Finite Group Databasebeta

← #13105390

Subgroups — #13105390

Order 1600.

572 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717205C211yesmaximal, characteristic, complemented
2C5×#4029C211yesmaximal, characteristic, complemented
3C5×#4027C211yesmaximal, characteristic, complemented
4#11717101C211yesmaximal, characteristic
5Dic5×C22⋊D5C211yesmaximal, characteristic, complemented
6#11717087C211yesmaximal, characteristic, complemented
7#11717084C211yesmaximal, characteristic, complemented
8D5×#685C211yesmaximal, characteristic, complemented
9D5×#685C211yesmaximal, characteristic, complemented
10#11717015C211yesmaximal, characteristic, complemented
11#11716996C211yesmaximal, characteristic, complemented
12#11716987C211yesmaximal, characteristic, complemented
13#11716975C211yesmaximal, characteristic, complemented
14#11716973C211yesmaximal, characteristic, complemented
15#11716966C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×#6045C2211yescharacteristic, complemented
19C2×C5×C22⋊D5C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C22×C5×Dic5C2211yescharacteristic, complemented
22C2×C4×C5×D5C2211yescharacteristic
23C2×C4×C5×D5C2211yescharacteristic
24C2×D5×Dic5C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic
26C52×C22⋊C4C2211yescharacteristic, complemented
27C22⋊#1039C2211yescharacteristic, complemented
28#87808C2211yescharacteristic
29#87808C2211yescharacteristic
30C4×#1039C2211yescharacteristic, complemented
31C5×C22⋊Dic5C2211yescharacteristic, complemented
32C5×#685C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×C4⋊Dic5C2211yescharacteristic, complemented
35C5×#683C2211yescharacteristic, complemented
36C5×#683C2211yescharacteristic, complemented
37C4×C5×Dic5C2211yescharacteristic, complemented
38#87781C2211yescharacteristic, complemented
39#87780C2211yescharacteristic, complemented
40#87780C2211yescharacteristic, complemented
41#87780C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87779C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50Dic52C2211yescharacteristic, complemented
51#65992—51nomaximal
52#65890—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5C2311yescharacteristic
55C22×C5×D5C2311yescharacteristic
56C22×C5×D5C2311yescharacteristic
57C2×D52—22no—
58C2×D52—22no—
59C2×D52—41no—
60C2×C4×C52C2311yescharacteristic
61C2×C4×C52C2311yescharacteristic
62C22⋊(C5⋊D5)—41no—
63C2×#1039C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66#6045—41no—
67C5×C22⋊D5—22no—
68C5×C22⋊D5—22no—
69C2×C5×Dic5—41no—
70C2×C5×Dic5C2311yescharacteristic
71C2×C5×Dic5C2311yescharacteristic
72C2×C5×Dic5C2311yescharacteristic
73C2×C5×Dic5C2311yescharacteristic
74C2×C5×Dic5C2311yescharacteristic
75C2×C5×Dic5C2311yescharacteristic
76C4×C5×D5—22no—
77C4×C5×D5—41no—
78D5×Dic5—22no—
79D5×Dic5—41no—
80C5×#123—51no—
81C4×C5×D4—51no—
82#4083—51no—
83#4083—51no—
84#4081—51no—
85D4×Dic5—51no—
86#4075—51no—
87C4×C22⋊D5—51no—
88C2×#685—51no—
89C2×#685—51no—
90#4072—51no—
91#4070—51no—
92#4044—51no—
93#4044—51no—
94#4044—51no—
95#4043—51no—
96#4042—51no—
97#4038—51no—
98#4032—51no—
99#4031—51no—
100#4031—51no—
101#4031—102no—
102#4029—51no—
103#4029D511yescharacteristic, complemented
104#4027D511yescharacteristic, complemented
105D5×C22⋊C4—51no—
106D5×C22⋊C4—51no—
107#4024—51no—
108#4023—51no—
109#4022—51no—
110#4018—51no—
111C22×C52—21no—
112C22×C52—21no—
113C22×C52—21no—
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#4112yes—
123C2×C5×D5#4112yes—
124D52—44no—
125C4×C52—21no—
126C4×C52—21no—
127#1039—21no—
128#1039—21no—
129#1039—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—
138C2×C5×D4—51no—
139C2×C5×D4—51no—
140C22×C4×C5—51no—
141C22×C4×C5—51no—
142C22×C4×C5—51no—
143C2×C22⋊D5—51no—
144C2×C22⋊D5—51no—
145C2×C22⋊D5—102no—
146C2×C22⋊D5C2×D511yescharacteristic
147C2×C22⋊D5C2×D511yescharacteristic
148C22×Dic5—51no—
149C22×Dic5—51no—
150C22×Dic5C2×D511yescharacteristic, complemented
151C2×C4×D5—51no—
152C2×C4×D5—51no—
153C2×C4×D5C2×D511yescharacteristic
154C2×C4×D5C2×D511yescharacteristic
155C2×Dic10—51no—
156C2×Dic10—51no—
157C2×Dic10—102no—
158C5×C4⋊C4—51no—
159C5×C4⋊C4—51no—
160C5×C4⋊C4—51no—
161C5×C22⋊C4—22no—
162C5×C22⋊C4—51no—
163C5×C22⋊C4—51no—
164C5×C22⋊C4—51no—
165C5×C22⋊C4C2×D511yescharacteristic, complemented
166C5×C22⋊C4C2×D511yescharacteristic, complemented
167C42×C5—51no—
168C22⋊Dic5—51no—
169C22⋊Dic5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—51no—
173C22⋊Dic5—51no—
174C22⋊Dic5—102no—
175C22⋊Dic5C2×D511yescharacteristic, complemented
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—102no—
191#685—102no—
192#685C2×D511yescharacteristic
193#685C2×D511yescharacteristic
194C4⋊Dic5—51no—
195C4⋊Dic5—51no—
196C4⋊Dic5—51no—
197C4⋊Dic5—51no—
198C4⋊Dic5C2×D511yescharacteristic, complemented
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683C2×D511yescharacteristic, complemented
206#683C2×D511yescharacteristic, complemented
207C4×Dic5—51no—
208C4×Dic5—51no—
209C4×Dic5—51no—
210C4×Dic5—51no—
211C4×Dic5—102no—
212C4×Dic5C2×D511yescharacteristic, complemented
213#551—251noSylow
214C2×C52—41no—
215C2×C52C2×#4111yescharacteristic
216C2×C52C2×#4111yescharacteristic
217C2×C52H(5,𝔽2)11yescharacteristic
218C5⋊D5—41no—
219C5×D5—22no—
220C5×D5—22no—
221C23×C5—22no—
222C23×C5—51no—
223C23×C5—51no—
224C23×C5C22×D511yescharacteristic
225C23×C5C22×D511yescharacteristic
226C22×D5—51no—
227C22×D5—51no—
228C22×D5—102no—
229C22×D5—102no—
230C22×D5—102no—
231C22×D5—102no—
232C22×D5—102no—
233C22×D5—201no—
234C22×D5—201no—
235C22×D5C22×D511yescharacteristic
236C22×D5C22×D511yescharacteristic
237C5×D4—102no—
238C5×D4—102no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—51no—
242C2×C4×C5—51no—
243C2×C4×C5—51no—
244C2×C4×C5—51no—
245C2×C4×C5—51no—
246C2×C4×C5—51no—
247C2×C4×C5—102no—
248C2×C4×C5—201no—
249C2×C4×C5—201no—
250C2×C4×C5C22×D511yescharacteristic
251C2×C4×C5C22×D511yescharacteristic
252C2×C4×C5C22×D511yescharacteristic
253C2×C4×C5C22×D511yescharacteristic
254C22⋊D5—22no—
255C22⋊D5—22no—
256C22⋊D5—201no—
257C22⋊D5—201no—
258C22⋊D5—204no—
259C2×Dic5—41no—
260C2×Dic5—51no—
261C2×Dic5—51no—
262C2×Dic5—51no—
263C2×Dic5—51no—
264C2×Dic5—51no—
265C2×Dic5—51no—
266C2×Dic5—102no—
267C2×Dic5—102no—
268C2×Dic5—102no—
269C2×Dic5—102no—
270C2×Dic5—201no—
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274C2×Dic5C22×D511yescharacteristic
275C2×Dic5C22×D511yescharacteristic
276C2×Dic5C22×D511yescharacteristic
277C4×D5—22no—
278C4×D5—41no—
279C4×D5—102no—
280C4×D5—201no—
281Dic10—201no—
282Dic10—201no—
283Dic10—204no—
284#126—251no—
285#126—251no—
286#124—251no—
287#123—51no—
288#123—251no—
289#123—251no—
290C22⋊Q8—251no—
291C22⋊Q8—251no—
292#120—251no—
293C4×D4—51no—
294C4×D4—251no—
295#117—251no—
296#117—251no—
297C2×C22⋊C4—251no—
298C2×C22⋊C4—251no—
299C52#55111yescharacteristic, complemented, Sylow
300C22×C5—21no—
301C22×C5—21no—
302C22×C5—21no—
303C22×C5—21no—
304C22×C5—21no—
305C22×C5—21no—
306C22×C5—22no—
307C22×C5—42no—
308C22×C5—42no—
309C22×C5—42no—
310C22×C5—52no—
311C22×C5—52no—
312C22×C5—101no—
313C22×C5—101no—
314C22×C5—101no—
315C22×C5—101no—
316C22×C5C23×D511yescharacteristic
317C22×C5C23×D511yescharacteristic
318C2×D5—21no—
319C2×D5—21no—
320C2×D5—21no—
321C2×D5—21no—
322C2×D5—101no—
323C2×D5—101no—
324C2×D5—101no—
325C2×D5—101no—
326C2×D5—101no—
327C2×D5—101no—
328C2×D5—202no—
329C2×D5—202no—
330C2×D5—202no—
331C2×D5—204no—
332C2×D5—204no—
333C2×D5#70912yes—
334C2×D5#71112yes—
335C4×C5—21no—
336C4×C5—21no—
337C4×C5—21no—
338C4×C5—21no—
339C4×C5—42no—
340C4×C5—42no—
341C4×C5—101no—
342C4×C5—101no—
343C4×C5—101no—
344C4×C5—101no—
345C4×C5—101no—
346C4×C5—101no—
347Dic5—21no—
348Dic5—21no—
349Dic5—21no—
350Dic5—21no—
351Dic5—21no—
352Dic5—21no—
353Dic5—104no—
354Dic5—101no—
355Dic5—101no—
356Dic5—101no—
357Dic5—101no—
358Dic5—101no—
359Dic5—101no—
360Dic5—202no—
361Dic5—202no—
362C24—251no—
363C2×Q8—251no—
364C2×D4—51no—
365C2×D4—51no—
366C2×D4—251no—
367C22×C4—51no—
368C22×C4—51no—
369C22×C4—51no—
370C22×C4—251no—
371C22×C4—251no—
372C4⋊C4—51no—
373C4⋊C4—51no—
374C4⋊C4—51no—
375C4⋊C4—251no—
376C4⋊C4—251no—
377C4⋊C4—251no—
378C4⋊C4—251no—
379C4⋊C4—251no—
380C22⋊C4—51no—
381C22⋊C4—51no—
382C22⋊C4—51no—
383C22⋊C4—251no—
384C22⋊C4—251no—
385C22⋊C4—251no—
386C22⋊C4—251no—
387C22⋊C4—251no—
388C22⋊C4—251no—
389C22⋊C4—251no—
390C22⋊C4—251no—
391C22⋊C4—251no—
392C22⋊C4—251no—
393C22⋊C4D5211yescharacteristic, complemented
394C42—51no—
395C42—251no—
396C42—251no—
397C2×C5—22no—
398C2×C5—22no—
399C2×C5—22no—
400C2×C5—41no—
401C2×C5—41no—
402C2×C5—82no—
403C2×C5—102no—
404C2×C5—102no—
405C2×C5C2×#70911yescharacteristic
406C2×C5C2×#71111yescharacteristic
407C2×C5#414411yescharacteristic
408C2×C5D5×#4111yescharacteristic
409C2×C5D5×#4111yescharacteristic
410C2×C5#414911yescharacteristic
411D5—22no—
412D5—22no—
413D5—201no—
414D5—201no—
415D5—402no—
416C23—51no—
417C23—51no—
418C23—251no—
419C23—502no—
420C23—502no—
421C23—1001no—
422C23C2×D5211yescharacteristic
423Q8—1001no—
424D4—102no—
425D4—102no—
426D4—1001no—
427C2×C4—51no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—51no—
433C2×C4—102no—
434C2×C4—201no—
435C2×C4—201no—
436C2×C4—251no—
437C2×C4—251no—
438C2×C4—251no—
439C2×C4—502no—
440C2×C4—1001no—
441C2×C4C2×D5211yescharacteristic
442C2×C4C2×D5211yescharacteristic
443C5—22no—
444C5#6589011yescharacteristic, complemented
445C5#6599211yescharacteristic, complemented
446C22—21no—
447C22—21no—
448C22—21no—
449C22—52no—
450C22—52no—
451C22—101no—
452C22—101no—
453C22—101no—
454C22—101no—
455C22—501no—
456C22—501no—
457C22—501no—
458C22—1004no—
459C22C22×D5211yescharacteristic
460C4—21no—
461C4—21no—
462C4—101no—
463C4—101no—
464C4—101no—
465C4—101no—
466C4—101no—
467C4—101no—
468C4—501no—
469C4—501no—
470C4—501no—
471C2—41no—
472C2—102no—
473C2—102no—
474C2—1001no—
475C2D5×#70911yescharacteristic
476C2D5×#71111yescharacteristic
477C2#1171747911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radical#11717101C2
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
1. #11717205C2semidirect
2. C5×#4029C2semidirect
3. C5×#4027C2semidirect
5. Dic5×C22⋊D5C2semidirect
6. #11717087C2semidirect
7. #11717084C2semidirect
8. D5×#685C2semidirect
9. D5×#685C2semidirect
10. #11717015C2semidirect
11. #11716996C2semidirect
12. #11716987C2semidirect
13. #11716975C2semidirect
14. #11716973C2semidirect
15. #11716966C2semidirect
18. C2×#6045C22semidirect
21. C22×C5×Dic5C22semidirect
26. C52×C22⋊C4C22semidirect
27. C22⋊#1039C22semidirect
30. C4×#1039C22semidirect
31. C5×C22⋊Dic5C22semidirect
34. C5×C4⋊Dic5C22semidirect
35. C5×#683C22semidirect
36. C5×#683C22semidirect
37. C4×C5×Dic5C22semidirect
38. #87781C22semidirect
39. #87780C22semidirect
40. #87780C22semidirect
41. #87780C22semidirect
42. #87780C22semidirect
50. Dic52C22semidirect
103. #4029D5semidirect
104. #4027D5semidirect
150. C22×Dic5C2×D5semidirect
165. C5×C22⋊C4C2×D5semidirect
166. C5×C22⋊C4C2×D5semidirect
175. C22⋊Dic5C2×D5semidirect
198. C4⋊Dic5C2×D5semidirect
205. #683C2×D5semidirect
206. #683C2×D5semidirect
212. C4×Dic5C2×D5semidirect
299. C52#551semidirect
393. C22⋊C4D52semidirect
444. C5#65890semidirect
445. C5#65992semidirect

Maximal subgroups

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

subgroupindex
#117172052
C5×#40292
C5×#40272
#117171012
Dic5×C22⋊D52
#117170872
#117170842
D5×#6852
#117170152
#117169962
#117169872
#117169752
#117169732
#117169662
#659925
#658905