Finite Group Databasebeta

← #10609877

Subgroups — #10609877

Order 640.

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

normal subgroups only

This group has 584 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×#4039C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C5×#400C211yesmaximal, characteristic, complemented
4#65281C211yesmaximal, characteristic, complemented
5#65275C211yesmaximal, characteristic, complemented
6#64966C211yesmaximal, characteristic, complemented
7#64957C212yesmaximal, complemented
8#64956C212yesmaximal, complemented
9#64956C211yesmaximal, characteristic, complemented
10#64954C212yesmaximal
11#64936C212yesmaximal, complemented
12C22×D20C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×C4×D5C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×#685C2211yescharacteristic
19C2×#685C2211yescharacteristic
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×C4⋊Dic5C2212yes—
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×#683C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4039—24no—
30D5×C4⋊C4—24no—
31C5×#95C2212yes—
32C5×#95C2212yes—
33C5×#95C2211yescharacteristic
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2487—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×D20—22no—
50C2×D20—22no—
51C2×D20Q814yes—
52C2×C4×D5—24no—
53C2×C4×D5—24no—
54C2×C4×D5—22no—
55C2×C4×D5—24no—
56C2×C4×D5—42no—
57C2×C4×D5—42no—
58C2×C4×D5D414yes—
59C5×C4⋊C4—22no—
60C5×C4⋊C4—22no—
61#685—22no—
62#685—24no—
63#685—22no—
64#685—42no—
65#685—42no—
66C4⋊Dic5—22no—
67C4⋊Dic5—42no—
68#683—24no—
69#683—24no—
70#683—42no—
71C4×Dic5—22no—
72C2×C4×D4—51no—
73C22×C4⋊C4—51no—
74#400—51no—
75#400D511yescharacteristic, complemented
76#399—51no—
77#399—51no—
78#389—52no—
79#386—52no—
80#385—52no—
81#380—52no—
82#380—51no—
83C23×C5C2411yescharacteristic
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—21no—
87C22×D5—22no—
88C22×D5—21no—
89C22×D5—22no—
90C22×D5C2×D412yes—
91C22×D5C2×Q812yes—
92C22×D5C2×Q812yes—
93C22×D5#4112yes—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—21no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5C2×D412yes—
106C2×C4×C5C2×Q812yes—
107C2×Dic5—21no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5C2×D412yes—
116C2×Dic5#4114yes—
117D20—28no—
118C4×D5—28no—
119C4×D5—48no—
120C4×D5—44no—
121C22×D4—51no—
122C23×C4—51no—
123C23×C4—51no—
124C23×C4—52no—
125C4×D4—104no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—51no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4—52no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—104no—
132C2×C4⋊C4C2×D511yescharacteristic
133C2×C4⋊C4C2×D511yescharacteristic
134C2×C22⋊C4—51no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C42—51no—
140#95—52no—
141#95—51no—
142#95—52no—
143#95C2×D512yes—
144#95C2×D511yescharacteristic
145#95C2×D512yes—
146C22×C5C22×D411yescharacteristic
147C22×C5C22×Q811yescharacteristic
148C22×C5C2×#4111yescharacteristic
149C22×C5C2×#4112yes—
150C22×C5H(5,𝔽2)12yes—
151C2×D5—22no—
152C2×D5—22no—
153C2×D5—24no—
154C2×D5—24no—
155C2×D5—22no—
156C2×D5—24no—
157C2×D5—41no—
158C2×D5—41no—
159C2×D5—42no—
160C2×D5C22⋊Q814yes—
161C4×C5—22no—
162C4×C5—22no—
163C4×C5—41no—
164C4×C5—42no—
165C4×C5—42no—
166Dic5—42no—
167Dic5—41no—
168Dic5—42no—
169Dic5#12314yes—
170C24—51no—
171C24—51no—
172C2×D4—54no—
173C2×D4—102no—
174C2×D4—102no—
175C22×C4—51no—
176C22×C4—54no—
177C22×C4—52no—
178C22×C4—52no—
179C22×C4—51no—
180C22×C4—104no—
181C22×C4—102no—
182C22×C4—104no—
183C22×C4—104no—
184C22×C4—202no—
185C22×C4—202no—
186C22×C4C22×D511yescharacteristic
187C22×C4C22×D512yes—
188C22×C4C22×D512yes—
189C22×C4C22×D511yescharacteristic
190C22×C4C22×D511yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—104no—
194C4⋊C4—104no—
195C4⋊C4—102no—
196C4⋊C4—202no—
197C4⋊C4—202no—
198C22⋊C4—102no—
199C22⋊C4—104no—
200C22⋊C4—102no—
201C22⋊C4—202no—
202C22⋊C4—202no—
203C42—102no—
204C2×C5C2×C22⋊Q811yescharacteristic
205C2×C5C2×#12311yescharacteristic
206C2×C5#53511yescharacteristic
207C2×C5#55312yes—
208C2×C5#55412yes—
209D5—24no—
210D5—42no—
211C23—52no—
212C23—52no—
213C23—52no—
214C23—52no—
215C23—102no—
216C23—102no—
217C23—101no—
218C23—102no—
219C23—101no—
220C23—102no—
221C23C23×D511yescharacteristic
222D4—108no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—52no—
235C2×C4—54no—
236C2×C4—108no—
237C2×C4—101no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—204no—
246C2×C4—208no—
247C2×C4D4×D512yes—
248C2×C4Q8×D512yes—
249C5#248711yescharacteristic, complemented, Sylow
250C22—54no—
251C22—102no—
252C22—102no—
253C22—104no—
254C22—104no—
255C22—102no—
256C22—104no—
257C22—201no—
258C22—201no—
259C22—202no—
260C22C2×D4×D511yescharacteristic
261C22C2×Q8×D511yescharacteristic
262C22D5×#4111yescharacteristic
263C22D5×#4112yes—
264C22#414912yes—
265C4—22no—
266C4—22no—
267C4—41no—
268C4—42no—
269C4—42no—
270C4—54no—
271C4—202no—
272C4—201no—
273C4—202no—
274C2—104no—
275C2—202no—
276C2D5×C22⋊Q811yescharacteristic
277C2D5×#12311yescharacteristic
278C2#6600811yescharacteristic
279C2#6602412yes—
280C2#6604312yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC22×D20C22
hypercenter#400D5
hypocenterC5#2487
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊C4C2semidirect
3. C5×#400C2semidirect
4. #65281C2semidirect
5. #65275C2semidirect
6. #64966C2semidirect
7. #64957C2semidirect
8. #64956C2semidirect
9. #64956C2semidirect
11. #64936C2semidirect
75. #400D5semidirect
249. C5#2487semidirect

Maximal subgroups

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

subgroupindex
C2×#40392
C2×D5×C4⋊C42
C5×#4002
#652812
#652752
#649662
#649572
#649562
#649542
#649362
#24875