Finite Group Databasebeta

← #10608458

Subgroups — #10608458

Order 640.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#4082C212yesmaximal, complemented
2C5×#392C211yesmaximal, characteristic
3#65511C212yesmaximal, complemented
4#65509C212yesmaximal, complemented
5#65242C211yesmaximal, characteristic, complemented
6#65240C212yesmaximal, complemented
7#65236C212yesmaximal, complemented
8#65235C211yesmaximal, characteristic, complemented
9#64935C212yesmaximal, complemented
10C22×C5×D4C2212yes—
11C22×C5×D4C2211yescharacteristic
12C23×Dic5C2211yescharacteristic
13C22×Dic10C2212yes—
14C2×C5×C22⋊C4C2212yes—
15C2×C5×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic5C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2211yescharacteristic
22#4082—28no—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×#683C2212yes—
25C2×C4×Dic5C2212yes—
26C5×#95C2211yescharacteristic
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2211yescharacteristic
32#2712—51nomaximal, Sylow
33C24×C5C2312yes—
34C24×C5C2311yescharacteristic
35C2×C5×D4—24no—
36C2×C5×D4—24no—
37C2×C5×D4—42no—
38C2×C5×D4—42no—
39C2×C5×D4—41no—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×Dic5—42no—
44C22×Dic5—41no—
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C2×Dic10—24no—
50C2×Dic10—24no—
51C2×Dic10—42no—
52C5×C22⋊C4—24no—
53C5×C22⋊C4—41no—
54C22⋊Dic5—24no—
55C22⋊Dic5—24no—
56C22⋊Dic5—24no—
57C22⋊Dic5—42no—
58C22⋊Dic5—42no—
59C22⋊Dic5D414yescomplemented
60C4⋊Dic5—41no—
61#683—42no—
62C4×Dic5—24no—
63C2×#124—52no—
64#397—51no—
65#394—52no—
66#393—52no—
67#392D511yescharacteristic
68#391—52no—
69#388—52no—
70#386—52no—
71#385—51no—
72C23×C5—22no—
73C23×C5—21no—
74C23×C5—22no—
75C23×C5—21no—
76C23×C5—22no—
77C23×C5—22no—
78C23×C5—22no—
79C23×C5—22no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5C2×D412yes—
84C23×C5C2411yescharacteristic
85C5×D4—48no—
86C5×D4—44no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—21no—
92C2×C4×C5—21no—
93C2×C4×C5#4114yes—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—44no—
106C2×Dic5C2×D414yes—
107Dic10—48no—
108C22×Q8—52no—
109C22×D4C2×D512yes—
110C22×D4C2×D511yescharacteristic
111C23×C4—51no—
112#124—108no—
113C2×C4⋊C4—52no—
114C2×C4⋊C4—51no—
115C2×C22⋊C4—51no—
116C2×C22⋊C4—52no—
117C2×C22⋊C4—52no—
118C2×C22⋊C4—52no—
119C2×C22⋊C4—52no—
120C2×C22⋊C4—52no—
121C2×C22⋊C4C2×D512yes—
122C2×C22⋊C4C2×D511yescharacteristic
123C2×C42—52no—
124#95—52no—
125#95—52no—
126#95—52no—
127#95—52no—
128#95—51no—
129#95C2×D511yescharacteristic
130C22×C5—22no—
131C22×C5—24no—
132C22×C5—42no—
133C22×C5—41no—
134C22×C5—41no—
135C22×C5—42no—
136C22×C5—42no—
137C22×C5—42no—
138C22×C5—42no—
139C22×C5—42no—
140C22×C5—42no—
141C22×C5—42no—
142C22×C5C22×D411yescharacteristic
143C22×C5C2×#4112yes—
144C22×C5H(5,𝔽2)11yescharacteristic
145C22×C5H(5,𝔽2)12yes—
146C22×C52−1+411yescharacteristic
147C4×C5—24no—
148C4×C5—41no—
149C4×C5—41no—
150Dic5—24no—
151Dic5—42no—
152Dic5—42no—
153Dic5—42no—
154C24C22×D512yes—
155C24C22×D511yescharacteristic
156C2×Q8—104no—
157C2×Q8—104no—
158C2×Q8—202no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—42no—
164C22×C4—52no—
165C22×C4—52no—
166C22×C4—52no—
167C22×C4—52no—
168C22×C4—201no—
169C22×C4—202no—
170C22×C4C22×D511yescharacteristic
171C22×C4C22×D511yescharacteristic
172C22×C4C22×D512yes—
173C4⋊C4—202no—
174C4⋊C4—201no—
175C22⋊C4—24no—
176C22⋊C4—41no—
177C22⋊C4—54no—
178C22⋊C4—104no—
179C22⋊C4—104no—
180C22⋊C4—104no—
181C22⋊C4—202no—
182C22⋊C4—202no—
183C42—104no—
184C2×C5—42no—
185C2×C5—82no—
186C2×C5#53711yescharacteristic
187C2×C5#54212yes—
188C2×C5#54612yes—
189C2×C5#55911yescharacteristic
190C2×C5H(3,𝔽4)11yescharacteristic
191C23—22no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23D4×D512yes—
203C23C23×D511yescharacteristic
204Q8—208no—
205D4—48no—
206D4—44no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—54no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—102no—
224C2×C4—102no—
225C2×C4—204no—
226C2×C4#71114yes—
227C5#271211yescharacteristic, complemented, Sylow
228C22—24no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—41no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22C2×D4×D511yescharacteristic
241C22C2×#71112yes—
242C22#414412yes—
243C22#414411yescharacteristic
244C22#415011yescharacteristic
245C4—24no—
246C4—41no—
247C4—41no—
248C4—104no—
249C4—202no—
250C4—202no—
251C4—202no—
252C2—42no—
253C2—82no—
254C2#6592512yes—
255C2#6592811yescharacteristic
256C2#6593111yescharacteristic
257C2#6593712yes—
258C2#6604711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC5×#392C2
hypercenter#392D5
hypocenterC5#2712
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#4082C2semidirect
3. #65511C2semidirect
4. #65509C2semidirect
5. #65242C2semidirect
6. #65240C2semidirect
7. #65236C2semidirect
8. #65235C2semidirect
9. #64935C2semidirect
59. C22⋊Dic5D4semidirect
227. C5#2712semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
C5×#3922
#655112
#655092
#652422
#652402
#652362
#652352
#649352
#27125