Finite Group Databasebeta

← #10609245

Subgroups — #10609245

Order 640.

604 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 151 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×#4034C211yesmaximal, characteristic, complemented
2C2×#4031C212yesmaximal
3C2×#4027C211yesmaximal, characteristic
4C5×#397C211yesmaximal, characteristic, complemented
5#65282C211yesmaximal, characteristic, complemented
6#65242C211yesmaximal, characteristic, complemented
7#65231C212yesmaximal, complemented
8#64963C211yesmaximal, characteristic, complemented
9#64956C212yesmaximal, complemented
10#64938C211yesmaximal, characteristic, complemented
11#64935C212yesmaximal, complemented
12C22×C22⋊D5C2211yescharacteristic
13C23×Dic5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4034—24no—
30#4031—28no—
31#4027—24no—
32C5×#95C2212yes—
33C5×#95C2211yescharacteristic
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#3963C2212yes—
38#2547—51nomaximal, Sylow
39C24×C5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C2×C22⋊D5—22no—
46C2×C22⋊D5—22no—
47C2×C22⋊D5D414yes—
48C22×Dic5—22no—
49C22×Dic5—24no—
50C22×Dic5C2312yes—
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2311yescharacteristic
55C2×C4×D5—22no—
56C2×C4×D5—24no—
57C2×Dic10—24no—
58C2×Dic10—24no—
59C2×Dic10—24no—
60C5×C4⋊C4—22no—
61C5×C22⋊C4—24no—
62C5×C22⋊C4—22no—
63C22⋊Dic5—24no—
64#685—24no—
65#685—24no—
66#685—22no—
67C4⋊Dic5—22no—
68#683—22no—
69#683—24no—
70#683—24no—
71C4×Dic5—24no—
72C4×Dic5D414yes—
73C2×C4⋊Q8—51no—
74C2×#124—52no—
75C2×C4×D4—51no—
76#397—51no—
77#397—51no—
78#397—51no—
79#397D511yescharacteristic, complemented
80#391—52no—
81#380—52no—
82#380—52no—
83C4×C4⋊C4—51no—
84C23×C5—21no—
85C23×C5—22no—
86C23×C5—21no—
87C23×C5—22no—
88C23×C5C2×D412yes—
89C23×C5C2411yescharacteristic
90C22×D5—21no—
91C22×D5—22no—
92C22×D5—21no—
93C22×D5—22no—
94C22×D5C2×D412yes—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—21no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5C2×D412yes—
105C22⋊D5—28no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—21no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—44no—
115C2×Dic5C2×D412yes—
116C2×Dic5C2×D412yes—
117C2×Dic5C2×D412yes—
118C2×Dic5#4112yes—
119C2×Dic5#4114yes—
120C4×D5—44no—
121Dic10—48no—
122C22×Q8—52no—
123C22×D4—51no—
124C23×C4—51no—
125C23×C4—51no—
126C4⋊Q8—104no—
127#124—108no—
128C4×D4—104no—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—52no—
133C2×C4⋊C4C2×D511yescharacteristic
134C2×C22⋊C4—52no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4C2×D511yescharacteristic
139C2×C22⋊C4C2×D512yes—
140C2×C42—51no—
141C2×C42—52no—
142#95—51no—
143#95—52no—
144#95—52no—
145#95—52no—
146#95C2×D511yescharacteristic
147#95C2×D512yes—
148C22×C5—22no—
149C22×C5—24no—
150C22×C5—41no—
151C22×C5—41no—
152C22×C5—42no—
153C22×C5C22×D411yescharacteristic
154C22×C5C22×D411yescharacteristic
155C22×C5C2×#4111yescharacteristic
156C22×C5C2×#4112yes—
157C22×C52−1+412yes—
158C2×D5—22no—
159C2×D5—24no—
160C2×D5—41no—
161C2×D5—41no—
162C2×D5—42no—
163C4×C5—22no—
164C4×C5—42no—
165C4×C5—41no—
166C4×C5—42no—
167Dic5—42no—
168Dic5—41no—
169Dic5—42no—
170Dic5#12114yes—
171Dic5#12414yescomplemented
172C24—51no—
173C24C22×D511yescharacteristic
174C2×Q8—104no—
175C2×Q8—104no—
176C2×Q8—104no—
177C2×D4—54no—
178C2×D4—102no—
179C2×D4—102no—
180C22×C4—51no—
181C22×C4—51no—
182C22×C4—51no—
183C22×C4—52no—
184C22×C4—52no—
185C22×C4—102no—
186C22×C4—104no—
187C22×C4—102no—
188C22×C4—104no—
189C22×C4C22×D511yescharacteristic
190C22×C4C22×D512yes—
191C22×C4C22×D512yes—
192C22×C4C22×D511yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—102no—
195C4⋊C4—104no—
196C4⋊C4—104no—
197C4⋊C4—102no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—104no—
201C22⋊C4—104no—
202C22⋊C4—104no—
203C22⋊C4—102no—
204C42—54no—
205C42—104no—
206C2×C5—42no—
207C2×C5C2×#12111yescharacteristic
208C2×C5C2×#12411yescharacteristic
209C2×C5#53611yescharacteristic
210C2×C5#54712yes—
211C2×C5#55612yes—
212D5—42no—
213C23—21no—
214C23—22no—
215C23—21no—
216C23—22no—
217C23—52no—
218C23—101no—
219C23—102no—
220C23—101no—
221C23—102no—
222C23D4×D512yes—
223C23C23×D511yescharacteristic
224Q8—208no—
225D4—108no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—52no—
236C2×C4—52no—
237C2×C4—52no—
238C2×C4—54no—
239C2×C4—52no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—101no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—102no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—204no—
249C2×C4—204no—
250C2×C4D4×D512yes—
251C5#254711yescharacteristic, complemented, Sylow
252C22—22no—
253C22—24no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—102no—
258C22—104no—
259C22—201no—
260C22—201no—
261C22—202no—
262C22C2×D4×D511yescharacteristic
263C22C2×D4×D511yescharacteristic
264C22D5×#4111yescharacteristic
265C22D5×#4112yes—
266C22#415012yes—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—54no—
272C4—54no—
273C4—202no—
274C4—202no—
275C4—201no—
276C2—42no—
277C2—202no—
278C2D5×#12111yescharacteristic
279C2#6598211yescharacteristic
280C2#6598912yes—
281C2D5×#12411yescharacteristic
282C2#6603612yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC22×C22⋊D5C22
hypercenter#397D5
hypocenterC5#2547
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#4034C2semidirect
4. C5×#397C2semidirect
5. #65282C2semidirect
6. #65242C2semidirect
7. #65231C2semidirect
8. #64963C2semidirect
9. #64956C2semidirect
10. #64938C2semidirect
11. #64935C2semidirect
79. #397D5semidirect
171. Dic5#124semidirect
251. C5#2547semidirect

Maximal subgroups

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

subgroupindex
C2×#40342
C2×#40312
C2×#40272
C5×#3972
#652822
#652422
#652312
#649632
#649562
#649382
#649352
#25475