Finite Group Databasebeta

← #10609748

Subgroups — #10609748

Order 640.

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

normal subgroups only

This group has 560 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×#4038C211yesmaximal, characteristic, complemented
2C2×#4027C211yesmaximal, characteristic
3C5×#399C211yesmaximal, characteristic, complemented
4#65282C212yesmaximal, complemented
5#65271C211yesmaximal, characteristic, complemented
6C4⋊C4×Dic5C212yesmaximal, complemented
7#65250C212yesmaximal
8#65242C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic5C212yesmaximal, complemented
10#64955C211yesmaximal, characteristic, complemented
11#64950C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C23×Dic5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×C4⋊Dic5C2212yes—
25C2×#683C2211yescharacteristic
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2212yes—
29#4038—24no—
30#4027—24no—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2395—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C2×C22⋊D5—22no—
45C2×C22⋊D5—22no—
46C2×C22⋊D5—22no—
47C22×Dic5—22no—
48C22×Dic5—24no—
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C2×C4×D5—24no—
55C2×C4×D5D414yes—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—24no—
58C5×C22⋊C4—22no—
59C5×C22⋊C4—24no—
60C22⋊Dic5—24no—
61#685—22no—
62#685—24no—
63#685—24no—
64C4⋊Dic5—22no—
65C4⋊Dic5—24no—
66#683—22no—
67C4×Dic5—22no—
68C4×Dic5—24no—
69C4×Dic5—24no—
70C2×C4×D4—51no—
71C2×#117—51no—
72#400—51no—
73#399D511yescharacteristic, complemented
74#398—51no—
75#397—52no—
76#397—51no—
77#394—52no—
78C4×C4⋊C4—52no—
79C4×C22⋊C4—52no—
80C4×C22⋊C4—51no—
81C23×C5—21no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—21no—
85C23×C5#4112yes—
86C23×C5C2411yescharacteristic
87C22×D5—22no—
88C22×D5—22no—
89C22×D5—21no—
90C22×D5—21no—
91C22×D5C2×D412yes—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—21no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5C2×D412yes—
101C2×C4×C5#4114yes—
102C22⋊D5—28no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—21no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—44no—
114C2×Dic5C2×D412yes—
115C2×Dic5#4112yes—
116C2×Dic5#4114yes—
117C4×D5—28no—
118C22×D4—51no—
119C23×C4—51no—
120C23×C4—51no—
121C4×D4—104no—
122#117—104no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4C2×D511yescharacteristic
127C2×C4⋊C4C2×D512yes—
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—51no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4C2×D511yescharacteristic
133C2×C22⋊C4C2×D512yes—
134C2×C42—52no—
135C2×C42—51no—
136C2×C42—52no—
137#95—52no—
138#95—51no—
139#95—52no—
140#95—52no—
141#95—52no—
142#95C2×D511yescharacteristic
143C22×C5—22no—
144C22×C5—24no—
145C22×C5—41no—
146C22×C5—42no—
147C22×C5—41no—
148C22×C5C22×D411yescharacteristic
149C22×C5C2×#4112yes—
150C22×C5C2×#4111yescharacteristic
151C22×C5C2×#4112yes—
152C22×C5C2×#4111yescharacteristic
153C2×D5—22no—
154C2×D5—22no—
155C2×D5—22no—
156C2×D5—42no—
157C2×D5—42no—
158C4×C5—22no—
159C4×C5—24no—
160C4×C5—41no—
161C4×C5—42no—
162Dic5—22no—
163Dic5—41no—
164Dic5—42no—
165Dic5—42no—
166Dic5#12314yes—
167C24—51no—
168C24C22×D511yescharacteristic
169C2×D4—102no—
170C2×D4—102no—
171C2×D4—102no—
172C22×C4—51no—
173C22×C4—51no—
174C22×C4—54no—
175C22×C4—52no—
176C22×C4—51no—
177C22×C4—52no—
178C22×C4—104no—
179C22×C4—102no—
180C22×C4—104no—
181C22×C4C22×D511yescharacteristic
182C22×C4C22×D512yes—
183C22×C4C22×D511yescharacteristic
184C22×C4C22×D512yes—
185C4⋊C4—22no—
186C4⋊C4—24no—
187C4⋊C4—104no—
188C4⋊C4—102no—
189C4⋊C4—102no—
190C22⋊C4—22no—
191C22⋊C4—24no—
192C22⋊C4—104no—
193C22⋊C4—102no—
194C22⋊C4—104no—
195C22⋊C4—104no—
196C42—104no—
197C42—102no—
198C42—104no—
199C2×C5—42no—
200C2×C5C2×#12311yescharacteristic
201C2×C5#52511yescharacteristic
202C2×C5#52912yes—
203C2×C5#52912yes—
204C2×C5#53211yescharacteristic
205D5—42no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—52no—
211C23—102no—
212C23—101no—
213C23—102no—
214C23—101no—
215C23#71312yes—
216C23C23×D511yescharacteristic
217D4—108no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—52no—
227C2×C4—54no—
228C2×C4—52no—
229C2×C4—108no—
230C2×C4—101no—
231C2×C4—101no—
232C2×C4—102no—
233C2×C4—101no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—204no—
241C2×C4D4×D512yes—
242C2×C4#71114yes—
243C5#239511yescharacteristic, complemented, Sylow
244C22—24no—
245C22—22no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—102no—
250C22—102no—
251C22—102no—
252C22—202no—
253C22—202no—
254C22C2×D4×D511yescharacteristic
255C22C2×#71112yes—
256C22C2×#71311yescharacteristic
257C22D5×#4112yes—
258C22D5×#4111yescharacteristic
259C4—24no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—54no—
264C4—102no—
265C4—202no—
266C4—202no—
267C4—201no—
268C2—42no—
269C2—202no—
270C2#6595812yes—
271C2#6596111yescharacteristic
272C2#6598312yes—
273C2D5×#12311yescharacteristic
274C2#6605011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC22×C22⋊D5C22
hypercenter#399D5
hypocenterC5#2395
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4038C2semidirect
3. C5×#399C2semidirect
4. #65282C2semidirect
5. #65271C2semidirect
6. C4⋊C4×Dic5C2semidirect
8. #65242C2semidirect
9. C22⋊C4×Dic5C2semidirect
10. #64955C2semidirect
11. #64950C2semidirect
73. #399D5semidirect
243. C5#2395semidirect

Maximal subgroups

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

subgroupindex
C2×#40382
C2×#40272
C5×#3992
#652822
#652712
C4⋊C4×Dic52
#652502
#652422
C22⋊C4×Dic52
#649552
#649502
#23955