Finite Group Databasebeta

← #11998703

Subgroups — #11998703

Order 1280.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3167C211yesmaximal, characteristic
2#10602629C211yesmaximal, characteristic, complemented
3#10602615C211yesmaximal, characteristic, complemented
4#10602605C211yesmaximal, characteristic
5#10602591C211yesmaximal, characteristic
6#10600542C211yesmaximal, characteristic, complemented
7#10600525C211yesmaximal, characteristic, complemented
8C5×#533C2211yescharacteristic
9C5×#532C2211yescharacteristic
10C5×#498C411yescharacteristic, complemented
11C5×#492C411yescharacteristic, complemented
12C5×#487C411yescharacteristic, complemented
13C5×#486C411yescharacteristic, complemented
14C5×#465—21no—
15C5×#465—21no—
16C4×C5×SD16—21no—
17C4×C5×SD16—21no—
18C2×C4×C5×C8C2211yescharacteristic
19C2×#3947—21no—
20C2×C8×Dic5—21no—
21#65221C2211yescharacteristic
22#65215C2211yescharacteristic
23#65215C2211yescharacteristic
24C2×#3935C2211yescharacteristic
25#64763—22no—
26#64760—22no—
27#64758—22no—
28#64757—22no—
29#64678—21no—
30#64677—21no—
31#9761—51nomaximal, Sylow
32C2×C5×#41—21no—
33C2×C5×SD16—21no—
34C2×C5×SD16—21no—
35C22×C5×C8D411yescharacteristic
36C22×C5×C8D411yescharacteristic
37C5×C4⋊Q8C2×C411yescharacteristic
38C5×C4⋊Q8C2×C411yescharacteristic
39C5×C4⋊Q8C2×C411yescharacteristic
40C5×#127C2×C411yescharacteristic
41C5×#125—21no—
42C5×#124—21no—
43C5×C22⋊Q8—21no—
44C5×C22⋊Q8—21no—
45C5×#121—21no—
46C5×#121—21no—
47C4×C5×Q8—21no—
48C4×C5×Q8—21no—
49C4×C5×D4—21no—
50C4×C5×D4—21no—
51C5×#117—21no—
52C2×C42×C5C2311yescharacteristic
53C2×C4×Dic5—21no—
54C22×#168—21no—
55C5×#106—21no—
56C5×#106—21no—
57C5×#103—21no—
58C5×#103—21no—
59C5×#102—21no—
60C5×#102—21no—
61C4×C5×C8C2×C411yescharacteristic
62C4×C5×C8C2×C411yescharacteristic
63C4×C5×C8D411yescharacteristic
64C4×C5×C8D411yescharacteristic
65#3963—21no—
66#3963D412yes—
67C22⋊#168—21no—
68C22⋊#168—21no—
69#3947—24no—
70C8×Dic5—24no—
71C4⋊#168—21no—
72C4⋊#168—21no—
73#3935—21no—
74#3935—21no—
75#3935—21no—
76#3935D412yes—
77#3167D511yescharacteristic
78#1726—51no—
79#1568—51no—
80#1531—51no—
81#1530—51no—
82#1506—51no—
83#1505—51no—
84C5×#41—82no—
85C2×C5×Q8—21no—
86C2×C5×Q8—21no—
87C2×C5×Q8—21no—
88C2×C5×D4—21no—
89C2×C5×D4—21no—
90C2×C5×D4—21no—
91C2×C5×D4—41no—
92C22×C4×C5—41no—
93C22×C4×C5C2×D411yescharacteristic
94C22×C4×C5C2×D411yescharacteristic
95C22×C4×C5#4111yescharacteristic
96C22×Dic5—22no—
97C5×SD16—42no—
98C5×SD16—42no—
99C2×C5×C8—22no—
100C2×C5×C8—22no—
101C2×C5×C8C22⋊C411yescharacteristic
102C2×C5×C8C22⋊C411yescharacteristic
103C2×C5×C8C22⋊C411yescharacteristic
104C2×C5×C8C22⋊C411yescharacteristic
105C5×C4⋊C4—21no—
106C5×C4⋊C4—21no—
107C5×C4⋊C4—21no—
108C5×C4⋊C4—21no—
109C5×C4⋊C4—21no—
110C5×C4⋊C4—21no—
111C5×C4⋊C4—41no—
112C5×C4⋊C4—41no—
113C5×C22⋊C4—41no—
114C5×C22⋊C4—41no—
115C5×C22⋊C4—41no—
116C42×C5—41no—
117C42×C5C22×C411yescharacteristic
118C42×C5C2×D411yescharacteristic
119C42×C5C2×D411yescharacteristic
120C42×C5#4111yescharacteristic
121C4×Dic5—22no—
122C4×Dic5—22no—
123C2×#168—22no—
124C2×#168—22no—
125C2×#168—22no—
126C2×#168—41no—
127C2×#168—41no—
128#533C2×D511yescharacteristic
129#532C2×D511yescharacteristic
130#498Dic511yescharacteristic, complemented
131#492Dic511yescharacteristic, complemented
132#487Dic511yescharacteristic, complemented
133#486Dic511yescharacteristic, complemented
134#465—21no—
135#465—21no—
136C4×SD16—21no—
137C4×SD16—21no—
138#432—51no—
139#432—51no—
140C2×#97—51no—
141C2×#97—101no—
142C2×C4×C8—101no—
143C2×C4×C8C2×D511yescharacteristic
144#376—51no—
145#335—101no—
146#334—101no—
147#330—102no—
148#329—102no—
149#328—102no—
150#327—102no—
151C23×C5—41no—
152C23×C5#12111yescharacteristic
153C5×Q8—41no—
154C5×Q8—41no—
155C5×Q8—81no—
156C5×Q8—81no—
157C5×D4—41no—
158C5×D4—41no—
159C5×D4—81no—
160C5×D4—81no—
161C5×D4—82no—
162C2×C4×C5—22no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—41no—
166C2×C4×C5—41no—
167C2×C4×C5—41no—
168C2×C4×C5—41no—
169C2×C4×C5—82no—
170C2×C4×C5C2×C22⋊C411yescharacteristic
171C2×C4×C5C4×D411yescharacteristic
172C2×C4×C5C4×D411yescharacteristic
173C2×C4×C5#12111yescharacteristic
174C2×C4×C5#12411yescharacteristic
175C2×C4×C5#12711yescharacteristic
176C2×Dic5—24no—
177C2×Dic5—42no—
178C2×Dic5—42no—
179C5×C8—21no—
180C5×C8—21no—
181C5×C8#10412yes—
182C5×C8#10412yes—
183#168—24no—
184#168—81no—
185#168—81no—
186C2×#41—21no—
187C2×SD16—21no—
188C2×SD16—21no—
189C22×C8—101no—
190C22×C8C22⋊D511yescharacteristic
191C22×C8C22⋊D511yescharacteristic
192C4⋊Q8C2×Dic511yescharacteristic
193C4⋊Q8C2×Dic511yescharacteristic
194C4⋊Q8C2×Dic511yescharacteristic
195#127C2×Dic511yescharacteristic
196#125—21no—
197#124—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200#121—21no—
201#121—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×D4—21no—
205C4×D4—21no—
206#117—21no—
207C2×C42—101no—
208C2×C42C22×D511yescharacteristic
209#106—21no—
210#106—21no—
211C4⋊C8—101no—
212C4⋊C8—101no—
213#103—21no—
214#103—21no—
215#102—21no—
216#102—21no—
217C22⋊C8—101no—
218C22⋊C8—101no—
219#97—52no—
220#97—104no—
221#97—101no—
222#97—101no—
223#97—101no—
224C4×C8—104no—
225C4×C8C2×Dic511yescharacteristic
226C4×C8C2×Dic511yescharacteristic
227C4×C8C22⋊D511yescharacteristic
228C4×C8C22⋊D511yescharacteristic
229#95—52no—
230#95—101no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—81no—
234C22×C5—81no—
235C22×C5—81no—
236C22×C5#39011yescharacteristic
237C22×C5#47112yes—
238C4×C5—21no—
239C4×C5—21no—
240C4×C5—21no—
241C4×C5—21no—
242C4×C5—21no—
243C4×C5—81no—
244C4×C5—81no—
245C4×C5—81no—
246C4×C5C2×#10411yescharacteristic
247C4×C5C2×#10411yescharacteristic
248Dic5—44no—
249#41—82no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C22×C4—41no—
258C22×C4—102no—
259C22×C4D4×D511yescharacteristic
260C22×C4#71111yescharacteristic
261C22×C4C2×C22⋊D511yescharacteristic
262SD16—42no—
263SD16—42no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—102no—
267C2×C8—102no—
268C2×C8—102no—
269C2×C8—201no—
270C2×C8—201no—
271C2×C8C22⋊Dic511yescharacteristic
272C2×C8C22⋊Dic511yescharacteristic
273C2×C8C22⋊Dic511yescharacteristic
274C2×C8C22⋊Dic511yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C42—41no—
287C42—102no—
288C42—102no—
289C42D4×D511yescharacteristic
290C42#71111yescharacteristic
291C42C22×Dic511yescharacteristic
292C42C2×C22⋊D511yescharacteristic
293C2×C5—22no—
294C2×C5—161no—
295C2×C5#196211yescharacteristic
296C2×C5#197611yescharacteristic
297C2×C5#197711yescharacteristic
298C23—41no—
299C23#408411yescharacteristic
300Q8—41no—
301Q8—41no—
302Q8—81no—
303Q8—81no—
304D4—41no—
305D4—41no—
306D4—81no—
307D4—81no—
308D4—82no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—104no—
318C2×C4—202no—
319C2×C4—202no—
320C2×C4D4×Dic511yescharacteristic
321C2×C4D4×Dic511yescharacteristic
322C2×C4#408211yescharacteristic
323C2×C4#408411yescharacteristic
324C2×C4#408611yescharacteristic
325C2×C4C2×C22⋊Dic511yescharacteristic
326C8—21no—
327C8—21no—
328C8—104no—
329C8—401no—
330C8—401no—
331C8#396912yes—
332C8#396912yes—
333C5#976111yescharacteristic, complemented, Sylow
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#6546612yes—
340C22#6551011yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—204no—
350C4C2×#396911yescharacteristic
351C4C2×#396911yescharacteristic
352C2—22no—
353C2—161no—
354C2#1060260811yescharacteristic
355C2#1060284511yescharacteristic
356C2#1060287911yescharacteristic
subgroupG/H
centerC22#65510
commutatorC42×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#3167C2
socleC22×C5#390
prime radicalC5×#532C22
cosocleC2×C42×C5C23
hypercenter#3167D5
hypocenterC5#9761
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
2. #10602629C2semidirect
3. #10602615C2semidirect
6. #10600542C2semidirect
7. #10600525C2semidirect
10. C5×#498C4semidirect
11. C5×#492C4semidirect
12. C5×#487C4semidirect
13. C5×#486C4semidirect
130. #498Dic5semidirect
131. #492Dic5semidirect
132. #487Dic5semidirect
133. #486Dic5semidirect
333. C5#9761semidirect

Maximal subgroups

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

subgroupindex
C5×#31672
#106026292
#106026152
#106026052
#106025912
#106005422
#106005252
#97615