Finite Group Databasebeta

← #10608852

Subgroups — #10608852

Order 640.

596 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 139 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×#4084C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C5×#394C211yesmaximal, characteristic, complemented
4#65511C211yesmaximal, characteristic, complemented
5#65275C211yesmaximal, characteristic, complemented
6#65245C212yesmaximal, complemented
7#65245C212yesmaximal, complemented
8#65240C212yesmaximal, complemented
9#65233C212yesmaximal, complemented
10#64955C211yesmaximal, characteristic, complemented
11#64948C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C22×C4×D5C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C22×C4×D5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C5×C22⋊C4C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23#4084—24no—
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×#683C2211yescharacteristic
29C2×C4×Dic5C2212yes—
30D5×C4⋊C4—24no—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2591—51nomaximal, Sylow
38C24×C5C2312yes—
39C23×D5C2311yescharacteristic
40C2×C5×D4—22no—
41C2×C5×D4—42no—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C2×C22⋊D5—24no—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—24no—
49C22×Dic5C2311yescharacteristic
50C22×Dic5C2312yes—
51C22×Dic5C2312yes—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C2×C4×D5—22no—
55C2×C4×D5—22no—
56C2×C4×D5—22no—
57C2×C4×D5—22no—
58C2×C4×D5—42no—
59C2×C4×D5—42no—
60C2×C4×D5D414yes—
61C5×C4⋊C4—22no—
62C5×C22⋊C4—24no—
63C5×C22⋊C4—42no—
64C22⋊Dic5—24no—
65C22⋊Dic5—24no—
66C22⋊Dic5—42no—
67#685—24no—
68C4⋊Dic5—22no—
69C4⋊Dic5—22no—
70#683—22no—
71#683—22no—
72C4×Dic5—42no—
73C2×#121—51no—
74C22×C4⋊C4—51no—
75#400—51no—
76#400—51no—
77#394—51no—
78#394D511yescharacteristic, complemented
79#388—52no—
80#388—52no—
81#386—52no—
82#386—52no—
83C4×C22⋊C4—51no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5C2411yescharacteristic
92C22×D5—21no—
93C22×D5—21no—
94C22×D5—22no—
95C22×D5C2×D412yes—
96C22×D5#4114yes—
97C5×D4—44no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—22no—
101C2×C4×C5—21no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5C2×D412yes—
107C2×C4×C5#4112yes—
108C22⋊D5—48no—
109C2×Dic5—21no—
110C2×Dic5—21no—
111C2×Dic5—22no—
112C2×Dic5—21no—
113C2×Dic5—22no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5C2×D412yes—
121C2×Dic5#4112yes—
122C4×D5—28no—
123C4×D5—44no—
124C4×D5—44no—
125C22×D4—52no—
126C22×D4C2×D511yescharacteristic
127C23×C4—51no—
128C23×C4—51no—
129C23×C4—51no—
130#121—104no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—51no—
133C2×C4⋊C4—51no—
134C2×C4⋊C4—51no—
135C2×C4⋊C4—104no—
136C2×C4⋊C4C2×D511yescharacteristic
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4C2×D512yes—
142C2×C22⋊C4C2×D512yes—
143C2×C42—52no—
144#95—51no—
145#95—52no—
146#95—52no—
147#95—52no—
148#95—52no—
149#95C2×D511yescharacteristic
150C22×C5—42no—
151C22×C5—42no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5—42no—
155C22×C5—42no—
156C22×C5—42no—
157C22×C5C22×D411yescharacteristic
158C22×C5C2×#4111yescharacteristic
159C22×C5C2×#4111yescharacteristic
160C22×C5C2×#4112yes—
161C22×C5H(5,𝔽2)12yes—
162C2×D5—22no—
163C2×D5—22no—
164C2×D5—24no—
165C2×D5—24no—
166C2×D5#12414yes—
167C4×C5—22no—
168C4×C5—22no—
169C4×C5—41no—
170C4×C5—42no—
171Dic5—22no—
172Dic5—22no—
173Dic5—42no—
174Dic5—41no—
175Dic5—42no—
176C24—51no—
177C24C22×D512yes—
178C2×D4—22no—
179C2×D4—42no—
180C2×D4—104no—
181C2×D4—104no—
182C2×D4—104no—
183C22×C4—51no—
184C22×C4—51no—
185C22×C4—54no—
186C22×C4—52no—
187C22×C4—52no—
188C22×C4—51no—
189C22×C4—102no—
190C22×C4—102no—
191C22×C4—102no—
192C22×C4—102no—
193C22×C4—202no—
194C22×C4—202no—
195C22×C4C22×D512yes—
196C22×C4C22×D511yescharacteristic
197C22×C4C22×D511yescharacteristic
198C22×C4C22×D511yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—102no—
201C4⋊C4—102no—
202C4⋊C4—102no—
203C4⋊C4—102no—
204C22⋊C4—24no—
205C22⋊C4—42no—
206C22⋊C4—104no—
207C22⋊C4—104no—
208C22⋊C4—104no—
209C22⋊C4—202no—
210C42—202no—
211C2×C5—82no—
212C2×C5#52511yescharacteristic
213C2×C5C2×#12411yescharacteristic
214C2×C5#53411yescharacteristic
215C2×C5#55312yes—
216C2×C5#55312yes—
217D5—24no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—52no—
226C23—54no—
227C23—101no—
228C23—102no—
229C23—101no—
230C23C23×D511yescharacteristic
231D4—44no—
232D4—208no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—52no—
242C2×C4—52no—
243C2×C4—108no—
244C2×C4—101no—
245C2×C4—101no—
246C2×C4—102no—
247C2×C4—102no—
248C2×C4—101no—
249C2×C4—102no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—204no—
256C2×C4—204no—
257C2×C4D4×D512yes—
258C2×C4#71112yes—
259C5#259111yescharacteristic, complemented, Sylow
260C22—42no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—54no—
268C22—102no—
269C22—102no—
270C22—104no—
271C22—104no—
272C22C2×D4×D511yescharacteristic
273C22C2×#71111yescharacteristic
274C22#414412yes—
275C22D5×#4111yescharacteristic
276C22D5×#4112yes—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—42no—
281C4—102no—
282C4—102no—
283C4—201no—
284C4—202no—
285C4—202no—
286C2—82no—
287C2—104no—
288C2#6592311yescharacteristic
289C2#6594011yescharacteristic
290C2#6594812yes—
291C2#6599512yes—
292C2D5×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#394C2
prime radicalC2×#4084C2
hypercenter#394D5
hypocenterC5#2591
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊C4C2semidirect
3. C5×#394C2semidirect
4. #65511C2semidirect
5. #65275C2semidirect
6. #65245C2semidirect
7. #65245C2semidirect
8. #65240C2semidirect
9. #65233C2semidirect
10. #64955C2semidirect
11. #64948C2semidirect
78. #394D5semidirect
259. C5#2591semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×D5×C4⋊C42
C5×#3942
#655112
#652752
#652452
#652402
#652332
#649552
#649482
#25915