Finite Group Databasebeta

← #10609656

Subgroups — #10609656

Order 640.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 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×#4044C211yesmaximal, characteristic
2C2×#4034C211yesmaximal, characteristic, complemented
3C5×#398C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic
5#65281C212yesmaximal
6#65272C211yesmaximal, characteristic, complemented
7#65268C212yesmaximal, complemented
8#64963C211yesmaximal, characteristic
9#64962C212yesmaximal
10#64949C212yesmaximal, complemented
11#64940C211yesmaximal, characteristic, complemented
12C22×Dic10C2212yes—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×#685C2211yescharacteristic
17C2×#685C2211yescharacteristic
18C2×#685C2211yescharacteristic
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2212yes—
24C2×#683C2211yescharacteristic
25C2×#683C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27#4044—24no—
28#4034—24no—
29C5×#95C2211yescharacteristic
30C5×#95C2212yes—
31#3963C2212yes—
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2821—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×Dic10—24no—
50C2×Dic10—42no—
51C2×Dic10—42no—
52C5×C4⋊C4—22no—
53C5×C4⋊C4—22no—
54C5×C4⋊C4—42no—
55#685—22no—
56#685—22no—
57#685—22no—
58#685—42no—
59#685—42no—
60C4⋊Dic5—22no—
61C4⋊Dic5—22no—
62#683—24no—
63#683—22no—
64#683—42no—
65C4×Dic5D414yes—
66C2×C4⋊Q8—51no—
67C2×#126—51no—
68#400—52no—
69#399—52no—
70#398—51no—
71#398D511yescharacteristic, complemented
72#397—51no—
73#397—51no—
74#395—52no—
75#393—52no—
76#383—51no—
77C23×C5C2411yescharacteristic
78C22×D5—21no—
79C22×D5—21no—
80C22×D5—21no—
81C22×D5—22no—
82C22×D5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—21no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5C2×D412yes—
96C2×Dic5—21no—
97C2×Dic5—21no—
98C2×Dic5—22no—
99C2×Dic5—21no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5C2×D412yes—
108C2×Dic5C2×D412yes—
109Dic10—48no—
110C22×Q8—52no—
111C4⋊Q8—104no—
112#126—104no—
113C2×C4⋊C4—51no—
114C2×C4⋊C4—51no—
115C2×C4⋊C4—51no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4C2×D511yescharacteristic
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C4⋊C4C2×D512yes—
121C2×C22⋊C4—51no—
122C2×C22⋊C4—51no—
123C2×C22⋊C4—51no—
124C2×C22⋊C4—52no—
125C2×C22⋊C4—52no—
126C2×C42—51no—
127#95—52no—
128#95—51no—
129#95—52no—
130#95—52no—
131#95—52no—
132#95—52no—
133#95C2×D511yescharacteristic
134#95C2×D512yes—
135C22×C5C22×D411yescharacteristic
136C22×C5H(5,𝔽2)11yescharacteristic
137C22×C5H(5,𝔽2)11yescharacteristic
138C22×C52−1+412yes—
139C22×C52−1+412yes—
140C2×D5—41no—
141C2×D5—41no—
142C2×D5—41no—
143C2×D5—42no—
144C2×D5—42no—
145C4×C5—22no—
146C4×C5—42no—
147C4×C5—41no—
148C4×C5—41no—
149C4×C5—42no—
150Dic5—22no—
151Dic5—22no—
152Dic5—42no—
153Dic5—41no—
154Dic5—42no—
155C24—51no—
156C2×Q8—104no—
157C2×Q8—202no—
158C2×Q8—202no—
159C22×C4—51no—
160C22×C4—51no—
161C22×C4—52no—
162C22×C4—51no—
163C22×C4—52no—
164C22×C4C22×D512yes—
165C22×C4C22×D511yescharacteristic
166C22×C4C22×D512yes—
167C22×C4C22×D511yescharacteristic
168C22×C4C22×D511yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—42no—
172C4⋊C4—102no—
173C4⋊C4—102no—
174C4⋊C4—104no—
175C4⋊C4—102no—
176C4⋊C4—202no—
177C22⋊C4—102no—
178C22⋊C4—102no—
179C22⋊C4—102no—
180C22⋊C4—202no—
181C22⋊C4—202no—
182C42—54no—
183C2×C5#53511yescharacteristic
184C2×C5#53611yescharacteristic
185C2×C5#53611yescharacteristic
186C2×C5#56312yes—
187C2×C5#56312yes—
188D5—81no—
189C23—101no—
190C23—101no—
191C23—102no—
192C23—101no—
193C23—102no—
194C23C23×D511yescharacteristic
195Q8—208no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—52no—
209C2×C4—52no—
210C2×C4—101no—
211C2×C4—102no—
212C2×C4—101no—
213C2×C4—101no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4D4×D512yes—
222C5#282111yescharacteristic, complemented, Sylow
223C22—201no—
224C22—201no—
225C22—202no—
226C22—201no—
227C22—202no—
228C22C2×D4×D511yescharacteristic
229C22#414411yescharacteristic
230C22#414712yes—
231C22#414911yescharacteristic
232C22#415012yes—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4—102no—
239C4—102no—
240C4—201no—
241C4—202no—
242C4—202no—
243C2—401no—
244C2#6594111yescharacteristic
245C2#6598012yes—
246C2#6598211yescharacteristic
247C2#6603312yes—
248C2#6605811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#398C2
prime radicalC23×D5C23
hypercenter#398D5
hypocenterC5#2821
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#4034C2semidirect
3. C5×#398C2semidirect
6. #65272C2semidirect
7. #65268C2semidirect
10. #64949C2semidirect
11. #64940C2semidirect
71. #398D5semidirect
222. C5#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#40442
C2×#40342
C5×#3982
#652822
#652812
#652722
#652682
#649632
#649622
#649492
#649402
#28215