Finite Group Databasebeta

← #10614819

Subgroups — #10614819

Order 640.

470 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 141 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×C4×Dic10C211yesmaximal, characteristic, complemented
3#65275C211yesmaximal, characteristic, complemented
4#65268C211yesmaximal, characteristic, complemented
5#65267C212yesmaximal, complemented
6#65261C212yesmaximal, complemented
7#65238C211yesmaximal, characteristic
8#65236C211yesmaximal, characteristic
9#65234C212yesmaximal
10#65233C212yesmaximal
11#65221C211yesmaximal, characteristic, complemented
12C22×Dic10C2211yescharacteristic
13C5×#126—24no—
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C42×C5C2211yescharacteristic
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C4⋊Dic5C2211yescharacteristic
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2212yes—
29C4×Dic10—24no—
30#3963C2212yes—
31#3963C2211yescharacteristic
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2604—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—
49C2×Dic10—22no—
50C2×Dic10—24no—
51C5×C4⋊C4—22no—
52C5×C4⋊C4—22no—
53C5×C4⋊C4—22no—
54C5×C22⋊C4—22no—
55C5×C22⋊C4—22no—
56C5×C22⋊C4—22no—
57C42×C5D414yes—
58C22⋊Dic5—24no—
59C22⋊Dic5—24no—
60C4⋊Dic5—22no—
61#683—24no—
62#683—24no—
63#683—24no—
64C4×Dic5—24no—
65C4×Dic5—24no—
66C2×#126D511yescharacteristic
67C2×C4×Q8—51no—
68#400—51no—
69#399—51no—
70#395—51no—
71#393—51no—
72#389—52no—
73#388—52no—
74#388—52no—
75#387—52no—
76#376—51no—
77C23×C5—21no—
78C23×C5—22no—
79C23×C5—22no—
80C23×C5—21no—
81C23×C5—21no—
82C23×C5C2411yescharacteristic
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—21no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—21no—
89C2×C4×C5—21no—
90C2×C4×C5—22no—
91C2×C4×C5—22no—
92C2×C4×C5—21no—
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#4114yes—
108C2×Dic5#4114yes—
109Dic10—28no—
110C22×Q8—51no—
111#126—24no—
112C4×Q8—104no—
113C2×C4⋊C4—52no—
114C2×C4⋊C4—51no—
115C2×C4⋊C4—52no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4C2×D511yescharacteristic
118C2×C4⋊C4C2×D511yescharacteristic
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C22⋊C4—52no—
121C2×C22⋊C4—52no—
122C2×C22⋊C4C2×D511yescharacteristic
123C2×C22⋊C4C2×D511yescharacteristic
124C2×C22⋊C4C2×D511yescharacteristic
125C2×C42—52no—
126C2×C42—52no—
127C2×C42C2×D511yescharacteristic
128#95—52no—
129#95—51no—
130#95—51no—
131#95—52no—
132#95—52no—
133#95—52no—
134#95—52no—
135C22×C5—41no—
136C22×C5—41no—
137C22×C5—41no—
138C22×C5—42no—
139C22×C5—42no—
140C22×C5C22×D411yescharacteristic
141C22×C5C2×#4112yes—
142C22×C5C2×#4112yes—
143C22×C52−1+411yescharacteristic
144C22×C52−1+411yescharacteristic
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—22no—
148C4×C5—41no—
149C4×C5—41no—
150C4×C5—41no—
151Dic5—24no—
152Dic5—24no—
153Dic5—42no—
154Dic5—42no—
155C24C22×D511yescharacteristic
156C2×Q8—104no—
157C2×Q8—102no—
158C22×C4—52no—
159C22×C4—52no—
160C22×C4—52no—
161C22×C4—52no—
162C22×C4C22×D511yescharacteristic
163C22×C4C22×D511yescharacteristic
164C22×C4C22×D511yescharacteristic
165C22×C4C22×D511yescharacteristic
166C22×C4C22×D511yescharacteristic
167C22×C4C22×D511yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—104no—
172C4⋊C4—104no—
173C4⋊C4—104no—
174C4⋊C4—102no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—104no—
179C22⋊C4—104no—
180C42—104no—
181C42—104no—
182C42C22⋊D514yes—
183C2×C5—81no—
184C2×C5#52511yescharacteristic
185C2×C5#53211yescharacteristic
186C2×C5#53611yescharacteristic
187C2×C5#55212yes—
188C2×C5#55612yes—
189C23—22no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23C23×D511yescharacteristic
195Q8—108no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—54no—
208C2×C4—54no—
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×C4—102no—
218C2×C4—102no—
219C2×C4C2×C22⋊D512yes—
220C2×C4C2×C22⋊D512yes—
221C2×C4C2×C22⋊D512yes—
222C5#260411yescharacteristic, complemented, Sylow
223C22—41no—
224C22—41no—
225C22—41no—
226C22—42no—
227C22—42no—
228C22D5×#4112yes—
229C22D5×#4112yes—
230C22#415011yescharacteristic
231C22#415011yescharacteristic
232C22C22×C22⋊D511yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—104no—
240C4—104no—
241C4—202no—
242C4—202no—
243C2—81no—
244C2#6603612yes—
245C2#6604212yes—
246C2#6615911yescharacteristic
247C2#6616211yescharacteristic
248C2#6616611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#126C2
prime radicalC24×C5C23
hypercenterC2×#126D5
hypocenterC5#2604
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic10C2semidirect
3. #65275C2semidirect
4. #65268C2semidirect
5. #65267C2semidirect
6. #65261C2semidirect
11. #65221C2semidirect
222. C5#2604semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1262
C2×C4×Dic102
#652752
#652682
#652672
#652612
#652382
#652362
#652342
#652332
#652212
#26045