Finite Group Databasebeta

← #10606320

Subgroups — #10606320

Order 640.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C4×Dic5C211yesmaximal, characteristic
2C2×#4024C211yesmaximal, characteristic
3C5×#385C211yesmaximal, characteristic
4#65499C211yesmaximal, characteristic
5#65270C212yesmaximal, complemented
6#65261C212yesmaximal, complemented
7#65232C211yesmaximal, characteristic
8#65232C211yesmaximal, characteristic
9#65231C211yesmaximal, characteristic
10#64941C212yesmaximal, complemented
11#64935C212yesmaximal, complemented
12C23×C4×C5C2211yescharacteristic
13C23×Dic5C2211yescharacteristic
14C23×Dic5C2211yescharacteristic
15C22×Dic10C2211yescharacteristic
16C2×C5×C4⋊C4C2212yes—
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2211yescharacteristic
21C2×C4⋊Dic5C2211yescharacteristic
22C2×#683C2211yescharacteristic
23C2×#683C2211yescharacteristic
24C2×#683C2212yes—
25C2×#683C2212yes—
26C2×C4×Dic5—24no—
27C2×C4×Dic5C2212yes—
28C2×C4×Dic5C2212yes—
29#4024—24no—
30C5×#95C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2855—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C22×C4×C5—22no—
40C22×C4×C5—24no—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2312yes—
45C22×Dic5—22no—
46C22×Dic5—22no—
47C22×Dic5—24no—
48C22×Dic5D414yes—
49C22×Dic5Q814yescomplemented
50C22×Dic5C2311yescharacteristic
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2312yes—
55C22×Dic5C2311yescharacteristic
56C2×Dic10—22no—
57C2×Dic10—24no—
58C5×C4⋊C4—24no—
59C5×C22⋊C4—22no—
60C5×C22⋊C4—22no—
61C22⋊Dic5—22no—
62C22⋊Dic5—22no—
63C4⋊Dic5—22no—
64#683—24no—
65#683—22no—
66#683—22no—
67#683—24no—
68C4×Dic5—24no—
69C4×Dic5—24no—
70C2×C22⋊Q8—51no—
71C22×C42—51no—
72#391—52no—
73#389—52no—
74#387—52no—
75#387—52no—
76#385—51no—
77#385D511yescharacteristic
78#381—51no—
79#381—51no—
80#380—51no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5C2×D412yes—
84C23×C5C2×Q812yes—
85C23×C5#4112yes—
86C23×C5C2411yescharacteristic
87C2×C4×C5—28no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5#4114yes—
98C2×Dic5—28no—
99C2×Dic5—22no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—28no—
103C2×Dic5—21no—
104C2×Dic5—21no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5C2×D412yes—
111C2×Dic5C2×D412yes—
112C2×Dic5C2×Q814yes—
113C2×Dic5#4114yes—
114Dic10—44no—
115C22×Q8—51no—
116C23×C4—51no—
117C23×C4—51no—
118C23×C4C2×D511yescharacteristic
119C22⋊Q8—104no—
120C2×C4⋊C4—51no—
121C2×C4⋊C4—51no—
122C2×C4⋊C4—51no—
123C2×C4⋊C4—52no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4C2×D512yes—
126C2×C22⋊C4—51no—
127C2×C22⋊C4—51no—
128C2×C22⋊C4C2×D511yescharacteristic
129C2×C22⋊C4C2×D511yescharacteristic
130C2×C42—52no—
131C2×C42—52no—
132C2×C42—104no—
133#95—52no—
134#95—52no—
135#95—52no—
136#95—52no—
137#95—52no—
138#95—52no—
139#95C2×D512yes—
140C22×C5—22no—
141C22×C5—22no—
142C22×C5—24no—
143C22×C5—24no—
144C22×C5C22⋊Q814yes—
145C22×C5C22×D411yescharacteristic
146C22×C5C22×Q811yescharacteristic
147C22×C5C2×#4111yescharacteristic
148C22×C5C2×#4112yes—
149C22×C5C2×#4112yes—
150C4×C5—24no—
151C4×C5—41no—
152C4×C5—41no—
153C4×C5—42no—
154Dic5—24no—
155Dic5—22no—
156Dic5—41no—
157Dic5—41no—
158Dic5—42no—
159Dic5#12314yes—
160C24C22×D511yescharacteristic
161C2×Q8—102no—
162C2×Q8—104no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—51no—
166C22×C4—54no—
167C22×C4—54no—
168C22×C4—52no—
169C22×C4—51no—
170C22×C4—52no—
171C22×C4—51no—
172C22×C4—51no—
173C22×C4—104no—
174C22×C4—102no—
175C22×C4—102no—
176C22×C4C22×D512yes—
177C22×C4C22×D511yescharacteristic
178C22×C4C22×D511yescharacteristic
179C22×C4C22×D512yes—
180C4⋊C4—24no—
181C4⋊C4—102no—
182C4⋊C4—102no—
183C4⋊C4—102no—
184C4⋊C4—104no—
185C4⋊C4—104no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—102no—
189C22⋊C4—102no—
190C42—104no—
191C42—104no—
192C2×C5—24no—
193C2×C5C2×C22⋊Q811yescharacteristic
194C2×C5C2×#12311yescharacteristic
195C2×C5#52511yescharacteristic
196C2×C5#52912yes—
197C2×C5#53312yes—
198C23—22no—
199C23—22no—
200C23#70912yes—
201C23D4×D512yes—
202C23Q8×D512yes—
203C23C23×D511yescharacteristic
204Q8—204no—
205C2×C4—28no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—52no—
216C2×C4—54no—
217C2×C4—52no—
218C2×C4—54no—
219C2×C4—108no—
220C2×C4—102no—
221C2×C4—102no—
222C2×C4—102no—
223C2×C4—108no—
224C2×C4—102no—
225C2×C4—101no—
226C2×C4—101no—
227C2×C4—102no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—102no—
231C2×C4#71114yes—
232C5#285511yescharacteristic, complemented, Sylow
233C22—24no—
234C22—22no—
235C22—24no—
236C22—22no—
237C22#404214yes—
238C22C2×#70911yescharacteristic
239C22C2×D4×D511yescharacteristic
240C22C2×#71112yes—
241C22C2×Q8×D511yescharacteristic
242C22D5×#4112yes—
243C4—24no—
244C4—42no—
245C4—41no—
246C4—41no—
247C4—54no—
248C4—104no—
249C4—102no—
250C4—202no—
251C4—201no—
252C4—201no—
253C2—24no—
254C2C2×#404211yescharacteristic
255C2#6587212yes—
256C2#6592011yescharacteristic
257C2#6595612yes—
258C2D5×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC24×C5C23
hypercenter#385D5
hypocenterC5#2855
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
5. #65270C2semidirect
6. #65261C2semidirect
10. #64941C2semidirect
11. #64935C2semidirect
49. C22×Dic5Q8semidirect
232. C5#2855semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic52
C2×#40242
C5×#3852
#654992
#652702
#652612
#652322
#652312
#649412
#649352
#28555