Finite Group Databasebeta

← #10609860

Subgroups — #10609860

Order 640.

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

normal subgroups only

This group has 560 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×#4039C211yesmaximal, characteristic
2C2×#4038C211yesmaximal, characteristic, complemented
3C5×#400C211yesmaximal, characteristic, complemented
4#65281C211yesmaximal, characteristic, complemented
5#65272C211yesmaximal, characteristic, complemented
6#64964C212yesmaximal, complemented
7#64963C211yesmaximal, characteristic, complemented
8#64961C212yesmaximal
9#64955C212yesmaximal, complemented
10#64955C211yesmaximal, characteristic, complemented
11#64938C212yesmaximal, complemented
12C22×D20C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×#685C2211yescharacteristic
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28C2×C4×Dic5C2212yes—
29#4039—24no—
30#4038—24no—
31C5×#95C2212yes—
32C5×#95C2211yescharacteristic
33C5×#95C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2395—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C2×D20—22no—
50C2×D20—22no—
51C2×D20—22no—
52C2×C4×D5—24no—
53C2×C4×D5—22no—
54C2×C4×D5—24no—
55C2×C4×D5D414yes—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—22no—
58#685—24no—
59#685—22no—
60#685—24no—
61#685—22no—
62#685—24no—
63#685—24no—
64C4⋊Dic5—22no—
65#683—24no—
66#683—24no—
67C4×Dic5—24no—
68C4×Dic5—22no—
69C4×Dic5—24no—
70C2×C4×D4—51no—
71C2×#117—51no—
72#400D511yescharacteristic, complemented
73#399—51no—
74#398—51no—
75#397—51no—
76#397—52no—
77#394—52no—
78C4×C4⋊C4—52no—
79C4×C22⋊C4—52no—
80C4×C22⋊C4—51no—
81C23×C5C2411yescharacteristic
82C22×D5—21no—
83C22×D5—22no—
84C22×D5—22no—
85C22×D5—22no—
86C22×D5—22no—
87C22×D5—21no—
88C22×D5—21no—
89C22×D5—21no—
90C22×D5C2×D412yes—
91C22×D5#4112yes—
92C2×C4×C5—21no—
93C2×C4×C5—22no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5C2×D412yes—
104C2×C4×C5#4112yes—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—21no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5C2×D412yes—
113C2×Dic5#4114yes—
114C2×Dic5#4114yes—
115D20—28no—
116C4×D5—28no—
117C4×D5—44no—
118C22×D4—51no—
119C23×C4—51no—
120C23×C4—51no—
121C4×D4—104no—
122#117—104no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4C2×D511yescharacteristic
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C22⋊C4—52no—
129C2×C22⋊C4—51no—
130C2×C22⋊C4—51no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4—52no—
134C2×C42—52no—
135C2×C42—51no—
136C2×C42—52no—
137#95—52no—
138#95—51no—
139#95—52no—
140#95C2×D512yes—
141#95C2×D511yescharacteristic
142#95C2×D512yes—
143C22×C5C22×D411yescharacteristic
144C22×C5C2×#4111yescharacteristic
145C22×C5C2×#4112yes—
146C22×C5C2×#4111yescharacteristic
147C22×C5C2×#4112yes—
148C2×D5—22no—
149C2×D5—22no—
150C2×D5—22no—
151C2×D5—24no—
152C2×D5—22no—
153C2×D5—41no—
154C2×D5—42no—
155C2×D5—42no—
156C2×D5—42no—
157C2×D5—41no—
158C4×C5—22no—
159C4×C5—22no—
160C4×C5—41no—
161C4×C5—42no—
162C4×C5—42no—
163Dic5—24no—
164Dic5—42no—
165Dic5—41no—
166Dic5#12314yes—
167C24—51no—
168C24—51no—
169C2×D4—102no—
170C2×D4—102no—
171C2×D4—102no—
172C22×C4—51no—
173C22×C4—52no—
174C22×C4—54no—
175C22×C4—51no—
176C22×C4—52no—
177C22×C4—104no—
178C22×C4—102no—
179C22×C4—104no—
180C22×C4C22×D511yescharacteristic
181C22×C4C22×D512yes—
182C22×C4C22×D512yes—
183C22×C4C22×D511yescharacteristic
184C22×C4C22×D511yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—104no—
188C4⋊C4—102no—
189C4⋊C4—104no—
190C22⋊C4—104no—
191C22⋊C4—104no—
192C22⋊C4—104no—
193C22⋊C4—102no—
194C22⋊C4—104no—
195C22⋊C4—102no—
196C42—104no—
197C42—102no—
198C42—104no—
199C2×C5C2×#12311yescharacteristic
200C2×C5#52511yescharacteristic
201C2×C5#52912yes—
202C2×C5#52912yes—
203C2×C5#53211yescharacteristic
204D5—42no—
205D5—42no—
206C23—52no—
207C23—52no—
208C23—101no—
209C23—101no—
210C23—101no—
211C23—102no—
212C23—102no—
213C23—102no—
214C23—102no—
215C23—101no—
216C23C23×D511yescharacteristic
217D4—108no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
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—54no—
232C2×C4—108no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—101no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4—102no—
240C2×C4—204no—
241C2×C4D4×D512yes—
242C2×C4#71312yes—
243C5#239511yescharacteristic, complemented, Sylow
244C22—102no—
245C22—102no—
246C22—102no—
247C22—102no—
248C22—104no—
249C22—201no—
250C22—201no—
251C22—202no—
252C22—202no—
253C22—202no—
254C22C2×D4×D511yescharacteristic
255C22C2×#71311yescharacteristic
256C22D5×#4111yescharacteristic
257C22D5×#4112yes—
258C22D5×#4112yes—
259C4—22no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—42no—
264C4—54no—
265C4—104no—
266C4—202no—
267C4—201no—
268C2—202no—
269C2—202no—
270C2#6596111yescharacteristic
271C2D5×#12311yescharacteristic
272C2#6600211yescharacteristic
273C2#6602512yes—
274C2#6604012yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC22×D20C22
hypercenter#400D5
hypocenterC5#2395
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#4038C2semidirect
3. C5×#400C2semidirect
4. #65281C2semidirect
5. #65272C2semidirect
6. #64964C2semidirect
7. #64963C2semidirect
9. #64955C2semidirect
10. #64955C2semidirect
11. #64938C2semidirect
72. #400D5semidirect
243. C5#2395semidirect

Maximal subgroups

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

subgroupindex
C2×#40392
C2×#40382
C5×#4002
#652812
#652722
#649642
#649632
#649612
#649552
#649382
#23955