Finite Group Databasebeta

← #10607307

Subgroups — #10607307

Order 640.

420 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#388C211yesmaximal, characteristic
2#65271C211yesmaximal, characteristic, complemented
3#65267C211yesmaximal, characteristic, complemented
4#65241C211yesmaximal, characteristic
5#65236C211yesmaximal, characteristic
6#65235C211yesmaximal, characteristic
7#65234C211yesmaximal, characteristic
8#65232C211yesmaximal, characteristic
9C22⋊C4×Dic5C211yesmaximal, characteristic
10#65222C211yesmaximal, characteristic, complemented
11#65221C211yesmaximal, characteristic, complemented
12C5⋊#401C211yesmaximal, characteristic, complemented
13#64941C211yesmaximal, characteristic, complemented
14#64935C211yesmaximal, 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×#683C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×C4×Dic5C2211yescharacteristic
31C2×C4×Dic5C2211yescharacteristic
32C2×C4×Dic5C2211yescharacteristic
33C5×#95C2211yescharacteristic
34C5×#95C2211yescharacteristic
35#3963C2211yescharacteristic
36#3963C2211yescharacteristic
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#2527—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C22×C4×C5C2311yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×Dic5—22no—
60C22×Dic5—22no—
61C22×Dic5—41no—
62C22×Dic5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C2×Dic10—22no—
71C2×Dic10—22no—
72C2×Dic10—41no—
73C5×C4⋊C4—41no—
74C5×C22⋊C4—22no—
75C5×C22⋊C4—22no—
76C5×C22⋊C4—41no—
77C42×C5—22no—
78C22⋊Dic5—22no—
79C22⋊Dic5—22no—
80C22⋊Dic5—22no—
81#683—22no—
82#683—22no—
83#683—22no—
84#683—41no—
85C4×Dic5—22no—
86C4×Dic5—22no—
87C4×Dic5—41no—
88#401—51no—
89#400—51no—
90#397—51no—
91#393—51no—
92#391—51no—
93#391—51no—
94#388—51no—
95#388D511yescharacteristic
96#387—51no—
97#387—51no—
98#387—51no—
99#385—51no—
100#381—51no—
101C4×C22⋊C4—51no—
102#376—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#4112yes—
129C2×Dic5—21no—
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—44no—
151C2×Dic5#4112yes—
152C2×Dic5#4112yes—
153C2×Dic5#4112yes—
154Dic10—44no—
155C22×Q8—51no—
156C23×C4—51no—
157C2×C4⋊C4—51no—
158C2×C4⋊C4—51no—
159C2×C4⋊C4—51no—
160C2×C4⋊C4—51no—
161C2×C4⋊C4C2×D511yescharacteristic
162C2×C22⋊C4—51no—
163C2×C22⋊C4—51no—
164C2×C22⋊C4—51no—
165C2×C22⋊C4C2×D511yescharacteristic
166C2×C22⋊C4C2×D511yescharacteristic
167C2×C22⋊C4C2×D511yescharacteristic
168C2×C42—51no—
169C2×C42—51no—
170C2×C42—51no—
171C2×C42C2×D511yescharacteristic
172#95—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#95—51no—
188#95C2×D511yescharacteristic
189#95C2×D511yescharacteristic
190C22×C5—22no—
191C22×C5—22no—
192C22×C5—22no—
193C22×C5—41no—
194C22×C5—41no—
195C22×C5—41no—
196C22×C5—41no—
197C22×C5C2×#4111yescharacteristic
198C22×C5C2×#4111yescharacteristic
199C22×C5C2×#4111yescharacteristic
200C22×C5C2×#4111yescharacteristic
201C22×C5C2×#4111yescharacteristic
202C22×C5H(5,𝔽2)11yescharacteristic
203C22×C52−1+411yescharacteristic
204C4×C5—22no—
205C4×C5—41no—
206C4×C5—41no—
207C4×C5—41no—
208C4×C5—41no—
209C4×C5—41no—
210Dic5—22no—
211Dic5—22no—
212Dic5—22no—
213Dic5—41no—
214Dic5—41no—
215Dic5—41no—
216Dic5—41no—
217Dic5—41no—
218C24C22×D511yescharacteristic
219C2×Q8—102no—
220C2×Q8—102no—
221C2×Q8—201no—
222C22×C4—51no—
223C22×C4—51no—
224C22×C4—51no—
225C22×C4—51no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—102no—
231C22×C4—102no—
232C22×C4—201no—
233C22×C4C22×D511yescharacteristic
234C22×C4C22×D511yescharacteristic
235C22×C4C22×D511yescharacteristic
236C22×C4C22×D511yescharacteristic
237C22×C4C22×D511yescharacteristic
238C22×C4C22×D511yescharacteristic
239C4⋊C4—41no—
240C4⋊C4—102no—
241C4⋊C4—102no—
242C4⋊C4—102no—
243C4⋊C4—201no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—41no—
247C22⋊C4—102no—
248C22⋊C4—102no—
249C22⋊C4—102no—
250C42—22no—
251C42—102no—
252C42—102no—
253C42—201no—
254C2×C5—42no—
255C2×C5#52911yescharacteristic
256C2×C5#52911yescharacteristic
257C2×C5#54211yescharacteristic
258C2×C5#55111yescharacteristic
259C2×C5#55111yescharacteristic
260C2×C5#55211yescharacteristic
261C2×C5#55611yescharacteristic
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#70912yes—
269C23C23×D511yescharacteristic
270Q8—204no—
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—21no—
287C2×C4—21no—
288C2×C4—52no—
289C2×C4—52no—
290C2×C4—52no—
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—101no—
312C2×C4—204no—
313C2×C4#71112yes—
314C5#252711yescharacteristic, complemented, Sylow
315C22—22no—
316C22—22no—
317C22—22no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22C2×#70911yescharacteristic
323C22C2×#71111yescharacteristic
324C22#414411yescharacteristic
325C22D5×#4111yescharacteristic
326C22D5×#4111yescharacteristic
327C22D5×#4111yescharacteristic
328C22#415011yescharacteristic
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—102no—
336C4—102no—
337C4—102no—
338C4—201no—
339C4—201no—
340C4—201no—
341C4—201no—
342C4—201no—
343C2—42no—
344C2#6586511yescharacteristic
345C2#6587211yescharacteristic
346C2#6589011yescharacteristic
347C2#6593711yescharacteristic
348C2#6598311yescharacteristic
349C2#6601311yescharacteristic
350C2#6603611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#388C2
prime radicalC24×C5C23
hypercenter#388D5
hypocenterC5#2527
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #65271C2semidirect
3. #65267C2semidirect
10. #65222C2semidirect
11. #65221C2semidirect
12. C5⋊#401C2semidirect
13. #64941C2semidirect
14. #64935C2semidirect
15. #64935C2semidirect
314. C5#2527semidirect

Maximal subgroups

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

subgroupindex
C5×#3882
#652712
#652672
#652412
#652362
#652352
#652342
#652322
C22⋊C4×Dic52
#652222
#652212
C5⋊#4012
#649412
#649352
#25275