Finite Group Databasebeta

← #10614330

Subgroups — #10614330

Order 640.

666 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#120C211yesmaximal, characteristic
2C2×#4025C211yesmaximal, characteristic, complemented
3#65536C211yesmaximal, characteristic, complemented
4#65536C211yesmaximal, characteristic, complemented
5#65511C211yesmaximal, characteristic, complemented
6#65511C211yesmaximal, characteristic, complemented
7#65511C211yesmaximal, characteristic, complemented
8#65511C211yesmaximal, characteristic, complemented
9#65510C211yesmaximal, characteristic, complemented
10#65510C211yesmaximal, characteristic, complemented
11#65240C211yesmaximal, characteristic, complemented
12#65238C211yesmaximal, characteristic, complemented
13#65237C211yesmaximal, characteristic, complemented
14#65234C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C25×C5C2211yescharacteristic
17C22×C5×D4C2211yescharacteristic
18C22×C5×D4C2211yescharacteristic
19C22×C5×D4C2211yescharacteristic
20C5×#120—24no—
21C2×C5×C22⋊C4C2211yescharacteristic
22C2×C5×C22⋊C4C2211yescharacteristic
23C2×C5×C22⋊C4C2211yescharacteristic
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×C22⋊Dic5C2211yescharacteristic
36C2×C22⋊Dic5C2211yescharacteristic, complemented
37C2×C22⋊Dic5C2211yescharacteristic, complemented
38C2×C22⋊Dic5C2211yescharacteristic, complemented
39C2×C22⋊Dic5C2211yescharacteristic, complemented
40C2×C4⋊Dic5C2211yescharacteristic, complemented
41C2×C4⋊Dic5C2211yescharacteristic, complemented
42C2×#683C2211yescharacteristic, complemented
43C2×#683C2211yescharacteristic, complemented
44C2×#683C2211yescharacteristic, complemented
45C2×#683C2211yescharacteristic, complemented
46C2×C4×Dic5C2211yescharacteristic, complemented
47C2×C4×Dic5C2211yescharacteristic, complemented
48#4025—24no—
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#2685—51nomaximal, Sylow
54C24×C5—22no—
55C24×C5—22no—
56C24×C5—22no—
57C24×C5—41no—
58C24×C5—41no—
59C24×C5—41no—
60C24×C5—41no—
61C24×C5C2311yescharacteristic
62C24×C5C2311yescharacteristic
63C24×C5C2311yescharacteristic
64C24×C5C2311yescharacteristic
65C2×C5×D4—22no—
66C2×C5×D4—22no—
67C2×C5×D4—22no—
68C2×C5×D4—22no—
69C2×C5×D4—22no—
70C2×C5×D4—41no—
71C2×C5×D4—41no—
72C2×C5×D4—41no—
73C2×C5×D4—41no—
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C22×Dic5C2311yescharacteristic
78C22×Dic5C2311yescharacteristic
79C22×Dic5C2311yescharacteristic
80C22×Dic5C2311yescharacteristic
81C22×Dic5C2311yescharacteristic
82C22×Dic5C2311yescharacteristic
83C22×Dic5C2311yescharacteristic
84C22×Dic5C2311yescharacteristic
85C5×C22⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4D414yes—
88C22⋊Dic5—22no—
89C22⋊Dic5—22no—
90C22⋊Dic5—22no—
91C22⋊Dic5—22no—
92C22⋊Dic5—41no—
93C22⋊Dic5—41no—
94C22⋊Dic5—41no—
95C22⋊Dic5—41no—
96C22⋊Dic5—41no—
97C22⋊Dic5—41no—
98C22⋊Dic5—41no—
99C22⋊Dic5—41no—
100C22⋊Dic5—41no—
101C22⋊Dic5—41no—
102C22⋊Dic5—41no—
103C22⋊Dic5—41no—
104C4⋊Dic5—22no—
105C4⋊Dic5—41no—
106#683—22no—
107#683—22no—
108#683—41no—
109#683—41no—
110C4×Dic5—22no—
111C4×Dic5—41no—
112C2×#126—51no—
113C2×#120D511yescharacteristic
114#399—51no—
115#396—51no—
116#394—51no—
117#394—51no—
118#394—51no—
119#394—51no—
120#390—51no—
121#390—51no—
122#388—51no—
123#388—51no—
124#388—51no—
125#379—51no—
126#379—51no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—21no—
130C23×C5—21no—
131C23×C5—21no—
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×C5—21no—
145C23×C5—21no—
146C23×C5—21no—
147C23×C5—21no—
148C23×C5—21no—
149C23×C5—21no—
150C23×C5—21no—
151C23×C5—21no—
152C23×C5—44no—
153C23×C5—44no—
154C23×C5—44no—
155C23×C5—44no—
156C23×C5—44no—
157C23×C5—44no—
158C23×C5—44no—
159C23×C5C2×D412yes—
160C23×C5#4112yes—
161C23×C5#4112yes—
162C23×C5C2411yescharacteristic
163C5×D4—44no—
164C5×D4—44no—
165C5×D4—44no—
166C2×C4×C5—21no—
167C2×C4×C5—21no—
168C2×C4×C5—21no—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5C2×D412yes—
174C2×C4×C5C2×D412yes—
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—21no—
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—
199C25C2×D511yescharacteristic
200C22×D4C2×D511yescharacteristic
201C22×D4C2×D511yescharacteristic
202C22×D4C2×D511yescharacteristic
203#126—104no—
204#120—24no—
205C2×C4⋊C4—51no—
206C2×C4⋊C4—51no—
207C2×C4⋊C4—51no—
208C2×C4⋊C4—51no—
209C2×C4⋊C4—51no—
210C2×C4⋊C4—51no—
211C2×C22⋊C4—51no—
212C2×C22⋊C4—51no—
213C2×C22⋊C4—51no—
214C2×C22⋊C4—51no—
215C2×C22⋊C4—51no—
216C2×C22⋊C4—51no—
217C2×C22⋊C4—51no—
218C2×C22⋊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⋊C4C2×D511yescharacteristic
228C2×C22⋊C4C2×D511yescharacteristic
229C2×C22⋊C4C2×D511yescharacteristic
230C2×C42—51no—
231C2×C42—51no—
232#95—51no—
233#95—51no—
234#95—51no—
235#95—51no—
236C22×C5—22no—
237C22×C5—22no—
238C22×C5—22no—
239C22×C5—22no—
240C22×C5—22no—
241C22×C5—22no—
242C22×C5—22no—
243C22×C5—22no—
244C22×C5—22no—
245C22×C5—416no—
246C22×C5—41no—
247C22×C5—41no—
248C22×C5—41no—
249C22×C5—41no—
250C22×C5—41no—
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×C5—41no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5—41no—
264C22×C5—41no—
265C22×C5C22×D411yescharacteristic
266C22×C5C2×#4111yescharacteristic
267C22×C5C2×#4111yescharacteristic
268C22×C5H(5,𝔽2)11yescharacteristic
269C22×C5H(5,𝔽2)11yescharacteristic
270C22×C5H(5,𝔽2)11yescharacteristic
271C22×C5H(5,𝔽2)11yescharacteristic
272C4×C5—22no—
273C4×C5—22no—
274C4×C5—41no—
275Dic5—41no—
276Dic5—41no—
277Dic5—41no—
278Dic5—41no—
279Dic5—41no—
280Dic5—41no—
281Dic5—41no—
282Dic5—41no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24C22×D511yescharacteristic
291C24C22×D511yescharacteristic
292C24C22×D511yescharacteristic
293C24C22×D511yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C22×C4—51no—
304C22×C4—51no—
305C22×C4—51no—
306C22×C4—51no—
307C22×C4—51no—
308C22×C4—51no—
309C22×C4—51no—
310C22×C4—51no—
311C22×C4C22×D511yescharacteristic
312C22×C4C22×D511yescharacteristic
313C22×C4C22×D511yescharacteristic
314C4⋊C4—102no—
315C4⋊C4—102no—
316C4⋊C4—102no—
317C4⋊C4—201no—
318C4⋊C4—201no—
319C4⋊C4—201no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—102no—
323C22⋊C4—102no—
324C22⋊C4—102no—
325C22⋊C4—102no—
326C22⋊C4—201no—
327C22⋊C4—201no—
328C22⋊C4—201no—
329C22⋊C4—201no—
330C22⋊C4—201no—
331C22⋊C4—201no—
332C22⋊C4—201no—
333C22⋊C4—201no—
334C22⋊C4—201no—
335C22⋊C4—201no—
336C22⋊C4—201no—
337C22⋊C4—201no—
338C22⋊C4C22⋊D514yes—
339C42—102no—
340C42—201no—
341C2×C5—42no—
342C2×C5—42no—
343C2×C5—42no—
344C2×C5—81no—
345C2×C5#53511yescharacteristic
346C2×C5#53811yescharacteristic
347C2×C5#53811yescharacteristic
348C2×C5#54611yescharacteristic
349C2×C5#54611yescharacteristic
350C2×C5#55111yescharacteristic
351C2×C5#56011yescharacteristic
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
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—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23#71112yes—
385C23#71112yes—
386C23C2×C22⋊D512yes—
387C23C23×D511yescharacteristic
388D4—44no—
389D4—44no—
390D4—44no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—101no—
399C2×C4—101no—
400C2×C4—101no—
401C2×C4—101no—
402C2×C4—101no—
403C2×C4—101no—
404C2×C4—101no—
405C2×C4—101no—
406C2×C4—101no—
407C2×C4—101no—
408C2×C4—101no—
409C2×C4—101no—
410C2×C4—101no—
411C2×C4—101no—
412C2×C4—101no—
413C2×C4—101no—
414C2×C4—101no—
415C2×C4—101no—
416C2×C4—101no—
417C2×C4—101no—
418C2×C4—101no—
419C2×C4—101no—
420C2×C4—101no—
421C2×C4—101no—
422C2×C4C2×C22⋊D512yes—
423C2×C4C2×C22⋊D512yes—
424C5#268511yescharacteristic, complemented, Sylow
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—416no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22C2×#71111yescharacteristic
455C22C2×#71111yescharacteristic
456C22#414411yescharacteristic
457C22#414411yescharacteristic
458C22#414411yescharacteristic
459C22#414411yescharacteristic
460C22C22×C22⋊D511yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—201no—
465C4—201no—
466C4—201no—
467C4—201no—
468C4—201no—
469C4—201no—
470C4—201no—
471C4—201no—
472C2—42no—
473C2—42no—
474C2—42no—
475C2—81no—
476C2#6592111yescharacteristic
477C2#6592711yescharacteristic
478C2#6592711yescharacteristic
479C2#6592911yescharacteristic
480C2#6613911yescharacteristic
481C2#6614011yescharacteristic
482C2#6614011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
Fitting, prime radicalC2×C5×#120C2
hypercenterC2×#120D5
hypocenterC5#2685
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C2×#4025C2semidirect
3. #65536C2semidirect
4. #65536C2semidirect
5. #65511C2semidirect
6. #65511C2semidirect
7. #65511C2semidirect
8. #65511C2semidirect
9. #65510C2semidirect
10. #65510C2semidirect
11. #65240C2semidirect
12. #65238C2semidirect
13. #65237C2semidirect
14. #65234C2semidirect
15. #65233C2semidirect
36. C2×C22⋊Dic5C22semidirect
37. C2×C22⋊Dic5C22semidirect
38. C2×C22⋊Dic5C22semidirect
39. C2×C22⋊Dic5C22semidirect
40. C2×C4⋊Dic5C22semidirect
41. C2×C4⋊Dic5C22semidirect
42. C2×#683C22semidirect
43. C2×#683C22semidirect
44. C2×#683C22semidirect
45. C2×#683C22semidirect
46. C2×C4×Dic5C22semidirect
47. C2×C4×Dic5C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
424. C5#2685semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1202
C2×#40252
#655362
#655112
#655102
#652402
#652382
#652372
#652342
#652332
#26855