Finite Group Databasebeta

← #10614573

Subgroups — #10614573

Order 640.

610 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 161 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×#123C211yesmaximal, characteristic
2C2×#4081C211yesmaximal, characteristic
3C22×#683C211yesmaximal, characteristic, complemented
4C2×#4023C211yesmaximal, characteristic, complemented
5#65509C211yesmaximal, characteristic
6#65500C211yesmaximal, characteristic, complemented
7#65269C212yesmaximal, complemented
8#65262C212yesmaximal, complemented
9#65239C211yesmaximal, characteristic, complemented
10#65237C212yesmaximal, complemented
11#65233C212yesmaximal, complemented
12C22×C5×D4C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C23×Dic5C2211yescharacteristic
15C23×Dic5C2211yescharacteristic
16C5×#123—24no—
17C2×C5×C4⋊C4C2212yes—
18C2×C5×C22⋊C4C2212yes—
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic5C2211yescharacteristic
21C2×C22⋊Dic5C2211yescharacteristic
22C2×C22⋊Dic5C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2212yes—
25#4081—24no—
26C2×C4⋊Dic5C2212yes—
27C2×#683—24no—
28C2×#683C2212yes—
29C2×#683C2212yes—
30C2×#683C2212yes—
31C2×#683C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×C4×Dic5C2212yes—
34#4023—24no—
35#3963C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#3963C2212yes—
39#2413—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C24×C5C2311yescharacteristic
42C2×C5×D4—22no—
43C2×C5×D4—22no—
44C2×C5×D4—22no—
45C22×C4×C5—22no—
46C22×C4×C5—22no—
47C22×C4×C5—22no—
48C22×C4×C5C2312yes—
49C22×C4×C5C2311yescharacteristic
50C22×C4×C5C2312yes—
51C22×Dic5—22no—
52C22×Dic5—22no—
53C22×Dic5—22no—
54C22×Dic5—24no—
55C22×Dic5D414yes—
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2312yes—
58C22×Dic5C2311yescharacteristic
59C22×Dic5C2311yescharacteristic
60C22×Dic5C2311yescharacteristic
61C22×Dic5C2312yes—
62C5×C4⋊C4—24no—
63C5×C22⋊C4—24no—
64C5×C22⋊C4D414yes—
65C22⋊Dic5—22no—
66C22⋊Dic5—22no—
67C22⋊Dic5—22no—
68C22⋊Dic5—22no—
69C22⋊Dic5—24no—
70C4⋊Dic5—24no—
71#683—24no—
72#683—24no—
73#683—24no—
74#683—22no—
75#683—22no—
76C4×Dic5—24no—
77C2×#123—51no—
78C2×#123D511yescharacteristic
79C2×#117—51no—
80C22×C4⋊C4—51no—
81#398—52no—
82#396—52no—
83#389—52no—
84#388—52no—
85#386—51no—
86#385—51no—
87#381—51no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—21no—
91C23×C5—21no—
92C23×C5—22no—
93C23×C5—21no—
94C23×C5—22no—
95C23×C5C2×D412yes—
96C23×C5C2×D412yes—
97C23×C5#4112yes—
98C23×C5C2411yescharacteristic
99C5×D4—44no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—21no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—44no—
109C2×C4×C5C2×D414yes—
110C2×Dic5—28no—
111C2×Dic5—28no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—21no—
115C2×Dic5—21no—
116C2×Dic5—21no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—21no—
120C2×Dic5—21no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—22no—
124C2×Dic5—22no—
125C2×Dic5C2×D412yes—
126C2×Dic5C2×D412yes—
127C2×Dic5#4114yes—
128C22×D4C2×D511yescharacteristic
129C23×C4—51no—
130C23×C4—51no—
131C23×C4C2×D511yescharacteristic
132#123—24no—
133#123—104no—
134#117—104no—
135C2×C4⋊C4—52no—
136C2×C4⋊C4—52no—
137C2×C4⋊C4—52no—
138C2×C4⋊C4—51no—
139C2×C4⋊C4—51no—
140C2×C4⋊C4—52no—
141C2×C4⋊C4—104no—
142C2×C4⋊C4C2×D512yes—
143C2×C22⋊C4—51no—
144C2×C22⋊C4—51no—
145C2×C22⋊C4—51no—
146C2×C22⋊C4—51no—
147C2×C22⋊C4—52no—
148C2×C22⋊C4C2×D512yes—
149C2×C22⋊C4C2×D511yescharacteristic
150C2×C42—52no—
151#95—52no—
152#95—52no—
153#95—52no—
154#95—52no—
155C22×C5—22no—
156C22×C5—22no—
157C22×C5—24no—
158C22×C5—24no—
159C22×C5—41no—
160C22×C5—41no—
161C22×C5—41no—
162C22×C5—42no—
163C22×C5—42no—
164C22×C5#12114yes—
165C22×C5C22×D411yescharacteristic
166C22×C5C22×D411yescharacteristic
167C22×C5C2×#4111yescharacteristic
168C22×C5C2×#4112yes—
169C22×C52−1+412yes—
170C4×C5—24no—
171C4×C5—42no—
172C4×C5—41no—
173Dic5—22no—
174Dic5—22no—
175Dic5—24no—
176Dic5—41no—
177Dic5—41no—
178Dic5—42no—
179C24C22×D511yescharacteristic
180C24C22×D511yescharacteristic
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—54no—
188C22×C4—52no—
189C22×C4—51no—
190C22×C4—51no—
191C22×C4—51no—
192C22×C4—52no—
193C22×C4—51no—
194C22×C4—104no—
195C22×C4—102no—
196C22×C4—102no—
197C22×C4—102no—
198C22×C4C22×D512yes—
199C22×C4C22×D511yescharacteristic
200C22×C4C22×D512yes—
201C4⋊C4—24no—
202C4⋊C4—104no—
203C4⋊C4—104no—
204C4⋊C4—102no—
205C4⋊C4—102no—
206C4⋊C4—104no—
207C4⋊C4—104no—
208C22⋊C4—24no—
209C22⋊C4—102no—
210C22⋊C4—102no—
211C22⋊C4—104no—
212C22⋊C4—102no—
213C22⋊C4—102no—
214C22⋊C4C22⋊D514yes—
215C42—104no—
216C2×C5—24no—
217C2×C5—81no—
218C2×C5C2×#12111yescharacteristic
219C2×C5#52511yescharacteristic
220C2×C5#53611yescharacteristic
221C2×C5#54712yes—
222C2×C5#55212yes—
223C23—22no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23D4×D512yes—
231C23#71112yes—
232C23C2×C22⋊D512yes—
233C23C23×D511yescharacteristic
234D4—44no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—44no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—54no—
247C2×C4—108no—
248C2×C4—102no—
249C2×C4—101no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—101no—
253C2×C4—101no—
254C2×C4—108no—
255C2×C4—101no—
256C2×C4—101no—
257C2×C4—102no—
258C2×C4—102no—
259C2×C4—102no—
260C2×C4—102no—
261C2×C4—102no—
262C2×C4C2×C22⋊D514yes—
263C5#241311yescharacteristic, complemented, Sylow
264C22—24no—
265C22—22no—
266C22—24no—
267C22—22no—
268C22—41no—
269C22—42no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22#408414yes—
274C22C2×D4×D511yescharacteristic
275C22C2×#71111yescharacteristic
276C22D5×#4112yes—
277C22#415012yes—
278C22C22×C22⋊D511yescharacteristic
279C4—24no—
280C4—42no—
281C4—41no—
282C4—102no—
283C4—102no—
284C4—104no—
285C4—202no—
286C4—201no—
287C4—201no—
288C2—24no—
289C2—81no—
290C2#6598211yescharacteristic
291C2#6598412yes—
292C2#6599011yescharacteristic
293C2C2×#408411yescharacteristic
294C2#6615812yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC22×C5×D4C22
hypercenterC2×#123D5
hypocenterC5#2413
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. C22×#683C2semidirect
4. C2×#4023C2semidirect
6. #65500C2semidirect
7. #65269C2semidirect
8. #65262C2semidirect
9. #65239C2semidirect
10. #65237C2semidirect
11. #65233C2semidirect
263. C5#2413semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×#40812
C22×#6832
C2×#40232
#655092
#655002
#652692
#652622
#652392
#652372
#652332
#24135