Finite Group Databasebeta

← #13106156

Subgroups — #13106156

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×#709C211yesmaximal, characteristic, complemented
2C2×D5×C22⋊D5C211yesmaximal, characteristic, complemented
3#11717249C211yesmaximal, characteristic, complemented
4C2×C5×#685C211yesmaximal, characteristic, complemented
5#11717096C211yesmaximal, characteristic, complemented
6#11717094C211yesmaximal, characteristic, complemented
7Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
8#11717040C211yesmaximal, characteristic, complemented
9D5×#685C211yesmaximal, characteristic, complemented
10#11717025C211yesmaximal, characteristic, complemented
11#11717024C211yesmaximal, characteristic, complemented
12#11717023C211yesmaximal, characteristic, complemented
13#11717006C211yesmaximal, characteristic, complemented
14#11717004C211yesmaximal, characteristic, complemented
15#11716997C211yesmaximal, characteristic, complemented
16C23×C5×D5C2211yescharacteristic
17C22×D52C2211yescharacteristic
18C22×C4×C52C2211yescharacteristic, complemented
19C2×C22⋊(C5⋊D5)C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic, complemented
22C22×C5×Dic5C2211yescharacteristic, complemented
23C5×#709—24no—
24C2×C5×D20C2211yescharacteristic, complemented
25C2×C4×C5×D5C2211yescharacteristic, complemented
26C2×C4×C5×D5C2211yescharacteristic, complemented
27C2×C5×Dic10C2211yescharacteristic, complemented
28D5×C22⋊D5—24no—
29C2×#6038C2211yescharacteristic
30C2×#6038C2211yescharacteristic, complemented
31C2×#6037C2211yescharacteristic
32C2×#6037C2211yescharacteristic
33C2×#6037C2211yescharacteristic, complemented
34C2×D5×Dic5C2211yescharacteristic
35C2×D5×Dic5C2211yescharacteristic, complemented
36C22⋊#1039C2211yescharacteristic, complemented
37#87808C2211yescharacteristic, complemented
38#87808C2211yescharacteristic, complemented
39#87806C2211yescharacteristic, complemented
40#87806C2211yescharacteristic, complemented
41C5×#685C2211yescharacteristic
42C5×#685C2211yescharacteristic
43C5×#685C2211yescharacteristic, complemented
44C5×#685C2211yescharacteristic, complemented
45#87780C2211yescharacteristic, complemented
46#87780C2211yescharacteristic, complemented
47#87779C2211yescharacteristic, complemented
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic, complemented
50#87778C2211yescharacteristic, complemented
51#87778C2211yescharacteristic, complemented
52Dic52C2211yescharacteristic, complemented
53#66163—51nomaximal
54#65824—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×C52—22no—
68C2×C4×C52C2311yescharacteristic
69C2×C4×C52C2311yescharacteristic
70C22⋊(C5⋊D5)—22no—
71C2×#1039C2311yescharacteristic
72C2×#1039C2311yescharacteristic
73C2×#1039C2311yescharacteristic
74C5×C22⋊D5—22no—
75C5×C22⋊D5D414yescomplemented
76C2×C5×Dic5—41no—
77C2×C5×Dic5C2311yescharacteristic
78C2×C5×Dic5C2311yescharacteristic
79C2×C5×Dic5C2311yescharacteristic
80C2×C5×Dic5C2311yescharacteristic
81C5×D20—22no—
82C4×C5×D5—22no—
83C4×C5×D5—22no—
84C5×Dic10—22no—
85#6038—22no—
86#6038—41no—
87#6037—22no—
88#6037—22no—
89#6037—41no—
90D5×Dic5—22no—
91D5×Dic5—41no—
92C2×C5×#41—51no—
93C22×C22⋊D5—51no—
94C2×D4×D5—51no—
95C2×#709D511yescharacteristic, complemented
96C2×C5×C22⋊C4—51no—
97#4099—51no—
98#4093—51no—
99#4092—51no—
100#4085—51no—
101#4085—51no—
102#4083—51no—
103#4083—51no—
104D4×Dic5—51no—
105#4076—51no—
106#4076—51no—
107#4075—51no—
108#4075—51no—
109#4075—51no—
110#4075—102no—
111C4×C22⋊D5—51no—
112C4×C22⋊D5—51no—
113C2×#685—51no—
114C2×#685D511yescharacteristic, complemented
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×C52C2×D412yes—
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×D5C2×D412yes—
142C2×C5×D5#4112yescomplemented
143D52—44no—
144C4×C52—21no—
145C4×C52—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—
154C23×D5—51no—
155C23×D5—51no—
156C23×D5C2×D511yescharacteristic
157C5×#41—104no—
158C2×C5×Q8—51no—
159C2×C5×D4—51no—
160C2×C5×D4—51no—
161C2×C5×D4—51no—
162C22×C4×C5—22no—
163C22×C4×C5—51no—
164C22×C4×C5—51no—
165C22×C4×C5—51no—
166C22×C4×C5C2×D511yescharacteristic, complemented
167C22×C4×C5C2×D511yescharacteristic, complemented
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—51no—
176C2×C22⋊D5—51no—
177C2×C22⋊D5—51no—
178C2×C22⋊D5—102no—
179C2×C22⋊D5—104no—
180C2×C22⋊D5C2×D511yescharacteristic
181C2×C22⋊D5C2×D511yescharacteristic, complemented
182C22×Dic5—51no—
183C22×Dic5—51no—
184C22×Dic5C2×D511yescharacteristic, complemented
185D4×D5—104no—
186#709—24no—
187C2×D20—51no—
188C2×D20—51no—
189C2×D20C2×D511yescharacteristic, complemented
190C2×C4×D5—51no—
191C2×C4×D5—51no—
192C2×C4×D5C2×D511yescharacteristic, complemented
193C2×C4×D5C2×D511yescharacteristic, complemented
194C2×Dic10C2×D511yescharacteristic, complemented
195C5×C22⋊C4—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—102no—
206#685—51no—
207#685—51no—
208#685—51no—
209#685—51no—
210#685—51no—
211#685—51no—
212#685—51no—
213#685—51no—
214#685—51no—
215#685—51no—
216#685—102no—
217#685—102no—
218#685C2×D511yescharacteristic
219#685C2×D511yescharacteristic
220#685C2×D511yescharacteristic, complemented
221#685C2×D511yescharacteristic, complemented
222C4⋊Dic5—51no—
223C4⋊Dic5—51no—
224#683—51no—
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—41no—
240C5×D5—22no—
241C5×D5—22no—
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×Q8—102no—
271C5×D4—54no—
272C5×D4—102no—
273C5×D4—102no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—44no—
279C2×C4×C5—51no—
280C2×C4×C5—51no—
281C2×C4×C5—51no—
282C2×C4×C5—51no—
283C2×C4×C5—102no—
284C2×C4×C5—102no—
285C2×C4×C5—201no—
286C2×C4×C5C22×D511yescharacteristic
287C2×C4×C5C22×D511yescharacteristic
288C2×C4×C5C22×D511yescharacteristic
289C2×C4×C5C22×D511yescharacteristic
290C22⋊D5—22no—
291C22⋊D5—102no—
292C22⋊D5—102no—
293C22⋊D5—102no—
294C22⋊D5—102no—
295C22⋊D5—102no—
296C22⋊D5—102no—
297C22⋊D5—102no—
298C22⋊D5—204no—
299C22⋊D5—201no—
300C22⋊D5—201no—
301C22⋊D5—201no—
302C22⋊D5C22⋊D514yescomplemented
303C2×Dic5—41no—
304C2×Dic5—51no—
305C2×Dic5—51no—
306C2×Dic5—51no—
307C2×Dic5—51no—
308C2×Dic5—51no—
309C2×Dic5—51no—
310C2×Dic5—102no—
311C2×Dic5—102no—
312C2×Dic5—102no—
313C2×Dic5—102no—
314C2×Dic5—201no—
315C2×Dic5C22×D511yescharacteristic
316C2×Dic5C22×D511yescharacteristic
317C2×Dic5C22×D511yescharacteristic
318C2×Dic5C22×D511yescharacteristic
319D20—22no—
320D20—102no—
321D20—201no—
322C4×D5—22no—
323C4×D5—22no—
324C4×D5—102no—
325C4×D5—201no—
326Dic10—22no—
327C2×#41—51no—
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—251no—
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—204no—
364C22×C5D4×D512yes—
365C22×C5C2×C22⋊D512yes—
366C22×C5C23×D511yescharacteristic
367C22×C5C23×D511yescharacteristic
368C2×D5—21no—
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—44no—
379C2×D5—101no—
380C2×D5—101no—
381C2×D5—101no—
382C2×D5—101no—
383C2×D5—101no—
384C2×D5—101no—
385C2×D5—202no—
386C2×D5—202no—
387C2×D5—202no—
388C2×D5—204no—
389C2×D5—204no—
390C2×D5#70912yescomplemented
391C2×D5C2×C22⋊D512yes—
392C4×C5—21no—
393C4×C5—21no—
394C4×C5—21no—
395C4×C5—21no—
396C4×C5—42no—
397C4×C5—42no—
398C4×C5—52no—
399C4×C5—101no—
400C4×C5—101no—
401C4×C5—101no—
402Dic5—21no—
403Dic5—21no—
404Dic5—21no—
405Dic5—101no—
406Dic5—101no—
407Dic5—101no—
408Dic5—101no—
409Dic5—101no—
410Dic5—101no—
411Dic5—202no—
412Dic5—202no—
413Dic5—202no—
414Dic5C2×C22⋊D512yes—
415C24—51no—
416C24—251no—
417#41—104no—
418C2×Q8—51no—
419C2×D4—51no—
420C2×D4—51no—
421C2×D4—51no—
422C2×D4—251no—
423C2×D4—251no—
424C2×D4—251no—
425C2×D4—251no—
426C2×D4—251no—
427C2×D4—251no—
428C2×D4—504no—
429C22×C4—51no—
430C22×C4—51no—
431C22×C4—51no—
432C22×C4—251no—
433C22×C4—251no—
434C22×C4D5211yescharacteristic, complemented
435C4⋊C4—251no—
436C4⋊C4—251no—
437C4⋊C4—251no—
438C4⋊C4—251no—
439C22⋊C4—51no—
440C22⋊C4—51no—
441C22⋊C4—51no—
442C22⋊C4—51no—
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—102no—
459C2×C5—102no—
460C2×C5—201no—
461C2×C5—201no—
462C2×C5C2×#70911yescharacteristic
463C2×C5C2×D4×D511yescharacteristic
464C2×C5#414411yescharacteristic
465C2×C5D5×#4111yescharacteristic
466C2×C5#414911yescharacteristic
467C2×C5C22×C22⋊D511yescharacteristic
468D5—22no—
469D5—22no—
470D5—41no—
471D5—41no—
472D5—201no—
473D5—201no—
474D5—402no—
475C23—51no—
476C23—51no—
477C23—51no—
478C23—51no—
479C23—102no—
480C23—102no—
481C23—201no—
482C23—251no—
483C23—502no—
484C23—502no—
485C23—1001no—
486C23C2×D5211yescharacteristic
487Q8—102no—
488D4—54no—
489D4—102no—
490D4—102no—
491D4—502no—
492D4—502no—
493D4—502no—
494D4—502no—
495D4—1001no—
496D4—1001no—
497C2×C4—22no—
498C2×C4—51no—
499C2×C4—51no—
500C2×C4—51no—
501C2×C4—51no—
502C2×C4—102no—
503C2×C4—102no—
504C2×C4—201no—
505C2×C4—251no—
506C2×C4—251no—
507C2×C4—251no—
508C2×C4—502no—
509C2×C4—1001no—
510C2×C4C2×D5211yescharacteristic
511C2×C4C2×D5211yescharacteristic
512C5—22no—
513C5#6582411yescharacteristic, complemented
514C5#6616311yescharacteristic, complemented
515C22—21no—
516C22—21no—
517C22—52no—
518C22—52no—
519C22—101no—
520C22—101no—
521C22—101no—
522C22—101no—
523C22—101no—
524C22—101no—
525C22—101no—
526C22—101no—
527C22—101no—
528C22—101no—
529C22—204no—
530C22—501no—
531C22—501no—
532C22—501no—
533C22—1004no—
534C22D5×C22⋊D512yes—
535C22C22×D5211yescharacteristic
536C4—21no—
537C4—21no—
538C4—52no—
539C4—101no—
540C4—101no—
541C4—101no—
542C4—501no—
543C4—501no—
544C4—501no—
545C2—22no—
546C2—102no—
547C2—102no—
548C2—201no—
549C2—201no—
550C2—1001no—
551C2D5×#70911yescharacteristic
552C2#1171747211yescharacteristic
553C2C2×D5×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
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×C5×#709C2semidirect
2. C2×D5×C22⋊D5C2semidirect
3. #11717249C2semidirect
4. C2×C5×#685C2semidirect
5. #11717096C2semidirect
6. #11717094C2semidirect
7. Dic5×C22⋊D5C2semidirect
8. #11717040C2semidirect
9. D5×#685C2semidirect
10. #11717025C2semidirect
11. #11717024C2semidirect
12. #11717023C2semidirect
13. #11717006C2semidirect
14. #11717004C2semidirect
15. #11716997C2semidirect
18. C22×C4×C52C22semidirect
21. C2×C5×C22⋊D5C22semidirect
22. C22×C5×Dic5C22semidirect
24. C2×C5×D20C22semidirect
25. C2×C4×C5×D5C22semidirect
26. C2×C4×C5×D5C22semidirect
27. C2×C5×Dic10C22semidirect
30. C2×#6038C22semidirect
33. C2×#6037C22semidirect
35. C2×D5×Dic5C22semidirect
36. C22⋊#1039C22semidirect
37. #87808C22semidirect
38. #87808C22semidirect
39. #87806C22semidirect
40. #87806C22semidirect
43. C5×#685C22semidirect
44. C5×#685C22semidirect
45. #87780C22semidirect
46. #87780C22semidirect
47. #87779C22semidirect
49. #87778C22semidirect
50. #87778C22semidirect
51. #87778C22semidirect
52. Dic52C22semidirect
75. C5×C22⋊D5D4semidirect
95. C2×#709D5semidirect
114. C2×#685D5semidirect
142. C2×C5×D5#41semidirect
166. C22×C4×C5C2×D5semidirect
167. C22×C4×C5C2×D5semidirect
181. C2×C22⋊D5C2×D5semidirect
184. C22×Dic5C2×D5semidirect
189. C2×D20C2×D5semidirect
192. C2×C4×D5C2×D5semidirect
193. C2×C4×D5C2×D5semidirect
194. C2×Dic10C2×D5semidirect
220. #685C2×D5semidirect
221. #685C2×D5semidirect
302. C22⋊D5C22⋊D5semidirect
342. C52#546semidirect
390. C2×D5#709semidirect
434. C22×C4D52semidirect
513. C5#65824semidirect
514. C5#66163semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#7092
C2×D5×C22⋊D52
#117172492
C2×C5×#6852
#117170962
#117170942
Dic5×C22⋊D52
#117170402
D5×#6852
#117170252
#117170242
#117170232
#117170062
#117170042
#117169972
#661635
#658245