Finite Group Databasebeta

← #10609516

Subgroups — #10609516

Order 640.

540 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 171 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×#4034C212yesmaximal
2C2×#4034C211yesmaximal, characteristic
3C2×#4033C211yesmaximal, characteristic
4C5×#398C211yesmaximal, characteristic
5#65272C211yesmaximal, characteristic
6#65269C211yesmaximal, characteristic
7#65261C212yesmaximal
8#64950C211yesmaximal, characteristic
9#64938C211yesmaximal, characteristic
10#64936C212yesmaximal
11#64935C212yesmaximal
12C22×Dic10C2212yes—
13C22×Dic10C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C5×C4⋊C4C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C4⋊Dic5C2212yes—
18C2×C4⋊Dic5C2211yescharacteristic
19C2×#683C2211yescharacteristic
20C2×#683C2211yescharacteristic
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×#683C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26C2×C4×Dic5C2211yescharacteristic
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4034—28no—
30#4034—24no—
31#4033—24no—
32C5×#95C2212yes—
33C5×#95C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#3963C2211yescharacteristic
38#2546—51nomaximal, Sylow
39C22×C4×C5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×Dic5C2312yes—
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C2×Dic10—22no—
51C2×Dic10—22no—
52C2×Dic10—24no—
53C2×Dic10—24no—
54C2×Dic10—24no—
55C2×Dic10D414yes—
56C5×C4⋊C4—22no—
57C5×C4⋊C4—24no—
58C5×C4⋊C4—22no—
59C4⋊Dic5—22no—
60C4⋊Dic5—24no—
61#683—22no—
62#683—24no—
63#683—22no—
64#683—24no—
65#683—24no—
66#683Q818yescomplemented
67C4×Dic5—22no—
68C4×Dic5—22no—
69C4×Dic5—24no—
70C4×Dic5D414yes—
71C2×C4⋊Q8—51no—
72C2×C4⋊Q8—52no—
73C2×C4×Q8—51no—
74#398—51no—
75#398—51no—
76#398—51no—
77#398D511yescharacteristic
78#391—52no—
79#389—52no—
80#389—52no—
81C4×C4⋊C4—51no—
82C23×C5C2411yescharacteristic
83C2×C4×C5—22no—
84C2×C4×C5—21no—
85C2×C4×C5—21no—
86C2×C4×C5—22no—
87C2×C4×C5—22no—
88C2×C4×C5—21no—
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×Q814yes—
96C2×Dic5—22no—
97C2×Dic5—21no—
98C2×Dic5—21no—
99C2×Dic5—22no—
100C2×Dic5—21no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—22no—
104C2×Dic5—22no—
105C2×Dic5C2×D412yes—
106C2×Dic5C2×D412yes—
107C2×Dic5C2×D412yes—
108C2×Dic5C2×D412yes—
109C2×Dic5C2×Q814yes—
110C2×Dic5C2×Q814yes—
111C2×Dic5#4112yes—
112Dic10—28no—
113Dic10—48no—
114C22×Q8—51no—
115C22×Q8—52no—
116C4⋊Q8—104no—
117C4⋊Q8—108no—
118C4×Q8—104no—
119C2×C4⋊C4—52no—
120C2×C4⋊C4—51no—
121C2×C4⋊C4—51no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4—52no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—51no—
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C4⋊C4C2×D512yes—
129C2×C4⋊C4C2×D511yescharacteristic
130C2×C42—51no—
131C2×C42—51no—
132C2×C42—51no—
133C2×C42—52no—
134#95—51no—
135#95—52no—
136#95—52no—
137#95—52no—
138#95C2×D511yescharacteristic
139#95C2×D512yes—
140C22×C5C22×D411yescharacteristic
141C22×C5C22×D411yescharacteristic
142C22×C5C22×Q812yes—
143C22×C5C2×#4111yescharacteristic
144C22×C52−1+412yes—
145C4×C5—22no—
146C4×C5—22no—
147C4×C5—24no—
148C4×C5—42no—
149C4×C5—41no—
150Dic5—22no—
151Dic5—24no—
152Dic5—42no—
153Dic5—41no—
154Dic5#12114yes—
155Dic5C4⋊Q814yescomplemented
156C2×Q8—54no—
157C2×Q8—102no—
158C2×Q8—102no—
159C2×Q8—104no—
160C2×Q8—104no—
161C2×Q8—104no—
162C22×C4—51no—
163C22×C4—51no—
164C22×C4—52no—
165C22×C4—51no—
166C22×C4—51no—
167C22×C4—52no—
168C22×C4C22×D511yescharacteristic
169C22×C4C22×D512yes—
170C22×C4C22×D511yescharacteristic
171C22×C4C22×D511yescharacteristic
172C22×C4C22×D512yes—
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—58no—
177C4⋊C4—102no—
178C4⋊C4—104no—
179C4⋊C4—104no—
180C4⋊C4—104no—
181C4⋊C4—102no—
182C4⋊C4—104no—
183C4⋊C4—102no—
184C42—54no—
185C42—102no—
186C42—102no—
187C42—104no—
188C2×C5C2×#12111yescharacteristic
189C2×C5C2×C4⋊Q811yescharacteristic
190C2×C5#53611yescharacteristic
191C2×C5D4×Q812yes—
192C2×C5#55712yes—
193C23C23×D511yescharacteristic
194Q8—108no—
195Q8—208no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—52no—
207C2×C4—52no—
208C2×C4—52no—
209C2×C4—54no—
210C2×C4—52no—
211C2×C4—52no—
212C2×C4—54no—
213C2×C4—102no—
214C2×C4—101no—
215C2×C4—101no—
216C2×C4—102no—
217C2×C4—101no—
218C2×C4—102no—
219C2×C4—102no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4D4×D512yes—
223C2×C4D4×D512yes—
224C2×C4Q8×D514yes—
225C5#254611yescharacteristic, complemented, Sylow
226C22C2×D4×D511yescharacteristic
227C22C2×D4×D511yescharacteristic
228C22C2×Q8×D512yes—
229C22D5×#4111yescharacteristic
230C22#415012yes—
231C4—22no—
232C4—24no—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—54no—
237C4—54no—
238C4—102no—
239C4—104no—
240C4—202no—
241C4—201no—
242C2D5×#12111yescharacteristic
243C2#6596612yes—
244C2#6598211yescharacteristic
245C2#6601912yes—
246C2D5×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, prime radical, cosocleC23×C5C24
FittingC5×#398C2
hypercenter#398D5
hypocenterC5#2546
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
66. #683Q8semidirect
155. Dic5C4⋊Q8semidirect
225. C5#2546semidirect

Maximal subgroups

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

subgroupindex
C2×#40342
C2×#40332
C5×#3982
#652722
#652692
#652612
#649502
#649382
#649362
#649352
#25465