Finite Group Databasebeta

← #10614599

Subgroups — #10614599

Order 640.

498 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 137 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×#4070C211yesmaximal, characteristic, complemented
3#65511C212yesmaximal
4C4×C22⋊Dic5C211yesmaximal, characteristic, complemented
5#65275C211yesmaximal, characteristic, complemented
6#65268C211yesmaximal, characteristic, complemented
7#65261C212yesmaximal, complemented
8#65241C212yesmaximal, complemented
9#65238C211yesmaximal, characteristic, complemented
10#65236C211yesmaximal, characteristic, complemented
11#65234C212yesmaximal, complemented
12C22×C5×D4C2211yescharacteristic
13C23×C4×C5C2211yescharacteristic
14C22×Dic10C2211yescharacteristic
15C5×#123—24no—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24C2×C22⋊Dic5C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26#4070—24no—
27C2×#683C2212yes—
28C2×#683C2212yes—
29C2×#683C2212yes—
30C2×C4×Dic5C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2663—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C24×C5C2311yescharacteristic
40C2×C5×D4—22no—
41C2×C5×D4—42no—
42C22×C4×C5—42no—
43C22×C4×C5D414yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2311yescharacteristic
48C22×C4×C5C2311yescharacteristic
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2312yes—
52C22×Dic5C2312yes—
53C2×Dic10—22no—
54C2×Dic10—42no—
55C5×C4⋊C4—22no—
56C5×C4⋊C4—22no—
57C5×C22⋊C4—22no—
58C5×C22⋊C4—22no—
59C5×C22⋊C4—22no—
60C22⋊Dic5—24no—
61C22⋊Dic5—24no—
62C22⋊Dic5—24no—
63C22⋊Dic5—42no—
64C4⋊Dic5—22no—
65#683—24no—
66#683—24no—
67#683—42no—
68C4×Dic5—42no—
69C2×#123D511yescharacteristic
70C2×C22⋊Q8—51no—
71#400—51no—
72#399—51no—
73#397—52no—
74#395—51no—
75#394—52no—
76#393—51no—
77#389—52no—
78#388—52no—
79C4×C22⋊C4—51no—
80C23×C5—22no—
81C23×C5—22no—
82C23×C5—21no—
83C23×C5—21no—
84C23×C5—21no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—21no—
88C23×C5—21no—
89C23×C5C2×D412yes—
90C23×C5C2411yescharacteristic
91C5×D4—44no—
92C2×C4×C5—22no—
93C2×C4×C5—22no—
94C2×C4×C5—21no—
95C2×C4×C5—21no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5—22no—
99C2×C4×C5—21no—
100C2×C4×C5—21no—
101C2×C4×C5—44no—
102C2×C4×C5C2×D412yes—
103C2×C4×C5C2×D412yes—
104C2×Dic5—22no—
105C2×Dic5—22no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5#4114yes—
116Dic10—44no—
117C22×Q8—51no—
118C22×D4C2×D511yescharacteristic
119C23×C4C2×D511yescharacteristic
120#123—24no—
121C22⋊Q8—104no—
122C2×C4⋊C4—52no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—52no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4C2×D511yescharacteristic
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4C2×D511yescharacteristic
133C2×C22⋊C4C2×D511yescharacteristic
134C2×C22⋊C4C2×D511yescharacteristic
135C2×C42—52no—
136#95—51no—
137#95—51no—
138#95—52no—
139#95—52no—
140#95—52no—
141#95—52no—
142C22×C5—22no—
143C22×C5—22no—
144C22×C5—22no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5—41no—
148C22×C5—41no—
149C22×C5—41no—
150C22×C5—42no—
151C22×C5—42no—
152C22×C5C22×D411yescharacteristic
153C22×C5C2×#4112yes—
154C22×C5H(5,𝔽2)12yes—
155C22×C52−1+411yescharacteristic
156C22×C52−1+411yescharacteristic
157C4×C5—22no—
158C4×C5—22no—
159C4×C5—41no—
160C4×C5—41no—
161C4×C5—41no—
162Dic5—24no—
163Dic5—42no—
164Dic5—42no—
165Dic5—42no—
166C24C22×D511yescharacteristic
167C24C22×D511yescharacteristic
168C2×Q8—102no—
169C2×Q8—202no—
170C2×D4—22no—
171C2×D4—42no—
172C22×C4—42no—
173C22×C4—52no—
174C22×C4—52no—
175C22×C4—52no—
176C22×C4—52no—
177C22×C4C22⋊D514yes—
178C22×C4C22×D511yescharacteristic
179C22×C4C22×D511yescharacteristic
180C22×C4C22×D511yescharacteristic
181C22×C4C22×D511yescharacteristic
182C22×C4C22×D511yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—104no—
186C4⋊C4—104no—
187C4⋊C4—102no—
188C4⋊C4—202no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—104no—
193C22⋊C4—104no—
194C22⋊C4—104no—
195C22⋊C4—202no—
196C42—202no—
197C2×C5—42no—
198C2×C5—81no—
199C2×C5#53211yescharacteristic
200C2×C5#53411yescharacteristic
201C2×C5#53611yescharacteristic
202C2×C5#53912yes—
203C2×C5#54212yes—
204C23—22no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23C2×C22⋊D512yes—
214C23C23×D511yescharacteristic
215Q8—204no—
216D4—44no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—44no—
227C2×C4—54no—
228C2×C4—102no—
229C2×C4—102no—
230C2×C4—102no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—102no—
239C2×C4C2×C22⋊D512yes—
240C2×C4C2×C22⋊D512yes—
241C5#266311yescharacteristic, complemented, Sylow
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—42no—
252C22#414412yes—
253C22D5×#4112yes—
254C22#415011yescharacteristic
255C22#415011yescharacteristic
256C22C22×C22⋊D511yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—104no—
263C4—202no—
264C4—202no—
265C4—202no—
266C2—42no—
267C2—81no—
268C2#6599412yes—
269C2#6599612yes—
270C2#6613811yescharacteristic
271C2#6615911yescharacteristic
272C2#6616611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#123C2
prime radicalC22×C5×D4C22
hypercenterC2×#123D5
hypocenterC5#2663
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#4070C2semidirect
4. C4×C22⋊Dic5C2semidirect
5. #65275C2semidirect
6. #65268C2semidirect
7. #65261C2semidirect
8. #65241C2semidirect
9. #65238C2semidirect
10. #65236C2semidirect
11. #65234C2semidirect
241. C5#2663semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1232
C2×#40702
#655112
C4×C22⋊Dic52
#652752
#652682
#652612
#652412
#652382
#652362
#652342
#26635