Finite Group Databasebeta

← #10608423

Subgroups — #10608423

Order 640.

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

normal subgroups only

This group has 604 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×D4×Dic5C211yesmaximal, characteristic, complemented
2C2×#4023C211yesmaximal, characteristic, complemented
3C5×#392C211yesmaximal, characteristic
4#65511C211yesmaximal, characteristic, complemented
5#65509C212yesmaximal, complemented
6#65509C212yesmaximal, complemented
7#65238C211yesmaximal, characteristic, complemented
8#65235C212yesmaximal, complemented
9#65234C212yesmaximal, complemented
10#64948C211yesmaximal, characteristic, complemented
11#64940C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C5×D4C2212yes—
14C23×Dic5C2211yescharacteristic
15C23×Dic5C2212yes—
16C2×C5×C22⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2211yescharacteristic
19C2×C22⋊Dic5C2212yes—
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23D4×Dic5—24no—
24C2×C4⋊Dic5C2211yescharacteristic
25C2×#683C2211yescharacteristic
26C2×#683C2211yescharacteristic
27C2×#683C2212yes—
28C2×C4×Dic5C2211yescharacteristic
29C2×C4×Dic5C2211yescharacteristic
30#4023—24no—
31C5×#95C2211yescharacteristic
32#3963C2212yes—
33#3963C2212yes—
34#3963C2211yescharacteristic
35#3963C2212yes—
36#3963C2212yes—
37#2598—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C24×C5C2312yes—
40C2×C5×D4—24no—
41C2×C5×D4—24no—
42C2×C5×D4—24no—
43C2×C5×D4—22no—
44C2×C5×D4—42no—
45C22×C4×C5C2311yescharacteristic
46C22×C4×C5C2312yes—
47C22×C4×C5C2311yescharacteristic
48C22×Dic5—22no—
49C22×Dic5—22no—
50C22×Dic5—22no—
51C22×Dic5—24no—
52C22×Dic5—42no—
53C22×Dic5—42no—
54C22×Dic5C2312yes—
55C22×Dic5C2311yescharacteristic
56C22×Dic5C2312yes—
57C22×Dic5C2311yescharacteristic
58C22×Dic5C2311yescharacteristic
59C22×Dic5C2311yescharacteristic
60C5×C22⋊C4—22no—
61C5×C22⋊C4—42no—
62C22⋊Dic5—24no—
63C22⋊Dic5—22no—
64C22⋊Dic5—24no—
65C22⋊Dic5—24no—
66C22⋊Dic5—42no—
67C4⋊Dic5—22no—
68#683—22no—
69#683—22no—
70#683—42no—
71C4×Dic5—22no—
72C4×Dic5D414yescomplemented
73C2×C4×D4—51no—
74C2×#117—51no—
75#400—51no—
76#399—51no—
77#394—51no—
78#392D511yescharacteristic
79#388—52no—
80#386—52no—
81#386—52no—
82#385—52no—
83#383—51no—
84C23×C5—21no—
85C23×C5—22no—
86C23×C5—22no—
87C23×C5—21no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5#4114yes—
95C23×C5#4112yes—
96C23×C5C2411yescharacteristic
97C5×D4—28no—
98C5×D4—48no—
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×C5C2×D412yes—
106C2×Dic5—28no—
107C2×Dic5—21no—
108C2×Dic5—21no—
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×Dic5—48no—
121C2×Dic5C2×D412yes—
122C2×Dic5C2×D412yes—
123C2×Dic5#4112yes—
124C22×D4C2×D511yescharacteristic
125C22×D4C2×D512yes—
126C23×C4—51no—
127C23×C4—52no—
128C4×D4—104no—
129#117—104no—
130C2×C4⋊C4—51no—
131C2×C4⋊C4—51no—
132C2×C4⋊C4—51no—
133C2×C4⋊C4—52no—
134C2×C22⋊C4—51no—
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—
143#95—52no—
144#95—52no—
145#95—51no—
146#95—52no—
147#95—52no—
148#95C2×D511yescharacteristic
149C22×C5—22no—
150C22×C5—24no—
151C22×C5—24no—
152C22×C5—24no—
153C22×C5—24no—
154C22×C5—41no—
155C22×C5—42no—
156C22×C5—42no—
157C22×C5—41no—
158C22×C5—42no—
159C22×C5—42no—
160C22×C5—42no—
161C22×C5C22×D411yescharacteristic
162C22×C5C2×#4111yescharacteristic
163C22×C5C2×#4112yes—
164C22×C5C2×#4111yescharacteristic
165C22×C5H(5,𝔽2)12yes—
166C4×C5—22no—
167C4×C5—41no—
168C4×C5—42no—
169Dic5—22no—
170Dic5—22no—
171Dic5—22no—
172Dic5—41no—
173Dic5—42no—
174Dic5—42no—
175C24C22×D511yescharacteristic
176C24C22×D512yes—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—22no—
181C2×D4—42no—
182C22×C4—51no—
183C22×C4—51no—
184C22×C4—51no—
185C22×C4—52no—
186C22×C4—51no—
187C22×C4—52no—
188C22×C4—102no—
189C22×C4—102no—
190C22×C4—104no—
191C22×C4—102no—
192C22×C4—202no—
193C22×C4—202no—
194C22×C4C22×D512yes—
195C22×C4C22×D511yescharacteristic
196C22×C4C22×D511yescharacteristic
197C4⋊C4—102no—
198C4⋊C4—102no—
199C4⋊C4—102no—
200C4⋊C4—202no—
201C22⋊C4—22no—
202C22⋊C4—42no—
203C22⋊C4—102no—
204C22⋊C4—104no—
205C22⋊C4—104no—
206C22⋊C4—104no—
207C22⋊C4—202no—
208C42—54no—
209C42—102no—
210C2×C5—42no—
211C2×C5—44no—
212C2×C5#52511yescharacteristic
213C2×C5#52511yescharacteristic
214C2×C5#53511yescharacteristic
215C2×C5#55112yes—
216C2×C5#55312yes—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23#71114yes—
228C23#71112yes—
229C23C23×D511yescharacteristic
230D4—28no—
231D4—48no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—52no—
239C2×C4—52no—
240C2×C4—52no—
241C2×C4—108no—
242C2×C4—101no—
243C2×C4—101no—
244C2×C4—101no—
245C2×C4—102no—
246C2×C4—101no—
247C2×C4—101no—
248C2×C4—102no—
249C2×C4—102no—
250C2×C4—102no—
251C2×C4—102no—
252C2×C4—102no—
253C2×C4—102no—
254C2×C4—102no—
255C2×C4—208no—
256C2×C4D4×D512yes—
257C5#259811yescharacteristic, complemented, Sylow
258C22—24no—
259C22—24no—
260C22—24no—
261C22—24no—
262C22—22no—
263C22—42no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22C2×D4×D511yescharacteristic
271C22C2×#71112yes—
272C22C2×#71111yescharacteristic
273C22#414412yes—
274C22D5×#4111yescharacteristic
275C4—22no—
276C4—41no—
277C4—42no—
278C4—102no—
279C4—102no—
280C4—102no—
281C4—202no—
282C4—201no—
283C4—202no—
284C2—44no—
285C2—42no—
286C2#6592011yescharacteristic
287C2#6592112yes—
288C2#6593512yes—
289C2#6594011yescharacteristic
290C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC5×#392C2
hypercenter#392D5
hypocenterC5#2598
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic5C2semidirect
2. C2×#4023C2semidirect
4. #65511C2semidirect
5. #65509C2semidirect
6. #65509C2semidirect
7. #65238C2semidirect
8. #65235C2semidirect
9. #65234C2semidirect
10. #64948C2semidirect
11. #64940C2semidirect
72. C4×Dic5D4semidirect
257. C5#2598semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40232
C5×#3922
#655112
#655092
#652382
#652352
#652342
#649482
#649402
#25985