Finite Group Databasebeta

← #10608511

Subgroups — #10608511

Order 640.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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×#4027C211yesmaximal, characteristic
3C2×#4024C212yesmaximal, complemented
4C5×#393C211yesmaximal, characteristic, complemented
5#65518C211yesmaximal, characteristic, complemented
6#65245C212yesmaximal
7#65236C211yesmaximal, characteristic, complemented
8#64957C212yesmaximal, complemented
9#64956C211yesmaximal, characteristic, complemented
10#64945C211yesmaximal, characteristic, complemented
11#64935C212yesmaximal, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C23×Dic5C2211yescharacteristic
15C2×Q8×D5—24no—
16C2×Q8×D5—28no—
17C22×C4×D5C2212yes—
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic
20C22×Dic10C2212yes—
21C2×C5×C22⋊C4C2212yes—
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic5C2212yes—
24C2×#685C2211yescharacteristic
25C2×#685C2212yes—
26C2×C4⋊Dic5C2212yes—
27C2×#683C2212yes—
28C2×#683C2212yes—
29C2×#683C2211yescharacteristic
30C2×C4×Dic5C2211yescharacteristic
31#4027—24no—
32#4024—28no—
33C5×#95C2212yes—
34C5×#95C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#3963C2211yescharacteristic
39#2409—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C23×D5C2311yescharacteristic
42C2×C5×Q8—22no—
43C2×C5×Q8—24no—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2312yes—
48C2×C22⋊D5—22no—
49C2×C22⋊D5—22no—
50C2×C22⋊D5Q814yescomplemented
51C22×Dic5—24no—
52C22×Dic5—22no—
53C22×Dic5C2312yes—
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2312yes—
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2311yescharacteristic
58Q8×D5—232no—
59C2×C4×D5—24no—
60C2×C4×D5—24no—
61C2×C4×D5—24no—
62C2×C4×D5—24no—
63C2×C4×D5D414yes—
64C2×Dic10—22no—
65C2×Dic10—24no—
66C2×Dic10—24no—
67C2×Dic10—24no—
68C2×Dic10D418yescomplemented
69C5×C22⋊C4—24no—
70C5×C22⋊C4—22no—
71C22⋊Dic5—24no—
72#685—22no—
73#685—24no—
74C4⋊Dic5—24no—
75#683—24no—
76#683—22no—
77#683—24no—
78C4×Dic5—22no—
79C23×Q8—51no—
80C2×C22⋊Q8—52no—
81C2×C4×D4—51no—
82#395—51no—
83#395—51no—
84#393—51no—
85#393D511yescharacteristic, complemented
86#391—52no—
87#386—52no—
88#385—52no—
89#380—51no—
90C23×C5—22no—
91C23×C5—21no—
92C23×C5—21no—
93C23×C5—22no—
94C23×C5C2×Q812yescomplemented
95C23×C5C2411yescharacteristic
96C22×D5—22no—
97C22×D5—22no—
98C22×D5C2×D412yes—
99C22×D5C2×Q812yes—
100C22×D5#4112yes—
101C5×Q8—28no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5C2×D412yes—
111C2×C4×C5C2×D414yes—
112C22⋊D5—28no—
113C2×Dic5—21no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—21no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—44no—
123C2×Dic5C2×D414yes—
124C2×Dic5C2×D412yes—
125C2×Dic5C2×D414yes—
126C2×Dic5C2×Q812yescomplemented
127C4×D5—216no—
128C4×D5—28no—
129Dic10—28no—
130Dic10—216no—
131C22×Q8—52no—
132C22×Q8—51no—
133C22×Q8—104no—
134C22×Q8—108no—
135C22×Q8C2×D511yescharacteristic
136C22×D4—51no—
137C23×C4—51no—
138C23×C4—51no—
139C23×C4—52no—
140C22⋊Q8—108no—
141C4×D4—104no—
142C2×C4⋊C4—51no—
143C2×C4⋊C4—52no—
144C2×C4⋊C4—52no—
145C2×C4⋊C4—52no—
146C2×C22⋊C4—52no—
147C2×C22⋊C4—51no—
148C2×C22⋊C4—52no—
149C2×C22⋊C4C2×D511yescharacteristic
150C2×C22⋊C4C2×D512yes—
151C2×C42—51no—
152#95—51no—
153#95—52no—
154#95—52no—
155#95—52no—
156#95C2×D511yescharacteristic
157#95C2×D512yes—
158C22×C5—22no—
159C22×C5—24no—
160C22×C5—41no—
161C22×C5—41no—
162C22×C5—42no—
163C22×C5C22×D411yescharacteristic
164C22×C5C22×D412yes—
165C22×C5C22×Q811yescharacteristic
166C22×C5C2×#4111yescharacteristic
167C22×C52−1+412yes—
168C2×D5—22no—
169C2×D5—22no—
170C2×D5—24no—
171C2×D5—24no—
172C2×D5C22⋊Q814yescomplemented
173C4×C5—22no—
174C4×C5—24no—
175C4×C5—41no—
176C4×C5—42no—
177Dic5—24no—
178Dic5—22no—
179Dic5—41no—
180Dic5—42no—
181Dic5#12014yes—
182C24—51no—
183C24C22×D511yescharacteristic
184C2×Q8—22no—
185C2×Q8—24no—
186C2×Q8—58no—
187C2×Q8—104no—
188C2×Q8—104no—
189C2×Q8—104no—
190C2×Q8—1032no—
191C2×Q8—102no—
192C2×D4—54no—
193C2×D4—102no—
194C2×D4—102no—
195C22×C4—51no—
196C22×C4—51no—
197C22×C4—52no—
198C22×C4—54no—
199C22×C4—52no—
200C22×C4—51no—
201C22×C4—104no—
202C22×C4—104no—
203C22×C4—104no—
204C22×C4—102no—
205C22×C4—104no—
206C22×C4—104no—
207C22×C4C22×D511yescharacteristic
208C22×C4C22×D512yes—
209C22×C4C22×D511yescharacteristic
210C22×C4C22×D512yes—
211C4⋊C4—104no—
212C4⋊C4—104no—
213C4⋊C4—102no—
214C4⋊C4—104no—
215C22⋊C4—22no—
216C22⋊C4—24no—
217C22⋊C4—102no—
218C22⋊C4—104no—
219C22⋊C4—104no—
220C42—102no—
221C2×C5—42no—
222C2×C5C2×#12011yescharacteristic
223C2×C5C2×C22⋊Q811yescharacteristic
224C2×C5#53611yescharacteristic
225C2×C5#54812yes—
226C2×C5D4×Q812yes—
227D5—24no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—52no—
233C23—52no—
234C23—52no—
235C23—102no—
236C23—102no—
237C23Q8×D512yescomplemented
238C23C23×D511yescharacteristic
239Q8—28no—
240Q8—1016no—
241Q8—108no—
242D4—108no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—52no—
252C2×C4—54no—
253C2×C4—54no—
254C2×C4—52no—
255C2×C4—108no—
256C2×C4—1016no—
257C2×C4—101no—
258C2×C4—102no—
259C2×C4—102no—
260C2×C4—101no—
261C2×C4—102no—
262C2×C4—101no—
263C2×C4—102no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—204no—
267C2×C4D4×D512yes—
268C2×C4D4×D514yes—
269C5#240911yescharacteristic, complemented, Sylow
270C22—22no—
271C22—24no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—54no—
276C22—102no—
277C22—104no—
278C22—102no—
279C22—104no—
280C22C2×D4×D511yescharacteristic
281C22C2×D4×D512yes—
282C22C2×Q8×D511yescharacteristic
283C22D5×#4111yescharacteristic
284C22#415012yes—
285C4—22no—
286C4—24no—
287C4—41no—
288C4—42no—
289C4—54no—
290C4—104no—
291C4—102no—
292C4—201no—
293C4—202no—
294C2—42no—
295C2—104no—
296C2D5×#12011yescharacteristic
297C2D5×C22⋊Q811yescharacteristic
298C2#6596612yes—
299C2#6600911yescharacteristic
300C2#6601112yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC22×C22⋊D5C22
hypercenter#393D5
hypocenterC5#2409
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
3. C2×#4024C2semidirect
4. C5×#393C2semidirect
5. #65518C2semidirect
7. #65236C2semidirect
8. #64957C2semidirect
9. #64956C2semidirect
10. #64945C2semidirect
11. #64935C2semidirect
50. C2×C22⋊D5Q8semidirect
68. C2×Dic10D4semidirect
85. #393D5semidirect
94. C23×C5C2×Q8semidirect
126. C2×Dic5C2×Q8semidirect
172. C2×D5C22⋊Q8semidirect
237. C23Q8×D5semidirect
269. C5#2409semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D52
C2×#40272
C2×#40242
C5×#3932
#655182
#652452
#652362
#649572
#649562
#649452
#649352
#24095