Finite Group Databasebeta

← #13105385

Subgroups — #13105385

Order 1600.

762 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C22⋊D5C211yesmaximal, characteristic, complemented
2C2×#87879C211yesmaximal, characteristic, complemented
3#11717201C211yesmaximal, characteristic, complemented
4C5×#4032C211yesmaximal, characteristic, complemented
5C5×#4028C211yesmaximal, characteristic, complemented
6#11717098C211yesmaximal, characteristic, complemented
7C2×#87778C211yesmaximal, characteristic, complemented
8#11717040C211yesmaximal, characteristic, complemented
9#11717039C211yesmaximal, characteristic, complemented
10#11717034C211yesmaximal, characteristic, complemented
11#11717033C211yesmaximal, characteristic, complemented
12#11717017C211yesmaximal, characteristic, complemented
13#11717014C211yesmaximal, characteristic, complemented
14#11717004C211yesmaximal, characteristic, complemented
15#11716972C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C22×D52C2211yescharacteristic
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C22×#1039C2211yescharacteristic, complemented
20C2×C4×C5⋊D5C2211yescharacteristic, complemented
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C5×C22⋊D5C2211yescharacteristic, complemented
23C22×C5×Dic5C2211yescharacteristic, complemented
24C2×C5×D20C2211yescharacteristic
25C2×C5×D20C2211yescharacteristic, complemented
26D5×C22⋊D5—24no—
27C2×#6039C2211yescharacteristic, complemented
28C2×#6038C2211yescharacteristic
29C2×#6038C2211yescharacteristic, complemented
30C2×#6037C2211yescharacteristic
31C2×#6037C2211yescharacteristic, complemented
32C2×#6036C2211yescharacteristic, complemented
33#87879—24no—
34C2×D5×Dic5C2211yescharacteristic
35C2×D5×Dic5C2211yescharacteristic, complemented
36C52×C22⋊C4C2211yescharacteristic, complemented
37#87808C2211yescharacteristic, complemented
38#87806C2211yescharacteristic, complemented
39C4×#1039C2211yescharacteristic, complemented
40C5×#685C2211yescharacteristic
41C5×#685C2211yescharacteristic, complemented
42C5×#685C2211yescharacteristic, complemented
43C5×#685C2211yescharacteristic, complemented
44C5×C4⋊Dic5C2211yescharacteristic, complemented
45C5×C4⋊Dic5C2211yescharacteristic, complemented
46#87781C2211yescharacteristic, complemented
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52#87778C2211yescharacteristic, complemented
53#65988—51nomaximal
54#65925—51nomaximal
55C23×C52C2311yescharacteristic
56C22×C5⋊D5C2311yescharacteristic
57C22×C5×D5—22no—
58C22×C5×D5—22no—
59C22×C5×D5—41no—
60C22×C5×D5C2311yescharacteristic
61C22×C5×D5C2311yescharacteristic
62C22×C5×D5C2311yescharacteristic
63C22×C5×D5C2311yescharacteristic
64C2×D52—22no—
65C2×D52—22no—
66C2×D52—41no—
67C2×C4×C52C2311yescharacteristic
68C2×C4×C52C2311yescharacteristic
69C22⋊(C5⋊D5)D414yescomplemented
70C2×#1039—41no—
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C2×#1039C2311yescharacteristic
74C4×C5⋊D5—41no—
75C5×C22⋊D5—22no—
76C5×C22⋊D5—22no—
77C2×C5×Dic5—22no—
78C2×C5×Dic5C2311yescharacteristic
79C2×C5×Dic5C2311yescharacteristic
80C2×C5×Dic5C2311yescharacteristic
81C2×C5×Dic5C2311yescharacteristic
82C5×D20—22no—
83C5×D20—41no—
84#6039—22no—
85#6038—22no—
86#6038—22no—
87#6037—22no—
88#6037—41no—
89#6036—22no—
90D5×Dic5—22no—
91D5×Dic5—22no—
92C22×C22⋊D5—51no—
93C2×#711—51no—
94C2×D4×D5—51no—
95C2×#709—51no—
96C5×#123—51no—
97C5×#120—51no—
98C2×C22⋊Dic5—51no—
99#4085—51no—
100#4085—51no—
101#4085—51no—
102#4084—51no—
103#4083—51no—
104D4×Dic5—51no—
105C2×#685—51no—
106#4042—51no—
107#4041—51no—
108#4040—51no—
109#4039—51no—
110#4032D511yescharacteristic, complemented
111#4031—51no—
112#4031—51no—
113#4030—51no—
114#4029—51no—
115#4028—51no—
116#4028D511yescharacteristic, complemented
117#4027—51no—
118#4027—51no—
119#4027—102no—
120D5×C22⋊C4—51no—
121D5×C22⋊C4—51no—
122#4024—51no—
123C22×C52—21no—
124C22×C52—21no—
125C22×C52C2×D412yes—
126C22×C52C2411yescharacteristic
127C2×C5⋊D5—21no—
128C2×C5⋊D5—21no—
129C2×C5⋊D5C2×D412yes—
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—21no—
141C2×C5×D5—44no—
142C2×C5×D5#4112yescomplemented
143D52—44no—
144C4×C52—21no—
145C4×C52—21no—
146#1039—21no—
147#1039—21no—
148#1039C2×D412yes—
149C5×Dic5—21no—
150C5×Dic5—21no—
151C5×Dic5—21no—
152C5×Dic5—21no—
153C24×C5—51no—
154C23×D5—51no—
155C23×D5—51no—
156C23×D5C2×D511yescharacteristic
157C2×C5×D4—51no—
158C2×C5×D4—51no—
159C2×C5×D4—51no—
160C2×C5×D4—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—102no—
171C2×C22⋊D5—104no—
172C2×C22⋊D5C2×D511yescharacteristic
173C2×C22⋊D5C2×D511yescharacteristic, complemented
174C22×Dic5—51no—
175C22×Dic5—51no—
176C22×Dic5—51no—
177C22×Dic5—51no—
178C22×Dic5—102no—
179C22×Dic5C2×D511yescharacteristic, complemented
180#711—104no—
181D4×D5—104no—
182#709—104no—
183C2×D20—51no—
184C2×D20—51no—
185C2×D20C2×D511yescharacteristic
186C2×D20C2×D511yescharacteristic, complemented
187C2×C4×D5—51no—
188C2×C4×D5—51no—
189C2×C4×D5—51no—
190C2×C4×D5—51no—
191C2×C4×D5—51no—
192C2×C4×D5—51no—
193C2×C4×D5—102no—
194C2×Dic10—51no—
195C2×Dic10—51no—
196C5×C4⋊C4—51no—
197C5×C4⋊C4—51no—
198C5×C22⋊C4—22no—
199C5×C22⋊C4—51no—
200C5×C22⋊C4—51no—
201C5×C22⋊C4—51no—
202C5×C22⋊C4—51no—
203C5×C22⋊C4C2×D511yescharacteristic, complemented
204C5×C22⋊C4C2×D511yescharacteristic, complemented
205C22⋊Dic5—51no—
206C22⋊Dic5—51no—
207C22⋊Dic5—51no—
208C22⋊Dic5—51no—
209C22⋊Dic5—51no—
210C22⋊Dic5—51no—
211#685—51no—
212#685—51no—
213#685—51no—
214#685—51no—
215#685—51no—
216#685—51no—
217#685—51no—
218#685—51no—
219#685—102no—
220#685C2×D511yescharacteristic
221#685C2×D511yescharacteristic, complemented
222#685C2×D511yescharacteristic, complemented
223#685C2×D511yescharacteristic, complemented
224C4⋊Dic5C2×D511yescharacteristic, complemented
225C4⋊Dic5C2×D511yescharacteristic, complemented
226#683—51no—
227#683—51no—
228#683—51no—
229#683—51no—
230#683—102no—
231C4×Dic5—51no—
232C4×Dic5—51no—
233C4×Dic5—102no—
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—102no—
250C23×C5—102no—
251C23×C5—201no—
252C23×C5C22×D511yescharacteristic
253C23×C5C22×D511yescharacteristic
254C22×D5—22no—
255C22×D5—22no—
256C22×D5—41no—
257C22×D5—51no—
258C22×D5—51no—
259C22×D5—102no—
260C22×D5—102no—
261C22×D5—102no—
262C22×D5—102no—
263C22×D5—102no—
264C22×D5—201no—
265C22×D5—201no—
266C22×D5C22×D511yescharacteristic
267C22×D5C22×D511yescharacteristic
268C22×D5C22×D511yescharacteristic
269C22×D5C22×D511yescharacteristic
270C5×D4—102no—
271C5×D4—102no—
272C5×D4—102no—
273C5×D4—201no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—51no—
277C2×C4×C5—51no—
278C2×C4×C5—51no—
279C2×C4×C5—51no—
280C2×C4×C5—102no—
281C2×C4×C5C22×D511yescharacteristic
282C2×C4×C5C22×D511yescharacteristic
283C2×C4×C5C22×D511yescharacteristic
284C2×C4×C5C22×D511yescharacteristic
285C22⋊D5—22no—
286C22⋊D5—22no—
287C22⋊D5—54no—
288C22⋊D5—54no—
289C22⋊D5—108no—
290C22⋊D5—102no—
291C22⋊D5—102no—
292C22⋊D5—102no—
293C22⋊D5—102no—
294C22⋊D5—201no—
295C22⋊D5—201no—
296C2×Dic5—22no—
297C2×Dic5—51no—
298C2×Dic5—51no—
299C2×Dic5—51no—
300C2×Dic5—51no—
301C2×Dic5—51no—
302C2×Dic5—51no—
303C2×Dic5—102no—
304C2×Dic5—102no—
305C2×Dic5—102no—
306C2×Dic5—102no—
307C2×Dic5—102no—
308C2×Dic5—201no—
309C2×Dic5—201no—
310C2×Dic5—204no—
311C2×Dic5C22×D511yescharacteristic
312C2×Dic5C22×D511yescharacteristic
313C2×Dic5C22×D511yescharacteristic
314C2×Dic5C22×D511yescharacteristic
315D20—22no—
316D20—41no—
317D20—102no—
318D20—102no—
319C4×D5—102no—
320C4×D5—102no—
321C4×D5—102no—
322C4×D5—102no—
323C4×D5—201no—
324C4×D5—201no—
325C4×D5—204no—
326Dic10—102no—
327Dic10—102no—
328C2×#41—251no—
329C22×D4—251no—
330#124—251no—
331#123—51no—
332#123—251no—
333C22⋊Q8—251no—
334#121—251no—
335#121—251no—
336#121—251no—
337#120—51no—
338#120—251no—
339C4×D4—251no—
340C4×D4—251no—
341C2×C22⋊C4—251no—
342C2×C22⋊C4—251no—
343C52#54611yescharacteristic, complemented, Sylow
344C22×C5—21no—
345C22×C5—21no—
346C22×C5—21no—
347C22×C5—21no—
348C22×C5—24no—
349C22×C5—22no—
350C22×C5—42no—
351C22×C5—42no—
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—204no—
365C22×C5D4×D512yes—
366C22×C5D4×D512yes—
367C22×C5C23×D511yescharacteristic
368C22×C5C23×D511yescharacteristic
369C2×D5—21no—
370C2×D5—21no—
371C2×D5—21no—
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—44no—
381C2×D5—52no—
382C2×D5—52no—
383C2×D5—104no—
384C2×D5—101no—
385C2×D5—101no—
386C2×D5—101no—
387C2×D5—101no—
388C2×D5—202no—
389C2×D5—202no—
390C2×D5—204no—
391C2×D5—204no—
392C2×D5#71112yescomplemented
393C4×C5—21no—
394C4×C5—21no—
395C4×C5—21no—
396C4×C5—21no—
397C4×C5—42no—
398C4×C5—42no—
399C4×C5—101no—
400C4×C5—101no—
401C4×C5—101no—
402C4×C5—101no—
403Dic5—21no—
404Dic5—21no—
405Dic5—21no—
406Dic5—21no—
407Dic5—52no—
408Dic5—52no—
409Dic5—104no—
410Dic5—104no—
411Dic5—101no—
412Dic5—101no—
413Dic5—101no—
414Dic5—101no—
415Dic5—202no—
416C24—51no—
417C24—251no—
418#41—504no—
419C2×Q8—251no—
420C2×D4—51no—
421C2×D4—51no—
422C2×D4—51no—
423C2×D4—51no—
424C2×D4—251no—
425C2×D4—251no—
426C2×D4—251no—
427C2×D4—251no—
428C2×D4—251no—
429C2×D4—504no—
430C22×C4—51no—
431C22×C4—251no—
432C22×C4—251no—
433C22×C4—251no—
434C22×C4—251no—
435C22×C4—251no—
436C4⋊C4—51no—
437C4⋊C4—51no—
438C4⋊C4—251no—
439C4⋊C4—251no—
440C22⋊C4—51no—
441C22⋊C4—51no—
442C22⋊C4—51no—
443C22⋊C4—51no—
444C22⋊C4—251no—
445C22⋊C4—251no—
446C22⋊C4—251no—
447C22⋊C4—251no—
448C22⋊C4—251no—
449C22⋊C4—251no—
450C22⋊C4—251no—
451C22⋊C4D5211yescharacteristic, complemented
452C42—251no—
453C2×C5—22no—
454C2×C5—22no—
455C2×C5—22no—
456C2×C5—22no—
457C2×C5—22no—
458C2×C5—44no—
459C2×C5—102no—
460C2×C5—201no—
461C2×C5—201no—
462C2×C5—201no—
463C2×C5C2×D4×D511yescharacteristic
464C2×C5C2×D4×D511yescharacteristic
465C2×C5C2×#71111yescharacteristic
466C2×C5#414411yescharacteristic
467C2×C5D5×#4111yescharacteristic
468C2×C5#414911yescharacteristic
469D5—22no—
470D5—41no—
471D5—41no—
472D5—41no—
473D5—102no—
474D5—102no—
475D5—204no—
476C23—51no—
477C23—51no—
478C23—51no—
479C23—51no—
480C23—102no—
481C23—102no—
482C23—201no—
483C23—251no—
484C23—502no—
485C23—502no—
486C23—1001no—
487C23C2×D5211yescharacteristic
488Q8—502no—
489D4—102no—
490D4—102no—
491D4—102no—
492D4—201no—
493D4—254no—
494D4—502no—
495D4—502no—
496D4—502no—
497D4—1001no—
498C2×C4—51no—
499C2×C4—51no—
500C2×C4—51no—
501C2×C4—51no—
502C2×C4—102no—
503C2×C4—251no—
504C2×C4—251no—
505C2×C4—251no—
506C2×C4—502no—
507C2×C4—502no—
508C2×C4—502no—
509C2×C4—1001no—
510C2×C4—1001no—
511C2×C4C2×D5211yescharacteristic
512C2×C4C2×D5211yescharacteristic
513C5—22no—
514C5#6592511yescharacteristic, complemented
515C5#6598811yescharacteristic, complemented
516C22—21no—
517C22—21no—
518C22—52no—
519C22—101no—
520C22—101no—
521C22—101no—
522C22—101no—
523C22—101no—
524C22—101no—
525C22—101no—
526C22—101no—
527C22—101no—
528C22—101no—
529C22—101no—
530C22—204no—
531C22—252no—
532C22—501no—
533C22—501no—
534C22—1004no—
535C22#8787712yes—
536C22C22×D5211yescharacteristic
537C4—21no—
538C4—21no—
539C4—101no—
540C4—101no—
541C4—101no—
542C4—101no—
543C4—252no—
544C4—501no—
545C4—501no—
546C2—22no—
547C2—102no—
548C2—201no—
549C2—201no—
550C2—201no—
551C2—502no—
552C2C2×#8787711yescharacteristic
553C2D5×#71111yescharacteristic
554C2#1171747911yescharacteristic
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. C2×D5×C22⋊D5C2semidirect
2. C2×#87879C2semidirect
3. #11717201C2semidirect
4. C5×#4032C2semidirect
5. C5×#4028C2semidirect
6. #11717098C2semidirect
7. C2×#87778C2semidirect
8. #11717040C2semidirect
9. #11717039C2semidirect
10. #11717034C2semidirect
11. #11717033C2semidirect
12. #11717017C2semidirect
13. #11717014C2semidirect
14. #11717004C2semidirect
15. #11716972C2semidirect
19. C22×#1039C22semidirect
20. C2×C4×C5⋊D5C22semidirect
22. C2×C5×C22⋊D5C22semidirect
23. C22×C5×Dic5C22semidirect
25. C2×C5×D20C22semidirect
27. C2×#6039C22semidirect
29. C2×#6038C22semidirect
31. C2×#6037C22semidirect
32. C2×#6036C22semidirect
35. C2×D5×Dic5C22semidirect
36. C52×C22⋊C4C22semidirect
37. #87808C22semidirect
38. #87806C22semidirect
39. C4×#1039C22semidirect
41. C5×#685C22semidirect
42. C5×#685C22semidirect
43. C5×#685C22semidirect
44. C5×C4⋊Dic5C22semidirect
45. C5×C4⋊Dic5C22semidirect
46. #87781C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. #87778C22semidirect
69. C22⋊(C5⋊D5)D4semidirect
110. #4032D5semidirect
116. #4028D5semidirect
142. C2×C5×D5#41semidirect
173. C2×C22⋊D5C2×D5semidirect
179. C22×Dic5C2×D5semidirect
186. C2×D20C2×D5semidirect
203. C5×C22⋊C4C2×D5semidirect
204. C5×C22⋊C4C2×D5semidirect
221. #685C2×D5semidirect
222. #685C2×D5semidirect
223. #685C2×D5semidirect
224. C4⋊Dic5C2×D5semidirect
225. C4⋊Dic5C2×D5semidirect
343. C52#546semidirect
392. C2×D5#711semidirect
451. C22⋊C4D52semidirect
514. C5#65925semidirect
515. C5#65988semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C22⋊D52
C2×#878792
#117172012
C5×#40322
C5×#40282
#117170982
C2×#877782
#117170402
#117170392
#117170342
#117170332
#117170172
#117170142
#117170042
#117169722
#659885
#659255