C22semidirect
48. #87778C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
77. C5×C22⋊D5D4semidirect
95. C2×D4×D5D5semidirect
103. #4085D5semidirect
143. C2×C5×D5#41semidirect
176. C2×C22⋊D5C2×D5semidirect
177. C2×C22⋊D5C2×D5semidirect
184. C22×Dic5C2×D5semidirect
214. C22⋊Dic5C2×D5semidirect
222. #685C2×D5semidirect
231. #683C2×D5semidirect
296. C22⋊D5C22⋊D5semidirect
342. C52#546semidirect
390. C2×D5#711semidirect
513. C5#65944semidirect
514. C5#66140semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D4×D52
C2×#878802
#117172552
C5×#40852
#117170942
Dic5×C22⋊D52
#117170882
D5×C22⋊Dic52
C2×#877782
#117170772
#117170042
#117170022
#117169972
#117169692
#661405
#659445