Finite Group Databasebeta

← #10611822

Subgroups — #10611822

Order 640.

466 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66162C211yesmaximal, characteristic, complemented
2D5×#121C211yesmaximal, characteristic, complemented
3C5×#449C211yesmaximal, characteristic, complemented
4#65527C211yesmaximal, characteristic, complemented
5#65463C211yesmaximal, characteristic, complemented
6#65458C211yesmaximal, characteristic, complemented
7#65448C211yesmaximal, characteristic, complemented
8#65445C211yesmaximal, characteristic, complemented
9#65324C211yesmaximal, characteristic, complemented
10#65101C211yesmaximal, characteristic, complemented
11#65094C211yesmaximal, characteristic, complemented
12#65071C211yesmaximal, characteristic, complemented
13#65063C211yesmaximal, characteristic, complemented
14#65026C211yesmaximal, characteristic, complemented
15#65015C211yesmaximal, characteristic, complemented
16C2×C5×#41C2211yescharacteristic, complemented
17C2×D4×D5—21no—
18C2×D4×D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C2×C5×SD16C2211yescharacteristic, complemented
21C2×C5×D8C2211yescharacteristic, complemented
22C5×#121C2211yescharacteristic
23#4092C2211yescharacteristic
24#4092C2211yescharacteristic, complemented
25C2×#688C2211yescharacteristic, complemented
26#4085—21no—
27#4084C2211yescharacteristic
28#4084C2211yescharacteristic
29#4084C2211yescharacteristic, complemented
30#4083—21no—
31#4081—21no—
32C2×#686C2211yescharacteristic, complemented
33#4076C2211yescharacteristic
34C4×C22⋊D5—21no—
35C4×C22⋊D5—21no—
36#4072C2211yescharacteristic, complemented
37C2×#676C2211yescharacteristic, complemented
38C2×#676C2211yescharacteristic, complemented
39#4041C2211yescharacteristic
40D5×C4⋊C4C2211yescharacteristic
41#4030—21no—
42D5×C22⋊C4—21no—
43C5×#103C2211yescharacteristic, complemented
44C5×#102C2211yescharacteristic, complemented
45C5×C22⋊C8C2211yescharacteristic, complemented
46#3967C2211yescharacteristic, complemented
47#3964C2211yescharacteristic, complemented
48#3964C2211yescharacteristic, complemented
49#3964C2211yescharacteristic, complemented
50C22⋊#168C2211yescharacteristic, complemented
51#3953C2211yescharacteristic, complemented
52#3953C2211yescharacteristic, complemented
53#3952C2211yescharacteristic
54#3952C2211yescharacteristic
55#3950C2211yescharacteristic, complemented
56#3949C2211yescharacteristic, complemented
57#3940C2211yescharacteristic, complemented
58#3939C2211yescharacteristic, complemented
59#3194—51nomaximal, Sylow
60C23×D5—21no—
61C23×D5D411yescharacteristic
62C5×#41—41no—
63C5×#41—41no—
64C2×C5×Q8C2311yescharacteristic
65C2×C5×D4—21no—
66C2×C5×D4—21no—
67C2×C5×D4C2311yescharacteristic
68C2×C5×D4C2311yescharacteristic
69C22×C4×C5—21no—
70C22×C4×C5C2311yescharacteristic
71C2×C22⋊D5—21no—
72C2×C22⋊D5—21no—
73C2×C22⋊D5—21no—
74C2×C22⋊D5—21no—
75C2×C22⋊D5—21no—
76C22×Dic5D411yescharacteristic
77D4×D5—41no—
78D4×D5—41no—
79D4×D5—44no—
80C2×D20—21no—
81C2×D20C2311yescharacteristic
82C2×C4×D5—21no—
83C2×C4×D5—41no—
84C2×C4×D5—41no—
85C2×C4×D5D411yescharacteristic
86C2×C4×D5D411yescharacteristic
87C2×C4×D5C2311yescharacteristic
88C2×C4×D5C2311yescharacteristic
89C5×SD16—41no—
90C5×D8—41no—
91C2×C5×C8C2311yescharacteristic
92C2×C5×C8C2311yescharacteristic
93C5×C4⋊C4C2311yescharacteristic
94C5×C22⋊C4—21no—
95C22⋊Dic5—21no—
96C22⋊Dic5—21no—
97C22⋊Dic5—21no—
98C22⋊Dic5—21no—
99#688—41no—
100#686—41no—
101#685—21no—
102#685—21no—
103#685—21no—
104#685—21no—
105C4⋊Dic5C2311yescharacteristic
106C4⋊Dic5C2311yescharacteristic
107C4⋊Dic5C2311yescharacteristic
108#683—21no—
109#683—21no—
110#683—21no—
111C4×Dic5—21no—
112C2×#168C2311yescharacteristic
113C2×#168C2311yescharacteristic
114#676—41no—
115#676—41no—
116#525—51no—
117C2×#121—51no—
118#483—51no—
119#482—51no—
120#482—51no—
121#480—51no—
122#469—51no—
123#469—51no—
124#468—51no—
125#468—51no—
126#450—51no—
127#449—51no—
128#449D511yescharacteristic, complemented
129#447—51no—
130#407—51no—
131C23×C5—21no—
132C23×C5—21no—
133C23×C5C2×D411yescharacteristic
134C22×D5—22no—
135C22×D5—21no—
136C22×D5—41no—
137C22×D5—41no—
138C22×D5—42no—
139C22×D5—42no—
140C22×D5—42no—
141C22×D5C2×D411yescharacteristic
142C22×D5C2×D411yescharacteristic
143C5×Q8—21no—
144C5×Q8—21no—
145C5×D4—21no—
146C5×D4—21no—
147C5×D4—21no—
148C5×D4—21no—
149C5×D4—42no—
150C5×D4—42no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—42no—
156C2×C4×C5C2×D411yescharacteristic
157C2×C4×C5C2411yescharacteristic
158C22⋊D5—42no—
159C22⋊D5—42no—
160C22⋊D5—42no—
161C22⋊D5—42no—
162C22⋊D5—42no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—41no—
167C2×Dic5C2×D411yescharacteristic
168C2×Dic5C2×D411yescharacteristic
169D20—21no—
170D20—21no—
171D20—42no—
172C4×D5—21no—
173C4×D5—21no—
174C4×D5—21no—
175C4×D5—21no—
176C4×D5—41no—
177C4×D5—41no—
178C4×D5—42no—
179C5×C8—21no—
180C5×C8—21no—
181#168—21no—
182#168—21no—
183C2×#41C2×D511yescharacteristic, complemented
184C22×D4—51no—
185C22×D4—101no—
186C23×C4—51no—
187C2×SD16—51no—
188C2×SD16C2×D511yescharacteristic, complemented
189C2×D8—51no—
190C2×D8C2×D511yescharacteristic, complemented
191C2×M16—51no—
192C2×M16—51no—
193#123—101no—
194C22⋊Q8—51no—
195C22⋊Q8—51no—
196#121—51no—
197#121—51no—
198#121—51no—
199#121—51no—
200#121—51no—
201#121—101no—
202#121—101no—
203#121C2×D511yescharacteristic
204#120—101no—
205C4×D4—101no—
206C4×D4—101no—
207#117—51no—
208C2×C4⋊C4—51no—
209C2×C22⋊C4—101no—
210#107—51no—
211#107—51no—
212#106—51no—
213#106—51no—
214#103—51no—
215#103C2×D511yescharacteristic, complemented
216#102—51no—
217#102—51no—
218#102—51no—
219#102—51no—
220#102—51no—
221#102C2×D511yescharacteristic, complemented
222C22⋊C8—51no—
223C22⋊C8—51no—
224C22⋊C8—51no—
225C22⋊C8C2×D511yescharacteristic, complemented
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—41no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5—41no—
233C22×C5—41no—
234C22×C5—41no—
235C22×C5C22×D411yescharacteristic
236C2×D5—21no—
237C2×D5—21no—
238C2×D5—21no—
239C2×D5—21no—
240C2×D5—21no—
241C2×D5—44no—
242C2×D5—41no—
243C2×D5—41no—
244C2×D5—41no—
245C2×D5—84no—
246C2×D5AGL(1,ℤ/8)12yes—
247C4×C5—21no—
248C4×C5—41no—
249C4×C5—41no—
250C4×C5H(5,𝔽2)11yescharacteristic
251C4×C5H(5,𝔽2)11yescharacteristic
252Dic5—21no—
253Dic5—21no—
254Dic5—41no—
255Dic5—41no—
256Dic5—41no—
257C24—51no—
258C24—101no—
259#41—41no—
260#41—41no—
261C2×Q8C22×D511yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—51no—
265C2×D4—101no—
266C2×D4—101no—
267C2×D4—101no—
268C2×D4—101no—
269C2×D4—101no—
270C2×D4—101no—
271C2×D4—201no—
272C2×D4—201no—
273C2×D4—204no—
274C2×D4C22×D511yescharacteristic
275C2×D4C22×D511yescharacteristic
276C22×C4—21no—
277C22×C4—51no—
278C22×C4—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—101no—
283C22×C4—201no—
284C22×C4—201no—
285C22×C4C22×D511yescharacteristic
286SD16—41no—
287SD16—201no—
288D8—41no—
289D8—201no—
290M16—201no—
291M16—201no—
292C2×C8—51no—
293C2×C8—51no—
294C2×C8C22×D511yescharacteristic
295C2×C8C22×D511yescharacteristic
296C4⋊C4—51no—
297C4⋊C4—51no—
298C4⋊C4—51no—
299C4⋊C4—101no—
300C4⋊C4—101no—
301C4⋊C4—101no—
302C4⋊C4C22×D511yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—101no—
305C22⋊C4—101no—
306C22⋊C4—101no—
307C22⋊C4—101no—
308C22⋊C4—101no—
309C22⋊C4—101no—
310C22⋊C4—101no—
311C22⋊C4—101no—
312C42—101no—
313C2×C5—41no—
314C2×C5—81no—
315C2×C5—81no—
316C2×C5#53411yescharacteristic
317C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
318C2×C5#57511yescharacteristic
319D5—22no—
320D5—41no—
321D5—81no—
322C23—21no—
323C23—21no—
324C23—51no—
325C23—51no—
326C23—102no—
327C23—101no—
328C23—201no—
329C23—201no—
330C23—202no—
331C23—202no—
332C23—202no—
333C23D4×D511yescharacteristic
334Q8—21no—
335Q8—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—42no—
341D4—42no—
342D4—101no—
343D4—101no—
344D4—202no—
345D4—202no—
346D4—202no—
347D4—202no—
348D4—202no—
349D4—202no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—51no—
356C2×C4—51no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—201no—
365C2×C4—201no—
366C2×C4—202no—
367C2×C4—201no—
368C2×C4D4×D511yescharacteristic
369C2×C4C23×D511yescharacteristic
370C8—21no—
371C8—21no—
372C8—101no—
373C8—101no—
374C5#319411yescharacteristic, complemented, Sylow
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—52no—
385C22—101no—
386C22—101no—
387C22—101no—
388C22—101no—
389C22—101no—
390C22—204no—
391C22—201no—
392C22—201no—
393C22—201no—
394C22—404no—
395C22C2×D4×D511yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—101no—
400C4—101no—
401C4—201no—
402C4—201no—
403C4—201no—
404C4#414411yescharacteristic
405C4#414411yescharacteristic
406C2—41no—
407C2—81no—
408C2—81no—
409C2—102no—
410C2—201no—
411C2—401no—
412C2#6592311yescharacteristic
413C2#6610211yescharacteristic
414C2D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#449C2
socleC22×C5C22×D4
hypercenter#449D5
hypocenterC5#3194
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #66162C2semidirect
2. D5×#121C2semidirect
3. C5×#449C2semidirect
4. #65527C2semidirect
5. #65463C2semidirect
6. #65458C2semidirect
7. #65448C2semidirect
8. #65445C2semidirect
9. #65324C2semidirect
10. #65101C2semidirect
11. #65094C2semidirect
12. #65071C2semidirect
13. #65063C2semidirect
14. #65026C2semidirect
15. #65015C2semidirect
16. C2×C5×#41C22semidirect
20. C2×C5×SD16C22semidirect
21. C2×C5×D8C22semidirect
24. #4092C22semidirect
25. C2×#688C22semidirect
29. #4084C22semidirect
32. C2×#686C22semidirect
36. #4072C22semidirect
37. C2×#676C22semidirect
38. C2×#676C22semidirect
43. C5×#103C22semidirect
44. C5×#102C22semidirect
45. C5×C22⋊C8C22semidirect
46. #3967C22semidirect
47. #3964C22semidirect
48. #3964C22semidirect
49. #3964C22semidirect
50. C22⋊#168C22semidirect
51. #3953C22semidirect
52. #3953C22semidirect
55. #3950C22semidirect
56. #3949C22semidirect
57. #3940C22semidirect
58. #3939C22semidirect
128. #449D5semidirect
183. C2×#41C2×D5semidirect
188. C2×SD16C2×D5semidirect
190. C2×D8C2×D5semidirect
215. #103C2×D5semidirect
221. #102C2×D5semidirect
225. C22⋊C8C2×D5semidirect
374. C5#3194semidirect

Maximal subgroups

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

subgroupindex
#661622
D5×#1212
C5×#4492
#655272
#654632
#654582
#654482
#654452
#653242
#651012
#650942
#650712
#650632
#650262
#650152
#31945