Finite Group Databasebeta

← #10606138

Subgroups — #10606138

Order 640.

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

normal subgroups only

This group has 610 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, complemented
2C2×#4032C211yesmaximal, characteristic
3C2×#4032C211yesmaximal, characteristic
4C2×#4023C211yesmaximal, characteristic, complemented
5C5×#385C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic
7#65283C212yesmaximal, complemented
8#65274C212yesmaximal, complemented
9#65232C211yesmaximal, characteristic, complemented
10#64959C212yesmaximal, complemented
11#64950C212yesmaximal, complemented
12C23×C4×C5C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C23×Dic5C2211yescharacteristic
15C23×Dic5C2211yescharacteristic
16C2×C5×C4⋊C4C2212yes—
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C22⋊Dic5C2211yescharacteristic
20C2×#685C2211yescharacteristic
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#685C2211yescharacteristic
24C2×C4⋊Dic5—24no—
25C2×C4⋊Dic5C2212yes—
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2212yes—
28C2×C4⋊Dic5C2211yescharacteristic
29C2×C4⋊Dic5C2212yes—
30C2×#683C2212yes—
31C2×C4×Dic5C2212yes—
32#4032—24no—
33#4032—24no—
34#4023—24no—
35C5×#95C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#3963C2212yes—
39#2413—51nomaximal, Sylow
40C24×C5C2311yescharacteristic
41C23×D5C2311yescharacteristic
42C22×C4×C5—22no—
43C22×C4×C5—24no—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C22×C4×C5C2312yes—
48C2×C22⋊D5—22no—
49C2×C22⋊D5—22no—
50C2×C22⋊D5—22no—
51C22×Dic5—22no—
52C22×Dic5—22no—
53C22×Dic5—22no—
54C22×Dic5—22no—
55C22×Dic5—22no—
56C22×Dic5D414yes—
57C22×Dic5C2312yes—
58C22×Dic5C2312yes—
59C22×Dic5C2311yescharacteristic
60C22×Dic5C2311yescharacteristic
61C22×Dic5C2311yescharacteristic
62C5×C4⋊C4—24no—
63C5×C22⋊C4—22no—
64C5×C22⋊C4D414yes—
65C22⋊Dic5—22no—
66#685—24no—
67#685—24no—
68#685—22no—
69#685—22no—
70C4⋊Dic5—24no—
71C4⋊Dic5—22no—
72C4⋊Dic5—22no—
73C4⋊Dic5—24no—
74C4⋊Dic5—24no—
75#683—24no—
76C4×Dic5—24no—
77C2×#123—51no—
78C2×#123—51no—
79C2×#117—51no—
80C22×C4⋊C4—51no—
81#398—52no—
82#396—52no—
83#389—52no—
84#388—52no—
85#386—51no—
86#385D511yescharacteristic, complemented
87#381—51no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5C2×D412yes—
91C23×C5C2×D412yes—
92C23×C5#4112yes—
93C23×C5C2411yescharacteristic
94C22×D5—21no—
95C22×D5—21no—
96C22×D5—22no—
97C22×D5—21no—
98C22×D5—22no—
99C2×C4×C5—28no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—21no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5C2×D414yes—
109C2×C4×C5#4114yes—
110C22⋊D5—44no—
111C2×Dic5—28no—
112C2×Dic5—21no—
113C2×Dic5—21no—
114C2×Dic5—21no—
115C2×Dic5—21no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—21no—
119C2×Dic5—21no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—21no—
124C2×Dic5—22no—
125C2×Dic5—44no—
126C2×Dic5C2×D412yes—
127C2×Dic5C2×D412yes—
128C22×D4—51no—
129C23×C4—51no—
130C23×C4—51no—
131C23×C4C2×D511yescharacteristic
132#123—104no—
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—52no—
140C2×C4⋊C4—51no—
141C2×C4⋊C4—104no—
142C2×C4⋊C4C2×D512yes—
143C2×C22⋊C4—51no—
144C2×C22⋊C4—51no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4—51no—
147C2×C22⋊C4—52no—
148C2×C22⋊C4C2×D511yescharacteristic
149C2×C22⋊C4C2×D511yescharacteristic
150C2×C42—52no—
151#95—52no—
152#95—52no—
153#95—52no—
154#95C2×D512yes—
155C22×C5—22no—
156C22×C5—22no—
157C22×C5—24no—
158C22×C5—24no—
159C22×C5#12114yes—
160C22×C5C22×D411yescharacteristic
161C22×C5C22×D411yescharacteristic
162C22×C5C2×#4112yes—
163C22×C5C2×#4111yescharacteristic
164C22×C52−1+412yes—
165C2×D5—41no—
166C2×D5—41no—
167C2×D5—41no—
168C2×D5—42no—
169C2×D5—42no—
170C4×C5—24no—
171C4×C5—24no—
172C4×C5—41no—
173C4×C5—41no—
174Dic5—22no—
175Dic5—22no—
176Dic5—42no—
177Dic5—41no—
178Dic5—42no—
179C24—51no—
180C24C22×D511yescharacteristic
181C2×D4—102no—
182C2×D4—102no—
183C2×D4—102no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—54no—
187C22×C4—51no—
188C22×C4—51no—
189C22×C4—52no—
190C22×C4—52no—
191C22×C4—51no—
192C22×C4—102no—
193C22×C4—102no—
194C22×C4—102no—
195C22×C4—102no—
196C22×C4—102no—
197C22×C4C22×D511yescharacteristic
198C22×C4C22×D511yescharacteristic
199C22×C4C22×D512yes—
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—22no—
209C22⋊C4—102no—
210C22⋊C4—102no—
211C22⋊C4—102no—
212C22⋊C4—104no—
213C22⋊C4—104no—
214C22⋊C4D2014yes—
215C42—104no—
216C2×C5—24no—
217C2×C5C2×#12111yescharacteristic
218C2×C5#52511yescharacteristic
219C2×C5#53611yescharacteristic
220C2×C5#54712yes—
221C2×C5#55212yes—
222D5—81no—
223C23—22no—
224C23—22no—
225C23—101no—
226C23—101no—
227C23—102no—
228C23—101no—
229C23—102no—
230C23C2×D2012yes—
231C23D4×D512yes—
232C23#71312yes—
233C23C23×D511yescharacteristic
234D4—204no—
235C2×C4—28no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—101no—
247C2×C4—101no—
248C2×C4—101no—
249C2×C4—101no—
250C2×C4—108no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—101no—
254C2×C4—102no—
255C2×C4—101no—
256C2×C4—102no—
257C2×C4—102no—
258C2×C4—101no—
259C2×C4—102no—
260C2×C4—204no—
261C2×C4C2×D2014yes—
262C2×C4#71114yes—
263C5#241311yescharacteristic, complemented, Sylow
264C22—24no—
265C22—22no—
266C22—24no—
267C22—22no—
268C22—201no—
269C22—202no—
270C22—201no—
271C22—202no—
272C22—201no—
273C22#404114yes—
274C22C22×D2011yescharacteristic
275C22C2×D4×D511yescharacteristic
276C22C2×#71112yes—
277C22C2×#71311yescharacteristic
278C22#415012yes—
279C4—24no—
280C4—24no—
281C4—41no—
282C4—41no—
283C4—102no—
284C4—102no—
285C4—202no—
286C4—201no—
287C4—202no—
288C2—24no—
289C2—401no—
290C2C2×#404111yescharacteristic
291C2#6588912yes—
292C2#6592011yescharacteristic
293C2#6597612yes—
294C2#6598211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC22×C22⋊D5C22
hypercenter#385D5
hypocenterC5#2413
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic5C2semidirect
4. C2×#4023C2semidirect
5. C5×#385C2semidirect
7. #65283C2semidirect
8. #65274C2semidirect
9. #65232C2semidirect
10. #64959C2semidirect
11. #64950C2semidirect
86. #385D5semidirect
263. C5#2413semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic52
C2×#40322
C2×#40232
C5×#3852
#655022
#652832
#652742
#652322
#649592
#649502
#24135