Finite Group Databasebeta

← #13105144

Subgroups — #13105144

Order 1600.

738 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 152 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×#87879C211yesmaximal, characteristic, complemented
2C22×D5×Dic5C211yesmaximal, characteristic, complemented
3#11717206C211yesmaximal, characteristic, complemented
4C5×#4028C211yesmaximal, characteristic, complemented
5C5×#4023C211yesmaximal, characteristic, complemented
6#11717095C211yesmaximal, characteristic, complemented
7#11717080C211yesmaximal, characteristic, complemented
8Dic5×D20C212yesmaximal, complemented
9#11717007C212yesmaximal, complemented
10#11717005C212yesmaximal, complemented
11#11716998C212yesmaximal, complemented
12C23×C5×D5C2211yescharacteristic
13C2×C22⋊(C5⋊D5)C2211yescharacteristic
14C22×#1039C2211yescharacteristic
15C2×C5×C22⋊D5C2211yescharacteristic
16C22×C5×Dic5C2211yescharacteristic
17C2×C5×D20C2212yes—
18C2×#6039C2211yescharacteristic, complemented
19C2×#6038C2212yes—
20C2×#6038C2211yescharacteristic, complemented
21C2×#6036C2211yescharacteristic, complemented
22#87879—24no—
23C2×D5×Dic5—24no—
24C2×D5×Dic5C2212yes—
25C2×D5×Dic5C2212yes—
26C2×D5×Dic5C2211yescharacteristic, complemented
27C52×C22⋊C4C2211yescharacteristic
28#87808C2212yescomplemented
29C4⋊#1039C2212yescomplemented
30C5×C22⋊Dic5C2211yescharacteristic
31C5×#685C2212yes—
32C5×#683C2212yescomplemented
33C4×C5×Dic5C2212yescomplemented
34#87780C2212yescomplemented
35#87778C2212yes—
36#87778C2212yescomplemented
37#65920—51nomaximal
38#65858—51nomaximal
39C23×C52C2311yescharacteristic
40C22×C5⋊D5C2311yescharacteristic
41C22×C5×D5—22no—
42C22×C5×D5—24no—
43C22×C5×D5C2311yescharacteristic
44C22×C5×D5C2312yes—
45C2×C4×C52C2312yes—
46C22⋊(C5⋊D5)—22no—
47C2×#1039—22no—
48C2×#1039C2312yes—
49C2×#1039C2311yescharacteristic
50C5×C22⋊D5—22no—
51C2×C5×Dic5D414yescomplemented
52C2×C5×Dic5C2311yescharacteristic
53C2×C5×Dic5C2312yes—
54C2×C5×Dic5C2311yescharacteristic
55C2×C5×Dic5C2311yescharacteristic
56C5×D20—24no—
57#6039—22no—
58#6038—24no—
59#6038—22no—
60#6036—22no—
61D5×Dic5—24no—
62D5×Dic5—24no—
63D5×Dic5—22no—
64C23×Dic5—51no—
65C2×#711—51no—
66C2×#709—51no—
67C22×C4×D5—51no—
68C5×#120—51no—
69C5×#117—51no—
70#4099—51no—
71#4085—52no—
72#4081—51no—
73D4×Dic5—52no—
74#4043—52no—
75#4041—52no—
76#4032—51no—
77#4032—51no—
78#4032—51no—
79#4032—102no—
80#4028—51no—
81#4028D511yescharacteristic, complemented
82#4027—52no—
83#4024—52no—
84#4023D511yescharacteristic, complemented
85C4×D20—52no—
86C4×D20—52no—
87C22×C52—22no—
88C22×C52C2×D412yes—
89C22×C52C2411yescharacteristic
90C2×C5⋊D5—21no—
91C2×C5⋊D5—22no—
92C2×C5×D5—28no—
93C2×C5×D5—22no—
94C2×C5×D5—21no—
95C2×C5×D5—22no—
96C2×C5×D5—22no—
97C2×C5×D5#4114yescomplemented
98C4×C52—22no—
99#1039—22no—
100#1039—21no—
101C5×Dic5—21no—
102C5×Dic5—22no—
103C5×Dic5C2×D412yes—
104C5×Dic5C2×D412yes—
105C24×C5—51no—
106C23×D5C2×D511yescharacteristic
107C2×C5×D4—51no—
108C2×C5×D4—52no—
109C22×C4×C5—51no—
110C2×C22⋊D5—51no—
111C2×C22⋊D5—51no—
112C2×C22⋊D5—52no—
113C2×C22⋊D5—51no—
114C2×C22⋊D5—102no—
115C2×C22⋊D5C2×D511yescharacteristic
116C22×Dic5—51no—
117C22×Dic5—51no—
118C22×Dic5—51no—
119C22×Dic5—52no—
120C22×Dic5—52no—
121C22×Dic5—102no—
122C22×Dic5—104no—
123C22×Dic5C2×D511yescharacteristic
124#711—104no—
125#709—104no—
126C2×D20—51no—
127C2×D20—52no—
128C2×D20C2×D512yes—
129C2×C4×D5—52no—
130C2×C4×D5—52no—
131C2×C4×D5—51no—
132C2×C4×D5—51no—
133C2×C4×D5—51no—
134C2×C4×D5—104no—
135C2×Dic10—51no—
136C2×Dic10—51no—
137C5×C4⋊C4—52no—
138C5×C22⋊C4—22no—
139C5×C22⋊C4—51no—
140C5×C22⋊C4—52no—
141C5×C22⋊C4C2×D511yescharacteristic
142C5×C22⋊C4C2×D511yescharacteristic
143C42×C5—52no—
144C22⋊Dic5—52no—
145C22⋊Dic5—52no—
146C22⋊Dic5C2×D511yescharacteristic
147#685—52no—
148#685—52no—
149#685—52no—
150#685—52no—
151#685—104no—
152#685C2×D512yes—
153C4⋊Dic5—52no—
154C4⋊Dic5—52no—
155C4⋊Dic5—52no—
156C4⋊Dic5—104no—
157#683—52no—
158#683C2×D512yescomplemented
159C4×Dic5C2×D512yescomplemented
160#525—251noSylow
161C2×C52—22no—
162C2×C52C22×D411yescharacteristic
163C2×C52C2×#4112yes—
164C5⋊D5—41no—
165C5×D5—24no—
166C5×D5—41no—
167C23×C5—22no—
168C23×C5—52no—
169C23×C5—51no—
170C23×C5—104no—
171C23×C5—102no—
172C23×C5C22×D511yescharacteristic
173C23×C5C22×D511yescharacteristic
174C22×D5—22no—
175C22×D5—24no—
176C22×D5—51no—
177C22×D5—51no—
178C22×D5—102no—
179C22×D5C22×D511yescharacteristic
180C22×D5C22×D512yes—
181C5×D4—104no—
182C5×D4—102no—
183C2×C4×C5—24no—
184C2×C4×C5—54no—
185C2×C4×C5—51no—
186C2×C4×C5—51no—
187C2×C4×C5—51no—
188C2×C4×C5—52no—
189C2×C4×C5C22×D512yes—
190C2×C4×C5C22×D512yes—
191C22⋊D5—22no—
192C22⋊D5—104no—
193C22⋊D5—102no—
194C22⋊D5—102no—
195C22⋊D5—102no—
196C22⋊D5—204no—
197C2×Dic5—52no—
198C2×Dic5—51no—
199C2×Dic5—52no—
200C2×Dic5—51no—
201C2×Dic5—102no—
202C2×Dic5—104no—
203C2×Dic5—102no—
204C2×Dic5—102no—
205C2×Dic5—102no—
206C2×Dic5—104no—
207C2×Dic5—104no—
208C2×Dic5—204no—
209C2×Dic5D2014yescomplemented
210C2×Dic5C22×D512yes—
211C2×Dic5C22×D511yescharacteristic
212C2×Dic5C22×D511yescharacteristic
213C2×Dic5C22×D511yescharacteristic
214D20—24no—
215D20—102no—
216D20—104no—
217C4×D5—102no—
218C4×D5—104no—
219C4×D5—104no—
220C4×D5—102no—
221C4×D5—102no—
222Dic10—102no—
223Dic10—102no—
224C2×#41—251no—
225C23×C4—251no—
226#123—251no—
227#123—251no—
228C22⋊Q8—252no—
229#121—252no—
230#120—51no—
231#120—251no—
232C4×D4—252no—
233C4×D4—252no—
234#117—51no—
235C52#52511yescharacteristic, complemented, Sylow
236C22×C5—22no—
237C22×C5—22no—
238C22×C5—24no—
239C22×C5—22no—
240C22×C5—44no—
241C22×C5—54no—
242C22×C5—108no—
243C22×C5—102no—
244C22×C5—102no—
245C22×C5—101no—
246C22×C5—102no—
247C22×C5C2×D2012yes—
248C22×C5D4×D512yes—
249C22×C5C23×D511yescharacteristic
250C22×C5C23×D511yescharacteristic
251C2×D5—28no—
252C2×D5—22no—
253C2×D5—22no—
254C2×D5—21no—
255C2×D5—22no—
256C2×D5—101no—
257C2×D5—101no—
258C2×D5—102no—
259C2×D5—102no—
260C2×D5—202no—
261C2×D5—204no—
262C2×D5#71114yescomplemented
263C4×C5—22no—
264C4×C5—22no—
265C4×C5—44no—
266C4×C5—52no—
267C4×C5—52no—
268C4×C5—101no—
269C4×C5—102no—
270Dic5—21no—
271Dic5—22no—
272Dic5—101no—
273Dic5—102no—
274Dic5—101no—
275Dic5—102no—
276Dic5—204no—
277Dic5—202no—
278Dic5C2×D2012yes—
279Dic5C2×D2012yes—
280C24—51no—
281#41—504no—
282C2×Q8—251no—
283C2×D4—51no—
284C2×D4—52no—
285C2×D4—252no—
286C2×D4—251no—
287C2×D4—251no—
288C22×C4—51no—
289C22×C4—251no—
290C22×C4—252no—
291C22×C4—251no—
292C22×C4—252no—
293C22×C4—251no—
294C22×C4—504no—
295C4⋊C4—52no—
296C4⋊C4—252no—
297C4⋊C4—252no—
298C22⋊C4—51no—
299C22⋊C4—52no—
300C22⋊C4—252no—
301C22⋊C4—252no—
302C22⋊C4—252no—
303C22⋊C4D5211yescharacteristic
304C42—52no—
305C2×C5—22no—
306C2×C5—22no—
307C2×C5—22no—
308C2×C5—24no—
309C2×C5—44no—
310C2×C5—104no—
311C2×C5—201no—
312C2×C5C22×D2011yescharacteristic
313C2×C5C2×D4×D511yescharacteristic
314C2×C5C2×#71112yes—
315C2×C5D5×#4112yes—
316D5—24no—
317D5—41no—
318D5—201no—
319D5—201no—
320D5—402no—
321C23—52no—
322C23—51no—
323C23—104no—
324C23—102no—
325C23—251no—
326C23C2×D5211yescharacteristic
327Q8—502no—
328D4—104no—
329D4—102no—
330D4—502no—
331D4—504no—
332D4—502no—
333C2×C4—54no—
334C2×C4—51no—
335C2×C4—51no—
336C2×C4—52no—
337C2×C4—51no—
338C2×C4—252no—
339C2×C4—251no—
340C2×C4—502no—
341C2×C4—504no—
342C2×C4—504no—
343C2×C4—502no—
344C2×C4—502no—
345C2×C4C2×D5212yes—
346C5—22no—
347C5#6585811yescharacteristic, complemented
348C5#6592011yescharacteristic, complemented
349C22—22no—
350C22—54no—
351C22—108no—
352C22—102no—
353C22—102no—
354C22—101no—
355C22—102no—
356C22—501no—
357C22—502no—
358C22D5×D2012yes—
359C22C22×D5211yescharacteristic
360C4—22no—
361C4—52no—
362C4—52no—
363C4—101no—
364C4—102no—
365C4—502no—
366C4—501no—
367C2—22no—
368C2—104no—
369C2—201no—
370C2—1001no—
371C2C2×D5×D2011yescharacteristic
372C2D5×#71112yes—
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC52×C22⋊C4C22
hypercenterC22⋊C4D52
hypocenterC52#525
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#87879C2semidirect
2. C22×D5×Dic5C2semidirect
3. #11717206C2semidirect
4. C5×#4028C2semidirect
5. C5×#4023C2semidirect
6. #11717095C2semidirect
7. #11717080C2semidirect
8. Dic5×D20C2semidirect
9. #11717007C2semidirect
10. #11717005C2semidirect
11. #11716998C2semidirect
18. C2×#6039C22semidirect
20. C2×#6038C22semidirect
21. C2×#6036C22semidirect
26. C2×D5×Dic5C22semidirect
28. #87808C22semidirect
29. C4⋊#1039C22semidirect
32. C5×#683C22semidirect
33. C4×C5×Dic5C22semidirect
34. #87780C22semidirect
36. #87778C22semidirect
51. C2×C5×Dic5D4semidirect
81. #4028D5semidirect
84. #4023D5semidirect
97. C2×C5×D5#41semidirect
158. #683C2×D5semidirect
159. C4×Dic5C2×D5semidirect
209. C2×Dic5D20semidirect
235. C52#525semidirect
262. C2×D5#711semidirect
347. C5#65858semidirect
348. C5#65920semidirect

Maximal subgroups

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

subgroupindex
C2×#878792
C22×D5×Dic52
#117172062
C5×#40282
C5×#40232
#117170952
#117170802
Dic5×D202
#117170072
#117170052
#117169982
#659205
#658585