Finite Group Databasebeta

← #10606184

Subgroups — #10606184

Order 640.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#685C211yesmaximal, characteristic
2C2×#4044C211yesmaximal, characteristic, complemented
3C5×#385C211yesmaximal, characteristic, complemented
4#65501C211yesmaximal, characteristic, complemented
5#65283C211yesmaximal, characteristic, complemented
6#65282C211yesmaximal, characteristic, complemented
7#65279C211yesmaximal, characteristic, complemented
8#65244C211yesmaximal, characteristic
9#65244C211yesmaximal, characteristic
10#65235C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic5C211yesmaximal, characteristic, complemented
12#64964C211yesmaximal, characteristic, complemented
13#64963C211yesmaximal, characteristic, complemented
14#64959C211yesmaximal, characteristic, complemented
15#64958C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×Dic5C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic, complemented
20C2×C5×C4⋊C4C2211yescharacteristic, complemented
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×#685—24no—
26C2×#685C2211yescharacteristic
27C2×#685C2211yescharacteristic
28C2×#685C2211yescharacteristic
29C2×#685C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×C4⋊Dic5C2211yescharacteristic, complemented
39C2×C4⋊Dic5C2211yescharacteristic, complemented
40C2×#683C2211yescharacteristic, complemented
41C2×#683C2211yescharacteristic, complemented
42C2×C4×Dic5C2211yescharacteristic, complemented
43C2×C4×Dic5C2211yescharacteristic, complemented
44#4044—24no—
45C5×#95C2211yescharacteristic, complemented
46C5×#95C2211yescharacteristic, complemented
47#3963C2211yescharacteristic, complemented
48#3963C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#2480—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×C5—22no—
65C22×C4×C5—22no—
66C22×C4×C5—41no—
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—41no—
76C22×Dic5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C5×C4⋊C4—22no—
83C5×C4⋊C4—41no—
84C5×C22⋊C4—22no—
85C5×C22⋊C4—22no—
86C22⋊Dic5—22no—
87C22⋊Dic5—22no—
88#685—22no—
89#685—22no—
90#685—22no—
91#685—22no—
92#685—22no—
93#685—22no—
94#685—22no—
95#685—41no—
96#685—41no—
97#685—41no—
98#685—41no—
99#685D414yes—
100C4⋊Dic5—22no—
101C4⋊Dic5—41no—
102#683—22no—
103#683—41no—
104C4×Dic5—22no—
105C4×Dic5—41no—
106C2×#126—51no—
107C22×C22⋊C4—51no—
108#397—51no—
109#397—51no—
110#397—51no—
111#396—51no—
112#388—51no—
113#388—51no—
114#388—51no—
115#385—51no—
116#385D511yescharacteristic, complemented
117#380—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#4112yes—
126C23×C5#4112yes—
127C23×C5#4112yes—
128C23×C5C2411yescharacteristic
129C22×D5—28no—
130C22×D5—21no—
131C22×D5—21no—
132C22×D5—21no—
133C22×D5—21no—
134C22×D5—21no—
135C22×D5—21no—
136C22×D5—21no—
137C22×D5—21no—
138C22×D5—21no—
139C22×D5—21no—
140C22×D5—21no—
141C22×D5—21no—
142C22×D5—44no—
143C22×D5—44no—
144C22×D5—44no—
145C22×D5—44no—
146C22×D5—44no—
147C22×D5—44no—
148C22×D5C2×D412yes—
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×C5—44no—
168C2×C4×C5C2×D412yes—
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×Dic5—21no—
186C2×Dic5—44no—
187C2×Dic5C2×D412yes—
188C25—51no—
189C23×C4—51no—
190C23×C4C2×D511yescharacteristic
191#126—104no—
192C2×C4⋊C4—51no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4—51no—
196C2×C4⋊C4C2×D511yescharacteristic, complemented
197C2×C4⋊C4C2×D511yescharacteristic, complemented
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—51no—
211C2×C22⋊C4—51no—
212C2×C22⋊C4—104no—
213C2×C22⋊C4C2×D511yescharacteristic
214C2×C22⋊C4C2×D511yescharacteristic
215C2×C42—51no—
216C2×C42—51no—
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222#95—51no—
223#95C2×D511yescharacteristic, complemented
224#95C2×D511yescharacteristic, complemented
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—22no—
229C22×C5—22no—
230C22×C5—22no—
231C22×C5#12614yescomplemented
232C22×C5C22×D411yescharacteristic
233C22×C5C2×#4111yescharacteristic
234C22×C5C2×#4111yescharacteristic
235C22×C5C2×#4111yescharacteristic
236C22×C5C2×#4111yescharacteristic
237C22×C5H(5,𝔽2)11yescharacteristic
238C22×C5H(5,𝔽2)11yescharacteristic
239C2×D5—22no—
240C2×D5—22no—
241C2×D5—22no—
242C2×D5—22no—
243C2×D5—22no—
244C2×D5—22no—
245C2×D5—416no—
246C2×D5—41no—
247C2×D5—41no—
248C2×D5—41no—
249C2×D5—41no—
250C2×D5—41no—
251C2×D5—41no—
252C2×D5—41no—
253C2×D5—41no—
254C4×C5—22no—
255C4×C5—41no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5—41no—
259C4×C5—41no—
260Dic5—22no—
261Dic5—41no—
262Dic5—41no—
263Dic5—41no—
264Dic5—41no—
265Dic5—41no—
266C24—51no—
267C24—51no—
268C24—102no—
269C24—102no—
270C24—102no—
271C24—102no—
272C24—102no—
273C24—201no—
274C24—201no—
275C24C22×D511yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—51no—
285C22×C4—102no—
286C22×C4—102no—
287C22×C4—201no—
288C22×C4C22×D511yescharacteristic
289C22×C4C22×D511yescharacteristic
290C22×C4C22×D511yescharacteristic
291C22×C4C22×D511yescharacteristic
292C22×C4C22×D511yescharacteristic
293C22×C4C22×D511yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—102no—
297C4⋊C4—102no—
298C4⋊C4—201no—
299C4⋊C4—201no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—54no—
303C22⋊C4—102no—
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—201no—
313C22⋊C4—201no—
314C22⋊C4—201no—
315C22⋊C4—201no—
316C42—102no—
317C42—201no—
318C2×C5—24no—
319C2×C5#52511yescharacteristic
320C2×C5C2×#12611yescharacteristic
321C2×C5#53811yescharacteristic
322C2×C5#54611yescharacteristic
323C2×C5#54611yescharacteristic
324C2×C5#55111yescharacteristic
325C2×C5#55111yescharacteristic
326D5—42no—
327D5—42no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—52no—
333C23—52no—
334C23—108no—
335C23—101no—
336C23—101no—
337C23—101no—
338C23—101no—
339C23—101no—
340C23—101no—
341C23—101no—
342C23—101no—
343C23—101no—
344C23—101no—
345C23—101no—
346C23—101no—
347C23—204no—
348C23—204no—
349C23—204no—
350C23—204no—
351C23—204no—
352C23—204no—
353C23#70912yes—
354C23#71112yes—
355C23#71312yes—
356C23C23×D511yescharacteristic
357C2×C4—21no—
358C2×C4—21no—
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—44no—
375C2×C4—52no—
376C2×C4—101no—
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—204no—
394C2×C4D4×D512yes—
395C5#248011yescharacteristic, complemented, Sylow
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—102no—
403C22—102no—
404C22—102no—
405C22—102no—
406C22—102no—
407C22—102no—
408C22—2016no—
409C22—201no—
410C22—201no—
411C22—201no—
412C22—201no—
413C22—201no—
414C22—201no—
415C22—201no—
416C22—201no—
417C22#404414yescomplemented
418C22C2×#70911yescharacteristic
419C22C2×D4×D511yescharacteristic
420C22C2×#71111yescharacteristic
421C22#414411yescharacteristic
422C22C2×#71311yescharacteristic
423C22D5×#4111yescharacteristic
424C22#414911yescharacteristic
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—102no—
432C4—201no—
433C4—201no—
434C4—201no—
435C4—201no—
436C4—201no—
437C2—24no—
438C2—202no—
439C2—202no—
440C2C2×#404411yescharacteristic
441C2#6588411yescharacteristic
442C2#6589011yescharacteristic
443C2#6592511yescharacteristic
444C2#6596111yescharacteristic
445C2#6597811yescharacteristic
446C2#6599211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#385C2
prime radicalC24×D5C22
hypercenter#385D5
hypocenterC5#2480
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#4044C2semidirect
3. C5×#385C2semidirect
4. #65501C2semidirect
5. #65283C2semidirect
6. #65282C2semidirect
7. #65279C2semidirect
10. #65235C2semidirect
11. C22⋊C4×Dic5C2semidirect
12. #64964C2semidirect
13. #64963C2semidirect
14. #64959C2semidirect
15. #64958C2semidirect
19. C2×C5×C4⋊C4C22semidirect
20. C2×C5×C4⋊C4C22semidirect
38. C2×C4⋊Dic5C22semidirect
39. C2×C4⋊Dic5C22semidirect
40. C2×#683C22semidirect
41. C2×#683C22semidirect
42. C2×C4×Dic5C22semidirect
43. C2×C4×Dic5C22semidirect
45. C5×#95C22semidirect
46. C5×#95C22semidirect
47. #3963C22semidirect
48. #3963C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
116. #385D5semidirect
196. C2×C4⋊C4C2×D5semidirect
197. C2×C4⋊C4C2×D5semidirect
223. #95C2×D5semidirect
224. #95C2×D5semidirect
231. C22×C5#126semidirect
395. C5#2480semidirect
417. C22#4044semidirect

Maximal subgroups

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

subgroupindex
C22×#6852
C2×#40442
C5×#3852
#655012
#652832
#652822
#652792
#652442
#652352
C22⋊C4×Dic52
#649642
#649632
#649592
#649582
#24805