Finite Group Databasebeta

← #10609417

Subgroups — #10609417

Order 640.

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

normal subgroups only

This group has 526 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×D5×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#4029C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic, complemented
4#65269C211yesmaximal, characteristic, complemented
5#65245C212yesmaximal
6#65242C211yesmaximal, characteristic, complemented
7#65238C212yesmaximal, complemented
8#64957C212yesmaximal, complemented
9#64953C211yesmaximal, characteristic, complemented
10#64950C211yesmaximal, characteristic, complemented
11#64949C212yesmaximal, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×C4×D5C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×C4⋊Dic5C2211yescharacteristic
24C2×C4⋊Dic5C2212yes—
25C2×#683C2211yescharacteristic
26C2×#683C2212yes—
27C2×#683C2212yes—
28D5×C4⋊C4—24no—
29#4029—24no—
30C5×#95C2212yes—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2211yescharacteristic
37#2623—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C2×C22⋊D5—22no—
45C2×C22⋊D5—24no—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×C4×D5—24no—
52C2×C4×D5—24no—
53C2×C4×D5—42no—
54C2×C4×D5—42no—
55C2×C4×D5D414yes—
56C5×C4⋊C4—22no—
57C5×C22⋊C4—22no—
58C5×C22⋊C4—42no—
59C22⋊Dic5—22no—
60C22⋊Dic5—42no—
61#685—22no—
62#685—24no—
63C4⋊Dic5—22no—
64C4⋊Dic5—22no—
65C4⋊Dic5—42no—
66#683—22no—
67#683—24no—
68#683—42no—
69C2×#123—51no—
70C22×C4⋊C4—51no—
71#400—52no—
72#399—52no—
73#398—51no—
74#398—51no—
75#397—51no—
76#397D511yescharacteristic, complemented
77#386—52no—
78#385—52no—
79#381—51no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5—21no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5C2411yescharacteristic
86C22×D5—21no—
87C22×D5—21no—
88C22×D5—22no—
89C22×D5C2×D412yes—
90C22×D5#4114yes—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5C2×D412yes—
101C22⋊D5—44no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—21no—
105C2×Dic5—21no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—21no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5C2×D412yes—
115C4×D5—28no—
116C4×D5—48no—
117C22×D4—51no—
118C23×C4—51no—
119C23×C4—52no—
120#123—104no—
121C2×C4⋊C4—51no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—104no—
128C2×C4⋊C4C2×D511yescharacteristic
129C2×C22⋊C4—51no—
130C2×C22⋊C4—51no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4C2×D511yescharacteristic
134C2×C22⋊C4C2×D512yes—
135#95—51no—
136#95—52no—
137#95—52no—
138#95—52no—
139#95—52no—
140#95C2×D511yescharacteristic
141#95C2×D512yes—
142C22×C5—41no—
143C22×C5—41no—
144C22×C5—41no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5C22×D411yescharacteristic
148C22×C5C2×#4112yes—
149C22×C5H(5,𝔽2)11yescharacteristic
150C22×C5H(5,𝔽2)11yescharacteristic
151C22×C52−1+412yes—
152C2×D5—22no—
153C2×D5—22no—
154C2×D5—24no—
155C2×D5—24no—
156C2×D5#12314yes—
157C4×C5—22no—
158C4×C5—42no—
159C4×C5—41no—
160C4×C5—42no—
161Dic5—22no—
162Dic5—41no—
163Dic5—42no—
164Dic5—41no—
165Dic5—42no—
166C24—51no—
167C24C22×D511yescharacteristic
168C2×D4—102no—
169C2×D4—104no—
170C22×C4—51no—
171C22×C4—54no—
172C22×C4—52no—
173C22×C4—51no—
174C22×C4—52no—
175C22×C4—51no—
176C22×C4—104no—
177C22×C4—104no—
178C22×C4—202no—
179C22×C4—202no—
180C22×C4C22×D511yescharacteristic
181C22×C4C22×D512yes—
182C22×C4C22×D512yes—
183C22×C4C22×D511yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—104no—
186C4⋊C4—102no—
187C4⋊C4—102no—
188C4⋊C4—102no—
189C4⋊C4—202no—
190C4⋊C4—202no—
191C22⋊C4—22no—
192C22⋊C4—42no—
193C22⋊C4—102no—
194C22⋊C4—104no—
195C22⋊C4—102no—
196C22⋊C4—202no—
197C2×C5—81no—
198C2×C5C2×#12311yescharacteristic
199C2×C5#53511yescharacteristic
200C2×C5#53611yescharacteristic
201C2×C5#53912yes—
202C2×C5#53912yes—
203D5—24no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—52no—
210C23—54no—
211C23—101no—
212C23—101no—
213C23—102no—
214C23C23×D511yescharacteristic
215D4—204no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—52no—
226C2×C4—108no—
227C2×C4—102no—
228C2×C4—101no—
229C2×C4—102no—
230C2×C4—101no—
231C2×C4—101no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—101no—
238C2×C4—102no—
239C2×C4—208no—
240C2×C4D4×D512yes—
241C5#262311yescharacteristic, complemented, Sylow
242C22—41no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—42no—
247C22—54no—
248C22—102no—
249C22—104no—
250C22—102no—
251C22—104no—
252C22C2×D4×D511yescharacteristic
253C22#414411yescharacteristic
254C22D5×#4112yes—
255C22#414911yescharacteristic
256C22#415012yes—
257C4—22no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—102no—
262C4—202no—
263C4—201no—
264C4—201no—
265C4—202no—
266C2—81no—
267C2—104no—
268C2#6594111yescharacteristic
269C2D5×#12311yescharacteristic
270C2#6599412yes—
271C2#6600911yescharacteristic
272C2#6604112yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC22×C22⋊D5C22
hypercenter#397D5
hypocenterC5#2623
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×D5×C4⋊C4C2semidirect
3. C5×#397C2semidirect
4. #65269C2semidirect
6. #65242C2semidirect
7. #65238C2semidirect
8. #64957C2semidirect
9. #64953C2semidirect
10. #64950C2semidirect
11. #64949C2semidirect
76. #397D5semidirect
241. C5#2623semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊C42
C2×#40292
C5×#3972
#652692
#652452
#652422
#652382
#649572
#649532
#649502
#649492
#26235