Finite Group Databasebeta

← #10609822

Subgroups — #10609822

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×#4036C211yesmaximal, characteristic, complemented
2C2×#4031C211yesmaximal, characteristic
3C5×#399C211yesmaximal, characteristic, complemented
4#65283C212yesmaximal, complemented
5#65281C211yesmaximal, characteristic, complemented
6#65268C212yesmaximal, complemented
7#65248C212yesmaximal
8#65242C211yesmaximal, characteristic, complemented
9#65237C212yesmaximal, complemented
10#64963C211yesmaximal, characteristic, complemented
11#64939C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×#683C2212yes—
27C2×#683C2212yes—
28C2×#683C2212yes—
29C2×C4×Dic5C2211yescharacteristic
30#4036—24no—
31#4031—24no—
32C5×#95C2211yescharacteristic
33#3963C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2820—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C2×C22⋊D5—22no—
45C2×C22⋊D5—42no—
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×Dic10—22no—
52C2×Dic10—42no—
53C5×C4⋊C4—22no—
54C5×C4⋊C4—42no—
55C5×C22⋊C4—22no—
56C5×C22⋊C4—42no—
57C22⋊Dic5—22no—
58C22⋊Dic5—42no—
59#685—22no—
60#685—22no—
61#685—42no—
62#685—42no—
63C4⋊Dic5—22no—
64C4⋊Dic5—24no—
65#683—24no—
66#683—42no—
67#683—42no—
68C4×Dic5D414yes—
69C2×#125—51no—
70C2×#124—51no—
71#399—51no—
72#399D511yescharacteristic, complemented
73#397—51no—
74#397—51no—
75#396—52no—
76#396—52no—
77#395—52no—
78#394—52no—
79#382—51no—
80C23×C5—21no—
81C23×C5—21no—
82C23×C5—22no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5C2411yescharacteristic
86C22×D5—21no—
87C22×D5—21no—
88C22×D5—22no—
89C22×D5—21no—
90C22×D5—22no—
91C2×C4×C5—22no—
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×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5C2×D412yes—
101C22⋊D5—44no—
102C2×Dic5—21no—
103C2×Dic5—22no—
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×Dic5C2×D412yes—
114C2×Dic5C2×D412yes—
115Dic10—44no—
116C22×Q8—51no—
117C22×D4—51no—
118#125—104no—
119#124—104no—
120C2×C4⋊C4—52no—
121C2×C4⋊C4—52no—
122C2×C4⋊C4—51no—
123C2×C4⋊C4—52no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4C2×D511yescharacteristic
126C2×C4⋊C4C2×D512yes—
127C2×C22⋊C4—51no—
128C2×C22⋊C4—51no—
129C2×C22⋊C4—51no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4—52no—
133C2×C22⋊C4C2×D511yescharacteristic
134C2×C22⋊C4C2×D512yes—
135C2×C42—51no—
136#95—52no—
137#95—51no—
138#95—52no—
139#95—52no—
140#95C2×D511yescharacteristic
141C22×C5—41no—
142C22×C5—41no—
143C22×C5—41no—
144C22×C5—42no—
145C22×C5—42no—
146C22×C5C22×D411yescharacteristic
147C22×C5H(5,𝔽2)12yes—
148C22×C5H(5,𝔽2)11yescharacteristic
149C22×C52−1+411yescharacteristic
150C22×C52−1+412yes—
151C2×D5—41no—
152C2×D5—41no—
153C2×D5—41no—
154C2×D5—42no—
155C2×D5—42no—
156C4×C5—22no—
157C4×C5—41no—
158C4×C5—42no—
159C4×C5—42no—
160Dic5—22no—
161Dic5—22no—
162Dic5—41no—
163Dic5—42no—
164Dic5—42no—
165C24—51no—
166C24C22×D511yescharacteristic
167C2×Q8—102no—
168C2×Q8—202no—
169C2×D4—102no—
170C2×D4—202no—
171C22×C4—51no—
172C22×C4—51no—
173C22×C4—52no—
174C22×C4—52no—
175C22×C4—51no—
176C22×C4C22×D512yes—
177C22×C4C22×D511yescharacteristic
178C22×C4C22×D512yes—
179C22×C4C22×D511yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—104no—
183C4⋊C4—104no—
184C4⋊C4—102no—
185C4⋊C4—202no—
186C4⋊C4—202no—
187C22⋊C4—22no—
188C22⋊C4—42no—
189C22⋊C4—102no—
190C22⋊C4—102no—
191C22⋊C4—102no—
192C22⋊C4—202no—
193C22⋊C4—202no—
194C22⋊C4—202no—
195C42—54no—
196C2×C5—81no—
197C2×C5#53511yescharacteristic
198C2×C5#53611yescharacteristic
199C2×C5#53711yescharacteristic
200C2×C5#56212yes—
201C2×C5#56312yes—
202D5—81no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—101no—
209C23—102no—
210C23—101no—
211C23—101no—
212C23—102no—
213C23C23×D511yescharacteristic
214Q8—204no—
215D4—204no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—52no—
226C2×C4—52no—
227C2×C4—102no—
228C2×C4—101no—
229C2×C4—101no—
230C2×C4—101no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4D4×D512yes—
239C5#282011yescharacteristic, complemented, Sylow
240C22—41no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—201no—
246C22—201no—
247C22—202no—
248C22—201no—
249C22—202no—
250C22C2×D4×D511yescharacteristic
251C22#414412yes—
252C22#414711yescharacteristic
253C22#414911yescharacteristic
254C22#415012yes—
255C4—22no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—102no—
260C4—102no—
261C4—201no—
262C4—202no—
263C4—202no—
264C2—81no—
265C2—401no—
266C2#6596311yescharacteristic
267C2#6598012yes—
268C2#6598211yescharacteristic
269C2#6600112yes—
270C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#399C2
prime radicalC22×C22⋊D5C22
hypercenter#399D5
hypocenterC5#2820
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#4036C2semidirect
3. C5×#399C2semidirect
4. #65283C2semidirect
5. #65281C2semidirect
6. #65268C2semidirect
8. #65242C2semidirect
9. #65237C2semidirect
10. #64963C2semidirect
11. #64939C2semidirect
72. #399D5semidirect
239. C5#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#40362
C2×#40312
C5×#3992
#652832
#652812
#652682
#652482
#652422
#652372
#649632
#649392
#28205