Finite Group Databasebeta

← #13105391

Subgroups — #13105391

Order 1600.

574 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#11717203C211yesmaximal, characteristic, complemented
2C5×#4031C211yesmaximal, characteristic, complemented
3C5×#4027C211yesmaximal, characteristic, complemented
4#11717101C211yesmaximal, characteristic
5#11717087C211yesmaximal, characteristic, complemented
6#11717079C211yesmaximal, characteristic, complemented
7#11717070C211yesmaximal, characteristic, complemented
8#11717039C211yesmaximal, characteristic, complemented
9D5×#685C211yesmaximal, characteristic, complemented
10#11717008C211yesmaximal, characteristic, complemented
11#11716995C211yesmaximal, characteristic, complemented
12#11716992C211yesmaximal, characteristic, complemented
13#11716978C211yesmaximal, characteristic, complemented
14#11716977C211yesmaximal, characteristic, complemented
15#11716974C211yesmaximal, characteristic, complemented
16C22×D52C2211yescharacteristic
17C2×C22⋊(C5⋊D5)C2211yescharacteristic
18C2×C4×C5⋊D5C2211yescharacteristic
19C2×C5×C22⋊D5C2211yescharacteristic
20C2×C5×C22⋊D5C2211yescharacteristic
21C22×C5×Dic5C2211yescharacteristic, complemented
22C2×C4×C5×D5C2211yescharacteristic
23C2×C5×Dic10C2211yescharacteristic, complemented
24C2×#6036C2211yescharacteristic
25C2×D5×Dic5C2211yescharacteristic
26C52×C22⋊C4C2211yescharacteristic, complemented
27C22⋊#1039C2211yescharacteristic, complemented
28#87808C2211yescharacteristic
29C4⋊#1039C2211yescharacteristic, complemented
30#87806C2211yescharacteristic, complemented
31C5×C22⋊Dic5C2211yescharacteristic, complemented
32C5×#685C2211yescharacteristic
33C5×#685C2211yescharacteristic
34C5×#685C2211yescharacteristic
35C5×#683C2211yescharacteristic, complemented
36C4×C5×Dic5C2211yescharacteristic, complemented
37C4×C5×Dic5C2211yescharacteristic, complemented
38#87781C2211yescharacteristic, complemented
39#87781C2211yescharacteristic, complemented
40#87781C2211yescharacteristic, complemented
41#87781C2211yescharacteristic, complemented
42#87780C2211yescharacteristic, complemented
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50Dic52C2211yescharacteristic, complemented
51#66013—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)—22no—
63C2×#1039C2311yescharacteristic
64C2×#1039C2311yescharacteristic
65C2×#1039C2311yescharacteristic
66C4×C5⋊D5—22no—
67C5×C22⋊D5—22no—
68C5×C22⋊D5—41no—
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—41no—
77C5×Dic10—41no—
78#6036—41no—
79D5×Dic5—22no—
80C5×#124—51no—
81C4×C5×D4—51no—
82#4092—51no—
83#4092—51no—
84#4083—51no—
85#4083—51no—
86#4081—51no—
87#4081—51no—
88#4075—51no—
89C4×C22⋊D5—51no—
90C2×#685—51no—
91#4070—51no—
92#4044—51no—
93#4042—51no—
94#4031D511yescharacteristic, complemented
95#4029—51no—
96#4029—51no—
97#4029—51no—
98#4029—102no—
99#4027—51no—
100#4027D511yescharacteristic, complemented
101D5×C22⋊C4—51no—
102D5×C22⋊C4—51no—
103D5×C22⋊C4—51no—
104#4025—51no—
105#4025—51no—
106#4023—51no—
107#4022—51no—
108#4018—51no—
109#4018—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#4112yes—
118C2×C5×D5—21no—
119C2×C5×D5—21no—
120C2×C5×D5—21no—
121C2×C5×D5—21no—
122C2×C5×D5—21no—
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×Q8—51no—
139C2×C5×D4—51no—
140C2×C5×D4—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×Dic5C2×D511yescharacteristic, complemented
150C2×C4×D5—51no—
151C2×C4×D5—51no—
152C2×C4×D5—51no—
153C2×C4×D5—51no—
154C2×C4×D5—51no—
155C2×C4×D5—102no—
156C2×C4×D5C2×D511yescharacteristic
157C2×Dic10C2×D511yescharacteristic, complemented
158C5×C4⋊C4—51no—
159C5×C22⋊C4—22no—
160C5×C22⋊C4—51no—
161C5×C22⋊C4—51no—
162C5×C22⋊C4—51no—
163C5×C22⋊C4—51no—
164C5×C22⋊C4C2×D511yescharacteristic, complemented
165C5×C22⋊C4C2×D511yescharacteristic, complemented
166C42×C5—51no—
167C42×C5—51no—
168C22⋊Dic5—51no—
169C22⋊Dic5—51no—
170C22⋊Dic5—51no—
171C22⋊Dic5—51no—
172C22⋊Dic5—51no—
173C22⋊Dic5—51no—
174C22⋊Dic5—51no—
175C22⋊Dic5—102no—
176C22⋊Dic5C2×D511yescharacteristic, complemented
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—102no—
189#685C2×D511yescharacteristic
190#685C2×D511yescharacteristic
191#685C2×D511yescharacteristic
192C4⋊Dic5—51no—
193C4⋊Dic5—51no—
194C4⋊Dic5—51no—
195C4⋊Dic5—102no—
196#683—51no—
197#683—51no—
198#683—51no—
199#683—51no—
200#683—51no—
201#683—51no—
202#683—51no—
203#683—51no—
204#683—51no—
205#683—51no—
206#683—51no—
207#683—102no—
208#683C2×D511yescharacteristic, complemented
209C4×Dic5—51no—
210C4×Dic5—51no—
211C4×Dic5C2×D511yescharacteristic, complemented
212C4×Dic5C2×D511yescharacteristic, complemented
213#551—251noSylow
214C2×C52—41no—
215C2×C52C2×#4111yescharacteristic
216C2×C52C2×#4111yescharacteristic
217C2×C52H(5,𝔽2)11yescharacteristic
218C5⋊D5—22no—
219C5×D5—22no—
220C5×D5—41no—
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×Q8—201no—
238C5×D4—102no—
239C5×D4—201no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—51no—
243C2×C4×C5—51no—
244C2×C4×C5—51no—
245C2×C4×C5—51no—
246C2×C4×C5—51no—
247C2×C4×C5—51no—
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—41no—
256C22⋊D5—102no—
257C22⋊D5—102no—
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×Dic5C22×D511yescharacteristic
271C2×Dic5C22×D511yescharacteristic
272C2×Dic5C22×D511yescharacteristic
273C2×Dic5C22×D511yescharacteristic
274C2×Dic5C22×D511yescharacteristic
275C2×Dic5C22×D511yescharacteristic
276C4×D5—41no—
277C4×D5—102no—
278C4×D5—102no—
279C4×D5—102no—
280C4×D5—204no—
281C4×D5—201no—
282C4×D5—201no—
283Dic10—41no—
284#126—251no—
285#126—251no—
286#124—51no—
287#123—251no—
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—101no—
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—21no—
323C2×D5—52no—
324C2×D5—52no—
325C2×D5—104no—
326C2×D5—101no—
327C2×D5—101no—
328C2×D5—101no—
329C2×D5—101no—
330C2×D5—202no—
331C2×D5—202no—
332C2×D5—204no—
333C2×D5—204no—
334C2×D5#70912yes—
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—101no—
354Dic5—101no—
355Dic5—101no—
356Dic5—101no—
357Dic5—101no—
358Dic5—101no—
359Dic5—202no—
360Dic5—202no—
361Dic5—202no—
362C24—251no—
363C2×Q8—51no—
364C2×D4—51no—
365C2×D4—51no—
366C2×D4—251no—
367C22×C4—51no—
368C22×C4—51no—
369C22×C4—251no—
370C22×C4—251no—
371C22×C4—251no—
372C4⋊C4—51no—
373C4⋊C4—251no—
374C4⋊C4—251no—
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—51no—
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—51no—
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—201no—
405C2×C5C2×#70911yescharacteristic
406C2×C5#414411yescharacteristic
407C2×C5#414411yescharacteristic
408C2×C5D5×#4111yescharacteristic
409C2×C5D5×#4111yescharacteristic
410C2×C5D5×#4111yescharacteristic
411D5—22no—
412D5—41no—
413D5—102no—
414D5—102no—
415D5—204no—
416C23—51no—
417C23—51no—
418C23—251no—
419C23—502no—
420C23—502no—
421C23—1001no—
422C23C2×D5211yescharacteristic
423Q8—201no—
424D4—102no—
425D4—201no—
426D4—502no—
427C2×C4—51no—
428C2×C4—51no—
429C2×C4—51no—
430C2×C4—51no—
431C2×C4—51no—
432C2×C4—51no—
433C2×C4—201no—
434C2×C4—201no—
435C2×C4—251no—
436C2×C4—251no—
437C2×C4—251no—
438C2×C4—502no—
439C2×C4—502no—
440C2×C4—1001no—
441C2×C4C2×D5211yescharacteristic
442C2×C4C2×D5211yescharacteristic
443C5—22no—
444C5#6589011yescharacteristic, complemented
445C5#6601311yescharacteristic, complemented
446C22—21no—
447C22—21no—
448C22—21no—
449C22—52no—
450C22—101no—
451C22—101no—
452C22—101no—
453C22—101no—
454C22—101no—
455C22—252no—
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—201no—
474C2—502no—
475C2D5×#70911yescharacteristic
476C2#1171747711yescharacteristic
477C2#1171747811yescharacteristic
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. #11717203C2semidirect
2. C5×#4031C2semidirect
3. C5×#4027C2semidirect
5. #11717087C2semidirect
6. #11717079C2semidirect
7. #11717070C2semidirect
8. #11717039C2semidirect
9. D5×#685C2semidirect
10. #11717008C2semidirect
11. #11716995C2semidirect
12. #11716992C2semidirect
13. #11716978C2semidirect
14. #11716977C2semidirect
15. #11716974C2semidirect
21. C22×C5×Dic5C22semidirect
23. C2×C5×Dic10C22semidirect
26. C52×C22⋊C4C22semidirect
27. C22⋊#1039C22semidirect
29. C4⋊#1039C22semidirect
30. #87806C22semidirect
31. C5×C22⋊Dic5C22semidirect
35. C5×#683C22semidirect
36. C4×C5×Dic5C22semidirect
37. C4×C5×Dic5C22semidirect
38. #87781C22semidirect
39. #87781C22semidirect
40. #87781C22semidirect
41. #87781C22semidirect
42. #87780C22semidirect
50. Dic52C22semidirect
94. #4031D5semidirect
100. #4027D5semidirect
149. C22×Dic5C2×D5semidirect
157. C2×Dic10C2×D5semidirect
164. C5×C22⋊C4C2×D5semidirect
165. C5×C22⋊C4C2×D5semidirect
176. C22⋊Dic5C2×D5semidirect
208. #683C2×D5semidirect
211. C4×Dic5C2×D5semidirect
212. C4×Dic5C2×D5semidirect
299. C52#551semidirect
393. C22⋊C4D52semidirect
444. C5#65890semidirect
445. C5#66013semidirect

Maximal subgroups

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

subgroupindex
#117172032
C5×#40312
C5×#40272
#117171012
#117170872
#117170792
#117170702
#117170392
D5×#6852
#117170082
#117169952
#117169922
#117169782
#117169772
#117169742
#660135
#658905