Finite Group Databasebeta

← #13105392

Subgroups — #13105392

Order 1600.

576 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717205C211yesmaximal, characteristic, complemented
2C5×#4032C211yesmaximal, characteristic, complemented
3C5×#4030C211yesmaximal, characteristic, complemented
4#11717101C211yesmaximal, characteristic, complemented
5#11717096C211yesmaximal, characteristic, complemented
6Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
7#11717088C211yesmaximal, characteristic, complemented
8#11717040C211yesmaximal, characteristic, complemented
9D5×#685C211yesmaximal, characteristic, complemented
10#11717031C211yesmaximal, characteristic, complemented
11#11717017C211yesmaximal, characteristic, complemented
12#11717011C211yesmaximal, characteristic, complemented
13#11716980C211yesmaximal, characteristic, complemented
14#11716975C211yesmaximal, characteristic, complemented
15#11716966C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×#6045C2211yescharacteristic, complemented
19C2×C5×C22⋊D5C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C2×C5×C22⋊D5C2211yescharacteristic, complemented
22C22×C5×Dic5C2211yescharacteristic, complemented
23C2×C5×D20C2211yescharacteristic
24C2×C4×C5×D5C2211yescharacteristic
25C2×#6038C2211yescharacteristic
26C2×#6037C2211yescharacteristic
27C2×D5×Dic5C2211yescharacteristic
28C2×D5×Dic5C2211yescharacteristic, complemented
29C52×C22⋊C4C2211yescharacteristic, complemented
30C22⋊#1039C2211yescharacteristic, complemented
31#87808C2211yescharacteristic
32#87808C2211yescharacteristic
33C4×#1039C2211yescharacteristic, complemented
34C5×#685C2211yescharacteristic
35C5×#685C2211yescharacteristic
36C5×#685C2211yescharacteristic, complemented
37C5×C4⋊Dic5C2211yescharacteristic, complemented
38C5×C4⋊Dic5C2211yescharacteristic, complemented
39C5×#683C2211yescharacteristic, complemented
40#87780C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65991—51nomaximal
52#65951—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C22×C5×D5C2311yescharacteristic
57C22×C5×D5C2311yescharacteristic
58C2×D52—22no—
59C2×D52—41no—
60C2×D52—41no—
61C2×C4×C52C2311yescharacteristic
62C2×C4×C52C2311yescharacteristic
63C22⋊(C5⋊D5)—41no—
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C2×#1039C2311yescharacteristic
67#6045—41no—
68C5×C22⋊D5—22no—
69C5×C22⋊D5—41no—
70C5×C22⋊D5—41no—
71C2×C5×Dic5—41no—
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C2×C5×Dic5C2311yescharacteristic
77C5×D20—22no—
78C4×C5×D5—41no—
79#6038—41no—
80#6037—41no—
81D5×Dic5—41no—
82D5×Dic5—41no—
83C5×#123—51no—
84C5×#121—51no—
85#4085—51no—
86#4085—51no—
87#4084—51no—
88#4083—51no—
89#4083—51no—
90#4083—51no—
91D4×Dic5—51no—
92D4×Dic5—51no—
93#4076—51no—
94C4×C22⋊D5—51no—
95C2×#685—51no—
96#4070—51no—
97#4044—51no—
98#4044—51no—
99#4044—51no—
100#4043—51no—
101#4040—51no—
102#4039—51no—
103#4032—51no—
104#4032D511yescharacteristic, complemented
105#4031—51no—
106#4031—51no—
107#4031—51no—
108#4031—51no—
109#4031—102no—
110#4030D511yescharacteristic, complemented
111#4028—51no—
112D5×C22⋊C4—51no—
113#4025—51no—
114C22×C52—21no—
115C22×C52—21no—
116C22×C52—21no—
117C22×C52C2411yescharacteristic
118C2×C5⋊D5—21no—
119C2×C5⋊D5—21no—
120C2×C5⋊D5—21no—
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—
132C4×C52—21no—
133#1039—21no—
134#1039—21no—
135#1039—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—51no—
144C2×C5×D4—51no—
145C2×C5×D4—51no—
146C2×C5×D4—51no—
147C22×C4×C5—51no—
148C22×C4×C5—51no—
149C2×C22⋊D5—51no—
150C2×C22⋊D5—51no—
151C2×C22⋊D5—51no—
152C2×C22⋊D5—51no—
153C2×C22⋊D5—51no—
154C2×C22⋊D5—102no—
155C2×C22⋊D5C2×D511yescharacteristic
156C2×C22⋊D5C2×D511yescharacteristic
157C2×C22⋊D5C2×D511yescharacteristic, complemented
158C22×Dic5—51no—
159C22×Dic5—51no—
160C22×Dic5C2×D511yescharacteristic, complemented
161C2×D20—51no—
162C2×D20C2×D511yescharacteristic
163C2×C4×D5—51no—
164C2×C4×D5—51no—
165C2×C4×D5C2×D511yescharacteristic
166C2×Dic10—51no—
167C2×Dic10—51no—
168C2×Dic10—102no—
169C5×C4⋊C4—51no—
170C5×C4⋊C4—51no—
171C5×C4⋊C4—51no—
172C5×C22⋊C4—22no—
173C5×C22⋊C4—51no—
174C5×C22⋊C4—51no—
175C5×C22⋊C4—51no—
176C5×C22⋊C4C2×D511yescharacteristic, complemented
177C5×C22⋊C4C2×D511yescharacteristic, complemented
178C22⋊Dic5—51no—
179C22⋊Dic5—51no—
180C22⋊Dic5—51no—
181C22⋊Dic5—51no—
182C22⋊Dic5—51no—
183C22⋊Dic5—51no—
184C22⋊Dic5—51no—
185C22⋊Dic5—102no—
186#685—51no—
187#685—51no—
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—102no—
200#685—102no—
201#685C2×D511yescharacteristic
202#685C2×D511yescharacteristic
203#685C2×D511yescharacteristic, complemented
204C4⋊Dic5—51no—
205C4⋊Dic5—51no—
206C4⋊Dic5—51no—
207C4⋊Dic5C2×D511yescharacteristic, complemented
208C4⋊Dic5C2×D511yescharacteristic, complemented
209#683—51no—
210#683—51no—
211#683—51no—
212#683C2×D511yescharacteristic, complemented
213C4×Dic5—51no—
214C4×Dic5—51no—
215C4×Dic5—51no—
216C4×Dic5—51no—
217C4×Dic5—102no—
218#538—251noSylow
219C2×C52—41no—
220C2×C52C2×#4111yescharacteristic
221C2×C52H(5,𝔽2)11yescharacteristic
222C2×C52H(5,𝔽2)11yescharacteristic
223C5⋊D5—41no—
224C5×D5—22no—
225C5×D5—41no—
226C5×D5—41no—
227C23×C5—22no—
228C23×C5—51no—
229C23×C5—51no—
230C23×C5—51no—
231C23×C5C22×D511yescharacteristic
232C23×C5C22×D511yescharacteristic
233C22×D5—51no—
234C22×D5—51no—
235C22×D5—102no—
236C22×D5—102no—
237C22×D5—102no—
238C22×D5—201no—
239C22×D5—201no—
240C22×D5—201no—
241C22×D5—201no—
242C22×D5C22×D511yescharacteristic
243C22×D5C22×D511yescharacteristic
244C22×D5C22×D511yescharacteristic
245C5×D4—102no—
246C5×D4—102no—
247C5×D4—201no—
248C5×D4—201no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—51no—
252C2×C4×C5—51no—
253C2×C4×C5—51no—
254C2×C4×C5—51no—
255C2×C4×C5—51no—
256C2×C4×C5—201no—
257C2×C4×C5—201no—
258C2×C4×C5C22×D511yescharacteristic
259C2×C4×C5C22×D511yescharacteristic
260C2×C4×C5C22×D511yescharacteristic
261C2×C4×C5C22×D511yescharacteristic
262C22⋊D5—22no—
263C22⋊D5—41no—
264C22⋊D5—41no—
265C22⋊D5—201no—
266C22⋊D5—201no—
267C22⋊D5—201no—
268C22⋊D5—201no—
269C22⋊D5—201no—
270C22⋊D5—204no—
271C2×Dic5—41no—
272C2×Dic5—51no—
273C2×Dic5—51no—
274C2×Dic5—51no—
275C2×Dic5—51no—
276C2×Dic5—51no—
277C2×Dic5—51no—
278C2×Dic5—102no—
279C2×Dic5—102no—
280C2×Dic5—102no—
281C2×Dic5—201no—
282C2×Dic5—201no—
283C2×Dic5C22×D511yescharacteristic
284C2×Dic5C22×D511yescharacteristic
285C2×Dic5C22×D511yescharacteristic
286C2×Dic5C22×D511yescharacteristic
287C2×Dic5C22×D511yescharacteristic
288D20—22no—
289D20—201no—
290C4×D5—41no—
291C4×D5—201no—
292C4×D5—201no—
293Dic10—201no—
294Dic10—201no—
295Dic10—204no—
296#126—251no—
297#126—251no—
298#124—251no—
299#124—251no—
300#123—51no—
301#123—251no—
302C22⋊Q8—251no—
303#121—51no—
304#121—251no—
305#121—251no—
306#120—251no—
307#120—251no—
308C4×D4—251no—
309C4×D4—251no—
310C2×C22⋊C4—251no—
311C52#53811yescharacteristic, complemented, Sylow
312C22×C5—21no—
313C22×C5—21no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—21no—
317C22×C5—21no—
318C22×C5—22no—
319C22×C5—42no—
320C22×C5—42no—
321C22×C5—42no—
322C22×C5—52no—
323C22×C5—101no—
324C22×C5—101no—
325C22×C5—101no—
326C22×C5—101no—
327C22×C5—101no—
328C22×C5—101no—
329C22×C5—101no—
330C22×C5—101no—
331C22×C5C23×D511yescharacteristic
332C22×C5C23×D511yescharacteristic
333C2×D5—21no—
334C2×D5—21no—
335C2×D5—21no—
336C2×D5—21no—
337C2×D5—21no—
338C2×D5—21no—
339C2×D5—21no—
340C2×D5—21no—
341C2×D5—101no—
342C2×D5—101no—
343C2×D5—101no—
344C2×D5—101no—
345C2×D5—101no—
346C2×D5—101no—
347C2×D5—202no—
348C2×D5—202no—
349C2×D5—202no—
350C2×D5—204no—
351C2×D5—204no—
352C2×D5#71112yes—
353C4×C5—21no—
354C4×C5—21no—
355C4×C5—21no—
356C4×C5—21no—
357C4×C5—42no—
358C4×C5—42no—
359C4×C5—101no—
360C4×C5—101no—
361C4×C5—101no—
362C4×C5—101no—
363C4×C5—101no—
364Dic5—21no—
365Dic5—21no—
366Dic5—21no—
367Dic5—21no—
368Dic5—21no—
369Dic5—104no—
370Dic5—101no—
371Dic5—101no—
372Dic5—101no—
373Dic5—101no—
374Dic5—101no—
375Dic5—101no—
376Dic5—202no—
377Dic5—202no—
378C24—251no—
379C2×Q8—251no—
380C2×D4—51no—
381C2×D4—51no—
382C2×D4—51no—
383C2×D4—51no—
384C2×D4—251no—
385C2×D4—251no—
386C2×D4—251no—
387C22×C4—51no—
388C22×C4—51no—
389C22×C4—251no—
390C22×C4—251no—
391C4⋊C4—51no—
392C4⋊C4—51no—
393C4⋊C4—51no—
394C4⋊C4—251no—
395C4⋊C4—251no—
396C4⋊C4—251no—
397C22⋊C4—51no—
398C22⋊C4—51no—
399C22⋊C4—51no—
400C22⋊C4—251no—
401C22⋊C4—251no—
402C22⋊C4—251no—
403C22⋊C4—251no—
404C22⋊C4—251no—
405C22⋊C4—251no—
406C22⋊C4—251no—
407C22⋊C4—251no—
408C22⋊C4—251no—
409C22⋊C4—251no—
410C22⋊C4D5211yescharacteristic, complemented
411C42—251no—
412C42—251no—
413C2×C5—22no—
414C2×C5—22no—
415C2×C5—22no—
416C2×C5—41no—
417C2×C5—41no—
418C2×C5—82no—
419C2×C5—102no—
420C2×C5—201no—
421C2×C5—201no—
422C2×C5C2×#71111yescharacteristic
423C2×C5#414411yescharacteristic
424C2×C5#414411yescharacteristic
425C2×C5D5×#4111yescharacteristic
426C2×C5#414911yescharacteristic
427C2×C5#414911yescharacteristic
428D5—22no—
429D5—41no—
430D5—41no—
431D5—201no—
432D5—201no—
433D5—402no—
434C23—51no—
435C23—51no—
436C23—51no—
437C23—251no—
438C23—502no—
439C23—1001no—
440C23—1001no—
441C23C2×D5211yescharacteristic
442Q8—1001no—
443D4—102no—
444D4—102no—
445D4—201no—
446D4—201no—
447D4—1001no—
448D4—1001no—
449D4—1001no—
450C2×C4—51no—
451C2×C4—51no—
452C2×C4—51no—
453C2×C4—51no—
454C2×C4—51no—
455C2×C4—201no—
456C2×C4—201no—
457C2×C4—251no—
458C2×C4—251no—
459C2×C4—251no—
460C2×C4—1001no—
461C2×C4—1001no—
462C2×C4C2×D5211yescharacteristic
463C2×C4C2×D5211yescharacteristic
464C5—22no—
465C5#6595111yescharacteristic, complemented
466C5#6599111yescharacteristic, complemented
467C22—21no—
468C22—21no—
469C22—21no—
470C22—52no—
471C22—101no—
472C22—101no—
473C22—101no—
474C22—101no—
475C22—101no—
476C22—101no—
477C22—101no—
478C22—101no—
479C22—501no—
480C22—501no—
481C22—501no—
482C22—1004no—
483C22C22×D5211yescharacteristic
484C4—21no—
485C4—21no—
486C4—101no—
487C4—101no—
488C4—101no—
489C4—101no—
490C4—101no—
491C4—501no—
492C4—501no—
493C4—501no—
494C2—41no—
495C2—102no—
496C2—201no—
497C2—201no—
498C2—1001no—
499C2#1171747211yescharacteristic
500C2D5×#71111yescharacteristic
501C2#1171747911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#538
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #11717205C2semidirect
2. C5×#4032C2semidirect
3. C5×#4030C2semidirect
4. #11717101C2semidirect
5. #11717096C2semidirect
6. Dic5×C22⋊D5C2semidirect
7. #11717088C2semidirect
8. #11717040C2semidirect
9. D5×#685C2semidirect
10. #11717031C2semidirect
11. #11717017C2semidirect
12. #11717011C2semidirect
13. #11716980C2semidirect
14. #11716975C2semidirect
15. #11716966C2semidirect
18. C2×#6045C22semidirect
21. C2×C5×C22⋊D5C22semidirect
22. C22×C5×Dic5C22semidirect
28. C2×D5×Dic5C22semidirect
29. C52×C22⋊C4C22semidirect
30. C22⋊#1039C22semidirect
33. C4×#1039C22semidirect
36. C5×#685C22semidirect
37. C5×C4⋊Dic5C22semidirect
38. C5×C4⋊Dic5C22semidirect
39. C5×#683C22semidirect
40. #87780C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
49. #87778C22semidirect
50. Dic52C22semidirect
104. #4032D5semidirect
110. #4030D5semidirect
157. C2×C22⋊D5C2×D5semidirect
160. C22×Dic5C2×D5semidirect
176. C5×C22⋊C4C2×D5semidirect
177. C5×C22⋊C4C2×D5semidirect
203. #685C2×D5semidirect
207. C4⋊Dic5C2×D5semidirect
208. C4⋊Dic5C2×D5semidirect
212. #683C2×D5semidirect
311. C52#538semidirect
410. C22⋊C4D52semidirect
465. C5#65951semidirect
466. C5#65991semidirect

Maximal subgroups

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

subgroupindex
#117172052
C5×#40322
C5×#40302
#117171012
#117170962
Dic5×C22⋊D52
#117170882
#117170402
D5×#6852
#117170312
#117170172
#117170112
#117169802
#117169752
#117169662
#659915
#659515