Finite Group Databasebeta

← #10614812

Subgroups — #10614812

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
1C2×C5×#126C211yesmaximal, characteristic, complemented
2C2×D5×C22⋊C4C211yesmaximal, characteristic
3#65283C211yesmaximal, characteristic, complemented
4#65282C211yesmaximal, characteristic, complemented
5#65279C211yesmaximal, characteristic, complemented
6#65278C211yesmaximal, characteristic, complemented
7#65278C211yesmaximal, characteristic, complemented
8#65276C211yesmaximal, characteristic, complemented
9#65244C211yesmaximal, characteristic
10#65244C211yesmaximal, characteristic
11#65241C211yesmaximal, characteristic, complemented
12#65241C211yesmaximal, characteristic, complemented
13#65240C211yesmaximal, characteristic, complemented
14#65229C211yesmaximal, characteristic, complemented
15C4×#685C211yesmaximal, characteristic, complemented
16C24×D5C2211yescharacteristic
17C22×C4×D5C2211yescharacteristic
18C22×C4×D5C2211yescharacteristic
19C5×#126—24no—
20C2×C5×C4⋊C4C2211yescharacteristic, complemented
21C2×C5×C4⋊C4C2211yescharacteristic, complemented
22C2×C5×C4⋊C4C2211yescharacteristic, complemented
23C2×C5×C22⋊C4C2211yescharacteristic
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C42×C5C2211yescharacteristic, complemented
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×#685C2211yescharacteristic
31C2×#685C2211yescharacteristic
32C2×#685C2211yescharacteristic
33C2×#685C2211yescharacteristic
34C2×#685C2211yescharacteristic
35C2×#685C2211yescharacteristic
36C2×#685C2211yescharacteristic
37C2×#685C2211yescharacteristic
38C2×#685C2211yescharacteristic
39C2×#685C2211yescharacteristic
40C2×C4⋊Dic5C2211yescharacteristic, complemented
41C2×#683C2211yescharacteristic, complemented
42C2×#683C2211yescharacteristic, complemented
43C2×C4×Dic5C2211yescharacteristic, complemented
44D5×C22⋊C4—24no—
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#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×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—22no—
80C2×C4×D5—41no—
81C2×C4×D5—41no—
82C5×C4⋊C4—22no—
83C5×C4⋊C4—22no—
84C5×C4⋊C4—22no—
85C5×C22⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4D414yes—
88C42×C5—22no—
89C22⋊Dic5—22no—
90C22⋊Dic5—41no—
91C22⋊Dic5—41no—
92#685—22no—
93#685—22no—
94#685—22no—
95#685—22no—
96#685—22no—
97#685—22no—
98#685—22no—
99#685—22no—
100#685—41no—
101#685—41no—
102C4⋊Dic5—41no—
103#683—41no—
104#683—41no—
105C4×Dic5—41no—
106C2×#126D511yescharacteristic, complemented
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#385—51no—
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—21no—
126C23×C5—21no—
127C23×C5C2×D412yes—
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×D5#4112yes—
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×C5C2×D412yes—
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×Dic5—21no—
186C4×D5—44no—
187C4×D5—44no—
188C25—51no—
189C23×C4—51no—
190C23×C4—51no—
191#126—24no—
192C2×C4⋊C4—51no—
193C2×C4⋊C4—51no—
194C2×C4⋊C4—51no—
195C2×C4⋊C4C2×D511yescharacteristic, complemented
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—104no—
212C2×C22⋊C4C2×D511yescharacteristic
213C2×C22⋊C4C2×D511yescharacteristic
214C2×C22⋊C4C2×D511yescharacteristic
215C2×C42—51no—
216C2×C42C2×D511yescharacteristic, complemented
217#95—51no—
218#95—51no—
219#95—51no—
220#95—51no—
221#95—51no—
222#95—51no—
223#95—51no—
224#95—51no—
225C22×C5—22no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—41no—
229C22×C5—41no—
230C22×C5—41no—
231C22×C5—41no—
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—22no—
246C2×D5—22no—
247C2×D5—22no—
248C2×D5—416no—
249C2×D5—41no—
250C2×D5—41no—
251C2×D5—41no—
252C2×D5—41no—
253C2×D5#12614yescomplemented
254C4×C5—22no—
255C4×C5—22no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5—41no—
259C4×C5—41no—
260Dic5—41no—
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—51no—
277C22×C4—51no—
278C22×C4—51no—
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—102no—
283C22×C4—102no—
284C22×C4—102no—
285C22×C4—102no—
286C22×C4—201no—
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—22no—
296C4⋊C4—22no—
297C4⋊C4—201no—
298C4⋊C4—201no—
299C4⋊C4—201no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—102no—
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—201no—
312C22⋊C4—201no—
313C22⋊C4—201no—
314C22⋊C4—201no—
315C22⋊C4C22⋊D514yes—
316C42—22no—
317C42—201no—
318C2×C5—42no—
319C2×C5#52511yescharacteristic
320C2×C5C2×#12611yescharacteristic
321C2×C5#53811yescharacteristic
322C2×C5#54611yescharacteristic
323C2×C5#54611yescharacteristic
324C2×C5#55111yescharacteristic
325C2×C5#55111yescharacteristic
326D5—24no—
327D5—42no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—52no—
335C23—52no—
336C23—52no—
337C23—52no—
338C23—108no—
339C23—101no—
340C23—101no—
341C23—101no—
342C23—101no—
343C23—101no—
344C23—101no—
345C23—101no—
346C23—101no—
347C23—101no—
348C23—101no—
349C23—204no—
350C23—204no—
351C23—204no—
352C23—204no—
353C23—204no—
354C23—204no—
355C23C2×C22⋊D512yes—
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—101no—
374C2×C4—101no—
375C2×C4—101no—
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—204no—
392C2×C4—204no—
393C2×C4C2×C22⋊D512yes—
394C2×C4C2×C22⋊D512yes—
395C5#248011yescharacteristic, complemented, Sylow
396C22—22no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—54no—
404C22—102no—
405C22—102no—
406C22—102no—
407C22—102no—
408C22—102no—
409C22—102no—
410C22—102no—
411C22—102no—
412C22—102no—
413C22—2016no—
414C22—201no—
415C22—201no—
416C22—201no—
417C22—201no—
418C22D5×#4111yescharacteristic
419C22D5×#4111yescharacteristic
420C22D5×#4111yescharacteristic
421C22D5×#4111yescharacteristic
422C22#414911yescharacteristic
423C22#414911yescharacteristic
424C22C22×C22⋊D511yescharacteristic
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—201no—
432C4—201no—
433C4—201no—
434C4—201no—
435C4—201no—
436C4—201no—
437C2—42no—
438C2—104no—
439C2—202no—
440C2D5×#12611yescharacteristic
441C2#6603811yescharacteristic
442C2#6603911yescharacteristic
443C2#6603911yescharacteristic
444C2#6616211yescharacteristic
445C2#6616311yescharacteristic
446C2#6616311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#126C2
prime radicalC24×D5C22
hypercenterC2×#126D5
hypocenterC5#2480
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C5×#126C2semidirect
3. #65283C2semidirect
4. #65282C2semidirect
5. #65279C2semidirect
6. #65278C2semidirect
7. #65278C2semidirect
8. #65276C2semidirect
11. #65241C2semidirect
12. #65241C2semidirect
13. #65240C2semidirect
14. #65229C2semidirect
15. C4×#685C2semidirect
20. C2×C5×C4⋊C4C22semidirect
21. C2×C5×C4⋊C4C22semidirect
22. C2×C5×C4⋊C4C22semidirect
26. C2×C42×C5C22semidirect
40. C2×C4⋊Dic5C22semidirect
41. C2×#683C22semidirect
42. C2×#683C22semidirect
43. C2×C4×Dic5C22semidirect
45. #3963C22semidirect
46. #3963C22semidirect
47. #3963C22semidirect
48. #3963C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
106. C2×#126D5semidirect
195. C2×C4⋊C4C2×D5semidirect
196. C2×C4⋊C4C2×D5semidirect
197. C2×C4⋊C4C2×D5semidirect
216. C2×C42C2×D5semidirect
253. C2×D5#126semidirect
395. C5#2480semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1262
C2×D5×C22⋊C42
#652832
#652822
#652792
#652782
#652762
#652442
#652412
#652402
#652292
C4×#6852
#24805