Finite Group Databasebeta

← #13105381

Subgroups — #13105381

Order 1600.

770 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C22⋊D5C211yesmaximal, characteristic, complemented
2C22×#6037C211yesmaximal, characteristic, complemented
3#11717206C211yesmaximal, characteristic, complemented
4C5×#4028C211yesmaximal, characteristic, complemented
5C5×D5×C22⋊C4C211yesmaximal, characteristic, complemented
6#11717095C211yesmaximal, characteristic, complemented
7#11717080C211yesmaximal, characteristic, complemented
8#11717040C212yesmaximal, complemented
9#11717026C212yesmaximal, complemented
10#11717024C212yesmaximal, complemented
11#11717003C212yesmaximal, complemented
12C23×C5×D5C2211yescharacteristic
13C23×C5×D5C2211yescharacteristic, complemented
14C22×D52C2211yescharacteristic, complemented
15C2×C22⋊(C5⋊D5)C2211yescharacteristic
16C22×#1039C2211yescharacteristic, complemented
17C2×C5×C22⋊D5C2211yescharacteristic
18C2×C5×D20C2212yescomplemented
19C2×C4×C5×D5C2212yescomplemented
20D5×C22⋊D5—24no—
21C2×#6038C2211yescharacteristic, complemented
22C2×#6038C2212yescomplemented
23C2×#6037—24no—
24C2×#6037C2212yes—
25C2×#6037C2212yescomplemented
26C2×#6037C2211yescharacteristic, complemented
27C2×D5×Dic5C2211yescharacteristic, complemented
28C52×C22⋊C4C2211yescharacteristic, complemented
29#87808C2212yescomplemented
30C4⋊#1039C2212yescomplemented
31C5×C22⋊Dic5C2211yescharacteristic, complemented
32C5×#685C2212yescomplemented
33C5×#685C2212yescomplemented
34#87780C2212yescomplemented
35#87778C2212yescomplemented
36#87778C2212yescomplemented
37#65923—51nomaximal
38#65827—51nomaximal
39C23×C52C2311yescharacteristic
40C22×C5⋊D5C2311yescharacteristic
41C22×C5×D5—22no—
42C22×C5×D5—42no—
43C22×C5×D5—42no—
44C22×C5×D5D414yescomplemented
45C22×C5×D5C2312yes—
46C22×C5×D5C2311yescharacteristic
47C22×C5×D5C2311yescharacteristic
48C22×C5×D5C2311yescharacteristic
49C2×D52—22no—
50C2×D52—42no—
51C2×C4×C52C2312yes—
52C22⋊(C5⋊D5)—22no—
53C2×#1039—22no—
54C2×#1039C2312yes—
55C2×#1039C2311yescharacteristic
56C5×C22⋊D5—22no—
57C2×C5×Dic5C2311yescharacteristic
58C2×C5×Dic5C2312yes—
59C5×D20—42no—
60C4×C5×D5—42no—
61#6038—22no—
62#6038—42no—
63#6037—24no—
64#6037—24no—
65#6037—22no—
66D5×Dic5—22no—
67C22×C22⋊D5—51no—
68C22×C22⋊D5—51no—
69C22×C22⋊D5—51no—
70C2×D4×D5—51no—
71C5×#120—51no—
72C2×C5×C22⋊C4—51no—
73#4099—51no—
74#4084—52no—
75#4083—52no—
76#4081—51no—
77#4076—52no—
78#4076—52no—
79#4032—51no—
80#4032—51no—
81#4032—51no—
82#4032—52no—
83#4032—102no—
84#4030—52no—
85#4029—52no—
86#4028—51no—
87#4028—52no—
88#4028D511yescharacteristic, complemented
89D5×C22⋊C4D511yescharacteristic, complemented
90C22×C52—22no—
91C22×C52C2×D412yes—
92C22×C52C2411yescharacteristic
93C2×C5⋊D5—21no—
94C2×C5⋊D5—22no—
95C2×C5×D5—22no—
96C2×C5×D5—21no—
97C2×C5×D5—22no—
98C2×C5×D5—22no—
99C2×C5×D5—22no—
100C2×C5×D5—22no—
101C2×C5×D5—22no—
102C2×C5×D5—44no—
103C2×C5×D5—44no—
104C2×C5×D5C2×D412yes—
105C2×C5×D5C2×D412yes—
106D52—44no—
107C4×C52—22no—
108#1039—21no—
109#1039—22no—
110C5×Dic5—21no—
111C5×Dic5—22no—
112C24×C5—51no—
113C24×C5—51no—
114C23×D5—51no—
115C23×D5—51no—
116C23×D5C2×D511yescharacteristic
117C23×D5C2×D511yescharacteristic, complemented
118C2×C5×D4—51no—
119C2×C5×D4—52no—
120C22×C4×C5—52no—
121C2×C22⋊D5—51no—
122C2×C22⋊D5—51no—
123C2×C22⋊D5—51no—
124C2×C22⋊D5—52no—
125C2×C22⋊D5—52no—
126C2×C22⋊D5—52no—
127C2×C22⋊D5—51no—
128C2×C22⋊D5—52no—
129C2×C22⋊D5—51no—
130C2×C22⋊D5—52no—
131C2×C22⋊D5—102no—
132C2×C22⋊D5—104no—
133C2×C22⋊D5—104no—
134C2×C22⋊D5—104no—
135C2×C22⋊D5C2×D511yescharacteristic
136C22×Dic5—51no—
137C22×Dic5—51no—
138C22×Dic5—51no—
139C22×Dic5—102no—
140D4×D5—104no—
141C2×D20—51no—
142C2×D20—52no—
143C2×D20C2×D512yescomplemented
144C2×C4×D5—51no—
145C2×C4×D5C2×D512yescomplemented
146C5×C22⋊C4—22no—
147C5×C22⋊C4—51no—
148C5×C22⋊C4—52no—
149C5×C22⋊C4—52no—
150C5×C22⋊C4C2×D511yescharacteristic, complemented
151C5×C22⋊C4C2×D511yescharacteristic, complemented
152C22⋊Dic5—52no—
153C22⋊Dic5—52no—
154C22⋊Dic5C2×D511yescharacteristic, complemented
155#685—52no—
156#685—52no—
157#685—52no—
158#685—52no—
159#685—104no—
160#685C2×D512yescomplemented
161#685C2×D512yescomplemented
162C4⋊Dic5—52no—
163C4⋊Dic5—52no—
164C4⋊Dic5—52no—
165C4⋊Dic5—104no—
166#683—52no—
167#534—251noSylow
168C2×C52—22no—
169C2×C52C22×D411yescharacteristic
170C2×C52H(5,𝔽2)12yes—
171C5⋊D5—41no—
172C5×D5—22no—
173C5×D5—22no—
174C5×D5—41no—
175C5×D5—42no—
176C23×C5—22no—
177C23×C5—51no—
178C23×C5—51no—
179C23×C5—54no—
180C23×C5—51no—
181C23×C5—52no—
182C23×C5—102no—
183C23×C5—202no—
184C23×C5—202no—
185C23×C5C22×D511yescharacteristic
186C23×C5C22×D511yescharacteristic
187C22×D5—22no—
188C22×D5—42no—
189C22×D5—42no—
190C22×D5—51no—
191C22×D5—51no—
192C22×D5—102no—
193C22×D5—102no—
194C22×D5—102no—
195C22×D5—202no—
196C22×D5—202no—
197C22×D5D2014yescomplemented
198C22×D5C22×D511yescharacteristic
199C22×D5C22×D512yes—
200C22×D5C22×D511yescharacteristic
201C22×D5C22×D511yescharacteristic
202C5×D4—102no—
203C5×D4—202no—
204C2×C4×C5—24no—
205C2×C4×C5—51no—
206C2×C4×C5—52no—
207C2×C4×C5—202no—
208C2×C4×C5C22×D512yes—
209C2×C4×C5C22×D512yes—
210C22⋊D5—22no—
211C22⋊D5—102no—
212C22⋊D5—102no—
213C22⋊D5—104no—
214C22⋊D5—104no—
215C22⋊D5—102no—
216C22⋊D5—102no—
217C22⋊D5—102no—
218C22⋊D5—104no—
219C22⋊D5—104no—
220C22⋊D5—204no—
221C22⋊D5—202no—
222C2×Dic5—52no—
223C2×Dic5—52no—
224C2×Dic5—51no—
225C2×Dic5—51no—
226C2×Dic5—102no—
227C2×Dic5—104no—
228C2×Dic5—102no—
229C2×Dic5—102no—
230C2×Dic5—102no—
231C2×Dic5—204no—
232C2×Dic5C22×D512yes—
233C2×Dic5C22×D511yescharacteristic
234D20—42no—
235D20—102no—
236D20—202no—
237C4×D5—42no—
238C4×D5—102no—
239C22×D4—251no—
240C22×D4—251no—
241#123—252no—
242#123—251no—
243#123—251no—
244#121—252no—
245#121—252no—
246#120—51no—
247#120—252no—
248#120—251no—
249C2×C22⋊C4—51no—
250C52#53411yescharacteristic, complemented, Sylow
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—24no—
254C22×C5—22no—
255C22×C5—44no—
256C22×C5—52no—
257C22×C5—52no—
258C22×C5—102no—
259C22×C5—102no—
260C22×C5—102no—
261C22×C5—101no—
262C22×C5—102no—
263C22×C5—102no—
264C22×C5—102no—
265C22×C5—204no—
266C22×C5—204no—
267C22×C5C2×D2012yes—
268C22×C5D4×D512yes—
269C22×C5C23×D511yescharacteristic
270C22×C5C23×D511yescharacteristic
271C2×D5—22no—
272C2×D5—22no—
273C2×D5—21no—
274C2×D5—22no—
275C2×D5—22no—
276C2×D5—22no—
277C2×D5—22no—
278C2×D5—44no—
279C2×D5—44no—
280C2×D5—101no—
281C2×D5—101no—
282C2×D5—102no—
283C2×D5—102no—
284C2×D5—202no—
285C2×D5—204no—
286C2×D5—204no—
287C2×D5—204no—
288C2×D5C2×D2012yes—
289C2×D5C2×D2012yes—
290C4×C5—22no—
291C4×C5—22no—
292C4×C5—44no—
293C4×C5—101no—
294C4×C5—102no—
295Dic5—21no—
296Dic5—22no—
297Dic5—101no—
298Dic5—102no—
299Dic5—101no—
300Dic5—102no—
301Dic5—204no—
302Dic5—202no—
303C24—51no—
304C24—51no—
305C24—251no—
306C2×D4—51no—
307C2×D4—52no—
308C2×D4—251no—
309C2×D4—252no—
310C2×D4—252no—
311C2×D4—252no—
312C2×D4—251no—
313C2×D4—251no—
314C2×D4—504no—
315C2×D4—504no—
316C22×C4—52no—
317C22×C4—251no—
318C22×C4—251no—
319C4⋊C4—252no—
320C4⋊C4—252no—
321C22⋊C4—51no—
322C22⋊C4—52no—
323C22⋊C4—52no—
324C22⋊C4—252no—
325C22⋊C4—252no—
326C22⋊C4—252no—
327C22⋊C4D5211yescharacteristic, complemented
328C2×C5—22no—
329C2×C5—22no—
330C2×C5—22no—
331C2×C5—24no—
332C2×C5—44no—
333C2×C5—102no—
334C2×C5—102no—
335C2×C5—201no—
336C2×C5—202no—
337C2×C5C22×D2011yescharacteristic
338C2×C5C2×D4×D511yescharacteristic
339C2×C5#414412yes—
340C2×C5#414412yes—
341D5—22no—
342D5—22no—
343D5—42no—
344D5—41no—
345D5—201no—
346D5—201no—
347D5—402no—
348C23—51no—
349C23—51no—
350C23—54no—
351C23—52no—
352C23—51no—
353C23—102no—
354C23—202no—
355C23—202no—
356C23—251no—
357C23—502no—
358C23—1002no—
359C23C2×D5211yescharacteristic
360D4—102no—
361D4—202no—
362D4—502no—
363D4—504no—
364D4—502no—
365D4—504no—
366D4—502no—
367D4—1002no—
368C2×C4—52no—
369C2×C4—51no—
370C2×C4—202no—
371C2×C4—251no—
372C2×C4—252no—
373C2×C4—502no—
374C2×C4—502no—
375C2×C4C2×D5212yes—
376C5—22no—
377C5#6582711yescharacteristic, complemented
378C5#6592311yescharacteristic, complemented
379C22—22no—
380C22—52no—
381C22—52no—
382C22—102no—
383C22—102no—
384C22—101no—
385C22—102no—
386C22—102no—
387C22—102no—
388C22—102no—
389C22—204no—
390C22—204no—
391C22—501no—
392C22—502no—
393C22—1004no—
394C22D5×D2012yes—
395C22C22×D5211yescharacteristic
396C4—22no—
397C4—101no—
398C4—102no—
399C4—501no—
400C4—502no—
401C2—22no—
402C2—102no—
403C2—102no—
404C2—202no—
405C2—201no—
406C2—1001no—
407C2C2×D5×D2011yescharacteristic
408C2#1171747812yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#534
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D5×C22⋊D5C2semidirect
2. C22×#6037C2semidirect
3. #11717206C2semidirect
4. C5×#4028C2semidirect
5. C5×D5×C22⋊C4C2semidirect
6. #11717095C2semidirect
7. #11717080C2semidirect
8. #11717040C2semidirect
9. #11717026C2semidirect
10. #11717024C2semidirect
11. #11717003C2semidirect
13. C23×C5×D5C22semidirect
14. C22×D52C22semidirect
16. C22×#1039C22semidirect
18. C2×C5×D20C22semidirect
19. C2×C4×C5×D5C22semidirect
21. C2×#6038C22semidirect
22. C2×#6038C22semidirect
25. C2×#6037C22semidirect
26. C2×#6037C22semidirect
27. C2×D5×Dic5C22semidirect
28. C52×C22⋊C4C22semidirect
29. #87808C22semidirect
30. C4⋊#1039C22semidirect
31. C5×C22⋊Dic5C22semidirect
32. C5×#685C22semidirect
33. C5×#685C22semidirect
34. #87780C22semidirect
35. #87778C22semidirect
36. #87778C22semidirect
44. C22×C5×D5D4semidirect
88. #4028D5semidirect
89. D5×C22⋊C4D5semidirect
117. C23×D5C2×D5semidirect
143. C2×D20C2×D5semidirect
145. C2×C4×D5C2×D5semidirect
150. C5×C22⋊C4C2×D5semidirect
151. C5×C22⋊C4C2×D5semidirect
154. C22⋊Dic5C2×D5semidirect
160. #685C2×D5semidirect
161. #685C2×D5semidirect
197. C22×D5D20semidirect
250. C52#534semidirect
327. C22⋊C4D52semidirect
377. C5#65827semidirect
378. C5#65923semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C22⋊D52
C22×#60372
#117172062
C5×#40282
C5×D5×C22⋊C42
#117170952
#117170802
#117170402
#117170262
#117170242
#117170032
#659235
#658275