Finite Group Databasebeta

← #10607222

Subgroups — #10607222

Order 640.

464 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×#4024C211yesmaximal, characteristic
2C5×#388C211yesmaximal, characteristic
3#65263C211yesmaximal, characteristic, complemented
4#65261C211yesmaximal, characteristic, complemented
5#65240C211yesmaximal, characteristic
6#65233C211yesmaximal, characteristic
7#65233C211yesmaximal, characteristic
8#65232C211yesmaximal, characteristic
9#65231C211yesmaximal, characteristic
10#65222C211yesmaximal, characteristic, complemented
11#65217C211yesmaximal, characteristic, complemented
12#64944C211yesmaximal, characteristic, complemented
13#64943C211yesmaximal, characteristic, complemented
14#64941C211yesmaximal, 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×C4×Dic5C2211yescharacteristic
35C2×C4×Dic5C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37#4024C418yescomplemented
38C5×#95C2211yescharacteristic
39C5×#95C2211yescharacteristic
40#3963C2211yescharacteristic
41#3963C2211yescharacteristic
42#3963C2211yescharacteristic
43#3963C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#2378—51nomaximal, Sylow
53C24×C5C2311yescharacteristic
54C22×C4×C5C2311yescharacteristic
55C22×C4×C5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×Dic5—41no—
61C22×Dic5—41no—
62C22×Dic5C2×C414yes—
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C2×Dic10—41no—
72C2×Dic10—41no—
73C2×Dic10C2×C414yes—
74C5×C4⋊C4—41no—
75C5×C22⋊C4—41no—
76C5×C22⋊C4—41no—
77C5×C22⋊C4C2×C414yes—
78C42×C5—41no—
79C22⋊Dic5—41no—
80C22⋊Dic5—41no—
81C22⋊Dic5C2×C414yes—
82C4⋊Dic5—41no—
83C4⋊Dic5C2×C414yes—
84#683—41no—
85#683—41no—
86#683—41no—
87#683—41no—
88#683C2×C414yes—
89#683C2×C414yes—
90C4×Dic5—41no—
91C4×Dic5—41no—
92C4×Dic5—41no—
93C2×C22⋊Q8—51no—
94#391—51no—
95#391—51no—
96#389—51no—
97#389—51no—
98#388—51no—
99#388—51no—
100#388—51no—
101#388D511yescharacteristic
102#387—51no—
103#387—51no—
104#387—51no—
105#387—51no—
106#381—51no—
107#380—51no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5C22×C412yes—
115C23×C5C2411yescharacteristic
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—21no—
132C2×C4×C5C22×C412yes—
133C2×C4×C5C22×C412yes—
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—44no—
155C2×Dic5C22×C412yes—
156C2×Dic5C22×C412yes—
157C2×Dic5C22×C412yes—
158C2×Dic5C22×C412yes—
159Dic10—44no—
160C22×Q8—51no—
161C23×C4—51no—
162C22⋊Q8—58no—
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⋊C4C2×D511yescharacteristic
172C2×C22⋊C4—51no—
173C2×C22⋊C4—51no—
174C2×C22⋊C4—51no—
175C2×C22⋊C4C2×D511yescharacteristic
176C2×C22⋊C4C2×D511yescharacteristic
177C2×C22⋊C4C2×D511yescharacteristic
178C2×C42—51no—
179C2×C42—51no—
180C2×C42—51no—
181C2×C42C2×D511yescharacteristic
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#95—51no—
193#95—51no—
194#95C2×D511yescharacteristic
195#95C2×D511yescharacteristic
196C22×C5—22no—
197C22×C5—22no—
198C22×C5—22no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5C23×C411yescharacteristic
204C22×C5H(5,𝔽2)11yescharacteristic
205C22×C5H(5,𝔽2)11yescharacteristic
206C22×C5H(5,𝔽2)11yescharacteristic
207C22×C52−1+411yescharacteristic
208C22×C52−1+411yescharacteristic
209C22×C52−1+411yescharacteristic
210C4×C5—22no—
211C4×C5—22no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—41no—
215C4×C5—41no—
216Dic5—22no—
217Dic5—22no—
218Dic5—22no—
219Dic5—22no—
220Dic5—41no—
221Dic5—41no—
222Dic5—41no—
223Dic5—41no—
224C24C22×D511yescharacteristic
225C2×Q8—54no—
226C2×Q8—201no—
227C2×Q8—201no—
228C22×C4—51no—
229C22×C4—51no—
230C22×C4—51no—
231C22×C4—51no—
232C22×C4—54no—
233C22×C4—51no—
234C22×C4—51no—
235C22×C4—51no—
236C22×C4—51no—
237C22×C4—201no—
238C22×C4—201no—
239C22×C4C22×D511yescharacteristic
240C22×C4C22×D511yescharacteristic
241C22×C4C22×D511yescharacteristic
242C22×C4C22×D511yescharacteristic
243C22×C4C22×D511yescharacteristic
244C22×C4C22×D511yescharacteristic
245C4⋊C4—41no—
246C4⋊C4—54no—
247C4⋊C4—54no—
248C4⋊C4—54no—
249C4⋊C4—201no—
250C4⋊C4—201no—
251C4⋊C4—201no—
252C4⋊C4—201no—
253C4⋊C4—201no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—54no—
257C22⋊C4—201no—
258C22⋊C4—201no—
259C22⋊C4C4×D514yes—
260C42—41no—
261C42—201no—
262C42—201no—
263C42—201no—
264C2×C5—42no—
265C2×C5#51811yescharacteristic
266C2×C5#51911yescharacteristic
267C2×C5#52011yescharacteristic
268C2×C5#56211yescharacteristic
269C2×C5#56211yescharacteristic
270C2×C5#56311yescharacteristic
271C2×C5#56311yescharacteristic
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D512yes—
279C23C23×D511yescharacteristic
280Q8—204no—
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—52no—
298C2×C4—52no—
299C2×C4—52no—
300C2×C4—52no—
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×C4—101no—
321C2×C4—204no—
322C2×C4C2×C4×D512yes—
323C2×C4C2×C4×D512yes—
324C5#237811yescharacteristic, complemented, Sylow
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D511yescharacteristic
333C22#414411yescharacteristic
334C22#414411yescharacteristic
335C22#414911yescharacteristic
336C22#415011yescharacteristic
337C22#415011yescharacteristic
338C22#415011yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—102no—
346C4—102no—
347C4—102no—
348C4—102no—
349C4—201no—
350C4—201no—
351C4—201no—
352C4—201no—
353C2—42no—
354C2#6585011yescharacteristic
355C2#6587911yescharacteristic
356C2#6588011yescharacteristic
357C2#6595511yescharacteristic
358C2#6600111yescharacteristic
359C2#6600611yescharacteristic
360C2#6604611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#388C2
socle, cosocleC23×C5C24
prime radicalC24×C5C23
hypercenter#388D5
hypocenterC5#2378
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
3. #65263C2semidirect
4. #65261C2semidirect
10. #65222C2semidirect
11. #65217C2semidirect
12. #64944C2semidirect
13. #64943C2semidirect
14. #64941C2semidirect
15. #64935C2semidirect
37. #4024C4semidirect
324. C5#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#40242
C5×#3882
#652632
#652612
#652402
#652332
#652322
#652312
#652222
#652172
#649442
#649432
#649412
#649352
#23785