Finite Group Databasebeta

← #13105212

Subgroups — #13105212

Order 1600.

626 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717202C211yesmaximal, characteristic
2C5×#4024C211yesmaximal, characteristic, complemented
3C5×#4023C211yesmaximal, characteristic, complemented
4#11717092C211yesmaximal, characteristic, complemented
5#11717084C211yesmaximal, characteristic, complemented
6C2×#87779C211yesmaximal, characteristic, complemented
7#11717070C211yesmaximal, characteristic, complemented
8#11717016C211yesmaximal, characteristic, complemented
9#11717016C211yesmaximal, characteristic, complemented
10#11717013C211yesmaximal, characteristic, complemented
11#11717008C211yesmaximal, characteristic, complemented
12#11716987C211yesmaximal, characteristic, complemented
13#11716982C211yesmaximal, characteristic, complemented
14#11716966C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic, complemented
16C23×C5⋊D5C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C4⋊(C5⋊D5)C2211yescharacteristic
19C2×C4⋊(C5⋊D5)C2211yescharacteristic
20C22×C5×Dic5C2211yescharacteristic
21C22×C5×Dic5C2211yescharacteristic
22C2×C5×Dic10C2211yescharacteristic, complemented
23C2×#6036C2211yescharacteristic
24C2×#6036C2211yescharacteristic
25C2×#6036C2211yescharacteristic, complemented
26C52×C22⋊C4C2211yescharacteristic
27#87808C2211yescharacteristic
28#87808C2211yescharacteristic
29C5×C22⋊Dic5C2211yescharacteristic
30C5×C22⋊Dic5C2211yescharacteristic
31C5×C4⋊Dic5C2211yescharacteristic, complemented
32C5×#683C2211yescharacteristic, complemented
33C5×#683C2211yescharacteristic, complemented
34C5×#683C2211yescharacteristic, complemented
35C5×#683C2211yescharacteristic, complemented
36C4×C5×Dic5C2211yescharacteristic, complemented
37C4×C5×Dic5C2211yescharacteristic, complemented
38#87781C2211yescharacteristic, complemented
39#87780C2211yescharacteristic, complemented
40#87780C2211yescharacteristic, complemented
41#87779C2211yescharacteristic
42#87779C2211yescharacteristic
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87779C2211yescharacteristic
46#87779C2211yescharacteristic
47#87779C2211yescharacteristic, complemented
48#87779C2211yescharacteristic, complemented
49#87779C2211yescharacteristic, complemented
50Dic52C2211yescharacteristic, complemented
51#65971—51nomaximal
52#65861—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5—22no—
55C22×C5⋊D5—22no—
56C22×C5⋊D5—41no—
57C22×C5⋊D5C2311yescharacteristic
58C22×C5⋊D5C2311yescharacteristic
59C22×C5⋊D5C2311yescharacteristic
60C2×C4×C52C2311yescharacteristic
61C2×C4×C52C2311yescharacteristic
62C22⋊(C5⋊D5)—22no—
63C2×#1039C2311yescharacteristic
64C4⋊(C5⋊D5)—22no—
65C4⋊(C5⋊D5)—41no—
66C2×C5×Dic5—22no—
67C2×C5×Dic5—41no—
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C5×Dic10—41no—
77#6036—22no—
78#6036—41no—
79#6036—41no—
80C5×C22⋊Q8—51no—
81C5×#117—51no—
82#4093—51no—
83#4092—51no—
84#4075—51no—
85C4×C22⋊D5—51no—
86C2×#685—51no—
87C2×#685—51no—
88#4044—51no—
89#4044—51no—
90#4044—51no—
91#4042—51no—
92#4040—51no—
93#4040—51no—
94#4040—51no—
95#4040—51no—
96#4039—51no—
97#4038—51no—
98#4032—51no—
99#4028—51no—
100#4028—51no—
101#4028—102no—
102#4027—51no—
103D5×C22⋊C4—51no—
104D5×C22⋊C4—51no—
105#4024D511yescharacteristic, complemented
106#4023D511yescharacteristic, complemented
107#4022—51no—
108#4021—51no—
109C4×D20—51no—
110#4018—51no—
111C22×C52—21no—
112C22×C52—21no—
113C22×C52#4112yes—
114C22×C52C2411yescharacteristic
115C2×C5⋊D5—21no—
116C2×C5⋊D5—21no—
117C2×C5⋊D5—21no—
118C2×C5⋊D5—21no—
119C2×C5⋊D5—21no—
120C2×C5⋊D5—21no—
121C2×C5⋊D5—21no—
122C2×C5⋊D5—21no—
123C2×C5⋊D5—44no—
124C2×C5⋊D5#4112yes—
125C4×C52—21no—
126C4×C52—21no—
127#1039—21no—
128C5×Dic5—21no—
129C5×Dic5—21no—
130C5×Dic5—21no—
131C5×Dic5—21no—
132C5×Dic5—21no—
133C5×Dic5—21no—
134C5×Dic5—21no—
135C5×Dic5—21no—
136C23×D5—51no—
137C23×D5—51no—
138C23×D5—102no—
139C2×C5×Q8—51no—
140C22×C4×C5—51no—
141C22×C4×C5—51no—
142C2×C22⋊D5—51no—
143C2×C22⋊D5—51no—
144C2×C22⋊D5—102no—
145C22×Dic5C2×D511yescharacteristic
146C22×Dic5C2×D511yescharacteristic
147C2×D20—51no—
148C2×D20—51no—
149C2×D20—51no—
150C2×D20—51no—
151C2×D20—102no—
152C2×D20—102no—
153C2×C4×D5—51no—
154C2×C4×D5—51no—
155C2×C4×D5—51no—
156C2×C4×D5—51no—
157C2×C4×D5—51no—
158C2×C4×D5—51no—
159C2×Dic10C2×D511yescharacteristic, complemented
160C5×C4⋊C4—51no—
161C5×C4⋊C4—51no—
162C5×C4⋊C4—51no—
163C5×C4⋊C4—51no—
164C5×C4⋊C4—51no—
165C5×C22⋊C4—22no—
166C5×C22⋊C4—51no—
167C5×C22⋊C4—51no—
168C5×C22⋊C4C2×D511yescharacteristic
169C5×C22⋊C4C2×D511yescharacteristic
170C42×C5—51no—
171C42×C5—51no—
172C22⋊Dic5C2×D511yescharacteristic
173C22⋊Dic5C2×D511yescharacteristic
174#685—51no—
175#685—51no—
176#685—51no—
177#685—51no—
178#685—51no—
179#685—51no—
180#685—51no—
181#685—51no—
182#685—51no—
183#685—51no—
184#685—51no—
185#685—51no—
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—102no—
197#685—102no—
198C4⋊Dic5—51no—
199C4⋊Dic5—51no—
200C4⋊Dic5C2×D511yescharacteristic, complemented
201#683—51no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683C2×D511yescharacteristic, complemented
206#683C2×D511yescharacteristic, complemented
207#683C2×D511yescharacteristic, complemented
208#683C2×D511yescharacteristic, complemented
209C4×Dic5—51no—
210C4×Dic5—51no—
211C4×Dic5C2×D511yescharacteristic, complemented
212C4×Dic5C2×D511yescharacteristic, complemented
213#551—251noSylow
214C2×C52—22no—
215C2×C52C2×#4111yescharacteristic
216C2×C52C2×#4111yescharacteristic
217C2×C52H(5,𝔽2)11yescharacteristic
218C5⋊D5—22no—
219C5⋊D5—41no—
220C5⋊D5—41no—
221C23×C5—22no—
222C23×C5C22×D511yescharacteristic
223C23×C5C22×D511yescharacteristic
224C22×D5—51no—
225C22×D5—51no—
226C22×D5—51no—
227C22×D5—51no—
228C22×D5—51no—
229C22×D5—51no—
230C22×D5—102no—
231C22×D5—102no—
232C22×D5—102no—
233C22×D5—102no—
234C22×D5—102no—
235C22×D5—102no—
236C22×D5—102no—
237C22×D5—204no—
238C22×D5—204no—
239C22×D5—201no—
240C22×D5—201no—
241C22×D5—204no—
242C5×Q8—201no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—51no—
246C2×C4×C5—51no—
247C2×C4×C5—51no—
248C2×C4×C5—51no—
249C2×C4×C5—51no—
250C2×C4×C5—51no—
251C2×C4×C5—51no—
252C2×C4×C5—51no—
253C2×C4×C5—102no—
254C2×C4×C5—201no—
255C2×C4×C5C22×D511yescharacteristic
256C2×C4×C5C22×D511yescharacteristic
257C2×C4×C5C22×D511yescharacteristic
258C2×C4×C5C22×D511yescharacteristic
259C22⋊D5—102no—
260C22⋊D5—102no—
261C22⋊D5—204no—
262C2×Dic5—22no—
263C2×Dic5—41no—
264C2×Dic5—51no—
265C2×Dic5—51no—
266C2×Dic5—102no—
267C2×Dic5C22×D511yescharacteristic
268C2×Dic5C22×D511yescharacteristic
269C2×Dic5C22×D511yescharacteristic
270C2×Dic5C22×D511yescharacteristic
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274C2×Dic5C22×D511yescharacteristic
275D20—102no—
276D20—102no—
277D20—204no—
278D20—201no—
279D20—201no—
280D20—204no—
281C4×D5—102no—
282C4×D5—102no—
283C4×D5—201no—
284C4×D5—201no—
285C4×D5—201no—
286C4×D5—201no—
287Dic10—41no—
288#126—251no—
289#126—251no—
290#124—251no—
291#123—251no—
292#123—251no—
293#123—251no—
294C22⋊Q8—51no—
295C22⋊Q8—251no—
296#120—251no—
297C4×D4—251no—
298C4×D4—251no—
299#117—51no—
300#117—251no—
301C2×C22⋊C4—251no—
302C2×C22⋊C4—251no—
303C52#55111yescharacteristic, complemented, Sylow
304C22×C5—21no—
305C22×C5—21no—
306C22×C5—21no—
307C22×C5—21no—
308C22×C5—24no—
309C22×C5—22no—
310C22×C5—42no—
311C22×C5—42no—
312C22×C5#70912yes—
313C22×C5#71312yes—
314C22×C5C23×D511yescharacteristic
315C22×C5C23×D511yescharacteristic
316C2×D5—52no—
317C2×D5—52no—
318C2×D5—104no—
319C2×D5—101no—
320C2×D5—101no—
321C2×D5—101no—
322C2×D5—101no—
323C2×D5—104no—
324C2×D5—101no—
325C2×D5—101no—
326C2×D5—101no—
327C2×D5—101no—
328C2×D5—101no—
329C2×D5—101no—
330C2×D5—101no—
331C2×D5—101no—
332C2×D5—101no—
333C2×D5—101no—
334C2×D5—101no—
335C2×D5—101no—
336C2×D5—202no—
337C2×D5—202no—
338C2×D5—202no—
339C2×D5—202no—
340C2×D5—202no—
341C2×D5—202no—
342C2×D5—202no—
343C2×D5—204no—
344C2×D5—204no—
345C2×D5—2016no—
346C4×C5—21no—
347C4×C5—21no—
348C4×C5—21no—
349C4×C5—21no—
350C4×C5—42no—
351C4×C5—42no—
352C4×C5—101no—
353C4×C5—101no—
354C4×C5—101no—
355C4×C5—101no—
356C4×C5—101no—
357C4×C5—101no—
358C4×C5—101no—
359C4×C5—101no—
360Dic5—21no—
361Dic5—21no—
362Dic5—21no—
363Dic5—21no—
364Dic5—21no—
365Dic5—21no—
366Dic5—21no—
367Dic5—21no—
368Dic5—101no—
369Dic5—101no—
370Dic5—202no—
371C24—251no—
372C2×Q8—51no—
373C2×D4—251no—
374C2×D4—251no—
375C2×D4—251no—
376C22×C4—51no—
377C22×C4—51no—
378C22×C4—251no—
379C22×C4—251no—
380C22×C4—251no—
381C4⋊C4—51no—
382C4⋊C4—51no—
383C4⋊C4—51no—
384C4⋊C4—51no—
385C4⋊C4—51no—
386C4⋊C4—251no—
387C4⋊C4—251no—
388C4⋊C4—251no—
389C22⋊C4—51no—
390C22⋊C4—51no—
391C22⋊C4—251no—
392C22⋊C4—251no—
393C22⋊C4—251no—
394C22⋊C4—251no—
395C22⋊C4—251no—
396C22⋊C4—251no—
397C22⋊C4—251no—
398C22⋊C4—251no—
399C22⋊C4—251no—
400C22⋊C4—251no—
401C22⋊C4—251no—
402C22⋊C4D5211yescharacteristic
403C42—51no—
404C42—51no—
405C42—251no—
406C2×C5—22no—
407C2×C5—22no—
408C2×C5—22no—
409C2×C5—22no—
410C2×C5—22no—
411C2×C5—44no—
412C2×C5C2×#70911yescharacteristic
413C2×C5#414411yescharacteristic
414C2×C5C2×#71311yescharacteristic
415C2×C5D5×#4111yescharacteristic
416C2×C5D5×#4111yescharacteristic
417C2×C5#414911yescharacteristic
418D5—102no—
419D5—102no—
420D5—204no—
421D5—204no—
422D5—201no—
423D5—201no—
424D5—201no—
425D5—201no—
426D5—402no—
427C23—251no—
428C23—251no—
429C23—251no—
430C23—502no—
431C23—502no—
432C23—1001no—
433C23C2×D5211yescharacteristic
434Q8—201no—
435D4—502no—
436D4—502no—
437D4—1001no—
438C2×C4—51no—
439C2×C4—51no—
440C2×C4—51no—
441C2×C4—51no—
442C2×C4—51no—
443C2×C4—51no—
444C2×C4—51no—
445C2×C4—51no—
446C2×C4—102no—
447C2×C4—201no—
448C2×C4—251no—
449C2×C4—502no—
450C2×C4—1001no—
451C2×C4—1001no—
452C2×C4C2×D5211yescharacteristic
453C2×C4C2×D5211yescharacteristic
454C5—22no—
455C5#6586111yescharacteristic, complemented
456C5#6597111yescharacteristic, complemented
457C22—21no—
458C22—21no—
459C22—252no—
460C22—501no—
461C22—501no—
462C22—501no—
463C22—501no—
464C22—501no—
465C22—501no—
466C22—501no—
467C22—501no—
468C22—1004no—
469C22#8787412yes—
470C22C22×D5211yescharacteristic
471C4—21no—
472C4—21no—
473C4—101no—
474C4—101no—
475C4—101no—
476C4—101no—
477C4—101no—
478C4—101no—
479C4—101no—
480C4—101no—
481C4—501no—
482C2—22no—
483C2—502no—
484C2—1001no—
485C2—1001no—
486C2C2×#8787411yescharacteristic
487C2#1171747711yescharacteristic
488C2#1171747911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radical#11717202C2
hypercenterC22⋊C4D52
hypocenterC52#551
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#4024C2semidirect
3. C5×#4023C2semidirect
4. #11717092C2semidirect
5. #11717084C2semidirect
6. C2×#87779C2semidirect
7. #11717070C2semidirect
8. #11717016C2semidirect
9. #11717016C2semidirect
10. #11717013C2semidirect
11. #11717008C2semidirect
12. #11716987C2semidirect
13. #11716982C2semidirect
14. #11716966C2semidirect
15. #11716954C2semidirect
22. C2×C5×Dic10C22semidirect
25. C2×#6036C22semidirect
31. C5×C4⋊Dic5C22semidirect
32. C5×#683C22semidirect
33. C5×#683C22semidirect
34. C5×#683C22semidirect
35. C5×#683C22semidirect
36. C4×C5×Dic5C22semidirect
37. C4×C5×Dic5C22semidirect
38. #87781C22semidirect
39. #87780C22semidirect
40. #87780C22semidirect
47. #87779C22semidirect
48. #87779C22semidirect
49. #87779C22semidirect
50. Dic52C22semidirect
105. #4024D5semidirect
106. #4023D5semidirect
159. C2×Dic10C2×D5semidirect
200. C4⋊Dic5C2×D5semidirect
205. #683C2×D5semidirect
206. #683C2×D5semidirect
207. #683C2×D5semidirect
208. #683C2×D5semidirect
211. C4×Dic5C2×D5semidirect
212. C4×Dic5C2×D5semidirect
303. C52#551semidirect
455. C5#65861semidirect
456. C5#65971semidirect

Maximal subgroups

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

subgroupindex
#117172022
C5×#40242
C5×#40232
#117170922
#117170842
C2×#877792
#117170702
#117170162
#117170132
#117170082
#117169872
#117169822
#117169662
#117169542
#659715
#658615