Finite Group Databasebeta

← #10617218

Subgroups — #10617218

Order 640.

412 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#65920C211yesmaximal, characteristic
2#65798C211yesmaximal, characteristic, complemented
3C2×C22⋊#170C211yesmaximal, characteristic, complemented
4#65776C211yesmaximal, characteristic, complemented
5#65775C211yesmaximal, characteristic, complemented
6#65774C211yesmaximal, characteristic, complemented
7#65773C211yesmaximal, characteristic, complemented
8#65773C211yesmaximal, characteristic, complemented
9D4×#170C211yesmaximal, characteristic, complemented
10#65697C211yesmaximal, characteristic, complemented
11#65697C211yesmaximal, characteristic, complemented
12#65695C211yesmaximal, characteristic, complemented
13#65694C211yesmaximal, characteristic, complemented
14#65693C211yesmaximal, characteristic, complemented
15#65692C211yesmaximal, characteristic, complemented
16C23×Dic5C2211yescharacteristic
17C2×#711C2211yescharacteristic
18C2×#704C2211yescharacteristic
19C2×#704C2211yescharacteristic
20C2×#704C2211yescharacteristic, complemented
21C22×#170C2211yescharacteristic
22C22×#170C2211yescharacteristic
23C22×#170C2211yescharacteristic, complemented
24C2×#700C2211yescharacteristic, complemented
25C2×D5⋊C8C2211yescharacteristic, complemented
26C5×#120C411yescharacteristic
27#4099C411yescharacteristic
28#4085C411yescharacteristic
29#4085C411yescharacteristic
30#4081C411yescharacteristic
31D4×Dic5C2211yescharacteristic
32D4×Dic5C2211yescharacteristic
33#4032C411yescharacteristic
34#4027C2211yescharacteristic
35#4027C2211yescharacteristic
36#4024C411yescharacteristic
37#4024C411yescharacteristic
38#4023C2211yescharacteristic
39C22⋊#170C2211yescharacteristic
40C22⋊#170C2211yescharacteristic
41C22⋊#170C2211yescharacteristic
42C22⋊#170C2211yescharacteristic
43C22⋊#170C2211yescharacteristic
44C22⋊#170C2211yescharacteristic
45C22⋊#170C2211yescharacteristic
46C22⋊#170C2211yescharacteristic
47C22⋊#170C2211yescharacteristic, complemented
48C22⋊#170C2211yescharacteristic, complemented
49#4004C2211yescharacteristic, complemented
50#4004C2211yescharacteristic, complemented
51#4003C2211yescharacteristic, complemented
52#4003C2211yescharacteristic, complemented
53#4002C2211yescharacteristic, complemented
54#4002C2211yescharacteristic, complemented
55C4⋊#170C2211yescharacteristic, complemented
56C4⋊#170C2211yescharacteristic, complemented
57C4×#170C2211yescharacteristic, complemented
58C4×#170C2211yescharacteristic, complemented
59#2976—51nomaximal, Sylow
60C24×C5C2×C411yescharacteristic
61C2×C5×D4C2×C411yescharacteristic
62C2×C5×D4C2×C411yescharacteristic
63C2×C5×D4C2×C411yescharacteristic
64C2×C22⋊D5C2×C411yescharacteristic
65C2×C22⋊D5C2×C411yescharacteristic
66C2×C22⋊D5C2×C411yescharacteristic
67C2×C22⋊D5C2×C411yescharacteristic
68C22×Dic5—41no—
69C22×Dic5—41no—
70C22×Dic5C2×C411yescharacteristic
71C22×Dic5C2×C411yescharacteristic
72C22×Dic5C2×C411yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77#711—41no—
78#711—41no—
79C2×C4×D5C2311yescharacteristic
80C2×Dic10C2×C411yescharacteristic
81#704—41no—
82#704—41no—
83#704—41no—
84C2×#170—41no—
85C2×#170—41no—
86C2×#170—41no—
87C2×#170C2311yescharacteristic
88C2×#170C2311yescharacteristic
89C2×#170C2311yescharacteristic
90C2×#170C2311yescharacteristic
91C2×#170C2311yescharacteristic
92C2×#170C2311yescharacteristic
93C2×#170C2311yescharacteristic
94C2×#170C2311yescharacteristic
95#700—41no—
96D5⋊C8—41no—
97C5×C22⋊C4C2×C411yescharacteristic
98C5×C22⋊C4C2×C411yescharacteristic
99C5×C22⋊C4C2×C411yescharacteristic
100C22⋊Dic5C2×C411yescharacteristic
101C22⋊Dic5C2×C411yescharacteristic
102C22⋊Dic5C2×C411yescharacteristic
103C22⋊Dic5C2×C411yescharacteristic
104C22⋊Dic5C2×C411yescharacteristic
105#685C2×C411yescharacteristic
106#685C2×C411yescharacteristic
107C4⋊Dic5C2×C411yescharacteristic
108C4⋊Dic5C2×C411yescharacteristic
109#683C2×C411yescharacteristic
110#683C2×C411yescharacteristic
111#683C2×C411yescharacteristic
112#683C2×C411yescharacteristic
113C4×Dic5C2311yescharacteristic
114C4×Dic5C2311yescharacteristic
115#525—51no—
116#436—51no—
117#436—51no—
118#435—51no—
119#435—51no—
120#435—51no—
121#435—51no—
122C8×D4—51no—
123C8×D4—51no—
124#433—51no—
125#433—51no—
126#408—51no—
127#408—51no—
128#407—51no—
129C2×C22⋊C8—51no—
130C23×C5—22no—
131C23×C5—41no—
132C23×C5—41no—
133C23×C5C22×C411yescharacteristic
134C23×C5C22×C411yescharacteristic
135C23×C5C22×C411yescharacteristic
136C23×C5C22×C411yescharacteristic
137C22×D5C22×C411yescharacteristic
138C5×D4—22no—
139C5×D4—41no—
140C5×D4—41no—
141C2×C4×C5C22×C411yescharacteristic
142C2×C4×C5C22×C411yescharacteristic
143C2×C4×C5C22×C411yescharacteristic
144C22⋊D5—22no—
145C22⋊D5—41no—
146C22⋊D5—41no—
147C22⋊D5—41no—
148C2×Dic5—21no—
149C2×Dic5—21no—
150C2×Dic5—21no—
151C2×Dic5—21no—
152C2×Dic5—21no—
153C2×Dic5—21no—
154C2×Dic5—21no—
155C2×Dic5—21no—
156C2×Dic5—41no—
157C2×Dic5—41no—
158C2×Dic5—41no—
159C2×Dic5C22×C411yescharacteristic
160C2×Dic5C22×C411yescharacteristic
161C2×Dic5C22×C411yescharacteristic
162C2×Dic5C22×C411yescharacteristic
163C2×Dic5C22×C411yescharacteristic
164C2×Dic5C22×C411yescharacteristic
165C2×Dic5C2411yescharacteristic
166C4×D5—21no—
167C4×D5—21no—
168Dic10—41no—
169#170—21no—
170#170—21no—
171#170—21no—
172#170—21no—
173#170—21no—
174#170—21no—
175#170—21no—
176#170—21no—
177C2×#41—51no—
178C23×C4—51no—
179C2×M16—51no—
180C2×M16—51no—
181C2×M16—51no—
182C2×M16—51no—
183C22×C8—51no—
184C22×C8—51no—
185C22×C8—51no—
186C22×C8—51no—
187#123—51no—
188#123—51no—
189C22⋊Q8—51no—
190C22⋊Q8—51no—
191#121—51no—
192#121—51no—
193#120—51no—
194#120AGL(1,𝔽5)11yescharacteristic
195C4×D4—51no—
196C4×D4—51no—
197C4×D4—51no—
198C4×D4—51no—
199#117—51no—
200C4⋊C8—51no—
201C4⋊C8—51no—
202C4⋊C8—51no—
203C4⋊C8—51no—
204C22⋊C8—51no—
205C22⋊C8—51no—
206C22⋊C8—51no—
207C22⋊C8—51no—
208C22⋊C8—51no—
209C22⋊C8—51no—
210C22⋊C8—51no—
211C22⋊C8—51no—
212C22⋊C8—51no—
213C22⋊C8—51no—
214C22⋊C8—51no—
215C22⋊C8—51no—
216#97—51no—
217#97—51no—
218C4×C8—51no—
219C4×C8—51no—
220C22×C5—21no—
221C22×C5—21no—
222C22×C5—21no—
223C22×C5—21no—
224C22×C5—21no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—21no—
228C22×C5—21no—
229C22×C5—21no—
230C22×C5—21no—
231C22×C5—44no—
232C22×C5#13112yes—
233C22×C5C23×C411yescharacteristic
234C2×D5—21no—
235C2×D5—21no—
236C2×D5—21no—
237C4×C5—21no—
238C4×C5—21no—
239C4×C5—21no—
240Dic5—21no—
241Dic5—21no—
242Dic5—21no—
243Dic5—21no—
244Dic5—21no—
245Dic5—21no—
246Dic5H(5,𝔽2)11yescharacteristic
247Dic5H(5,𝔽2)11yescharacteristic
248C24C2×AGL(1,𝔽5)11yescharacteristic
249#41—201no—
250#41—201no—
251C2×Q8—51no—
252C2×D4—51no—
253C2×D4—51no—
254C2×D4—51no—
255C2×D4—51no—
256C2×D4C2×AGL(1,𝔽5)11yescharacteristic
257C2×D4C2×AGL(1,𝔽5)11yescharacteristic
258C2×D4C2×AGL(1,𝔽5)11yescharacteristic
259C22×C4—51no—
260C22×C4—51no—
261C22×C4—51no—
262C22×C4—51no—
263C22×C4—51no—
264C22×C4—51no—
265C22×C4—51no—
266C22×C4—51no—
267C22×C4—201no—
268C22×C4—201no—
269M16—201no—
270M16—201no—
271M16—201no—
272M16—201no—
273C2×C8—51no—
274C2×C8—51no—
275C2×C8—51no—
276C2×C8—51no—
277C2×C8—51no—
278C2×C8—51no—
279C2×C8—51no—
280C2×C8—51no—
281C2×C8—201no—
282C2×C8—201no—
283C2×C8—201no—
284C2×C8—201no—
285C4⋊C4—51no—
286C4⋊C4—51no—
287C4⋊C4—51no—
288C4⋊C4—51no—
289C4⋊C4—51no—
290C4⋊C4—51no—
291C22⋊C4—51no—
292C22⋊C4—51no—
293C22⋊C4—51no—
294C22⋊C4—51no—
295C22⋊C4—51no—
296C22⋊C4—51no—
297C22⋊C4—51no—
298C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
299C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
300C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
301C42—51no—
302C42—51no—
303C2×C5—22no—
304C2×C5—41no—
305C2×C5—41no—
306C2×C5—41no—
307C2×C5#51811yescharacteristic
308C2×C5C2×#13111yescharacteristic
309C2×C5#56811yescharacteristic
310D5—41no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—51no—
315C23C22×AGL(1,𝔽5)11yescharacteristic
316C23C22×AGL(1,𝔽5)11yescharacteristic
317C23C22×AGL(1,𝔽5)11yescharacteristic
318C23C22×AGL(1,𝔽5)11yescharacteristic
319Q8—201no—
320D4—22no—
321D4—41no—
322D4—41no—
323D4—102no—
324D4—201no—
325D4—201no—
326D4—201no—
327C2×C4—51no—
328C2×C4—51no—
329C2×C4—51no—
330C2×C4—51no—
331C2×C4—51no—
332C2×C4—51no—
333C2×C4—51no—
334C2×C4—101no—
335C2×C4—101no—
336C2×C4—101no—
337C2×C4—101no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—101no—
341C2×C4—101no—
342C2×C4—101no—
343C2×C4—101no—
344C2×C4—201no—
345C2×C4—201no—
346C2×C4—201no—
347C2×C4C22×AGL(1,𝔽5)11yescharacteristic
348C2×C4C22×AGL(1,𝔽5)11yescharacteristic
349C2×C4C22×AGL(1,𝔽5)11yescharacteristic
350C8—101no—
351C8—101no—
352C8—101no—
353C8—101no—
354C8—101no—
355C8—101no—
356C8—101no—
357C8—101no—
358C5#297611yescharacteristic, complemented, Sylow
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22—101no—
372C22—101no—
373C22—101no—
374C22#413112yes—
375C22C23×AGL(1,𝔽5)11yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—51no—
380C4—51no—
381C4—101no—
382C4—101no—
383C4—101no—
384C4—101no—
385C4—101no—
386C4—101no—
387C2—22no—
388C2—41no—
389C2—41no—
390C2—41no—
391C2—201no—
392C2C2×#413111yescharacteristic
393C2#6625611yescharacteristic
394C2#6625811yescharacteristic
subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽5)
commutator, socleC22×C5C23×C4
FittingC5×#120C4
prime radical#65920C2
cosocleC2×Dic5C24
hypercenter#120AGL(1,𝔽5)
hypocenterC5#2976
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
2. #65798C2semidirect
3. C2×C22⋊#170C2semidirect
4. #65776C2semidirect
5. #65775C2semidirect
6. #65774C2semidirect
7. #65773C2semidirect
8. #65773C2semidirect
9. D4×#170C2semidirect
10. #65697C2semidirect
11. #65697C2semidirect
12. #65695C2semidirect
13. #65694C2semidirect
14. #65693C2semidirect
15. #65692C2semidirect
20. C2×#704C22semidirect
23. C22×#170C22semidirect
24. C2×#700C22semidirect
25. C2×D5⋊C8C22semidirect
47. C22⋊#170C22semidirect
48. C22⋊#170C22semidirect
49. #4004C22semidirect
50. #4004C22semidirect
51. #4003C22semidirect
52. #4003C22semidirect
53. #4002C22semidirect
54. #4002C22semidirect
55. C4⋊#170C22semidirect
56. C4⋊#170C22semidirect
57. C4×#170C22semidirect
58. C4×#170C22semidirect
358. C5#2976semidirect

Maximal subgroups

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

subgroupindex
#659202
#657982
C2×C22⋊#1702
#657762
#657752
#657742
#657732
D4×#1702
#656972
#656952
#656942
#656932
#656922
#29765