Finite Group Databasebeta

← #10609934

Subgroups — #10609934

Order 640.

604 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 135 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×#4039C211yesmaximal, characteristic, complemented
2C2×#4038C211yesmaximal, characteristic, complemented
3C5×#400C211yesmaximal, characteristic, complemented
4#65281C211yesmaximal, characteristic, complemented
5#65280C211yesmaximal, characteristic, complemented
6#64960C211yesmaximal, characteristic
7#64958C212yesmaximal, complemented
8#64957C212yesmaximal, complemented
9#64954C212yesmaximal, complemented
10#64954C212yesmaximal, complemented
11#64940C211yesmaximal, characteristic, complemented
12C22×D20C2211yescharacteristic
13C22×D20C2212yes—
14C22×C4×D5C2211yescharacteristic
15C22×C4×D5C2212yes—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2211yescharacteristic
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×#685C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×#683C2212yes—
27C2×C4×Dic5C2211yescharacteristic
28C2×C4×Dic5C2211yescharacteristic
29#4039—24no—
30#4038—24no—
31C5×#95C2211yescharacteristic
32C5×#95C2212yes—
33C5×#95C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2598—51nomaximal, Sylow
38C23×D5C2311yescharacteristic
39C23×D5C2312yes—
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×C4×C5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2311yescharacteristic
48C22×Dic5C2311yescharacteristic
49C2×D20—24no—
50C2×D20—22no—
51C2×D20—24no—
52C2×D20—24no—
53C2×D20—42no—
54C2×C4×D5—22no—
55C2×C4×D5—22no—
56C2×C4×D5—22no—
57C2×C4×D5—24no—
58C2×C4×D5—42no—
59C2×C4×D5—42no—
60C5×C4⋊C4—22no—
61C5×C4⋊C4—22no—
62#685—24no—
63#685—22no—
64#685—24no—
65#685—22no—
66#685—24no—
67#685—42no—
68#685—42no—
69C4⋊Dic5—22no—
70#683—42no—
71C4×Dic5—22no—
72C4×Dic5D414yescomplemented
73C2×C4×D4—51no—
74C2×#117—51no—
75#400D511yescharacteristic, complemented
76#399—51no—
77#394—51no—
78#392—51no—
79#388—52no—
80#386—52no—
81#386—52no—
82#385—52no—
83#383—51no—
84C23×C5C2411yescharacteristic
85C22×D5—21no—
86C22×D5—22no—
87C22×D5—22no—
88C22×D5—21no—
89C22×D5—22no—
90C22×D5—22no—
91C22×D5—22no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5#4114yes—
96C22×D5#4112yes—
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×Dic5C2×D412yes—
120D20—28no—
121D20—48no—
122C4×D5—28no—
123C4×D5—48no—
124C22×D4—51no—
125C22×D4—52no—
126C23×C4—51no—
127C23×C4—52no—
128C4×D4—104no—
129#117—104no—
130C2×C4⋊C4—51no—
131C2×C4⋊C4—52no—
132C2×C4⋊C4C2×D511yescharacteristic
133C2×C4⋊C4C2×D511yescharacteristic
134C2×C22⋊C4—51no—
135C2×C22⋊C4—51no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4—52no—
140C2×C22⋊C4—52no—
141C2×C42—51no—
142C2×C42—51no—
143#95—52no—
144#95—52no—
145#95—51no—
146#95C2×D511yescharacteristic
147#95C2×D512yes—
148#95C2×D512yes—
149C22×C5C22×D411yescharacteristic
150C22×C5C2×#4111yescharacteristic
151C22×C5C2×#4111yescharacteristic
152C22×C5C2×#4112yes—
153C22×C5H(5,𝔽2)12yes—
154C2×D5—22no—
155C2×D5—24no—
156C2×D5—24no—
157C2×D5—24no—
158C2×D5—24no—
159C2×D5—42no—
160C2×D5—41no—
161C2×D5—42no—
162C2×D5—41no—
163C2×D5—42no—
164C2×D5—42no—
165C2×D5—42no—
166C4×C5—22no—
167C4×C5—22no—
168C4×C5—41no—
169C4×C5—42no—
170C4×C5—42no—
171Dic5—22no—
172Dic5—22no—
173Dic5—41no—
174Dic5—42no—
175C24—52no—
176C24—51no—
177C2×D4—104no—
178C2×D4—104no—
179C2×D4—104no—
180C2×D4—102no—
181C2×D4—202no—
182C22×C4—51no—
183C22×C4—51no—
184C22×C4—52no—
185C22×C4—51no—
186C22×C4—102no—
187C22×C4—104no—
188C22×C4—102no—
189C22×C4—102no—
190C22×C4—202no—
191C22×C4—202no—
192C22×C4C22×D512yes—
193C22×C4C22×D511yescharacteristic
194C22×C4C22×D512yes—
195C22×C4C22×D511yescharacteristic
196C22×C4C22×D511yescharacteristic
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—102no—
200C4⋊C4—202no—
201C22⋊C4—102no—
202C22⋊C4—104no—
203C22⋊C4—104no—
204C22⋊C4—102no—
205C22⋊C4—104no—
206C22⋊C4—202no—
207C22⋊C4—202no—
208C42—54no—
209C42—102no—
210C2×C5#52511yescharacteristic
211C2×C5#52511yescharacteristic
212C2×C5#53511yescharacteristic
213C2×C5#55112yes—
214C2×C5#55312yes—
215D5—42no—
216D5—44no—
217C23—54no—
218C23—52no—
219C23—101no—
220C23—102no—
221C23—102no—
222C23—101no—
223C23—102no—
224C23—102no—
225C23—102no—
226C23—102no—
227C23—102no—
228C23—102no—
229C23C23×D511yescharacteristic
230D4—108no—
231D4—208no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—52no—
244C2×C4—52no—
245C2×C4—108no—
246C2×C4—101no—
247C2×C4—101no—
248C2×C4—102no—
249C2×C4—101no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—208no—
255C2×C4D4×D512yes—
256C2×C4#71312yes—
257C5#259811yescharacteristic, complemented, Sylow
258C22—102no—
259C22—104no—
260C22—104no—
261C22—104no—
262C22—104no—
263C22—202no—
264C22—201no—
265C22—202no—
266C22—201no—
267C22—202no—
268C22—202no—
269C22—202no—
270C22C2×D4×D511yescharacteristic
271C22C2×#71311yescharacteristic
272C22D5×#4111yescharacteristic
273C22D5×#4112yes—
274C22#414912yes—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—42no—
280C4—102no—
281C4—102no—
282C4—201no—
283C4—202no—
284C2—202no—
285C2—204no—
286C2#6596111yescharacteristic
287C2#6599011yescharacteristic
288C2#6600811yescharacteristic
289C2#6602912yes—
290C2#6603912yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radical#64960C2
hypercenter#400D5
hypocenterC5#2598
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#4039C2semidirect
2. C2×#4038C2semidirect
3. C5×#400C2semidirect
4. #65281C2semidirect
5. #65280C2semidirect
7. #64958C2semidirect
8. #64957C2semidirect
9. #64954C2semidirect
10. #64954C2semidirect
11. #64940C2semidirect
72. C4×Dic5D4semidirect
75. #400D5semidirect
257. C5#2598semidirect

Maximal subgroups

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

subgroupindex
C2×#40392
C2×#40382
C5×#4002
#652812
#652802
#649602
#649582
#649572
#649542
#649402
#25985