Finite Group Databasebeta

← #10608512

Subgroups — #10608512

Order 640.

852 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C22×Q8×D5C211yesmaximal, characteristic, complemented
2C2×#4031C212yesmaximal, complemented
3C2×#4030C211yesmaximal, characteristic
4C5×#393C211yesmaximal, characteristic, complemented
5#65520C211yesmaximal, characteristic, complemented
6#65245C212yesmaximal, complemented
7#65236C211yesmaximal, characteristic, complemented
8#64962C211yesmaximal, characteristic, complemented
9#64955C211yesmaximal, characteristic, complemented
10#64954C212yesmaximal, complemented
11#64935C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C22×C22⋊D5C2211yescharacteristic
15C2×Q8×D5—24no—
16C2×Q8×D5—28no—
17C22×D20C2211yescharacteristic
18C22×C4×D5C2212yes—
19C22×C4×D5C2211yescharacteristic
20C22×Dic10C2211yescharacteristic
21C22×Dic10C2212yes—
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2212yes—
24C2×C22⋊Dic5C2212yes—
25C2×#685C2211yescharacteristic
26C2×#685C2212yes—
27C2×#685C2212yes—
28C2×#685C2212yes—
29C2×#683C2211yescharacteristic
30C2×C4×Dic5C2212yes—
31#4031—28no—
32#4030—24no—
33C5×#95C2212yes—
34C5×#95C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#3963C2211yescharacteristic
39#2436—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C23×D5C2311yescharacteristic
42C23×D5C2311yescharacteristic
43C2×C5×Q8—22no—
44C2×C5×Q8—24no—
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C22×C4×C5C2311yescharacteristic
48C22×C4×C5C2312yes—
49C2×C22⋊D5—22no—
50C2×C22⋊D5—24no—
51C2×C22⋊D5—22no—
52C2×C22⋊D5—24no—
53C22×Dic5C2312yes—
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2312yes—
56C22×Dic5C2311yescharacteristic
57Q8×D5—232no—
58C2×D20—22no—
59C2×D20—24no—
60C2×C4×D5—24no—
61C2×C4×D5—24no—
62C2×C4×D5—24no—
63C2×C4×D5—24no—
64C2×C4×D5D414yes—
65C2×Dic10—22no—
66C2×Dic10—24no—
67C2×Dic10—24no—
68C2×Dic10—24no—
69C2×Dic10D418yescomplemented
70C5×C22⋊C4—22no—
71C5×C22⋊C4—24no—
72C22⋊Dic5—24no—
73#685—24no—
74#685—22no—
75#685—24no—
76#685—24no—
77#683—22no—
78C4×Dic5—24no—
79C23×Q8—51no—
80C2×#124—52no—
81C2×#121—51no—
82#393—51no—
83#393—51no—
84#393—51no—
85#393D511yescharacteristic, complemented
86#391—52no—
87#386—52no—
88#386—52no—
89C4×C22⋊C4—51no—
90C23×C5—22no—
91C23×C5—21no—
92C23×C5—22no—
93C23×C5—21no—
94C23×C5—21no—
95C23×C5C2411yescharacteristic
96C22×D5—22no—
97C22×D5—21no—
98C22×D5—21no—
99C22×D5—22no—
100C22×D5—21no—
101C22×D5—22no—
102C22×D5—21no—
103C22×D5—21no—
104C22×D5C2×D412yes—
105C22×D5#4114yes—
106C5×Q8—28no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—21no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5C2×D412yes—
116C2×C4×C5C2×D414yes—
117C22⋊D5—44no—
118C22⋊D5—44no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—21no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5C2×D414yes—
127C2×Dic5C2×D414yes—
128C2×Dic5C2×D412yes—
129D20—44no—
130C4×D5—216no—
131C4×D5—28no—
132Dic10—28no—
133Dic10—216no—
134C22×Q8—52no—
135C22×Q8—51no—
136C22×Q8—104no—
137C22×Q8—108no—
138C22×Q8C2×D511yescharacteristic
139C22×D4—51no—
140C22×D4—51no—
141C22×D4—51no—
142C23×C4—51no—
143C23×C4—52no—
144#124—108no—
145#121—104no—
146C2×C4⋊C4—51no—
147C2×C22⋊C4—52no—
148C2×C22⋊C4—51no—
149C2×C22⋊C4—52no—
150C2×C22⋊C4—52no—
151C2×C22⋊C4—52no—
152C2×C22⋊C4C2×D511yescharacteristic
153C2×C22⋊C4C2×D512yes—
154C2×C42—52no—
155#95—51no—
156#95—52no—
157#95—52no—
158#95—52no—
159#95C2×D511yescharacteristic
160#95C2×D512yes—
161C22×C5—41no—
162C22×C5—41no—
163C22×C5—41no—
164C22×C5—42no—
165C22×C5—42no—
166C22×C5C22×D411yescharacteristic
167C22×C5C22×D412yes—
168C22×C5C2×#4112yes—
169C22×C52−1+411yescharacteristic
170C22×C52−1+411yescharacteristic
171C2×D5—22no—
172C2×D5—22no—
173C2×D5—24no—
174C2×D5—24no—
175C2×D5—41no—
176C2×D5—41no—
177C2×D5—41no—
178C2×D5—42no—
179C2×D5—42no—
180C2×D5#12414yescomplemented
181C4×C5—22no—
182C4×C5—24no—
183C4×C5—41no—
184C4×C5—42no—
185Dic5—24no—
186Dic5—41no—
187Dic5—42no—
188Dic5#12014yes—
189C24—51no—
190C24—51no—
191C24C22×D511yescharacteristic
192C2×Q8—22no—
193C2×Q8—24no—
194C2×Q8—58no—
195C2×Q8—104no—
196C2×Q8—104no—
197C2×Q8—104no—
198C2×Q8—1032no—
199C2×Q8—102no—
200C2×D4—102no—
201C2×D4—104no—
202C2×D4—102no—
203C2×D4—104no—
204C2×D4—102no—
205C2×D4—104no—
206C22×C4—51no—
207C22×C4—52no—
208C22×C4—54no—
209C22×C4—51no—
210C22×C4—52no—
211C22×C4—104no—
212C22×C4—104no—
213C22×C4—104no—
214C22×C4—104no—
215C22×C4C22×D511yescharacteristic
216C22×C4C22×D512yes—
217C22×C4C22×D511yescharacteristic
218C22×C4C22×D512yes—
219C4⋊C4—102no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—104no—
223C22⋊C4—104no—
224C22⋊C4—104no—
225C22⋊C4—102no—
226C22⋊C4—104no—
227C42—104no—
228C2×C5—81no—
229C2×C5C2×#12011yescharacteristic
230C2×C5C2×#12411yescharacteristic
231C2×C5#53611yescharacteristic
232C2×C5#54812yes—
233C2×C5#54812yes—
234D5—24no—
235D5—81no—
236C23—22no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—52no—
242C23—54no—
243C23—102no—
244C23—101no—
245C23—101no—
246C23—102no—
247C23—101no—
248C23—101no—
249C23—102no—
250C23—101no—
251C23C23×D511yescharacteristic
252Q8—28no—
253Q8—1016no—
254Q8—108no—
255D4—204no—
256D4—204no—
257D4—204no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—54no—
267C2×C4—52no—
268C2×C4—54no—
269C2×C4—108no—
270C2×C4—1016no—
271C2×C4—102no—
272C2×C4—102no—
273C2×C4—102no—
274C2×C4—101no—
275C2×C4—102no—
276C2×C4—102no—
277C2×C4—102no—
278C2×C4D4×D512yes—
279C2×C4D4×D514yes—
280C5#243611yescharacteristic, complemented, Sylow
281C22—41no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—54no—
287C22—102no—
288C22—102no—
289C22—104no—
290C22—104no—
291C22—201no—
292C22—201no—
293C22—202no—
294C22—201no—
295C22—202no—
296C22C2×D4×D511yescharacteristic
297C22C2×D4×D512yes—
298C22#414711yescharacteristic
299C22D5×#4112yes—
300C22#415011yescharacteristic
301C4—22no—
302C4—24no—
303C4—41no—
304C4—42no—
305C4—54no—
306C4—104no—
307C4—202no—
308C4—201no—
309C2—81no—
310C2—104no—
311C2—401no—
312C2D5×#12011yescharacteristic
313C2#6596211yescharacteristic
314C2#6596712yes—
315C2D5×#12411yescharacteristic
316C2#6601112yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC2×#4030C2
hypercenter#393D5
hypocenterC5#2436
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D5C2semidirect
2. C2×#4031C2semidirect
4. C5×#393C2semidirect
5. #65520C2semidirect
6. #65245C2semidirect
7. #65236C2semidirect
8. #64962C2semidirect
9. #64955C2semidirect
10. #64954C2semidirect
11. #64935C2semidirect
69. C2×Dic10D4semidirect
85. #393D5semidirect
180. C2×D5#124semidirect
280. C5#2436semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D52
C2×#40312
C2×#40302
C5×#3932
#655202
#652452
#652362
#649622
#649552
#649542
#649352
#24365