Finite Group Databasebeta

← #13106217

Subgroups — #13106217

Order 1600.

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

normal subgroups only

This group has 824 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×#87902C211yesmaximal, characteristic, complemented
2C22×D5×Dic5C211yesmaximal, characteristic, complemented
3C5×#4083C211yesmaximal, characteristic, complemented
4C5×#4081C211yesmaximal, characteristic, complemented
5#11717095C211yesmaximal, characteristic, complemented
6#11717091C212yesmaximal, complemented
7#11717089C212yesmaximal, complemented
8#11717084C211yesmaximal, characteristic, complemented
9#11717075C211yesmaximal, characteristic, complemented
10#11717015C212yesmaximal, complemented
11#11716999C212yesmaximal, complemented
12C23×C5×D5C2211yescharacteristic
13C2×C52×D4C2211yescharacteristic
14C2×C22⋊(C5⋊D5)C2211yescharacteristic
15C2×C22⋊(C5⋊D5)C2211yescharacteristic, complemented
16C22×#1039C2211yescharacteristic
17C22×#1039C2211yescharacteristic, complemented
18#87902—24no—
19C2×C4×C5⋊D5C2211yescharacteristic, complemented
20C2×#6045C2211yescharacteristic, complemented
21C2×C5×C22⋊D5C2212yes—
22C22×C5×Dic5C2211yescharacteristic
23C2×#6038C2212yes—
24C2×D5×Dic5—24no—
25C2×D5×Dic5C2212yes—
26C2×D5×Dic5C2212yes—
27C5×C22⋊Dic5C2211yescharacteristic
28C5×C22⋊Dic5C2211yescharacteristic
29C5×C22⋊Dic5C2212yescomplemented
30C5×#685C2212yes—
31C5×#683C2212yescomplemented
32#87781C2212yescomplemented
33#87780C2212yescomplemented
34#87779C2212yescomplemented
35#87778C2212yes—
36Dic52C2212yescomplemented
37#65990—51nomaximal
38#65920—51nomaximal
39C23×C52C2311yescharacteristic
40C23×C52C2311yescharacteristic
41C22×C5⋊D5C2311yescharacteristic
42C22×C5×D5—22no—
43C22×C5×D5—24no—
44C22×C5×D5C2312yes—
45C52×D4—22no—
46C2×C4×C52C2311yescharacteristic
47C22⋊(C5⋊D5)—22no—
48C22⋊(C5⋊D5)—22no—
49C2×#1039—22no—
50C2×#1039D414yescomplemented
51C2×#1039C2311yescharacteristic
52C2×#1039C2311yescharacteristic
53C2×#1039C2311yescharacteristic
54C4×C5⋊D5—22no—
55#6045—22no—
56C5×C22⋊D5—24no—
57C2×C5×Dic5—22no—
58C2×C5×Dic5C2312yes—
59C2×C5×Dic5C2312yes—
60C2×C5×Dic5C2312yes—
61#6038—24no—
62D5×Dic5—24no—
63D5×Dic5—24no—
64C23×Dic5—51no—
65C2×#711—51no—
66C2×#711—51no—
67C2×#711—102no—
68C22×C4×D5—51no—
69C5×#123—51no—
70C5×#120—51no—
71#4099—51no—
72#4085—52no—
73#4083D511yescharacteristic, complemented
74#4081D511yescharacteristic, complemented
75D4×Dic5—52no—
76#4075—51no—
77#4042—52no—
78#4039—52no—
79#4032—51no—
80#4030—52no—
81#4028—51no—
82#4027—52no—
83#4027—52no—
84#4024—52no—
85#4023—51no—
86#4023—51no—
87C22×C52—22no—
88C22×C52—21no—
89C22×C52—22no—
90C22×C52C2×D412yes—
91C22×C52C2411yescharacteristic
92C2×C5⋊D5—21no—
93C2×C5⋊D5—22no—
94C2×C5×D5—28no—
95C2×C5×D5—22no—
96C2×C5×D5—22no—
97C2×C5×D5#4114yescomplemented
98C4×C52—21no—
99#1039—21no—
100#1039C2×D412yes—
101#1039C2×D412yes—
102C5×Dic5—22no—
103C5×Dic5—22no—
104C5×Dic5—22no—
105C24×C5—51no—
106C23×D5C2×D511yescharacteristic
107C2×C5×D4—22no—
108C2×C5×D4—52no—
109C2×C5×D4C2×D511yescharacteristic
110C2×C5×D4C2×D511yescharacteristic
111C22×C4×C5—51no—
112C2×C22⋊D5—51no—
113C2×C22⋊D5—51no—
114C2×C22⋊D5—51no—
115C2×C22⋊D5—51no—
116C2×C22⋊D5—52no—
117C2×C22⋊D5—102no—
118C2×C22⋊D5—102no—
119C2×C22⋊D5C2×D512yes—
120C22×Dic5—51no—
121C22×Dic5—51no—
122C22×Dic5—51no—
123C22×Dic5—51no—
124C22×Dic5—52no—
125C22×Dic5—52no—
126C22×Dic5—102no—
127C22×Dic5—102no—
128C22×Dic5—104no—
129C22×Dic5C2×D511yescharacteristic
130#711—104no—
131#711—104no—
132#711—1016no—
133C2×D20—52no—
134C2×C4×D5—52no—
135C2×C4×D5—52no—
136C2×C4×D5—51no—
137C2×C4×D5—51no—
138C2×C4×D5—102no—
139C2×C4×D5—104no—
140C2×Dic10—51no—
141C2×Dic10—51no—
142C2×Dic10—102no—
143C5×C4⋊C4—52no—
144C5×C22⋊C4—51no—
145C5×C22⋊C4—51no—
146C5×C22⋊C4—52no—
147C5×C22⋊C4—52no—
148C22⋊Dic5—52no—
149C22⋊Dic5C2×D511yescharacteristic
150C22⋊Dic5C2×D511yescharacteristic
151C22⋊Dic5C2×D512yescomplemented
152#685—52no—
153#685—52no—
154#685—52no—
155#685C2×D512yes—
156C4⋊Dic5—52no—
157#683—52no—
158#683—52no—
159#683—52no—
160#683C2×D512yescomplemented
161C4×Dic5—52no—
162C4×Dic5—52no—
163#525—251noSylow
164C2×C52—22no—
165C2×C52—41no—
166C2×C52C22×D411yescharacteristic
167C2×C52C2×#4112yes—
168C5⋊D5—41no—
169C5×D5—24no—
170C23×C5—22no—
171C23×C5—22no—
172C23×C5—52no—
173C23×C5—104no—
174C23×C5—102no—
175C23×C5C22×D511yescharacteristic
176C23×C5C22×D511yescharacteristic
177C23×C5C22×D511yescharacteristic
178C23×C5C22×D511yescharacteristic
179C22×D5—22no—
180C22×D5—24no—
181C22×D5—51no—
182C22×D5—51no—
183C22×D5—102no—
184C22×D5C22×D512yes—
185C5×D4—28no—
186C5×D4—22no—
187C5×D4—22no—
188C5×D4—104no—
189C2×C4×C5—22no—
190C2×C4×C5—52no—
191C2×C4×C5—52no—
192C2×C4×C5—52no—
193C2×C4×C5—102no—
194C2×C4×C5C22×D511yescharacteristic
195C2×C4×C5C22×D511yescharacteristic
196C22⋊D5—24no—
197C22⋊D5—104no—
198C22⋊D5—102no—
199C22⋊D5—102no—
200C22⋊D5—108no—
201C22⋊D5—108no—
202C22⋊D5—102no—
203C22⋊D5—102no—
204C2×Dic5—22no—
205C2×Dic5—54no—
206C2×Dic5—54no—
207C2×Dic5—51no—
208C2×Dic5—51no—
209C2×Dic5—51no—
210C2×Dic5—51no—
211C2×Dic5—51no—
212C2×Dic5—51no—
213C2×Dic5—108no—
214C2×Dic5—102no—
215C2×Dic5—102no—
216C2×Dic5—102no—
217C2×Dic5—104no—
218C2×Dic5—104no—
219C2×Dic5—102no—
220C2×Dic5—102no—
221C2×Dic5—108no—
222C2×Dic5C22×D512yes—
223C2×Dic5C22×D512yes—
224C2×Dic5C22×D512yes—
225D20—104no—
226C4×D5—102no—
227C4×D5—102no—
228C4×D5—104no—
229C4×D5—104no—
230C4×D5—108no—
231Dic10—102no—
232Dic10—102no—
233Dic10—108no—
234C2×#41—251no—
235C23×C4—251no—
236#123—51no—
237#123—251no—
238C22⋊Q8—252no—
239#121—252no—
240#120—51no—
241#120—251no—
242C4×D4—252no—
243C4×D4—252no—
244#117—251no—
245C52#52511yescharacteristic, complemented, Sylow
246C22×C5—22no—
247C22×C5—22no—
248C22×C5—24no—
249C22×C5—21no—
250C22×C5—21no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—24no—
254C22×C5—22no—
255C22×C5—44no—
256C22×C5—44no—
257C22×C5—54no—
258C22×C5—108no—
259C22×C5—102no—
260C22×C5—102no—
261C22×C5D4×D512yes—
262C22×C5D4×D512yes—
263C22×C5C23×D511yescharacteristic
264C22×C5C23×D511yescharacteristic
265C2×D5—28no—
266C2×D5—22no—
267C2×D5—22no—
268C2×D5—104no—
269C2×D5—101no—
270C2×D5—101no—
271C2×D5—102no—
272C2×D5—102no—
273C2×D5—204no—
274C2×D5#71114yescomplemented
275C4×C5—24no—
276C4×C5—21no—
277C4×C5—21no—
278C4×C5—102no—
279C4×C5—102no—
280C4×C5—102no—
281Dic5—22no—
282Dic5—22no—
283Dic5—22no—
284Dic5—52no—
285Dic5—52no—
286Dic5—52no—
287Dic5—52no—
288Dic5—104no—
289Dic5—104no—
290Dic5—104no—
291Dic5—101no—
292Dic5—101no—
293C24—51no—
294#41—504no—
295C2×Q8—251no—
296C2×D4—52no—
297C2×D4—252no—
298C2×D4—251no—
299C2×D4—251no—
300C2×D4D5211yescharacteristic
301C22×C4—51no—
302C22×C4—252no—
303C22×C4—252no—
304C22×C4—251no—
305C22×C4—251no—
306C22×C4—251no—
307C22×C4—504no—
308C4⋊C4—52no—
309C4⋊C4—252no—
310C4⋊C4—252no—
311C22⋊C4—51no—
312C22⋊C4—51no—
313C22⋊C4—52no—
314C22⋊C4—52no—
315C22⋊C4—252no—
316C22⋊C4—252no—
317C42—252no—
318C2×C5—22no—
319C2×C5—22no—
320C2×C5—24no—
321C2×C5—22no—
322C2×C5—44no—
323C2×C5—44no—
324C2×C5—41no—
325C2×C5—41no—
326C2×C5—104no—
327C2×C5C2×D4×D511yescharacteristic
328C2×C5C2×D4×D511yescharacteristic
329C2×C5C2×#71112yes—
330C2×C5D5×#4112yes—
331D5—24no—
332D5—204no—
333D5—201no—
334D5—201no—
335C23—52no—
336C23—104no—
337C23—102no—
338C23—251no—
339C23C2×D5211yescharacteristic
340C23C2×D5211yescharacteristic
341Q8—502no—
342D4—22no—
343D4—104no—
344D4—504no—
345D4—502no—
346D4—502no—
347C2×C4—52no—
348C2×C4—52no—
349C2×C4—52no—
350C2×C4—102no—
351C2×C4—254no—
352C2×C4—251no—
353C2×C4—251no—
354C2×C4—251no—
355C2×C4—502no—
356C2×C4—504no—
357C2×C4—504no—
358C2×C4—502no—
359C2×C4C2×D5211yescharacteristic
360C5—22no—
361C5#6592011yescharacteristic, complemented
362C5#6599011yescharacteristic, complemented
363C22—22no—
364C22—22no—
365C22—21no—
366C22—54no—
367C22—108no—
368C22—102no—
369C22—102no—
370C22—502no—
371C22—501no—
372C22#8788612yes—
373C22C22×D5211yescharacteristic
374C4—21no—
375C4—102no—
376C4—102no—
377C4—102no—
378C4—252no—
379C4—252no—
380C4—501no—
381C2—22no—
382C2—41no—
383C2—104no—
384C2—1001no—
385C2D5×#71112yes—
386C2C2×#8788611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC2×C52×D4C22
hypercenterC2×D4D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#87902C2semidirect
2. C22×D5×Dic5C2semidirect
3. C5×#4083C2semidirect
4. C5×#4081C2semidirect
5. #11717095C2semidirect
6. #11717091C2semidirect
7. #11717089C2semidirect
8. #11717084C2semidirect
9. #11717075C2semidirect
10. #11717015C2semidirect
11. #11716999C2semidirect
15. C2×C22⋊(C5⋊D5)C22semidirect
17. C22×#1039C22semidirect
19. C2×C4×C5⋊D5C22semidirect
20. C2×#6045C22semidirect
29. C5×C22⋊Dic5C22semidirect
31. C5×#683C22semidirect
32. #87781C22semidirect
33. #87780C22semidirect
34. #87779C22semidirect
36. Dic52C22semidirect
50. C2×#1039D4semidirect
73. #4083D5semidirect
74. #4081D5semidirect
97. C2×C5×D5#41semidirect
151. C22⋊Dic5C2×D5semidirect
160. #683C2×D5semidirect
245. C52#525semidirect
274. C2×D5#711semidirect
361. C5#65920semidirect
362. C5#65990semidirect

Maximal subgroups

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

subgroupindex
C2×#879022
C22×D5×Dic52
C5×#40832
C5×#40812
#117170952
#117170912
#117170892
#117170842
#117170752
#117170152
#117169992
#659905
#659205