Finite Group Databasebeta

← #10609474

Subgroups — #10609474

Order 640.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic
2C2×#4033C211yesmaximal, characteristic, complemented
3C5×#398C211yesmaximal, characteristic, complemented
4#65281C211yesmaximal, characteristic
5#65281C211yesmaximal, characteristic
6#65279C211yesmaximal, characteristic
7#65267C211yesmaximal, characteristic, complemented
8#65265C211yesmaximal, characteristic, complemented
9#65261C211yesmaximal, characteristic, complemented
10#64959C211yesmaximal, characteristic
11#64956C211yesmaximal, characteristic
12#64955C211yesmaximal, characteristic
13#64946C211yesmaximal, 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×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39C2×C4×Dic5C2211yescharacteristic
40C2×C4×Dic5C2211yescharacteristic
41#4042—24no—
42#4033—24no—
43C5×#95C2211yescharacteristic
44C5×#95C2211yescharacteristic
45C5×#95C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2423—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—22no—
72C2×Dic10—22no—
73C2×Dic10—22no—
74C2×Dic10D414yes—
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—41no—
83#685—41no—
84#685Q814yescomplemented
85C4⋊Dic5—41no—
86#683—22no—
87#683—22no—
88#683—22no—
89#683—22no—
90#683—22no—
91#683—22no—
92#683—22no—
93#683—41no—
94C4×Dic5—22no—
95C4×Dic5—22no—
96C4×Dic5—22no—
97C4×Dic5—22no—
98C2×C22⋊Q8—51no—
99C2×C4×Q8—51no—
100#399—51no—
101#399—51no—
102#398—51no—
103#398D511yescharacteristic, complemented
104#391—51no—
105#389—51no—
106#388—51no—
107#388—51no—
108#387—51no—
109#387—51no—
110#384—51no—
111#380—51no—
112C4×C22⋊C4—51no—
113C23×C5C2411yescharacteristic
114C22×D5—21no—
115C22×D5—21no—
116C22×D5—21no—
117C22×D5—21no—
118C22×D5—21no—
119C22×D5—21no—
120C22×D5C2×Q812yes—
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×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5C2×D412yes—
141C2×C4×C5C2×Q812yes—
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×Dic5—21no—
157C2×Dic5C2×D412yes—
158C2×Dic5C2×D412yes—
159C2×Dic5C2×Q812yes—
160C2×Dic5#4112yes—
161C2×Dic5#4112yes—
162C2×Dic5#4112yes—
163C4×D5—44no—
164Dic10—28no—
165C22×Q8—51no—
166C23×C4—51no—
167C22⋊Q8—104no—
168C4×Q8—104no—
169C2×C4⋊C4—51no—
170C2×C4⋊C4—51no—
171C2×C4⋊C4—51no—
172C2×C4⋊C4—51no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—51no—
175C2×C4⋊C4—51no—
176C2×C4⋊C4—51no—
177C2×C4⋊C4—51no—
178C2×C4⋊C4C2×D511yescharacteristic
179C2×C4⋊C4C2×D511yescharacteristic
180C2×C4⋊C4C2×D511yescharacteristic
181C2×C4⋊C4C2×D511yescharacteristic
182C2×C22⋊C4—51no—
183C2×C22⋊C4—51no—
184C2×C22⋊C4—51no—
185C2×C22⋊C4—51no—
186C2×C22⋊C4—51no—
187C2×C22⋊C4—51no—
188C2×C42—51no—
189C2×C42—51no—
190C2×C42—51no—
191C2×C42—51no—
192#95—51no—
193#95—51no—
194#95—51no—
195#95—51no—
196#95—51no—
197#95—51no—
198#95—51no—
199#95C2×D511yescharacteristic
200#95C2×D511yescharacteristic
201#95C2×D511yescharacteristic
202C22×C5C22×D411yescharacteristic
203C22×C5C22×Q811yescharacteristic
204C22×C5C2×#4111yescharacteristic
205C22×C5C2×#4111yescharacteristic
206C22×C5C2×#4111yescharacteristic
207C22×C5H(5,𝔽2)11yescharacteristic
208C22×C52−1+411yescharacteristic
209C2×D5—22no—
210C2×D5—22no—
211C2×D5—22no—
212C2×D5—41no—
213C2×D5—41no—
214C2×D5—41no—
215C2×D5—41no—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5—41no—
219C4×C5—41no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223Dic5—22no—
224Dic5—22no—
225Dic5—22no—
226Dic5—41no—
227Dic5—41no—
228Dic5—41no—
229Dic5#12314yes—
230C24—51no—
231C2×Q8—54no—
232C2×Q8—102no—
233C2×Q8—102no—
234C22×C4—51no—
235C22×C4—51no—
236C22×C4—51no—
237C22×C4—51no—
238C22×C4—51no—
239C22×C4—51no—
240C22×C4—51no—
241C22×C4—102no—
242C22×C4—102no—
243C22×C4—102no—
244C22×C4C22×D511yescharacteristic
245C22×C4C22×D511yescharacteristic
246C22×C4C22×D511yescharacteristic
247C22×C4C22×D511yescharacteristic
248C22×C4C22×D511yescharacteristic
249C22×C4C22×D511yescharacteristic
250C22×C4C22×D511yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—102no—
256C4⋊C4—102no—
257C4⋊C4—102no—
258C4⋊C4—102no—
259C4⋊C4—102no—
260C4⋊C4—102no—
261C4⋊C4—102no—
262C4⋊C4—201no—
263C4⋊C4—201no—
264C22⋊C4—54no—
265C22⋊C4—102no—
266C22⋊C4—102no—
267C22⋊C4—102no—
268C22⋊C4—201no—
269C22⋊C4—201no—
270C42—102no—
271C42—102no—
272C42—102no—
273C42—102no—
274C2×C5C2×#12311yescharacteristic
275C2×C5#53311yescharacteristic
276C2×C5#54211yescharacteristic
277C2×C5D4×Q811yescharacteristic
278C2×C5#55011yescharacteristic
279C2×C5#55211yescharacteristic
280C2×C5#55411yescharacteristic
281D5—42no—
282C23—52no—
283C23—101no—
284C23—101no—
285C23—101no—
286C23—101no—
287C23—101no—
288C23—101no—
289C23C23×D511yescharacteristic
290Q8—108no—
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—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—52no—
311C2×C4—52no—
312C2×C4—52no—
313C2×C4—52no—
314C2×C4—52no—
315C2×C4—52no—
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—101no—
325C2×C4—101no—
326C2×C4—101no—
327C2×C4—101no—
328C2×C4—101no—
329C2×C4—101no—
330C2×C4—101no—
331C2×C4—204no—
332C2×C4D4×D512yes—
333C2×C4Q8×D512yes—
334C5#242311yescharacteristic, complemented, Sylow
335C22—102no—
336C22—102no—
337C22—102no—
338C22—201no—
339C22—201no—
340C22—201no—
341C22—201no—
342C22C2×D4×D511yescharacteristic
343C22#414411yescharacteristic
344C22C2×Q8×D511yescharacteristic
345C22#414711yescharacteristic
346C22D5×#4111yescharacteristic
347C22D5×#4111yescharacteristic
348C22D5×#4111yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—54no—
357C4—102no—
358C4—102no—
359C4—102no—
360C4—201no—
361C4—201no—
362C4—201no—
363C2—202no—
364C2#6593311yescharacteristic
365C2#6597011yescharacteristic
366C2#6597411yescharacteristic
367C2D5×#12311yescharacteristic
368C2#6602211yescharacteristic
369C2#6602711yescharacteristic
370C2#6606111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC23×D5C23
hypercenter#398D5
hypocenterC5#2423
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#4033C2semidirect
3. C5×#398C2semidirect
7. #65267C2semidirect
8. #65265C2semidirect
9. #65261C2semidirect
13. #64946C2semidirect
14. #64941C2semidirect
15. #64935C2semidirect
84. #685Q8semidirect
103. #398D5semidirect
334. C5#2423semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40332
C5×#3982
#652812
#652792
#652672
#652652
#652612
#649592
#649562
#649552
#649462
#649412
#649352
#24235