Finite Group Databasebeta

← #10609209

Subgroups — #10609209

Order 640.

476 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4025C211yesmaximal, characteristic
2C2×#4023C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic
4#65270C211yesmaximal, characteristic, complemented
5#65262C211yesmaximal, characteristic, complemented
6#65241C211yesmaximal, characteristic
7#65239C211yesmaximal, characteristic
8#65235C211yesmaximal, characteristic
9#65233C211yesmaximal, characteristic
10#64946C211yesmaximal, characteristic, complemented
11#64946C211yesmaximal, characteristic, complemented
12#64943C211yesmaximal, characteristic, complemented
13#64941C211yesmaximal, characteristic, complemented
14#64938C211yesmaximal, characteristic, complemented
15#64936C211yesmaximal, characteristic, complemented
16C23×Dic5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C4⋊Dic5C2211yescharacteristic
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×C4×Dic5C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37C2×C4×Dic5C2211yescharacteristic
38C2×C4×Dic5C2211yescharacteristic
39#4025—24no—
40#4023—24no—
41C5×#95C2211yescharacteristic
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2478—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—22no—
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C5×C4⋊C4—22no—
73C5×C22⋊C4—22no—
74C5×C22⋊C4—22no—
75C5×C22⋊C4—22no—
76C22⋊Dic5—22no—
77C22⋊Dic5—22no—
78C22⋊Dic5D414yes—
79C4⋊Dic5—22no—
80C4⋊Dic5—22no—
81C4⋊Dic5—22no—
82#683—22no—
83#683—22no—
84#683—22no—
85#683—22no—
86#683—22no—
87#683—22no—
88#683—22no—
89#683—22no—
90C4×Dic5—22no—
91C4×Dic5—22no—
92C4×Dic5—22no—
93C4×Dic5—22no—
94C2×#126—51no—
95C2×#117—51no—
96#398—51no—
97#398—51no—
98#397—51no—
99#397D511yescharacteristic
100#389—51no—
101#389—51no—
102#388—51no—
103#387—51no—
104#387—51no—
105#387—51no—
106#385—51no—
107#385—51no—
108C4×C4⋊C4—51no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5C2×D412yes—
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×C5#4112yes—
135C2×Dic5—28no—
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×Dic5C2×D412yes—
156C2×Dic5C2×D412yes—
157C2×Dic5#4112yes—
158C2×Dic5#4112yes—
159C2×Dic5#4112yes—
160C23×C4—51no—
161#126—104no—
162#117—104no—
163C2×C4⋊C4—51no—
164C2×C4⋊C4—51no—
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⋊C4C2×D511yescharacteristic
175C2×C22⋊C4—51no—
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C22⋊C4C2×D511yescharacteristic
179C2×C22⋊C4C2×D511yescharacteristic
180C2×C22⋊C4C2×D511yescharacteristic
181C2×C42—51no—
182C2×C42—51no—
183C2×C42—51no—
184C2×C42—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#95C2×D511yescharacteristic
195#95C2×D511yescharacteristic
196#95C2×D511yescharacteristic
197C22×C5—22no—
198C22×C5—22no—
199C22×C5—22no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—41no—
204C22×C5C22×D411yescharacteristic
205C22×C5C2×#4111yescharacteristic
206C22×C5C2×#4111yescharacteristic
207C22×C5C2×#4111yescharacteristic
208C22×C5C2×#4111yescharacteristic
209C22×C52−1+411yescharacteristic
210C22×C52−1+411yescharacteristic
211C4×C5—22no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—41no—
215C4×C5—41no—
216C4×C5—41no—
217Dic5—22no—
218Dic5—22no—
219Dic5—22no—
220Dic5—22no—
221Dic5—22no—
222Dic5—41no—
223Dic5—41no—
224Dic5—41no—
225C24C22×D511yescharacteristic
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—51no—
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—51no—
234C22×C4—102no—
235C22×C4—102no—
236C22×C4—102no—
237C22×C4C22×D511yescharacteristic
238C22×C4C22×D511yescharacteristic
239C22×C4C22×D511yescharacteristic
240C22×C4C22×D511yescharacteristic
241C22×C4C22×D511yescharacteristic
242C22×C4C22×D511yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—102no—
245C4⋊C4—102no—
246C4⋊C4—102no—
247C4⋊C4—102no—
248C4⋊C4—102no—
249C4⋊C4—102no—
250C4⋊C4—102no—
251C4⋊C4—102no—
252C4⋊C4—102no—
253C4⋊C4—102no—
254C4⋊C4—102no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—54no—
259C22⋊C4—102no—
260C22⋊C4—102no—
261C42—102no—
262C42—102no—
263C42—102no—
264C42—102no—
265C2×C5—42no—
266C2×C5#52511yescharacteristic
267C2×C5#52911yescharacteristic
268C2×C5#54111yescharacteristic
269C2×C5#54711yescharacteristic
270C2×C5#54711yescharacteristic
271C2×C5#55211yescharacteristic
272C2×C5#55611yescharacteristic
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23D4×D512yes—
280C23C23×D511yescharacteristic
281C2×C4—21no—
282C2×C4—21no—
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—52no—
299C2×C4—52no—
300C2×C4—52no—
301C2×C4—52no—
302C2×C4—52no—
303C2×C4—108no—
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#71112yes—
324C5#247811yescharacteristic, complemented, Sylow
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C2×D4×D511yescharacteristic
333C22C2×#71111yescharacteristic
334C22D5×#4111yescharacteristic
335C22D5×#4111yescharacteristic
336C22D5×#4111yescharacteristic
337C22#415011yescharacteristic
338C22#415011yescharacteristic
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—102no—
346C4—102no—
347C4—102no—
348C4—102no—
349C4—102no—
350C4—201no—
351C4—201no—
352C4—201no—
353C2—42no—
354C2#6594511yescharacteristic
355C2#6598411yescharacteristic
356C2#6598911yescharacteristic
357C2#6599011yescharacteristic
358C2#6601411yescharacteristic
359C2#6603511yescharacteristic
360C2#6603611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC24×C5C23
hypercenter#397D5
hypocenterC5#2478
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
4. #65270C2semidirect
5. #65262C2semidirect
10. #64946C2semidirect
11. #64946C2semidirect
12. #64943C2semidirect
13. #64941C2semidirect
14. #64938C2semidirect
15. #64936C2semidirect
324. C5#2478semidirect

Maximal subgroups

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

subgroupindex
C2×#40252
C2×#40232
C5×#3972
#652702
#652622
#652412
#652392
#652352
#652332
#649462
#649432
#649412
#649382
#649362
#24785