Finite Group Databasebeta

← #10609278

Subgroups — #10609278

Order 640.

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

normal subgroups only

This group has 500 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×#4034C211yesmaximal, characteristic, complemented
2C2×#4031C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65246C212yesmaximal
6#65242C211yesmaximal, characteristic, complemented
7#65236C212yesmaximal, complemented
8#64963C211yesmaximal, characteristic, complemented
9#64962C212yesmaximal, complemented
10#64937C211yesmaximal, characteristic, complemented
11#64935C212yesmaximal, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C22×Dic10C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×#683C2212yes—
26C2×C4×Dic5C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28#4034—24no—
29#4031—24no—
30C5×#95C2212yes—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2640—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×Dic5C2311yescharacteristic
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×Dic10—22no—
52C2×Dic10—24no—
53C2×Dic10—24no—
54C2×Dic10—42no—
55C2×Dic10—42no—
56C5×C4⋊C4—22no—
57C5×C22⋊C4—22no—
58C5×C22⋊C4—42no—
59C22⋊Dic5—22no—
60C22⋊Dic5—42no—
61#685—22no—
62#685—22no—
63#685—42no—
64#685—42no—
65C4⋊Dic5—22no—
66#683—24no—
67C4×Dic5—24no—
68C4×Dic5D414yes—
69C2×C4⋊Q8—51no—
70C2×#124—51no—
71#397—51no—
72#397—51no—
73#397—51no—
74#397D511yescharacteristic, complemented
75#393—52no—
76#393—52no—
77#391—52no—
78#390—52no—
79#376—51no—
80C23×C5—21no—
81C23×C5—22no—
82C23×C5—21no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5C2411yescharacteristic
86C22×D5—21no—
87C22×D5—22no—
88C22×D5—21no—
89C22×D5—21no—
90C22×D5—22no—
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—21no—
103C2×Dic5—21no—
104C2×Dic5—22no—
105C2×Dic5—21no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5C2×D412yes—
113C2×Dic5C2×D412yes—
114C2×Dic5#4114yes—
115Dic10—48no—
116Dic10—44no—
117C22×Q8—51no—
118C22×Q8—52no—
119C22×D4—51no—
120C4⋊Q8—104no—
121#124—104no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4C2×D511yescharacteristic
125C2×C22⋊C4—51no—
126C2×C22⋊C4—51no—
127C2×C22⋊C4—51no—
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4C2×D511yescharacteristic
132C2×C22⋊C4C2×D512yes—
133C2×C42—51no—
134C2×C42—52no—
135#95—52no—
136#95—51no—
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—41no—
153C2×D5—41no—
154C2×D5—41no—
155C2×D5—42no—
156C2×D5—42no—
157C4×C5—22no—
158C4×C5—41no—
159C4×C5—42no—
160C4×C5—42no—
161Dic5—22no—
162Dic5—41no—
163Dic5—42no—
164Dic5—42no—
165Dic5#12414yes—
166C24—51no—
167C24C22×D511yescharacteristic
168C2×Q8—102no—
169C2×Q8—104no—
170C2×Q8—104no—
171C2×Q8—202no—
172C2×Q8—202no—
173C2×D4—102no—
174C2×D4—104no—
175C22×C4—51no—
176C22×C4—51no—
177C22×C4—52no—
178C22×C4—51no—
179C22×C4—52no—
180C22×C4C22×D512yes—
181C22×C4C22×D511yescharacteristic
182C22×C4C22×D512yes—
183C22×C4C22×D511yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—104no—
186C4⋊C4—102no—
187C22⋊C4—22no—
188C22⋊C4—42no—
189C22⋊C4—102no—
190C22⋊C4—102no—
191C22⋊C4—102no—
192C22⋊C4—202no—
193C22⋊C4—202no—
194C22⋊C4—202no—
195C42—54no—
196C42—104no—
197C2×C5—81no—
198C2×C5C2×#12411yescharacteristic
199C2×C5#53511yescharacteristic
200C2×C5#53611yescharacteristic
201C2×C5#54212yes—
202C2×C5#54212yes—
203D5—81no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—101no—
210C23—101no—
211C23—102no—
212C23—101no—
213C23—102no—
214C23C23×D511yescharacteristic
215Q8—204no—
216Q8—208no—
217D4—204no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—52no—
228C2×C4—54no—
229C2×C4—52no—
230C2×C4—101no—
231C2×C4—101no—
232C2×C4—101no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4D4×D512yes—
241C5#264011yescharacteristic, complemented, Sylow
242C22—41no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—42no—
247C22—201no—
248C22—201no—
249C22—202no—
250C22—201no—
251C22—202no—
252C22C2×D4×D511yescharacteristic
253C22#414411yescharacteristic
254C22D5×#4112yes—
255C22#414911yescharacteristic
256C22#415012yes—
257C4—22no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—54no—
262C4—102no—
263C4—202no—
264C4—201no—
265C4—202no—
266C2—81no—
267C2—401no—
268C2#6594111yescharacteristic
269C2#6598211yescharacteristic
270C2#6599612yes—
271C2D5×#12411yescharacteristic
272C2#6604412yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC22×C22⋊D5C22
hypercenter#397D5
hypocenterC5#2640
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#4034C2semidirect
3. C5×#397C2semidirect
4. #65282C2semidirect
6. #65242C2semidirect
7. #65236C2semidirect
8. #64963C2semidirect
9. #64962C2semidirect
10. #64937C2semidirect
11. #64935C2semidirect
74. #397D5semidirect
241. C5#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#40342
C2×#40312
C5×#3972
#652822
#652462
#652422
#652362
#649632
#649622
#649372
#649352
#26405