Finite Group Databasebeta

← #10607621

Subgroups — #10607621

Order 640.

672 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4042C211yesmaximal, characteristic, complemented
2C2×D5×C22⋊C4C211yesmaximal, characteristic
3C5×#388C211yesmaximal, characteristic, complemented
4#65279C211yesmaximal, characteristic, complemented
5#65244C211yesmaximal, characteristic
6#65244C211yesmaximal, characteristic
7#65241C211yesmaximal, characteristic, complemented
8#65236C211yesmaximal, characteristic, complemented
9#65234C211yesmaximal, characteristic, complemented
10#65229C211yesmaximal, characteristic, complemented
11C4×#685C211yesmaximal, characteristic, complemented
12#64964C211yesmaximal, characteristic, complemented
13#64962C211yesmaximal, characteristic, complemented
14#64953C211yesmaximal, characteristic, complemented
15#64953C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C22×Dic10C2211yescharacteristic, complemented
20C2×C5×C4⋊C4C2211yescharacteristic, complemented
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C42×C5C2211yescharacteristic, complemented
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#683C2211yescharacteristic, complemented
39C2×#683C2211yescharacteristic, complemented
40C2×C4×Dic5C2211yescharacteristic, complemented
41#4042—24no—
42D5×C22⋊C4—24no—
43C5×#95C2211yescharacteristic, complemented
44C5×#95C2211yescharacteristic, complemented
45#3963C2211yescharacteristic, complemented
46#3963C2211yescharacteristic, complemented
47#3963C2211yescharacteristic, complemented
48#3963C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#2561—51nomaximal, Sylow
54C24×C5C2311yescharacteristic
55C23×D5—22no—
56C23×D5—22no—
57C23×D5—22no—
58C23×D5—22no—
59C23×D5—22no—
60C23×D5—41no—
61C23×D5—41no—
62C23×D5C2311yescharacteristic
63C23×D5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C22×Dic5C2311yescharacteristic
75C22×Dic5C2311yescharacteristic
76C2×C4×D5—22no—
77C2×C4×D5—22no—
78C2×C4×D5—22no—
79C2×C4×D5—41no—
80C2×C4×D5—41no—
81C2×C4×D5D414yes—
82C2×Dic10—22no—
83C2×Dic10—41no—
84C2×Dic10—41no—
85C5×C4⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—22no—
88C5×C22⋊C4—41no—
89C42×C5—41no—
90C22⋊Dic5—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—41no—
93#685—22no—
94#685—22no—
95#685—22no—
96#685—22no—
97#685—22no—
98#685—22no—
99#685—22no—
100#685—22no—
101#685—41no—
102#685—41no—
103#683—22no—
104#683—22no—
105C4×Dic5—41no—
106C2×C22⋊Q8—51no—
107C22×C22⋊C4—51no—
108#397—51no—
109#397—51no—
110#393—51no—
111#393—51no—
112#388—51no—
113#388—51no—
114#388—51no—
115#388D511yescharacteristic, complemented
116#381—51no—
117#381—51no—
118#379—51no—
119#379—51no—
120C4×C22⋊C4—51no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5#4112yes—
128C23×C5C2411yescharacteristic
129C22×D5—21no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—28no—
134C22×D5—21no—
135C22×D5—21no—
136C22×D5—21no—
137C22×D5—21no—
138C22×D5—21no—
139C22×D5—21no—
140C22×D5—44no—
141C22×D5—44no—
142C22×D5—44no—
143C22×D5—44no—
144C22×D5—44no—
145C22×D5—44no—
146C22×D5C2×D412yes—
147C22×D5#4112yes—
148C22×D5#4112yes—
149C22×D5#4112yes—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
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×C5C2×D412yes—
168C2×Dic5—21no—
169C2×Dic5—21no—
170C2×Dic5—21no—
171C2×Dic5—21no—
172C2×Dic5—21no—
173C2×Dic5—21no—
174C2×Dic5—21no—
175C2×Dic5—21no—
176C2×Dic5—21no—
177C2×Dic5—21no—
178C2×Dic5—21no—
179C2×Dic5—21no—
180C2×Dic5—21no—
181C2×Dic5—21no—
182C2×Dic5—21no—
183C2×Dic5—21no—
184C2×Dic5—21no—
185C2×Dic5C2×D412yes—
186C4×D5—44no—
187C4×D5—44no—
188Dic10—44no—
189C25—51no—
190C22×Q8—51no—
191C23×C4—51no—
192C23×C4—51no—
193C22⋊Q8—104no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4C2×D511yescharacteristic, complemented
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4—51no—
200C2×C22⋊C4—51no—
201C2×C22⋊C4—51no—
202C2×C22⋊C4—51no—
203C2×C22⋊C4—51no—
204C2×C22⋊C4—51no—
205C2×C22⋊C4—51no—
206C2×C22⋊C4—51no—
207C2×C22⋊C4—51no—
208C2×C22⋊C4—51no—
209C2×C22⋊C4—51no—
210C2×C22⋊C4—104no—
211C2×C22⋊C4C2×D511yescharacteristic
212C2×C22⋊C4C2×D511yescharacteristic
213C2×C22⋊C4C2×D511yescharacteristic
214C2×C42—51no—
215C2×C42C2×D511yescharacteristic, complemented
216#95—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222#95—51no—
223#95—51no—
224#95C2×D511yescharacteristic, complemented
225#95C2×D511yescharacteristic, complemented
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—22no—
229C22×C5—41no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5—41no—
233C22×C5C22×D411yescharacteristic
234C22×C5C2×#4111yescharacteristic
235C22×C5C2×#4111yescharacteristic
236C22×C5C2×#4111yescharacteristic
237C22×C5C2×#4111yescharacteristic
238C22×C5H(5,𝔽2)11yescharacteristic
239C22×C5H(5,𝔽2)11yescharacteristic
240C2×D5—22no—
241C2×D5—22no—
242C2×D5—22no—
243C2×D5—22no—
244C2×D5—22no—
245C2×D5—22no—
246C2×D5—22no—
247C2×D5—22no—
248C2×D5—22no—
249C2×D5—416no—
250C2×D5—41no—
251C2×D5—41no—
252C2×D5—41no—
253C2×D5—41no—
254C2×D5#12414yes—
255C4×C5—22no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5—41no—
259C4×C5—41no—
260C4×C5—41no—
261Dic5—22no—
262Dic5—41no—
263Dic5—41no—
264Dic5—41no—
265Dic5—41no—
266Dic5—41no—
267C24—51no—
268C24—51no—
269C24—102no—
270C24—102no—
271C24—102no—
272C24—102no—
273C24—102no—
274C24—201no—
275C24—201no—
276C24C22×D511yescharacteristic
277C2×Q8—102no—
278C2×Q8—201no—
279C2×Q8—201no—
280C22×C4—51no—
281C22×C4—54no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—51no—
285C22×C4—51no—
286C22×C4—51no—
287C22×C4—102no—
288C22×C4—102no—
289C22×C4—102no—
290C22×C4—201no—
291C22×C4—201no—
292C22×C4C22×D511yescharacteristic
293C22×C4C22×D511yescharacteristic
294C22×C4C22×D511yescharacteristic
295C22×C4C22×D511yescharacteristic
296C22×C4C22×D511yescharacteristic
297C22×C4C22×D511yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—102no—
300C4⋊C4—102no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—41no—
304C22⋊C4—102no—
305C22⋊C4—102no—
306C22⋊C4—102no—
307C22⋊C4—102no—
308C22⋊C4—102no—
309C22⋊C4—102no—
310C22⋊C4—102no—
311C22⋊C4—102no—
312C22⋊C4—102no—
313C22⋊C4—102no—
314C22⋊C4—201no—
315C22⋊C4—201no—
316C22⋊C4—201no—
317C42—41no—
318C42—201no—
319C2×C5—42no—
320C2×C5#52511yescharacteristic
321C2×C5C2×#12411yescharacteristic
322C2×C5#53411yescharacteristic
323C2×C5#55111yescharacteristic
324C2×C5#55111yescharacteristic
325C2×C5#55111yescharacteristic
326C2×C5#55111yescharacteristic
327D5—24no—
328D5—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—52no—
336C23—52no—
337C23—52no—
338C23—52no—
339C23—108no—
340C23—101no—
341C23—101no—
342C23—101no—
343C23—101no—
344C23—101no—
345C23—101no—
346C23—101no—
347C23—101no—
348C23—101no—
349C23—101no—
350C23—204no—
351C23—204no—
352C23—204no—
353C23—204no—
354C23—204no—
355C23—204no—
356C23#70912yes—
357C23C23×D511yescharacteristic
358Q8—204no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—52no—
377C2×C4—101no—
378C2×C4—101no—
379C2×C4—101no—
380C2×C4—101no—
381C2×C4—101no—
382C2×C4—101no—
383C2×C4—101no—
384C2×C4—101no—
385C2×C4—101no—
386C2×C4—101no—
387C2×C4—101no—
388C2×C4—101no—
389C2×C4—101no—
390C2×C4—101no—
391C2×C4—101no—
392C2×C4—101no—
393C2×C4—101no—
394C2×C4—204no—
395C2×C4—204no—
396C2×C4D4×D512yes—
397C5#256111yescharacteristic, complemented, Sylow
398C22—22no—
399C22—22no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—54no—
406C22—102no—
407C22—102no—
408C22—102no—
409C22—102no—
410C22—102no—
411C22—102no—
412C22—102no—
413C22—102no—
414C22—102no—
415C22—2016no—
416C22—201no—
417C22—201no—
418C22—201no—
419C22—201no—
420C22C2×#70911yescharacteristic
421C22C2×D4×D511yescharacteristic
422C22#414411yescharacteristic
423C22D5×#4111yescharacteristic
424C22D5×#4111yescharacteristic
425C22D5×#4111yescharacteristic
426C22#414911yescharacteristic
427C4—22no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—102no—
434C4—201no—
435C4—201no—
436C4—201no—
437C4—201no—
438C4—201no—
439C2—42no—
440C2—104no—
441C2—202no—
442C2#6586111yescharacteristic
443C2#6588111yescharacteristic
444C2#6589011yescharacteristic
445C2#6593911yescharacteristic
446C2#6599311yescharacteristic
447C2D5×#12411yescharacteristic
448C2#6603911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#388C2
prime radicalC24×D5C22
hypercenter#388D5
hypocenterC5#2561
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#4042C2semidirect
3. C5×#388C2semidirect
4. #65279C2semidirect
7. #65241C2semidirect
8. #65236C2semidirect
9. #65234C2semidirect
10. #65229C2semidirect
11. C4×#685C2semidirect
12. #64964C2semidirect
13. #64962C2semidirect
14. #64953C2semidirect
15. #64953C2semidirect
19. C22×Dic10C22semidirect
20. C2×C5×C4⋊C4C22semidirect
24. C2×C42×C5C22semidirect
38. C2×#683C22semidirect
39. C2×#683C22semidirect
40. C2×C4×Dic5C22semidirect
43. C5×#95C22semidirect
44. C5×#95C22semidirect
45. #3963C22semidirect
46. #3963C22semidirect
47. #3963C22semidirect
48. #3963C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
115. #388D5semidirect
196. C2×C4⋊C4C2×D5semidirect
215. C2×C42C2×D5semidirect
224. #95C2×D5semidirect
225. #95C2×D5semidirect
397. C5#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#40422
C2×D5×C22⋊C42
C5×#3882
#652792
#652442
#652412
#652362
#652342
#652292
C4×#6852
#649642
#649622
#649532
#25615