Finite Group Databasebeta

← #10608624

Subgroups — #10608624

Order 640.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×D4×Dic5C211yesmaximal, characteristic
2C2×#4033C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic
4#65511C211yesmaximal, characteristic
5#65275C211yesmaximal, characteristic, complemented
6#65241C212yesmaximal, complemented
7#65236C212yesmaximal, complemented
8C22⋊C4×Dic5C212yesmaximal, complemented
9C22⋊C4×Dic5C212yesmaximal, complemented
10#64948C211yesmaximal, characteristic, complemented
11#64937C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C23×Dic5C2212yes—
14C22×Dic10C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21D4×Dic5—24no—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×#683C2211yescharacteristic
25C2×C4×Dic5C2211yescharacteristic
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4033—24no—
30C5×#95C2211yescharacteristic
31#3963C2212yes—
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2398—51nomaximal, Sylow
38C24×C5C2312yes—
39C2×C5×D4—24no—
40C2×C5×D4—22no—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5—24no—
46C22×Dic5—24no—
47C22×Dic5—24no—
48C22×Dic5C2312yes—
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C2×Dic10—22no—
55C2×Dic10—22no—
56C2×Dic10—22no—
57C5×C4⋊C4—22no—
58C5×C22⋊C4—24no—
59C5×C22⋊C4—24no—
60C22⋊Dic5—24no—
61C22⋊Dic5—24no—
62C22⋊Dic5—24no—
63C4⋊Dic5—22no—
64#683—22no—
65#683—22no—
66C4×Dic5—22no—
67C4×Dic5—22no—
68C4×Dic5—24no—
69C4×Dic5D414yes—
70C2×C4×Q8—51no—
71C2×C4×D4—51no—
72#400—51no—
73#400—51no—
74#397—52no—
75#394—51no—
76#394D511yescharacteristic
77#393—52no—
78C4×C22⋊C4—52no—
79C4×C22⋊C4—52no—
80#376—51no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5#4114yes—
88C23×C5C2411yescharacteristic
89C5×D4—28no—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
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×C5C2×D412yes—
99C2×C4×C5#4112yes—
100C2×Dic5—21no—
101C2×Dic5—21no—
102C2×Dic5—21no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—48no—
113C2×Dic5C2×D412yes—
114C2×Dic5C2×D412yes—
115C2×Dic5#4112yes—
116C2×Dic5#4114yes—
117Dic10—28no—
118C22×Q8—51no—
119C22×D4C2×D511yescharacteristic
120C23×C4—52no—
121C4×Q8—104no—
122C4×D4—104no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—51no—
126C2×C4⋊C4C2×D511yescharacteristic
127C2×C22⋊C4—52no—
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4C2×D512yes—
131C2×C22⋊C4C2×D512yes—
132C2×C42—51no—
133C2×C42—51no—
134C2×C42—51no—
135C2×C42—52no—
136#95—52no—
137#95—51no—
138#95—52no—
139#95—52no—
140#95—52no—
141#95—52no—
142#95C2×D511yescharacteristic
143C22×C5—24no—
144C22×C5—24no—
145C22×C5—24no—
146C22×C5—42no—
147C22×C5—42no—
148C22×C5—42no—
149C22×C5—42no—
150C22×C5C22×D411yescharacteristic
151C22×C5C2×#4112yes—
152C22×C5C2×#4111yescharacteristic
153C22×C5C2×#4111yescharacteristic
154C22×C5C2×#4112yes—
155C4×C5—22no—
156C4×C5—22no—
157C4×C5—41no—
158C4×C5—42no—
159Dic5—22no—
160Dic5—22no—
161Dic5—41no—
162Dic5—42no—
163Dic5—42no—
164Dic5#12414yes—
165C24C22×D512yes—
166C2×Q8—102no—
167C2×Q8—102no—
168C2×Q8—102no—
169C2×D4—24no—
170C2×D4—22no—
171C22×C4—51no—
172C22×C4—51no—
173C22×C4—51no—
174C22×C4—52no—
175C22×C4—52no—
176C22×C4—51no—
177C22×C4—104no—
178C22×C4—104no—
179C22×C4—104no—
180C22×C4C22×D512yes—
181C22×C4C22×D511yescharacteristic
182C22×C4C22×D511yescharacteristic
183C22×C4C22×D511yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—102no—
186C4⋊C4—102no—
187C4⋊C4—102no—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—104no—
191C22⋊C4—104no—
192C22⋊C4—104no—
193C42—54no—
194C42—102no—
195C42—102no—
196C42—104no—
197C2×C5—44no—
198C2×C5#52511yescharacteristic
199C2×C5#52511yescharacteristic
200C2×C5C2×#12411yescharacteristic
201C2×C5#52912yes—
202C2×C5#52912yes—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23#71114yes—
210C23C23×D511yescharacteristic
211Q8—108no—
212D4—28no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—52no—
222C2×C4—54no—
223C2×C4—52no—
224C2×C4—52no—
225C2×C4—101no—
226C2×C4—101no—
227C2×C4—101no—
228C2×C4—101no—
229C2×C4—101no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—208no—
238C2×C4D4×D512yes—
239C2×C4#71112yes—
240C5#239811yescharacteristic, complemented, Sylow
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D511yescharacteristic
249C22C2×#71112yes—
250C22C2×#71111yescharacteristic
251C22D5×#4112yes—
252C22D5×#4111yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—54no—
258C4—102no—
259C4—102no—
260C4—201no—
261C4—202no—
262C4—202no—
263C2—44no—
264C2#6592011yescharacteristic
265C2#6593412yes—
266C2#6594011yescharacteristic
267C2#6598312yes—
268C2D5×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2398
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4033C2semidirect
5. #65275C2semidirect
6. #65241C2semidirect
7. #65236C2semidirect
8. C22⋊C4×Dic5C2semidirect
9. C22⋊C4×Dic5C2semidirect
10. #64948C2semidirect
11. #64937C2semidirect
240. C5#2398semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40332
C5×#3942
#655112
#652752
#652412
#652362
C22⋊C4×Dic52
#649482
#649372
#23985