Finite Group Databasebeta

← #10606421

Subgroups — #10606421

Order 640.

814 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 157 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×#4083C211yesmaximal, characteristic
2C2×#4081C211yesmaximal, characteristic, complemented
3C22×#685C211yesmaximal, characteristic, complemented
4C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
5C5×#386C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic, complemented
7#65248C212yesmaximal, complemented
8#65245C212yesmaximal, complemented
9#65232C211yesmaximal, characteristic, complemented
10#64964C212yesmaximal, complemented
11#64956C212yesmaximal, complemented
12C24×D5C2211yescharacteristic
13C22×C5×D4C2211yescharacteristic
14C23×C4×C5C2211yescharacteristic
15C22×C22⋊D5C2211yescharacteristic
16C22×C22⋊D5C2211yescharacteristic
17C23×Dic5C2211yescharacteristic
18C22×C4×D5C2212yes—
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2212yescomplemented
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2211yescharacteristic
23C2×C22⋊Dic5C2212yescomplemented
24#4083—24no—
25#4081—24no—
26C2×#685—24no—
27C2×#685C2212yes—
28C2×#685C2212yes—
29C2×#685C2212yes—
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#683C2212yescomplemented
33C2×#683C2212yescomplemented
34D5×C22⋊C4—24no—
35C5×#95C2212yescomplemented
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—42no—
54C22×C4×C5C2312yes—
55C22×C4×C5C2312yes—
56C22×C4×C5C2311yescharacteristic
57C2×C22⋊D5—22no—
58C2×C22⋊D5—22no—
59C2×C22⋊D5—24no—
60C2×C22⋊D5—42no—
61C22×Dic5—22no—
62C22×Dic5—22no—
63C22×Dic5—22no—
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2312yes—
67C22×Dic5C2312yes—
68C2×C4×D5—24no—
69C2×C4×D5—24no—
70C2×C4×D5—42no—
71C5×C22⋊C4—22no—
72C5×C22⋊C4—42no—
73C22⋊Dic5—22no—
74C22⋊Dic5—24no—
75C22⋊Dic5D414yescomplemented
76#685—24no—
77#685—24no—
78#685—22no—
79#685—22no—
80#685—24no—
81#683—24no—
82#683—42no—
83C2×#123—51no—
84C2×#120—51no—
85C22×C22⋊C4—51no—
86C22×C22⋊C4—51no—
87#397—52no—
88#394—52no—
89#386—52no—
90#386D511yescharacteristic, complemented
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—21no—
122C2×C4×C5—22no—
123C2×C4×C5—21no—
124C2×C4×C5—22no—
125C2×C4×C5—22no—
126C2×C4×C5—21no—
127C2×C4×C5—22no—
128C2×C4×C5—22no—
129C2×C4×C5—22no—
130C2×C4×C5—44no—
131C22⋊D5—44no—
132C22⋊D5—44no—
133C2×Dic5—22no—
134C2×Dic5—22no—
135C2×Dic5—22no—
136C2×Dic5—22no—
137C2×Dic5—21no—
138C2×Dic5—21no—
139C2×Dic5—22no—
140C2×Dic5—22no—
141C2×Dic5—22no—
142C2×Dic5—44no—
143C2×Dic5C2×D414yes—
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—104no—
153#120—104no—
154C2×C4⋊C4—52no—
155C2×C4⋊C4—52no—
156C2×C22⋊C4—51no—
157C2×C22⋊C4—51no—
158C2×C22⋊C4—52no—
159C2×C22⋊C4—52no—
160C2×C22⋊C4—52no—
161C2×C22⋊C4—52no—
162C2×C22⋊C4—51no—
163C2×C22⋊C4—51no—
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#95C2×D512yescomplemented
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—41no—
199C4×C5—42no—
200C4×C5—42no—
201Dic5—24no—
202Dic5—42no—
203Dic5—41no—
204Dic5—41no—
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—42no—
223C22×C4—52no—
224C22×C4—51no—
225C22×C4—52no—
226C22×C4—51no—
227C22×C4—104no—
228C22×C4—104no—
229C22×C4—102no—
230C22×C4—102no—
231C22×C4—102no—
232C22×C4—202no—
233C22×C4C22×D512yes—
234C22×C4C22×D512yes—
235C22×C4C22×D511yescharacteristic
236C4⋊C4—104no—
237C4⋊C4—202no—
238C22⋊C4—22no—
239C22⋊C4—42no—
240C22⋊C4—54no—
241C22⋊C4—104no—
242C22⋊C4—104no—
243C22⋊C4—104no—
244C22⋊C4—102no—
245C22⋊C4—102no—
246C22⋊C4—104no—
247C22⋊C4—102no—
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—21no—
262C23—22no—
263C23—22no—
264C23—52no—
265C23—54no—
266C23—52no—
267C23—108no—
268C23—101no—
269C23—101no—
270C23—102no—
271C23—101no—
272C23—101no—
273C23—102no—
274C23—102no—
275C23—108no—
276C23—204no—
277C23—208no—
278C23—208no—
279C23#70912yes—
280C23D4×D512yes—
281C23D4×D512yes—
282C23C23×D511yescharacteristic
283D4—44no—
284D4—204no—
285D4—204no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—54no—
297C2×C4—102no—
298C2×C4—102no—
299C2×C4—102no—
300C2×C4—102no—
301C2×C4—101no—
302C2×C4—101no—
303C2×C4—102no—
304C2×C4—102no—
305C2×C4—102no—
306C2×C4—208no—
307C2×C4—204no—
308C5#241211yescharacteristic, complemented, Sylow
309C22—24no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—54no—
319C22—102no—
320C22—104no—
321C22—102no—
322C22—104no—
323C22—102no—
324C22—102no—
325C22—102no—
326C22—2016no—
327C22—202no—
328C22—202no—
329C22#403014yes—
330C22C2×#70911yescharacteristic
331C22C2×D4×D511yescharacteristic
332C22C2×D4×D511yescharacteristic
333C22#414412yes—
334C22D5×#4112yes—
335C4—41no—
336C4—42no—
337C4—42no—
338C4—104no—
339C4—202no—
340C4—201no—
341C4—201no—
342C2—24no—
343C2—81no—
344C2—104no—
345C2—202no—
346C2C2×#403011yescharacteristic
347C2#6589012yes—
348C2#6592311yescharacteristic
349C2#6594412yes—
350C2D5×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#386C2
prime radicalC2×#4083C2
hypercenter#386D5
hypocenterC5#2412
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#4081C2semidirect
3. C22×#685C2semidirect
4. C2×D5×C22⋊C4C2semidirect
5. C5×#386C2semidirect
6. #65499C2semidirect
7. #65248C2semidirect
8. #65245C2semidirect
9. #65232C2semidirect
10. #64964C2semidirect
11. #64956C2semidirect
20. C2×C5×C22⋊C4C22semidirect
23. C2×C22⋊Dic5C22semidirect
32. C2×#683C22semidirect
33. C2×#683C22semidirect
35. C5×#95C22semidirect
36. #3963C22semidirect
37. #3963C22semidirect
38. #3963C22semidirect
75. C22⋊Dic5D4semidirect
90. #386D5semidirect
167. C2×C22⋊C4C2×D5semidirect
171. #95C2×D5semidirect
197. C2×D5#123semidirect
308. C5#2412semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C2×#40812
C22×#6852
C2×D5×C22⋊C42
C5×#3862
#654992
#652482
#652452
#652322
#649642
#649562
#24125