Finite Group Databasebeta

← #10614821

Subgroups — #10614821

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×#126C211yesmaximal, characteristic
2C2×#4016C211yesmaximal, characteristic, complemented
3#65275C211yesmaximal, characteristic, complemented
4#65273C212yesmaximal, complemented
5#65272C211yesmaximal, characteristic, complemented
6#65261C212yesmaximal, complemented
7#65242C211yesmaximal, characteristic
8#65238C211yesmaximal, characteristic
9#65237C212yesmaximal
10#65234C212yesmaximal
11C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
12C5×#126—24no—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C42×C5C2211yescharacteristic
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×C4⋊Dic5C2211yescharacteristic
24C2×C4⋊Dic5C2212yes—
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×#683C2212yes—
28C2×#683C2212yes—
29C2×C4×Dic5C2212yes—
30#4016—24no—
31#3963C2211yescharacteristic
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2664—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C5×C4⋊C4—22no—
50C5×C4⋊C4—22no—
51C5×C4⋊C4—22no—
52C5×C22⋊C4—22no—
53C5×C22⋊C4—22no—
54C5×C22⋊C4—22no—
55C42×C5D414yes—
56C22⋊Dic5—24no—
57C22⋊Dic5—42no—
58C4⋊Dic5—22no—
59C4⋊Dic5—22no—
60C4⋊Dic5—24no—
61#683—24no—
62#683—24no—
63#683—24no—
64#683—42no—
65C4×Dic5—42no—
66C2×#126D511yescharacteristic
67C2×#125—51no—
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×C5—21no—
78C23×C5—22no—
79C23×C5—21no—
80C23×C5—22no—
81C23×C5—21no—
82C23×C5C2411yescharacteristic
83C2×C4×C5—22no—
84C2×C4×C5—21no—
85C2×C4×C5—22no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5C2×D412yes—
95C2×C4×C5C2×D412yes—
96C2×C4×C5C2×D412yes—
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—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5#4114yes—
109#126—24no—
110#125—104no—
111C2×C4⋊C4—52no—
112C2×C4⋊C4—52no—
113C2×C4⋊C4—51no—
114C2×C4⋊C4—51no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4C2×D511yescharacteristic
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C4⋊C4C2×D511yescharacteristic
121C2×C22⋊C4—52no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4C2×D511yescharacteristic
124C2×C22⋊C4C2×D511yescharacteristic
125C2×C22⋊C4C2×D511yescharacteristic
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×C5—41no—
135C22×C5—41no—
136C22×C5—41no—
137C22×C5—42no—
138C22×C5—42no—
139C22×C5C22×D411yescharacteristic
140C22×C5C2×#4112yes—
141C22×C5H(5,𝔽2)11yescharacteristic
142C22×C52−1+412yes—
143C22×C52−1+411yescharacteristic
144C4×C5—22no—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—41no—
148C4×C5—41no—
149C4×C5—41no—
150Dic5—24no—
151Dic5—42no—
152Dic5—42no—
153Dic5—42no—
154C24C22×D511yescharacteristic
155C22×C4—52no—
156C22×C4—52no—
157C22×C4—52no—
158C22×C4—52no—
159C22×C4C22×D511yescharacteristic
160C22×C4C22×D511yescharacteristic
161C22×C4C22×D511yescharacteristic
162C22×C4C22×D511yescharacteristic
163C22×C4C22×D511yescharacteristic
164C22×C4C22×D511yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—104no—
169C4⋊C4—104no—
170C4⋊C4—104no—
171C4⋊C4—104no—
172C4⋊C4—102no—
173C4⋊C4—102no—
174C4⋊C4—202no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—104no—
179C22⋊C4—202no—
180C42—202no—
181C42C22⋊D514yes—
182C2×C5—81no—
183C2×C5#53211yescharacteristic
184C2×C5#53611yescharacteristic
185C2×C5#53711yescharacteristic
186C2×C5#53912yes—
187C2×C5#54112yes—
188C23—22no—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23C23×D511yescharacteristic
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—54no—
206C2×C4—102no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—102no—
210C2×C4—102no—
211C2×C4—102no—
212C2×C4—102no—
213C2×C4—102no—
214C2×C4—102no—
215C2×C4—102no—
216C2×C4—102no—
217C2×C4C2×C22⋊D512yes—
218C2×C4C2×C22⋊D512yes—
219C2×C4C2×C22⋊D512yes—
220C5#266411yescharacteristic, complemented, Sylow
221C22—41no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22D5×#4112yes—
227C22#414911yescharacteristic
228C22#415012yes—
229C22#415011yescharacteristic
230C22C22×C22⋊D511yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—104no—
238C4—202no—
239C4—202no—
240C4—202no—
241C2—81no—
242C2#6603512yes—
243C2#6604112yes—
244C2#6615911yescharacteristic
245C2#6616611yescharacteristic
246C2#6616711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#126C2
prime radicalC24×C5C23
hypercenterC2×#126D5
hypocenterC5#2664
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. C2×#4016C2semidirect
3. #65275C2semidirect
4. #65273C2semidirect
5. #65272C2semidirect
6. #65261C2semidirect
11. C4×C4⋊Dic5C2semidirect
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×#1262
C2×#40162
#652752
#652732
#652722
#652612
#652422
#652382
#652372
#652342
C4×C4⋊Dic52
#26645