Finite Group Databasebeta

← #13105446

Subgroups — #13105446

Order 1600.

480 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717208C211yesmaximal, characteristic, complemented
2C5×#4042C211yesmaximal, characteristic
3C5×#4035C211yesmaximal, characteristic, complemented
4#11717015C211yesmaximal, characteristic
5#11717007C211yesmaximal, characteristic
6#11716996C211yesmaximal, characteristic
7#11716995C211yesmaximal, characteristic
8D5×#683C211yesmaximal, characteristic
9#11716989C211yesmaximal, characteristic, complemented
10#11716983C211yesmaximal, characteristic, complemented
11#11716975C211yesmaximal, characteristic
12#11716965C211yesmaximal, characteristic, complemented
13#11716963C211yesmaximal, characteristic, complemented
14#11716958C211yesmaximal, characteristic, complemented
15#11716957C211yesmaximal, characteristic, complemented
16C2×#6045C2211yescharacteristic
17C2×#6045C2211yescharacteristic
18C2×C4×C5×D5C2211yescharacteristic
19C2×C5×Dic10C2211yescharacteristic
20C2×#6039C2211yescharacteristic
21C2×D5×Dic5C2211yescharacteristic
22C2×D5×Dic5C2211yescharacteristic
23C52×C4⋊C4C2211yescharacteristic
24C4⋊#1039C2211yescharacteristic
25#87806C2211yescharacteristic
26#87806C2211yescharacteristic
27C4×#1039C2211yescharacteristic
28C5×#685C2211yescharacteristic
29C5×#685C2211yescharacteristic
30C5×C4⋊Dic5C2211yescharacteristic
31C5×#683C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C5×#683C2211yescharacteristic
37C4×C5×Dic5C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87780C2211yescharacteristic
46#87780C2211yescharacteristic
47#87778C2211yescharacteristic
48#87778C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#66023—51nomaximal
52#65957—51nomaximal
53C22×C5×D5C2311yescharacteristic
54C2×C4×C52C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×#1039C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61#6045—41no—
62#6045—41no—
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C4×C5×D5—41no—
71C5×Dic10—41no—
72#6039—41no—
73D5×Dic5—41no—
74D5×Dic5Q814yescomplemented
75C5×#125—51no—
76C5×C22⋊Q8—51no—
77#4090—51no—
78#4090—51no—
79#4070—51no—
80C2×#683—51no—
81#4043—51no—
82#4043—51no—
83#4042—51no—
84#4042D511yescharacteristic
85#4038—51no—
86D5×C4⋊C4—51no—
87#4036—51no—
88#4035—51no—
89#4035—51no—
90#4035—51no—
91#4035—51no—
92#4035—51no—
93#4035D511yescharacteristic, complemented
94#4034—51no—
95#4034—51no—
96#4034—51no—
97#4034—51no—
98#4034—51no—
99#4034—102no—
100#4024—51no—
101#4024—51no—
102#4023—51no—
103#4023—51no—
104#4018—51no—
105#4015—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×Dic5—21no—
122C5×Dic5—21no—
123C5×Dic5C2×Q812yes—
124C2×C5×Q8—51no—
125C22×C4×C5—51no—
126C22×Dic5—51no—
127C22×Dic5—51no—
128C2×C4×D5—51no—
129C2×C4×D5—51no—
130C2×C4×D5C2×D511yescharacteristic
131C2×Dic10—51no—
132C2×Dic10—51no—
133C2×Dic10—51no—
134C2×Dic10—51no—
135C2×Dic10—51no—
136C2×Dic10—51no—
137C2×Dic10—102no—
138C2×Dic10—102no—
139C2×Dic10C2×D511yescharacteristic
140C5×C4⋊C4—22no—
141C5×C4⋊C4—51no—
142C5×C4⋊C4—51no—
143C5×C4⋊C4—51no—
144C5×C4⋊C4—51no—
145C5×C4⋊C4—51no—
146C5×C4⋊C4—51no—
147C5×C4⋊C4—51no—
148C5×C4⋊C4C2×D511yescharacteristic
149C5×C4⋊C4C2×D511yescharacteristic
150C5×C22⋊C4—51no—
151C5×C22⋊C4—51no—
152C42×C5—51no—
153C22⋊Dic5—51no—
154C22⋊Dic5—51no—
155#685—51no—
156#685—51no—
157#685C2×D511yescharacteristic
158#685C2×D511yescharacteristic
159C4⋊Dic5—51no—
160C4⋊Dic5—51no—
161C4⋊Dic5—51no—
162C4⋊Dic5—51no—
163C4⋊Dic5—51no—
164C4⋊Dic5—51no—
165C4⋊Dic5—51no—
166C4⋊Dic5—51no—
167C4⋊Dic5—102no—
168C4⋊Dic5C2×D511yescharacteristic
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—51no—
180#683—51no—
181#683—51no—
182#683—51no—
183#683—51no—
184#683—51no—
185#683—102no—
186#683—102no—
187#683C2×D511yescharacteristic
188#683C2×D511yescharacteristic
189#683C2×D511yescharacteristic
190#683C2×D511yescharacteristic
191#683C2×D511yescharacteristic
192#683C2×D511yescharacteristic
193C4×Dic5—51no—
194C4×Dic5—51no—
195C4×Dic5—51no—
196C4×Dic5—51no—
197C4×Dic5—51no—
198C4×Dic5—51no—
199C4×Dic5—102no—
200C4×Dic5C2×D511yescharacteristic
201#544—251noSylow
202C2×C52C22×Q811yescharacteristic
203C2×C52H(5,𝔽2)11yescharacteristic
204C2×C522−1+411yescharacteristic
205C5×D5—22no—
206C23×C5—51no—
207C22×D5C22×D511yescharacteristic
208C5×Q8—201no—
209C2×C4×C5—22no—
210C2×C4×C5—22no—
211C2×C4×C5—22no—
212C2×C4×C5—51no—
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—201no—
220C2×C4×C5C22×D511yescharacteristic
221C2×C4×C5C22×D511yescharacteristic
222C2×C4×C5C22×D511yescharacteristic
223C2×C4×C5C22×D511yescharacteristic
224C2×C4×C5C22×D511yescharacteristic
225C2×C4×C5C22×D511yescharacteristic
226C2×Dic5—51no—
227C2×Dic5—51no—
228C2×Dic5—54no—
229C2×Dic5—51no—
230C2×Dic5—51no—
231C2×Dic5—51no—
232C2×Dic5—51no—
233C2×Dic5—51no—
234C2×Dic5—51no—
235C2×Dic5—102no—
236C2×Dic5—102no—
237C2×Dic5—102no—
238C2×Dic5—102no—
239C2×Dic5—201no—
240C2×Dic5C22×D511yescharacteristic
241C2×Dic5C22×D511yescharacteristic
242C2×Dic5C22×D511yescharacteristic
243C2×Dic5C22×D511yescharacteristic
244C2×Dic5C22×D511yescharacteristic
245C2×Dic5C22×D511yescharacteristic
246C2×Dic5C22×D511yescharacteristic
247C4×D5—41no—
248C4×D5—54no—
249C4×D5—201no—
250Dic10—41no—
251Dic10—201no—
252Dic10—201no—
253Dic10—201no—
254Dic10—201no—
255Dic10—201no—
256Dic10—201no—
257Dic10—204no—
258Dic10—204no—
259C4⋊Q8—251no—
260C4⋊Q8—251no—
261C4⋊Q8—251no—
262C4⋊Q8—251no—
263#125—51no—
264#125—251no—
265#125—251no—
266#125—251no—
267C22⋊Q8—51no—
268C22⋊Q8—251no—
269C22⋊Q8—251no—
270C22⋊Q8—251no—
271#117—251no—
272#117—251no—
273C2×C4⋊C4—251no—
274C52#54411yescharacteristic, complemented, Sylow
275C22×C5—22no—
276C22×C5—52no—
277C22×C5—101no—
278C22×C5—101no—
279C22×C5C23×D511yescharacteristic
280C22×C5C23×D511yescharacteristic
281C2×D5—21no—
282C2×D5—21no—
283C2×D5Q8×D512yes—
284C4×C5—24no—
285C4×C5—21no—
286C4×C5—21no—
287C4×C5—21no—
288C4×C5—21no—
289C4×C5—21no—
290C4×C5—21no—
291C4×C5—42no—
292C4×C5—42no—
293C4×C5—52no—
294C4×C5—101no—
295C4×C5—101no—
296C4×C5—101no—
297C4×C5—101no—
298C4×C5—101no—
299C4×C5—101no—
300Dic5—21no—
301Dic5—21no—
302Dic5—21no—
303Dic5—21no—
304Dic5—21no—
305Dic5—21no—
306Dic5—52no—
307Dic5—52no—
308Dic5—104no—
309Dic5—104no—
310Dic5—101no—
311Dic5—101no—
312Dic5—101no—
313Dic5—101no—
314Dic5—101no—
315Dic5—101no—
316Dic5—202no—
317Dic5—202no—
318Dic5Q8×D512yes—
319C2×Q8—51no—
320C2×Q8—251no—
321C2×Q8—251no—
322C2×Q8—251no—
323C22×C4—51no—
324C22×C4—251no—
325C22×C4—251no—
326C4⋊C4—51no—
327C4⋊C4—51no—
328C4⋊C4—51no—
329C4⋊C4—51no—
330C4⋊C4—51no—
331C4⋊C4—51no—
332C4⋊C4—51no—
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⋊C4—251no—
343C4⋊C4—251no—
344C4⋊C4—251no—
345C4⋊C4D5211yescharacteristic
346C22⋊C4—51no—
347C22⋊C4—51no—
348C22⋊C4—251no—
349C22⋊C4—251no—
350C42—51no—
351C42—251no—
352C42—251no—
353C42—251no—
354C2×C5—22no—
355C2×C5—22no—
356C2×C5—22no—
357C2×C5—102no—
358C2×C5#414411yescharacteristic
359C2×C5C2×Q8×D511yescharacteristic
360C2×C5C2×Q8×D511yescharacteristic
361C2×C5#414911yescharacteristic
362C2×C5#415011yescharacteristic
363C2×C5#415011yescharacteristic
364D5—22no—
365C23—51no—
366Q8—201no—
367Q8—1001no—
368Q8—1001no—
369Q8—1001no—
370C2×C4—51no—
371C2×C4—51no—
372C2×C4—51no—
373C2×C4—51no—
374C2×C4—51no—
375C2×C4—51no—
376C2×C4—51no—
377C2×C4—201no—
378C2×C4—254no—
379C2×C4—251no—
380C2×C4—251no—
381C2×C4—251no—
382C2×C4—251no—
383C2×C4—1001no—
384C2×C4C2×D5211yescharacteristic
385C2×C4C2×D5211yescharacteristic
386C2×C4C2×D5211yescharacteristic
387C5—22no—
388C5#6595711yescharacteristic, complemented
389C5#6602311yescharacteristic, complemented
390C22—52no—
391C22—101no—
392C22—101no—
393C22C22×D5211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—52no—
398C4—101no—
399C4—101no—
400C4—101no—
401C4—101no—
402C4—101no—
403C4—101no—
404C4—252no—
405C4—501no—
406C4—501no—
407C4—501no—
408C2—102no—
409C2#1171746311yescharacteristic
410C2#1171747911yescharacteristic
411C2Q8×D5211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C4⋊C4C22
prime radicalC22×C5×D5C23
hypercenterC4⋊C4D52
hypocenterC52#544
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11717208C2semidirect
3. C5×#4035C2semidirect
9. #11716989C2semidirect
10. #11716983C2semidirect
12. #11716965C2semidirect
13. #11716963C2semidirect
14. #11716958C2semidirect
15. #11716957C2semidirect
74. D5×Dic5Q8semidirect
93. #4035D5semidirect
274. C52#544semidirect
388. C5#65957semidirect
389. C5#66023semidirect

Maximal subgroups

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

subgroupindex
#117172082
C5×#40422
C5×#40352
#117170152
#117170072
#117169962
#117169952
D5×#6832
#117169892
#117169832
#117169752
#117169652
#117169632
#117169582
#117169572
#660235
#659575