Finite Group Databasebeta

← #10608819

Subgroups — #10608819

Order 640.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#4082C211yesmaximal, characteristic
2C2×#4035C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic
4#65511C211yesmaximal, characteristic
5#65268C211yesmaximal, characteristic, complemented
6#65241C212yesmaximal, complemented
7#65238C212yesmaximal, complemented
8#65237C212yesmaximal, complemented
9#65237C212yesmaximal, complemented
10#64945C211yesmaximal, characteristic, complemented
11#64939C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C22⋊C4C2212yes—
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21#4082—24no—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×C4⋊Dic5C2212yes—
24C2×#683C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2212yes—
29C2×#683C2212yes—
30C2×C4×Dic5C2211yescharacteristic
31#4035—24no—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2820—51nomaximal, Sylow
38C24×C5C2312yes—
39C2×C5×D4—22no—
40C2×C5×D4—42no—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×Dic10—22no—
52C2×Dic10—42no—
53C5×C4⋊C4—22no—
54C5×C22⋊C4—42no—
55C5×C22⋊C4—42no—
56C22⋊Dic5—24no—
57C22⋊Dic5—24no—
58C22⋊Dic5—42no—
59C22⋊Dic5—42no—
60C4⋊Dic5—22no—
61C4⋊Dic5—42no—
62#683—22no—
63#683—22no—
64#683—22no—
65#683—22no—
66#683—42no—
67#683—42no—
68C4×Dic5D414yes—
69C2×#125—51no—
70C2×#124—51no—
71#399—52no—
72#397—52no—
73#396—52no—
74#396—52no—
75#395—51no—
76#395—51no—
77#394—51no—
78#394D511yescharacteristic
79#382—51no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5C2411yescharacteristic
88C5×D4—44no—
89C2×C4×C5—22no—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5C2×D412yes—
99C2×Dic5—21no—
100C2×Dic5—22no—
101C2×Dic5—21no—
102C2×Dic5—21no—
103C2×Dic5—21no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111C2×Dic5—21no—
112C2×Dic5—22no—
113C2×Dic5C2×D412yes—
114C2×Dic5C2×D412yes—
115Dic10—44no—
116C22×Q8—51no—
117C22×D4C2×D511yescharacteristic
118#125—104no—
119#124—104no—
120C2×C4⋊C4—51no—
121C2×C4⋊C4—51no—
122C2×C4⋊C4—51no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4—52no—
128C2×C4⋊C4C2×D511yescharacteristic
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4C2×D512yes—
134C2×C22⋊C4C2×D512yes—
135C2×C42—51no—
136#95—51no—
137#95—52no—
138#95—52no—
139#95—52no—
140#95C2×D511yescharacteristic
141C22×C5—42no—
142C22×C5—42no—
143C22×C5—42no—
144C22×C5—42no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5—42no—
148C22×C5C22×D411yescharacteristic
149C22×C5H(5,𝔽2)12yes—
150C22×C5H(5,𝔽2)11yescharacteristic
151C22×C52−1+412yes—
152C22×C52−1+411yescharacteristic
153C4×C5—22no—
154C4×C5—41no—
155C4×C5—41no—
156C4×C5—42no—
157Dic5—22no—
158Dic5—22no—
159Dic5—42no—
160Dic5—41no—
161Dic5—41no—
162Dic5—42no—
163C24C22×D512yes—
164C2×Q8—102no—
165C2×Q8—202no—
166C2×D4—22no—
167C2×D4—42no—
168C22×C4—51no—
169C22×C4—51no—
170C22×C4—52no—
171C22×C4—51no—
172C22×C4—52no—
173C22×C4—51no—
174C22×C4C22×D512yes—
175C22×C4C22×D511yescharacteristic
176C22×C4C22×D511yescharacteristic
177C22×C4C22×D511yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—102no—
180C4⋊C4—102no—
181C4⋊C4—102no—
182C4⋊C4—102no—
183C4⋊C4—102no—
184C4⋊C4—202no—
185C4⋊C4—202no—
186C4⋊C4—202no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—104no—
190C22⋊C4—104no—
191C22⋊C4—202no—
192C22⋊C4—202no—
193C42—54no—
194C2×C5—82no—
195C2×C5#53511yescharacteristic
196C2×C5#53611yescharacteristic
197C2×C5#53711yescharacteristic
198C2×C5#56212yes—
199C2×C5#56312yes—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23C23×D511yescharacteristic
208Q8—204no—
209D4—44no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—52no—
220C2×C4—52no—
221C2×C4—101no—
222C2×C4—101no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—101no—
226C2×C4—101no—
227C2×C4—101no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—101no—
234C2×C4—102no—
235C2×C4D4×D512yes—
236C5#282011yescharacteristic, complemented, Sylow
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D511yescharacteristic
245C22#414412yes—
246C22#414411yescharacteristic
247C22#415011yescharacteristic
248C22#415012yes—
249C4—22no—
250C4—41no—
251C4—41no—
252C4—42no—
253C4—102no—
254C4—102no—
255C4—202no—
256C4—201no—
257C4—201no—
258C4—202no—
259C2—82no—
260C2#6592611yescharacteristic
261C2#6594211yescharacteristic
262C2#6595512yes—
263C2#6598512yes—
264C2#6600911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2820
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4035C2semidirect
5. #65268C2semidirect
6. #65241C2semidirect
7. #65238C2semidirect
8. #65237C2semidirect
9. #65237C2semidirect
10. #64945C2semidirect
11. #64939C2semidirect
236. C5#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
C2×#40352
C5×#3942
#655112
#652682
#652412
#652382
#652372
#649452
#649392
#28205