Finite Group Databasebeta

← #10609149

Subgroups — #10609149

Order 640.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4043C211yesmaximal, characteristic, complemented
2C2×#4042C211yesmaximal, characteristic, complemented
3C2×D5×C22⋊C4C211yesmaximal, characteristic
4C2×D5×C22⋊C4C211yesmaximal, characteristic
5C5×#396C211yesmaximal, characteristic, complemented
6#65278C211yesmaximal, characteristic, complemented
7#65278C211yesmaximal, characteristic, complemented
8#65278C211yesmaximal, characteristic, complemented
9#65276C211yesmaximal, characteristic, complemented
10#65244C211yesmaximal, characteristic
11#65238C211yesmaximal, characteristic, complemented
12#65237C211yesmaximal, characteristic, complemented
13#65236C211yesmaximal, characteristic, complemented
14#64957C211yesmaximal, characteristic, complemented
15#64956C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×C4×D5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×Dic10C2211yescharacteristic, complemented
22C2×C5×C4⋊C4C2211yescharacteristic, complemented
23C2×C5×C4⋊C4C2211yescharacteristic, complemented
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×C4⋊Dic5C2211yescharacteristic, complemented
40C2×C4⋊Dic5C2211yescharacteristic, complemented
41C2×#683C2211yescharacteristic, complemented
42C2×#683C2211yescharacteristic, complemented
43C2×#683C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45#4043—24no—
46#4042—24no—
47D5×C22⋊C4—24no—
48D5×C22⋊C4—24no—
49C5×#95C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2437—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C23×D5—22no—
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—41no—
62C23×D5—41no—
63C23×D5Q814yescomplemented
64C23×D5C2311yescharacteristic
65C23×D5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C2×C4×D5—22no—
79C2×C4×D5—22no—
80C2×C4×D5—22no—
81C2×C4×D5—22no—
82C2×C4×D5—22no—
83C2×C4×D5—22no—
84C2×C4×D5—22no—
85C2×C4×D5—22no—
86C2×C4×D5—41no—
87C2×C4×D5—41no—
88C2×C4×D5—41no—
89C2×C4×D5—41no—
90C2×Dic10—22no—
91C2×Dic10—22no—
92C2×Dic10—41no—
93C5×C4⋊C4—22no—
94C5×C4⋊C4—22no—
95C5×C4⋊C4—41no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—41no—
99C22⋊Dic5—22no—
100C22⋊Dic5—22no—
101C22⋊Dic5—41no—
102#685—22no—
103#685—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—22no—
108#685D414yes—
109#685D414yes—
110C4⋊Dic5—22no—
111C4⋊Dic5—22no—
112#683—22no—
113#683—22no—
114#683—41no—
115#683—41no—
116C2×C22⋊Q8—51no—
117C2×C22⋊Q8—51no—
118C22×C22⋊C4—51no—
119C22×C22⋊C4—51no—
120#399—51no—
121#396—51no—
122#396D511yescharacteristic, complemented
123#393—51no—
124#385—51no—
125#385—51no—
126#385—51no—
127#385—51no—
128#380—51no—
129#380—51no—
130#379—51no—
131C23×C5—21no—
132C23×C5—21no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5C2×Q812yes—
138C23×C5C2411yescharacteristic
139C22×D5—21no—
140C22×D5—21no—
141C22×D5—21no—
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—21no—
146C22×D5—21no—
147C22×D5—28no—
148C22×D5—28no—
149C22×D5—44no—
150C22×D5—44no—
151C22×D5—44no—
152C22×D5—44no—
153C22×D5—44no—
154C22×D5C2×D412yes—
155C22×D5C2×D412yes—
156C22×D5C2×Q812yes—
157C22×D5C2×Q812yes—
158C22×D5#4112yes—
159C22×D5#4112yes—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—21no—
166C2×C4×C5—21no—
167C2×C4×C5—21no—
168C2×C4×C5—21no—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—21no—
175C2×C4×C5—21no—
176C2×C4×C5C2×D412yes—
177C2×C4×C5C2×D412yes—
178C2×Dic5—21no—
179C2×Dic5—21no—
180C2×Dic5—21no—
181C2×Dic5—21no—
182C2×Dic5—21no—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5—21no—
186C2×Dic5—21no—
187C2×Dic5—21no—
188C2×Dic5—21no—
189C2×Dic5—21no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5C2×D412yes—
195C2×Dic5C2×D412yes—
196C4×D5—44no—
197C4×D5—44no—
198C4×D5—44no—
199C4×D5—44no—
200Dic10—44no—
201C25—51no—
202C22×Q8—51no—
203C23×C4—51no—
204C23×C4—51no—
205C23×C4—51no—
206C23×C4—51no—
207C22⋊Q8—104no—
208C22⋊Q8—104no—
209C2×C4⋊C4—51no—
210C2×C4⋊C4—51no—
211C2×C4⋊C4—51no—
212C2×C4⋊C4—51no—
213C2×C4⋊C4—51no—
214C2×C4⋊C4—51no—
215C2×C4⋊C4C2×D511yescharacteristic, complemented
216C2×C4⋊C4C2×D511yescharacteristic, complemented
217C2×C4⋊C4C2×D511yescharacteristic, complemented
218C2×C22⋊C4—51no—
219C2×C22⋊C4—51no—
220C2×C22⋊C4—51no—
221C2×C22⋊C4—51no—
222C2×C22⋊C4—51no—
223C2×C22⋊C4—51no—
224C2×C22⋊C4—51no—
225C2×C22⋊C4—51no—
226C2×C22⋊C4—51no—
227C2×C22⋊C4—51no—
228C2×C22⋊C4—51no—
229C2×C22⋊C4—104no—
230C2×C22⋊C4—104no—
231C2×C22⋊C4C2×D511yescharacteristic
232C2×C22⋊C4C2×D511yescharacteristic
233C2×C22⋊C4C2×D511yescharacteristic
234#95—51no—
235#95—51no—
236#95—51no—
237#95—51no—
238#95—51no—
239#95C2×D511yescharacteristic, complemented
240C22×C5—22no—
241C22×C5—22no—
242C22×C5—22no—
243C22×C5—41no—
244C22×C5—41no—
245C22×C5—41no—
246C22×C5—41no—
247C22×C5C22×D411yescharacteristic
248C22×C5C22×D411yescharacteristic
249C22×C5C22×Q811yescharacteristic
250C22×C5C2×#4111yescharacteristic
251C22×C5C2×#4111yescharacteristic
252C22×C5H(5,𝔽2)11yescharacteristic
253C22×C5H(5,𝔽2)11yescharacteristic
254C2×D5—22no—
255C2×D5—22no—
256C2×D5—22no—
257C2×D5—22no—
258C2×D5—22no—
259C2×D5—22no—
260C2×D5—22no—
261C2×D5—22no—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—22no—
265C2×D5—22no—
266C2×D5—416no—
267C2×D5C22⋊Q814yescomplemented
268C2×D5C22⋊Q814yescomplemented
269C4×C5—22no—
270C4×C5—22no—
271C4×C5—41no—
272C4×C5—41no—
273C4×C5—41no—
274C4×C5—41no—
275Dic5—22no—
276Dic5—22no—
277Dic5—41no—
278Dic5—41no—
279Dic5—41no—
280Dic5—41no—
281C24—51no—
282C24—51no—
283C24—54no—
284C24—102no—
285C24—102no—
286C24—102no—
287C24—102no—
288C24—201no—
289C24—201no—
290C24C22×D511yescharacteristic
291C2×Q8—102no—
292C2×Q8—102no—
293C2×Q8—201no—
294C22×C4—51no—
295C22×C4—51no—
296C22×C4—51no—
297C22×C4—51no—
298C22×C4—51no—
299C22×C4—51no—
300C22×C4—102no—
301C22×C4—102no—
302C22×C4—102no—
303C22×C4—102no—
304C22×C4—102no—
305C22×C4—102no—
306C22×C4—102no—
307C22×C4—102no—
308C22×C4—201no—
309C22×C4—201no—
310C22×C4—201no—
311C22×C4—201no—
312C22×C4C22×D511yescharacteristic
313C22×C4C22×D511yescharacteristic
314C22×C4C22×D511yescharacteristic
315C22×C4C22×D511yescharacteristic
316C22×C4C22×D511yescharacteristic
317C22×C4C22×D511yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—102no—
322C4⋊C4—102no—
323C4⋊C4—102no—
324C4⋊C4—102no—
325C4⋊C4—201no—
326C4⋊C4—201no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—41no—
330C22⋊C4—54no—
331C22⋊C4—54no—
332C22⋊C4—102no—
333C22⋊C4—102no—
334C22⋊C4—102no—
335C22⋊C4—102no—
336C22⋊C4—102no—
337C22⋊C4—102no—
338C22⋊C4—102no—
339C22⋊C4—102no—
340C22⋊C4—201no—
341C2×C5—42no—
342C2×C5C2×C22⋊Q811yescharacteristic
343C2×C5C2×C22⋊Q811yescharacteristic
344C2×C5#54311yescharacteristic
345C2×C5D4211yescharacteristic
346C2×C5#54611yescharacteristic
347C2×C5#54611yescharacteristic
348C2×C5#55111yescharacteristic
349D5—24no—
350D5—24no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—52no—
358C23—52no—
359C23—52no—
360C23—52no—
361C23—52no—
362C23—52no—
363C23—108no—
364C23—108no—
365C23—101no—
366C23—101no—
367C23—101no—
368C23—101no—
369C23—101no—
370C23—101no—
371C23—101no—
372C23—101no—
373C23—204no—
374C23—204no—
375C23—204no—
376C23—204no—
377C23—204no—
378C23Q8×D512yes—
379C23C23×D511yescharacteristic
380Q8—204no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—52no—
398C2×C4—52no—
399C2×C4—101no—
400C2×C4—101no—
401C2×C4—101no—
402C2×C4—101no—
403C2×C4—101no—
404C2×C4—101no—
405C2×C4—101no—
406C2×C4—101no—
407C2×C4—101no—
408C2×C4—101no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—101no—
414C2×C4—101no—
415C2×C4—204no—
416C2×C4—204no—
417C2×C4—204no—
418C2×C4—204no—
419C2×C4D4×D512yes—
420C2×C4D4×D512yes—
421C5#243711yescharacteristic, complemented, Sylow
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—54no—
430C22—54no—
431C22—102no—
432C22—102no—
433C22—102no—
434C22—102no—
435C22—102no—
436C22—102no—
437C22—102no—
438C22—102no—
439C22—102no—
440C22—102no—
441C22—102no—
442C22—102no—
443C22—2016no—
444C22C2×D4×D511yescharacteristic
445C22C2×D4×D511yescharacteristic
446C22#414411yescharacteristic
447C22C2×Q8×D511yescharacteristic
448C22D5×#4111yescharacteristic
449C22D5×#4111yescharacteristic
450C22#414911yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—102no—
458C4—102no—
459C4—201no—
460C4—201no—
461C4—201no—
462C4—201no—
463C2—42no—
464C2—104no—
465C2—104no—
466C2#6594311yescharacteristic
467C2#6594411yescharacteristic
468C2#6594611yescharacteristic
469C2D5×C22⋊Q811yescharacteristic
470C2D5×C22⋊Q811yescharacteristic
471C2#6596811yescharacteristic
472C2#6603911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#396C2
prime radicalC24×D5C22
hypercenter#396D5
hypocenterC5#2437
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#4043C2semidirect
2. C2×#4042C2semidirect
5. C5×#396C2semidirect
6. #65278C2semidirect
7. #65278C2semidirect
8. #65278C2semidirect
9. #65276C2semidirect
11. #65238C2semidirect
12. #65237C2semidirect
13. #65236C2semidirect
14. #64957C2semidirect
15. #64956C2semidirect
21. C22×Dic10C22semidirect
22. C2×C5×C4⋊C4C22semidirect
23. C2×C5×C4⋊C4C22semidirect
24. C2×C5×C4⋊C4C22semidirect
39. C2×C4⋊Dic5C22semidirect
40. C2×C4⋊Dic5C22semidirect
41. C2×#683C22semidirect
42. C2×#683C22semidirect
43. C2×#683C22semidirect
44. C2×#683C22semidirect
49. C5×#95C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
63. C23×D5Q8semidirect
122. #396D5semidirect
215. C2×C4⋊C4C2×D5semidirect
216. C2×C4⋊C4C2×D5semidirect
217. C2×C4⋊C4C2×D5semidirect
239. #95C2×D5semidirect
267. C2×D5C22⋊Q8semidirect
268. C2×D5C22⋊Q8semidirect
421. C5#2437semidirect

Maximal subgroups

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

subgroupindex
C2×#40432
C2×#40422
C2×D5×C22⋊C42
C5×#3962
#652782
#652762
#652442
#652382
#652372
#652362
#649572
#649562
#24375