Finite Group Databasebeta

← #13105237

Subgroups — #13105237

Order 1600.

742 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6037C211yesmaximal, characteristic, complemented
2C2×#87870C211yesmaximal, characteristic, complemented
3#11717198C211yesmaximal, characteristic, complemented
4C5×#4028C211yesmaximal, characteristic, complemented
5C5×D5×C22⋊C4C211yesmaximal, characteristic, complemented
6#11717094C211yesmaximal, characteristic, complemented
7#11717091C211yesmaximal, characteristic, complemented
8D5×C22⋊Dic5C211yesmaximal, characteristic, complemented
9#11717031C211yesmaximal, characteristic, complemented
10#11717026C211yesmaximal, characteristic, complemented
11#11717023C211yesmaximal, characteristic, complemented
12#11717004C211yesmaximal, characteristic, complemented
13#11717004C211yesmaximal, characteristic, complemented
14#11717003C211yesmaximal, characteristic, complemented
15#11716997C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C23×C5×D5C2211yescharacteristic
18C22×#1039C2211yescharacteristic
19C2×#6045C2211yescharacteristic, complemented
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×D20C2211yescharacteristic, complemented
22C2×C5×D20C2211yescharacteristic, complemented
23C2×C4×C5×D5C2211yescharacteristic, complemented
24C2×C4×C5×D5C2211yescharacteristic, complemented
25C2×#6037—24no—
26C2×#6037C2211yescharacteristic
27C2×#6037C2211yescharacteristic
28C2×#6037C2211yescharacteristic
29C2×#6037C2211yescharacteristic
30C2×#6037C2211yescharacteristic, complemented
31C2×#6037C2211yescharacteristic, complemented
32C2×D5×Dic5C2211yescharacteristic, complemented
33C2×D5×Dic5C2211yescharacteristic, complemented
34C2×D5×Dic5C2211yescharacteristic, complemented
35#87870—24no—
36C52×C22⋊C4C2211yescharacteristic
37C22⋊#1039C2211yescharacteristic
38C4⋊#1039C2211yescharacteristic, complemented
39#87806C2211yescharacteristic, complemented
40#87806C2211yescharacteristic, complemented
41C5×C22⋊Dic5C2211yescharacteristic
42C5×#685C2211yescharacteristic, complemented
43C5×#685C2211yescharacteristic, complemented
44C5×#685C2211yescharacteristic, complemented
45C5×#685C2211yescharacteristic, complemented
46#87780C2211yescharacteristic, complemented
47#87778C2211yescharacteristic, complemented
48#87778C2211yescharacteristic, complemented
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#65925—51nomaximal
54#65824—51nomaximal
55C23×C52C2311yescharacteristic
56C22×C5×D5—22no—
57C22×C5×D5—22no—
58C22×C5×D5—22no—
59C22×C5×D5—22no—
60C22×C5×D5—41no—
61C22×C5×D5—41no—
62C22×C5×D5C2311yescharacteristic
63C22×C5×D5C2311yescharacteristic
64C22×C5×D5C2311yescharacteristic
65C22×C5×D5C2311yescharacteristic
66C22×C5×D5C2311yescharacteristic
67C2×C4×C52C2311yescharacteristic
68C2×C4×C52C2311yescharacteristic
69C2×#1039—22no—
70C2×#1039C2311yescharacteristic
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C2×#1039C2311yescharacteristic
74#6045—22no—
75C5×C22⋊D5—22no—
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78C2×C5×Dic5C2311yescharacteristic
79C5×D20—22no—
80C5×D20—41no—
81C4×C5×D5—22no—
82C4×C5×D5—41no—
83#6037—22no—
84#6037—22no—
85#6037—22no—
86#6037—22no—
87#6037—41no—
88#6037D414yescomplemented
89D5×Dic5—22no—
90D5×Dic5—22no—
91D5×Dic5—41no—
92C22×C22⋊D5—51no—
93C22×C22⋊D5—51no—
94C2×#711—51no—
95C2×#709—51no—
96C5×#120—51no—
97C2×C5×C22⋊C4—51no—
98#4099—51no—
99C2×C22⋊Dic5—51no—
100#4085—51no—
101#4085—51no—
102#4084—51no—
103#4083—51no—
104D4×Dic5—51no—
105#4076—51no—
106#4075—51no—
107C4×C22⋊D5—51no—
108#4032—51no—
109#4031—51no—
110#4031—51no—
111#4030—51no—
112#4030—51no—
113#4029—51no—
114#4029—51no—
115#4028D511yescharacteristic, complemented
116#4027—51no—
117#4027—51no—
118D5×C22⋊C4—51no—
119D5×C22⋊C4D511yescharacteristic, complemented
120#4024—51no—
121#4024—51no—
122#4024—102no—
123C22×C52—21no—
124C22×C52—21no—
125C22×C52#4112yescomplemented
126C22×C52C2411yescharacteristic
127C2×C5×D5—21no—
128C2×C5×D5—21no—
129C2×C5×D5—21no—
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×D5C2×D412yes—
144C4×C52—21no—
145C4×C52—21no—
146#1039—21no—
147#1039—21no—
148#1039—21no—
149#1039C2×D412yes—
150C5×Dic5—21no—
151C5×Dic5—21no—
152C5×Dic5—21no—
153C24×C5—51no—
154C24×C5—51no—
155C23×D5C2×D511yescharacteristic
156C23×D5C2×D511yescharacteristic
157C2×C5×D4—51no—
158C2×C5×D4—51no—
159C2×C5×D4—51no—
160C22×C4×C5—51no—
161C22×C4×C5—51no—
162C2×C22⋊D5—51no—
163C2×C22⋊D5—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—104no—
175C2×C22⋊D5—104no—
176C2×C22⋊D5C2×D511yescharacteristic
177C22×Dic5—51no—
178C22×Dic5—51no—
179C22×Dic5—51no—
180C22×Dic5—51no—
181C22×Dic5—51no—
182C22×Dic5—102no—
183#711—104no—
184#709—104no—
185C2×D20C2×D511yescharacteristic, complemented
186C2×D20C2×D511yescharacteristic, complemented
187C2×C4×D5—51no—
188C2×C4×D5—51no—
189C2×C4×D5—51no—
190C2×C4×D5C2×D511yescharacteristic, complemented
191C2×C4×D5C2×D511yescharacteristic, complemented
192C2×Dic10—51no—
193C2×Dic10—51no—
194C2×Dic10—102no—
195C5×C22⋊C4—22no—
196C5×C22⋊C4—51no—
197C5×C22⋊C4—51no—
198C5×C22⋊C4—51no—
199C5×C22⋊C4—51no—
200C5×C22⋊C4—51no—
201C5×C22⋊C4C2×D511yescharacteristic
202C5×C22⋊C4C2×D511yescharacteristic
203C22⋊Dic5—51no—
204C22⋊Dic5—51no—
205C22⋊Dic5—51no—
206C22⋊Dic5—51no—
207C22⋊Dic5—51no—
208C22⋊Dic5—51no—
209C22⋊Dic5—51no—
210C22⋊Dic5—102no—
211C22⋊Dic5C2×D511yescharacteristic
212#685—51no—
213#685—51no—
214#685—51no—
215#685—51no—
216#685—51no—
217#685C2×D511yescharacteristic, complemented
218#685C2×D511yescharacteristic, complemented
219#685C2×D511yescharacteristic, complemented
220#685C2×D511yescharacteristic, complemented
221C4⋊Dic5—51no—
222C4⋊Dic5—51no—
223C4⋊Dic5—51no—
224C4⋊Dic5—102no—
225#683—51no—
226#683—51no—
227#683—51no—
228#683—51no—
229#683—51no—
230#683—102no—
231#683—102no—
232C4×Dic5—51no—
233C4×Dic5—51no—
234#546—251noSylow
235C2×C52—22no—
236C2×C52C22×D411yescharacteristic
237C2×C52C2×#4111yescharacteristic
238C2×C52H(5,𝔽2)11yescharacteristic
239C5×D5—22no—
240C5×D5—22no—
241C5×D5—41no—
242C5×D5—41no—
243C5×D5—41no—
244C23×C5—22no—
245C23×C5—51no—
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
258C22×D5—22no—
259C22×D5—22no—
260C22×D5—22no—
261C22×D5—22no—
262C22×D5—41no—
263C22×D5—41no—
264C22×D5C22×D511yescharacteristic
265C22×D5C22×D511yescharacteristic
266C22×D5C22×D511yescharacteristic
267C22×D5C22×D511yescharacteristic
268C22×D5C22×D511yescharacteristic
269C5×D4—102no—
270C5×D4—102no—
271C5×D4—201no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—51no—
275C2×C4×C5—51no—
276C2×C4×C5—51no—
277C2×C4×C5—102no—
278C2×C4×C5—201no—
279C2×C4×C5C22×D511yescharacteristic
280C2×C4×C5C22×D511yescharacteristic
281C2×C4×C5C22×D511yescharacteristic
282C2×C4×C5C22×D511yescharacteristic
283C22⋊D5—22no—
284C22⋊D5—54no—
285C22⋊D5—54no—
286C22⋊D5—102no—
287C22⋊D5—102no—
288C22⋊D5—102no—
289C22⋊D5—102no—
290C22⋊D5—102no—
291C22⋊D5—102no—
292C22⋊D5—102no—
293C22⋊D5—102no—
294C22⋊D5—201no—
295C22⋊D5—201no—
296C2×Dic5—51no—
297C2×Dic5—51no—
298C2×Dic5—51no—
299C2×Dic5—51no—
300C2×Dic5—51no—
301C2×Dic5—51no—
302C2×Dic5—51no—
303C2×Dic5—51no—
304C2×Dic5—102no—
305C2×Dic5—102no—
306C2×Dic5—102no—
307C2×Dic5—102no—
308C2×Dic5—102no—
309C2×Dic5—102no—
310C2×Dic5—102no—
311C2×Dic5—102no—
312C2×Dic5—204no—
313C2×Dic5—201no—
314C2×Dic5C22×D511yescharacteristic
315C2×Dic5C22×D511yescharacteristic
316C2×Dic5C22×D511yescharacteristic
317D20—22no—
318D20—41no—
319C4×D5—22no—
320C4×D5—41no—
321C4×D5—102no—
322C4×D5—102no—
323C4×D5—201no—
324Dic10—102no—
325Dic10—102no—
326Dic10—204no—
327C2×#41—251no—
328C22×D4—251no—
329#124—251no—
330#123—251no—
331#123—251no—
332C22⋊Q8—251no—
333#121—251no—
334#121—251no—
335#121—251no—
336#120—51no—
337#120—251no—
338C4×D4—251no—
339C4×D4—251no—
340C2×C22⋊C4—51no—
341C2×C22⋊C4—251no—
342C52#54611yescharacteristic, complemented, Sylow
343C22×C5—21no—
344C22×C5—21no—
345C22×C5—21no—
346C22×C5—21no—
347C22×C5—24no—
348C22×C5—22no—
349C22×C5—42no—
350C22×C5—42no—
351C22×C5—52no—
352C22×C5—52no—
353C22×C5—101no—
354C22×C5—101no—
355C22×C5—101no—
356C22×C5—101no—
357C22×C5—101no—
358C22×C5—101no—
359C22×C5—101no—
360C22×C5—101no—
361C22×C5—101no—
362C22×C5—101no—
363C22×C5—101no—
364C22×C5—101no—
365C22×C5—101no—
366C22×C5—204no—
367C22×C5—204no—
368C22×C5#70912yescomplemented
369C22×C5#71112yescomplemented
370C22×C5C23×D511yescharacteristic
371C22×C5C23×D511yescharacteristic
372C2×D5—21no—
373C2×D5—21no—
374C2×D5—21no—
375C2×D5—21no—
376C2×D5—21no—
377C2×D5—21no—
378C2×D5—21no—
379C2×D5—21no—
380C2×D5—21no—
381C2×D5—21no—
382C2×D5—21no—
383C2×D5—21no—
384C2×D5—21no—
385C2×D5—44no—
386C2×D5—44no—
387C2×D5D4×D512yes—
388C2×D5D4×D512yes—
389C4×C5—21no—
390C4×C5—21no—
391C4×C5—21no—
392C4×C5—21no—
393C4×C5—42no—
394C4×C5—42no—
395C4×C5—101no—
396C4×C5—101no—
397C4×C5—101no—
398Dic5—21no—
399Dic5—21no—
400Dic5—21no—
401Dic5—52no—
402Dic5—52no—
403Dic5—104no—
404Dic5—101no—
405Dic5—101no—
406Dic5—101no—
407Dic5—101no—
408Dic5—101no—
409Dic5—101no—
410Dic5—202no—
411Dic5—202no—
412Dic5—202no—
413C24—51no—
414C24—51no—
415#41—504no—
416C2×Q8—251no—
417C2×D4—51no—
418C2×D4—51no—
419C2×D4—51no—
420C2×D4—251no—
421C2×D4—251no—
422C2×D4—251no—
423C2×D4—251no—
424C2×D4—251no—
425C2×D4—251no—
426C2×D4—504no—
427C22×C4—51no—
428C22×C4—51no—
429C22×C4—251no—
430C22×C4—251no—
431C22×C4—251no—
432C22×C4—251no—
433C4⋊C4—251no—
434C4⋊C4—251no—
435C4⋊C4—251no—
436C4⋊C4—251no—
437C22⋊C4—51no—
438C22⋊C4—51no—
439C22⋊C4—51no—
440C22⋊C4—51no—
441C22⋊C4—51no—
442C22⋊C4—251no—
443C22⋊C4—251no—
444C22⋊C4—251no—
445C22⋊C4—251no—
446C22⋊C4—251no—
447C22⋊C4—251no—
448C22⋊C4D5211yescharacteristic
449C42—251no—
450C2×C5—22no—
451C2×C5—22no—
452C2×C5—22no—
453C2×C5—22no—
454C2×C5—22no—
455C2×C5—44no—
456C2×C5—102no—
457C2×C5—102no—
458C2×C5—201no—
459C2×C5—201no—
460C2×C5—201no—
461C2×C5C2×#70911yescharacteristic
462C2×C5C2×D4×D511yescharacteristic
463C2×C5C2×D4×D511yescharacteristic
464C2×C5C2×#71111yescharacteristic
465C2×C5#414411yescharacteristic
466C2×C5#414411yescharacteristic
467D5—22no—
468D5—22no—
469D5—41no—
470D5—41no—
471D5—41no—
472C23—51no—
473C23—51no—
474C23—51no—
475C23—51no—
476C23—51no—
477C23—102no—
478C23—102no—
479C23—102no—
480C23—102no—
481C23—201no—
482C23—201no—
483C23C2×D5211yescharacteristic
484Q8—502no—
485D4—102no—
486D4—102no—
487D4—201no—
488D4—254no—
489D4—502no—
490D4—502no—
491D4—502no—
492D4—502no—
493D4—1001no—
494C2×C4—51no—
495C2×C4—51no—
496C2×C4—51no—
497C2×C4—102no—
498C2×C4—201no—
499C2×C4—251no—
500C2×C4—251no—
501C2×C4—251no—
502C2×C4—251no—
503C2×C4—502no—
504C2×C4—502no—
505C2×C4—502no—
506C2×C4—1001no—
507C2×C4C2×D5211yescharacteristic
508C2×C4C2×D5211yescharacteristic
509C5—22no—
510C5#6582411yescharacteristic, complemented
511C5#6592511yescharacteristic, complemented
512C22—21no—
513C22—21no—
514C22—52no—
515C22—52no—
516C22—101no—
517C22—101no—
518C22—101no—
519C22—101no—
520C22—101no—
521C22—101no—
522C22—101no—
523C22—101no—
524C22—101no—
525C22—101no—
526C22—101no—
527C22—101no—
528C22—101no—
529C22—204no—
530C22—204no—
531C22#8787012yescomplemented
532C22C22×D5211yescharacteristic
533C4—21no—
534C4—21no—
535C4—101no—
536C4—101no—
537C4—101no—
538C4—252no—
539C4—501no—
540C4—501no—
541C4—501no—
542C2—22no—
543C2—102no—
544C2—102no—
545C2—201no—
546C2—201no—
547C2—201no—
548C2C2×#8787011yescharacteristic
549C2D4×D5211yescharacteristic
550C2#1171747811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#546
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6037C2semidirect
2. C2×#87870C2semidirect
3. #11717198C2semidirect
4. C5×#4028C2semidirect
5. C5×D5×C22⋊C4C2semidirect
6. #11717094C2semidirect
7. #11717091C2semidirect
8. D5×C22⋊Dic5C2semidirect
9. #11717031C2semidirect
10. #11717026C2semidirect
11. #11717023C2semidirect
12. #11717004C2semidirect
13. #11717004C2semidirect
14. #11717003C2semidirect
15. #11716997C2semidirect
19. C2×#6045C22semidirect
21. C2×C5×D20C22semidirect
22. C2×C5×D20C22semidirect
23. C2×C4×C5×D5C22semidirect
24. C2×C4×C5×D5C22semidirect
30. C2×#6037C22semidirect
31. C2×#6037C22semidirect
32. C2×D5×Dic5C22semidirect
33. C2×D5×Dic5C22semidirect
34. C2×D5×Dic5C22semidirect
38. C4⋊#1039C22semidirect
39. #87806C22semidirect
40. #87806C22semidirect
42. C5×#685C22semidirect
43. C5×#685C22semidirect
44. C5×#685C22semidirect
45. C5×#685C22semidirect
46. #87780C22semidirect
47. #87778C22semidirect
48. #87778C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
88. #6037D4semidirect
115. #4028D5semidirect
119. D5×C22⋊C4D5semidirect
125. C22×C52#41semidirect
185. C2×D20C2×D5semidirect
186. C2×D20C2×D5semidirect
190. C2×C4×D5C2×D5semidirect
191. C2×C4×D5C2×D5semidirect
217. #685C2×D5semidirect
218. #685C2×D5semidirect
219. #685C2×D5semidirect
220. #685C2×D5semidirect
342. C52#546semidirect
368. C22×C5#709semidirect
369. C22×C5#711semidirect
510. C5#65824semidirect
511. C5#65925semidirect
531. C22#87870semidirect

Maximal subgroups

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

subgroupindex
C22×#60372
C2×#878702
#117171982
C5×#40282
C5×D5×C22⋊C42
#117170942
#117170912
D5×C22⋊Dic52
#117170312
#117170262
#117170232
#117170042
#117170032
#117169972
#659255
#658245