Finite Group Databasebeta

← #10608708

Subgroups — #10608708

Order 640.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#4082C211yesmaximal, characteristic
2C2×#4033C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic
4#65511C211yesmaximal, characteristic
5#65268C211yesmaximal, characteristic, complemented
6#65240C212yesmaximal, complemented
7#65238C212yesmaximal, complemented
8#65236C212yesmaximal, complemented
9#65234C212yesmaximal, complemented
10#64945C211yesmaximal, characteristic, complemented
11#64940C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×Dic10C2211yescharacteristic
14C22×Dic10C2211yescharacteristic
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2212yes—
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22#4082—24no—
23C2×C4⋊Dic5C2212yes—
24C2×#683C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28C2×C4×Dic5C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30#4033—24no—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2632—51nomaximal, Sylow
38C24×C5C2312yes—
39C2×C5×D4—22no—
40C2×C5×D4—42no—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2311yescharacteristic
51C2×Dic10—22no—
52C2×Dic10—22no—
53C2×Dic10—22no—
54C2×Dic10—22no—
55C2×Dic10—42no—
56C5×C4⋊C4—22no—
57C5×C22⋊C4—24no—
58C5×C22⋊C4—42no—
59C22⋊Dic5—24no—
60C22⋊Dic5—24no—
61C22⋊Dic5—24no—
62C22⋊Dic5—42no—
63C4⋊Dic5—42no—
64#683—22no—
65#683—22no—
66#683—42no—
67C4×Dic5—22no—
68C4×Dic5—22no—
69C4×Dic5D414yes—
70C2×#124—51no—
71C2×C4×Q8—51no—
72#399—52no—
73#395—51no—
74#395—51no—
75#394—51no—
76#394D511yescharacteristic
77#393—52no—
78#388—52no—
79#388—52no—
80#383—51no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5C2411yescharacteristic
89C5×D4—44no—
90C2×C4×C5—21no—
91C2×C4×C5—21no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5—22no—
95C2×C4×C5—22no—
96C2×C4×C5—22no—
97C2×C4×C5—22no—
98C2×C4×C5C2×D412yes—
99C2×C4×C5#4112yes—
100C2×Dic5—21no—
101C2×Dic5—21no—
102C2×Dic5—21no—
103C2×Dic5—21no—
104C2×Dic5—22no—
105C2×Dic5—21no—
106C2×Dic5—22no—
107C2×Dic5—22no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5C2×D412yes—
114C2×Dic5C2×D412yes—
115C2×Dic5#4112yes—
116Dic10—28no—
117Dic10—44no—
118C22×Q8—51no—
119C22×Q8—51no—
120C22×D4C2×D511yescharacteristic
121#124—104no—
122C4×Q8—104no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—52no—
126C2×C4⋊C4—52no—
127C2×C4⋊C4C2×D511yescharacteristic
128C2×C22⋊C4—52no—
129C2×C22⋊C4—52no—
130C2×C22⋊C4—52no—
131C2×C22⋊C4—52no—
132C2×C22⋊C4C2×D512yes—
133C2×C22⋊C4C2×D512yes—
134C2×C42—51no—
135C2×C42—51no—
136C2×C42—51no—
137#95—52no—
138#95—52no—
139#95—51no—
140#95—52no—
141#95—52no—
142#95C2×D511yescharacteristic
143C22×C5—42no—
144C22×C5—42no—
145C22×C5—42no—
146C22×C5—42no—
147C22×C5—42no—
148C22×C5—42no—
149C22×C5—42no—
150C22×C5C22×D411yescharacteristic
151C22×C5C2×#4111yescharacteristic
152C22×C5C2×#4111yescharacteristic
153C22×C5H(5,𝔽2)12yes—
154C22×C52−1+412yes—
155C4×C5—22no—
156C4×C5—22no—
157C4×C5—41no—
158C4×C5—42no—
159Dic5—22no—
160Dic5—22no—
161Dic5—22no—
162Dic5—42no—
163Dic5—41no—
164Dic5—42no—
165C24C22×D512yes—
166C2×Q8—102no—
167C2×Q8—102no—
168C2×Q8—102no—
169C2×Q8—102no—
170C2×Q8—202no—
171C2×D4—22no—
172C2×D4—42no—
173C22×C4—51no—
174C22×C4—51no—
175C22×C4—51no—
176C22×C4—52no—
177C22×C4—52no—
178C22×C4—51no—
179C22×C4C22×D512yes—
180C22×C4C22×D511yescharacteristic
181C22×C4C22×D511yescharacteristic
182C22×C4C22×D511yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—102no—
185C4⋊C4—102no—
186C4⋊C4—202no—
187C4⋊C4—202no—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—104no—
191C22⋊C4—104no—
192C22⋊C4—104no—
193C22⋊C4—202no—
194C42—54no—
195C42—102no—
196C42—102no—
197C2×C5—82no—
198C2×C5#52511yescharacteristic
199C2×C5#53511yescharacteristic
200C2×C5#53611yescharacteristic
201C2×C5#54212yes—
202C2×C5#54212yes—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23C23×D511yescharacteristic
211Q8—108no—
212Q8—204no—
213D4—44no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—52no—
223C2×C4—52no—
224C2×C4—52no—
225C2×C4—101no—
226C2×C4—101no—
227C2×C4—101no—
228C2×C4—101no—
229C2×C4—102no—
230C2×C4—101no—
231C2×C4—102no—
232C2×C4—102no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4D4×D512yes—
239C2×C4#71112yes—
240C5#263211yescharacteristic, complemented, Sylow
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D511yescharacteristic
249C22C2×#71111yescharacteristic
250C22#414412yes—
251C22D5×#4111yescharacteristic
252C22#415012yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—102no—
258C4—102no—
259C4—102no—
260C4—202no—
261C4—201no—
262C4—202no—
263C2—82no—
264C2#6592611yescharacteristic
265C2#6593712yes—
266C2#6594011yescharacteristic
267C2#6599612yes—
268C2#6600911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC22×C5×D4C22
hypercenter#394D5
hypocenterC5#2632
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#4033C2semidirect
5. #65268C2semidirect
6. #65240C2semidirect
7. #65238C2semidirect
8. #65236C2semidirect
9. #65234C2semidirect
10. #64945C2semidirect
11. #64940C2semidirect
240. C5#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#40822
C2×#40332
C5×#3942
#655112
#652682
#652402
#652382
#652362
#652342
#649452
#649402
#26325