Finite Group Databasebeta

← #10606351

Subgroups — #10606351

Order 640.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 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×#683C211yesmaximal, characteristic, complemented
2C2×#4032C211yesmaximal, characteristic
3C5×#385C211yesmaximal, characteristic, complemented
4#65502C211yesmaximal, characteristic
5#65281C212yesmaximal, complemented
6#65273C212yesmaximal, complemented
7#65247C211yesmaximal, characteristic
8#65235C211yesmaximal, characteristic, complemented
9#65232C211yesmaximal, characteristic, complemented
10#64964C212yesmaximal, complemented
11#64949C212yesmaximal, 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×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×C4⋊Dic5C2211yescharacteristic
26C2×#683—24no—
27C2×#683C2212yes—
28C2×#683C2212yes—
29C2×#683C2212yes—
30#4032—24no—
31C5×#95C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#2623—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C22×C4×C5—22no—
41C22×C4×C5—42no—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2312yes—
46C2×C22⋊D5—22no—
47C2×C22⋊D5—22no—
48C2×C22⋊D5—22no—
49C22×Dic5—22no—
50C22×Dic5—22no—
51C22×Dic5—22no—
52C22×Dic5—42no—
53C22×Dic5D414yes—
54C22×Dic5C2312yes—
55C22×Dic5C2312yes—
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2311yescharacteristic
58C22×Dic5C2311yescharacteristic
59C5×C4⋊C4—42no—
60C5×C22⋊C4—22no—
61C5×C22⋊C4—22no—
62C22⋊Dic5—22no—
63#685—22no—
64#685—22no—
65#685—42no—
66#685—42no—
67C4⋊Dic5—22no—
68C4⋊Dic5—22no—
69#683—24no—
70#683—24no—
71#683—42no—
72C2×#123—51no—
73C22×C4⋊C4—51no—
74#400—52no—
75#399—52no—
76#398—52no—
77#397—52no—
78#386—51no—
79#386—51no—
80#385—51no—
81#385D511yescharacteristic, complemented
82#381—51no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5C2×D412yes—
86C23×C5#4112yes—
87C23×C5#4112yes—
88C23×C5C2411yescharacteristic
89C22×D5—21no—
90C22×D5—21no—
91C22×D5—21no—
92C22×D5—22no—
93C22×D5—22no—
94C2×C4×C5—22no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—22no—
100C2×C4×C5—22no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—44no—
105C22⋊D5—44no—
106C2×Dic5—28no—
107C2×Dic5—21no—
108C2×Dic5—21no—
109C2×Dic5—21no—
110C2×Dic5—21no—
111C2×Dic5—22no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—21no—
115C2×Dic5—21no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—44no—
121C2×Dic5C2×D412yes—
122C2×Dic5C2×D412yes—
123C22×D4—51no—
124C23×C4—51no—
125C23×C4—51no—
126C23×C4C2×D511yescharacteristic
127#123—104no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4—52no—
130C2×C4⋊C4—51no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—52no—
133C2×C4⋊C4—104no—
134C2×C4⋊C4C2×D512yes—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—51no—
137C2×C22⋊C4—51no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4C2×D511yescharacteristic
141C2×C22⋊C4C2×D511yescharacteristic
142#95—52no—
143#95—52no—
144#95—52no—
145#95—52no—
146#95—52no—
147#95C2×D512yes—
148C22×C5—22no—
149C22×C5—22no—
150C22×C5—24no—
151C22×C5—24no—
152C22×C5#12314yes—
153C22×C5C22×D411yescharacteristic
154C22×C5C2×#4111yescharacteristic
155C22×C5C2×#4111yescharacteristic
156C22×C5H(5,𝔽2)12yes—
157C22×C52−1+412yes—
158C2×D5—41no—
159C2×D5—41no—
160C2×D5—41no—
161C2×D5—42no—
162C2×D5—42no—
163C4×C5—41no—
164C4×C5—42no—
165C4×C5—41no—
166C4×C5—42no—
167Dic5—22no—
168Dic5—22no—
169Dic5—41no—
170Dic5—42no—
171Dic5—42no—
172C24—51no—
173C24C22×D511yescharacteristic
174C2×D4—102no—
175C2×D4—102no—
176C2×D4—102no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—54no—
180C22×C4—51no—
181C22×C4—51no—
182C22×C4—52no—
183C22×C4—51no—
184C22×C4—52no—
185C22×C4—102no—
186C22×C4—102no—
187C22×C4—102no—
188C22×C4—202no—
189C22×C4C22×D512yes—
190C22×C4C22×D511yescharacteristic
191C22×C4C22×D512yes—
192C22×C4C22×D511yescharacteristic
193C4⋊C4—42no—
194C4⋊C4—104no—
195C4⋊C4—104no—
196C4⋊C4—102no—
197C4⋊C4—102no—
198C4⋊C4—202no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—102no—
202C22⋊C4—102no—
203C22⋊C4—102no—
204C22⋊C4—202no—
205C22⋊C4—202no—
206C2×C5—24no—
207C2×C5C2×#12311yescharacteristic
208C2×C5#53511yescharacteristic
209C2×C5#53611yescharacteristic
210C2×C5#53912yes—
211C2×C5#53912yes—
212D5—81no—
213C23—22no—
214C23—22no—
215C23—101no—
216C23—101no—
217C23—102no—
218C23—101no—
219C23—102no—
220C23#70912yes—
221C23D4×D512yes—
222C23#71312yes—
223C23C23×D511yescharacteristic
224D4—204no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—52no—
237C2×C4—52no—
238C2×C4—101no—
239C2×C4—101no—
240C2×C4—101no—
241C2×C4—101no—
242C2×C4—108no—
243C2×C4—101no—
244C2×C4—102no—
245C2×C4—101no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—102no—
249C2×C4—102no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—204no—
253C5#262311yescharacteristic, complemented, Sylow
254C22—24no—
255C22—22no—
256C22—24no—
257C22—22no—
258C22—201no—
259C22—202no—
260C22—201no—
261C22—202no—
262C22—201no—
263C22#404014yes—
264C22C2×#70911yescharacteristic
265C22C2×D4×D511yescharacteristic
266C22#414412yes—
267C22C2×#71311yescharacteristic
268C22#415012yes—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—102no—
274C4—102no—
275C4—201no—
276C4—202no—
277C4—202no—
278C2—24no—
279C2—401no—
280C2C2×#404011yescharacteristic
281C2#6589812yes—
282C2#6592611yescharacteristic
283C2#6597912yes—
284C2#6598211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC22×C22⋊D5C22
hypercenter#385D5
hypocenterC5#2623
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×#683C2semidirect
3. C5×#385C2semidirect
5. #65281C2semidirect
6. #65273C2semidirect
8. #65235C2semidirect
9. #65232C2semidirect
10. #64964C2semidirect
11. #64949C2semidirect
81. #385D5semidirect
253. C5#2623semidirect

Maximal subgroups

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

subgroupindex
C22×#6832
C2×#40322
C5×#3852
#655022
#652812
#652732
#652472
#652352
#652322
#649642
#649492
#26235