Finite Group Databasebeta

← #10608539

Subgroups — #10608539

Order 640.

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

normal subgroups only

This group has 576 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×#4093C211yesmaximal, characteristic, complemented
2C2×#4023C211yesmaximal, characteristic, complemented
3C5×#393C211yesmaximal, characteristic, complemented
4#65518C211yesmaximal, characteristic, complemented
5#65249C211yesmaximal, characteristic
6#65247C212yesmaximal, complemented
7#65246C212yesmaximal, complemented
8#64961C211yesmaximal, characteristic, complemented
9#64958C212yesmaximal, complemented
10#64958C212yesmaximal, complemented
11#64937C211yesmaximal, characteristic, complemented
12C22×C5×Q8C2211yescharacteristic
13C22×C22⋊D5C2212yes—
14C23×Dic5C2211yescharacteristic
15C22×D20C2211yescharacteristic
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19#4093—24no—
20C2×#685C2212yes—
21C2×#685C2212yes—
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29C2×C4×Dic5C2212yes—
30C2×C4×Dic5C2211yescharacteristic
31#4023—24no—
32C5×#95C2211yescharacteristic
33C5×#95C2212yes—
34#3963C2212yes—
35#3963C2211yescharacteristic
36#3963C2212yes—
37#2610—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C23×D5C2312yes—
40C2×C5×Q8—22no—
41C2×C5×Q8—42no—
42C22×C4×C5C2312yes—
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C2×C22⋊D5—24no—
47C2×C22⋊D5—24no—
48C2×C22⋊D5—42no—
49C22×Dic5—22no—
50C22×Dic5—22no—
51C22×Dic5—22no—
52C22×Dic5C2311yescharacteristic
53C22×Dic5C2311yescharacteristic
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2312yes—
57C2×D20—22no—
58C2×D20—42no—
59C5×C22⋊C4—22no—
60C5×C22⋊C4—42no—
61C22⋊Dic5—22no—
62#685—24no—
63#685—24no—
64#685—24no—
65#685—24no—
66#685—42no—
67#683—22no—
68#683—22no—
69#683—42no—
70C4×Dic5—22no—
71C4×Dic5—24no—
72C4×Dic5D414yescomplemented
73C2×#124—51no—
74C2×#117—51no—
75#395—51no—
76#394—51no—
77#393D511yescharacteristic, complemented
78#392—51no—
79#390—52no—
80#388—52no—
81#388—52no—
82#386—52no—
83#376—51no—
84C23×C5—21no—
85C23×C5—21no—
86C23×C5—22no—
87C23×C5—22no—
88C23×C5#4112yes—
89C23×C5C2411yescharacteristic
90C22×D5—22no—
91C22×D5—22no—
92C22×D5—22no—
93C22×D5—22no—
94C22×D5—22no—
95C22×D5—22no—
96C22×D5—22no—
97C5×Q8—44no—
98C2×C4×C5—22no—
99C2×C4×C5—21no—
100C2×C4×C5—22no—
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×C5C2×D412yes—
108C22⋊D5—48no—
109C2×Dic5—28no—
110C2×Dic5—21no—
111C2×Dic5—22no—
112C2×Dic5—21no—
113C2×Dic5—21no—
114C2×Dic5—21no—
115C2×Dic5—22no—
116C2×Dic5—22no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5C2×D412yes—
120C2×Dic5C2×D412yes—
121C2×Dic5#4114yes—
122C2×Dic5#4112yes—
123D20—44no—
124C22×Q8C2×D511yescharacteristic
125C22×D4—52no—
126C22×D4—51no—
127C23×C4—51no—
128#124—104no—
129#117—104no—
130C2×C4⋊C4—51no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—52no—
133C2×C22⋊C4—51no—
134C2×C22⋊C4—52no—
135C2×C22⋊C4—52no—
136C2×C22⋊C4—52no—
137C2×C22⋊C4—52no—
138C2×C22⋊C4—52no—
139C2×C22⋊C4C2×D511yescharacteristic
140C2×C22⋊C4C2×D512yes—
141C2×C42—51no—
142C2×C42—51no—
143C2×C42—52no—
144#95—52no—
145#95—51no—
146#95—52no—
147#95C2×D511yescharacteristic
148#95C2×D512yes—
149C22×C5—22no—
150C22×C5—24no—
151C22×C5—41no—
152C22×C5—41no—
153C22×C5—42no—
154C22×C5C22×D411yescharacteristic
155C22×C5C2×#4111yescharacteristic
156C22×C5C2×#4111yescharacteristic
157C22×C5C2×#4112yes—
158C22×C5H(5,𝔽2)12yes—
159C2×D5—42no—
160C2×D5—42no—
161C2×D5—42no—
162C2×D5—42no—
163C2×D5—42no—
164C2×D5—42no—
165C2×D5—42no—
166C4×C5—22no—
167C4×C5—41no—
168C4×C5—42no—
169C4×C5—42no—
170Dic5—22no—
171Dic5—22no—
172Dic5—22no—
173Dic5—24no—
174Dic5—41no—
175C24—52no—
176C24C22×D511yescharacteristic
177C2×Q8—22no—
178C2×Q8—42no—
179C2×D4—104no—
180C2×D4—104no—
181C2×D4—102no—
182C2×D4—202no—
183C2×D4—202no—
184C22×C4—52no—
185C22×C4—51no—
186C22×C4—51no—
187C22×C4—51no—
188C22×C4—51no—
189C22×C4—102no—
190C22×C4—102no—
191C22×C4—102no—
192C22×C4C22×D512yes—
193C22×C4C22×D512yes—
194C22×C4C22×D511yescharacteristic
195C22×C4C22×D511yescharacteristic
196C4⋊C4—102no—
197C4⋊C4—102no—
198C4⋊C4—202no—
199C22⋊C4—22no—
200C22⋊C4—42no—
201C22⋊C4—104no—
202C22⋊C4—104no—
203C22⋊C4—104no—
204C22⋊C4—104no—
205C22⋊C4—102no—
206C22⋊C4—202no—
207C42—54no—
208C42—102no—
209C42—104no—
210C2×C5—42no—
211C2×C5#52511yescharacteristic
212C2×C5#53211yescharacteristic
213C2×C5#53511yescharacteristic
214C2×C5#55112yes—
215C2×C5#55512yes—
216D5—82no—
217C23—21no—
218C23—22no—
219C23—21no—
220C23—22no—
221C23—102no—
222C23—102no—
223C23—102no—
224C23—102no—
225C23—102no—
226C23—102no—
227C23—102no—
228C23#71312yes—
229C23C23×D511yescharacteristic
230Q8—44no—
231D4—204no—
232D4—208no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—52no—
243C2×C4—52no—
244C2×C4—54no—
245C2×C4—52no—
246C2×C4—108no—
247C2×C4—101no—
248C2×C4—102no—
249C2×C4—101no—
250C2×C4—101no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—101no—
254C2×C4—101no—
255C2×C4—102no—
256C2×C4D4×D512yes—
257C5#261011yescharacteristic, complemented, Sylow
258C22—22no—
259C22—24no—
260C22—41no—
261C22—41no—
262C22—42no—
263C22—202no—
264C22—202no—
265C22—202no—
266C22—202no—
267C22—202no—
268C22—202no—
269C22—202no—
270C22C2×D4×D511yescharacteristic
271C22#414412yes—
272C22C2×#71311yescharacteristic
273C22D5×#4111yescharacteristic
274C22D5×#4112yes—
275C4—22no—
276C4—42no—
277C4—41no—
278C4—42no—
279C4—102no—
280C4—102no—
281C4—104no—
282C4—102no—
283C4—201no—
284C2—42no—
285C2—402no—
286C2#6592611yescharacteristic
287C2#6596111yescharacteristic
288C2#6597112yes—
289C2#6600211yescharacteristic
290C2#6600412yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#393C2
prime radical#65249C2
hypercenter#393D5
hypocenterC5#2610
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#4093C2semidirect
2. C2×#4023C2semidirect
3. C5×#393C2semidirect
4. #65518C2semidirect
6. #65247C2semidirect
7. #65246C2semidirect
8. #64961C2semidirect
9. #64958C2semidirect
10. #64958C2semidirect
11. #64937C2semidirect
72. C4×Dic5D4semidirect
77. #393D5semidirect
257. C5#2610semidirect

Maximal subgroups

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

subgroupindex
C2×#40932
C2×#40232
C5×#3932
#655182
#652492
#652472
#652462
#649612
#649582
#649372
#26105