Finite Group Databasebeta

← #10614818

Subgroups — #10614818

Order 640.

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

normal subgroups only

This group has 582 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
1C2×C5×#126C211yesmaximal, characteristic, complemented
2C2×C4×D20C211yesmaximal, characteristic, complemented
3#65280C211yesmaximal, characteristic, complemented
4#65279C212yesmaximal, complemented
5#65278C212yesmaximal, complemented
6#65272C211yesmaximal, characteristic, complemented
7#65249C211yesmaximal, characteristic
8#65246C212yesmaximal, complemented
9#65245C212yesmaximal, complemented
10#65242C211yesmaximal, characteristic, complemented
11#65221C211yesmaximal, characteristic, complemented
12C22×C22⋊D5C2212yes—
13C22×D20C2211yescharacteristic
14C22×C4×D5C2212yes—
15C5×#126—24no—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C42×C5C2211yescharacteristic
23C2×C22⋊Dic5C2212yes—
24C2×#685C2212yes—
25C2×#685C2212yes—
26C2×#685C2212yes—
27C2×#685C2212yes—
28C2×C4⋊Dic5C2211yescharacteristic
29C2×#683C2212yes—
30C2×C4×Dic5C2212yes—
31C4×D20—24no—
32#3963C2211yescharacteristic
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2603—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2312yes—
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C2×C22⋊D5—24no—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—42no—
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2312yes—
52C2×D20—24no—
53C2×D20—22no—
54C2×C4×D5—24no—
55C2×C4×D5—42no—
56C2×C4×D5—42no—
57C5×C4⋊C4—22no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—22no—
60C5×C22⋊C4—22no—
61C5×C22⋊C4—22no—
62C5×C22⋊C4—22no—
63C42×C5D414yescomplemented
64C22⋊Dic5—42no—
65#685—24no—
66#685—24no—
67#685—24no—
68#685—42no—
69C4⋊Dic5—22no—
70#683—42no—
71C4×Dic5—24no—
72C2×#126D511yescharacteristic, complemented
73C2×C4×D4—51no—
74#398—51no—
75#397—51no—
76#394—51no—
77#392—51no—
78#390—52no—
79#388—52no—
80#386—52no—
81#385—52no—
82#376—51no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—21no—
87C23×C5—21no—
88C23×C5C2411yescharacteristic
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—22no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5#4114yes—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—21no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—21no—
107C2×C4×C5C2×D412yes—
108C2×C4×C5C2×D412yes—
109C2×C4×C5C2×D412yes—
110C22⋊D5—48no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5#4114yes—
120D20—28no—
121C4×D5—48no—
122C22×D4—52no—
123C22×D4—51no—
124C23×C4—52no—
125#126—24no—
126C4×D4—104no—
127C2×C4⋊C4—51no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4C2×D511yescharacteristic
130C2×C4⋊C4C2×D511yescharacteristic
131C2×C4⋊C4C2×D511yescharacteristic
132C2×C22⋊C4—52no—
133C2×C22⋊C4—52no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4—52no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4C2×D511yescharacteristic
138C2×C22⋊C4C2×D511yescharacteristic
139C2×C22⋊C4C2×D511yescharacteristic
140C2×C42—52no—
141C2×C42C2×D511yescharacteristic
142#95—52no—
143#95—51no—
144#95—51no—
145#95—52no—
146#95—52no—
147C22×C5—41no—
148C22×C5—41no—
149C22×C5—41no—
150C22×C5—42no—
151C22×C5—42no—
152C22×C5C22×D411yescharacteristic
153C22×C5C2×#4112yes—
154C22×C5C2×#4112yes—
155C22×C5H(5,𝔽2)11yescharacteristic
156C22×C5H(5,𝔽2)11yescharacteristic
157C2×D5—24no—
158C2×D5—24no—
159C2×D5—24no—
160C2×D5—42no—
161C2×D5—42no—
162C2×D5—42no—
163C2×D5—42no—
164C4×C5—22no—
165C4×C5—22no—
166C4×C5—22no—
167C4×C5—41no—
168C4×C5—41no—
169C4×C5—41no—
170Dic5—24no—
171Dic5—42no—
172Dic5—42no—
173C24—52no—
174C24C22×D511yescharacteristic
175C2×D4—104no—
176C2×D4—104no—
177C2×D4—104no—
178C2×D4—102no—
179C2×D4—202no—
180C22×C4—52no—
181C22×C4—52no—
182C22×C4—52no—
183C22×C4—104no—
184C22×C4—202no—
185C22×C4—202no—
186C22×C4C22×D511yescharacteristic
187C22×C4C22×D511yescharacteristic
188C22×C4C22×D511yescharacteristic
189C22×C4C22×D511yescharacteristic
190C22×C4C22×D511yescharacteristic
191C22×C4C22×D511yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—102no—
196C4⋊C4—202no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—104no—
201C22⋊C4—104no—
202C22⋊C4—104no—
203C22⋊C4—202no—
204C22⋊C4—202no—
205C42—104no—
206C42C22⋊D514yescomplemented
207C2×C5—81no—
208C2×C5#52511yescharacteristic
209C2×C5#53211yescharacteristic
210C2×C5#53511yescharacteristic
211C2×C5#55112yes—
212C2×C5#55312yes—
213D5—44no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—54no—
220C23—102no—
221C23—102no—
222C23—102no—
223C23—102no—
224C23—102no—
225C23—102no—
226C23C23×D511yescharacteristic
227D4—108no—
228D4—208no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—54no—
241C2×C4—102no—
242C2×C4—102no—
243C2×C4—102no—
244C2×C4—102no—
245C2×C4—102no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—102no—
249C2×C4—208no—
250C2×C4C2×C22⋊D512yes—
251C2×C4C2×C22⋊D512yes—
252C2×C4C2×C22⋊D512yes—
253C5#260311yescharacteristic, complemented, Sylow
254C22—41no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—104no—
260C22—104no—
261C22—104no—
262C22—202no—
263C22—202no—
264C22—202no—
265C22—202no—
266C22D5×#4112yes—
267C22D5×#4112yes—
268C22#414911yescharacteristic
269C22#414911yescharacteristic
270C22C22×C22⋊D511yescharacteristic
271C4—22no—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—104no—
278C4—202no—
279C4—202no—
280C2—81no—
281C2—204no—
282C2#6603912yes—
283C2#6604312yes—
284C2#6616211yescharacteristic
285C2#6616411yescharacteristic
286C2#6616611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#126C2
prime radical#65249C2
hypercenterC2×#126D5
hypocenterC5#2603
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C5×#126C2semidirect
2. C2×C4×D20C2semidirect
3. #65280C2semidirect
4. #65279C2semidirect
5. #65278C2semidirect
6. #65272C2semidirect
8. #65246C2semidirect
9. #65245C2semidirect
10. #65242C2semidirect
11. #65221C2semidirect
63. C42×C5D4semidirect
72. C2×#126D5semidirect
206. C42C22⋊D5semidirect
253. C5#2603semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1262
C2×C4×D202
#652802
#652792
#652782
#652722
#652492
#652462
#652452
#652422
#652212
#26035