Finite Group Databasebeta

← #10609223

Subgroups — #10609223

Order 640.

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

normal subgroups only

This group has 596 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×D5×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#4030C211yesmaximal, characteristic
3C5×#397C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65246C212yesmaximal, complemented
6#65245C212yesmaximal, complemented
7#65242C211yesmaximal, characteristic, complemented
8#64963C211yesmaximal, characteristic, complemented
9#64955C211yesmaximal, characteristic, complemented
10#64954C212yesmaximal, complemented
11#64939C212yesmaximal, complemented
12C22×C22⋊D5C2211yescharacteristic
13C22×C22⋊D5C2211yescharacteristic
14C22×D20C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C22×C4×D5C2212yes—
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2212yes—
28C2×C4×Dic5C2212yes—
29D5×C4⋊C4—24no—
30#4030—24no—
31C5×#95C2212yes—
32C5×#95C2211yescharacteristic
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2504—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2311yescharacteristic
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C2×C22⋊D5—22no—
46C2×C22⋊D5—24no—
47C2×C22⋊D5—22no—
48C2×C22⋊D5—24no—
49C22×Dic5C2312yes—
50C22×Dic5C2312yes—
51C22×Dic5C2311yescharacteristic
52C22×Dic5C2311yescharacteristic
53C2×D20—22no—
54C2×D20—24no—
55C2×C4×D5—24no—
56C2×C4×D5—24no—
57C2×C4×D5—42no—
58C2×C4×D5—42no—
59C2×C4×D5D414yes—
60C5×C4⋊C4—22no—
61C5×C22⋊C4—22no—
62C5×C22⋊C4—42no—
63C22⋊Dic5—42no—
64#685—24no—
65#685—22no—
66#685—42no—
67#685—42no—
68C4⋊Dic5—22no—
69#683—22no—
70#683—24no—
71C4×Dic5—24no—
72C2×#121—51no—
73C22×C4⋊C4—51no—
74#397—51no—
75#397—51no—
76#397—51no—
77#397D511yescharacteristic, complemented
78#390—52no—
79#386—52no—
80#386—52no—
81#382—52no—
82C4×C22⋊C4—51no—
83C23×C5—21no—
84C23×C5—22no—
85C23×C5—22no—
86C23×C5—21no—
87C23×C5—21no—
88C23×C5C2411yescharacteristic
89C22×D5—22no—
90C22×D5—21no—
91C22×D5—21no—
92C22×D5—21no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5—21no—
96C22×D5—21no—
97C22×D5C2×D412yes—
98C22×D5#4114yes—
99C2×C4×C5—22no—
100C2×C4×C5—21no—
101C2×C4×C5—22no—
102C2×C4×C5—22no—
103C2×C4×C5—22no—
104C2×C4×C5—22no—
105C2×C4×C5—22no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5C2×D412yes—
109C22⋊D5—44no—
110C22⋊D5—44no—
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×Dic5C2×D412yes—
120C2×Dic5#4114yes—
121D20—44no—
122C4×D5—28no—
123C4×D5—48no—
124C22×D4—51no—
125C22×D4—51no—
126C22×D4—51no—
127C23×C4—51no—
128C23×C4—52no—
129#121—104no—
130C2×C4⋊C4—52no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—51no—
133C2×C4⋊C4—104no—
134C2×C4⋊C4C2×D511yescharacteristic
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4C2×D511yescharacteristic
141C2×C22⋊C4C2×D512yes—
142C2×C42—52no—
143#95—51no—
144#95—52no—
145#95—52no—
146#95—52no—
147#95C2×D511yescharacteristic
148#95C2×D512yes—
149C22×C5—41no—
150C22×C5—41no—
151C22×C5—41no—
152C22×C5—42no—
153C22×C5—42no—
154C22×C5C22×D411yescharacteristic
155C22×C5C2×#4112yes—
156C22×C5C2×#4112yes—
157C22×C5H(5,𝔽2)11yescharacteristic
158C22×C5H(5,𝔽2)11yescharacteristic
159C2×D5—22no—
160C2×D5—22no—
161C2×D5—24no—
162C2×D5—24no—
163C2×D5—41no—
164C2×D5—41no—
165C2×D5—41no—
166C2×D5—42no—
167C2×D5—42no—
168C2×D5#12414yes—
169C4×C5—22no—
170C4×C5—41no—
171C4×C5—42no—
172C4×C5—42no—
173Dic5—41no—
174Dic5—42no—
175Dic5—42no—
176Dic5#12314yes—
177C24—51no—
178C24—51no—
179C24C22×D511yescharacteristic
180C2×D4—102no—
181C2×D4—104no—
182C2×D4—102no—
183C2×D4—104no—
184C2×D4—102no—
185C2×D4—104no—
186C22×C4—51no—
187C22×C4—54no—
188C22×C4—51no—
189C22×C4—52no—
190C22×C4—52no—
191C22×C4—104no—
192C22×C4—104no—
193C22×C4—202no—
194C22×C4—202no—
195C22×C4C22×D511yescharacteristic
196C22×C4C22×D512yes—
197C22×C4C22×D512yes—
198C22×C4C22×D511yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—104no—
201C4⋊C4—102no—
202C4⋊C4—102no—
203C22⋊C4—22no—
204C22⋊C4—42no—
205C22⋊C4—104no—
206C22⋊C4—102no—
207C22⋊C4—202no—
208C22⋊C4—202no—
209C22⋊C4—202no—
210C42—104no—
211C2×C5—81no—
212C2×C5C2×#12311yescharacteristic
213C2×C5C2×#12411yescharacteristic
214C2×C5#53511yescharacteristic
215C2×C5#55312yes—
216C2×C5#55312yes—
217D5—24no—
218D5—81no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—52no—
225C23—54no—
226C23—101no—
227C23—102no—
228C23—101no—
229C23—101no—
230C23—101no—
231C23—102no—
232C23—102no—
233C23—101no—
234C23C23×D511yescharacteristic
235D4—204no—
236D4—204no—
237D4—204no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—52no—
248C2×C4—54no—
249C2×C4—108no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—102no—
256C2×C4—102no—
257C2×C4—102no—
258C2×C4—208no—
259C2×C4D4×D512yes—
260C5#250411yescharacteristic, complemented, Sylow
261C22—41no—
262C22—41no—
263C22—42no—
264C22—41no—
265C22—42no—
266C22—54no—
267C22—102no—
268C22—102no—
269C22—104no—
270C22—104no—
271C22—201no—
272C22—201no—
273C22—202no—
274C22—201no—
275C22—202no—
276C22C2×D4×D511yescharacteristic
277C22#414411yescharacteristic
278C22D5×#4112yes—
279C22D5×#4112yes—
280C22#414911yescharacteristic
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—54no—
286C4—202no—
287C4—201no—
288C4—202no—
289C2—81no—
290C2—104no—
291C2—401no—
292C2#6594111yescharacteristic
293C2D5×#12311yescharacteristic
294C2#6599512yes—
295C2D5×#12411yescharacteristic
296C2#6604312yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#397C2
prime radicalC2×#4030C2
hypercenter#397D5
hypocenterC5#2504
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×D5×C4⋊C4C2semidirect
3. C5×#397C2semidirect
4. #65282C2semidirect
5. #65246C2semidirect
6. #65245C2semidirect
7. #65242C2semidirect
8. #64963C2semidirect
9. #64955C2semidirect
10. #64954C2semidirect
11. #64939C2semidirect
77. #397D5semidirect
260. C5#2504semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊C42
C2×#40302
C5×#3972
#652822
#652462
#652452
#652422
#649632
#649552
#649542
#649392
#25045