Finite Group Databasebeta

← #10614500

Subgroups — #10614500

Order 640.

786 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4093C211yesmaximal, characteristic, complemented
3C22×#685C211yesmaximal, characteristic, complemented
4C2×#4028C211yesmaximal, characteristic
5#65520C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65282C211yesmaximal, characteristic, complemented
8#65280C211yesmaximal, characteristic, complemented
9#65280C211yesmaximal, characteristic, complemented
10#65279C211yesmaximal, characteristic, complemented
11#65279C211yesmaximal, characteristic, complemented
12#65277C211yesmaximal, characteristic, complemented
13#65247C211yesmaximal, characteristic, complemented
14#65244C211yesmaximal, characteristic, complemented
15C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×Q8C2211yescharacteristic, complemented
18C23×C4×C5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C23×Dic5C2211yescharacteristic
21C22×D20C2211yescharacteristic
22C22×D20C2211yescharacteristic
23C5×C22⋊Q8—24no—
24C2×C5×C4⋊C4C2211yescharacteristic, complemented
25C2×C5×C4⋊C4C2211yescharacteristic, complemented
26C2×C5×C4⋊C4C2211yescharacteristic, complemented
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C5×C22⋊C4C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30#4093—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×#685C2211yescharacteristic, complemented
43C2×#685C2211yescharacteristic, complemented
44C2×#685C2211yescharacteristic, complemented
45C2×#685C2211yescharacteristic, complemented
46C2×C4×Dic5C2211yescharacteristic, complemented
47C2×C4×Dic5C2211yescharacteristic, complemented
48#4028—24no—
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2472—51nomaximal, Sylow
56C24×C5C2311yescharacteristic
57C23×D5—22no—
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—22no—
61C23×D5—22no—
62C23×D5—41no—
63C23×D5—41no—
64C23×D5C2311yescharacteristic
65C23×D5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C2×C5×Q8—22no—
68C2×C5×Q8—22no—
69C2×C5×Q8—41no—
70C22×C4×C5—22no—
71C22×C4×C5—22no—
72C22×C4×C5—41no—
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C22×C4×C5C2311yescharacteristic
78C22×C4×C5C2311yescharacteristic
79C2×C22⋊D5—22no—
80C2×C22⋊D5—22no—
81C2×C22⋊D5—22no—
82C22×Dic5—22no—
83C22×Dic5—22no—
84C22×Dic5—41no—
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C2×D20—22no—
91C2×D20—22no—
92C2×D20—22no—
93C2×D20—22no—
94C2×D20—41no—
95C2×D20—41no—
96C5×C4⋊C4—22no—
97C5×C4⋊C4—22no—
98C5×C4⋊C4—22no—
99C5×C22⋊C4—22no—
100C5×C22⋊C4D414yes—
101C22⋊Dic5—22no—
102#685—22no—
103#685—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—22no—
108#685—22no—
109#685—22no—
110#685—22no—
111#685—41no—
112#685—41no—
113#685—41no—
114#685—41no—
115#685D414yescomplemented
116C4×Dic5—22no—
117C4×Dic5—41no—
118C2×#124—51no—
119C2×C22⋊Q8D511yescharacteristic, complemented
120C2×#120—51no—
121C22×C22⋊C4—51no—
122#397—51no—
123#394—51no—
124#394—51no—
125#393—51no—
126#390—51no—
127#388—51no—
128#388—51no—
129#386—51no—
130#386—51no—
131#379—51no—
132C4×C22⋊C4—51no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5C2×D412yes—
138C23×C5#4112yes—
139C23×C5#4112yes—
140C23×C5C2411yescharacteristic
141C22×D5—28no—
142C22×D5—21no—
143C22×D5—21no—
144C22×D5—21no—
145C22×D5—21no—
146C22×D5—21no—
147C22×D5—21no—
148C22×D5—21no—
149C22×D5—21no—
150C22×D5—21no—
151C22×D5—21no—
152C22×D5—21no—
153C22×D5—21no—
154C22×D5—21no—
155C22×D5—21no—
156C22×D5—21no—
157C22×D5—21no—
158C22×D5—21no—
159C22×D5—21no—
160C22×D5—21no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5—44no—
164C22×D5—44no—
165C22×D5—44no—
166C22×D5—44no—
167C22×D5C2×D412yes—
168C22×D5#4112yes—
169C5×Q8—44no—
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—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—21no—
185C2×C4×C5—44no—
186C2×C4×C5C2×D412yes—
187C2×C4×C5C2×D412yes—
188C2×C4×C5C2×D412yes—
189C22⋊D5—44no—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5—21no—
195C2×Dic5—21no—
196C2×Dic5—21no—
197C2×Dic5—21no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—21no—
204C2×Dic5—44no—
205C2×Dic5C2×D412yes—
206D20—44no—
207D20—44no—
208C25—51no—
209C22×Q8C2×D511yescharacteristic, complemented
210C22×D4—51no—
211C22×D4—51no—
212C22×D4—51no—
213C23×C4—51no—
214C23×C4C2×D511yescharacteristic
215#124—104no—
216C22⋊Q8—24no—
217#120—104no—
218C2×C4⋊C4C2×D511yescharacteristic, complemented
219C2×C4⋊C4C2×D511yescharacteristic, complemented
220C2×C4⋊C4C2×D511yescharacteristic, complemented
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—51no—
231C2×C22⋊C4—51no—
232C2×C22⋊C4—51no—
233C2×C22⋊C4—51no—
234C2×C22⋊C4—51no—
235C2×C22⋊C4—51no—
236C2×C22⋊C4—104no—
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C42—51no—
240C2×C42—51no—
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95—51no—
246#95—51no—
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5#12414yescomplemented
254C22×C5C22×D411yescharacteristic
255C22×C5C22×D411yescharacteristic
256C22×C5C2×#4111yescharacteristic
257C22×C5C2×#4111yescharacteristic
258C22×C5C2×#4111yescharacteristic
259C22×C5H(5,𝔽2)11yescharacteristic
260C22×C5H(5,𝔽2)11yescharacteristic
261C2×D5—22no—
262C2×D5—22no—
263C2×D5—22no—
264C2×D5—22no—
265C2×D5—22no—
266C2×D5—22no—
267C2×D5—416no—
268C2×D5—41no—
269C2×D5—41no—
270C2×D5—41no—
271C2×D5—41no—
272C2×D5—41no—
273C2×D5—41no—
274C2×D5—41no—
275C2×D5—41no—
276C2×D5—41no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C2×D5—41no—
282C2×D5—41no—
283C4×C5—22no—
284C4×C5—22no—
285C4×C5—22no—
286C4×C5—41no—
287C4×C5—41no—
288C4×C5—41no—
289Dic5—22no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294C24—51no—
295C24—51no—
296C24—51no—
297C24—102no—
298C24—102no—
299C24—102no—
300C24—102no—
301C24—102no—
302C24—201no—
303C24—201no—
304C24C22×D511yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—102no—
312C2×D4—102no—
313C2×D4—102no—
314C2×D4—102no—
315C2×D4—201no—
316C2×D4—201no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—51no—
323C22×C4—51no—
324C22×C4—51no—
325C22×C4—102no—
326C22×C4—102no—
327C22×C4—201no—
328C22×C4C22×D511yescharacteristic
329C22×C4C22×D511yescharacteristic
330C22×C4C22×D511yescharacteristic
331C22×C4C22×D511yescharacteristic
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—54no—
339C22⋊C4—102no—
340C22⋊C4—102no—
341C22⋊C4—102no—
342C22⋊C4—102no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4—102no—
348C22⋊C4—102no—
349C22⋊C4—201no—
350C22⋊C4—201no—
351C22⋊C4—201no—
352C22⋊C4—201no—
353C22⋊C4C22⋊D514yes—
354C42—102no—
355C42—201no—
356C2×C5—24no—
357C2×C5#52511yescharacteristic
358C2×C5C2×#12411yescharacteristic
359C2×C5#53811yescharacteristic
360C2×C5D4211yescharacteristic
361C2×C5#54611yescharacteristic
362C2×C5#54611yescharacteristic
363C2×C5#55111yescharacteristic
364D5—42no—
365D5—42no—
366D5—81no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—52no—
372C23—52no—
373C23—108no—
374C23—101no—
375C23—101no—
376C23—101no—
377C23—101no—
378C23—101no—
379C23—101no—
380C23—101no—
381C23—101no—
382C23—101no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—101no—
393C23—204no—
394C23—204no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23—204no—
399C23#71312yes—
400C23#71312yes—
401C23C2×C22⋊D512yes—
402C23C23×D511yescharacteristic
403Q8—44no—
404D4—204no—
405D4—204no—
406D4—204no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—44no—
423C2×C4—52no—
424C2×C4—101no—
425C2×C4—101no—
426C2×C4—101no—
427C2×C4—101no—
428C2×C4—101no—
429C2×C4—101no—
430C2×C4—101no—
431C2×C4—101no—
432C2×C4—101no—
433C2×C4—101no—
434C2×C4—101no—
435C2×C4—101no—
436C2×C4—101no—
437C2×C4—101no—
438C2×C4—204no—
439C2×C4D4×D512yes—
440C2×C4C2×C22⋊D512yes—
441C2×C4C2×C22⋊D512yes—
442C5#247211yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—102no—
450C22—102no—
451C22—102no—
452C22—102no—
453C22—102no—
454C22—102no—
455C22—2016no—
456C22—201no—
457C22—201no—
458C22—201no—
459C22—201no—
460C22—201no—
461C22—201no—
462C22—201no—
463C22—201no—
464C22—201no—
465C22—201no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22—201no—
471C22#409314yescomplemented
472C22C2×D4×D511yescharacteristic
473C22#414411yescharacteristic
474C22C2×#71311yescharacteristic
475C22C2×#71311yescharacteristic
476C22D5×#4111yescharacteristic
477C22#414911yescharacteristic
478C22C22×C22⋊D511yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—102no—
486C4—201no—
487C4—201no—
488C4—201no—
489C4—201no—
490C2—24no—
491C2—202no—
492C2—202no—
493C2—401no—
494C2#6596111yescharacteristic
495C2#6596411yescharacteristic
496C2#6597111yescharacteristic
497C2#6597811yescharacteristic
498C2D4×C22⋊D511yescharacteristic
499C2C2×#409311yescharacteristic
500C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC2×#4028C2
hypercenterC2×C22⋊Q8D5
hypocenterC5#2472
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×#4093C2semidirect
3. C22×#685C2semidirect
5. #65520C2semidirect
6. #65502C2semidirect
7. #65282C2semidirect
8. #65280C2semidirect
9. #65280C2semidirect
10. #65279C2semidirect
11. #65279C2semidirect
12. #65277C2semidirect
13. #65247C2semidirect
14. #65244C2semidirect
15. C22⋊C4×Dic5C2semidirect
17. C22×C5×Q8C22semidirect
24. C2×C5×C4⋊C4C22semidirect
25. C2×C5×C4⋊C4C22semidirect
26. C2×C5×C4⋊C4C22semidirect
42. C2×#685C22semidirect
43. C2×#685C22semidirect
44. C2×#685C22semidirect
45. C2×#685C22semidirect
46. C2×C4×Dic5C22semidirect
47. C2×C4×Dic5C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
115. #685D4semidirect
119. C2×C22⋊Q8D5semidirect
209. C22×Q8C2×D5semidirect
218. C2×C4⋊C4C2×D5semidirect
219. C2×C4⋊C4C2×D5semidirect
220. C2×C4⋊C4C2×D5semidirect
253. C22×C5#124semidirect
442. C5#2472semidirect
471. C22#4093semidirect

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×#40932
C22×#6852
C2×#40282
#655202
#655022
#652822
#652802
#652792
#652772
#652472
#652442
C22⋊C4×Dic52
#24725