Finite Group Databasebeta

← #10610089

Subgroups — #10610089

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×#4041C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C5×#400C211yesmaximal, characteristic, complemented
4#65280C211yesmaximal, characteristic, complemented
5#65271C211yesmaximal, characteristic, complemented
6#64961C211yesmaximal, characteristic, complemented
7#64959C212yesmaximal, complemented
8#64958C212yesmaximal, complemented
9#64955C211yesmaximal, characteristic, complemented
10#64954C212yesmaximal, complemented
11#64954C212yesmaximal, complemented
12C22×D20C2212yes—
13C22×D20C2211yescharacteristic
14C22×C4×D5C2211yescharacteristic
15C22×C4×D5C2211yescharacteristic
16C22×C4×D5C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×#685C2212yes—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2211yescharacteristic
28C2×C4×Dic5C2212yes—
29#4041—24no—
30D5×C4⋊C4—24no—
31C5×#95C2212yes—
32C5×#95C2212yes—
33C5×#95C2211yescharacteristic
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2591—51nomaximal, Sylow
38C23×D5C2312yes—
39C23×D5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2312yes—
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×D20—24no—
50C2×D20—24no—
51C2×D20—24no—
52C2×D20—22no—
53C2×D20—42no—
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×C4⋊C4—22no—
63#685—24no—
64#685—24no—
65#685—24no—
66#685—24no—
67#685—42no—
68#685—42no—
69C4⋊Dic5—22no—
70#683—22no—
71#683—22no—
72C4×Dic5—42no—
73C2×#121—51no—
74C22×C4⋊C4—51no—
75#400—51no—
76#400D511yescharacteristic, complemented
77#394—51no—
78#394—51no—
79#388—52no—
80#388—52no—
81#386—52no—
82#386—52no—
83C4×C22⋊C4—51no—
84C23×C5C2411yescharacteristic
85C22×D5—21no—
86C22×D5—21no—
87C22×D5—22no—
88C22×D5—22no—
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—22no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5C2×D412yes—
96C22×D5#4114yes—
97C2×C4×C5—21no—
98C2×C4×C5—21no—
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—
109C2×C4×C5#4112yes—
110C2×Dic5—21no—
111C2×Dic5—21no—
112C2×Dic5—22no—
113C2×Dic5—21no—
114C2×Dic5—22no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—22no—
118C2×Dic5C2×D412yes—
119C2×Dic5#4112yes—
120D20—44no—
121D20—48no—
122C4×D5—28no—
123C4×D5—44no—
124C4×D5—44no—
125C22×D4—52no—
126C22×D4—51no—
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—104no—
135C2×C4⋊C4C2×D511yescharacteristic
136C2×C4⋊C4C2×D511yescharacteristic
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4—52no—
143C2×C42—52no—
144#95—51no—
145#95—52no—
146#95—52no—
147#95C2×D511yescharacteristic
148#95C2×D512yes—
149#95C2×D512yes—
150C22×C5C22×D411yescharacteristic
151C22×C5C2×#4111yescharacteristic
152C22×C5C2×#4111yescharacteristic
153C22×C5C2×#4112yes—
154C22×C5H(5,𝔽2)12yes—
155C2×D5—22no—
156C2×D5—22no—
157C2×D5—24no—
158C2×D5—24no—
159C2×D5—42no—
160C2×D5—42no—
161C2×D5—42no—
162C2×D5—42no—
163C2×D5—42no—
164C2×D5—42no—
165C2×D5—42no—
166C2×D5#12414yes—
167C4×C5—22no—
168C4×C5—22no—
169C4×C5—42no—
170C4×C5—41no—
171C4×C5—42no—
172Dic5—22no—
173Dic5—22no—
174Dic5—41no—
175Dic5—42no—
176C24—51no—
177C24—52no—
178C2×D4—104no—
179C2×D4—104no—
180C2×D4—104no—
181C2×D4—102no—
182C2×D4—202no—
183C22×C4—51no—
184C22×C4—51no—
185C22×C4—54no—
186C22×C4—52no—
187C22×C4—51no—
188C22×C4—102no—
189C22×C4—102no—
190C22×C4—102no—
191C22×C4—102no—
192C22×C4—202no—
193C22×C4—202no—
194C22×C4C22×D512yes—
195C22×C4C22×D512yes—
196C22×C4C22×D511yescharacteristic
197C22×C4C22×D511yescharacteristic
198C22×C4C22×D511yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—102no—
202C4⋊C4—102no—
203C4⋊C4—102no—
204C22⋊C4—104no—
205C22⋊C4—104no—
206C22⋊C4—104no—
207C22⋊C4—104no—
208C22⋊C4—202no—
209C22⋊C4—202no—
210C42—202no—
211C2×C5#52511yescharacteristic
212C2×C5C2×#12411yescharacteristic
213C2×C5#53411yescharacteristic
214C2×C5#55312yes—
215C2×C5#55312yes—
216D5—24no—
217D5—82no—
218C23—52no—
219C23—54no—
220C23—101no—
221C23—101no—
222C23—102no—
223C23—102no—
224C23—102no—
225C23—102no—
226C23—102no—
227C23—102no—
228C23—102no—
229C23—102no—
230C23C23×D511yescharacteristic
231D4—204no—
232D4—208no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—52no—
245C2×C4—52no—
246C2×C4—108no—
247C2×C4—101no—
248C2×C4—101no—
249C2×C4—102no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—204no—
256C2×C4—204no—
257C2×C4D4×D512yes—
258C2×C4#71312yes—
259C5#259111yescharacteristic, complemented, Sylow
260C22—54no—
261C22—102no—
262C22—102no—
263C22—104no—
264C22—104no—
265C22—202no—
266C22—202no—
267C22—202no—
268C22—202no—
269C22—202no—
270C22—202no—
271C22—202no—
272C22C2×D4×D511yescharacteristic
273C22C2×#71311yescharacteristic
274C22D5×#4111yescharacteristic
275C22D5×#4112yes—
276C22#414912yes—
277C4—22no—
278C4—22no—
279C4—41no—
280C4—42no—
281C4—42no—
282C4—102no—
283C4—102no—
284C4—201no—
285C4—202no—
286C2—104no—
287C2—402no—
288C2#6596111yescharacteristic
289C2#6598711yescharacteristic
290C2D5×#12411yescharacteristic
291C2#6602912yes—
292C2#6604312yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC2×#4041C2
hypercenter#400D5
hypocenterC5#2591
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊C4C2semidirect
3. C5×#400C2semidirect
4. #65280C2semidirect
5. #65271C2semidirect
6. #64961C2semidirect
7. #64959C2semidirect
8. #64958C2semidirect
9. #64955C2semidirect
10. #64954C2semidirect
11. #64954C2semidirect
76. #400D5semidirect
259. C5#2591semidirect

Maximal subgroups

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

subgroupindex
C2×#40412
C2×D5×C4⋊C42
C5×#4002
#652802
#652712
#649612
#649592
#649582
#649552
#649542
#25915