Finite Group Databasebeta

← #10614776

Subgroups — #10614776

Order 640.

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

normal subgroups only

This group has 434 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×C5×#125C211yesmaximal, characteristic, complemented
2C2×#4022C211yesmaximal, characteristic
3#65283C212yesmaximal
4#65282C211yesmaximal, characteristic
5#65281C211yesmaximal, characteristic
6#65279C212yesmaximal
7#65273C212yesmaximal, complemented
8#65271C211yesmaximal, characteristic, complemented
9#65269C211yesmaximal, characteristic, complemented
10#65261C212yesmaximal, complemented
11C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
12C5×#125—24no—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×C5×C4⋊C4C2212yes—
17C2×C42×C5C2211yescharacteristic
18C2×#685C2211yescharacteristic
19C2×#685C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×C4⋊Dic5C2212yes—
24C2×#683C2212yes—
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2212yes—
29C2×C4×Dic5C2212yes—
30#4022—24no—
31#3963C2211yescharacteristic
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2664—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×Dic5C2312yes—
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2312yes—
49C5×C4⋊C4—22no—
50C5×C4⋊C4—22no—
51C5×C4⋊C4—24no—
52C5×C4⋊C4—24no—
53C42×C5D414yes—
54#685—22no—
55#685—22no—
56#685—22no—
57#685—24no—
58#685—42no—
59C4⋊Dic5—24no—
60#683—22no—
61#683—22no—
62#683—22no—
63#683—24no—
64#683—42no—
65C4×Dic5—42no—
66C2×#126—51no—
67C2×#125D511yescharacteristic, complemented
68#400—51no—
69#399—51no—
70#398—51no—
71#398—52no—
72#397—51no—
73#396—52no—
74#389—52no—
75#388—52no—
76C4×C4⋊C4—51no—
77C23×C5C2411yescharacteristic
78C22×D5—21no—
79C22×D5—22no—
80C22×D5—21no—
81C22×D5—22no—
82C22×D5—21no—
83C2×C4×C5—22no—
84C2×C4×C5—21no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
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—21no—
99C2×Dic5—21no—
100C2×Dic5—21no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—21no—
108C2×Dic5#4114yes—
109#126—104no—
110#125—24no—
111C2×C4⋊C4—51no—
112C2×C4⋊C4—51no—
113C2×C4⋊C4—51no—
114C2×C4⋊C4—52no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4C2×D512yes—
118C2×C4⋊C4C2×D512yes—
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C4⋊C4C2×D511yescharacteristic
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—51no—
129#95—51no—
130#95—52no—
131#95—52no—
132#95—52no—
133#95—52no—
134C22×C5C22×D411yescharacteristic
135C22×C5C2×#4112yes—
136C22×C5H(5,𝔽2)11yescharacteristic
137C22×C52−1+412yes—
138C22×C52−1+411yescharacteristic
139C2×D5—41no—
140C2×D5—41no—
141C2×D5—41no—
142C2×D5—42no—
143C2×D5—42no—
144C4×C5—22no—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—42no—
148C4×C5—42no—
149Dic5—24no—
150Dic5—41no—
151Dic5—41no—
152Dic5—41no—
153Dic5—42no—
154C24—51no—
155C22×C4—52no—
156C22×C4—52no—
157C22×C4—51no—
158C22×C4—51no—
159C22×C4—51no—
160C22×C4C22×D512yes—
161C22×C4C22×D512yes—
162C22×C4C22×D511yescharacteristic
163C22×C4C22×D511yescharacteristic
164C22×C4C22×D511yescharacteristic
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—104no—
170C4⋊C4—104no—
171C4⋊C4—102no—
172C4⋊C4—102no—
173C4⋊C4—102no—
174C4⋊C4—202no—
175C22⋊C4—104no—
176C22⋊C4—102no—
177C22⋊C4—102no—
178C22⋊C4—102no—
179C22⋊C4—202no—
180C42—202no—
181C42C22⋊D514yes—
182C2×C5#53211yescharacteristic
183C2×C5#53611yescharacteristic
184C2×C5#53711yescharacteristic
185C2×C5#53912yes—
186C2×C5#54112yes—
187D5—81no—
188C23—102no—
189C23—102no—
190C23—101no—
191C23—101no—
192C23—101no—
193C23C23×D511yescharacteristic
194C2×C4—22no—
195C2×C4—22no—
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—54no—
205C2×C4—102no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—102no—
210C2×C4—102no—
211C2×C4—102no—
212C2×C4—101no—
213C2×C4—101no—
214C2×C4—101no—
215C2×C4—101no—
216C2×C4—101no—
217C2×C4C2×C22⋊D512yes—
218C2×C4C2×C22⋊D512yes—
219C2×C4C2×C22⋊D512yes—
220C5#266411yescharacteristic, complemented, Sylow
221C22—201no—
222C22—201no—
223C22—201no—
224C22—202no—
225C22—202no—
226C22#414712yes—
227C22D5×#4112yes—
228C22#414911yescharacteristic
229C22#415011yescharacteristic
230C22C22×C22⋊D511yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—42no—
236C4—104no—
237C4—201no—
238C4—201no—
239C4—201no—
240C4—202no—
241C2—401no—
242C2#6602612yes—
243C2#6603012yes—
244C2#6615011yescharacteristic
245C2#6616611yescharacteristic
246C2#6616711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#125C2
prime radicalC23×D5C23
hypercenterC2×#125D5
hypocenterC5#2664
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C5×#125C2semidirect
7. #65273C2semidirect
8. #65271C2semidirect
9. #65269C2semidirect
10. #65261C2semidirect
11. C4×C4⋊Dic5C2semidirect
67. C2×#125D5semidirect
220. C5#2664semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1252
C2×#40222
#652832
#652822
#652812
#652792
#652732
#652712
#652692
#652612
C4×C4⋊Dic52
#26645