Finite Group Databasebeta

← #10609270

Subgroups — #10609270

Order 640.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 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×#4038C211yesmaximal, characteristic, complemented
2C2×#4030C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65250C212yesmaximal, complemented
6#65246C212yesmaximal, complemented
7#65238C211yesmaximal, characteristic, complemented
8#64966C211yesmaximal, characteristic, complemented
9#64961C212yesmaximal, complemented
10#64956C211yesmaximal, characteristic, complemented
11#64936C212yesmaximal, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C22×D20C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2212yes—
27C2×#683C2211yescharacteristic
28C2×#683C2212yes—
29C2×#683C2212yes—
30C2×C4×Dic5C2212yes—
31#4038—24no—
32#4030—24no—
33C5×#95C2212yes—
34C5×#95C2211yescharacteristic
35#3963C2212yes—
36#3963C2211yescharacteristic
37#2624—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C2×C22⋊D5—22no—
46C2×C22⋊D5—22no—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—42no—
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C2×D20—22no—
54C2×D20—42no—
55C2×C4×D5—24no—
56C2×C4×D5D414yes—
57C5×C4⋊C4—22no—
58C5×C22⋊C4—22no—
59C5×C22⋊C4—42no—
60C22⋊Dic5—42no—
61#685—24no—
62#685—22no—
63#685—42no—
64#685—42no—
65#685—42no—
66C4⋊Dic5—22no—
67C4⋊Dic5—42no—
68#683—22no—
69#683—24no—
70#683—42no—
71C4×Dic5—24no—
72C2×#121—51no—
73C2×#117—51no—
74#399—51no—
75#399—51no—
76#397—51no—
77#397D511yescharacteristic, complemented
78#394—52no—
79#394—52no—
80#390—52no—
81#389—52no—
82#380—51no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—21no—
87C23×C5—21no—
88C23×C5C2411yescharacteristic
89C22×D5—22no—
90C22×D5—21no—
91C22×D5—22no—
92C22×D5—21no—
93C22×D5—21no—
94C22×D5—22no—
95C22×D5—21no—
96C22×D5—22no—
97C22×D5—21no—
98C22×D5C2×D412yes—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—21no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5C2×D412yes—
109C22⋊D5—44no—
110C22⋊D5—44no—
111C2×Dic5—22no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5C2×D412yes—
120C2×Dic5#4114yes—
121D20—44no—
122C4×D5—28no—
123C22×D4—51no—
124C22×D4—51no—
125C22×D4—51no—
126C23×C4—51no—
127#121—104no—
128#117—104no—
129C2×C4⋊C4—51no—
130C2×C4⋊C4—51no—
131C2×C4⋊C4—52no—
132C2×C4⋊C4—52no—
133C2×C4⋊C4—52no—
134C2×C4⋊C4C2×D511yescharacteristic
135C2×C22⋊C4—52no—
136C2×C22⋊C4—51no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4C2×D511yescharacteristic
142C2×C22⋊C4C2×D512yes—
143C2×C42—52no—
144#95—51no—
145#95—52no—
146#95C2×D511yescharacteristic
147#95C2×D512yes—
148C22×C5—41no—
149C22×C5—41no—
150C22×C5—41no—
151C22×C5—42no—
152C22×C5—42no—
153C22×C5C22×D411yescharacteristic
154C22×C5C2×#4112yes—
155C22×C5H(5,𝔽2)11yescharacteristic
156C22×C5H(5,𝔽2)12yes—
157C22×C52−1+411yescharacteristic
158C2×D5—22no—
159C2×D5—22no—
160C2×D5—22no—
161C2×D5—42no—
162C2×D5—42no—
163C2×D5—41no—
164C2×D5—41no—
165C2×D5—41no—
166C2×D5—42no—
167C2×D5—42no—
168C4×C5—22no—
169C4×C5—41no—
170C4×C5—42no—
171C4×C5—42no—
172Dic5—42no—
173Dic5—41no—
174Dic5—42no—
175Dic5#12314yes—
176C24—51no—
177C24—51no—
178C24C22×D511yescharacteristic
179C2×D4—102no—
180C2×D4—102no—
181C2×D4—104no—
182C2×D4—102no—
183C2×D4—202no—
184C2×D4—202no—
185C22×C4—51no—
186C22×C4—54no—
187C22×C4—52no—
188C22×C4—51no—
189C22×C4—52no—
190C22×C4—104no—
191C22×C4C22×D512yes—
192C22×C4C22×D512yes—
193C22×C4C22×D511yescharacteristic
194C22×C4C22×D511yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—102no—
197C4⋊C4—104no—
198C4⋊C4—102no—
199C4⋊C4—202no—
200C4⋊C4—202no—
201C22⋊C4—22no—
202C22⋊C4—42no—
203C22⋊C4—104no—
204C22⋊C4—102no—
205C22⋊C4—202no—
206C22⋊C4—202no—
207C22⋊C4—202no—
208C22⋊C4—202no—
209C42—104no—
210C2×C5—81no—
211C2×C5C2×#12311yescharacteristic
212C2×C5#53511yescharacteristic
213C2×C5#53711yescharacteristic
214C2×C5#54012yes—
215C2×C5#54212yes—
216D5—42no—
217D5—81no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—21no—
223C23—52no—
224C23—101no—
225C23—102no—
226C23—102no—
227C23—101no—
228C23—101no—
229C23—101no—
230C23—102no—
231C23—102no—
232C23—101no—
233C23C23×D511yescharacteristic
234D4—204no—
235D4—204no—
236D4—204no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—52no—
247C2×C4—54no—
248C2×C4—108no—
249C2×C4—102no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4—102no—
257C2×C4D4×D512yes—
258C5#262411yescharacteristic, complemented, Sylow
259C22—41no—
260C22—41no—
261C22—42no—
262C22—41no—
263C22—42no—
264C22—102no—
265C22—102no—
266C22—102no—
267C22—202no—
268C22—202no—
269C22—201no—
270C22—201no—
271C22—202no—
272C22—201no—
273C22—202no—
274C22C2×D4×D511yescharacteristic
275C22#414411yescharacteristic
276C22D5×#4112yes—
277C22#414912yes—
278C22#415011yescharacteristic
279C4—22no—
280C4—42no—
281C4—42no—
282C4—41no—
283C4—54no—
284C4—202no—
285C4—201no—
286C4—202no—
287C2—81no—
288C2—202no—
289C2—401no—
290C2#6594211yescharacteristic
291C2D5×#12311yescharacteristic
292C2#6599712yes—
293C2#6600811yescharacteristic
294C2#6604412yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC2×#4030C2
hypercenter#397D5
hypocenterC5#2624
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#4038C2semidirect
3. C5×#397C2semidirect
4. #65282C2semidirect
5. #65250C2semidirect
6. #65246C2semidirect
7. #65238C2semidirect
8. #64966C2semidirect
9. #64961C2semidirect
10. #64956C2semidirect
11. #64936C2semidirect
77. #397D5semidirect
258. C5#2624semidirect

Maximal subgroups

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

subgroupindex
C2×#40382
C2×#40302
C5×#3972
#652822
#652502
#652462
#652382
#649662
#649612
#649562
#649362
#26245