Finite Group Databasebeta

← #10609418

Subgroups — #10609418

Order 640.

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

normal subgroups only

This group has 576 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×#4043C211yesmaximal, characteristic, complemented
2C2×#4030C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65250C212yesmaximal, complemented
6#65245C212yesmaximal, complemented
7#65236C211yesmaximal, characteristic, complemented
8#64962C211yesmaximal, characteristic, complemented
9#64954C212yesmaximal, complemented
10D5×#95C211yesmaximal, characteristic, complemented
11#64949C212yesmaximal, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C22×D20C2211yescharacteristic
15C22×C4×D5C2212yes—
16C22×C4×D5C2211yescharacteristic
17C22×Dic10C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×#685C2211yescharacteristic
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×C4⋊Dic5C2212yes—
27C2×#683C2211yescharacteristic
28#4043—24no—
29#4030—24no—
30C5×#95C2212yes—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2639—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C2×C22⋊D5—22no—
46C2×C22⋊D5—24no—
47C2×C22⋊D5—22no—
48C2×C22⋊D5—42no—
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C2×D20—22no—
54C2×D20—24no—
55C2×C4×D5—24no—
56C2×C4×D5—42no—
57C2×C4×D5—42no—
58C2×C4×D5—42no—
59C2×C4×D5D414yes—
60C2×Dic10—22no—
61C2×Dic10—42no—
62C5×C4⋊C4—22no—
63C5×C22⋊C4—22no—
64C5×C22⋊C4—42no—
65C22⋊Dic5—42no—
66#685—24no—
67#685—22no—
68#685—42no—
69#685—42no—
70C4⋊Dic5—24no—
71#683—22no—
72C2×C22⋊Q8—51no—
73C2×#121—51no—
74#400—52no—
75#397—51no—
76#397D511yescharacteristic, complemented
77#394—52no—
78#393—51no—
79#393—51no—
80#386—52no—
81#386—52no—
82C2×#95—51no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—21no—
87C23×C5—21no—
88C23×C5C2411yescharacteristic
89C22×D5—21no—
90C22×D5—22no—
91C22×D5—21no—
92C22×D5—21no—
93C22×D5—22no—
94C22×D5—21no—
95C22×D5—22no—
96C22×D5—21no—
97C22×D5C2×D412yes—
98C22×D5#4114yes—
99C2×C4×C5—22no—
100C2×C4×C5—21no—
101C2×C4×C5—22no—
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×Dic5—22no—
120C2×Dic5C2×D412yes—
121D20—44no—
122C4×D5—44no—
123C4×D5—48no—
124Dic10—44no—
125C22×Q8—51no—
126C22×D4—51no—
127C22×D4—51no—
128C22×D4—51no—
129C23×C4—51no—
130C23×C4—52no—
131C22⋊Q8—104no—
132#121—104no—
133C2×C4⋊C4—51no—
134C2×C4⋊C4—52no—
135C2×C4⋊C4C2×D511yescharacteristic
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—
143#95—51no—
144#95—52no—
145#95—52no—
146#95—52no—
147#95—52no—
148#95C2×D511yescharacteristic
149#95C2×D512yes—
150C22×C5—41no—
151C22×C5—41no—
152C22×C5—41no—
153C22×C5—42no—
154C22×C5—42no—
155C22×C5C22×D411yescharacteristic
156C22×C5C2×#4112yes—
157C22×C5H(5,𝔽2)11yescharacteristic
158C22×C5H(5,𝔽2)12yes—
159C22×C52−1+411yescharacteristic
160C2×D5—22no—
161C2×D5—22no—
162C2×D5—24no—
163C2×D5—24no—
164C2×D5—41no—
165C2×D5—41no—
166C2×D5—41no—
167C2×D5—42no—
168C2×D5—42no—
169C2×D5#12414yes—
170C4×C5—22no—
171C4×C5—41no—
172C4×C5—42no—
173C4×C5—42no—
174Dic5—22no—
175Dic5—41no—
176Dic5—42no—
177Dic5—42no—
178C24—51no—
179C24—51no—
180C24C22×D511yescharacteristic
181C2×Q8—102no—
182C2×Q8—202no—
183C2×D4—102no—
184C2×D4—104no—
185C2×D4—102no—
186C2×D4—102no—
187C2×D4—104no—
188C2×D4—202no—
189C22×C4—51no—
190C22×C4—54no—
191C22×C4—52no—
192C22×C4—51no—
193C22×C4—52no—
194C22×C4—104no—
195C22×C4—202no—
196C22×C4—202no—
197C22×C4—202no—
198C22×C4C22×D512yes—
199C22×C4C22×D511yescharacteristic
200C22×C4C22×D512yes—
201C22×C4C22×D511yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—102no—
204C4⋊C4—104no—
205C22⋊C4—22no—
206C22⋊C4—42no—
207C22⋊C4—104no—
208C22⋊C4—102no—
209C22⋊C4—202no—
210C22⋊C4—202no—
211C22⋊C4—202no—
212C2×C5—81no—
213C2×C5C2×#12411yescharacteristic
214C2×C5#53411yescharacteristic
215C2×C5#53711yescharacteristic
216C2×C5#53812yes—
217C2×C5#53912yes—
218D5—24no—
219D5—81no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—52no—
226C23—54no—
227C23—101no—
228C23—102no—
229C23—101no—
230C23—101no—
231C23—101no—
232C23—102no—
233C23—102no—
234C23—101no—
235C23C23×D511yescharacteristic
236Q8—204no—
237D4—204no—
238D4—204no—
239D4—204no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—52no—
250C2×C4—102no—
251C2×C4—101no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4—102no—
257C2×C4—102no—
258C2×C4—102no—
259C2×C4—204no—
260C2×C4—208no—
261C2×C4D4×D512yes—
262C5#263911yescharacteristic, complemented, Sylow
263C22—41no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—42no—
268C22—54no—
269C22—102no—
270C22—102no—
271C22—104no—
272C22—104no—
273C22—201no—
274C22—201no—
275C22—202no—
276C22—201no—
277C22—202no—
278C22C2×D4×D511yescharacteristic
279C22#414411yescharacteristic
280C22D5×#4112yes—
281C22#414912yes—
282C22#415011yescharacteristic
283C4—22no—
284C4—42no—
285C4—42no—
286C4—41no—
287C4—102no—
288C4—202no—
289C4—202no—
290C4—201no—
291C2—81no—
292C2—104no—
293C2—401no—
294C2#6594211yescharacteristic
295C2#6598711yescharacteristic
296C2#6599112yes—
297C2D5×#12411yescharacteristic
298C2#6604112yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC2×#4030C2
hypercenter#397D5
hypocenterC5#2639
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#4043C2semidirect
3. C5×#397C2semidirect
4. #65282C2semidirect
5. #65250C2semidirect
6. #65245C2semidirect
7. #65236C2semidirect
8. #64962C2semidirect
9. #64954C2semidirect
10. D5×#95C2semidirect
11. #64949C2semidirect
76. #397D5semidirect
262. C5#2639semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×#40302
C5×#3972
#652822
#652502
#652452
#652362
#649622
#649542
D5×#952
#649492
#26395