Finite Group Databasebeta

← #10607962

Subgroups — #10607962

Order 640.

624 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4040C212yesmaximal
2C2×D5×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C4×D20C211yesmaximal, characteristic
4C5×#389C211yesmaximal, characteristic, complemented
5#65276C211yesmaximal, characteristic, complemented
6#65276C211yesmaximal, characteristic, complemented
7#65276C211yesmaximal, characteristic, complemented
8#65273C212yesmaximal, complemented
9#65217C211yesmaximal, characteristic, complemented
10#64964C212yesmaximal, complemented
11#64953C212yesmaximal, complemented
12C22×D20C2211yescharacteristic
13C22×C4×D5C2211yescharacteristic
14C22×C4×D5C2212yes—
15C22×C4×D5C2211yescharacteristic
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C4⋊C4C2212yes—
18C2×C5×C4⋊C4C2211yescharacteristic
19C2×C42×C5C2211yescharacteristic
20C2×#685C2211yescharacteristic
21C2×#685C2211yescharacteristic
22C2×#685C2212yes—
23C2×#685C2212yes—
24C2×#685C2212yes—
25C2×C4⋊Dic5C2211yescharacteristic
26C2×C4⋊Dic5C2211yescharacteristic
27C2×C4⋊Dic5C2211yescharacteristic
28C2×#683C2212yes—
29C2×#683C2212yes—
30#4040—28no—
31D5×C4⋊C4—24no—
32C4×D20—24no—
33C5×#95C2212yes—
34#3963C2212yes—
35#3963C2212yes—
36#3963C2212yes—
37#3963C2212yes—
38#2486—51nomaximal, Sylow
39C23×D5C2311yescharacteristic
40C23×D5C2311yescharacteristic
41C22×C4×C5C2311yescharacteristic
42C22×C4×C5C2312yes—
43C22×C4×C5C2312yes—
44C22×C4×C5C2311yescharacteristic
45C22×C4×C5C2311yescharacteristic
46C22×Dic5C2311yescharacteristic
47C22×Dic5C2312yes—
48C22×Dic5C2312yes—
49C22×Dic5C2311yescharacteristic
50C2×D20—22no—
51C2×D20—22no—
52C2×D20D414yes—
53C2×C4×D5—22no—
54C2×C4×D5—22no—
55C2×C4×D5—24no—
56C2×C4×D5—24no—
57C2×C4×D5—24no—
58C2×C4×D5—22no—
59C2×C4×D5—22no—
60C2×C4×D5—22no—
61C2×C4×D5D414yes—
62C5×C4⋊C4—22no—
63C5×C4⋊C4—24no—
64C5×C4⋊C4—22no—
65C42×C5—22no—
66#685—22no—
67#685—24no—
68#685—24no—
69#685—24no—
70#685—22no—
71C4⋊Dic5—22no—
72C4⋊Dic5—22no—
73C4⋊Dic5—22no—
74#683—24no—
75#683—24no—
76C2×#123—52no—
77C2×C4×D4—51no—
78C22×C4⋊C4—51no—
79#398—52no—
80#397—52no—
81#389—51no—
82#389D511yescharacteristic, complemented
83#381—52no—
84#380—51no—
85#380—51no—
86#380—51no—
87C23×C5C2411yescharacteristic
88C22×D5—21no—
89C22×D5—21no—
90C22×D5—21no—
91C22×D5—21no—
92C22×D5—21no—
93C22×D5—21no—
94C22×D5—21no—
95C22×D5—21no—
96C22×D5—21no—
97C22×D5—21no—
98C22×D5C2×D412yes—
99C22×D5C2×D412yes—
100C22×D5C2×D412yes—
101C22×D5#4112yes—
102C2×C4×C5—21no—
103C2×C4×C5—21no—
104C2×C4×C5—22no—
105C2×C4×C5—21no—
106C2×C4×C5—22no—
107C2×C4×C5—22no—
108C2×C4×C5—22no—
109C2×C4×C5—22no—
110C2×C4×C5—21no—
111C2×C4×C5—21no—
112C2×C4×C5—22no—
113C2×C4×C5C2×D412yes—
114C2×C4×C5C2×D412yes—
115C2×C4×C5#4112yes—
116C2×C4×C5#4112yes—
117C2×Dic5—21no—
118C2×Dic5—21no—
119C2×Dic5—22no—
120C2×Dic5—22no—
121C2×Dic5—22no—
122C2×Dic5—21no—
123C2×Dic5—22no—
124C2×Dic5—21no—
125C2×Dic5—21no—
126C2×Dic5—22no—
127C2×Dic5—22no—
128C2×Dic5C2×D412yes—
129D20—28no—
130C4×D5—28no—
131C4×D5—48no—
132C4×D5—44no—
133C22×D4—51no—
134C23×C4—51no—
135C23×C4—52no—
136C23×C4—51no—
137#123—108no—
138C4×D4—104no—
139C2×C4⋊C4—51no—
140C2×C4⋊C4—52no—
141C2×C4⋊C4—51no—
142C2×C4⋊C4—52no—
143C2×C4⋊C4—51no—
144C2×C4⋊C4—104no—
145C2×C4⋊C4C2×D511yescharacteristic
146C2×C4⋊C4C2×D512yes—
147C2×C4⋊C4C2×D511yescharacteristic
148C2×C22⋊C4—51no—
149C2×C22⋊C4—51no—
150C2×C22⋊C4—52no—
151C2×C22⋊C4—52no—
152C2×C22⋊C4—52no—
153C2×C42C2×D511yescharacteristic
154#95—52no—
155#95—52no—
156#95—52no—
157#95—52no—
158#95C2×D512yes—
159C22×C5C22×D411yescharacteristic
160C22×C5C22×D411yescharacteristic
161C22×C5C2×#4111yescharacteristic
162C22×C5C2×#4111yescharacteristic
163C22×C5C2×#4111yescharacteristic
164C22×C52−1+411yescharacteristic
165C22×C52−1+411yescharacteristic
166C2×D5—22no—
167C2×D5—22no—
168C2×D5—22no—
169C2×D5—22no—
170C2×D5—22no—
171C2×D5—22no—
172C2×D5—22no—
173C2×D5—22no—
174C2×D5—22no—
175C2×D5—41no—
176C2×D5—41no—
177C2×D5—41no—
178C2×D5—41no—
179C2×D5#12114yes—
180C4×C5—22no—
181C4×C5—42no—
182C4×C5—41no—
183C4×C5—42no—
184C4×C5#12314yescomplemented
185Dic5—22no—
186Dic5—42no—
187Dic5—41no—
188Dic5—42no—
189C24—51no—
190C24—51no—
191C2×D4—54no—
192C2×D4—102no—
193C2×D4—102no—
194C22×C4—51no—
195C22×C4—54no—
196C22×C4—51no—
197C22×C4—52no—
198C22×C4—52no—
199C22×C4—104no—
200C22×C4—104no—
201C22×C4—104no—
202C22×C4—102no—
203C22×C4—102no—
204C22×C4—102no—
205C22×C4—102no—
206C22×C4—102no—
207C22×C4C22×D511yescharacteristic
208C22×C4C22×D512yes—
209C22×C4C22×D512yes—
210C22×C4C22×D511yescharacteristic
211C22×C4C22×D511yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—24no—
215C4⋊C4—104no—
216C4⋊C4—102no—
217C4⋊C4—102no—
218C4⋊C4—104no—
219C4⋊C4—102no—
220C22⋊C4—102no—
221C22⋊C4—102no—
222C22⋊C4—104no—
223C22⋊C4—104no—
224C22⋊C4—104no—
225C42—22no—
226C2×C5C2×#12111yescharacteristic
227C2×C5C2×#12311yescharacteristic
228C2×C5#53611yescharacteristic
229C2×C5#54711yescharacteristic
230C2×C5#54711yescharacteristic
231C2×C5#55211yescharacteristic
232C2×C5#55211yescharacteristic
233D5—24no—
234D5—42no—
235C23—52no—
236C23—52no—
237C23—52no—
238C23—52no—
239C23—101no—
240C23—101no—
241C23—101no—
242C23—101no—
243C23—101no—
244C23—101no—
245C23—101no—
246C23—101no—
247C23—101no—
248C23—101no—
249C23C23×D511yescharacteristic
250D4—108no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—52no—
263C2×C4—101no—
264C2×C4—101no—
265C2×C4—108no—
266C2×C4—101no—
267C2×C4—102no—
268C2×C4—101no—
269C2×C4—102no—
270C2×C4—102no—
271C2×C4—102no—
272C2×C4—101no—
273C2×C4—102no—
274C2×C4—102no—
275C2×C4—208no—
276C2×C4—204no—
277C2×C4#70912yes—
278C2×C4D4×D512yes—
279C2×C4D4×D512yes—
280C2×C4#71312yes—
281C5#248611yescharacteristic, complemented, Sylow
282C22—54no—
283C22—102no—
284C22—102no—
285C22—102no—
286C22—102no—
287C22—102no—
288C22—102no—
289C22—102no—
290C22—102no—
291C22—102no—
292C22—201no—
293C22—201no—
294C22—201no—
295C22—201no—
296C22C2×#70911yescharacteristic
297C22C2×D4×D511yescharacteristic
298C22C2×D4×D511yescharacteristic
299C22C2×#71311yescharacteristic
300C22#414711yescharacteristic
301C22D5×#4111yescharacteristic
302C22#415011yescharacteristic
303C4—22no—
304C4—42no—
305C4—42no—
306C4—41no—
307C4—102no—
308C4—202no—
309C4—201no—
310C4—202no—
311C4#404014yescomplemented
312C2—104no—
313C2—202no—
314C2C2×#404011yescharacteristic
315C2#6588511yescharacteristic
316C2#6591511yescharacteristic
317C2D5×#12111yescharacteristic
318C2#6596211yescharacteristic
319C2#6602811yescharacteristic
320C2#6606011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#389C2
prime radicalC22×D20C22
hypercenter#389D5
hypocenterC5#2486
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊C4C2semidirect
4. C5×#389C2semidirect
5. #65276C2semidirect
6. #65276C2semidirect
7. #65276C2semidirect
8. #65273C2semidirect
9. #65217C2semidirect
10. #64964C2semidirect
11. #64953C2semidirect
82. #389D5semidirect
184. C4×C5#123semidirect
281. C5#2486semidirect
311. C4#4040semidirect

Maximal subgroups

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

subgroupindex
C2×#40402
C2×D5×C4⋊C42
C2×C4×D202
C5×#3892
#652762
#652732
#652172
#649642
#649532
#24865