Finite Group Databasebeta

← #13105238

Subgroups — #13105238

Order 1600.

950 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 150 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⋊D5C212yesmaximal, complemented
2C22×#6037C211yesmaximal, characteristic, complemented
3C2×#87877C211yesmaximal, characteristic, complemented
4#11717201C211yesmaximal, characteristic, complemented
5C5×#4028C212yesmaximal, complemented
6#11717098C211yesmaximal, characteristic, complemented
7#11717040C212yesmaximal, complemented
8#11717032C211yesmaximal, characteristic, complemented
9#11717030C211yesmaximal, characteristic, complemented
10#11717020C211yesmaximal, characteristic, complemented
11#11717004C212yesmaximal, complemented
12C23×C5×D5C2212yescomplemented
13C22×D52C2212yescomplemented
14C2×C22⋊(C5⋊D5)C2211yescharacteristic
15C22×#1039C2211yescharacteristic, complemented
16C2×C4×C5⋊D5C2211yescharacteristic, complemented
17C2×C5×C22⋊D5C2212yescomplemented
18C2×C5×D20C2212yescomplemented
19C2×C5×D20C2212yescomplemented
20D5×C22⋊D5—28no—
21C2×#6038C2212yescomplemented
22C2×#6037—24no—
23C2×#6037C2211yescharacteristic
24C2×#6037C2211yescharacteristic, complemented
25C2×#6037C2212yescomplemented
26C2×#6037C2211yescharacteristic, complemented
27C2×#6037C2212yescomplemented
28C2×D5×Dic5C2212yescomplemented
29#87877—24no—
30C52×C22⋊C4C2211yescharacteristic, complemented
31#87808C2211yescharacteristic, complemented
32#87806C2211yescharacteristic, complemented
33C4×#1039C2211yescharacteristic, complemented
34C5×#685C2212yescomplemented
35C5×#685C2212yescomplemented
36#87781C2211yescharacteristic, complemented
37#87778C2212yescomplemented
38#65924—52nomaximal
39C23×C52C2311yescharacteristic
40C22×C5⋊D5C2311yescharacteristic
41C22×C5×D5—24no—
42C22×C5×D5—24no—
43C22×C5×D5—42no—
44C22×C5×D5C2312yes—
45C22×C5×D5C2312yes—
46C22×C5×D5C2312yes—
47C2×D52—24no—
48C2×D52—24no—
49C2×D52—42no—
50C2×C4×C52C2311yescharacteristic
51C2×C4×C52C2311yescharacteristic
52C22⋊(C5⋊D5)D414yescomplemented
53C2×#1039—22no—
54C2×#1039C2311yescharacteristic
55C2×#1039C2311yescharacteristic
56C2×#1039C2311yescharacteristic
57C4×C5⋊D5—22no—
58C5×C22⋊D5—24no—
59C2×C5×Dic5C2312yes—
60C5×D20—24no—
61C5×D20—42no—
62#6038—24no—
63#6037—24no—
64#6037—22no—
65#6037—22no—
66#6037—24no—
67#6037D414yescomplemented
68D5×Dic5—24no—
69C22×C22⋊D5—52no—
70C22×C22⋊D5—52no—
71C2×D4×D5—52no—
72C2×D4×D5—52no—
73C5×#120—52no—
74#4086—52no—
75#4085—52no—
76#4085—52no—
77#4083—52no—
78#4030—52no—
79#4030—52no—
80#4028—52no—
81#4028D512yescomplemented
82#4027—52no—
83#4027—52no—
84#4027—102no—
85C22×C52—21no—
86C22×C52—21no—
87C22×C52C2×D412yescomplemented
88C22×C52C2411yescharacteristic
89C2×C5⋊D5—21no—
90C2×C5⋊D5—21no—
91C2×C5⋊D5C2×D412yescomplemented
92C2×C5×D5—22no—
93C2×C5×D5—22no—
94C2×C5×D5—22no—
95C2×C5×D5—22no—
96C2×C5×D5—22no—
97C2×C5×D5—22no—
98C2×C5×D5—22no—
99C2×C5×D5—22no—
100C2×C5×D5—48no—
101C2×C5×D5C2×D414yescomplemented
102D52—48no—
103C4×C52—21no—
104C4×C52—21no—
105#1039—21no—
106#1039C2×D412yescomplemented
107#1039C2×D412yescomplemented
108C5×Dic5—22no—
109C24×C5—52no—
110C23×D5—52no—
111C23×D5—52no—
112C23×D5C2×D512yescomplemented
113C2×C5×D4—52no—
114C2×C5×D4—52no—
115C2×C5×D4—52no—
116C2×C22⋊D5—52no—
117C2×C22⋊D5—52no—
118C2×C22⋊D5—52no—
119C2×C22⋊D5—52no—
120C2×C22⋊D5—52no—
121C2×C22⋊D5—52no—
122C2×C22⋊D5—52no—
123C2×C22⋊D5—52no—
124C2×C22⋊D5—52no—
125C2×C22⋊D5—102no—
126C2×C22⋊D5—108no—
127C2×C22⋊D5—108no—
128C2×C22⋊D5C2×D512yescomplemented
129C22×Dic5—52no—
130C22×Dic5—52no—
131C22×Dic5—102no—
132D4×D5—108no—
133D4×D5—108no—
134C2×D20—52no—
135C2×D20C2×D512yescomplemented
136C2×D20C2×D512yescomplemented
137C2×C4×D5—52no—
138C2×C4×D5—52no—
139C2×C4×D5—102no—
140C5×C22⋊C4—22no—
141C5×C22⋊C4—52no—
142C5×C22⋊C4—52no—
143C5×C22⋊C4C2×D512yescomplemented
144C22⋊Dic5—52no—
145#685—52no—
146#685—52no—
147#685—102no—
148#685C2×D512yescomplemented
149#685C2×D512yescomplemented
150#683—52no—
151#683—52no—
152#683—102no—
153C4×Dic5—52no—
154C4×Dic5—102no—
155D42—251noSylow
156C2×C52—22no—
157C2×C52C22×D411yescharacteristic
158C2×C52C22×D411yescharacteristic
159C2×C52H(5,𝔽2)11yescharacteristic
160C5⋊D5—22no—
161C5×D5—24no—
162C5×D5—42no—
163C5×D5—42no—
164C23×C5—22no—
165C23×C5—52no—
166C23×C5—52no—
167C23×C5—52no—
168C23×C5—104no—
169C23×C5—104no—
170C23×C5—202no—
171C23×C5C22×D512yes—
172C22×D5—24no—
173C22×D5—24no—
174C22×D5—42no—
175C22×D5—52no—
176C22×D5—102no—
177C22×D5—104no—
178C22×D5—104no—
179C22×D5—104no—
180C22×D5—104no—
181C22×D5—202no—
182C22×D5—202no—
183C22×D5C22×D512yes—
184C22×D5C22×D512yes—
185C22×D5C22×D512yes—
186C5×D4—104no—
187C5×D4—104no—
188C5×D4—202no—
189C2×C4×C5—22no—
190C2×C4×C5—22no—
191C2×C4×C5—52no—
192C2×C4×C5C22×D512yes—
193C2×C4×C5C22×D512yes—
194C22⋊D5—24no—
195C22⋊D5—58no—
196C22⋊D5—58no—
197C22⋊D5—108no—
198C22⋊D5—104no—
199C22⋊D5—104no—
200C22⋊D5—104no—
201C22⋊D5—104no—
202C22⋊D5—104no—
203C22⋊D5—104no—
204C22⋊D5—104no—
205C2×Dic5—52no—
206C2×Dic5—52no—
207C2×Dic5—52no—
208C2×Dic5—102no—
209C2×Dic5—102no—
210C2×Dic5—102no—
211C2×Dic5—104no—
212C2×Dic5—104no—
213C2×Dic5—204no—
214C2×Dic5C22×D512yes—
215D20—24no—
216D20—42no—
217D20—104no—
218C4×D5—104no—
219C4×D5—104no—
220C4×D5—204no—
221C22×D4—251no—
222C22×D4—252no—
223C22×D4—251no—
224#127—251no—
225#121—251no—
226#121—252no—
227#121—251no—
228#120—52no—
229#120—252no—
230C4×D4—251no—
231C4×D4—251no—
232C52D4211yescharacteristic, complemented, Sylow
233C22×C5—22no—
234C22×C5—22no—
235C22×C5—24no—
236C22×C5—22no—
237C22×C5—42no—
238C22×C5—42no—
239C22×C5—54no—
240C22×C5—102no—
241C22×C5—102no—
242C22×C5—102no—
243C22×C5—102no—
244C22×C5—102no—
245C22×C5—102no—
246C22×C5—102no—
247C22×C5—102no—
248C22×C5—208no—
249C22×C5D4×D514yescomplemented
250C22×C5C23×D512yes—
251C2×D5—22no—
252C2×D5—22no—
253C2×D5—22no—
254C2×D5—22no—
255C2×D5—22no—
256C2×D5—22no—
257C2×D5—22no—
258C2×D5—22no—
259C2×D5—48no—
260C2×D5—54no—
261C2×D5—104no—
262C2×D5—102no—
263C2×D5—102no—
264C2×D5—202no—
265C2×D5—202no—
266C2×D5—208no—
267C2×D5—208no—
268C2×D5D4×D514yescomplemented
269C4×C5—22no—
270C4×C5—22no—
271C4×C5—42no—
272C4×C5—42no—
273C4×C5—102no—
274Dic5—22no—
275Dic5—54no—
276Dic5—54no—
277Dic5—104no—
278Dic5—104no—
279Dic5—102no—
280Dic5—202no—
281C24—52no—
282C24—252no—
283C2×D4—52no—
284C2×D4—52no—
285C2×D4—52no—
286C2×D4—252no—
287C2×D4—251no—
288C2×D4—251no—
289C2×D4—252no—
290C2×D4—251no—
291C2×D4—252no—
292C2×D4—251no—
293C2×D4—504no—
294C2×D4—508no—
295C2×D4—504no—
296C22×C4—252no—
297C22×C4—251no—
298C22×C4—251no—
299C4⋊C4—251no—
300C4⋊C4—251no—
301C22⋊C4—52no—
302C22⋊C4—52no—
303C22⋊C4—252no—
304C22⋊C4—251no—
305C22⋊C4D5211yescharacteristic, complemented
306C42—251no—
307C2×C5—24no—
308C2×C5—22no—
309C2×C5—22no—
310C2×C5—22no—
311C2×C5—44no—
312C2×C5—104no—
313C2×C5—202no—
314C2×C5—202no—
315C2×C5C2×D4×D512yes—
316C2×C5C2×D4×D512yes—
317C2×C5#414412yes—
318D5—24no—
319D5—42no—
320D5—42no—
321D5—104no—
322D5—204no—
323C23—52no—
324C23—52no—
325C23—52no—
326C23—104no—
327C23—104no—
328C23—202no—
329C23—251no—
330C23—504no—
331C23—504no—
332C23—1002no—
333C23C2×D5211yescharacteristic
334D4—104no—
335D4—104no—
336D4—202no—
337D4—254no—
338D4—254no—
339D4—504no—
340D4—504no—
341D4—502no—
342D4—502no—
343D4—504no—
344C2×C4—52no—
345C2×C4—251no—
346C2×C4—251no—
347C2×C4—251no—
348C2×C4—502no—
349C2×C4—504no—
350C2×C4—502no—
351C2×C4C2×D5211yescharacteristic
352C2×C4C2×D5211yescharacteristic
353C5—22no—
354C5#6592412yescomplemented
355C22—21no—
356C22—21no—
357C22—54no—
358C22—102no—
359C22—102no—
360C22—102no—
361C22—102no—
362C22—102no—
363C22—102no—
364C22—102no—
365C22—102no—
366C22—208no—
367C22—252no—
368C22—501no—
369C22—501no—
370C22—1008no—
371C22#8787712yescomplemented
372C22C22×D5211yescharacteristic
373C4—21no—
374C4—21no—
375C4—102no—
376C4—252no—
377C4—252no—
378C4—501no—
379C2—22no—
380C2—104no—
381C2—202no—
382C2—202no—
383C2—502no—
384C2C2×#8787711yescharacteristic
385C2D4×D5211yescharacteristic
386C2#1171747811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52D42
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. C2×#87877C2semidirect
4. #11717201C2semidirect
5. C5×#4028C2semidirect
6. #11717098C2semidirect
7. #11717040C2semidirect
8. #11717032C2semidirect
9. #11717030C2semidirect
10. #11717020C2semidirect
11. #11717004C2semidirect
12. C23×C5×D5C22semidirect
13. C22×D52C22semidirect
15. C22×#1039C22semidirect
16. C2×C4×C5⋊D5C22semidirect
17. C2×C5×C22⋊D5C22semidirect
18. C2×C5×D20C22semidirect
19. C2×C5×D20C22semidirect
21. C2×#6038C22semidirect
24. C2×#6037C22semidirect
25. C2×#6037C22semidirect
26. C2×#6037C22semidirect
27. C2×#6037C22semidirect
28. C2×D5×Dic5C22semidirect
30. C52×C22⋊C4C22semidirect
31. #87808C22semidirect
32. #87806C22semidirect
33. C4×#1039C22semidirect
34. C5×#685C22semidirect
35. C5×#685C22semidirect
36. #87781C22semidirect
37. #87778C22semidirect
52. C22⋊(C5⋊D5)D4semidirect
67. #6037D4semidirect
81. #4028D5semidirect
87. C22×C52C2×D4semidirect
91. C2×C5⋊D5C2×D4semidirect
101. C2×C5×D5C2×D4semidirect
106. #1039C2×D4semidirect
107. #1039C2×D4semidirect
112. C23×D5C2×D5semidirect
128. C2×C22⋊D5C2×D5semidirect
135. C2×D20C2×D5semidirect
136. C2×D20C2×D5semidirect
143. C5×C22⋊C4C2×D5semidirect
148. #685C2×D5semidirect
149. #685C2×D5semidirect
232. C52D42semidirect
249. C22×C5D4×D5semidirect
268. C2×D5D4×D5semidirect
305. C22⋊C4D52semidirect
354. C5#65924semidirect
371. C22#87877semidirect

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
C2×#878772
#117172012
C5×#40282
#117170982
#117170402
#117170322
#117170302
#117170202
#117170042
#659245