Finite Group Databasebeta

← #10610072

Subgroups — #10610072

Order 640.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4040C211yesmaximal, characteristic
2C2×#4039C211yesmaximal, characteristic
3C5×#400C211yesmaximal, characteristic, complemented
4#65269C211yesmaximal, characteristic, complemented
5#65267C211yesmaximal, characteristic, complemented
6#64964C211yesmaximal, characteristic, complemented
7#64964C211yesmaximal, characteristic, complemented
8#64957C211yesmaximal, characteristic, complemented
9#64957C211yesmaximal, characteristic, complemented
10#64956C211yesmaximal, characteristic, complemented
11#64954C211yesmaximal, characteristic
12#64953C211yesmaximal, characteristic, complemented
13#64953C211yesmaximal, characteristic, complemented
14D5×#95C211yesmaximal, characteristic, complemented
15#64942C211yesmaximal, characteristic, complemented
16C22×D20C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×#685C2211yescharacteristic
24C2×#685C2211yescharacteristic
25C2×#685C2211yescharacteristic
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×C4⋊Dic5C2211yescharacteristic
32C2×#683C2211yescharacteristic
33C2×#683C2211yescharacteristic
34C2×#683C2211yescharacteristic
35C2×#683C2211yescharacteristic
36C2×C4×Dic5C2211yescharacteristic
37#4040—24no—
38#4039—24no—
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41C5×#95C2211yescharacteristic
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44#3963C2211yescharacteristic
45#3963C2211yescharacteristic
46#3963C2211yescharacteristic
47#3963C2211yescharacteristic
48#3963C2211yescharacteristic
49#3963C2211yescharacteristic
50#3963C2211yescharacteristic
51#3963C2211yescharacteristic
52#3963C2211yescharacteristic
53#2419—51nomaximal, Sylow
54C23×D5C2311yescharacteristic
55C23×D5C2311yescharacteristic
56C22×C4×C5C2311yescharacteristic
57C22×C4×C5C2311yescharacteristic
58C22×C4×C5C2311yescharacteristic
59C22×C4×C5C2311yescharacteristic
60C22×C4×C5C2311yescharacteristic
61C22×C4×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×Dic5C2311yescharacteristic
64C22×Dic5C2311yescharacteristic
65C22×Dic5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C2×D20—22no—
70C2×D20—22no—
71C2×D20D414yes—
72C2×C4×D5—22no—
73C2×C4×D5—22no—
74C2×C4×D5—22no—
75C2×C4×D5—22no—
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—22no—
79C2×C4×D5—22no—
80C2×C4×D5—22no—
81C2×C4×D5—41no—
82C2×C4×D5—41no—
83C2×C4×D5—41no—
84C5×C4⋊C4—22no—
85C5×C4⋊C4—22no—
86#685—22no—
87#685—22no—
88#685—22no—
89#685—22no—
90#685—22no—
91#685—22no—
92#685—22no—
93#685—41no—
94C4⋊Dic5—22no—
95#683—22no—
96#683—22no—
97#683—41no—
98#683—41no—
99C4×Dic5—22no—
100C2×#123—51no—
101C2×C4×D4—51no—
102#400D511yescharacteristic, complemented
103#398—51no—
104#397—51no—
105#397—51no—
106#387—51no—
107#387—51no—
108#386—51no—
109#385—51no—
110#385—51no—
111#381—51no—
112#381—51no—
113#380—51no—
114C2×#95—51no—
115C23×C5C2411yescharacteristic
116C22×D5—21no—
117C22×D5—21no—
118C22×D5—21no—
119C22×D5—21no—
120C22×D5—21no—
121C22×D5—21no—
122C22×D5—21no—
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—21no—
126C22×D5C2×D412yes—
127C22×D5C2×D412yes—
128C22×D5#4112yes—
129C22×D5#4112yes—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5C2×D412yes—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5#4112yes—
168C2×Dic5#4112yes—
169D20—28no—
170C4×D5—44no—
171C4×D5—44no—
172C4×D5—44no—
173C4×D5—44no—
174C22×D4—51no—
175C23×C4—51no—
176C23×C4—51no—
177C23×C4—51no—
178C23×C4—51no—
179#123—104no—
180C4×D4—104no—
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4C2×D511yescharacteristic
187C2×C4⋊C4C2×D511yescharacteristic
188C2×C22⋊C4—51no—
189C2×C22⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4—51no—
194C2×C22⋊C4—51no—
195C2×C22⋊C4—51no—
196C2×C42—51no—
197#95—51no—
198#95—51no—
199#95—51no—
200#95—51no—
201#95—51no—
202#95—51no—
203#95—51no—
204#95—51no—
205#95—51no—
206#95C2×D511yescharacteristic
207#95C2×D511yescharacteristic
208#95C2×D511yescharacteristic
209#95C2×D511yescharacteristic
210#95C2×D511yescharacteristic
211C22×C5C22×D411yescharacteristic
212C22×C5C2×#4111yescharacteristic
213C22×C5C2×#4111yescharacteristic
214C22×C5C2×#4111yescharacteristic
215C22×C5C2×#4111yescharacteristic
216C22×C5H(5,𝔽2)11yescharacteristic
217C22×C52−1+411yescharacteristic
218C2×D5—22no—
219C2×D5—22no—
220C2×D5—22no—
221C2×D5—22no—
222C2×D5—22no—
223C2×D5—22no—
224C2×D5—22no—
225C2×D5—22no—
226C2×D5—22no—
227C2×D5—41no—
228C2×D5—41no—
229C2×D5—41no—
230C2×D5—41no—
231C2×D5#12314yes—
232C4×C5—22no—
233C4×C5—41no—
234C4×C5—41no—
235C4×C5—41no—
236C4×C5—41no—
237C4×C5—41no—
238C4×C5—41no—
239Dic5—22no—
240Dic5—22no—
241Dic5—41no—
242Dic5—41no—
243Dic5—41no—
244Dic5—41no—
245C24—51no—
246C24—51no—
247C2×D4—54no—
248C2×D4—102no—
249C2×D4—102no—
250C22×C4—51no—
251C22×C4—51no—
252C22×C4—51no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—51no—
256C22×C4—102no—
257C22×C4—102no—
258C22×C4—102no—
259C22×C4—102no—
260C22×C4—102no—
261C22×C4—102no—
262C22×C4—102no—
263C22×C4—102no—
264C22×C4—102no—
265C22×C4—201no—
266C22×C4—201no—
267C22×C4—201no—
268C22×C4C22×D511yescharacteristic
269C22×C4C22×D511yescharacteristic
270C22×C4C22×D511yescharacteristic
271C22×C4C22×D511yescharacteristic
272C22×C4C22×D511yescharacteristic
273C22×C4C22×D511yescharacteristic
274C22×C4C22×D511yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—102no—
278C4⋊C4—102no—
279C4⋊C4—102no—
280C4⋊C4—201no—
281C4⋊C4—201no—
282C22⋊C4—102no—
283C22⋊C4—102no—
284C22⋊C4—102no—
285C22⋊C4—102no—
286C22⋊C4—102no—
287C22⋊C4—102no—
288C22⋊C4—102no—
289C22⋊C4—201no—
290C42—102no—
291C2×C5C2×#12311yescharacteristic
292C2×C5#52911yescharacteristic
293C2×C5#53911yescharacteristic
294C2×C5#54611yescharacteristic
295C2×C5#54711yescharacteristic
296C2×C5#55111yescharacteristic
297C2×C5#55211yescharacteristic
298D5—24no—
299D5—42no—
300C23—52no—
301C23—52no—
302C23—52no—
303C23—52no—
304C23—101no—
305C23—101no—
306C23—101no—
307C23—101no—
308C23—101no—
309C23—101no—
310C23—101no—
311C23—101no—
312C23—101no—
313C23—101no—
314C23C23×D511yescharacteristic
315D4—108no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—52no—
337C2×C4—52no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—101no—
342C2×C4—101no—
343C2×C4—101no—
344C2×C4—101no—
345C2×C4—101no—
346C2×C4—101no—
347C2×C4—101no—
348C2×C4—101no—
349C2×C4—101no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—101no—
353C2×C4—101no—
354C2×C4—204no—
355C2×C4—204no—
356C2×C4—204no—
357C2×C4—204no—
358C2×C4D4×D512yes—
359C5#241911yescharacteristic, complemented, Sylow
360C22—54no—
361C22—102no—
362C22—102no—
363C22—102no—
364C22—102no—
365C22—102no—
366C22—102no—
367C22—102no—
368C22—102no—
369C22—102no—
370C22—201no—
371C22—201no—
372C22—201no—
373C22—201no—
374C22C2×D4×D511yescharacteristic
375C22#414711yescharacteristic
376C22D5×#4111yescharacteristic
377C22D5×#4111yescharacteristic
378C22D5×#4111yescharacteristic
379C22D5×#4111yescharacteristic
380C22#414911yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—102no—
389C4—102no—
390C4—201no—
391C4—201no—
392C4—201no—
393C4—201no—
394C2—104no—
395C2—202no—
396C2#6596511yescharacteristic
397C2D5×#12311yescharacteristic
398C2#6601011yescharacteristic
399C2#6602611yescharacteristic
400C2#6602711yescharacteristic
401C2#6603911yescharacteristic
402C2#6604011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC22×D20C22
hypercenter#400D5
hypocenterC5#2419
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C5×#400C2semidirect
4. #65269C2semidirect
5. #65267C2semidirect
6. #64964C2semidirect
7. #64964C2semidirect
8. #64957C2semidirect
9. #64957C2semidirect
10. #64956C2semidirect
12. #64953C2semidirect
13. #64953C2semidirect
14. D5×#95C2semidirect
15. #64942C2semidirect
102. #400D5semidirect
359. C5#2419semidirect

Maximal subgroups

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

subgroupindex
C2×#40402
C2×#40392
C5×#4002
#652692
#652672
#649642
#649572
#649562
#649542
#649532
D5×#952
#649422
#24195