Finite Group Databasebeta

← #11991479

Subgroups — #11991479

Order 1280.

718 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C5×#3493C211yesmaximal, characteristic
2#10602503C212yesmaximal, complemented
3#10602503C212yesmaximal, complemented
4#10602387C211yesmaximal, characteristic, complemented
5#10602377C211yesmaximal, characteristic, complemented
6C5×#556—22no—
7C5×#555—22no—
8C5×#553—22no—
9C5×#552—22no—
10C5×#544C411yescharacteristic, complemented
11C5×#538—22no—
12C5×#537C411yescharacteristic, complemented
13C5×#529—22no—
14C5×#525C2212yes—
15C5×#520C412yescomplemented
16C2×C5×#117C2211yescharacteristic
17#65500C2212yes—
18C2×C22⋊#168C2211yescharacteristic
19C2×C22⋊#168C2211yescharacteristic
20#64758—22no—
21#64758—22no—
22#64757—22no—
23#64757—22no—
24#64746—22no—
25#64744—22no—
26#8463—51nomaximal, Sylow
27C2×C5×#41C2×C412yes—
28C23×C4×C5C2311yescharacteristic
29C5×C4⋊Q8—22no—
30C5×#126—24no—
31C5×#126—24no—
32C5×#125—22no—
33C5×#125—22no—
34C5×#124—22no—
35C5×#124—24no—
36C5×#123—24no—
37C5×#123—22no—
38C5×C22⋊Q8—22no—
39C5×C22⋊Q8—22no—
40C5×C22⋊Q8D412yes—
41C5×C22⋊Q8D412yes—
42C5×#121—24no—
43C5×#121—22no—
44C5×#121—22no—
45C5×#121D412yes—
46C5×#121D412yes—
47C5×#120—22no—
48C4×C5×Q8—22no—
49C4×C5×Q8—22no—
50C4×C5×D4—22no—
51C4×C5×D4—22no—
52C4×C5×D4—24no—
53C4×C5×D4—22no—
54C4×C5×D4—22no—
55C5×#117—24no—
56C5×#117—22no—
57C5×#117—22no—
58C5×#117—22no—
59C5×#117C2×C412yes—
60C2×C5×C4⋊C4—22no—
61C2×C5×C4⋊C4C2×C411yescharacteristic
62C2×C5×C4⋊C4C2×C411yescharacteristic
63C2×C5×C22⋊C4D412yes—
64C2×C42×C5D412yes—
65C2×C22⋊Dic5—22no—
66C22×#168—21no—
67C22×#168—21no—
68#3963—22no—
69#3963—22no—
70C22⋊#168—22no—
71C22⋊#168—22no—
72#3493D511yescharacteristic
73#1820—51no—
74#1818—51no—
75#1516—52no—
76#1516—52no—
77C24×C5C2×D411yescharacteristic
78C5×#41—44no—
79C2×C5×Q8—22no—
80C2×C5×Q8C22⋊C412yes—
81C2×C5×D4—22no—
82C2×C5×D4—22no—
83C2×C5×D4—22no—
84C2×C5×D4—22no—
85C2×C5×D4C22⋊C412yes—
86C22×C4×C5—24no—
87C22×C4×C5—22no—
88C22×C4×C5—42no—
89C22×C4×C5C22⋊C412yes—
90C22×C4×C5C22⋊C412yes—
91C22×C4×C5C22×C411yescharacteristic
92C22×C4×C5C2×D411yescharacteristic
93C22×C4×C5C2×D412yes—
94C22×C4×C5C2×D412yes—
95C22×Dic5—42no—
96C5×C4⋊C4—24no—
97C5×C4⋊C4—24no—
98C5×C4⋊C4—22no—
99C5×C4⋊C4—22no—
100C5×C4⋊C4—22no—
101C5×C4⋊C4—22no—
102C5×C4⋊C4—22no—
103C5×C4⋊C4—22no—
104C5×C4⋊C4C22⋊C412yes—
105C5×C4⋊C4C22⋊C412yes—
106C5×C22⋊C4—24no—
107C5×C22⋊C4—24no—
108C5×C22⋊C4—22no—
109C5×C22⋊C4—24no—
110C5×C22⋊C4—22no—
111C5×C22⋊C4—22no—
112C5×C22⋊C4—22no—
113C5×C22⋊C4—22no—
114C42×C5—24no—
115C42×C5—22no—
116C42×C5—22no—
117C42×C5—22no—
118C22⋊Dic5—44no—
119C2×#168—42no—
120C2×#168—42no—
121C2×#168—41no—
122C2×#168—41no—
123#556—22no—
124#555—22no—
125#553—22no—
126#552—22no—
127#544Dic511yescharacteristic, complemented
128#538—22no—
129#537Dic511yescharacteristic, complemented
130#529—22no—
131#525C2×D512yes—
132#520Dic512yescomplemented
133C2×#117C2×D511yescharacteristic
134C2×C22⋊C8—51no—
135C2×C22⋊C8—51no—
136#381—52no—
137#342—102no—
138#336—102no—
139#328—102no—
140#328—102no—
141#327—102no—
142#327—102no—
143C23×C5—22no—
144C23×C5—22no—
145C23×C5—41no—
146C23×C5—41no—
147C23×C5C2×C22⋊C411yescharacteristic
148C23×C5#12012yes—
149C5×Q8—42no—
150C5×Q8—82no—
151C5×D4—42no—
152C5×D4—44no—
153C5×D4—82no—
154C5×D4—82no—
155C5×D4—82no—
156C2×C4×C5—22no—
157C2×C4×C5—22no—
158C2×C4×C5—22no—
159C2×C4×C5—22no—
160C2×C4×C5—22no—
161C2×C4×C5—22no—
162C2×C4×C5—22no—
163C2×C4×C5—42no—
164C2×C4×C5—42no—
165C2×C4×C5—44no—
166C2×C4×C5—42no—
167C2×C4×C5—42no—
168C2×C4×C5—84no—
169C2×C4×C5C2×C22⋊C412yes—
170C2×C4×C5#12011yescharacteristic
171C2×C4×C5#12011yescharacteristic
172C2×Dic5—82no—
173C2×Dic5—82no—
174C2×Dic5—82no—
175#168—81no—
176#168—81no—
177C2×#41C2×Dic512yes—
178C23×C4C22×D511yescharacteristic
179C22×C8—101no—
180C22×C8—101no—
181C4⋊Q8—22no—
182#126—24no—
183#126—24no—
184#125—22no—
185#125—22no—
186#124—22no—
187#124—24no—
188#123—24no—
189#123—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8C22⋊D512yes—
193C22⋊Q8C22⋊D512yes—
194#121—24no—
195#121—22no—
196#121—22no—
197#121C22⋊D512yes—
198#121C22⋊D512yes—
199#120—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—22no—
207#117—24no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117C2×Dic512yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C2×Dic511yescharacteristic
214C2×C4⋊C4C2×Dic511yescharacteristic
215C2×C22⋊C4—102no—
216C2×C22⋊C4C22⋊D512yes—
217C2×C42C22⋊D512yes—
218C22⋊C8—102no—
219C22⋊C8—102no—
220#95—102no—
221#95—102no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—22no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—42no—
228C22×C5—42no—
229C22×C5—42no—
230C22×C5—82no—
231C22×C5#37911yescharacteristic
232C22×C5#41012yes—
233C4×C5—21no—
234C4×C5—21no—
235C4×C5—22no—
236C4×C5—42no—
237C4×C5—42no—
238C4×C5—42no—
239C4×C5—42no—
240C4×C5—42no—
241Dic5—162no—
242C24C2×C22⋊D511yescharacteristic
243#41—44no—
244C2×Q8—22no—
245C2×Q8C22⋊Dic512yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4C22⋊Dic512yes—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—202no—
255C22×C4C22⋊Dic512yes—
256C22×C4C22⋊Dic512yes—
257C22×C4C22×Dic511yescharacteristic
258C22×C4C2×C22⋊D511yescharacteristic
259C22×C4C2×C22⋊D512yes—
260C22×C4C2×C22⋊D512yes—
261C2×C8—202no—
262C2×C8—202no—
263C2×C8—201no—
264C2×C8—201no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4C22⋊Dic512yes—
274C4⋊C4C22⋊Dic512yes—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—22no—
278C22⋊C4—24no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—204no—
284C42—24no—
285C42—22no—
286C42—22no—
287C42—22no—
288C2×C5—22no—
289C2×C5—42no—
290C2×C5—82no—
291C2×C5#178211yescharacteristic
292C2×C5#179411yescharacteristic
293C2×C5#179411yescharacteristic
294C23—22no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23C2×C22⋊Dic511yescharacteristic
299C23#409912yes—
300Q8—42no—
301Q8—82no—
302D4—42no—
303D4—44no—
304D4—82no—
305D4—82no—
306D4—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—84no—
320C2×C4—402no—
321C2×C4—402no—
322C2×C4—402no—
323C2×C4C2×C22⋊Dic512yes—
324C2×C4#409911yescharacteristic
325C2×C4#409911yescharacteristic
326C8—401no—
327C8—401no—
328C5#846311yescharacteristic, complemented, Sylow
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—82no—
338C22#6552912yes—
339C22#6553611yescharacteristic
340C4—21no—
341C4—21no—
342C4—22no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—802no—
349C2—22no—
350C2—42no—
351C2—82no—
352C2#1060293211yescharacteristic
353C2#1060293911yescharacteristic
354C2#1060293911yescharacteristic
subgroupG/H
centerC22#65536
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
Fitting, prime radicalC5×#3493C2
socleC22×C5#379
cosocleC23×C4×C5C23
hypercenter#3493D5
hypocenterC5#8463
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3493

Complements

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

NH ≅ G/Ntype
2. #10602503C2semidirect
3. #10602503C2semidirect
4. #10602387C2semidirect
5. #10602377C2semidirect
10. C5×#544C4semidirect
12. C5×#537C4semidirect
15. C5×#520C4semidirect
127. #544Dic5semidirect
129. #537Dic5semidirect
132. #520Dic5semidirect
328. C5#8463semidirect

Maximal subgroups

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

subgroupindex
C5×#34932
#106025032
#106023872
#106023772
#84635