Finite Group Databasebeta

← #10609705

Subgroups — #10609705

Order 640.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic
2C2×#4035C211yesmaximal, characteristic, complemented
3C5×#398C211yesmaximal, characteristic, complemented
4#65279C211yesmaximal, characteristic
5#65278C211yesmaximal, characteristic
6#65276C211yesmaximal, characteristic
7#65275C211yesmaximal, characteristic, complemented
8#65268C211yesmaximal, characteristic, complemented
9#65267C211yesmaximal, characteristic, complemented
10#64964C211yesmaximal, characteristic
11#64964C211yesmaximal, characteristic
12#64958C211yesmaximal, characteristic
13#64947C211yesmaximal, characteristic, complemented
14#64941C211yesmaximal, characteristic, complemented
15#64935C211yesmaximal, characteristic, complemented
16C22×C4×D5C2211yescharacteristic
17C22×Dic10C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×#685C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×C4⋊Dic5C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×C4×Dic5C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37#4042—24no—
38#4035—24no—
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41C5×#95C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2690—51nomaximal, Sylow
54C23×D5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C2×C4×D5—22no—
70C2×C4×D5—22no—
71C2×C4×D5—41no—
72C2×Dic10—22no—
73C2×Dic10—22no—
74C2×Dic10—41no—
75C5×C4⋊C4—22no—
76C5×C4⋊C4—22no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—41no—
79#685—22no—
80#685—22no—
81#685—22no—
82#685—22no—
83#685—22no—
84#685—41no—
85C4⋊Dic5—22no—
86#683—22no—
87#683—22no—
88#683—22no—
89#683—22no—
90#683—22no—
91#683D414yes—
92C4×Dic5—22no—
93C4×Dic5—41no—
94C2×#125—51no—
95C2×C22⋊Q8—51no—
96#400—51no—
97#400—51no—
98#398D511yescharacteristic, complemented
99#397—51no—
100#397—51no—
101#395—51no—
102#391—51no—
103#388—51no—
104#388—51no—
105#387—51no—
106#387—51no—
107#385—51no—
108#380—51no—
109C23×C5C2411yescharacteristic
110C22×D5—21no—
111C22×D5—21no—
112C22×D5—21no—
113C22×D5—21no—
114C22×D5—21no—
115C22×D5—21no—
116C22×D5#4112yes—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5C2×D412yes—
138C2×Dic5—21no—
139C2×Dic5—21no—
140C2×Dic5—21no—
141C2×Dic5—21no—
142C2×Dic5—21no—
143C2×Dic5—21no—
144C2×Dic5—21no—
145C2×Dic5—21no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5C2×D412yes—
157C2×Dic5C2×D412yes—
158C2×Dic5#4112yes—
159C4×D5—44no—
160Dic10—44no—
161C22×Q8—51no—
162C23×C4—51no—
163#125—104no—
164C22⋊Q8—104no—
165C2×C4⋊C4—51no—
166C2×C4⋊C4—51no—
167C2×C4⋊C4—51no—
168C2×C4⋊C4—51no—
169C2×C4⋊C4—51no—
170C2×C4⋊C4—51no—
171C2×C4⋊C4—51no—
172C2×C4⋊C4C2×D511yescharacteristic
173C2×C4⋊C4C2×D511yescharacteristic
174C2×C4⋊C4C2×D511yescharacteristic
175C2×C4⋊C4C2×D511yescharacteristic
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C22⋊C4—51no—
179C2×C22⋊C4—51no—
180C2×C22⋊C4—51no—
181C2×C22⋊C4—51no—
182C2×C42—51no—
183C2×C42—51no—
184#95—51no—
185#95—51no—
186#95—51no—
187#95—51no—
188#95—51no—
189#95—51no—
190#95—51no—
191#95—51no—
192#95—51no—
193#95—51no—
194#95—51no—
195#95C2×D511yescharacteristic
196#95C2×D511yescharacteristic
197#95C2×D511yescharacteristic
198C22×C5C22×D411yescharacteristic
199C22×C5C2×#4111yescharacteristic
200C22×C5C2×#4111yescharacteristic
201C22×C5H(5,𝔽2)11yescharacteristic
202C22×C52−1+411yescharacteristic
203C22×C52−1+411yescharacteristic
204C22×C52−1+411yescharacteristic
205C2×D5—22no—
206C2×D5—22no—
207C2×D5—22no—
208C2×D5—41no—
209C2×D5—41no—
210C2×D5—41no—
211C2×D5—41no—
212C4×C5—22no—
213C4×C5—41no—
214C4×C5—41no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5—41no—
218C4×C5—41no—
219Dic5—22no—
220Dic5—22no—
221Dic5—22no—
222Dic5—41no—
223Dic5—41no—
224Dic5—41no—
225Dic5—41no—
226C24—51no—
227C2×Q8—102no—
228C2×Q8—102no—
229C2×Q8—201no—
230C22×C4—51no—
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—51no—
234C22×C4—51no—
235C22×C4—51no—
236C22×C4—51no—
237C22×C4—102no—
238C22×C4—102no—
239C22×C4—201no—
240C22×C4C22×D511yescharacteristic
241C22×C4C22×D511yescharacteristic
242C22×C4C22×D511yescharacteristic
243C22×C4C22×D511yescharacteristic
244C22×C4C22×D511yescharacteristic
245C22×C4C22×D511yescharacteristic
246C22×C4C22×D511yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—54no—
252C4⋊C4—102no—
253C4⋊C4—102no—
254C4⋊C4—102no—
255C4⋊C4—102no—
256C4⋊C4—102no—
257C4⋊C4—102no—
258C22⋊C4—102no—
259C22⋊C4—102no—
260C22⋊C4—102no—
261C22⋊C4—102no—
262C22⋊C4—102no—
263C22⋊C4—201no—
264C42—102no—
265C42—201no—
266C2×C5#53611yescharacteristic
267C2×C5#53911yescharacteristic
268C2×C5#54111yescharacteristic
269C2×C5#54611yescharacteristic
270C2×C5#54811yescharacteristic
271C2×C5#55211yescharacteristic
272C2×C5#56311yescharacteristic
273D5—42no—
274C23—52no—
275C23—101no—
276C23—101no—
277C23—101no—
278C23—101no—
279C23—101no—
280C23—101no—
281C23C23×D511yescharacteristic
282Q8—204no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—52no—
304C2×C4—52no—
305C2×C4—52no—
306C2×C4—101no—
307C2×C4—101no—
308C2×C4—101no—
309C2×C4—101no—
310C2×C4—101no—
311C2×C4—101no—
312C2×C4—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—101no—
318C2×C4—101no—
319C2×C4—101no—
320C2×C4—101no—
321C2×C4—101no—
322C2×C4—101no—
323C2×C4—101no—
324C2×C4—204no—
325C2×C4D4×D512yes—
326C5#269011yescharacteristic, complemented, Sylow
327C22—102no—
328C22—102no—
329C22—102no—
330C22—201no—
331C22—201no—
332C22—201no—
333C22—201no—
334C22C2×D4×D511yescharacteristic
335C22#414411yescharacteristic
336C22#414711yescharacteristic
337C22#414711yescharacteristic
338C22D5×#4111yescharacteristic
339C22D5×#4111yescharacteristic
340C22#415011yescharacteristic
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—102no—
349C4—102no—
350C4—102no—
351C4—201no—
352C4—201no—
353C4—201no—
354C4—201no—
355C2—202no—
356C2#6594411yescharacteristic
357C2#6596211yescharacteristic
358C2#6596711yescharacteristic
359C2#6599411yescharacteristic
360C2#6602711yescharacteristic
361C2#6603011yescharacteristic
362C2#6606911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC23×D5C23
hypercenter#398D5
hypocenterC5#2690
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#4035C2semidirect
3. C5×#398C2semidirect
7. #65275C2semidirect
8. #65268C2semidirect
9. #65267C2semidirect
13. #64947C2semidirect
14. #64941C2semidirect
15. #64935C2semidirect
98. #398D5semidirect
326. C5#2690semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40352
C5×#3982
#652792
#652782
#652762
#652752
#652682
#652672
#649642
#649582
#649472
#649412
#649352
#26905