Finite Group Databasebeta

← #10608917

Subgroups — #10608917

Order 640.

556 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 181 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
2C2×#4034C212yesmaximal
3C5×#395C211yesmaximal, characteristic
4#65268C211yesmaximal, characteristic
5#65265C211yesmaximal, characteristic
6#65261C212yesmaximal
7#64945C212yesmaximal
8#64938C212yesmaximal
9#64936C212yesmaximal
10C22×C5×Q8C2211yescharacteristic
11C22×Dic10C2211yescharacteristic
12C22×Dic10C2212yes—
13C2×C5×C4⋊C4C2212yes—
14C2×C5×C4⋊C4C2211yescharacteristic
15#4090—28no—
16C2×C4⋊Dic5C2212yes—
17C2×C4⋊Dic5C2211yescharacteristic
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×C4×Dic5C2212yes—
25C2×C4×Dic5C2212yes—
26C2×C4×Dic5C2211yescharacteristic
27#4034—28no—
28C5×#95C2211yescharacteristic
29C5×#95C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2211yescharacteristic
33#2551—51nomaximal, Sylow
34C2×C5×Q8—24no—
35C2×C5×Q8—41no—
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2312yes—
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×Dic5C2312yes—
42C22×Dic5C2312yes—
43C22×Dic5C2312yes—
44C22×Dic5C2312yes—
45C2×Dic10—24no—
46C2×Dic10—24no—
47C2×Dic10—24no—
48C2×Dic10—41no—
49C2×Dic10—42no—
50C5×C4⋊C4—24no—
51C5×C4⋊C4—41no—
52C4⋊Dic5—24no—
53C4⋊Dic5—41no—
54#683—24no—
55#683—24no—
56#683—24no—
57#683—24no—
58#683D418yes—
59#683Q818yescomplemented
60C4×Dic5—24no—
61C4×Dic5—24no—
62C4×Dic5Q814yescomplemented
63C2×C4⋊Q8—52no—
64C2×C4⋊Q8—52no—
65#395—51no—
66#395—52no—
67#395D511yescharacteristic
68#389—52no—
69#389—52no—
70#384—51no—
71C4×C4⋊C4—52no—
72C23×C5C2411yescharacteristic
73C5×Q8—44no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—21no—
77C2×C4×C5—22no—
78C2×C4×C5—21no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—21no—
83C2×C4×C5—21no—
84C2×C4×C5C2×D414yes—
85C2×C4×C5C2×Q812yes—
86C2×C4×C5C2×Q814yes—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5C2×D414yes—
95C2×Dic5C2×D414yes—
96C2×Dic5C2×Q814yes—
97C2×Dic5C2×Q814yes—
98C2×Dic5C2×Q814yes—
99Dic10—44no—
100Dic10—48no—
101C22×Q8—51no—
102C22×Q8—52no—
103C22×Q8C2×D511yescharacteristic
104C4⋊Q8—108no—
105C4⋊Q8—108no—
106C2×C4⋊C4—52no—
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×C4⋊C4C2×D512yes—
115C2×C4⋊C4C2×D511yescharacteristic
116C2×C42—51no—
117C2×C42—52no—
118C2×C42—52no—
119#95—52no—
120#95—51no—
121#95—52no—
122#95C2×D512yes—
123#95C2×D511yescharacteristic
124C22×C5C22×D412yes—
125C22×C5C22×Q811yescharacteristic
126C22×C5C22×Q812yes—
127C22×C5H(5,𝔽2)11yescharacteristic
128C22×C52−1+411yescharacteristic
129C4×C5—22no—
130C4×C5—24no—
131C4×C5—24no—
132C4×C5—41no—
133C4×C5—41no—
134Dic5—24no—
135Dic5—24no—
136Dic5—42no—
137Dic5C4⋊Q818yescomplemented
138C2×Q8—24no—
139C2×Q8—41no—
140C2×Q8—104no—
141C2×Q8—104no—
142C2×Q8—104no—
143C2×Q8—202no—
144C2×Q8—201no—
145C22×C4—52no—
146C22×C4—52no—
147C22×C4—52no—
148C22×C4—52no—
149C22×C4C22×D511yescharacteristic
150C22×C4C22×D511yescharacteristic
151C22×C4C22×D512yes—
152C22×C4C22×D511yescharacteristic
153C22×C4C22×D512yes—
154C4⋊C4—24no—
155C4⋊C4—41no—
156C4⋊C4—58no—
157C4⋊C4—58no—
158C4⋊C4—104no—
159C4⋊C4—104no—
160C4⋊C4—104no—
161C4⋊C4—104no—
162C4⋊C4—104no—
163C4⋊C4—201no—
164C42—54no—
165C42—104no—
166C42—104no—
167C2×C5C2×C4⋊Q812yes—
168C2×C5#54411yescharacteristic
169C2×C5D4211yescharacteristic
170C2×C5D4×Q812yes—
171C2×C5Q8211yescharacteristic
172C23C23×D511yescharacteristic
173Q8—44no—
174Q8—208no—
175Q8—204no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—54no—
187C2×C4—54no—
188C2×C4—54no—
189C2×C4—54no—
190C2×C4—54no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4D4×D514yes—
199C2×C4Q8×D512yes—
200C2×C4Q8×D514yes—
201C5#255111yescharacteristic, complemented, Sylow
202C22C2×D4×D512yes—
203C22#414411yescharacteristic
204C22C2×Q8×D512yes—
205C22C2×Q8×D511yescharacteristic
206C22#415011yescharacteristic
207C4—24no—
208C4—22no—
209C4—24no—
210C4—41no—
211C4—41no—
212C4—58no—
213C4—104no—
214C4—104no—
215C4—202no—
216C2#6592411yescharacteristic
217C2#6595711yescharacteristic
218C2#6596612yes—
219C2#6605611yescharacteristic
220C2D5×C4⋊Q812yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, prime radical, cosocleC23×C5C24
FittingC5×#395C2
hypercenter#395D5
hypocenterC5#2551
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
59. #683Q8semidirect
62. C4×Dic5Q8semidirect
137. Dic5C4⋊Q8semidirect
201. C5#2551semidirect

Maximal subgroups

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

subgroupindex
C2×#40902
C2×#40342
C5×#3952
#652682
#652652
#652612
#649452
#649382
#649362
#25515