Finite Group Databasebeta

← #13105467

Subgroups — #13105467

Order 1600.

484 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717209C211yesmaximal, characteristic, complemented
2C5×#4040C211yesmaximal, characteristic
3C5×#4033C211yesmaximal, characteristic, complemented
4Dic5×D20C211yesmaximal, characteristic
5#11717006C211yesmaximal, characteristic, complemented
6#11717006C211yesmaximal, characteristic, complemented
7#11717002C211yesmaximal, characteristic
8#11716996C211yesmaximal, characteristic, complemented
9#11716995C211yesmaximal, characteristic, complemented
10D5×#683C211yesmaximal, characteristic, complemented
11#11716986C211yesmaximal, characteristic, complemented
12#11716978C211yesmaximal, characteristic, complemented
13#11716978C211yesmaximal, characteristic, complemented
14#11716973C211yesmaximal, characteristic, complemented
15#11716963C211yesmaximal, characteristic, complemented
16C2×C5×D20C2211yescharacteristic
17C2×C4×C5×D5C2211yescharacteristic
18C2×C5×Dic10C2211yescharacteristic
19C2×D5×Dic5C2211yescharacteristic
20C2×D5×Dic5C2211yescharacteristic
21C2×D5×Dic5C2211yescharacteristic
22C52×C4⋊C4C2211yescharacteristic
23C4⋊#1039C2211yescharacteristic
24#87806C2211yescharacteristic
25#87806C2211yescharacteristic
26#87806C2211yescharacteristic
27#87806C2211yescharacteristic
28C4×#1039C2211yescharacteristic
29C5×#685C2211yescharacteristic
30C5×#685C2211yescharacteristic
31C5×#685C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C4×C5×Dic5C2211yescharacteristic
36C4×C5×Dic5C2211yescharacteristic
37C4×C5×Dic5C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87780C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49#87778C2211yescharacteristic
50Dic52C2211yescharacteristic
51#65984—51nomaximal
52#65915—51nomaximal
53C22×C5×D5C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×C4×C52C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×#1039C2311yescharacteristic
62C2×C5×Dic5C2311yescharacteristic
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C5×D20—22no—
69C4×C5×D5—22no—
70C5×Dic10—22no—
71D5×Dic5—22no—
72D5×Dic5—22no—
73D5×Dic5—22no—
74C5×#123—51no—
75C4×C5×Q8—51no—
76#4092—51no—
77#4092—51no—
78#4081—51no—
79D4×Dic5—51no—
80#4072—51no—
81C2×#683—51no—
82#4040—51no—
83#4040D511yescharacteristic
84#4038—51no—
85D5×C4⋊C4—51no—
86#4036—51no—
87#4035—51no—
88#4035—51no—
89#4035—51no—
90#4035—51no—
91#4035—102no—
92#4033D511yescharacteristic, complemented
93#4025—51no—
94#4025—51no—
95#4024—51no—
96#4024—51no—
97#4023—51no—
98#4023—51no—
99#4022—51no—
100#4022—51no—
101C4×D20—51no—
102#4018—51no—
103#4018—51no—
104#4016—51no—
105C22×C52C2411yescharacteristic
106C2×C5×D5—21no—
107C2×C5×D5—21no—
108C2×C5×D5—21no—
109C2×C5×D5—21no—
110C2×C5×D5—21no—
111C2×C5×D5#4112yes—
112C4×C52—21no—
113C4×C52—21no—
114C4×C52—21no—
115#1039—21no—
116#1039—21no—
117#1039—21no—
118#1039—21no—
119C5×Dic5—21no—
120C5×Dic5—21no—
121C5×Dic5—21no—
122C5×Dic5—21no—
123C5×Dic5—21no—
124C5×Dic5#4112yes—
125C2×C5×Q8—51no—
126C2×C5×D4—51no—
127C22×C4×C5—51no—
128C22×Dic5—51no—
129C22×Dic5—51no—
130C22×Dic5—51no—
131C2×D20C2×D511yescharacteristic
132C2×C4×D5—51no—
133C2×C4×D5—51no—
134C2×C4×D5—51no—
135C2×C4×D5C2×D511yescharacteristic
136C2×Dic10C2×D511yescharacteristic
137C5×C4⋊C4—22no—
138C5×C4⋊C4—51no—
139C5×C4⋊C4—51no—
140C5×C4⋊C4—51no—
141C5×C4⋊C4C2×D511yescharacteristic
142C5×C4⋊C4C2×D511yescharacteristic
143C5×C22⋊C4—51no—
144C5×C22⋊C4—51no—
145C5×C22⋊C4—51no—
146C42×C5—51no—
147C42×C5—51no—
148C42×C5—51no—
149C22⋊Dic5—51no—
150C22⋊Dic5—51no—
151C22⋊Dic5—51no—
152C22⋊Dic5—51no—
153C22⋊Dic5—51no—
154#685—51no—
155#685—51no—
156#685—51no—
157#685—51no—
158#685—51no—
159#685C2×D511yescharacteristic
160#685C2×D511yescharacteristic
161#685C2×D511yescharacteristic
162C4⋊Dic5—51no—
163C4⋊Dic5—51no—
164C4⋊Dic5—51no—
165C4⋊Dic5—51no—
166C4⋊Dic5—51no—
167C4⋊Dic5—51no—
168C4⋊Dic5—51no—
169C4⋊Dic5—102no—
170#683—51no—
171#683—51no—
172#683—51no—
173#683—51no—
174#683—51no—
175#683—51no—
176#683—51no—
177#683—51no—
178#683—51no—
179#683—51no—
180#683—51no—
181#683—51no—
182#683—51no—
183#683—51no—
184#683—51no—
185#683—51no—
186#683—51no—
187#683—102no—
188#683—102no—
189#683—102no—
190#683—102no—
191#683C2×D511yescharacteristic
192#683C2×D511yescharacteristic
193#683C2×D511yescharacteristic
194C4×Dic5—51no—
195C4×Dic5—51no—
196C4×Dic5—51no—
197C4×Dic5—51no—
198C4×Dic5—102no—
199C4×Dic5C2×D511yescharacteristic
200C4×Dic5C2×D511yescharacteristic
201C4×Dic5C2×D511yescharacteristic
202#552—251noSylow
203C2×C52C2×#4111yescharacteristic
204C2×C52C2×#4111yescharacteristic
205C2×C522−1+411yescharacteristic
206C5×D5—22no—
207C5×D5—41no—
208C23×C5—51no—
209C23×C5—51no—
210C22×D5C22×D511yescharacteristic
211C22×D5C22×D511yescharacteristic
212C5×Q8—102no—
213C5×D4—102no—
214C2×C4×C5—22no—
215C2×C4×C5—22no—
216C2×C4×C5—22no—
217C2×C4×C5—51no—
218C2×C4×C5—51no—
219C2×C4×C5—51no—
220C2×C4×C5—51no—
221C2×C4×C5—51no—
222C2×C4×C5—51no—
223C2×C4×C5—102no—
224C2×C4×C5C22×D511yescharacteristic
225C2×C4×C5C22×D511yescharacteristic
226C2×C4×C5C22×D511yescharacteristic
227C2×C4×C5C22×D511yescharacteristic
228C2×C4×C5C22×D511yescharacteristic
229C2×C4×C5C22×D511yescharacteristic
230C2×Dic5—51no—
231C2×Dic5—51no—
232C2×Dic5—51no—
233C2×Dic5—51no—
234C2×Dic5—51no—
235C2×Dic5—51no—
236C2×Dic5—51no—
237C2×Dic5—51no—
238C2×Dic5—102no—
239C2×Dic5—102no—
240C2×Dic5—102no—
241C2×Dic5—102no—
242C2×Dic5—102no—
243C2×Dic5—102no—
244C2×Dic5—102no—
245C2×Dic5C22×D511yescharacteristic
246C2×Dic5C22×D511yescharacteristic
247C2×Dic5C22×D511yescharacteristic
248C2×Dic5C22×D511yescharacteristic
249C2×Dic5C22×D511yescharacteristic
250C2×Dic5C22×D511yescharacteristic
251D20—22no—
252C4×D5—22no—
253C4×D5—102no—
254C4×D5—102no—
255C4×D5—102no—
256Dic10—22no—
257#126—251no—
258#126—251no—
259#125—251no—
260#125—251no—
261#125—251no—
262#123—51no—
263#123—251no—
264C22⋊Q8—251no—
265C22⋊Q8—251no—
266C4×Q8—51no—
267C4×D4—251no—
268#117—251no—
269#117—251no—
270#117—251no—
271C2×C4⋊C4—251no—
272C52#55211yescharacteristic, complemented, Sylow
273C22×C5—22no—
274C22×C5—52no—
275C22×C5—101no—
276C22×C5—101no—
277C22×C5—101no—
278C22×C5—101no—
279C22×C5—101no—
280C22×C5C23×D511yescharacteristic
281C22×C5C23×D511yescharacteristic
282C2×D5—21no—
283C2×D5—21no—
284C2×D5—21no—
285C2×D5—21no—
286C2×D5—21no—
287C2×D5#71112yes—
288C4×C5—21no—
289C4×C5—21no—
290C4×C5—21no—
291C4×C5—21no—
292C4×C5—21no—
293C4×C5—21no—
294C4×C5—42no—
295C4×C5—42no—
296C4×C5—42no—
297C4×C5—52no—
298C4×C5—101no—
299C4×C5—101no—
300C4×C5—101no—
301C4×C5—101no—
302C4×C5—101no—
303Dic5—21no—
304Dic5—21no—
305Dic5—21no—
306Dic5—21no—
307Dic5—21no—
308Dic5—104no—
309Dic5—101no—
310Dic5—101no—
311Dic5—101no—
312Dic5—101no—
313Dic5—101no—
314Dic5—101no—
315Dic5—101no—
316Dic5—101no—
317Dic5—202no—
318Dic5—202no—
319Dic5—202no—
320Dic5#70912yes—
321C2×Q8—51no—
322C2×D4—51no—
323C22×C4—51no—
324C22×C4—251no—
325C22×C4—251no—
326C22×C4—251no—
327C4⋊C4—51no—
328C4⋊C4—51no—
329C4⋊C4—51no—
330C4⋊C4—251no—
331C4⋊C4—251no—
332C4⋊C4—251no—
333C4⋊C4—251no—
334C4⋊C4—251no—
335C4⋊C4—251no—
336C4⋊C4—251no—
337C4⋊C4—251no—
338C4⋊C4—251no—
339C4⋊C4—251no—
340C4⋊C4—251no—
341C4⋊C4—251no—
342C4⋊C4D5211yescharacteristic
343C22⋊C4—51no—
344C22⋊C4—51no—
345C22⋊C4—51no—
346C22⋊C4—251no—
347C22⋊C4—251no—
348C22⋊C4—251no—
349C22⋊C4—251no—
350C22⋊C4—251no—
351C42—51no—
352C42—51no—
353C42—51no—
354C42—251no—
355C42—251no—
356C2×C5—22no—
357C2×C5—22no—
358C2×C5—22no—
359C2×C5—102no—
360C2×C5—201no—
361C2×C5C2×#70911yescharacteristic
362C2×C5C2×#71111yescharacteristic
363C2×C5#414711yescharacteristic
364C2×C5D5×#4111yescharacteristic
365C2×C5D5×#4111yescharacteristic
366C2×C5#415011yescharacteristic
367D5—22no—
368D5—41no—
369C23—51no—
370C23—51no—
371Q8—102no—
372D4—102no—
373C2×C4—51no—
374C2×C4—51no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—51no—
378C2×C4—51no—
379C2×C4—102no—
380C2×C4—251no—
381C2×C4—251no—
382C2×C4—251no—
383C2×C4—251no—
384C2×C4—502no—
385C2×C4—502no—
386C2×C4—502no—
387C2×C4C2×D5211yescharacteristic
388C2×C4C2×D5211yescharacteristic
389C2×C4C2×D5211yescharacteristic
390C5—22no—
391C5#6591511yescharacteristic, complemented
392C5#6598411yescharacteristic, complemented
393C22—52no—
394C22—101no—
395C22—101no—
396C22—101no—
397C22—101no—
398C22—101no—
399C22C22×D5211yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—52no—
404C4—101no—
405C4—101no—
406C4—101no—
407C4—101no—
408C4—101no—
409C4—501no—
410C4—501no—
411C4—501no—
412C4—501no—
413C2—102no—
414C2—201no—
415C2D5×#70911yescharacteristic
416C2D5×#71111yescharacteristic
417C2#1171748011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C4⋊C4C22
prime radicalC2×C5×D20C22
hypercenterC4⋊C4D52
hypocenterC52#552
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11717209C2semidirect
3. C5×#4033C2semidirect
5. #11717006C2semidirect
6. #11717006C2semidirect
8. #11716996C2semidirect
9. #11716995C2semidirect
10. D5×#683C2semidirect
11. #11716986C2semidirect
12. #11716978C2semidirect
13. #11716978C2semidirect
14. #11716973C2semidirect
15. #11716963C2semidirect
92. #4033D5semidirect
272. C52#552semidirect
391. C5#65915semidirect
392. C5#65984semidirect

Maximal subgroups

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

subgroupindex
#117172092
C5×#40402
C5×#40332
Dic5×D202
#117170062
#117170022
#117169962
#117169952
D5×#6832
#117169862
#117169782
#117169732
#117169632
#659845
#659155