Finite Group Databasebeta

← #10608675

Subgroups — #10608675

Order 640.

788 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 161 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×#4085C212yesmaximal, complemented
2C2×#4030C212yesmaximal, complemented
3C2×#4027C212yesmaximal, complemented
4C5×#394C211yesmaximal, characteristic, complemented
5#65280C211yesmaximal, characteristic, complemented
6#65265C211yesmaximal, characteristic, complemented
7#65250C212yesmaximal, complemented
8#65231C212yesmaximal, complemented
9#64956C212yesmaximal, complemented
10C22×C5×D4C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×C22⋊D5C2212yes—
13C23×Dic5C2212yes—
14C22×D20C2211yescharacteristic
15C22×C4×D5C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2212yes—
20#4085—28no—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×#683C2212yes—
26C2×#683C2212yes—
27C2×#683C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29#4030—28no—
30#4027—28no—
31C5×#95C2211yescharacteristic
32#3963C2211yescharacteristic
33#3963C2212yes—
34#2537—51nomaximal, Sylow
35C24×C5C2312yes—
36C23×D5C2312yes—
37C2×C5×D4—24no—
38C2×C5×D4—41no—
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2311yescharacteristic
43C2×C22⋊D5—24no—
44C2×C22⋊D5—24no—
45C2×C22⋊D5—24no—
46C2×C22⋊D5—24no—
47C2×C22⋊D5—42no—
48C2×C22⋊D5D418yescomplemented
49C22×Dic5—24no—
50C22×Dic5—24no—
51C22×Dic5—24no—
52C22×Dic5C2312yes—
53C22×Dic5C2312yes—
54C22×Dic5C2312yes—
55C2×D20—24no—
56C2×D20—41no—
57C2×C4×D5—24no—
58C2×C4×D5—24no—
59C2×C4×D5—24no—
60C5×C4⋊C4—41no—
61C5×C22⋊C4—24no—
62C5×C22⋊C4—24no—
63C22⋊Dic5—24no—
64#685—24no—
65#685—24no—
66#685—24no—
67C4⋊Dic5—41no—
68#683—24no—
69#683—24no—
70#683—24no—
71C4×Dic5D414yescomplemented
72C2×#121—52no—
73C2×#121—52no—
74C2×C4×D4—52no—
75#394—51no—
76#394—52no—
77#394D511yescharacteristic, complemented
78#384—51no—
79#380—52no—
80#380—52no—
81C23×C5—22no—
82C23×C5—22no—
83C23×C5—22no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5C2×D414yes—
88C23×C5C2411yescharacteristic
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—22no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5C2×D414yes—
96C5×D4—44no—
97C2×C4×C5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—22no—
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×C5C2×D412yes—
107C22⋊D5—216no—
108C22⋊D5—48no—
109C2×Dic5—22no—
110C2×Dic5—22no—
111C2×Dic5—22no—
112C2×Dic5—22no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—48no—
116C2×Dic5C2×D414yes—
117C2×Dic5C2×D414yes—
118C2×Dic5#4114yes—
119D20—44no—
120C4×D5—48no—
121C22×D4—52no—
122C22×D4—52no—
123C22×D4—51no—
124C22×D4C2×D511yescharacteristic
125C23×C4—52no—
126C23×C4—52no—
127#121—108no—
128#121—108no—
129C4×D4—108no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—52no—
132C2×C4⋊C4—52no—
133C2×C4⋊C4—51no—
134C2×C4⋊C4C2×D511yescharacteristic
135C2×C22⋊C4—52no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4C2×D512yes—
140C2×C22⋊C4C2×D512yes—
141C2×C42—51no—
142#95—51no—
143#95—52no—
144#95C2×D511yescharacteristic
145C22×C5—24no—
146C22×C5—24no—
147C22×C5—24no—
148C22×C5—42no—
149C22×C5—42no—
150C22×C5—42no—
151C22×C5—42no—
152C22×C5C22×D411yescharacteristic
153C22×C5C22×D412yes—
154C22×C5C2×#4112yes—
155C22×C5H(5,𝔽2)11yescharacteristic
156C22×C52−1+411yescharacteristic
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—42no—
166C4×C5—41no—
167C4×C5—41no—
168Dic5—42no—
169Dic5—42no—
170Dic5#12118yescomplemented
171C24—52no—
172C24C22×D512yes—
173C2×D4—24no—
174C2×D4—41no—
175C2×D4—58no—
176C2×D4—104no—
177C2×D4—104no—
178C2×D4—104no—
179C2×D4—104no—
180C2×D4—104no—
181C2×D4—201no—
182C2×D4—202no—
183C22×C4—52no—
184C22×C4—52no—
185C22×C4—52no—
186C22×C4—104no—
187C22×C4—104no—
188C22×C4—104no—
189C22×C4—104no—
190C22×C4—104no—
191C22×C4—104no—
192C22×C4C22×D512yes—
193C22×C4C22×D511yescharacteristic
194C22×C4C22×D511yescharacteristic
195C22×C4C22×D511yescharacteristic
196C4⋊C4—41no—
197C4⋊C4—104no—
198C4⋊C4—104no—
199C4⋊C4—104no—
200C4⋊C4—201no—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—104no—
204C22⋊C4—104no—
205C22⋊C4—104no—
206C22⋊C4—104no—
207C42—54no—
208C2×C5—44no—
209C2×C5C2×#12112yes—
210C2×C5#53711yescharacteristic
211C2×C5D4211yescharacteristic
212C2×C5#54712yes—
213C2×C5#55511yescharacteristic
214D5—44no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—54no—
222C23—102no—
223C23—102no—
224C23—102no—
225C23—102no—
226C23—102no—
227C23—102no—
228C23D4×D514yes—
229C23C23×D511yescharacteristic
230D4—44no—
231D4—1016no—
232D4—204no—
233D4—208no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—54no—
244C2×C4—54no—
245C2×C4—54no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—102no—
249C2×C4—102no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4—208no—
253C2×C4—208no—
254C2×C4D4×D512yes—
255C5#253711yescharacteristic, complemented, Sylow
256C22—24no—
257C22—24no—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—104no—
264C22—104no—
265C22—104no—
266C22—202no—
267C22—202no—
268C22—202no—
269C22—202no—
270C22C2×D4×D512yes—
271C22C2×D4×D511yescharacteristic
272C22#414411yescharacteristic
273C22D5×#4112yes—
274C22#415011yescharacteristic
275C4—22no—
276C4—41no—
277C4—41no—
278C4—42no—
279C4—58no—
280C4—202no—
281C4—202no—
282C2—44no—
283C2—204no—
284C2#6592411yescharacteristic
285C2#6593111yescharacteristic
286C2D5×#12112yes—
287C2#6598912yes—
288C2#6600411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
hypercenter#394D5
hypocenterC5#2537
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#4085C2semidirect
2. C2×#4030C2semidirect
3. C2×#4027C2semidirect
4. C5×#394C2semidirect
5. #65280C2semidirect
6. #65265C2semidirect
7. #65250C2semidirect
8. #65231C2semidirect
9. #64956C2semidirect
48. C2×C22⋊D5D4semidirect
71. C4×Dic5D4semidirect
77. #394D5semidirect
170. Dic5#121semidirect
255. C5#2537semidirect

Maximal subgroups

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

subgroupindex
C2×#40852
C2×#40302
C2×#40272
C5×#3942
#652802
#652652
#652502
#652312
#649562
#25375