Finite Group Databasebeta

← #10605977

Subgroups — #10605977

Order 640.

450 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 137 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×#4022C211yesmaximal, characteristic
2C2×C4×Dic10C211yesmaximal, characteristic, complemented
3C5×#382C211yesmaximal, characteristic, complemented
4#65283C212yesmaximal
5#65272C212yesmaximal, complemented
6#64963C211yesmaximal, characteristic
7#64962C211yesmaximal, characteristic
8#64959C212yesmaximal
9#64950C211yesmaximal, characteristic, complemented
10#64945C211yesmaximal, characteristic, complemented
11#64942C212yesmaximal, complemented
12C22×Dic10C2211yescharacteristic
13C2×C5×C4⋊C4C2212yes—
14C2×C42×C5C2211yescharacteristic
15C2×#685C2211yescharacteristic
16C2×#685C2212yes—
17C2×#685C2211yescharacteristic
18C2×#685C2211yescharacteristic
19C2×#685C2212yes—
20C2×C4⋊Dic5C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2212yes—
23C2×#683C2212yes—
24C2×#683C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×C4×Dic5C2212yes—
28#4022—24no—
29C4×Dic10—24no—
30C5×#95C2212yes—
31C5×#95C2211yescharacteristic
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2637—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×Dic5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2312yes—
49C2×Dic10—22no—
50C2×Dic10—24no—
51C5×C4⋊C4—42no—
52C42×C5D414yes—
53#685—22no—
54#685—22no—
55#685—22no—
56#685—24no—
57#685—42no—
58C4⋊Dic5—22no—
59C4⋊Dic5—24no—
60C4⋊Dic5—42no—
61#683—24no—
62#683—22no—
63#683—22no—
64#683—22no—
65C4×Dic5—24no—
66C2×#126—51no—
67C2×C4×Q8—51no—
68#398—52no—
69#398—51no—
70#397—51no—
71#396—52no—
72#395—51no—
73#393—51no—
74#388—52no—
75#387—52no—
76#382D511yescharacteristic, complemented
77C23×C5C2411yescharacteristic
78C22×D5—21no—
79C22×D5—22no—
80C22×D5—21no—
81C22×D5—21no—
82C22×D5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—21no—
85C2×C4×C5—21no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5C2×D412yes—
94C2×C4×C5C2×D412yes—
95C2×C4×C5C2×D412yes—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—21no—
100C2×Dic5—21no—
101C2×Dic5—21no—
102C2×Dic5—22no—
103C2×Dic5—21no—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—21no—
107C2×Dic5—22no—
108C2×Dic5#4114yes—
109Dic10—28no—
110C22×Q8—51no—
111#126—104no—
112C4×Q8—104no—
113C2×C4⋊C4—52no—
114C2×C4⋊C4—51no—
115C2×C4⋊C4—51no—
116C2×C4⋊C4—51no—
117C2×C4⋊C4—51no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4—52no—
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—52no—
127C2×C42C2×D511yescharacteristic
128#95—52no—
129#95—52no—
130#95—52no—
131#95—52no—
132#95C2×D512yes—
133#95C2×D511yescharacteristic
134#95C2×D511yescharacteristic
135C22×C5C22×D411yescharacteristic
136C22×C5C2×#4112yes—
137C22×C5H(5,𝔽2)11yescharacteristic
138C22×C52−1+411yescharacteristic
139C22×C52−1+412yes—
140C2×D5—41no—
141C2×D5—41no—
142C2×D5—41no—
143C2×D5—42no—
144C2×D5—42no—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—22no—
148C4×C5—42no—
149C4×C5—42no—
150Dic5—24no—
151Dic5—41no—
152Dic5—41no—
153Dic5—42no—
154Dic5—41no—
155C24—51no—
156C2×Q8—104no—
157C2×Q8—102no—
158C22×C4—52no—
159C22×C4—51no—
160C22×C4—52no—
161C22×C4—51no—
162C22×C4—51no—
163C22×C4C22×D512yes—
164C22×C4C22×D512yes—
165C22×C4C22×D511yescharacteristic
166C22×C4C22×D511yescharacteristic
167C22×C4C22×D511yescharacteristic
168C4⋊C4—42no—
169C4⋊C4—104no—
170C4⋊C4—104no—
171C4⋊C4—102no—
172C4⋊C4—102no—
173C4⋊C4—102no—
174C4⋊C4—102no—
175C4⋊C4—202no—
176C22⋊C4—104no—
177C22⋊C4—102no—
178C22⋊C4—102no—
179C22⋊C4—102no—
180C22⋊C4—202no—
181C42—104no—
182C42D2014yes—
183C2×C5#52511yescharacteristic
184C2×C5#53611yescharacteristic
185C2×C5#53711yescharacteristic
186C2×C5#54112yes—
187C2×C5#54212yes—
188D5—81no—
189C23—101no—
190C23—101no—
191C23—101no—
192C23—102no—
193C23—102no—
194C23C23×D511yescharacteristic
195Q8—108no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—54no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—102no—
210C2×C4—101no—
211C2×C4—101no—
212C2×C4—101no—
213C2×C4—102no—
214C2×C4—101no—
215C2×C4—102no—
216C2×C4—101no—
217C2×C4—102no—
218C2×C4—102no—
219C2×C4C2×D2012yes—
220C2×C4C2×D2012yes—
221C2×C4C2×D2012yes—
222C5#263711yescharacteristic, complemented, Sylow
223C22—201no—
224C22—201no—
225C22—201no—
226C22—202no—
227C22—202no—
228C22C22×D2011yescharacteristic
229C22#414411yescharacteristic
230C22#414711yescharacteristic
231C22D5×#4112yes—
232C22#415012yes—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—104no—
239C4—202no—
240C4—201no—
241C4—201no—
242C4—201no—
243C2—401no—
244C2#6584411yescharacteristic
245C2#6585811yescharacteristic
246C2#6586011yescharacteristic
247C2#6600312yes—
248C2#6603012yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#382C2
prime radicalC23×D5C23
hypercenter#382D5
hypocenterC5#2637
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic10C2semidirect
3. C5×#382C2semidirect
5. #65272C2semidirect
9. #64950C2semidirect
10. #64945C2semidirect
11. #64942C2semidirect
76. #382D5semidirect
222. C5#2637semidirect

Maximal subgroups

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

subgroupindex
C2×#40222
C2×C4×Dic102
C5×#3822
#652832
#652722
#649632
#649622
#649592
#649502
#649452
#649422
#26375