Finite Group Databasebeta

← #10609191

Subgroups — #10609191

Order 640.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#4033C211yesmaximal, characteristic, complemented
2C2×#4027C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65248C212yesmaximal
6#65236C211yesmaximal, characteristic, complemented
7C22⋊C4×Dic5C212yesmaximal, complemented
8#64962C211yesmaximal, characteristic, complemented
9#64955C212yesmaximal, complemented
10#64947C212yesmaximal, complemented
11#64937C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C23×Dic5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×Dic10C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#683C2211yescharacteristic
24C2×#683C2212yes—
25C2×C4×Dic5C2212yes—
26C2×C4×Dic5C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28#4033—24no—
29#4027—24no—
30C5×#95C2211yescharacteristic
31C5×#95C2212yes—
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2398—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—22no—
46C2×C22⋊D5—22no—
47C22×Dic5—22no—
48C22×Dic5—24no—
49C22×Dic5C2312yes—
50C22×Dic5C2311yescharacteristic
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2312yes—
53C22×Dic5C2311yescharacteristic
54C2×C4×D5—22no—
55C2×C4×D5—24no—
56C2×Dic10—22no—
57C2×Dic10—24no—
58C5×C4⋊C4—22no—
59C5×C22⋊C4—24no—
60C5×C22⋊C4—22no—
61C22⋊Dic5—24no—
62#685—24no—
63#685—22no—
64#685—24no—
65#683—22no—
66#683—24no—
67C4×Dic5—24no—
68C4×Dic5—24no—
69C4×Dic5D414yes—
70C2×C4×Q8—51no—
71C2×C4×D4—51no—
72#400—52no—
73#397—51no—
74#397D511yescharacteristic, complemented
75#394—52no—
76#393—51no—
77#393—51no—
78C4×C22⋊C4—52no—
79C4×C22⋊C4—52no—
80#376—51no—
81C23×C5—21no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—21no—
85C23×C5#4112yes—
86C23×C5C2411yescharacteristic
87C22×D5—21no—
88C22×D5—22no—
89C22×D5—22no—
90C22×D5—21no—
91C22×D5#4112yes—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5C2×D412yes—
102C22⋊D5—28no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—21no—
106C2×Dic5—21no—
107C2×Dic5—21no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—44no—
113C2×Dic5C2×D412yes—
114C2×Dic5C2×D412yes—
115C2×Dic5#4114yes—
116C2×Dic5#4114yes—
117C4×D5—44no—
118Dic10—28no—
119C22×Q8—51no—
120C22×D4—51no—
121C23×C4—51no—
122C23×C4—51no—
123C4×Q8—104no—
124C4×D4—104no—
125C2×C4⋊C4—51no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C22⋊C4—52no—
129C2×C22⋊C4—51no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4C2×D511yescharacteristic
133C2×C22⋊C4C2×D512yes—
134C2×C42—51no—
135C2×C42—52no—
136C2×C42—52no—
137#95—51no—
138#95—52no—
139#95—52no—
140#95—52no—
141#95—52no—
142#95C2×D511yescharacteristic
143#95C2×D512yes—
144C22×C5—22no—
145C22×C5—24no—
146C22×C5—41no—
147C22×C5—41no—
148C22×C5—42no—
149C22×C5C22×D411yescharacteristic
150C22×C5C2×#4111yescharacteristic
151C22×C5C2×#4111yescharacteristic
152C22×C5C2×#4112yes—
153C22×C5C2×#4112yes—
154C2×D5—22no—
155C2×D5—24no—
156C2×D5—41no—
157C2×D5—41no—
158C2×D5—42no—
159C4×C5—22no—
160C4×C5—42no—
161C4×C5—41no—
162C4×C5—42no—
163Dic5—24no—
164Dic5—22no—
165Dic5—41no—
166Dic5—42no—
167Dic5#12414yes—
168C24—51no—
169C24C22×D511yescharacteristic
170C2×Q8—104no—
171C2×Q8—102no—
172C2×D4—102no—
173C2×D4—102no—
174C2×D4—102no—
175C22×C4—51no—
176C22×C4—52no—
177C22×C4—51no—
178C22×C4—52no—
179C22×C4—51no—
180C22×C4—102no—
181C22×C4—104no—
182C22×C4—102no—
183C22×C4—104no—
184C22×C4C22×D512yes—
185C22×C4C22×D511yescharacteristic
186C22×C4C22×D512yes—
187C22×C4C22×D511yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—102no—
190C4⋊C4—104no—
191C22⋊C4—24no—
192C22⋊C4—22no—
193C22⋊C4—104no—
194C22⋊C4—104no—
195C22⋊C4—104no—
196C22⋊C4—102no—
197C42—54no—
198C42—104no—
199C42—104no—
200C2×C5—42no—
201C2×C5#52511yescharacteristic
202C2×C5#52511yescharacteristic
203C2×C5C2×#12411yescharacteristic
204C2×C5#52912yes—
205C2×C5#52912yes—
206D5—42no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—52no—
212C23—101no—
213C23—101no—
214C23—102no—
215C23—102no—
216C23#71112yes—
217C23C23×D511yescharacteristic
218Q8—108no—
219D4—108no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—52no—
230C2×C4—54no—
231C2×C4—52no—
232C2×C4—54no—
233C2×C4—102no—
234C2×C4—101no—
235C2×C4—101no—
236C2×C4—102no—
237C2×C4—101no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—102no—
241C2×C4—102no—
242C2×C4—204no—
243C2×C4—204no—
244C2×C4D4×D512yes—
245C5#239811yescharacteristic, complemented, Sylow
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—102no—
252C22—104no—
253C22—202no—
254C22—201no—
255C22—201no—
256C22C2×D4×D511yescharacteristic
257C22C2×#71111yescharacteristic
258C22D5×#4111yescharacteristic
259C22D5×#4112yes—
260C22D5×#4112yes—
261C4—22no—
262C4—42no—
263C4—42no—
264C4—41no—
265C4—54no—
266C4—102no—
267C4—104no—
268C4—201no—
269C4—202no—
270C2—42no—
271C2—202no—
272C2#6594011yescharacteristic
273C2#6598312yes—
274C2#6599011yescharacteristic
275C2D5×#12411yescharacteristic
276C2#6604012yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC22×C22⋊D5C22
hypercenter#397D5
hypocenterC5#2398
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#4033C2semidirect
3. C5×#397C2semidirect
4. #65282C2semidirect
6. #65236C2semidirect
7. C22⋊C4×Dic5C2semidirect
8. #64962C2semidirect
9. #64955C2semidirect
10. #64947C2semidirect
11. #64937C2semidirect
74. #397D5semidirect
245. C5#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#40332
C2×#40272
C5×#3972
#652822
#652482
#652362
C22⋊C4×Dic52
#649622
#649552
#649472
#649372
#23985