Finite Group Databasebeta

← #13105178

Subgroups — #13105178

Order 1600.

506 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11717203C211yesmaximal, characteristic
2C5×#4025C211yesmaximal, characteristic, complemented
3C5×#4023C211yesmaximal, characteristic, complemented
4#11717093C211yesmaximal, characteristic
5#11717082C211yesmaximal, characteristic, complemented
6#11717074C211yesmaximal, characteristic, complemented
7#11717070C211yesmaximal, characteristic
8#11717009C211yesmaximal, characteristic, complemented
9#11717001C211yesmaximal, characteristic, complemented
10#11716992C211yesmaximal, characteristic, complemented
11#11716988C211yesmaximal, characteristic, complemented
12#11716986C211yesmaximal, characteristic, complemented
13#11716963C211yesmaximal, characteristic, complemented
14#11716955C211yesmaximal, characteristic, complemented
15#11716954C211yesmaximal, characteristic, complemented
16C2×C22⋊(C5⋊D5)C2211yescharacteristic
17C2×C4×C5⋊D5C2211yescharacteristic
18C22×C5×Dic5C2211yescharacteristic
19C2×#6039C2211yescharacteristic
20C2×#6036C2211yescharacteristic
21C2×#6036C2211yescharacteristic
22C52×C22⋊C4C2211yescharacteristic
23C22⋊#1039C2211yescharacteristic
24#87808C2211yescharacteristic
25C4⋊#1039C2211yescharacteristic
26#87806C2211yescharacteristic
27C5×C22⋊Dic5C2211yescharacteristic
28C5×C22⋊Dic5C2211yescharacteristic
29C5×C22⋊Dic5C2211yescharacteristic
30C5×C4⋊Dic5C2211yescharacteristic
31C5×#683C2211yescharacteristic
32C5×#683C2211yescharacteristic
33C5×#683C2211yescharacteristic
34C5×#683C2211yescharacteristic
35C4×C5×Dic5C2211yescharacteristic
36C4×C5×Dic5C2211yescharacteristic
37C4×C5×Dic5C2211yescharacteristic
38#87781C2211yescharacteristic
39#87781C2211yescharacteristic
40#87781C2211yescharacteristic
41#87780C2211yescharacteristic
42#87780C2211yescharacteristic
43#87780C2211yescharacteristic
44#87780C2211yescharacteristic
45#87780C2211yescharacteristic
46#87780C2211yescharacteristic
47#87779C2211yescharacteristic
48#87779C2211yescharacteristic
49Dic52C2211yescharacteristic
50Dic52C2211yescharacteristic
51#66042—51nomaximal
52#65863—51nomaximal
53C23×C52C2311yescharacteristic
54C22×C5⋊D5C2311yescharacteristic
55C2×C4×C52C2311yescharacteristic
56C2×C4×C52C2311yescharacteristic
57C22⋊(C5⋊D5)—22no—
58C2×#1039C2311yescharacteristic
59C2×#1039C2311yescharacteristic
60C2×#1039C2311yescharacteristic
61C4×C5⋊D5—22no—
62C2×C5×Dic5—22no—
63C2×C5×Dic5C2311yescharacteristic
64C2×C5×Dic5C2311yescharacteristic
65C2×C5×Dic5C2311yescharacteristic
66C2×C5×Dic5C2311yescharacteristic
67C2×C5×Dic5C2311yescharacteristic
68C2×C5×Dic5C2311yescharacteristic
69C2×C5×Dic5C2311yescharacteristic
70C2×C5×Dic5C2311yescharacteristic
71#6039—22no—
72#6036—22no—
73#6036—22no—
74C5×#126—51no—
75C5×#117—51no—
76C4×C22⋊D5—51no—
77#4070—51no—
78#4043—51no—
79#4042—51no—
80#4038—51no—
81D5×C4⋊C4—51no—
82D5×C4⋊C4—51no—
83#4036—51no—
84#4036—51no—
85#4035—51no—
86#4035—51no—
87#4033—51no—
88#4029—51no—
89#4029—51no—
90#4029—51no—
91#4029—51no—
92#4029—102no—
93#4027—51no—
94#4025—51no—
95#4025—51no—
96#4025D511yescharacteristic, complemented
97#4024—51no—
98#4023D511yescharacteristic, complemented
99#4018—51no—
100#4018—51no—
101#4018—51no—
102#4016—51no—
103#4016—51no—
104C4×Dic10—51no—
105C22×C52—21no—
106C22×C52—21no—
107C22×C52—21no—
108C22×C52C2411yescharacteristic
109C2×C5⋊D5—21no—
110C2×C5⋊D5—21no—
111C2×C5⋊D5#4112yes—
112C4×C52—21no—
113C4×C52—21no—
114#1039—21no—
115#1039—21no—
116#1039—21no—
117C5×Dic5—21no—
118C5×Dic5—21no—
119C5×Dic5—21no—
120C5×Dic5—21no—
121C5×Dic5—21no—
122C5×Dic5—21no—
123C5×Dic5—21no—
124C5×Dic5#4112yes—
125C22×C4×C5—51no—
126C2×C22⋊D5—51no—
127C2×C22⋊D5—51no—
128C2×C22⋊D5—102no—
129C22×Dic5C2×D511yescharacteristic
130C2×C4×D5—51no—
131C2×C4×D5—51no—
132C2×C4×D5—51no—
133C2×C4×D5—51no—
134C2×C4×D5—51no—
135C2×C4×D5—51no—
136C2×C4×D5—102no—
137C2×Dic10—51no—
138C2×Dic10—51no—
139C5×C4⋊C4—51no—
140C5×C4⋊C4—51no—
141C5×C4⋊C4—51no—
142C5×C4⋊C4—51no—
143C5×C4⋊C4—51no—
144C5×C22⋊C4—22no—
145C5×C22⋊C4—51no—
146C5×C22⋊C4—51no—
147C5×C22⋊C4—51no—
148C5×C22⋊C4C2×D511yescharacteristic
149C5×C22⋊C4C2×D511yescharacteristic
150C42×C5—51no—
151C42×C5—51no—
152C42×C5—51no—
153C22⋊Dic5—51no—
154C22⋊Dic5—51no—
155C22⋊Dic5—102no—
156C22⋊Dic5C2×D511yescharacteristic
157C22⋊Dic5C2×D511yescharacteristic
158C22⋊Dic5C2×D511yescharacteristic
159#685—51no—
160#685—51no—
161#685—51no—
162#685—51no—
163#685—51no—
164#685—51no—
165#685—102no—
166C4⋊Dic5—51no—
167C4⋊Dic5—51no—
168C4⋊Dic5—51no—
169C4⋊Dic5—51no—
170C4⋊Dic5—51no—
171C4⋊Dic5—51no—
172C4⋊Dic5—51no—
173C4⋊Dic5—51no—
174C4⋊Dic5—102no—
175C4⋊Dic5C2×D511yescharacteristic
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—51no—
188#683—51no—
189#683—51no—
190#683—102no—
191#683C2×D511yescharacteristic
192#683C2×D511yescharacteristic
193#683C2×D511yescharacteristic
194#683C2×D511yescharacteristic
195C4×Dic5—51no—
196C4×Dic5—51no—
197C4×Dic5—51no—
198C4×Dic5—51no—
199C4×Dic5C2×D511yescharacteristic
200C4×Dic5C2×D511yescharacteristic
201C4×Dic5C2×D511yescharacteristic
202#552—251noSylow
203C2×C52—41no—
204C2×C52C2×#4111yescharacteristic
205C2×C52C2×#4111yescharacteristic
206C2×C522−1+411yescharacteristic
207C5⋊D5—22no—
208C23×C5—22no—
209C23×C5C22×D511yescharacteristic
210C23×C5C22×D511yescharacteristic
211C22×D5—51no—
212C22×D5—51no—
213C22×D5—102no—
214C2×C4×C5—22no—
215C2×C4×C5—22no—
216C2×C4×C5—51no—
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×C5—102no—
225C2×C4×C5C22×D511yescharacteristic
226C2×C4×C5C22×D511yescharacteristic
227C2×C4×C5C22×D511yescharacteristic
228C2×C4×C5C22×D511yescharacteristic
229C22⋊D5—102no—
230C22⋊D5—102no—
231C22⋊D5—204no—
232C2×Dic5—22no—
233C2×Dic5—51no—
234C2×Dic5—51no—
235C2×Dic5—51no—
236C2×Dic5—51no—
237C2×Dic5—51no—
238C2×Dic5—51no—
239C2×Dic5—102no—
240C2×Dic5—102no—
241C2×Dic5—102no—
242C2×Dic5C22×D511yescharacteristic
243C2×Dic5C22×D511yescharacteristic
244C2×Dic5C22×D511yescharacteristic
245C2×Dic5C22×D511yescharacteristic
246C2×Dic5C22×D511yescharacteristic
247C2×Dic5C22×D511yescharacteristic
248C2×Dic5C22×D511yescharacteristic
249C2×Dic5C22×D511yescharacteristic
250C4×D5—102no—
251C4×D5—102no—
252C4×D5—102no—
253C4×D5—102no—
254C4×D5—102no—
255C4×D5—102no—
256C4×D5—204no—
257Dic10—102no—
258Dic10—102no—
259#126—51no—
260#126—251no—
261#125—251no—
262#125—251no—
263#125—251no—
264#123—251no—
265#123—251no—
266C22⋊Q8—251no—
267C22⋊Q8—251no—
268C4×Q8—251no—
269C4×D4—251no—
270#117—51no—
271#117—251no—
272#117—251no—
273C2×C4⋊C4—251no—
274C52#55211yescharacteristic, complemented, Sylow
275C22×C5—21no—
276C22×C5—21no—
277C22×C5—21no—
278C22×C5—21no—
279C22×C5—21no—
280C22×C5—21no—
281C22×C5—22no—
282C22×C5—42no—
283C22×C5—42no—
284C22×C5—42no—
285C22×C5C23×D511yescharacteristic
286C22×C5C23×D511yescharacteristic
287C2×D5—52no—
288C2×D5—52no—
289C2×D5—104no—
290C2×D5—101no—
291C2×D5—101no—
292C2×D5—101no—
293C2×D5—101no—
294C2×D5—202no—
295C2×D5—202no—
296C4×C5—21no—
297C4×C5—21no—
298C4×C5—21no—
299C4×C5—21no—
300C4×C5—42no—
301C4×C5—42no—
302C4×C5—52no—
303C4×C5—101no—
304C4×C5—101no—
305C4×C5—101no—
306C4×C5—101no—
307C4×C5—101no—
308C4×C5—101no—
309C4×C5—101no—
310Dic5—21no—
311Dic5—21no—
312Dic5—21no—
313Dic5—21no—
314Dic5—21no—
315Dic5—21no—
316Dic5—21no—
317Dic5—101no—
318Dic5—101no—
319Dic5—101no—
320Dic5—101no—
321Dic5—101no—
322Dic5—101no—
323Dic5—202no—
324Dic5—202no—
325Dic5—202no—
326Dic5#70912yes—
327C2×Q8—251no—
328C2×D4—251no—
329C22×C4—51no—
330C22×C4—251no—
331C22×C4—251no—
332C22×C4—251no—
333C4⋊C4—51no—
334C4⋊C4—51no—
335C4⋊C4—51no—
336C4⋊C4—51no—
337C4⋊C4—51no—
338C4⋊C4—251no—
339C4⋊C4—251no—
340C4⋊C4—251no—
341C4⋊C4—251no—
342C4⋊C4—251no—
343C4⋊C4—251no—
344C4⋊C4—251no—
345C4⋊C4—251no—
346C4⋊C4—251no—
347C4⋊C4—251no—
348C4⋊C4—251no—
349C22⋊C4—51no—
350C22⋊C4—51no—
351C22⋊C4—51no—
352C22⋊C4—251no—
353C22⋊C4—251no—
354C22⋊C4—251no—
355C22⋊C4—251no—
356C22⋊C4D5211yescharacteristic
357C42—51no—
358C42—51no—
359C42—51no—
360C42—251no—
361C42—251no—
362C2×C5—22no—
363C2×C5—22no—
364C2×C5—22no—
365C2×C5—41no—
366C2×C5—41no—
367C2×C5—82no—
368C2×C5C2×#70911yescharacteristic
369C2×C5D5×#4111yescharacteristic
370C2×C5D5×#4111yescharacteristic
371C2×C5D5×#4111yescharacteristic
372C2×C5#415011yescharacteristic
373C2×C5#415011yescharacteristic
374D5—102no—
375D5—102no—
376D5—204no—
377C23—251no—
378C23C2×D5211yescharacteristic
379Q8—502no—
380D4—502no—
381C2×C4—51no—
382C2×C4—51no—
383C2×C4—51no—
384C2×C4—51no—
385C2×C4—51no—
386C2×C4—51no—
387C2×C4—51no—
388C2×C4—51no—
389C2×C4—102no—
390C2×C4—251no—
391C2×C4—251no—
392C2×C4—251no—
393C2×C4—502no—
394C2×C4—502no—
395C2×C4—502no—
396C2×C4C2×D5211yescharacteristic
397C2×C4C2×D5211yescharacteristic
398C5—22no—
399C5#6586311yescharacteristic, complemented
400C5#6604211yescharacteristic, complemented
401C22—21no—
402C22—21no—
403C22—21no—
404C22—252no—
405C22—501no—
406C22—501no—
407C22C22×D5211yescharacteristic
408C4—21no—
409C4—21no—
410C4—52no—
411C4—101no—
412C4—101no—
413C4—101no—
414C4—101no—
415C4—101no—
416C4—101no—
417C4—101no—
418C4—501no—
419C4—501no—
420C4—501no—
421C2—41no—
422C2—502no—
423C2D5×#70911yescharacteristic
424C2#1171747411yescharacteristic
425C2#1171747711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
prime radicalC2×C22⋊(C5⋊D5)C22
hypercenterC22⋊C4D52
hypocenterC52#552
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×#4025C2semidirect
3. C5×#4023C2semidirect
5. #11717082C2semidirect
6. #11717074C2semidirect
8. #11717009C2semidirect
9. #11717001C2semidirect
10. #11716992C2semidirect
11. #11716988C2semidirect
12. #11716986C2semidirect
13. #11716963C2semidirect
14. #11716955C2semidirect
15. #11716954C2semidirect
96. #4025D5semidirect
98. #4023D5semidirect
274. C52#552semidirect
399. C5#65863semidirect
400. C5#66042semidirect

Maximal subgroups

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

subgroupindex
#117172032
C5×#40252
C5×#40232
#117170932
#117170822
#117170742
#117170702
#117170092
#117170012
#117169922
#117169882
#117169862
#117169632
#117169552
#117169542
#660425
#658635