Finite Group Databasebeta

← #13105145

Subgroups — #13105145

Order 1600.

790 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C22×#6036C211yesmaximal, characteristic, complemented
2C2×#87879C211yesmaximal, characteristic, complemented
3#11717202C211yesmaximal, characteristic, complemented
4C5×#4032C211yesmaximal, characteristic, complemented
5C5×#4023C211yesmaximal, characteristic, complemented
6#11717099C211yesmaximal, characteristic, complemented
7#11717084C211yesmaximal, characteristic, complemented
8#11717028C212yesmaximal, complemented
9#11717013C212yesmaximal, complemented
10#11717009C212yesmaximal, complemented
11#11716998C212yesmaximal, complemented
12C23×C5⋊D5C2211yescharacteristic
13C2×C22⋊(C5⋊D5)C2211yescharacteristic
14C2×C4⋊(C5⋊D5)C2212yes—
15C2×C5×C22⋊D5C2211yescharacteristic
16C22×C5×Dic5C2211yescharacteristic
17C22×C5×Dic5C2211yescharacteristic
18C2×#6039C2211yescharacteristic, complemented
19C2×#6038C2212yes—
20C2×#6038C2211yescharacteristic, complemented
21C2×#6036—24no—
22C2×#6036C2212yes—
23C2×#6036C2212yes—
24C2×#6036C2211yescharacteristic, complemented
25#87879—24no—
26C2×D5×Dic5C2211yescharacteristic, complemented
27C52×C22⋊C4C2211yescharacteristic
28#87808C2212yes—
29C5×C22⋊Dic5C2211yescharacteristic
30C5×#685C2212yescomplemented
31C5×C4⋊Dic5C2212yescomplemented
32C5×#683C2212yescomplemented
33C4×C5×Dic5C2212yescomplemented
34#87780C2212yescomplemented
35#87779C2212yes—
36#87779C2212yescomplemented
37#65990—51nomaximal
38#65858—51nomaximal
39C23×C52C2311yescharacteristic
40C22×C5⋊D5—22no—
41C22×C5⋊D5—24no—
42C22×C5⋊D5C2311yescharacteristic
43C22×C5⋊D5C2312yes—
44C22×C5×D5C2311yescharacteristic
45C2×C4×C52C2312yes—
46C22⋊(C5⋊D5)—22no—
47C2×#1039C2311yescharacteristic
48C4⋊(C5⋊D5)—24no—
49C5×C22⋊D5—22no—
50C2×C5×Dic5—22no—
51C2×C5×Dic5D414yescomplemented
52C2×C5×Dic5C2312yes—
53C2×C5×Dic5C2311yescharacteristic
54C2×C5×Dic5C2312yes—
55C2×C5×Dic5C2311yescharacteristic
56C2×C5×Dic5C2311yescharacteristic
57#6039—22no—
58#6038—24no—
59#6038—22no—
60#6036—22no—
61#6036—24no—
62#6036—24no—
63D5×Dic5—22no—
64C2×#711—51no—
65C2×#709—51no—
66C22×C4×D5—51no—
67C22×C4×D5—51no—
68C5×#123—51no—
69C5×#117—51no—
70#4083—51no—
71#4075—51no—
72#4043—52no—
73#4042—52no—
74#4041—52no—
75#4039—52no—
76#4032—51no—
77#4032D511yescharacteristic, complemented
78#4030—52no—
79#4028—51no—
80#4028—51no—
81#4028—51no—
82#4028—102no—
83#4027—52no—
84#4023D511yescharacteristic, complemented
85C4×D20—52no—
86C4×D20—52no—
87C22×C52—22no—
88C22×C52C2×D412yes—
89C22×C52C2411yescharacteristic
90C2×C5⋊D5—28no—
91C2×C5⋊D5—22no—
92C2×C5⋊D5—22no—
93C2×C5⋊D5—21no—
94C2×C5⋊D5—22no—
95C2×C5⋊D5#4114yescomplemented
96C2×C5×D5—21no—
97C2×C5×D5—22no—
98C4×C52—22no—
99#1039—21no—
100C5×Dic5—22no—
101C5×Dic5—21no—
102C5×Dic5—22no—
103C5×Dic5C2×D412yes—
104C5×Dic5C2×D412yes—
105C23×D5—51no—
106C23×D5—51no—
107C23×D5—102no—
108C2×C5×D4—51no—
109C22×C4×C5—51no—
110C22×C4×C5—51no—
111C2×C22⋊D5—51no—
112C2×C22⋊D5—51no—
113C2×C22⋊D5—51no—
114C2×C22⋊D5—52no—
115C2×C22⋊D5—102no—
116C2×C22⋊D5C2×D511yescharacteristic
117C22×Dic5—51no—
118C22×Dic5C2×D511yescharacteristic
119C22×Dic5C2×D511yescharacteristic
120#711—104no—
121#709—104no—
122C2×D20—51no—
123C2×D20—52no—
124C2×D20—52no—
125C2×D20—52no—
126C2×D20—104no—
127C2×C4×D5—51no—
128C2×C4×D5—52no—
129C2×C4×D5—52no—
130C2×C4×D5—52no—
131C2×C4×D5—52no—
132C2×C4×D5—51no—
133C2×C4×D5—51no—
134C2×C4×D5—104no—
135C2×C4×D5—104no—
136C2×Dic10—51no—
137C2×Dic10—51no—
138C5×C4⋊C4—52no—
139C5×C4⋊C4—52no—
140C5×C22⋊C4—22no—
141C5×C22⋊C4—51no—
142C5×C22⋊C4—52no—
143C5×C22⋊C4C2×D511yescharacteristic
144C5×C22⋊C4C2×D511yescharacteristic
145C42×C5—52no—
146C22⋊Dic5C2×D511yescharacteristic
147#685—52no—
148#685—52no—
149#685—52no—
150#685—52no—
151#685—52no—
152#685—52no—
153#685—104no—
154#685C2×D512yescomplemented
155C4⋊Dic5—52no—
156C4⋊Dic5C2×D512yescomplemented
157#683—52no—
158#683C2×D512yescomplemented
159C4×Dic5C2×D512yescomplemented
160#525—251noSylow
161C2×C52—22no—
162C2×C52C22×D411yescharacteristic
163C2×C52C2×#4112yes—
164C5⋊D5—24no—
165C5⋊D5—41no—
166C5×D5—41no—
167C23×C5—22no—
168C23×C5—51no—
169C23×C5C22×D511yescharacteristic
170C23×C5C22×D511yescharacteristic
171C22×D5—52no—
172C22×D5—52no—
173C22×D5—51no—
174C22×D5—51no—
175C22×D5—104no—
176C22×D5—102no—
177C22×D5—104no—
178C22×D5—104no—
179C22×D5—102no—
180C22×D5—102no—
181C22×D5—208no—
182C22×D5—204no—
183C22×D5C22×D511yescharacteristic
184C5×D4—102no—
185C2×C4×C5—24no—
186C2×C4×C5—54no—
187C2×C4×C5—51no—
188C2×C4×C5—51no—
189C2×C4×C5—52no—
190C2×C4×C5—52no—
191C2×C4×C5—51no—
192C2×C4×C5—102no—
193C2×C4×C5C22×D512yes—
194C2×C4×C5C22×D512yes—
195C22⋊D5—22no—
196C22⋊D5—104no—
197C22⋊D5—102no—
198C22⋊D5—102no—
199C22⋊D5—102no—
200C22⋊D5—204no—
201C2×Dic5—22no—
202C2×Dic5—51no—
203C2×Dic5—51no—
204C2×Dic5—102no—
205C2×Dic5—102no—
206C2×Dic5D2014yescomplemented
207C2×Dic5C22×D512yes—
208C2×Dic5C22×D512yes—
209C2×Dic5C22×D511yescharacteristic
210C2×Dic5C22×D511yescharacteristic
211C2×Dic5C22×D511yescharacteristic
212D20—104no—
213D20—104no—
214D20—102no—
215D20—104no—
216D20—208no—
217C4×D5—104no—
218C4×D5—104no—
219C4×D5—102no—
220C4×D5—102no—
221C4×D5—104no—
222C4×D5—104no—
223C4×D5—102no—
224Dic10—102no—
225Dic10—102no—
226C2×#41—251no—
227C23×C4—251no—
228#123—51no—
229#123—251no—
230C22⋊Q8—252no—
231#121—252no—
232#120—251no—
233#120—251no—
234C4×D4—252no—
235C4×D4—252no—
236#117—51no—
237C52#52511yescharacteristic, complemented, Sylow
238C22×C5—22no—
239C22×C5—22no—
240C22×C5—24no—
241C22×C5—22no—
242C22×C5—44no—
243C22×C5—101no—
244C22×C5—102no—
245C22×C5C2×D2012yes—
246C22×C5D4×D512yes—
247C22×C5C23×D511yescharacteristic
248C22×C5C23×D511yescharacteristic
249C2×D5—21no—
250C2×D5—22no—
251C2×D5—54no—
252C2×D5—54no—
253C2×D5—108no—
254C2×D5—108no—
255C2×D5—108no—
256C2×D5—102no—
257C2×D5—102no—
258C2×D5—102no—
259C2×D5—102no—
260C2×D5—101no—
261C2×D5—101no—
262C2×D5—102no—
263C2×D5—102no—
264C2×D5—204no—
265C2×D5—204no—
266C2×D5—2016no—
267C2×D5—204no—
268C2×D5—202no—
269C4×C5—22no—
270C4×C5—22no—
271C4×C5—44no—
272C4×C5—52no—
273C4×C5—52no—
274C4×C5—101no—
275C4×C5—102no—
276C4×C5—102no—
277Dic5—21no—
278Dic5—22no—
279Dic5—22no—
280Dic5—101no—
281Dic5—101no—
282Dic5—202no—
283Dic5C2×D2012yes—
284Dic5C2×D2012yes—
285C24—251no—
286#41—504no—
287C2×Q8—251no—
288C2×D4—51no—
289C2×D4—252no—
290C2×D4—251no—
291C2×D4—251no—
292C2×D4—252no—
293C22×C4—51no—
294C22×C4—51no—
295C22×C4—252no—
296C22×C4—251no—
297C22×C4—252no—
298C22×C4—251no—
299C22×C4—504no—
300C4⋊C4—52no—
301C4⋊C4—52no—
302C4⋊C4—252no—
303C22⋊C4—51no—
304C22⋊C4—52no—
305C22⋊C4—252no—
306C22⋊C4—252no—
307C22⋊C4—252no—
308C22⋊C4D5211yescharacteristic
309C42—52no—
310C2×C5—22no—
311C2×C5—22no—
312C2×C5—22no—
313C2×C5—24no—
314C2×C5—44no—
315C2×C5—201no—
316C2×C5C22×D2011yescharacteristic
317C2×C5C2×D4×D511yescharacteristic
318C2×C5D5×#4112yes—
319C2×C5D5×#4112yes—
320D5—41no—
321D5—104no—
322D5—104no—
323D5—208no—
324D5—201no—
325D5—201no—
326D5—402no—
327C23—51no—
328C23—252no—
329C23—251no—
330C23—504no—
331C23—502no—
332C23C2×D5211yescharacteristic
333Q8—502no—
334D4—102no—
335D4—504no—
336D4—502no—
337D4—504no—
338D4—502no—
339C2×C4—54no—
340C2×C4—51no—
341C2×C4—51no—
342C2×C4—52no—
343C2×C4—52no—
344C2×C4—51no—
345C2×C4—102no—
346C2×C4—251no—
347C2×C4—504no—
348C2×C4—504no—
349C2×C4—502no—
350C2×C4—502no—
351C2×C4C2×D5212yes—
352C5—22no—
353C5#6585811yescharacteristic, complemented
354C5#6599011yescharacteristic, complemented
355C22—22no—
356C22—101no—
357C22—102no—
358C22—254no—
359C22—508no—
360C22—502no—
361C22—502no—
362C22—501no—
363C22—502no—
364C22D5×D2012yes—
365C22C22×D5211yescharacteristic
366C4—22no—
367C4—52no—
368C4—52no—
369C4—102no—
370C4—101no—
371C4—102no—
372C4—501no—
373C2—22no—
374C2—201no—
375C2—504no—
376C2—1001no—
377C2C2×D5×D2011yescharacteristic
378C2#1171747712yes—
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. C22×#6036C2semidirect
2. C2×#87879C2semidirect
3. #11717202C2semidirect
4. C5×#4032C2semidirect
5. C5×#4023C2semidirect
6. #11717099C2semidirect
7. #11717084C2semidirect
8. #11717028C2semidirect
9. #11717013C2semidirect
10. #11717009C2semidirect
11. #11716998C2semidirect
18. C2×#6039C22semidirect
20. C2×#6038C22semidirect
24. C2×#6036C22semidirect
26. C2×D5×Dic5C22semidirect
30. C5×#685C22semidirect
31. C5×C4⋊Dic5C22semidirect
32. C5×#683C22semidirect
33. C4×C5×Dic5C22semidirect
34. #87780C22semidirect
36. #87779C22semidirect
51. C2×C5×Dic5D4semidirect
77. #4032D5semidirect
84. #4023D5semidirect
95. C2×C5⋊D5#41semidirect
154. #685C2×D5semidirect
156. C4⋊Dic5C2×D5semidirect
158. #683C2×D5semidirect
159. C4×Dic5C2×D5semidirect
206. C2×Dic5D20semidirect
237. C52#525semidirect
353. C5#65858semidirect
354. C5#65990semidirect

Maximal subgroups

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

subgroupindex
C22×#60362
C2×#878792
#117172022
C5×#40322
C5×#40232
#117170992
#117170842
#117170282
#117170132
#117170092
#117169982
#659905
#658585