Finite Group Databasebeta

← #10605572

Subgroups — #10605572

Order 640.

552 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#379C211yesmaximal, characteristic
2#65536C211yesmaximal, characteristic
3#65536C211yesmaximal, characteristic
4#65242C211yesmaximal, characteristic, complemented
5#65241C211yesmaximal, characteristic, complemented
6#65241C211yesmaximal, characteristic, complemented
7#65240C211yesmaximal, characteristic, complemented
8#65240C211yesmaximal, characteristic, complemented
9#65238C211yesmaximal, characteristic, complemented
10#65237C211yesmaximal, characteristic, complemented
11#65237C211yesmaximal, characteristic, complemented
12#65234C211yesmaximal, characteristic, complemented
13#65234C211yesmaximal, characteristic, complemented
14#65233C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C25×C5C2211yescharacteristic
17C2×C5×C22⋊C4C2211yescharacteristic
18C2×C5×C22⋊C4C2211yescharacteristic
19C2×C5×C22⋊C4C2211yescharacteristic
20C2×C5×C22⋊C4C2211yescharacteristic
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊Dic5C2211yescharacteristic
25C2×C22⋊Dic5C2211yescharacteristic
26C2×C22⋊Dic5C2211yescharacteristic
27C2×C22⋊Dic5C2211yescharacteristic
28C2×C22⋊Dic5C2211yescharacteristic
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C4⋊Dic5C2211yescharacteristic, complemented
36C2×C4⋊Dic5C2211yescharacteristic, complemented
37C2×#683C2211yescharacteristic, complemented
38C2×#683C2211yescharacteristic, complemented
39C2×#683C2211yescharacteristic, complemented
40C2×#683C2211yescharacteristic, complemented
41C2×C4×Dic5C2211yescharacteristic, complemented
42C2×C4×Dic5C2211yescharacteristic, complemented
43#3963C2211yescharacteristic, complemented
44#3963C2211yescharacteristic, complemented
45#3963C2211yescharacteristic, complemented
46#3963C2211yescharacteristic, complemented
47#3963C2211yescharacteristic, complemented
48#3963C2211yescharacteristic, complemented
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#2735—51nomaximal, Sylow
52C24×C5—22no—
53C24×C5—22no—
54C24×C5—22no—
55C24×C5—41no—
56C24×C5—41no—
57C24×C5—41no—
58C24×C5—41no—
59C24×C5C2311yescharacteristic
60C24×C5C2311yescharacteristic
61C24×C5C2311yescharacteristic
62C22×C4×C5C2311yescharacteristic
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×Dic5C2311yescharacteristic
67C22×Dic5C2311yescharacteristic
68C22×Dic5C2311yescharacteristic
69C22×Dic5C2311yescharacteristic
70C22×Dic5C2311yescharacteristic
71C22×Dic5C2311yescharacteristic
72C22×Dic5C2311yescharacteristic
73C22×Dic5C2311yescharacteristic
74C5×C22⋊C4—22no—
75C5×C22⋊C4—22no—
76C5×C22⋊C4—41no—
77C5×C22⋊C4—41no—
78C5×C22⋊C4—41no—
79C5×C22⋊C4—41no—
80C22⋊Dic5—22no—
81C22⋊Dic5—22no—
82C22⋊Dic5—22no—
83C22⋊Dic5—22no—
84C22⋊Dic5—41no—
85C22⋊Dic5—41no—
86C22⋊Dic5—41no—
87C22⋊Dic5—41no—
88C22⋊Dic5—41no—
89C22⋊Dic5—41no—
90C22⋊Dic5—41no—
91C22⋊Dic5—41no—
92C4⋊Dic5—41no—
93C4⋊Dic5—41no—
94#683—41no—
95#683—41no—
96#683—41no—
97#683—41no—
98C4×Dic5—41no—
99C4×Dic5—41no—
100#399—51no—
101#397—51no—
102#397—51no—
103#397—51no—
104#396—51no—
105#396—51no—
106#388—51no—
107#388—51no—
108#388—51no—
109#388—51no—
110#388—51no—
111#388—51no—
112#379—51no—
113#379—51no—
114#379D511yescharacteristic
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—21no—
132C23×C5—21no—
133C23×C5—44no—
134C23×C5—44no—
135C23×C5—44no—
136C23×C5—44no—
137C23×C5—44no—
138C23×C5—44no—
139C23×C5—44no—
140C23×C5#4112yes—
141C23×C5#4112yes—
142C23×C5#4112yes—
143C23×C5C2411yescharacteristic
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
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×Dic5—21no—
157C2×Dic5—21no—
158C2×Dic5—21no—
159C2×Dic5—21no—
160C2×Dic5—21no—
161C2×Dic5—21no—
162C2×Dic5—21no—
163C2×Dic5—21no—
164C2×Dic5—21no—
165C2×Dic5—21no—
166C2×Dic5—21no—
167C2×Dic5—21no—
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—
180C25C2×D511yescharacteristic
181C2×C4⋊C4—51no—
182C2×C4⋊C4—51no—
183C2×C4⋊C4—51no—
184C2×C4⋊C4—51no—
185C2×C4⋊C4—51no—
186C2×C4⋊C4—51no—
187C2×C22⋊C4—51no—
188C2×C22⋊C4—51no—
189C2×C22⋊C4—51no—
190C2×C22⋊C4—51no—
191C2×C22⋊C4—51no—
192C2×C22⋊C4—51no—
193C2×C22⋊C4—51no—
194C2×C22⋊C4—51no—
195C2×C22⋊C4—51no—
196C2×C22⋊C4—51no—
197C2×C22⋊C4—51no—
198C2×C22⋊C4—51no—
199C2×C22⋊C4C2×D511yescharacteristic
200C2×C22⋊C4C2×D511yescharacteristic
201C2×C22⋊C4C2×D511yescharacteristic
202C2×C22⋊C4C2×D511yescharacteristic
203C2×C22⋊C4C2×D511yescharacteristic
204C2×C22⋊C4C2×D511yescharacteristic
205C2×C42—51no—
206C2×C42—51no—
207#95—51no—
208#95—51no—
209#95—51no—
210#95—51no—
211#95—51no—
212#95—51no—
213#95—51no—
214#95—51no—
215C22×C5—22no—
216C22×C5—22no—
217C22×C5—22no—
218C22×C5—22no—
219C22×C5—22no—
220C22×C5—22no—
221C22×C5—22no—
222C22×C5—22no—
223C22×C5—22no—
224C22×C5—416no—
225C22×C5—41no—
226C22×C5—41no—
227C22×C5—41no—
228C22×C5—41no—
229C22×C5—41no—
230C22×C5—41no—
231C22×C5—41no—
232C22×C5—41no—
233C22×C5—41no—
234C22×C5—41no—
235C22×C5—41no—
236C22×C5—41no—
237C22×C5C2×#4111yescharacteristic
238C22×C5C2×#4111yescharacteristic
239C22×C5C2×#4111yescharacteristic
240C22×C5H(5,𝔽2)11yescharacteristic
241C22×C5H(5,𝔽2)11yescharacteristic
242C22×C5H(5,𝔽2)11yescharacteristic
243C22×C5H(5,𝔽2)11yescharacteristic
244C4×C5—41no—
245C4×C5—41no—
246C4×C5—41no—
247C4×C5—41no—
248Dic5—41no—
249Dic5—41no—
250Dic5—41no—
251Dic5—41no—
252Dic5—41no—
253Dic5—41no—
254Dic5—41no—
255Dic5—41no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24C22×D511yescharacteristic
264C24C22×D511yescharacteristic
265C24C22×D511yescharacteristic
266C22×C4—51no—
267C22×C4—51no—
268C22×C4—51no—
269C22×C4—51no—
270C22×C4—51no—
271C22×C4—51no—
272C22×C4—51no—
273C22×C4—51no—
274C22×C4C22×D511yescharacteristic
275C22×C4C22×D511yescharacteristic
276C22×C4C22×D511yescharacteristic
277C22×C4C22×D511yescharacteristic
278C4⋊C4—201no—
279C4⋊C4—201no—
280C4⋊C4—201no—
281C4⋊C4—201no—
282C4⋊C4—201no—
283C4⋊C4—201no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—102no—
291C22⋊C4—102no—
292C22⋊C4—102no—
293C22⋊C4—102no—
294C22⋊C4—201no—
295C22⋊C4—201no—
296C22⋊C4—201no—
297C22⋊C4—201no—
298C22⋊C4—201no—
299C22⋊C4—201no—
300C22⋊C4—201no—
301C22⋊C4—201no—
302C42—201no—
303C42—201no—
304C2×C5—42no—
305C2×C5—42no—
306C2×C5—42no—
307C2×C5#53811yescharacteristic
308C2×C5#53811yescharacteristic
309C2×C5#53811yescharacteristic
310C2×C5#55111yescharacteristic
311C2×C5#55111yescharacteristic
312C2×C5#55111yescharacteristic
313C2×C5#56011yescharacteristic
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23#70912yes—
340C23#71112yes—
341C23#71112yes—
342C23C23×D511yescharacteristic
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—101no—
358C2×C4—101no—
359C2×C4—101no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—101no—
363C2×C4—101no—
364C2×C4—101no—
365C2×C4—101no—
366C2×C4—101no—
367C2×C4—101no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—101no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—101no—
374C2×C4—101no—
375C2×C4—101no—
376C2×C4—101no—
377C2×C4—101no—
378C2×C4—101no—
379C5#273511yescharacteristic, complemented, Sylow
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—416no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22C2×#70911yescharacteristic
403C22C2×#71111yescharacteristic
404C22C2×#71111yescharacteristic
405C22#414411yescharacteristic
406C22#414411yescharacteristic
407C22#414411yescharacteristic
408C22#414411yescharacteristic
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—201no—
414C4—201no—
415C4—201no—
416C4—201no—
417C4—201no—
418C4—201no—
419C4—201no—
420C4—201no—
421C2—42no—
422C2—42no—
423C2—42no—
424C2#6582811yescharacteristic
425C2#6582911yescharacteristic
426C2#6582911yescharacteristic
427C2#6592111yescharacteristic
428C2#6592711yescharacteristic
429C2#6592711yescharacteristic
430C2#6592911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#379C2
prime radicalC25×C5C22
hypercenter#379D5
hypocenterC5#2735
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
4. #65242C2semidirect
5. #65241C2semidirect
6. #65241C2semidirect
7. #65240C2semidirect
8. #65240C2semidirect
9. #65238C2semidirect
10. #65237C2semidirect
11. #65237C2semidirect
12. #65234C2semidirect
13. #65234C2semidirect
14. #65233C2semidirect
15. #65233C2semidirect
35. C2×C4⋊Dic5C22semidirect
36. C2×C4⋊Dic5C22semidirect
37. C2×#683C22semidirect
38. C2×#683C22semidirect
39. C2×#683C22semidirect
40. C2×#683C22semidirect
41. C2×C4×Dic5C22semidirect
42. C2×C4×Dic5C22semidirect
43. #3963C22semidirect
44. #3963C22semidirect
45. #3963C22semidirect
46. #3963C22semidirect
47. #3963C22semidirect
48. #3963C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
379. C5#2735semidirect

Maximal subgroups

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

subgroupindex
C5×#3792
#655362
#652422
#652412
#652402
#652382
#652372
#652342
#652332
#27355