Finite Group Databasebeta

← #10605728

Subgroups — #10605728

Order 640.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#380C211yesmaximal, characteristic
2#65501C211yesmaximal, characteristic
3#65500C211yesmaximal, characteristic
4#65275C211yesmaximal, characteristic, complemented
5#65273C211yesmaximal, characteristic, complemented
6#65273C211yesmaximal, characteristic, complemented
7#65268C211yesmaximal, characteristic, complemented
8#65238C211yesmaximal, characteristic
9#65237C211yesmaximal, characteristic
10#65237C211yesmaximal, characteristic
11#65236C211yesmaximal, characteristic
12#64950C211yesmaximal, characteristic, complemented
13#64949C211yesmaximal, characteristic, complemented
14#64946C211yesmaximal, characteristic, complemented
15#64945C211yesmaximal, characteristic, complemented
16C23×C4×C5C2211yescharacteristic
17C22×Dic10C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C22⋊Dic5C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2211yescharacteristic
28C2×C4⋊Dic5C2211yescharacteristic
29C2×C4⋊Dic5C2211yescharacteristic
30C2×C4⋊Dic5C2211yescharacteristic
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×#683C2211yescharacteristic
37C5×#95C2211yescharacteristic
38C5×#95C2211yescharacteristic
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#2841—51nomaximal, Sylow
52C24×C5C2311yescharacteristic
53C22×C4×C5—41no—
54C22×C4×C5—41no—
55C22×C4×C5Q814yescomplemented
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
70C2×Dic10—41no—
71C2×Dic10—41no—
72C2×Dic10—41no—
73C5×C4⋊C4—41no—
74C5×C4⋊C4—41no—
75C5×C22⋊C4—41no—
76C5×C22⋊C4—41no—
77C22⋊Dic5—41no—
78C22⋊Dic5—41no—
79C22⋊Dic5—41no—
80C22⋊Dic5—41no—
81C4⋊Dic5—41no—
82C4⋊Dic5—41no—
83C4⋊Dic5—41no—
84C4⋊Dic5—41no—
85C4⋊Dic5—41no—
86#683—41no—
87#683—41no—
88#683—41no—
89#683—41no—
90#683—41no—
91#683—41no—
92#400—51no—
93#400—51no—
94#399—51no—
95#398—51no—
96#398—51no—
97#398—51no—
98#398—51no—
99#396—51no—
100#396—51no—
101#395—51no—
102#395—51no—
103#393—51no—
104#381—51no—
105#380—51no—
106#380D511yescharacteristic
107C23×C5—21no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5C2×Q812yes—
114C23×C5C2411yescharacteristic
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—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—44no—
132C2×C4×C5C2×Q812yes—
133C2×C4×C5C2×Q812yes—
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—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158Dic10—44no—
159C22×Q8—51no—
160C23×C4C2×D511yescharacteristic
161C2×C4⋊C4—51no—
162C2×C4⋊C4—51no—
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⋊C4C2×D511yescharacteristic
173C2×C4⋊C4C2×D511yescharacteristic
174C2×C22⋊C4—51no—
175C2×C22⋊C4—51no—
176C2×C22⋊C4—51no—
177C2×C22⋊C4—51no—
178C2×C22⋊C4C2×D511yescharacteristic
179C2×C22⋊C4C2×D511yescharacteristic
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#95—51no—
189#95—51no—
190#95—51no—
191#95—51no—
192#95C2×D511yescharacteristic
193#95C2×D511yescharacteristic
194C22×C5—22no—
195C22×C5—22no—
196C22×C5—22no—
197C22×C5—41no—
198C22×C5—41no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5C22×Q811yescharacteristic
202C22×C5H(5,𝔽2)11yescharacteristic
203C22×C5H(5,𝔽2)11yescharacteristic
204C22×C5H(5,𝔽2)11yescharacteristic
205C22×C5H(5,𝔽2)11yescharacteristic
206C22×C52−1+411yescharacteristic
207C22×C52−1+411yescharacteristic
208C4×C5—22no—
209C4×C5—22no—
210C4×C5—41no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214Dic5—41no—
215Dic5—41no—
216Dic5—41no—
217Dic5—41no—
218Dic5—41no—
219Dic5—41no—
220Dic5—41no—
221Dic5—41no—
222C24C22×D511yescharacteristic
223C2×Q8—201no—
224C2×Q8—201no—
225C2×Q8—201no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—51no—
231C22×C4—51no—
232C22×C4—51no—
233C22×C4—51no—
234C22×C4—51no—
235C22×C4—51no—
236C22×C4Dic1014yescomplemented
237C22×C4C22×D511yescharacteristic
238C22×C4C22×D511yescharacteristic
239C22×C4C22×D511yescharacteristic
240C22×C4C22×D511yescharacteristic
241C22×C4C22×D511yescharacteristic
242C22×C4C22×D511yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—201no—
246C4⋊C4—201no—
247C4⋊C4—201no—
248C4⋊C4—201no—
249C4⋊C4—201no—
250C4⋊C4—201no—
251C4⋊C4—201no—
252C4⋊C4—201no—
253C4⋊C4—201no—
254C4⋊C4—201no—
255C4⋊C4—201no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—201no—
259C22⋊C4—201no—
260C22⋊C4—201no—
261C22⋊C4—201no—
262C2×C5—42no—
263C2×C5#54311yescharacteristic
264C2×C5#54411yescharacteristic
265C2×C5#54411yescharacteristic
266C2×C5#56011yescharacteristic
267C2×C5#56211yescharacteristic
268C2×C5#56211yescharacteristic
269C2×C5#56311yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23C2×Dic1012yes—
277C23C23×D511yescharacteristic
278Q8—204no—
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—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—44no—
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—101no—
313C2×C4—101no—
314C2×C4—101no—
315C2×C4—101no—
316C2×C4—101no—
317C2×C4—101no—
318C2×C4—101no—
319C2×C4—101no—
320C2×C4C2×Dic1012yes—
321C2×C4C2×Dic1012yes—
322C5#284111yescharacteristic, complemented, Sylow
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×Dic1011yescharacteristic
331C22#414411yescharacteristic
332C22#414411yescharacteristic
333C22#414411yescharacteristic
334C22#414711yescharacteristic
335C22#414911yescharacteristic
336C22#415011yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—201no—
344C4—201no—
345C4—201no—
346C4—201no—
347C4—201no—
348C4—201no—
349C4—201no—
350C4—201no—
351C2—42no—
352C2#6581711yescharacteristic
353C2#6583411yescharacteristic
354C2#6585311yescharacteristic
355C2#6595411yescharacteristic
356C2#6595511yescharacteristic
357C2#6598011yescharacteristic
358C2#6598111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#380C2
prime radicalC24×C5C23
hypercenter#380D5
hypocenterC5#2841
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
4. #65275C2semidirect
5. #65273C2semidirect
6. #65273C2semidirect
7. #65268C2semidirect
12. #64950C2semidirect
13. #64949C2semidirect
14. #64946C2semidirect
15. #64945C2semidirect
55. C22×C4×C5Q8semidirect
236. C22×C4Dic10semidirect
322. C5#2841semidirect

Maximal subgroups

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

subgroupindex
C5×#3802
#655012
#655002
#652752
#652732
#652682
#652382
#652372
#652362
#649502
#649492
#649462
#649452
#28415