Finite Group Databasebeta

← #13105442

Subgroups — #13105442

Order 1600.

500 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717209C211yesmaximal, characteristic, complemented
2C5×#4042C211yesmaximal, characteristic
3C5×#4033C211yesmaximal, characteristic, complemented
4#11717007C211yesmaximal, characteristic
5#11717006C211yesmaximal, characteristic
6#11716995C211yesmaximal, characteristic
7#11716995C211yesmaximal, characteristic
8C4×D5×Dic5C211yesmaximal, characteristic
9#11716989C211yesmaximal, characteristic, complemented
10#11716969C211yesmaximal, characteristic
11Dic5×Dic10C211yesmaximal, characteristic, complemented
12#11716957C211yesmaximal, characteristic, complemented
13#11716957C211yesmaximal, characteristic, complemented
14#11716955C211yesmaximal, characteristic, complemented
15#11716955C211yesmaximal, characteristic, complemented
16C2×C4×C5×D5C2211yescharacteristic
17C2×C5×Dic10C2211yescharacteristic
18C2×C5×Dic10C2211yescharacteristic
19C2×#6039C2211yescharacteristic
20C2×#6039C2211yescharacteristic
21C2×D5×Dic5C2211yescharacteristic
22C2×D5×Dic5C2211yescharacteristic
23C52×C4⋊C4C2211yescharacteristic
24C4⋊#1039C2211yescharacteristic
25#87806C2211yescharacteristic
26#87806C2211yescharacteristic
27#87806C2211yescharacteristic
28#87806C2211yescharacteristic
29C4×#1039C2211yescharacteristic
30C5×#685C2211yescharacteristic
31C5×#685C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C4×C5×Dic5C2211yescharacteristic
37C4×C5×Dic5C2211yescharacteristic
38C4×C5×Dic5C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87778C2211yescharacteristic
46#87778C2211yescharacteristic
47Dic52C2211yescharacteristic
48Dic52C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#65956—51nomaximal
52#65912—51nomaximal
53C22×C5×D5C2311yescharacteristic
54C2×C4×C52C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×#1039C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C2×C5×Dic5C2311yescharacteristic
62C2×C5×Dic5C2311yescharacteristic
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C4×C5×D5—22no—
69C5×Dic10—22no—
70C5×Dic10—22no—
71#6039—22no—
72#6039—22no—
73D5×Dic5—22no—
74D5×Dic5Q814yescomplemented
75C5×C22⋊Q8—51no—
76C4×C5×Q8—51no—
77#4092—51no—
78Q8×Dic5—51no—
79#4090—51no—
80#4090—51no—
81#4070—51no—
82C2×C4×Dic5—51no—
83#4043—51no—
84#4042—51no—
85#4042D511yescharacteristic
86#4035—51no—
87#4035—51no—
88#4035—51no—
89#4035—51no—
90#4035—102no—
91#4034—51no—
92#4034—51no—
93#4033—51no—
94#4033—51no—
95#4033D511yescharacteristic, complemented
96#4024—51no—
97#4024—51no—
98#4023—51no—
99#4023—51no—
100#4018—51no—
101#4018—51no—
102C42×D5—51no—
103C4×Dic10—51no—
104C4×Dic10—51no—
105C4×Dic10—51no—
106C22×C52C2411yescharacteristic
107C2×C5×D5—21no—
108C2×C5×D5—21no—
109C2×C5×D5C2×Q812yes—
110C4×C52—21no—
111C4×C52—21no—
112C4×C52—21no—
113#1039—21no—
114#1039—21no—
115#1039—21no—
116#1039C2×Q812yes—
117C5×Dic5—21no—
118C5×Dic5—21no—
119C5×Dic5—21no—
120C5×Dic5—21no—
121C5×Dic5C2×Q812yes—
122C5×Dic5#4112yes—
123C5×Dic5#4112yes—
124C2×C5×Q8—51no—
125C2×C5×Q8—51no—
126C22×C4×C5—51no—
127C22×Dic5—51no—
128C22×Dic5—51no—
129C2×C4×D5—51no—
130C2×C4×D5—51no—
131C2×C4×D5C2×D511yescharacteristic
132C2×Dic10—51no—
133C2×Dic10—51no—
134C2×Dic10—51no—
135C2×Dic10—51no—
136C2×Dic10C2×D511yescharacteristic
137C2×Dic10C2×D511yescharacteristic
138C5×C4⋊C4—22no—
139C5×C4⋊C4—51no—
140C5×C4⋊C4—51no—
141C5×C4⋊C4—51no—
142C5×C4⋊C4—51no—
143C5×C4⋊C4C2×D511yescharacteristic
144C5×C4⋊C4C2×D511yescharacteristic
145C5×C22⋊C4—51no—
146C5×C22⋊C4—51no—
147C42×C5—51no—
148C42×C5—51no—
149C42×C5—51no—
150C22⋊Dic5—51no—
151C22⋊Dic5—51no—
152#685—51no—
153#685—51no—
154#685C2×D511yescharacteristic
155#685C2×D511yescharacteristic
156C4⋊Dic5—51no—
157C4⋊Dic5—51no—
158C4⋊Dic5—51no—
159C4⋊Dic5—51no—
160C4⋊Dic5—51no—
161C4⋊Dic5—51no—
162C4⋊Dic5—102no—
163#683—51no—
164#683—51no—
165#683—51no—
166#683—51no—
167#683—51no—
168#683—51no—
169#683—51no—
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—102no—
180#683—102no—
181#683—102no—
182#683—102no—
183#683C2×D511yescharacteristic
184#683C2×D511yescharacteristic
185#683C2×D511yescharacteristic
186#683C2×D511yescharacteristic
187C4×Dic5—51no—
188C4×Dic5—51no—
189C4×Dic5—51no—
190C4×Dic5—51no—
191C4×Dic5—51no—
192C4×Dic5—51no—
193C4×Dic5—51no—
194C4×Dic5—51no—
195C4×Dic5—51no—
196C4×Dic5—51no—
197C4×Dic5—102no—
198C4×Dic5C2×D511yescharacteristic
199C4×Dic5C2×D511yescharacteristic
200C4×Dic5C2×D511yescharacteristic
201#533—251noSylow
202C2×C52C22×Q811yescharacteristic
203C2×C52C2×#4111yescharacteristic
204C2×C52C2×#4111yescharacteristic
205C5×D5—22no—
206C23×C5—51no—
207C22×D5C22×D511yescharacteristic
208C5×Q8—102no—
209C5×Q8—102no—
210C2×C4×C5—22no—
211C2×C4×C5—22no—
212C2×C4×C5—22no—
213C2×C4×C5—51no—
214C2×C4×C5—51no—
215C2×C4×C5—51no—
216C2×C4×C5—51no—
217C2×C4×C5—51no—
218C2×C4×C5—51no—
219C2×C4×C5—51no—
220C2×C4×C5—102no—
221C2×C4×C5C22×D511yescharacteristic
222C2×C4×C5C22×D511yescharacteristic
223C2×C4×C5C22×D511yescharacteristic
224C2×C4×C5C22×D511yescharacteristic
225C2×C4×C5C22×D511yescharacteristic
226C2×C4×C5C22×D511yescharacteristic
227C2×Dic5—51no—
228C2×Dic5—51no—
229C2×Dic5—54no—
230C2×Dic5—51no—
231C2×Dic5—51no—
232C2×Dic5—51no—
233C2×Dic5—51no—
234C2×Dic5—51no—
235C2×Dic5—51no—
236C2×Dic5—102no—
237C2×Dic5—102no—
238C2×Dic5—102no—
239C2×Dic5—102no—
240C2×Dic5—102no—
241C2×Dic5C22×D511yescharacteristic
242C2×Dic5C22×D511yescharacteristic
243C2×Dic5C22×D511yescharacteristic
244C2×Dic5C22×D511yescharacteristic
245C2×Dic5C22×D511yescharacteristic
246C2×Dic5C22×D511yescharacteristic
247C2×Dic5C22×D511yescharacteristic
248C4×D5—22no—
249C4×D5—54no—
250C4×D5—102no—
251Dic10—22no—
252Dic10—22no—
253Dic10—102no—
254Dic10—102no—
255Dic10—102no—
256Dic10—102no—
257C4⋊Q8—251no—
258C4⋊Q8—251no—
259#125—251no—
260#125—251no—
261C22⋊Q8—51no—
262C22⋊Q8—251no—
263C22⋊Q8—251no—
264C22⋊Q8—251no—
265C4×Q8—51no—
266C4×Q8—251no—
267C4×Q8—251no—
268C4×Q8—251no—
269#117—251no—
270#117—251no—
271C2×C42—251no—
272C52#53311yescharacteristic, complemented, Sylow
273C22×C5—22no—
274C22×C5—52no—
275C22×C5—101no—
276C22×C5—101no—
277C22×C5C23×D511yescharacteristic
278C22×C5C23×D511yescharacteristic
279C2×D5—21no—
280C2×D5—21no—
281C2×D5Q8×D512yes—
282C4×C5—21no—
283C4×C5—21no—
284C4×C5—21no—
285C4×C5—21no—
286C4×C5—21no—
287C4×C5—21no—
288C4×C5—42no—
289C4×C5—42no—
290C4×C5—42no—
291C4×C5—52no—
292C4×C5—52no—
293C4×C5—52no—
294C4×C5—101no—
295C4×C5—101no—
296C4×C5—101no—
297C4×C5—101no—
298Dic5—21no—
299Dic5—21no—
300Dic5—21no—
301Dic5—21no—
302Dic5—52no—
303Dic5—52no—
304Dic5—104no—
305Dic5—101no—
306Dic5—101no—
307Dic5—101no—
308Dic5—101no—
309Dic5—101no—
310Dic5—101no—
311Dic5—202no—
312Dic5—202no—
313Dic5—202no—
314Dic5#70912yes—
315Dic5#71112yes—
316Dic5Q8×D512yes—
317C2×Q8—51no—
318C2×Q8—51no—
319C2×Q8—251no—
320C2×Q8—251no—
321C22×C4—51no—
322C22×C4—251no—
323C22×C4—251no—
324C4⋊C4—51no—
325C4⋊C4—51no—
326C4⋊C4—51no—
327C4⋊C4—51no—
328C4⋊C4—251no—
329C4⋊C4—251no—
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⋊C4D5211yescharacteristic
340C22⋊C4—51no—
341C22⋊C4—51no—
342C22⋊C4—251no—
343C22⋊C4—251no—
344C42—51no—
345C42—51no—
346C42—51no—
347C42—251no—
348C42—251no—
349C42—251no—
350C42—251no—
351C42—251no—
352C2×C5—22no—
353C2×C5—22no—
354C2×C5—22no—
355C2×C5—102no—
356C2×C5C2×#70911yescharacteristic
357C2×C5C2×#71111yescharacteristic
358C2×C5C2×Q8×D511yescharacteristic
359C2×C5C2×Q8×D511yescharacteristic
360C2×C5D5×#4111yescharacteristic
361C2×C5D5×#4111yescharacteristic
362D5—22no—
363C23—51no—
364Q8—102no—
365Q8—102no—
366Q8—502no—
367Q8—502no—
368C2×C4—51no—
369C2×C4—51no—
370C2×C4—51no—
371C2×C4—51no—
372C2×C4—51no—
373C2×C4—51no—
374C2×C4—51no—
375C2×C4—102no—
376C2×C4—254no—
377C2×C4—251no—
378C2×C4—251no—
379C2×C4—251no—
380C2×C4—251no—
381C2×C4—502no—
382C2×C4C2×D5211yescharacteristic
383C2×C4C2×D5211yescharacteristic
384C2×C4C2×D5211yescharacteristic
385C5—22no—
386C5#6591211yescharacteristic, complemented
387C5#6595611yescharacteristic, complemented
388C22—52no—
389C22—101no—
390C22—101no—
391C22C22×D5211yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—52no—
396C4—52no—
397C4—52no—
398C4—101no—
399C4—101no—
400C4—101no—
401C4—101no—
402C4—252no—
403C4—501no—
404C4—501no—
405C4—501no—
406C2—102no—
407C2D5×#70911yescharacteristic
408C2D5×#71111yescharacteristic
409C2Q8×D5211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C4⋊C4C22
prime radicalC22×C5×D5C23
hypercenterC4⋊C4D52
hypocenterC52#533
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
9. #11716989C2semidirect
11. Dic5×Dic10C2semidirect
12. #11716957C2semidirect
13. #11716957C2semidirect
14. #11716955C2semidirect
15. #11716955C2semidirect
74. D5×Dic5Q8semidirect
95. #4033D5semidirect
272. C52#533semidirect
386. C5#65912semidirect
387. C5#65956semidirect

Maximal subgroups

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

subgroupindex
#117172092
C5×#40422
C5×#40332
#117170072
#117170062
#117169952
C4×D5×Dic52
#117169892
#117169692
Dic5×Dic102
#117169572
#117169552
#659565
#659125