Finite Group Databasebeta

← #10609768

Subgroups — #10609768

Order 640.

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

normal subgroups only

This group has 676 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
1C2×#4042C212yesmaximal, complemented
2C2×#4034C211yesmaximal, characteristic, complemented
3C2×#4027C211yesmaximal, characteristic
4C2×#4024C212yesmaximal, complemented
5C5×#399C211yesmaximal, characteristic, complemented
6#65281C211yesmaximal, characteristic, complemented
7#65265C212yesmaximal, complemented
8#65246C212yesmaximal
9#65238C211yesmaximal, characteristic, complemented
10#64966C211yesmaximal, characteristic, complemented
11#64938C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C23×Dic5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2212yes—
18C2×C5×C22⋊C4C2212yes—
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic5C2212yes—
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×#683C2212yes—
27C2×#683C2211yescharacteristic
28C2×#683C2212yes—
29C2×#683C2212yes—
30C2×#683C2212yes—
31C2×C4×Dic5C2212yes—
32C2×C4×Dic5C2211yescharacteristic
33#4042—28no—
34#4034—24no—
35#4027—24no—
36#4024—28no—
37C5×#95C2211yescharacteristic
38#3963C2211yescharacteristic
39#2542—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C23×D5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C2×C22⋊D5—22no—
47C2×C22⋊D5—22no—
48C2×C22⋊D5Q814yescomplemented
49C22×Dic5—24no—
50C22×Dic5—22no—
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2312yes—
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2311yescharacteristic
56C2×C4×D5—22no—
57C2×C4×D5—24no—
58C2×Dic10—24no—
59C2×Dic10—24no—
60C2×Dic10—24no—
61C5×C4⋊C4—22no—
62C5×C4⋊C4—24no—
63C5×C22⋊C4—24no—
64C5×C22⋊C4—22no—
65C22⋊Dic5—24no—
66#685—22no—
67#685—24no—
68#685—24no—
69C4⋊Dic5—24no—
70C4⋊Dic5—22no—
71#683—24no—
72#683—22no—
73#683—24no—
74#683—24no—
75#683D418yescomplemented
76C4×Dic5—24no—
77C4×Dic5D414yes—
78C2×C4⋊Q8—51no—
79C2×C22⋊Q8—52no—
80C2×C22⋊Q8—52no—
81C2×C4×D4—51no—
82#399—51no—
83#399—51no—
84#399—51no—
85#399D511yescharacteristic, complemented
86#390—52no—
87#384—52no—
88C4×C4⋊C4—51no—
89C23×C5—22no—
90C23×C5—21no—
91C23×C5—21no—
92C23×C5—22no—
93C23×C5C2×Q812yes—
94C23×C5C2411yescharacteristic
95C22×D5—22no—
96C22×D5—21no—
97C22×D5—21no—
98C22×D5—22no—
99C22×D5C2×Q812yes—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—21no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5C2×D412yes—
109C2×C4×C5C2×D414yes—
110C22⋊D5—28no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—21no—
117C2×Dic5—22no—
118C2×Dic5—44no—
119C2×Dic5C2×D414yes—
120C2×Dic5C2×D412yes—
121C2×Dic5C2×D412yes—
122C2×Dic5C2×D414yes—
123C2×Dic5C2×Q812yes—
124C2×Dic5#4112yes—
125C4×D5—44no—
126Dic10—48no—
127C22×Q8—52no—
128C22×D4—51no—
129C23×C4—51no—
130C23×C4—51no—
131C4⋊Q8—104no—
132C22⋊Q8—108no—
133C22⋊Q8—108no—
134C4×D4—104no—
135C2×C4⋊C4—52no—
136C2×C4⋊C4—51no—
137C2×C4⋊C4—52no—
138C2×C4⋊C4—52no—
139C2×C4⋊C4—52no—
140C2×C4⋊C4—52no—
141C2×C4⋊C4—51no—
142C2×C4⋊C4C2×D512yes—
143C2×C4⋊C4C2×D511yescharacteristic
144C2×C22⋊C4—52no—
145C2×C22⋊C4—51no—
146C2×C22⋊C4—52no—
147C2×C22⋊C4—52no—
148C2×C22⋊C4C2×D511yescharacteristic
149C2×C22⋊C4C2×D512yes—
150C2×C42—51no—
151C2×C42—52no—
152#95—51no—
153#95C2×D511yescharacteristic
154C22×C5—22no—
155C22×C5—24no—
156C22×C5—41no—
157C22×C5—41no—
158C22×C5—42no—
159C22×C5C22×D411yescharacteristic
160C22×C5C22×D412yes—
161C22×C5C22×Q811yescharacteristic
162C22×C5C2×#4111yescharacteristic
163C22×C52−1+412yes—
164C2×D5—22no—
165C2×D5—24no—
166C2×D5—41no—
167C2×D5—41no—
168C2×D5—42no—
169C4×C5—22no—
170C4×C5—24no—
171C4×C5—42no—
172C4×C5—41no—
173Dic5—24no—
174Dic5—42no—
175Dic5—41no—
176Dic5C22⋊Q814yescomplemented
177Dic5#12714yes—
178C24—51no—
179C24C22×D511yescharacteristic
180C2×Q8—104no—
181C2×Q8—104no—
182C2×Q8—104no—
183C2×D4—54no—
184C2×D4—102no—
185C2×D4—102no—
186C22×C4—51no—
187C22×C4—51no—
188C22×C4—52no—
189C22×C4—52no—
190C22×C4—51no—
191C22×C4—102no—
192C22×C4—104no—
193C22×C4—102no—
194C22×C4—104no—
195C22×C4C22×D512yes—
196C22×C4C22×D511yescharacteristic
197C22×C4C22×D511yescharacteristic
198C22×C4C22×D512yes—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—58no—
202C4⋊C4—102no—
203C4⋊C4—104no—
204C4⋊C4—104no—
205C4⋊C4—104no—
206C4⋊C4—104no—
207C4⋊C4—102no—
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—104no—
211C22⋊C4—104no—
212C22⋊C4—102no—
213C22⋊C4—104no—
214C42—54no—
215C42—104no—
216C2×C5—42no—
217C2×C5C2×C22⋊Q811yescharacteristic
218C2×C5C2×#12711yescharacteristic
219C2×C5#53611yescharacteristic
220C2×C5#54712yes—
221C2×C5D4×Q812yes—
222D5—42no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—52no—
228C23—102no—
229C23—101no—
230C23—101no—
231C23—102no—
232C23Q8×D512yes—
233C23C23×D511yescharacteristic
234Q8—208no—
235D4—108no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—54no—
247C2×C4—52no—
248C2×C4—52no—
249C2×C4—54no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4—101no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4—102no—
257C2×C4—204no—
258C2×C4—204no—
259C2×C4D4×D512yes—
260C2×C4D4×D514yes—
261C5#254211yescharacteristic, complemented, Sylow
262C22—22no—
263C22—24no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—102no—
268C22—104no—
269C22—201no—
270C22—201no—
271C22—202no—
272C22C2×D4×D511yescharacteristic
273C22C2×D4×D512yes—
274C22C2×Q8×D511yescharacteristic
275C22D5×#4111yescharacteristic
276C22#415012yes—
277C4—22no—
278C4—24no—
279C4—41no—
280C4—42no—
281C4—54no—
282C4—54no—
283C4—104no—
284C4—201no—
285C4—202no—
286C2—42no—
287C2—202no—
288C2D5×C22⋊Q811yescharacteristic
289C2#6596612yes—
290C2#6598211yescharacteristic
291C2#6598912yes—
292C2D5×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC22×C22⋊D5C22
hypercenter#399D5
hypocenterC5#2542
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4042C2semidirect
2. C2×#4034C2semidirect
4. C2×#4024C2semidirect
5. C5×#399C2semidirect
6. #65281C2semidirect
7. #65265C2semidirect
9. #65238C2semidirect
10. #64966C2semidirect
11. #64938C2semidirect
48. C2×C22⋊D5Q8semidirect
75. #683D4semidirect
85. #399D5semidirect
176. Dic5C22⋊Q8semidirect
261. C5#2542semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×#40342
C2×#40272
C2×#40242
C5×#3992
#652812
#652652
#652462
#652382
#649662
#649382
#25425