Finite Group Databasebeta

← #10608554

Subgroups — #10608554

Order 640.

544 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 149 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×#4090C212yesmaximal, complemented
2C2×#4024C212yesmaximal
3C5×#393C211yesmaximal, characteristic
4#65238C211yesmaximal, characteristic
5#65235C211yesmaximal, characteristic
6#65234C212yesmaximal
7#64946C212yesmaximal, complemented
8#64945C212yesmaximal, complemented
9#64941C212yesmaximal, complemented
10C22×C5×Q8C2211yescharacteristic
11C23×Dic5C2211yescharacteristic
12C22×Dic10C2212yes—
13C2×C5×C22⋊C4C2212yes—
14C2×C5×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2211yescharacteristic
17#4090—28no—
18C2×C4⋊Dic5C2212yes—
19C2×C4⋊Dic5C2211yescharacteristic
20C2×#683C2212yes—
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×C4×Dic5C2212yes—
26#4024—28no—
27C5×#95C2212yes—
28C5×#95C2211yescharacteristic
29#3963C2211yescharacteristic
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#2677—51nomaximal, Sylow
34C24×C5C2311yescharacteristic
35C2×C5×Q8—24no—
36C2×C5×Q8—41no—
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×C4×C5C2312yes—
41C22×Dic5—24no—
42C22×Dic5—41no—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C2×Dic10—24no—
48C2×Dic10—24no—
49C2×Dic10—42no—
50C5×C22⋊C4—24no—
51C5×C22⋊C4—41no—
52C22⋊Dic5—24no—
53C22⋊Dic5Q814yescomplemented
54C4⋊Dic5—24no—
55C4⋊Dic5—41no—
56#683—24no—
57#683—24no—
58#683—24no—
59#683—24no—
60#683D418yes—
61C4×Dic5—24no—
62C2×C4⋊Q8—52no—
63C2×C22⋊Q8—52no—
64#399—51no—
65#398—52no—
66#395—52no—
67#393D511yescharacteristic
68#388—52no—
69#387—52no—
70#385—51no—
71C23×C5—22no—
72C23×C5—21no—
73C23×C5—21no—
74C23×C5—22no—
75C23×C5C2×Q812yes—
76C23×C5C2411yescharacteristic
77C5×Q8—44no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—21no—
83C2×C4×C5—21no—
84C2×C4×C5—22no—
85C2×C4×C5—22no—
86C2×C4×C5—22no—
87C2×C4×C5C2×D414yes—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—44no—
98C2×Dic5C2×D414yes—
99C2×Dic5C2×D414yes—
100C2×Dic5C2×Q814yes—
101Dic10—48no—
102C22×Q8—52no—
103C22×Q8C2×D511yescharacteristic
104C23×C4—51no—
105C4⋊Q8—108no—
106C22⋊Q8—108no—
107C2×C4⋊C4—52no—
108C2×C4⋊C4—52no—
109C2×C4⋊C4—52no—
110C2×C4⋊C4—52no—
111C2×C4⋊C4—52no—
112C2×C4⋊C4—52no—
113C2×C4⋊C4—51no—
114C2×C22⋊C4—51no—
115C2×C22⋊C4—52no—
116C2×C22⋊C4C2×D512yes—
117C2×C22⋊C4C2×D511yescharacteristic
118C2×C42—52no—
119#95—52no—
120#95—51no—
121#95—52no—
122#95—52no—
123#95C2×D512yes—
124#95C2×D511yescharacteristic
125C22×C5—22no—
126C22×C5—24no—
127C22×C5—42no—
128C22×C5—41no—
129C22×C5—41no—
130C22×C5C22×D412yes—
131C22×C5C22×Q811yescharacteristic
132C22×C5H(5,𝔽2)11yescharacteristic
133C22×C52−1+411yescharacteristic
134C22×C52−1+412yes—
135C4×C5—24no—
136C4×C5—42no—
137C4×C5—41no—
138C4×C5—41no—
139Dic5—24no—
140Dic5—24no—
141Dic5—24no—
142Dic5—42no—
143C24C22×D511yescharacteristic
144C2×Q8—24no—
145C2×Q8—41no—
146C2×Q8—104no—
147C2×Q8—104no—
148C2×Q8—202no—
149C22×C4—52no—
150C22×C4—52no—
151C22×C4—52no—
152C22×C4—52no—
153C22×C4—104no—
154C22×C4—201no—
155C22×C4C22×D511yescharacteristic
156C22×C4C22×D512yes—
157C22×C4C22×D511yescharacteristic
158C22×C4C22×D512yes—
159C4⋊C4—58no—
160C4⋊C4—104no—
161C4⋊C4—104no—
162C4⋊C4—104no—
163C4⋊C4—104no—
164C4⋊C4—104no—
165C4⋊C4—201no—
166C22⋊C4—24no—
167C22⋊C4—41no—
168C22⋊C4—54no—
169C22⋊C4—104no—
170C42—104no—
171C2×C5—42no—
172C2×C5#53612yes—
173C2×C5#54411yescharacteristic
174C2×C5D4211yescharacteristic
175C2×C5D4×Q812yes—
176C2×C5#56311yescharacteristic
177C23—22no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23Q8×D512yes—
182C23C23×D511yescharacteristic
183Q8—44no—
184Q8—208no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—54no—
195C2×C4—54no—
196C2×C4—54no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—204no—
207C2×C4D4×D514yes—
208C5#267711yescharacteristic, complemented, Sylow
209C22—22no—
210C22—24no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22C2×D4×D512yes—
215C22#414411yescharacteristic
216C22C2×Q8×D511yescharacteristic
217C22#415011yescharacteristic
218C22#415012yes—
219C4—24no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—104no—
224C4—104no—
225C4—104no—
226C4—202no—
227C2—42no—
228C2#6592411yescharacteristic
229C2#6595711yescharacteristic
230C2#6596612yes—
231C2#6600611yescharacteristic
232C2#6600912yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radicalC24×C5C23
hypercenter#393D5
hypocenterC5#2677
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#4090C2semidirect
7. #64946C2semidirect
8. #64945C2semidirect
9. #64941C2semidirect
53. C22⋊Dic5Q8semidirect
208. C5#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#40902
C2×#40242
C5×#3932
#652382
#652352
#652342
#649462
#649452
#649412
#26775