Finite Group Databasebeta

← #10614547

Subgroups — #10614547

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×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4092C211yesmaximal, characteristic, complemented
3C22×#685C211yesmaximal, characteristic
4C2×D5×C22⋊C4C211yesmaximal, characteristic
5#65520C211yesmaximal, characteristic, complemented
6#65499C211yesmaximal, characteristic, complemented
7#65283C211yesmaximal, characteristic, complemented
8#65283C211yesmaximal, characteristic, complemented
9#65281C211yesmaximal, characteristic, complemented
10#65278C211yesmaximal, characteristic, complemented
11#65278C211yesmaximal, characteristic, complemented
12#65276C211yesmaximal, characteristic, complemented
13#65244C211yesmaximal, characteristic
14#65239C211yesmaximal, characteristic, complemented
15#65231C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic, complemented
18C23×C4×C5C2211yescharacteristic
19C23×Dic5C2211yescharacteristic
20C22×C4×D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C5×C22⋊Q8—24no—
23C2×C5×C4⋊C4C2211yescharacteristic, complemented
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C4⋊C4C2211yescharacteristic, complemented
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30#4092—24no—
31C2×#685—24no—
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×#685C2211yescharacteristic
42C2×C4⋊Dic5C2211yescharacteristic, complemented
43C2×C4⋊Dic5C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45C2×#683C2211yescharacteristic, complemented
46C2×#683C2211yescharacteristic, complemented
47C2×#683C2211yescharacteristic, complemented
48D5×C22⋊C4—24no—
49#3963C2211yescharacteristic, 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
66C2×C5×Q8—22no—
67C2×C5×Q8—22no—
68C2×C5×Q8—41no—
69C22×C4×C5—22no—
70C22×C4×C5—22no—
71C22×C4×C5—41no—
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C22×C4×C5C2311yescharacteristic
78C22×Dic5—22no—
79C22×Dic5—22no—
80C22×Dic5—41no—
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C2×C4×D5—22no—
88C2×C4×D5—22no—
89C2×C4×D5—22no—
90C2×C4×D5—22no—
91C2×C4×D5—41no—
92C2×C4×D5—41no—
93C5×C4⋊C4—22no—
94C5×C4⋊C4—22no—
95C5×C4⋊C4—22no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4D414yes—
98C22⋊Dic5—22no—
99C22⋊Dic5—22no—
100#685—22no—
101#685—22no—
102#685—22no—
103#685—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—41no—
108#685—41no—
109#685D414yes—
110C4⋊Dic5—22no—
111C4⋊Dic5—41no—
112#683—22no—
113#683—22no—
114#683—41no—
115#683—41no—
116C2×C22⋊Q8—51no—
117C2×C22⋊Q8D511yescharacteristic, complemented
118C22×C22⋊C4—51no—
119C22×C22⋊C4—51no—
120#399—51no—
121#396—51no—
122#396—51no—
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×C5C2×D412yes—
136C23×C5C2×Q812yes—
137C23×C5#4112yes—
138C23×C5C2411yescharacteristic
139C22×D5—21no—
140C22×D5—21no—
141C22×D5—21no—
142C22×D5—21no—
143C22×D5—28no—
144C22×D5—21no—
145C22×D5—21no—
146C22×D5—21no—
147C22×D5—21no—
148C22×D5—21no—
149C22×D5—21no—
150C22×D5—28no—
151C22×D5—44no—
152C22×D5—44no—
153C22×D5—44no—
154C22×D5—44no—
155C22×D5—44no—
156C22×D5C2×D412yes—
157C22×D5C2×Q812yes—
158C22×D5C2×Q812yes—
159C22×D5#4112yes—
160C5×Q8—44no—
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×C5—44no—
177C2×C4×C5C2×D412yes—
178C2×C4×C5C2×D412yes—
179C2×C4×C5C2×D412yes—
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×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—44no—
198C2×Dic5C2×D412yes—
199C4×D5—44no—
200C4×D5—44no—
201C25—51no—
202C22×Q8C2×D511yescharacteristic, complemented
203C23×C4—51no—
204C23×C4—51no—
205C23×C4—51no—
206C23×C4C2×D511yescharacteristic
207C22⋊Q8—24no—
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—51no—
230C2×C22⋊C4—104no—
231C2×C22⋊C4—104no—
232C2×C22⋊C4C2×D511yescharacteristic
233C2×C22⋊C4C2×D511yescharacteristic
234#95—51no—
235#95—51no—
236#95—51no—
237#95—51no—
238#95—51no—
239#95—51no—
240C22×C5—22no—
241C22×C5—22no—
242C22×C5—22no—
243C22×C5—22no—
244C22×C5—22no—
245C22×C5—22no—
246C22×C5C22⋊Q814yescomplemented
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—416no—
264C2×D5—41no—
265C2×D5—41no—
266C2×D5—41no—
267C2×D5—41no—
268C2×D5C22⋊Q814yescomplemented
269C4×C5—22no—
270C4×C5—22no—
271C4×C5—22no—
272C4×C5—41no—
273C4×C5—41no—
274C4×C5—41no—
275Dic5—22no—
276Dic5—41no—
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—22no—
292C2×Q8—22no—
293C2×Q8—41no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—51no—
298C22×C4—51no—
299C22×C4—51no—
300C22×C4—51no—
301C22×C4—51no—
302C22×C4—51no—
303C22×C4—102no—
304C22×C4—102no—
305C22×C4—102no—
306C22×C4—102no—
307C22×C4—102no—
308C22×C4—102no—
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—22no—
321C4⋊C4—102no—
322C4⋊C4—102no—
323C4⋊C4—102no—
324C4⋊C4—201no—
325C4⋊C4—201no—
326C4⋊C4—201no—
327C22⋊C4—22no—
328C22⋊C4—54no—
329C22⋊C4—102no—
330C22⋊C4—102no—
331C22⋊C4—102no—
332C22⋊C4—102no—
333C22⋊C4—102no—
334C22⋊C4—102no—
335C22⋊C4—102no—
336C22⋊C4—102no—
337C22⋊C4—102no—
338C22⋊C4—201no—
339C22⋊C4—201no—
340C22⋊C4C22⋊D514yes—
341C2×C5—24no—
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—42no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—52no—
356C23—52no—
357C23—52no—
358C23—52no—
359C23—108no—
360C23—101no—
361C23—101no—
362C23—101no—
363C23—101no—
364C23—101no—
365C23—101no—
366C23—101no—
367C23—101no—
368C23—101no—
369C23—101no—
370C23—108no—
371C23—204no—
372C23—204no—
373C23—204no—
374C23—204no—
375C23—204no—
376C23Q8×D512yes—
377C23#71312yes—
378C23C2×C22⋊D512yes—
379C23C23×D511yescharacteristic
380Q8—44no—
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—44no—
397C2×C4—52no—
398C2×C4—101no—
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×C4D4×D512yes—
419C2×C4C2×C22⋊D512yes—
420C2×C4C2×C22⋊D512yes—
421C5#243711yescharacteristic, complemented, Sylow
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—54no—
429C22—102no—
430C22—102no—
431C22—102no—
432C22—102no—
433C22—102no—
434C22—102no—
435C22—102no—
436C22—102no—
437C22—102no—
438C22—2016no—
439C22—201no—
440C22—201no—
441C22—201no—
442C22—201no—
443C22#409214yescomplemented
444C22C2×D4×D511yescharacteristic
445C22#414411yescharacteristic
446C22C2×Q8×D511yescharacteristic
447C22C2×#71311yescharacteristic
448C22D5×#4111yescharacteristic
449C22#414911yescharacteristic
450C22C22×C22⋊D511yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—102no—
458C4—201no—
459C4—201no—
460C4—201no—
461C4—201no—
462C4—201no—
463C2—24no—
464C2—104no—
465C2—202no—
466C2D5×C22⋊Q811yescharacteristic
467C2#6596411yescharacteristic
468C2#6596811yescharacteristic
469C2#6597111yescharacteristic
470C2D4×C22⋊D511yescharacteristic
471C2C2×#409211yescharacteristic
472C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×D5C22
hypercenterC2×C22⋊Q8D5
hypocenterC5#2437
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C22⋊Q8C2semidirect
2. C2×#4092C2semidirect
5. #65520C2semidirect
6. #65499C2semidirect
7. #65283C2semidirect
8. #65283C2semidirect
9. #65281C2semidirect
10. #65278C2semidirect
11. #65278C2semidirect
12. #65276C2semidirect
14. #65239C2semidirect
15. #65231C2semidirect
17. C22×C5×Q8C22semidirect
23. C2×C5×C4⋊C4C22semidirect
24. C2×C5×C4⋊C4C22semidirect
25. C2×C5×C4⋊C4C22semidirect
42. C2×C4⋊Dic5C22semidirect
43. C2×C4⋊Dic5C22semidirect
44. C2×#683C22semidirect
45. C2×#683C22semidirect
46. C2×#683C22semidirect
47. C2×#683C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
63. C23×D5Q8semidirect
117. C2×C22⋊Q8D5semidirect
202. C22×Q8C2×D5semidirect
215. C2×C4⋊C4C2×D5semidirect
216. C2×C4⋊C4C2×D5semidirect
217. C2×C4⋊C4C2×D5semidirect
246. C22×C5C22⋊Q8semidirect
268. C2×D5C22⋊Q8semidirect
421. C5#2437semidirect
443. C22#4092semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C2×#40922
C22×#6852
C2×D5×C22⋊C42
#655202
#654992
#652832
#652812
#652782
#652762
#652442
#652392
#652312
#24375