Finite Group Databasebeta

← #13105056

Subgroups — #13105056

Order 1600.

500 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717192C211yesmaximal, characteristic, complemented
2C5×#4022C211yesmaximal, characteristic, complemented
3C4×C5×Dic10C211yesmaximal, characteristic, complemented
4#11717025C211yesmaximal, characteristic
5#11717014C211yesmaximal, characteristic, complemented
6#11716999C211yesmaximal, characteristic
7#11716995C211yesmaximal, characteristic, complemented
8#11716987C211yesmaximal, characteristic, complemented
9#11716984C211yesmaximal, characteristic, complemented
10#11716980C211yesmaximal, characteristic, complemented
11#11716970C211yesmaximal, characteristic
12#11716966C211yesmaximal, characteristic, complemented
13#11716957C211yesmaximal, characteristic, complemented
14#11716956C211yesmaximal, characteristic, complemented
15#11716955C211yesmaximal, characteristic, complemented
16C2×C4×C5⋊D5C2211yescharacteristic
17C2×C5×Dic10C2211yescharacteristic
18C2×#6039C2211yescharacteristic
19C2×#6039C2211yescharacteristic
20C2×#6038C2211yescharacteristic
21C2×D5×Dic5C2211yescharacteristic
22C42×C52C2211yescharacteristic
23#87808C2211yescharacteristic
24#87808C2211yescharacteristic
25#87806C2211yescharacteristic
26#87806C2211yescharacteristic
27C4×#1039C2211yescharacteristic
28C5×#685C2211yescharacteristic
29C5×#685C2211yescharacteristic
30C5×#685C2211yescharacteristic
31C5×C4⋊Dic5C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C5×#683C2211yescharacteristic
36C5×#683C2211yescharacteristic
37C4×C5×Dic5C2211yescharacteristic
38C4×C5×Dic5C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87779C2211yescharacteristic
44#87779C2211yescharacteristic
45#87779C2211yescharacteristic
46#87778C2211yescharacteristic
47#87778C2211yescharacteristic
48Dic52C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#66036—51nomaximal
52#65902—51nomaximal
53C22×C5⋊D5C2311yescharacteristic
54C22×C5×D5C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C2×C4×C52C2311yescharacteristic
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C4×C5⋊D5—22no—
62C2×C5×Dic5C2311yescharacteristic
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C5×Dic10—22no—
70#6039—22no—
71#6039—22no—
72#6038—22no—
73D5×Dic5—22no—
74C5×#126—51no—
75C4×C5×Q8—51no—
76#4093—51no—
77#4090—51no—
78#4044—51no—
79#4044—51no—
80#4044—51no—
81#4044—51no—
82#4042—51no—
83#4042—51no—
84#4042—51no—
85#4039—51no—
86#4034—51no—
87#4033—51no—
88#4033—51no—
89#4033—51no—
90#4031—51no—
91#4031—51no—
92#4027—51no—
93#4025—51no—
94#4024—51no—
95#4023—51no—
96#4022—51no—
97#4022D511yescharacteristic, complemented
98#4021—51no—
99#4018—51no—
100#4018—51no—
101#4018—51no—
102#4018—102no—
103C4×Dic10—51no—
104C4×Dic10D511yescharacteristic, complemented
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—21no—
112C4×C52—21no—
113C4×C52—21no—
114C4×C52—21no—
115#1039—21no—
116#1039—21no—
117#1039#4112yes—
118C5×Dic5—21no—
119C5×Dic5—21no—
120C5×Dic5—21no—
121C5×Dic5—21no—
122C5×Dic5—21no—
123C5×Dic5—21no—
124C5×Dic5#4112yes—
125C2×C5×Q8—51no—
126C2×C22⋊D5—51no—
127C22×Dic5—51no—
128C2×D20—51no—
129C2×C4×D5—51no—
130C2×C4×D5—51no—
131C2×C4×D5—51no—
132C2×C4×D5—102no—
133C2×Dic10—51no—
134C2×Dic10—51no—
135C2×Dic10—51no—
136C2×Dic10—51no—
137C2×Dic10C2×D511yescharacteristic
138C5×C4⋊C4—51no—
139C5×C4⋊C4—51no—
140C5×C4⋊C4—51no—
141C5×C4⋊C4—51no—
142C5×C4⋊C4—51no—
143C5×C4⋊C4—51no—
144C5×C22⋊C4—51no—
145C5×C22⋊C4—51no—
146C5×C22⋊C4—51no—
147C42×C5—22no—
148C42×C5—51no—
149C42×C5—51no—
150C42×C5C2×D511yescharacteristic
151C42×C5C2×D511yescharacteristic
152C22⋊Dic5—51no—
153C22⋊Dic5—51no—
154#685—51no—
155#685—51no—
156#685—51no—
157#685—51no—
158#685—51no—
159#685—51no—
160#685—51no—
161#685—51no—
162#685—51no—
163#685—51no—
164#685—51no—
165#685—51no—
166#685—102no—
167#685—102no—
168#685C2×D511yescharacteristic
169#685C2×D511yescharacteristic
170#685C2×D511yescharacteristic
171C4⋊Dic5—51no—
172C4⋊Dic5—51no—
173C4⋊Dic5C2×D511yescharacteristic
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—102no—
185#683—102no—
186#683C2×D511yescharacteristic
187#683C2×D511yescharacteristic
188#683C2×D511yescharacteristic
189#683C2×D511yescharacteristic
190#683C2×D511yescharacteristic
191C4×Dic5—51no—
192C4×Dic5—51no—
193C4×Dic5—51no—
194C4×Dic5—51no—
195C4×Dic5—51no—
196C4×Dic5—51no—
197C4×Dic5—51no—
198C4×Dic5—51no—
199C4×Dic5—102no—
200C4×Dic5C2×D511yescharacteristic
201C4×Dic5C2×D511yescharacteristic
202#556—251noSylow
203C2×C52C2×#4111yescharacteristic
204C2×C52C2×#4111yescharacteristic
205C2×C522−1+411yescharacteristic
206C5⋊D5—41no—
207C5×D5—41no—
208C23×C5—51no—
209C22×D5—51no—
210C22×D5—51no—
211C22×D5—102no—
212C22×D5C22×D511yescharacteristic
213C5×Q8—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—51no—
224C2×C4×C5C22×D511yescharacteristic
225C2×C4×C5C22×D511yescharacteristic
226C2×C4×C5C22×D511yescharacteristic
227C2×C4×C5C22×D511yescharacteristic
228C2×C4×C5C22×D511yescharacteristic
229C2×C4×C5C22×D511yescharacteristic
230C22⋊D5—102no—
231C2×Dic5—51no—
232C2×Dic5—51no—
233C2×Dic5—51no—
234C2×Dic5—51no—
235C2×Dic5—51no—
236C2×Dic5—51no—
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
248D20—102no—
249C4×D5—102no—
250C4×D5—102no—
251C4×D5—102no—
252C4×D5—108no—
253Dic10—22no—
254Dic10—102no—
255Dic10—102no—
256Dic10—102no—
257Dic10—102no—
258C4⋊Q8—251no—
259#126—51no—
260#126—251no—
261#126—251no—
262#126—251no—
263#124—251no—
264#124—251no—
265C22⋊Q8—251no—
266C22⋊Q8—251no—
267C4×Q8—51no—
268C4×Q8—251no—
269C4×Q8—251no—
270C4×D4—251no—
271#117—251no—
272#117—251no—
273C52#55611yescharacteristic, complemented, Sylow
274C22×C5—22no—
275C22×C5—101no—
276C22×C5—101no—
277C22×C5—101no—
278C22×C5C23×D511yescharacteristic
279C22×C5C23×D511yescharacteristic
280C2×D5—21no—
281C2×D5—21no—
282C2×D5—21no—
283C2×D5—104no—
284C2×D5—101no—
285C2×D5—101no—
286C2×D5—101no—
287C2×D5—101no—
288C2×D5—101no—
289C2×D5—101no—
290C2×D5—202no—
291C2×D5—202no—
292C4×C5—24no—
293C4×C5—21no—
294C4×C5—21no—
295C4×C5—21no—
296C4×C5—21no—
297C4×C5—21no—
298C4×C5—21no—
299C4×C5—42no—
300C4×C5—42no—
301C4×C5—52no—
302C4×C5—101no—
303C4×C5—101no—
304C4×C5—101no—
305C4×C5—101no—
306C4×C5—101no—
307C4×C5—101no—
308Dic5—21no—
309Dic5—21no—
310Dic5—21no—
311Dic5—21no—
312Dic5—21no—
313Dic5—21no—
314Dic5—52no—
315Dic5—52no—
316Dic5—104no—
317Dic5—101no—
318Dic5—101no—
319Dic5—101no—
320Dic5—101no—
321Dic5—202no—
322Dic5—202no—
323Dic5#70912yes—
324C2×Q8—51no—
325C2×Q8—251no—
326C2×Q8—251no—
327C2×D4—251no—
328C22×C4—251no—
329C22×C4—251no—
330C4⋊C4—51no—
331C4⋊C4—51no—
332C4⋊C4—51no—
333C4⋊C4—51no—
334C4⋊C4—51no—
335C4⋊C4—51no—
336C4⋊C4—251no—
337C4⋊C4—251no—
338C4⋊C4—251no—
339C4⋊C4—251no—
340C4⋊C4—251no—
341C4⋊C4—251no—
342C22⋊C4—51no—
343C22⋊C4—51no—
344C22⋊C4—51no—
345C22⋊C4—251no—
346C22⋊C4—251no—
347C22⋊C4—251no—
348C22⋊C4—251no—
349C22⋊C4—251no—
350C22⋊C4—251no—
351C22⋊C4—251no—
352C42—51no—
353C42—51no—
354C42—251no—
355C42—251no—
356C42—251no—
357C42—251no—
358C42D5211yescharacteristic
359C2×C5—22no—
360C2×C5—22no—
361C2×C5—22no—
362C2×C5—201no—
363C2×C5C2×#70911yescharacteristic
364C2×C5#414711yescharacteristic
365C2×C5D5×#4111yescharacteristic
366C2×C5D5×#4111yescharacteristic
367C2×C5D5×#4111yescharacteristic
368C2×C5#415011yescharacteristic
369D5—41no—
370D5—204no—
371D5—201no—
372D5—201no—
373C23—51no—
374C23—251no—
375Q8—102no—
376Q8—502no—
377Q8—502no—
378D4—502no—
379C2×C4—51no—
380C2×C4—51no—
381C2×C4—51no—
382C2×C4—51no—
383C2×C4—51no—
384C2×C4—51no—
385C2×C4—51no—
386C2×C4—251no—
387C2×C4—251no—
388C2×C4—251no—
389C2×C4—502no—
390C2×C4—502no—
391C2×C4C2×D5211yescharacteristic
392C2×C4C2×D5211yescharacteristic
393C2×C4C2×D5211yescharacteristic
394C5—22no—
395C5#6590211yescharacteristic, complemented
396C5#6603611yescharacteristic, complemented
397C22—101no—
398C22—101no—
399C22—101no—
400C22—501no—
401C22—501no—
402C22—501no—
403C22C22×D5211yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—52no—
408C4—101no—
409C4—101no—
410C4—101no—
411C4—101no—
412C4—101no—
413C4—101no—
414C4—252no—
415C4—501no—
416C4—501no—
417C2—201no—
418C2—1001no—
419C2#1171746211yescharacteristic
420C2D5×#70911yescharacteristic
421C2#1171747111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC42×C52C22
prime radicalC2×#6038C22
hypercenterC42D52
hypocenterC52#556
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11717192C2semidirect
2. C5×#4022C2semidirect
3. C4×C5×Dic10C2semidirect
5. #11717014C2semidirect
7. #11716995C2semidirect
8. #11716987C2semidirect
9. #11716984C2semidirect
10. #11716980C2semidirect
12. #11716966C2semidirect
13. #11716957C2semidirect
14. #11716956C2semidirect
15. #11716955C2semidirect
97. #4022D5semidirect
104. C4×Dic10D5semidirect
273. C52#556semidirect
395. C5#65902semidirect
396. C5#66036semidirect

Maximal subgroups

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

subgroupindex
#117171922
C5×#40222
C4×C5×Dic102
#117170252
#117170142
#117169992
#117169952
#117169872
#117169842
#117169802
#117169702
#117169662
#117169572
#117169562
#117169552
#660365
#659025