Finite Group Databasebeta

← #13106401

Subgroups — #13106401

Order 1600.

594 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717259C211yesmaximal, characteristic, complemented
2C5×#4085C211yesmaximal, characteristic, complemented
3C5×#4084C211yesmaximal, characteristic, complemented
4#11717101C211yesmaximal, characteristic, complemented
5#11717101C211yesmaximal, characteristic, complemented
6#11717091C211yesmaximal, characteristic, complemented
7Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
8#11717084C211yesmaximal, characteristic, complemented
9#11717080C211yesmaximal, characteristic, complemented
10#11717077C211yesmaximal, characteristic, complemented
11#11717071C211yesmaximal, characteristic, complemented
12D5×#685C211yesmaximal, characteristic, complemented
13#11716976C211yesmaximal, characteristic, complemented
14#11716969C211yesmaximal, characteristic, complemented
15#11716966C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C52×D4C2211yescharacteristic, complemented
18C2×C22⋊(C5⋊D5)C2211yescharacteristic
19C2×C22⋊(C5⋊D5)C2211yescharacteristic
20C22×#1039C2211yescharacteristic, complemented
21C2×C5×C22⋊D5C2211yescharacteristic
22C2×C5×C22⋊D5C2211yescharacteristic
23C2×C5×C22⋊D5C2211yescharacteristic
24C2×C5×C22⋊D5C2211yescharacteristic
25C22×C5×Dic5C2211yescharacteristic, complemented
26C2×C4×C5×D5C2211yescharacteristic
27C2×#6039C2211yescharacteristic, complemented
28C2×D5×Dic5C2211yescharacteristic
29C22⋊#1039C2211yescharacteristic, complemented
30#87808C2211yescharacteristic
31#87806C2211yescharacteristic, complemented
32C5×C22⋊Dic5C2211yescharacteristic, complemented
33C5×C22⋊Dic5C2211yescharacteristic, complemented
34C5×C22⋊Dic5C2211yescharacteristic, complemented
35C5×#685C2211yescharacteristic
36C5×C4⋊Dic5C2211yescharacteristic, complemented
37C5×#683C2211yescharacteristic, complemented
38#87780C2211yescharacteristic, complemented
39#87780C2211yescharacteristic, complemented
40#87780C2211yescharacteristic, complemented
41#87779C2211yescharacteristic
42#87779C2211yescharacteristic
43#87779C2211yescharacteristic
44#87778C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49Dic52C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65951—51nomaximal
52#65947—51nomaximal
53C23×C52C2311yescharacteristic
54C23×C52C2311yescharacteristic
55C22×C5⋊D5C2311yescharacteristic
56C22×C5×D5C2311yescharacteristic
57C22×C5×D5C2311yescharacteristic
58C2×D52—22no—
59C2×D52—41no—
60C2×D52—41no—
61C52×D4—41no—
62C2×C4×C52C2311yescharacteristic
63C22⋊(C5⋊D5)—41no—
64C22⋊(C5⋊D5)—41no—
65C2×#1039—41no—
66C2×#1039C2311yescharacteristic
67C2×#1039C2311yescharacteristic
68C2×#1039C2311yescharacteristic
69C5×C22⋊D5—22no—
70C5×C22⋊D5—22no—
71C5×C22⋊D5—41no—
72C5×C22⋊D5—41no—
73C2×C5×Dic5—41no—
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C2×C5×Dic5C2311yescharacteristic
77C2×C5×Dic5C2311yescharacteristic
78C2×C5×Dic5C2311yescharacteristic
79C2×C5×Dic5C2311yescharacteristic
80C4×C5×D5—41no—
81#6039—41no—
82D5×Dic5—41no—
83C5×#121—51no—
84C5×#121—51no—
85#4085—51no—
86#4085—51no—
87#4085—102no—
88#4085D511yescharacteristic, complemented
89#4084D511yescharacteristic, complemented
90#4083—51no—
91#4083—51no—
92#4083—51no—
93#4083—51no—
94#4082—51no—
95#4081—51no—
96D4×Dic5—51no—
97D4×Dic5—51no—
98#4075—51no—
99C4×C22⋊D5—51no—
100C2×#685—51no—
101#4070—51no—
102#4044—51no—
103#4044—51no—
104#4044—51no—
105#4042—51no—
106#4032—51no—
107#4032—51no—
108#4031—51no—
109#4031—51no—
110#4031—51no—
111#4027—51no—
112D5×C22⋊C4—51no—
113#4025—51no—
114C22×C52—21no—
115C22×C52—21no—
116C22×C52—21no—
117C22×C52—21no—
118C22×C52—21no—
119C22×C52—21no—
120C22×C52C2411yescharacteristic
121C2×C5⋊D5—21no—
122C2×C5⋊D5—21no—
123C2×C5⋊D5—21no—
124C2×C5×D5—21no—
125C2×C5×D5—21no—
126C2×C5×D5—21no—
127C2×C5×D5—21no—
128C2×C5×D5—21no—
129C2×C5×D5#4112yes—
130D52—44no—
131C4×C52—21no—
132#1039—21no—
133#1039—21no—
134#1039—21no—
135C5×Dic5—21no—
136C5×Dic5—21no—
137C5×Dic5—21no—
138C5×Dic5—21no—
139C5×Dic5—21no—
140C5×Dic5—21no—
141C23×D5—51no—
142C23×D5—51no—
143C2×C5×D4—22no—
144C2×C5×D4—51no—
145C2×C5×D4—51no—
146C2×C5×D4—51no—
147C2×C5×D4—51no—
148C2×C5×D4C2×D511yescharacteristic, complemented
149C2×C5×D4C2×D511yescharacteristic, complemented
150C22×C4×C5—51no—
151C22×C4×C5—51no—
152C2×C22⋊D5—51no—
153C2×C22⋊D5—51no—
154C2×C22⋊D5—51no—
155C2×C22⋊D5—51no—
156C2×C22⋊D5—102no—
157C2×C22⋊D5—102no—
158C2×C22⋊D5C2×D511yescharacteristic
159C2×C22⋊D5C2×D511yescharacteristic
160C2×C22⋊D5C2×D511yescharacteristic
161C2×C22⋊D5C2×D511yescharacteristic
162C22×Dic5—51no—
163C22×Dic5—51no—
164C22×Dic5—51no—
165C22×Dic5—102no—
166C22×Dic5C2×D511yescharacteristic, complemented
167C2×C4×D5—51no—
168C2×C4×D5C2×D511yescharacteristic
169C2×Dic10—51no—
170C2×Dic10—51no—
171C5×C4⋊C4—51no—
172C5×C4⋊C4—51no—
173C5×C22⋊C4—51no—
174C5×C22⋊C4—51no—
175C5×C22⋊C4—51no—
176C5×C22⋊C4—51no—
177C22⋊Dic5—51no—
178C22⋊Dic5—51no—
179C22⋊Dic5—51no—
180C22⋊Dic5—51no—
181C22⋊Dic5—51no—
182C22⋊Dic5—51no—
183C22⋊Dic5—51no—
184C22⋊Dic5—102no—
185C22⋊Dic5C2×D511yescharacteristic, complemented
186C22⋊Dic5C2×D511yescharacteristic, complemented
187C22⋊Dic5C2×D511yescharacteristic, complemented
188#685—51no—
189#685—51no—
190#685—51no—
191#685—51no—
192#685—51no—
193#685—51no—
194#685—51no—
195#685—51no—
196#685—51no—
197#685—51no—
198#685—51no—
199#685—51no—
200#685—51no—
201#685—102no—
202#685C2×D511yescharacteristic
203C4⋊Dic5—51no—
204C4⋊Dic5—51no—
205C4⋊Dic5—51no—
206C4⋊Dic5C2×D511yescharacteristic, complemented
207#683—51no—
208#683—51no—
209#683—51no—
210#683—51no—
211#683—51no—
212#683—102no—
213#683C2×D511yescharacteristic, complemented
214C4×Dic5—51no—
215C4×Dic5—51no—
216C4×Dic5—51no—
217C4×Dic5—51no—
218#538—251noSylow
219C2×C52—41no—
220C2×C52—41no—
221C2×C52C2×#4111yescharacteristic
222C2×C52H(5,𝔽2)11yescharacteristic
223C2×C52H(5,𝔽2)11yescharacteristic
224C5⋊D5—41no—
225C5×D5—22no—
226C5×D5—41no—
227C23×C5—22no—
228C23×C5—22no—
229C23×C5—51no—
230C23×C5—51no—
231C23×C5C22×D511yescharacteristic
232C23×C5C22×D511yescharacteristic
233C23×C5C22×D511yescharacteristic
234C23×C5C22×D511yescharacteristic
235C22×D5—51no—
236C22×D5—51no—
237C22×D5—102no—
238C22×D5—102no—
239C22×D5—102no—
240C22×D5—201no—
241C22×D5—201no—
242C22×D5—201no—
243C22×D5—201no—
244C22×D5C22×D511yescharacteristic
245C22×D5C22×D511yescharacteristic
246C5×D4—41no—
247C5×D4—41no—
248C5×D4—44no—
249C5×D4—102no—
250C5×D4—102no—
251C5×D4—201no—
252C5×D4—201no—
253C2×C4×C5—22no—
254C2×C4×C5—51no—
255C2×C4×C5—51no—
256C2×C4×C5—51no—
257C2×C4×C5—51no—
258C2×C4×C5—51no—
259C2×C4×C5—51no—
260C2×C4×C5—201no—
261C2×C4×C5—201no—
262C2×C4×C5C22×D511yescharacteristic
263C2×C4×C5C22×D511yescharacteristic
264C22⋊D5—22no—
265C22⋊D5—22no—
266C22⋊D5—41no—
267C22⋊D5—41no—
268C22⋊D5—201no—
269C22⋊D5—201no—
270C22⋊D5—201no—
271C22⋊D5—201no—
272C22⋊D5—204no—
273C22⋊D5—204no—
274C2×Dic5—41no—
275C2×Dic5—51no—
276C2×Dic5—51no—
277C2×Dic5—51no—
278C2×Dic5—51no—
279C2×Dic5—51no—
280C2×Dic5—51no—
281C2×Dic5—102no—
282C2×Dic5—102no—
283C2×Dic5—102no—
284C2×Dic5—201no—
285C2×Dic5—201no—
286C2×Dic5—201no—
287C2×Dic5—204no—
288C2×Dic5C22×D511yescharacteristic
289C2×Dic5C22×D511yescharacteristic
290C2×Dic5C22×D511yescharacteristic
291C2×Dic5C22×D511yescharacteristic
292C2×Dic5C22×D511yescharacteristic
293C2×Dic5C22×D511yescharacteristic
294C4×D5—41no—
295C4×D5—201no—
296Dic10—201no—
297Dic10—201no—
298#126—251no—
299#126—251no—
300#124—251no—
301#124—251no—
302#123—251no—
303#123—251no—
304C22⋊Q8—251no—
305#121—51no—
306#121—51no—
307#121—251no—
308#120—251no—
309#120—251no—
310C4×D4—251no—
311C4×D4—251no—
312C2×C22⋊C4—251no—
313C52#53811yescharacteristic, complemented, Sylow
314C22×C5—24no—
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—21no—
319C22×C5—21no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—21no—
325C22×C5—21no—
326C22×C5—21no—
327C22×C5—22no—
328C22×C5—42no—
329C22×C5—42no—
330C22×C5—42no—
331C22×C5—42no—
332C22×C5—42no—
333C22×C5—52no—
334C22×C5—101no—
335C22×C5—101no—
336C22×C5—101no—
337C22×C5—101no—
338C22×C5—101no—
339C22×C5C23×D511yescharacteristic
340C22×C5C23×D511yescharacteristic
341C2×D5—21no—
342C2×D5—21no—
343C2×D5—21no—
344C2×D5—21no—
345C2×D5—21no—
346C2×D5—101no—
347C2×D5—101no—
348C2×D5—101no—
349C2×D5—101no—
350C2×D5—101no—
351C2×D5—101no—
352C2×D5—202no—
353C2×D5—202no—
354C2×D5—202no—
355C2×D5—204no—
356C2×D5—204no—
357C2×D5#71112yes—
358C4×C5—21no—
359C4×C5—21no—
360C4×C5—42no—
361C4×C5—101no—
362C4×C5—101no—
363C4×C5—101no—
364C4×C5—101no—
365C4×C5—101no—
366C4×C5—101no—
367Dic5—21no—
368Dic5—21no—
369Dic5—21no—
370Dic5—21no—
371Dic5—21no—
372Dic5—21no—
373Dic5—104no—
374Dic5—101no—
375Dic5—101no—
376Dic5—101no—
377Dic5—101no—
378Dic5—101no—
379Dic5—101no—
380Dic5—202no—
381Dic5—202no—
382C24—251no—
383C2×Q8—251no—
384C2×D4—51no—
385C2×D4—51no—
386C2×D4—51no—
387C2×D4—51no—
388C2×D4—251no—
389C2×D4—251no—
390C2×D4D5211yescharacteristic, complemented
391C22×C4—51no—
392C22×C4—51no—
393C22×C4—251no—
394C22×C4—251no—
395C4⋊C4—51no—
396C4⋊C4—51no—
397C4⋊C4—251no—
398C4⋊C4—251no—
399C4⋊C4—251no—
400C4⋊C4—251no—
401C22⋊C4—51no—
402C22⋊C4—51no—
403C22⋊C4—51no—
404C22⋊C4—51no—
405C22⋊C4—251no—
406C22⋊C4—251no—
407C22⋊C4—251no—
408C22⋊C4—251no—
409C22⋊C4—251no—
410C22⋊C4—251no—
411C22⋊C4—251no—
412C22⋊C4—251no—
413C22⋊C4—251no—
414C22⋊C4—251no—
415C42—251no—
416C42—251no—
417C2×C5—22no—
418C2×C5—22no—
419C2×C5—22no—
420C2×C5—44no—
421C2×C5—41no—
422C2×C5—41no—
423C2×C5—41no—
424C2×C5—41no—
425C2×C5—82no—
426C2×C5—102no—
427C2×C5—201no—
428C2×C5C2×#71111yescharacteristic
429C2×C5#414411yescharacteristic
430C2×C5#414411yescharacteristic
431C2×C5#414411yescharacteristic
432C2×C5D5×#4111yescharacteristic
433C2×C5#414911yescharacteristic
434D5—22no—
435D5—41no—
436D5—201no—
437D5—201no—
438D5—402no—
439C23—51no—
440C23—51no—
441C23—251no—
442C23—502no—
443C23—1001no—
444C23—1001no—
445C23C2×D5211yescharacteristic
446C23C2×D5211yescharacteristic
447Q8—1001no—
448D4—41no—
449D4—102no—
450D4—102no—
451D4—201no—
452D4—201no—
453D4—1001no—
454D4—1001no—
455C2×C4—51no—
456C2×C4—51no—
457C2×C4—51no—
458C2×C4—51no—
459C2×C4—51no—
460C2×C4—51no—
461C2×C4—201no—
462C2×C4—201no—
463C2×C4—251no—
464C2×C4—251no—
465C2×C4—251no—
466C2×C4—1001no—
467C2×C4—1001no—
468C2×C4C2×D5211yescharacteristic
469C5—22no—
470C5#6594711yescharacteristic, complemented
471C5#6595111yescharacteristic, complemented
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—52no—
479C22—101no—
480C22—101no—
481C22—101no—
482C22—101no—
483C22—101no—
484C22—501no—
485C22—501no—
486C22—501no—
487C22—1004no—
488C22C22×D5211yescharacteristic
489C4—21no—
490C4—101no—
491C4—101no—
492C4—101no—
493C4—101no—
494C4—101no—
495C4—101no—
496C4—501no—
497C4—501no—
498C4—501no—
499C2—41no—
500C2—41no—
501C2—102no—
502C2—201no—
503C2—1001no—
504C2D5×#71111yescharacteristic
505C2#1171747911yescharacteristic
506C2#1171749511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#538
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11717259C2semidirect
2. C5×#4085C2semidirect
3. C5×#4084C2semidirect
4. #11717101C2semidirect
5. #11717101C2semidirect
6. #11717091C2semidirect
7. Dic5×C22⋊D5C2semidirect
8. #11717084C2semidirect
9. #11717080C2semidirect
10. #11717077C2semidirect
11. #11717071C2semidirect
12. D5×#685C2semidirect
13. #11716976C2semidirect
14. #11716969C2semidirect
15. #11716966C2semidirect
17. C2×C52×D4C22semidirect
20. C22×#1039C22semidirect
25. C22×C5×Dic5C22semidirect
27. C2×#6039C22semidirect
29. C22⋊#1039C22semidirect
31. #87806C22semidirect
32. C5×C22⋊Dic5C22semidirect
33. C5×C22⋊Dic5C22semidirect
34. C5×C22⋊Dic5C22semidirect
36. C5×C4⋊Dic5C22semidirect
37. C5×#683C22semidirect
38. #87780C22semidirect
39. #87780C22semidirect
40. #87780C22semidirect
49. Dic52C22semidirect
50. Dic52C22semidirect
88. #4085D5semidirect
89. #4084D5semidirect
148. C2×C5×D4C2×D5semidirect
149. C2×C5×D4C2×D5semidirect
166. C22×Dic5C2×D5semidirect
185. C22⋊Dic5C2×D5semidirect
186. C22⋊Dic5C2×D5semidirect
187. C22⋊Dic5C2×D5semidirect
206. C4⋊Dic5C2×D5semidirect
213. #683C2×D5semidirect
313. C52#538semidirect
390. C2×D4D52semidirect
470. C5#65947semidirect
471. C5#65951semidirect

Maximal subgroups

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

subgroupindex
#117172592
C5×#40852
C5×#40842
#117171012
#117170912
Dic5×C22⋊D52
#117170842
#117170802
#117170772
#117170712
D5×#6852
#117169762
#117169692
#117169662
#659515
#659475