Finite Group Databasebeta

← #10606607

Subgroups — #10606607

Order 640.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4083C211yesmaximal, characteristic
2C2×C4×C22⋊D5C211yesmaximal, characteristic, complemented
3C2×D5×C22⋊C4C211yesmaximal, characteristic, complemented
4C5×#386C211yesmaximal, characteristic, complemented
5#65510C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65246C211yesmaximal, characteristic, complemented
8#65245C211yesmaximal, characteristic, complemented
9#65244C211yesmaximal, characteristic, complemented
10#65233C211yesmaximal, characteristic, complemented
11#65233C211yesmaximal, characteristic, complemented
12#64959C211yesmaximal, characteristic, complemented
13#64958C211yesmaximal, characteristic, complemented
14#64957C211yesmaximal, characteristic, complemented
15#64955C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic, complemented
19C22×C22⋊D5C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C22×C4×D5C2211yescharacteristic
22C22×C4×D5C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic, complemented
30C2×C22⋊Dic5C2211yescharacteristic, complemented
31#4083C418yescomplemented
32C4×C22⋊D5—24no—
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×#685C2211yescharacteristic
41C2×C4⋊Dic5C2211yescharacteristic, complemented
42C2×#683C2211yescharacteristic, complemented
43C2×#683C2211yescharacteristic, complemented
44C2×C4×Dic5C2211yescharacteristic, complemented
45C2×C4×Dic5C2211yescharacteristic, complemented
46C2×C4×Dic5C2211yescharacteristic, complemented
47D5×C22⋊C4—24no—
48C5×#95C2211yescharacteristic, complemented
49C5×#95C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#2321—51nomaximal, Sylow
55C24×C5C2311yescharacteristic
56C24×C5C2311yescharacteristic
57C23×D5—22no—
58C23×D5—22no—
59C23×D5—41no—
60C23×D5—41no—
61C23×D5—41no—
62C23×D5—41no—
63C23×D5C2×C414yescomplemented
64C23×D5C2311yescharacteristic
65C23×D5C2311yescharacteristic
66C2×C5×D4—41no—
67C2×C5×D4—41no—
68C2×C5×D4C2×C414yescomplemented
69C22×C4×C5—22no—
70C22×C4×C5—41no—
71C22×C4×C5—41no—
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C2×C22⋊D5—22no—
78C2×C22⋊D5—22no—
79C2×C22⋊D5—41no—
80C2×C22⋊D5—41no—
81C2×C22⋊D5C2×C414yes—
82C2×C22⋊D5C2×C414yescomplemented
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C22×Dic5C2311yescharacteristic
86C22×Dic5C2311yescharacteristic
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C2×C4×D5—22no—
90C2×C4×D5—41no—
91C2×C4×D5—41no—
92C2×C4×D5—41no—
93C2×C4×D5—41no—
94C2×C4×D5D414yescomplemented
95C5×C22⋊C4—22no—
96C5×C22⋊C4—41no—
97C5×C22⋊C4—41no—
98C22⋊Dic5—22no—
99C22⋊Dic5—22no—
100C22⋊Dic5—41no—
101C22⋊Dic5—41no—
102C22⋊Dic5C2×C414yescomplemented
103#685—22no—
104#685—22no—
105#685—22no—
106#685—22no—
107#685—41no—
108#685—41no—
109#685C2×C414yescomplemented
110#685C2×C414yescomplemented
111C4⋊Dic5—41no—
112#683—22no—
113#683—41no—
114C4×Dic5—22no—
115C4×Dic5—41no—
116C4×Dic5—41no—
117C2×#120—51no—
118C2×C4×D4—51no—
119C22×C22⋊C4—51no—
120#390—51no—
121#390—51no—
122#388—51no—
123#388—51no—
124#388—51no—
125#388—51no—
126#386—51no—
127#386—51no—
128#386D511yescharacteristic, complemented
129#385—51no—
130#379—51no—
131C4×C22⋊C4—51no—
132C23×C5—21no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—21no—
138C23×C5—21no—
139C23×C5—21no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5C22×C412yes—
145C23×C5C22×C412yes—
146C23×C5C2411yescharacteristic
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—28no—
158C22×D5—44no—
159C22×D5—44no—
160C22×D5—44no—
161C22×D5—44no—
162C22×D5—44no—
163C22×D5—44no—
164C22×D5C22×C412yes—
165C22×D5C22×C412yes—
166C22×D5C2×D412yes—
167C22×D5#4112yes—
168C5×D4—44no—
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—21no—
177C2×C4×C5—21no—
178C2×C4×C5—21no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—44no—
183C2×C4×C5C22×C412yes—
184C2×C4×C5C2×D412yes—
185C22⋊D5—28no—
186C22⋊D5—44no—
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—21no—
198C2×Dic5—21no—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5C22×C412yes—
203C2×Dic5C22×C412yes—
204C2×Dic5C2×D412yes—
205C4×D5—44no—
206C4×D5—44no—
207C25—51no—
208C22×D4—51no—
209C22×D4—51no—
210C22×D4C2×D511yescharacteristic
211C23×C4—51no—
212C23×C4—51no—
213C23×C4C2×D511yescharacteristic, complemented
214#120—58no—
215C4×D4—104no—
216C2×C4⋊C4—51no—
217C2×C4⋊C4—51no—
218C2×C4⋊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—51no—
231C2×C22⋊C4—51no—
232C2×C22⋊C4—104no—
233C2×C22⋊C4C2×D511yescharacteristic
234C2×C22⋊C4C2×D511yescharacteristic
235C2×C22⋊C4C2×D511yescharacteristic, complemented
236C2×C42—51no—
237C2×C42—51no—
238C2×C42—51no—
239#95—51no—
240#95—51no—
241#95—51no—
242#95—51no—
243#95C2×D511yescharacteristic, complemented
244#95C2×D511yescharacteristic, complemented
245C22×C5—22no—
246C22×C5—22no—
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—41no—
252C22×C5—41no—
253C22×C5—41no—
254C22×C5—41no—
255C22×C5—41no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—41no—
259C22×C5C23×C411yescharacteristic
260C22×C5C22×D411yescharacteristic
261C22×C5C2×#4111yescharacteristic
262C22×C5H(5,𝔽2)11yescharacteristic
263C22×C5H(5,𝔽2)11yescharacteristic
264C22×C5H(5,𝔽2)11yescharacteristic
265C22×C5H(5,𝔽2)11yescharacteristic
266C2×D5—22no—
267C2×D5—22no—
268C2×D5—22no—
269C2×D5—22no—
270C2×D5—22no—
271C2×D5—22no—
272C2×D5—22no—
273C2×D5—22no—
274C2×D5—22no—
275C2×D5—416no—
276C2×D5—41no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5C4×D414yes—
281C4×C5—22no—
282C4×C5—22no—
283C4×C5—41no—
284C4×C5—41no—
285C4×C5—41no—
286Dic5—22no—
287Dic5—22no—
288Dic5—22no—
289Dic5—41no—
290Dic5—41no—
291Dic5—41no—
292C24—51no—
293C24—51no—
294C24—54no—
295C24—102no—
296C24—102no—
297C24—201no—
298C24—201no—
299C24—201no—
300C24—201no—
301C24C22×D511yescharacteristic
302C24C22×D511yescharacteristic
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—54no—
306C2×D4—54no—
307C2×D4—102no—
308C2×D4—102no—
309C2×D4—201no—
310C2×D4—201no—
311C2×D4C4×D514yescomplemented
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—51no—
316C22×C4—51no—
317C22×C4—51no—
318C22×C4—54no—
319C22×C4—51no—
320C22×C4—51no—
321C22×C4—51no—
322C22×C4—102no—
323C22×C4—201no—
324C22×C4—201no—
325C22×C4—201no—
326C22×C4—201no—
327C22×C4C22×D511yescharacteristic
328C22×C4C22×D511yescharacteristic
329C22×C4C22×D511yescharacteristic
330C22×C4C22×D511yescharacteristic
331C22×C4C22×D511yescharacteristic
332C4⋊C4—102no—
333C4⋊C4—201no—
334C4⋊C4—201no—
335C22⋊C4—22no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—54no—
339C22⋊C4—54no—
340C22⋊C4—54no—
341C22⋊C4—102no—
342C22⋊C4—102no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4—201no—
348C22⋊C4—201no—
349C22⋊C4—201no—
350C22⋊C4—201no—
351C42—102no—
352C42—201no—
353C42—201no—
354C2×C5—42no—
355C2×C5—42no—
356C2×C5C2×C4×D411yescharacteristic
357C2×C5#51811yescharacteristic
358C2×C5#51811yescharacteristic
359C2×C5#53411yescharacteristic
360C2×C5#53511yescharacteristic
361C2×C5#53811yescharacteristic
362C2×C5#53811yescharacteristic
363D5—24no—
364D5—42no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—52no—
378C23—52no—
379C23—52no—
380C23—52no—
381C23—108no—
382C23—101no—
383C23—101no—
384C23—101no—
385C23—101no—
386C23—101no—
387C23—101no—
388C23—101no—
389C23—101no—
390C23—101no—
391C23—101no—
392C23—204no—
393C23—204no—
394C23—204no—
395C23—204no—
396C23—204no—
397C23—204no—
398C23C2×C4×D512yes—
399C23C2×C4×D512yes—
400C23C23×D511yescharacteristic
401D4—44no—
402D4—108no—
403D4—204no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
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—44no—
418C2×C4—52no—
419C2×C4—52no—
420C2×C4—52no—
421C2×C4—101no—
422C2×C4—101no—
423C2×C4—101no—
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—204no—
437C2×C4—204no—
438C2×C4C2×C4×D512yes—
439C2×C4D4×D512yes—
440C5#232111yescharacteristic, complemented, Sylow
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—54no—
456C22—102no—
457C22—102no—
458C22—102no—
459C22—102no—
460C22—102no—
461C22—102no—
462C22—102no—
463C22—102no—
464C22—102no—
465C22—2016no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22C22×C4×D511yescharacteristic
471C22C2×D4×D511yescharacteristic
472C22#414411yescharacteristic
473C22#414411yescharacteristic
474C22#414411yescharacteristic
475C22D5×#4111yescharacteristic
476C22#414911yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—102no—
483C4—102no—
484C4—102no—
485C4—201no—
486C4—201no—
487C4—201no—
488C2—42no—
489C2—42no—
490C2—104no—
491C2—202no—
492C2#6582011yescharacteristic
493C2C4×D4×D511yescharacteristic
494C2#6587911yescharacteristic
495C2#6592311yescharacteristic
496C2#6594111yescharacteristic
497C2#6594711yescharacteristic
498C2#6599111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutatorC22×C5C23×C4
FittingC5×#386C2
socle, cosocleC23×C5C24
prime radicalC2×#4083C2
hypercenter#386D5
hypocenterC5#2321
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D5C2semidirect
3. C2×D5×C22⋊C4C2semidirect
4. C5×#386C2semidirect
5. #65510C2semidirect
6. #65502C2semidirect
7. #65246C2semidirect
8. #65245C2semidirect
9. #65244C2semidirect
10. #65233C2semidirect
11. #65233C2semidirect
12. #64959C2semidirect
13. #64958C2semidirect
14. #64957C2semidirect
15. #64955C2semidirect
18. C23×C4×C5C22semidirect
25. C2×C5×C22⋊C4C22semidirect
29. C2×C22⋊Dic5C22semidirect
30. C2×C22⋊Dic5C22semidirect
31. #4083C4semidirect
41. C2×C4⋊Dic5C22semidirect
42. C2×#683C22semidirect
43. C2×#683C22semidirect
44. C2×C4×Dic5C22semidirect
45. C2×C4×Dic5C22semidirect
46. C2×C4×Dic5C22semidirect
48. C5×#95C22semidirect
49. C5×#95C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
63. C23×D5C2×C4semidirect
68. C2×C5×D4C2×C4semidirect
82. C2×C22⋊D5C2×C4semidirect
94. C2×C4×D5D4semidirect
102. C22⋊Dic5C2×C4semidirect
109. #685C2×C4semidirect
110. #685C2×C4semidirect
128. #386D5semidirect
213. C23×C4C2×D5semidirect
235. C2×C22⋊C4C2×D5semidirect
243. #95C2×D5semidirect
244. #95C2×D5semidirect
311. C2×D4C4×D5semidirect
440. C5#2321semidirect

Maximal subgroups

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

subgroupindex
C2×#40832
C2×C4×C22⋊D52
C2×D5×C22⋊C42
C5×#3862
#655102
#655022
#652462
#652452
#652442
#652332
#649592
#649582
#649572
#649552
#23215