Finite Group Databasebeta

← #10608506

Subgroups — #10608506

Order 640.

578 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 125 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×#4084C211yesmaximal, characteristic, complemented
2C2×#4029C211yesmaximal, characteristic, complemented
3C5×#392C211yesmaximal, characteristic, complemented
4#65512C212yesmaximal, complemented
5#65511C212yesmaximal, complemented
6#65511C211yesmaximal, characteristic, complemented
7#65248C212yesmaximal, complemented
8#65238C211yesmaximal, characteristic, complemented
9#65237C212yesmaximal, complemented
10#64956C211yesmaximal, characteristic, complemented
11#64948C211yesmaximal, characteristic, complemented
12C22×C5×D4C2211yescharacteristic
13C22×C5×D4C2212yes—
14C22×C22⋊D5C2211yescharacteristic
15C22×C22⋊D5C2212yes—
16C22×C4×D5C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2212yes—
19C2×C22⋊Dic5C2211yescharacteristic
20C2×C22⋊Dic5C2212yes—
21C2×C22⋊Dic5C2212yes—
22C2×C22⋊Dic5C2212yes—
23C2×C22⋊Dic5C2212yes—
24#4084—24no—
25C2×#685C2211yescharacteristic
26C2×#685C2212yes—
27C2×C4⋊Dic5C2211yescharacteristic
28C2×C4⋊Dic5C2211yescharacteristic
29C2×#683C2211yescharacteristic
30C2×#683C2212yes—
31C2×#683C2212yes—
32#4029—24no—
33C5×#95C2211yescharacteristic
34#3963C2212yes—
35#3963C2212yes—
36#3963C2211yescharacteristic
37#2813—51nomaximal, Sylow
38C24×C5C2311yescharacteristic
39C24×C5C2312yes—
40C23×D5C2311yescharacteristic
41C2×C5×D4—22no—
42C2×C5×D4—42no—
43C2×C5×D4—42no—
44C2×C5×D4—42no—
45C2×C5×D4—42no—
46C22×C4×C5C2311yescharacteristic
47C22×C4×C5C2312yes—
48C22×C4×C5C2311yescharacteristic
49C2×C22⋊D5—22no—
50C2×C22⋊D5—24no—
51C2×C22⋊D5—42no—
52C2×C22⋊D5—42no—
53C2×C22⋊D5—42no—
54C22×Dic5C2311yescharacteristic
55C22×Dic5C2312yes—
56C22×Dic5C2311yescharacteristic
57C22×Dic5C2312yes—
58C22×Dic5C2311yescharacteristic
59C2×C4×D5—42no—
60C2×C4×D5D414yescomplemented
61C5×C22⋊C4—22no—
62C5×C22⋊C4—42no—
63C22⋊Dic5—22no—
64C22⋊Dic5—24no—
65C22⋊Dic5—42no—
66C22⋊Dic5—42no—
67C22⋊Dic5—42no—
68#685—22no—
69#685—42no—
70C4⋊Dic5—22no—
71C4⋊Dic5—22no—
72#683—22no—
73#683—42no—
74#683—42no—
75C2×#123—51no—
76C2×#121—51no—
77#400—51no—
78#399—51no—
79#396—52no—
80#394—51no—
81#394—52no—
82#394—52no—
83#392—52no—
84#392D511yescharacteristic, complemented
85#380—51no—
86C23×C5—21no—
87C23×C5—22no—
88C23×C5—22no—
89C23×C5—22no—
90C23×C5—22no—
91C23×C5—22no—
92C23×C5—22no—
93C23×C5—22no—
94C23×C5—22no—
95C23×C5—21no—
96C23×C5—21no—
97C23×C5—22no—
98C23×C5C2411yescharacteristic
99C22×D5—21no—
100C22×D5—22no—
101C22×D5—22no—
102C22×D5—21no—
103C22×D5C2×D412yes—
104C5×D4—44no—
105C5×D4—48no—
106C2×C4×C5—22no—
107C2×C4×C5—21no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5C2×D412yes—
113C22⋊D5—44no—
114C22⋊D5—48no—
115C2×Dic5—22no—
116C2×Dic5—21no—
117C2×Dic5—21no—
118C2×Dic5—22no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—22no—
123C2×Dic5—21no—
124C2×Dic5—22no—
125C2×Dic5—22no—
126C2×Dic5—21no—
127C2×Dic5C2×D412yes—
128C4×D5—44no—
129C22×D4—52no—
130C22×D4—51no—
131C22×D4C2×D511yescharacteristic
132C22×D4C2×D512yes—
133C23×C4—51no—
134#123—104no—
135#121—104no—
136C2×C4⋊C4—51no—
137C2×C4⋊C4—51no—
138C2×C4⋊C4—51no—
139C2×C4⋊C4—52no—
140C2×C4⋊C4—52no—
141C2×C22⋊C4—52no—
142C2×C22⋊C4—51no—
143C2×C22⋊C4—51no—
144C2×C22⋊C4—52no—
145C2×C22⋊C4—52no—
146C2×C22⋊C4—52no—
147C2×C22⋊C4—52no—
148C2×C22⋊C4C2×D511yescharacteristic
149C2×C22⋊C4C2×D512yes—
150#95—51no—
151#95—52no—
152#95—52no—
153#95C2×D511yescharacteristic
154C22×C5—41no—
155C22×C5—42no—
156C22×C5—42no—
157C22×C5—41no—
158C22×C5—41no—
159C22×C5—42no—
160C22×C5—42no—
161C22×C5—42no—
162C22×C5—42no—
163C22×C5—42no—
164C22×C5—42no—
165C22×C5—42no—
166C22×C5C22×D411yescharacteristic
167C22×C5H(5,𝔽2)12yes—
168C22×C5H(5,𝔽2)12yes—
169C22×C5H(5,𝔽2)11yescharacteristic
170C22×C52−1+411yescharacteristic
171C2×D5—22no—
172C2×D5—22no—
173C2×D5—22no—
174C2×D5—42no—
175C2×D5—42no—
176C4×C5—22no—
177C4×C5—41no—
178C4×C5—42no—
179Dic5—22no—
180Dic5—41no—
181Dic5—42no—
182Dic5—41no—
183Dic5—42no—
184C24—51no—
185C24C22×D511yescharacteristic
186C24C22×D512yes—
187C2×D4—22no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—104no—
193C2×D4—102no—
194C2×D4—202no—
195C2×D4—202no—
196C2×D4—202no—
197C22×C4—51no—
198C22×C4—54no—
199C22×C4—51no—
200C22×C4—52no—
201C22×C4—51no—
202C22×C4—52no—
203C22×C4—202no—
204C22×C4C22×D512yes—
205C22×C4C22×D511yescharacteristic
206C22×C4C22×D511yescharacteristic
207C4⋊C4—102no—
208C4⋊C4—102no—
209C4⋊C4—102no—
210C4⋊C4—202no—
211C4⋊C4—202no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—102no—
215C22⋊C4—104no—
216C22⋊C4—102no—
217C22⋊C4—202no—
218C22⋊C4—202no—
219C22⋊C4—202no—
220C22⋊C4—202no—
221C2×C5—81no—
222C2×C5—82no—
223C2×C5#53411yescharacteristic
224C2×C5#53511yescharacteristic
225C2×C5#53711yescharacteristic
226C2×C5#56012yes—
227C2×C5#56212yes—
228D5—42no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—21no—
240C23—22no—
241C23—52no—
242C23—101no—
243C23—102no—
244C23—102no—
245C23—101no—
246C23C23×D511yescharacteristic
247D4—44no—
248D4—48no—
249D4—208no—
250D4—204no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—52no—
258C2×C4—102no—
259C2×C4—101no—
260C2×C4—102no—
261C2×C4—101no—
262C2×C4—102no—
263C2×C4—101no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—102no—
269C2×C4—101no—
270C2×C4—204no—
271C2×C4D4×D512yes—
272C5#281311yescharacteristic, complemented, Sylow
273C22—42no—
274C22—41no—
275C22—42no—
276C22—41no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—102no—
286C22—102no—
287C22—102no—
288C22—202no—
289C22—202no—
290C22C2×D4×D511yescharacteristic
291C22#414412yes—
292C22#414412yes—
293C22#414411yescharacteristic
294C22#415011yescharacteristic
295C4—22no—
296C4—41no—
297C4—42no—
298C4—102no—
299C4—202no—
300C4—201no—
301C4—202no—
302C4—201no—
303C2—82no—
304C2—81no—
305C2—202no—
306C2#6592311yescharacteristic
307C2#6592912yes—
308C2#6594211yescharacteristic
309C2#6595512yes—
310C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#392C2
hypercenter#392D5
hypocenterC5#2813
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#4084C2semidirect
2. C2×#4029C2semidirect
3. C5×#392C2semidirect
4. #65512C2semidirect
5. #65511C2semidirect
6. #65511C2semidirect
7. #65248C2semidirect
8. #65238C2semidirect
9. #65237C2semidirect
10. #64956C2semidirect
11. #64948C2semidirect
60. C2×C4×D5D4semidirect
84. #392D5semidirect
272. C5#2813semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C2×#40292
C5×#3922
#655122
#655112
#652482
#652382
#652372
#649562
#649482
#28135