Finite Group Databasebeta

← #10606293

Subgroups — #10606293

Order 640.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#385C211yesmaximal, characteristic
2#65500C211yesmaximal, characteristic
3#65499C211yesmaximal, characteristic
4#65271C211yesmaximal, characteristic, complemented
5#65267C211yesmaximal, characteristic, complemented
6#65262C211yesmaximal, characteristic, complemented
7#65261C211yesmaximal, characteristic, complemented
8#65241C211yesmaximal, characteristic
9#65240C211yesmaximal, characteristic
10#65238C211yesmaximal, characteristic
11#65234C211yesmaximal, characteristic
12#64949C211yesmaximal, characteristic, complemented
13#64948C211yesmaximal, characteristic, complemented
14#64947C211yesmaximal, characteristic, complemented
15#64943C211yesmaximal, characteristic, complemented
16C23×C4×C5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×C4×Dic5C2211yescharacteristic
34C2×C4×Dic5C2211yescharacteristic
35C5×#95C2211yescharacteristic
36C5×#95C2211yescharacteristic
37#3963C2211yescharacteristic
38#3963C2211yescharacteristic
39#3963C2211yescharacteristic
40#3963C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#2771—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C22×C4×C5—22no—
54C22×C4×C5—41no—
55C22×C4×C5—41no—
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
69C22×Dic5C2311yescharacteristic
70C5×C4⋊C4—41no—
71C5×C4⋊C4—41no—
72C5×C22⋊C4—22no—
73C5×C22⋊C4—22no—
74C22⋊Dic5—22no—
75C22⋊Dic5—22no—
76C22⋊Dic5—41no—
77C22⋊Dic5—41no—
78C4⋊Dic5—22no—
79C4⋊Dic5—22no—
80#683—22no—
81#683—22no—
82#683—41no—
83#683—41no—
84#683—41no—
85#683—41no—
86C4×Dic5—41no—
87C4×Dic5—41no—
88#400—51no—
89#400—51no—
90#400—51no—
91#400—51no—
92#399—51no—
93#397—51no—
94#389—51no—
95#389—51no—
96#388—51no—
97#388—51no—
98#387—51no—
99#387—51no—
100#385—51no—
101#385D511yescharacteristic
102#381—51no—
103C23×C5—21no—
104C23×C5—21no—
105C23×C5—21no—
106C23×C5—21no—
107C23×C5—21no—
108C23×C5—21no—
109C23×C5#4112yes—
110C23×C5C2411yescharacteristic
111C2×C4×C5—21no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—21no—
115C2×C4×C5—21no—
116C2×C4×C5—21no—
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—44no—
129C2×C4×C5#4112yes—
130C2×Dic5—21no—
131C2×Dic5—21no—
132C2×Dic5—21no—
133C2×Dic5—21no—
134C2×Dic5—21no—
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#4112yes—
154C23×C4C2×D511yescharacteristic
155C2×C4⋊C4—51no—
156C2×C4⋊C4—51no—
157C2×C4⋊C4—51no—
158C2×C4⋊C4—51no—
159C2×C4⋊C4—51no—
160C2×C4⋊C4—51no—
161C2×C4⋊C4—51no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4C2×D511yescharacteristic
164C2×C4⋊C4C2×D511yescharacteristic
165C2×C22⋊C4—51no—
166C2×C22⋊C4—51no—
167C2×C22⋊C4—51no—
168C2×C22⋊C4—51no—
169C2×C22⋊C4C2×D511yescharacteristic
170C2×C22⋊C4C2×D511yescharacteristic
171C2×C42—51no—
172C2×C42—51no—
173#95—51no—
174#95—51no—
175#95—51no—
176#95—51no—
177#95—51no—
178#95—51no—
179#95—51no—
180#95—51no—
181#95—51no—
182#95—51no—
183#95—51no—
184#95—51no—
185#95—51no—
186#95—51no—
187#95C2×D511yescharacteristic
188#95C2×D511yescharacteristic
189C22×C5—22no—
190C22×C5—22no—
191C22×C5—22no—
192C22×C5—41no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5C2×#4111yescharacteristic
197C22×C5C2×#4111yescharacteristic
198C22×C5C2×#4111yescharacteristic
199C22×C5H(5,𝔽2)11yescharacteristic
200C22×C5H(5,𝔽2)11yescharacteristic
201C22×C52−1+411yescharacteristic
202C22×C52−1+411yescharacteristic
203C4×C5—22no—
204C4×C5—41no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5—41no—
209Dic5—22no—
210Dic5—41no—
211Dic5—41no—
212Dic5—41no—
213Dic5—41no—
214Dic5—41no—
215Dic5—41no—
216Dic5—41no—
217C24C22×D511yescharacteristic
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—51no—
222C22×C4—51no—
223C22×C4—51no—
224C22×C4—51no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4C22×D511yescharacteristic
230C22×C4C22×D511yescharacteristic
231C22×C4C22×D511yescharacteristic
232C22×C4C22×D511yescharacteristic
233C22×C4C22×D511yescharacteristic
234C22×C4C22×D511yescharacteristic
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—102no—
238C4⋊C4—102no—
239C4⋊C4—102no—
240C4⋊C4—102no—
241C4⋊C4—201no—
242C4⋊C4—201no—
243C4⋊C4—201no—
244C4⋊C4—201no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—102no—
248C22⋊C4—102no—
249C22⋊C4—201no—
250C22⋊C4—201no—
251C42—201no—
252C42—201no—
253C2×C5—42no—
254C2×C5#53811yescharacteristic
255C2×C5#53911yescharacteristic
256C2×C5#53911yescharacteristic
257C2×C5#55211yescharacteristic
258C2×C5#55211yescharacteristic
259C2×C5#55911yescharacteristic
260C2×C5#56311yescharacteristic
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23#71112yes—
268C23C23×D511yescharacteristic
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—44no—
287C2×C4—52no—
288C2×C4—101no—
289C2×C4—101no—
290C2×C4—101no—
291C2×C4—101no—
292C2×C4—101no—
293C2×C4—101no—
294C2×C4—101no—
295C2×C4—101no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—101no—
299C2×C4—101no—
300C2×C4—101no—
301C2×C4—101no—
302C2×C4—101no—
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#70912yes—
312C5#277111yescharacteristic, complemented, Sylow
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×#70911yescharacteristic
321C22C2×#71111yescharacteristic
322C22#414411yescharacteristic
323C22#414411yescharacteristic
324C22#414711yescharacteristic
325C22D5×#4111yescharacteristic
326C22#415011yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—102no—
334C4—201no—
335C4—201no—
336C4—201no—
337C4—201no—
338C4—201no—
339C4—201no—
340C4—201no—
341C2—42no—
342C2#6584711yescharacteristic
343C2#6589311yescharacteristic
344C2#6589811yescharacteristic
345C2#6592711yescharacteristic
346C2#6597211yescharacteristic
347C2#6598011yescharacteristic
348C2#6598411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC24×C5C23
hypercenter#385D5
hypocenterC5#2771
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #65271C2semidirect
5. #65267C2semidirect
6. #65262C2semidirect
7. #65261C2semidirect
12. #64949C2semidirect
13. #64948C2semidirect
14. #64947C2semidirect
15. #64943C2semidirect
312. C5#2771semidirect

Maximal subgroups

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

subgroupindex
C5×#3852
#655002
#654992
#652712
#652672
#652622
#652612
#652412
#652402
#652382
#652342
#649492
#649482
#649472
#649432
#27715