Finite Group Databasebeta

← #10607403

Subgroups — #10607403

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×#4024C211yesmaximal, characteristic
2C2×#4016C211yesmaximal, characteristic, complemented
3C5×#388C211yesmaximal, characteristic
4#65269C211yesmaximal, characteristic, complemented
5#65268C211yesmaximal, characteristic, complemented
6#65242C211yesmaximal, characteristic
7#65241C211yesmaximal, characteristic
8#65235C211yesmaximal, characteristic
9#65233C211yesmaximal, characteristic
10#65231C211yesmaximal, characteristic
11#65222C211yesmaximal, characteristic, complemented
12#64948C211yesmaximal, characteristic, complemented
13#64947C211yesmaximal, characteristic, complemented
14#64943C211yesmaximal, characteristic, complemented
15#64935C211yesmaximal, characteristic, complemented
16C23×Dic5C2211yescharacteristic
17C22×Dic10C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C42×C5C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37#4024—24no—
38#4016—24no—
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41#3963C2211yescharacteristic
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
54C24×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×Dic5—22no—
62C22×Dic5—22no—
63C22×Dic5—41no—
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C2×Dic10—22no—
73C2×Dic10—22no—
74C2×Dic10—41no—
75C5×C4⋊C4—41no—
76C5×C22⋊C4—22no—
77C5×C22⋊C4—22no—
78C5×C22⋊C4—41no—
79C42×C5—22no—
80C22⋊Dic5—22no—
81C22⋊Dic5—22no—
82C22⋊Dic5—22no—
83C4⋊Dic5—22no—
84C4⋊Dic5D414yes—
85#683—22no—
86#683—22no—
87#683—22no—
88#683—22no—
89#683—22no—
90#683—22no—
91#683—22no—
92#683—41no—
93C4×Dic5—41no—
94C2×#125—51no—
95C2×C22⋊Q8—51no—
96#400—51no—
97#400—51no—
98#398—51no—
99#397—51no—
100#397—51no—
101#395—51no—
102#391—51no—
103#388—51no—
104#388D511yescharacteristic
105#387—51no—
106#387—51no—
107#385—51no—
108#380—51no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5#4112yes—
116C23×C5C2411yescharacteristic
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×C5C2×D412yes—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
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×Dic5—44no—
157C2×Dic5C2×D412yes—
158C2×Dic5C2×D412yes—
159C2×Dic5#4112yes—
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⋊C4—51no—
173C2×C4⋊C4—51no—
174C2×C4⋊C4—51no—
175C2×C4⋊C4C2×D511yescharacteristic
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C22⋊C4—51no—
179C2×C22⋊C4C2×D511yescharacteristic
180C2×C22⋊C4C2×D511yescharacteristic
181C2×C22⋊C4C2×D511yescharacteristic
182C2×C42—51no—
183C2×C42C2×D511yescharacteristic
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#95—51no—
196#95C2×D511yescharacteristic
197#95C2×D511yescharacteristic
198C22×C5—22no—
199C22×C5—22no—
200C22×C5—22no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5C22×D411yescharacteristic
206C22×C5C2×#4111yescharacteristic
207C22×C5C2×#4111yescharacteristic
208C22×C5H(5,𝔽2)11yescharacteristic
209C22×C52−1+411yescharacteristic
210C22×C52−1+411yescharacteristic
211C22×C52−1+411yescharacteristic
212C4×C5—22no—
213C4×C5—41no—
214C4×C5—41no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5—41no—
218Dic5—22no—
219Dic5—22no—
220Dic5—22no—
221Dic5—41no—
222Dic5—41no—
223Dic5—41no—
224Dic5—41no—
225Dic5—41no—
226C24C22×D511yescharacteristic
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—51no—
238C22×C4—102no—
239C22×C4—102no—
240C22×C4—201no—
241C22×C4C22×D511yescharacteristic
242C22×C4C22×D511yescharacteristic
243C22×C4C22×D511yescharacteristic
244C22×C4C22×D511yescharacteristic
245C22×C4C22×D511yescharacteristic
246C22×C4C22×D511yescharacteristic
247C4⋊C4—41no—
248C4⋊C4—54no—
249C4⋊C4—102no—
250C4⋊C4—102no—
251C4⋊C4—102no—
252C4⋊C4—102no—
253C4⋊C4—102no—
254C4⋊C4—102no—
255C4⋊C4—102no—
256C4⋊C4—102no—
257C4⋊C4—201no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—41no—
261C22⋊C4—102no—
262C22⋊C4—102no—
263C22⋊C4—102no—
264C42—22no—
265C42—201no—
266C2×C5—42no—
267C2×C5#53611yescharacteristic
268C2×C5#53911yescharacteristic
269C2×C5#54111yescharacteristic
270C2×C5#54611yescharacteristic
271C2×C5#54811yescharacteristic
272C2×C5#55211yescharacteristic
273C2×C5#56311yescharacteristic
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#70912yes—
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—52no—
301C2×C4—52no—
302C2×C4—52no—
303C2×C4—101no—
304C2×C4—101no—
305C2×C4—101no—
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—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22C2×#70911yescharacteristic
335C22C2×D4×D511yescharacteristic
336C22#414411yescharacteristic
337C22D5×#4111yescharacteristic
338C22#415011yescharacteristic
339C22#415011yescharacteristic
340C22#415011yescharacteristic
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—102no—
348C4—102no—
349C4—102no—
350C4—201no—
351C4—201no—
352C4—201no—
353C4—201no—
354C4—201no—
355C2—42no—
356C2#6586511yescharacteristic
357C2#6588611yescharacteristic
358C2#6589811yescharacteristic
359C2#6594611yescharacteristic
360C2#6598211yescharacteristic
361C2#6600611yescharacteristic
362C2#6603511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#388C2
prime radicalC24×C5C23
hypercenter#388D5
hypocenterC5#2690
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#4016C2semidirect
4. #65269C2semidirect
5. #65268C2semidirect
11. #65222C2semidirect
12. #64948C2semidirect
13. #64947C2semidirect
14. #64943C2semidirect
15. #64935C2semidirect
326. C5#2690semidirect

Maximal subgroups

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

subgroupindex
C2×#40242
C2×#40162
C5×#3882
#652692
#652682
#652422
#652412
#652352
#652332
#652312
#652222
#649482
#649472
#649432
#649352
#26905