Finite Group Databasebeta

← #12009476

Subgroups — #12009476

Order 1280.

552 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2392C211yesmaximal, characteristic
2#10602385C211yesmaximal, characteristic, complemented
3#10602364C211yesmaximal, characteristic, complemented
4#10598140C212yesmaximal, complemented
5#10598136C212yesmaximal
6C2×C4×C5×D4—21no—
7C2×C4×C5×D4C411yescharacteristic, complemented
8C2×C5×#117C2211yescharacteristic
9C2×#4072—21no—
10C5×#398C411yescharacteristic, complemented
11C5×#397C2211yescharacteristic
12C5×#396—21no—
13C5×#394—21no—
14C5×#394C411yescharacteristic, complemented
15C5×#392C2211yescharacteristic
16C4×C5×C4⋊C4C411yescharacteristic, complemented
17C4×C5×C22⋊C4—21no—
18#65214—21no—
19#64748—21no—
20#64746—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11691—51nomaximal, Sylow
24C22×C5×D4—21no—
25C22×C5×D4C2×C411yescharacteristic
26C23×C4×C5—21no—
27C23×C4×C5D411yescharacteristic
28C4×C5×D4—41no—
29C4×C5×D4—41no—
30C4×C5×D4—42no—
31C4×C5×D4—42no—
32C5×#117—41no—
33C5×#117—41no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4—21no—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C4⋊C4C2×C411yescharacteristic
38C2×C5×C4⋊C4D411yescharacteristic
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4—21no—
43C2×C5×C22⋊C4—21no—
44C2×C5×C22⋊C4D411yescharacteristic
45C2×C5×C22⋊C4C2311yescharacteristic
46C2×C42×C5—21no—
47C2×C42×C5C2×C411yescharacteristic
48C2×C42×C5D411yescharacteristic
49C2×C22⋊Dic5—22no—
50#4072—24no—
51#4072—42no—
52C2×C4⋊Dic5—41no—
53C2×C4×Dic5—41no—
54C5×#95—21no—
55C5×#95—21no—
56C5×#95C2×C411yescharacteristic
57C5×#95C2×C411yescharacteristic
58C22⋊#168—22no—
59C4⋊#168—41no—
60C4×#168—41no—
61#2392D511yescharacteristic
62#1740—51no—
63#1458—51no—
64#1344—52no—
65#1343—52no—
66C24×C5—21no—
67C24×C5C2×D411yescharacteristic
68C2×C5×D4—41no—
69C2×C5×D4—42no—
70C2×C5×D4—42no—
71C2×C5×D4—41no—
72C2×C5×D4—41no—
73C2×C5×D4—42no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—22no—
77C22×C4×C5—21no—
78C22×C4×C5—21no—
79C22×C4×C5—21no—
80C22×C4×C5—41no—
81C22×C4×C5—41no—
82C22×C4×C5—42no—
83C22×C4×C5—42no—
84C22×C4×C5C22⋊C411yescharacteristic
85C22×C4×C5C22⋊C411yescharacteristic
86C22×C4×C5C22⋊C411yescharacteristic
87C22×C4×C5C22⋊C411yescharacteristic
88C22×C4×C5D812yes—
89C22×C4×C5SD1612yes—
90C22×C4×C5C22×C411yescharacteristic
91C22×C4×C5C2×D411yescharacteristic
92C22×Dic5—42no—
93C5×C4⋊C4—41no—
94C5×C4⋊C4—41no—
95C5×C4⋊C4—41no—
96C5×C4⋊C4—42no—
97C5×C4⋊C4—42no—
98C5×C22⋊C4—41no—
99C5×C22⋊C4—41no—
100C5×C22⋊C4—42no—
101C5×C22⋊C4—42no—
102C5×C22⋊C4—42no—
103C5×C22⋊C4—41no—
104C5×C22⋊C4—41no—
105C5×C22⋊C4—42no—
106C42×C5—21no—
107C42×C5—21no—
108C42×C5—21no—
109C42×C5—21no—
110C42×C5—42no—
111C22⋊Dic5—24no—
112C22⋊Dic5—42no—
113C4⋊Dic5—44no—
114C4×Dic5—44no—
115C2×#168—42no—
116C2×C4×D4—21no—
117C2×C4×D4Dic511yescharacteristic, complemented
118C2×#117—101no—
119C2×#117C2×D511yescharacteristic
120#431—101no—
121#398Dic511yescharacteristic, complemented
122#397C2×D511yescharacteristic
123#396—21no—
124#394—21no—
125#394Dic511yescharacteristic, complemented
126#392C2×D511yescharacteristic
127C4×C4⋊C4Dic511yescharacteristic, complemented
128C4×C22⋊C4—21no—
129#342—52no—
130#342—101no—
131#324—101no—
132#323—52no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—22no—
136C23×C5—41no—
137C23×C5—41no—
138C23×C5—41no—
139C23×C5—41no—
140C23×C5—41no—
141C23×C5—41no—
142C23×C5—41no—
143C23×C5—41no—
144C23×C5C2×C22⋊C411yescharacteristic
145C23×C5C2×D811yescharacteristic
146C23×C5C2×SD1611yescharacteristic
147C5×D4—42no—
148C5×D4—44no—
149C5×D4—44no—
150C5×D4—42no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—22no—
154C2×C4×C5—22no—
155C2×C4×C5—22no—
156C2×C4×C5—22no—
157C2×C4×C5—22no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—41no—
165C2×C4×C5—41no—
166C2×C4×C5—41no—
167C2×C4×C5—41no—
168C2×C4×C5—41no—
169C2×C4×C5—41no—
170C2×C4×C5—41no—
171C2×C4×C5—41no—
172C2×C4×C5—41no—
173C2×C4×C5—41no—
174C2×C4×C5—82no—
175C2×C4×C5—82no—
176C2×C4×C5#9911yescharacteristic
177C2×C4×C5#9911yescharacteristic
178C2×C4×C5#10212yes—
179C2×C4×C5#10212yes—
180C2×C4×C5#10412yes—
181C2×Dic5—44no—
182C2×Dic5—82no—
183C2×Dic5—82no—
184#168—82no—
185C22×D4—21no—
186C22×D4C2×Dic511yescharacteristic
187C23×C4—21no—
188C23×C4C22⋊D511yescharacteristic
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—42no—
192C4×D4—42no—
193#117—41no—
194#117—41no—
195#117—104no—
196#117—202no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—201no—
200C2×C4⋊C4C2×Dic511yescharacteristic
201C2×C4⋊C4C2×Dic511yescharacteristic
202C2×C4⋊C4C22⋊D511yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—102no—
209C2×C22⋊C4C22⋊D511yescharacteristic
210C2×C22⋊C4C22×D511yescharacteristic
211C2×C42—21no—
212C2×C42—201no—
213C2×C42C2×Dic511yescharacteristic
214C2×C42C22⋊D511yescharacteristic
215C4⋊C8—201no—
216C22⋊C8—102no—
217C4×C8—201no—
218#95—21no—
219#95—21no—
220#95C2×Dic511yescharacteristic
221#95C2×Dic511yescharacteristic
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5—21no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—42no—
229C22×C5—42no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5—42no—
233C22×C5—81no—
234C22×C5—81no—
235C22×C5—81no—
236C22×C5—81no—
237C22×C5—81no—
238C22×C5C2×#9911yescharacteristic
239C22×C5C2×#10211yescharacteristic
240C22×C5C2×#10411yescharacteristic
241C4×C5—22no—
242C4×C5—42no—
243C4×C5—41no—
244C4×C5—41no—
245C4×C5—41no—
246C4×C5—41no—
247C4×C5—81no—
248C4×C5—81no—
249C4×C5#32712yes—
250C4×C5#32712yes—
251Dic5—84no—
252C24—21no—
253C24C2×C22⋊D511yescharacteristic
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—202no—
271C22×C4#68612yes—
272C22×C4#68712yes—
273C22×C4C22⋊Dic511yescharacteristic
274C22×C4C22⋊Dic511yescharacteristic
275C22×C4C22⋊Dic511yescharacteristic
276C22×C4C22⋊Dic511yescharacteristic
277C22×C4C22×Dic511yescharacteristic
278C22×C4C2×C22⋊D511yescharacteristic
279C2×C8—202no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—204no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—42no—
294C22⋊C4—104no—
295C22⋊C4—202no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—42no—
301C42—204no—
302C2×C5—21no—
303C2×C5—21no—
304C2×C5—41no—
305C2×C5—41no—
306C2×C5—82no—
307C2×C5C2×#32711yescharacteristic
308C2×C5#212011yescharacteristic
309C2×C5#212611yescharacteristic
310C23—21no—
311C23—21no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C2×#68611yescharacteristic
322C23C2×#68711yescharacteristic
323C23C2×C22⋊Dic511yescharacteristic
324D4—42no—
325D4—44no—
326D4—44no—
327D4—42no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—204no—
354C2×C4—402no—
355C2×C4—402no—
356C2×C4#396412yes—
357C2×C4#396412yes—
358C2×C4#396611yescharacteristic
359C2×C4#396611yescharacteristic
360C2×C4#396912yes—
361C8—402no—
362C5#1169111yescharacteristic, complemented, Sylow
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#396411yescharacteristic
380C22C2×#396611yescharacteristic
381C22C2×#396911yescharacteristic
382C4—22no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—404no—
391C4#6475712yes—
392C4#6475712yes—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C2×#6475711yescharacteristic
399C2#1060250911yescharacteristic
400C2#1060261211yescharacteristic
subgroupG/H
centerC2×C4#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2392C2
socleC22×C5C2×#99
prime radicalC5×#392C22
cosocleC2×C5×C22⋊C4C23
hypercenter#2392D5
hypocenterC5#11691
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
2. #10602385C2semidirect
3. #10602364C2semidirect
4. #10598140C2semidirect
7. C2×C4×C5×D4C4semidirect
10. C5×#398C4semidirect
14. C5×#394C4semidirect
16. C4×C5×C4⋊C4C4semidirect
117. C2×C4×D4Dic5semidirect
121. #398Dic5semidirect
125. #394Dic5semidirect
127. C4×C4⋊C4Dic5semidirect
362. C5#11691semidirect

Maximal subgroups

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

subgroupindex
C5×#23922
#106023852
#106023642
#105981402
#105981362
#116915