Finite Group Databasebeta

← #10614626

Subgroups — #10614626

Order 640.

814 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#123C211yesmaximal, characteristic, complemented
2C2×#4083C211yesmaximal, characteristic
3C22×#685C211yesmaximal, characteristic, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5#65509C211yesmaximal, characteristic, complemented
6#65500C211yesmaximal, characteristic, complemented
7#65282C212yesmaximal, complemented
8#65276C212yesmaximal, complemented
9#65250C212yesmaximal, complemented
10#65245C212yesmaximal, complemented
11#65239C211yesmaximal, characteristic, complemented
12C24×D5C2211yescharacteristic
13C22×C5×D4C2211yescharacteristic
14C23×C4×C5C2211yescharacteristic
15C22×C22⋊D5C2211yescharacteristic
16C22×C22⋊D5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C22×C4×D5C2212yes—
19C5×#123—24no—
20C2×C5×C4⋊C4C2212yescomplemented
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2212yescomplemented
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2212yescomplemented
26#4083—24no—
27C2×#685—24no—
28C2×#685C2211yescharacteristic
29C2×#685C2212yes—
30C2×#685C2212yes—
31C2×#685C2212yes—
32C2×#685C2211yescharacteristic
33C2×C4⋊Dic5C2212yescomplemented
34D5×C22⋊C4—24no—
35#3963C2212yescomplemented
36#3963C2212yescomplemented
37#3963C2212yescomplemented
38#3963C2212yescomplemented
39#2412—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C24×C5C2311yescharacteristic
42C23×D5—22no—
43C23×D5—22no—
44C23×D5—24no—
45C23×D5—42no—
46C23×D5D414yes—
47C23×D5C2311yescharacteristic
48C23×D5C2311yescharacteristic
49C2×C5×D4—22no—
50C2×C5×D4—22no—
51C2×C5×D4—22no—
52C22×C4×C5—22no—
53C22×C4×C5—22no—
54C22×C4×C5—22no—
55C22×C4×C5C2312yes—
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2312yes—
58C2×C22⋊D5—22no—
59C2×C22⋊D5—22no—
60C2×C22⋊D5—24no—
61C2×C22⋊D5—42no—
62C22×Dic5—22no—
63C22×Dic5—42no—
64C22×Dic5C2312yes—
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2312yes—
68C2×C4×D5—24no—
69C2×C4×D5—24no—
70C2×C4×D5—42no—
71C5×C4⋊C4—24no—
72C5×C22⋊C4—24no—
73C5×C22⋊C4D414yescomplemented
74C22⋊Dic5—22no—
75C22⋊Dic5—22no—
76C22⋊Dic5—42no—
77#685—24no—
78#685—24no—
79#685—22no—
80#685—22no—
81#685—24no—
82C4⋊Dic5—42no—
83C2×#123D511yescharacteristic, complemented
84C2×#120—51no—
85C22×C22⋊C4—51no—
86C22×C22⋊C4—51no—
87#397—52no—
88#394—52no—
89#386—51no—
90#386—52no—
91#385—51no—
92#381—51no—
93#380—52no—
94C23×C5—22no—
95C23×C5—22no—
96C23×C5—21no—
97C23×C5—21no—
98C23×C5—22no—
99C23×C5—21no—
100C23×C5—22no—
101C23×C5C2×D412yes—
102C23×C5C2×D412yes—
103C23×C5#4112yes—
104C23×C5C2411yescharacteristic
105C22×D5—21no—
106C22×D5—21no—
107C22×D5—22no—
108C22×D5—28no—
109C22×D5—21no—
110C22×D5—21no—
111C22×D5—22no—
112C22×D5—22no—
113C22×D5—28no—
114C22×D5—44no—
115C22×D5—48no—
116C22×D5—48no—
117C22×D5C2×D412yes—
118C22×D5C2×D412yes—
119C22×D5#4114yes—
120C5×D4—44no—
121C2×C4×C5—22no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—21no—
125C2×C4×C5—22no—
126C2×C4×C5—21no—
127C2×C4×C5—22no—
128C2×C4×C5—21no—
129C2×C4×C5—44no—
130C2×C4×C5C2×D414yes—
131C22⋊D5—44no—
132C22⋊D5—44no—
133C2×Dic5—22no—
134C2×Dic5—22no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—22no—
138C2×Dic5—22no—
139C2×Dic5—22no—
140C2×Dic5—22no—
141C2×Dic5—22no—
142C2×Dic5—22no—
143C2×Dic5—44no—
144C4×D5—48no—
145C25—51no—
146C22×D4—51no—
147C22×D4—51no—
148C22×D4C2×D511yescharacteristic
149C23×C4—51no—
150C23×C4—52no—
151C23×C4C2×D511yescharacteristic
152#123—24no—
153#120—104no—
154C2×C4⋊C4—52no—
155C2×C4⋊C4C2×D512yescomplemented
156C2×C22⋊C4—51no—
157C2×C22⋊C4—51no—
158C2×C22⋊C4—52no—
159C2×C22⋊C4—52no—
160C2×C22⋊C4—52no—
161C2×C22⋊C4—51no—
162C2×C22⋊C4—51no—
163C2×C22⋊C4—52no—
164C2×C22⋊C4—104no—
165C2×C22⋊C4—104no—
166C2×C22⋊C4C2×D511yescharacteristic
167C2×C22⋊C4C2×D512yescomplemented
168#95—52no—
169#95—52no—
170#95—52no—
171#95—52no—
172C22×C5—22no—
173C22×C5—22no—
174C22×C5—24no—
175C22×C5—24no—
176C22×C5—41no—
177C22×C5—41no—
178C22×C5—41no—
179C22×C5—42no—
180C22×C5—42no—
181C22×C5#12114yes—
182C22×C5C22×D411yescharacteristic
183C22×C5C22×D411yescharacteristic
184C22×C5C2×#4111yescharacteristic
185C22×C5C2×#4112yes—
186C22×C5H(5,𝔽2)12yes—
187C2×D5—22no—
188C2×D5—22no—
189C2×D5—24no—
190C2×D5—24no—
191C2×D5—22no—
192C2×D5—22no—
193C2×D5—22no—
194C2×D5—416no—
195C2×D5—42no—
196C2×D5—42no—
197C2×D5#12314yescomplemented
198C4×C5—24no—
199C4×C5—42no—
200C4×C5—41no—
201Dic5—41no—
202Dic5—42no—
203Dic5—41no—
204Dic5—42no—
205C24—51no—
206C24—51no—
207C24—54no—
208C24—104no—
209C24—102no—
210C24—102no—
211C24—202no—
212C24C22×D511yescharacteristic
213C24C22×D511yescharacteristic
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—102no—
218C2×D4—102no—
219C2×D4—104no—
220C2×D4—202no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—51no—
225C22×C4—52no—
226C22×C4—52no—
227C22×C4—51no—
228C22×C4—104no—
229C22×C4—102no—
230C22×C4—104no—
231C22×C4—202no—
232C22×C4—202no—
233C22×C4C22×D512yes—
234C22×C4C22×D511yescharacteristic
235C22×C4C22×D512yes—
236C4⋊C4—24no—
237C4⋊C4—202no—
238C22⋊C4—24no—
239C22⋊C4—104no—
240C22⋊C4—104no—
241C22⋊C4—102no—
242C22⋊C4—102no—
243C22⋊C4—102no—
244C22⋊C4—102no—
245C22⋊C4—104no—
246C22⋊C4—202no—
247C22⋊C4C22⋊D514yescomplemented
248C2×C5—24no—
249C2×C5—81no—
250C2×C5C2×#12111yescharacteristic
251C2×C5C2×#12311yescharacteristic
252C2×C5#53411yescharacteristic
253C2×C5#54612yes—
254C2×C5#55112yes—
255D5—24no—
256D5—42no—
257C23—22no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—21no—
264C23—52no—
265C23—54no—
266C23—52no—
267C23—108no—
268C23—101no—
269C23—102no—
270C23—101no—
271C23—101no—
272C23—102no—
273C23—102no—
274C23—101no—
275C23—108no—
276C23—204no—
277C23—208no—
278C23—208no—
279C23D4×D512yes—
280C23#71112yes—
281C23C2×C22⋊D512yes—
282C23C23×D511yescharacteristic
283D4—44no—
284D4—204no—
285D4—204no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—102no—
296C2×C4—101no—
297C2×C4—101no—
298C2×C4—102no—
299C2×C4—102no—
300C2×C4—102no—
301C2×C4—102no—
302C2×C4—102no—
303C2×C4—102no—
304C2×C4—102no—
305C2×C4—208no—
306C2×C4—204no—
307C2×C4C2×C22⋊D514yes—
308C5#241211yescharacteristic, complemented, Sylow
309C22—24no—
310C22—22no—
311C22—22no—
312C22—24no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—54no—
319C22—102no—
320C22—104no—
321C22—102no—
322C22—104no—
323C22—102no—
324C22—102no—
325C22—102no—
326C22—2016no—
327C22—202no—
328C22—202no—
329C22#408414yes—
330C22C2×D4×D511yescharacteristic
331C22C2×#71111yescharacteristic
332C22D5×#4112yes—
333C22#414912yes—
334C22C22×C22⋊D511yescharacteristic
335C4—24no—
336C4—42no—
337C4—41no—
338C4—202no—
339C4—201no—
340C4—202no—
341C4—201no—
342C2—24no—
343C2—81no—
344C2—104no—
345C2—202no—
346C2D5×#12311yescharacteristic
347C2#6598711yescharacteristic
348C2#6599212yes—
349C2C2×#408411yescharacteristic
350C2#6616312yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC2×#4083C2
hypercenterC2×#123D5
hypocenterC5#2412
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C5×#123C2semidirect
3. C22×#685C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. #65509C2semidirect
6. #65500C2semidirect
7. #65282C2semidirect
8. #65276C2semidirect
9. #65250C2semidirect
10. #65245C2semidirect
11. #65239C2semidirect
20. C2×C5×C4⋊C4C22semidirect
22. C2×C5×C22⋊C4C22semidirect
25. C2×C22⋊Dic5C22semidirect
33. C2×C4⋊Dic5C22semidirect
35. #3963C22semidirect
36. #3963C22semidirect
37. #3963C22semidirect
38. #3963C22semidirect
73. C5×C22⋊C4D4semidirect
83. C2×#123D5semidirect
155. C2×C4⋊C4C2×D5semidirect
167. C2×C22⋊C4C2×D5semidirect
197. C2×D5#123semidirect
247. C22⋊C4C22⋊D5semidirect
308. C5#2412semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×#40832
C22×#6852
C2×D5×C22⋊C42
#655092
#655002
#652822
#652762
#652502
#652452
#652392
#24125