Finite Group Databasebeta

← #10613842

Subgroups — #10613842

Order 640.

786 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#4099C211yesmaximal, characteristic
3C2×#4070C211yesmaximal, characteristic, complemented
4C2×#4031C211yesmaximal, characteristic, complemented
5#65536C211yesmaximal, characteristic, complemented
6#65502C211yesmaximal, characteristic, complemented
7#65502C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic5C211yesmaximal, characteristic, complemented
9#65250C211yesmaximal, characteristic, complemented
10#65248C211yesmaximal, characteristic, complemented
11#65246C211yesmaximal, characteristic, complemented
12#65241C211yesmaximal, characteristic, complemented
13#65240C211yesmaximal, characteristic, complemented
14#65236C211yesmaximal, characteristic, complemented
15#65233C211yesmaximal, characteristic, complemented
16C25×C5C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic
19C22×C22⋊D5C2211yescharacteristic
20C22×C22⋊D5C2211yescharacteristic
21C22×C22⋊D5C2211yescharacteristic
22C22×Dic10C2211yescharacteristic, complemented
23C2×C5×C22⋊C4—24no—
24C2×C5×C22⋊C4C2211yescharacteristic
25C2×C5×C22⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28#4099—24no—
29C2×C22⋊Dic5C2211yescharacteristic
30C2×C22⋊Dic5C2211yescharacteristic
31C2×C22⋊Dic5C2211yescharacteristic
32C2×C22⋊Dic5C2211yescharacteristic
33C2×C22⋊Dic5C2211yescharacteristic
34C2×C22⋊Dic5C2211yescharacteristic
35C2×C22⋊Dic5C2211yescharacteristic
36C2×C22⋊Dic5C2211yescharacteristic
37C2×C22⋊Dic5C2211yescharacteristic
38C2×#685C2211yescharacteristic, complemented
39C2×#685C2211yescharacteristic, complemented
40C2×#685C2211yescharacteristic, complemented
41C2×#685C2211yescharacteristic, complemented
42C2×C4⋊Dic5C2211yescharacteristic, complemented
43#4070—24no—
44C2×#683C2211yescharacteristic, complemented
45C2×#683C2211yescharacteristic, complemented
46C2×C4×Dic5C2211yescharacteristic, complemented
47C2×C4×Dic5C2211yescharacteristic, complemented
48#4031—24no—
49#3963C2211yescharacteristic, complemented
50#3963C2211yescharacteristic, complemented
51#3963C2211yescharacteristic, complemented
52#3963C2211yescharacteristic, complemented
53#3963C2211yescharacteristic, complemented
54#3963C2211yescharacteristic, complemented
55#2472—51nomaximal, Sylow
56C24×C5—22no—
57C24×C5—22no—
58C24×C5—22no—
59C24×C5—22no—
60C24×C5—22no—
61C24×C5—41no—
62C24×C5—41no—
63C24×C5C2311yescharacteristic
64C24×C5C2311yescharacteristic
65C24×C5C2311yescharacteristic
66C23×D5C2311yescharacteristic
67C22×C4×C5—22no—
68C22×C4×C5—22no—
69C22×C4×C5—22no—
70C22×C4×C5—22no—
71C22×C4×C5—41no—
72C22×C4×C5—41no—
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C22×C4×C5C2311yescharacteristic
77C2×C22⋊D5—22no—
78C2×C22⋊D5—22no—
79C2×C22⋊D5—22no—
80C2×C22⋊D5—22no—
81C2×C22⋊D5—22no—
82C2×C22⋊D5—22no—
83C2×C22⋊D5—22no—
84C2×C22⋊D5—41no—
85C2×C22⋊D5—41no—
86C22×Dic5C2311yescharacteristic
87C22×Dic5C2311yescharacteristic
88C22×Dic5C2311yescharacteristic
89C22×Dic5C2311yescharacteristic
90C22×Dic5C2311yescharacteristic
91C22×Dic5C2311yescharacteristic
92C22×Dic5C2311yescharacteristic
93C2×Dic10—22no—
94C2×Dic10—22no—
95C2×Dic10—41no—
96C5×C22⋊C4—22no—
97C5×C22⋊C4—22no—
98C5×C22⋊C4—22no—
99C5×C22⋊C4D414yescomplemented
100C22⋊Dic5—22no—
101C22⋊Dic5—22no—
102C22⋊Dic5—22no—
103C22⋊Dic5—22no—
104C22⋊Dic5—22no—
105C22⋊Dic5—22no—
106C22⋊Dic5—41no—
107C22⋊Dic5—41no—
108C22⋊Dic5D414yes—
109#685—22no—
110#685—22no—
111#685—41no—
112#685—41no—
113C4⋊Dic5—22no—
114#683—22no—
115#683—22no—
116C4×Dic5—22no—
117C4×Dic5—41no—
118C2×#124—51no—
119C2×C22⋊Q8—51no—
120C2×#120—51no—
121C22×C22⋊C4D511yescharacteristic, complemented
122#397—51no—
123#394—51no—
124#394—51no—
125#393—51no—
126#390—51no—
127#388—51no—
128#388—51no—
129#386—51no—
130#386—51no—
131#379—51no—
132C4×C22⋊C4—51no—
133C23×C5—21no—
134C23×C5—21no—
135C23×C5—21no—
136C23×C5—21no—
137C23×C5—28no—
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—44no—
151C23×C5—44no—
152C23×C5—44no—
153C23×C5—44no—
154C23×C5—44no—
155C23×C5—44no—
156C23×C5C2×D412yes—
157C23×C5C2×D412yes—
158C23×C5#4112yes—
159C23×C5#4112yes—
160C23×C5#4112yes—
161C23×C5C2411yescharacteristic
162C22×D5—21no—
163C22×D5—21no—
164C22×D5—21no—
165C22×D5—21no—
166C22×D5—21no—
167C22×D5—21no—
168C22×D5—21no—
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—44no—
180C2×C4×C5—44no—
181C2×C4×C5C2×D412yes—
182C2×C4×C5C2×D412yes—
183C22⋊D5—44no—
184C22⋊D5—44no—
185C22⋊D5—44no—
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—
199C2×Dic5—21no—
200C2×Dic5—21no—
201C2×Dic5—21no—
202C2×Dic5—21no—
203C2×Dic5—21no—
204C2×Dic5—21no—
205C2×Dic5C2×D412yes—
206C2×Dic5C2×D412yes—
207Dic10—44no—
208C25C2×D511yescharacteristic
209C22×Q8—51no—
210C22×D4—51no—
211C22×D4—51no—
212C22×D4—51no—
213C23×C4C2×D511yescharacteristic
214C23×C4C2×D511yescharacteristic
215#124—104no—
216C22⋊Q8—104no—
217#120—104no—
218C2×C4⋊C4—51no—
219C2×C4⋊C4—51no—
220C2×C4⋊C4—51no—
221C2×C22⋊C4—24no—
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—51no—
233C2×C22⋊C4—51no—
234C2×C22⋊C4—51no—
235C2×C22⋊C4C2×D511yescharacteristic
236C2×C22⋊C4C2×D511yescharacteristic
237C2×C22⋊C4C2×D511yescharacteristic
238C2×C22⋊C4C2×D511yescharacteristic
239C2×C42—51no—
240C2×C42—51no—
241#95—51no—
242#95—51no—
243#95—51no—
244#95—51no—
245#95—51no—
246#95—51no—
247C22×C5—22no—
248C22×C5—22no—
249C22×C5—22no—
250C22×C5—22no—
251C22×C5—22no—
252C22×C5—22no—
253C22×C5—22no—
254C22×C5—22no—
255C22×C5—22no—
256C22×C5—22no—
257C22×C5—22no—
258C22×C5—22no—
259C22×C5—416no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5—41no—
264C22×C5—41no—
265C22×C5—41no—
266C22×C5—41no—
267C22×C5—41no—
268C22×C5#12414yescomplemented
269C22×C5C22×D411yescharacteristic
270C22×C5C22×D411yescharacteristic
271C22×C5C2×#4111yescharacteristic
272C22×C5C2×#4111yescharacteristic
273C22×C5C2×#4111yescharacteristic
274C22×C5H(5,𝔽2)11yescharacteristic
275C22×C5H(5,𝔽2)11yescharacteristic
276C2×D5—41no—
277C2×D5—41no—
278C2×D5—41no—
279C2×D5—41no—
280C2×D5—41no—
281C2×D5—41no—
282C2×D5—41no—
283C4×C5—22no—
284C4×C5—22no—
285C4×C5—41no—
286C4×C5—41no—
287Dic5—22no—
288Dic5—22no—
289Dic5—41no—
290Dic5—41no—
291Dic5—41no—
292Dic5—41no—
293Dic5—41no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—41no—
300C24—41no—
301C24—51no—
302C24C22×D511yescharacteristic
303C24C22×D511yescharacteristic
304C24C22×D511yescharacteristic
305C2×Q8—102no—
306C2×Q8—102no—
307C2×Q8—201no—
308C2×D4—102no—
309C2×D4—102no—
310C2×D4—102no—
311C2×D4—102no—
312C2×D4—102no—
313C2×D4—102no—
314C2×D4—102no—
315C2×D4—201no—
316C2×D4—201no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—51no—
324C22×C4—51no—
325C22×C4—51no—
326C22×C4—51no—
327C22×C4—51no—
328C22×C4—51no—
329C22×C4—51no—
330C22×C4C22×D511yescharacteristic
331C22×C4C22×D511yescharacteristic
332C22×C4C22×D511yescharacteristic
333C22×C4C22×D511yescharacteristic
334C4⋊C4—102no—
335C4⋊C4—102no—
336C4⋊C4—102no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—54no—
341C22⋊C4—102no—
342C22⋊C4—102no—
343C22⋊C4—102no—
344C22⋊C4—102no—
345C22⋊C4—102no—
346C22⋊C4—102no—
347C22⋊C4—102no—
348C22⋊C4—102no—
349C22⋊C4—201no—
350C22⋊C4—201no—
351C22⋊C4—201no—
352C22⋊C4—201no—
353C22⋊C4C22⋊D514yescomplemented
354C42—102no—
355C42—201no—
356C2×C5—24no—
357C2×C5—42no—
358C2×C5—42no—
359C2×C5#52511yescharacteristic
360C2×C5C2×#12411yescharacteristic
361C2×C5#53811yescharacteristic
362C2×C5D4211yescharacteristic
363C2×C5#54611yescharacteristic
364C2×C5#54611yescharacteristic
365C2×C5#55111yescharacteristic
366D5—81no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—28no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—101no—
391C23—101no—
392C23—101no—
393C23—101no—
394C23—101no—
395C23—101no—
396C23—101no—
397C23#70912yes—
398C23#70912yes—
399C23D4×D512yes—
400C23#71112yes—
401C23C2×C22⋊D512yes—
402C23C23×D511yescharacteristic
403Q8—204no—
404D4—204no—
405D4—204no—
406D4—204no—
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—44no—
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—101no—
437C2×C4—101no—
438C2×C4—101no—
439C2×C4—101no—
440C2×C4C2×C22⋊D512yes—
441C2×C4C2×C22⋊D512yes—
442C5#247211yescharacteristic, complemented, Sylow
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—416no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—201no—
465C22—201no—
466C22—201no—
467C22—201no—
468C22—201no—
469C22—201no—
470C22—201no—
471C22#403114yescomplemented
472C22C2×#70911yescharacteristic
473C22C2×#70911yescharacteristic
474C22C2×D4×D511yescharacteristic
475C22C2×#71111yescharacteristic
476C22#414411yescharacteristic
477C22#414411yescharacteristic
478C22C22×C22⋊D511yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—102no—
484C4—102no—
485C4—201no—
486C4—201no—
487C4—201no—
488C4—201no—
489C4—201no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—401no—
494C2#6582411yescharacteristic
495C2C2×#403111yescharacteristic
496C2#6582811yescharacteristic
497C2#6582911yescharacteristic
498C2#6612511yescharacteristic
499C2D4×C22⋊D511yescharacteristic
500C2#6614011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC22×C5×C22⋊C4C2
prime radicalC2×#4099C2
hypercenterC22×C22⋊C4D5
hypocenterC5#2472
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C5×C22⋊C4C2semidirect
3. C2×#4070C2semidirect
4. C2×#4031C2semidirect
5. #65536C2semidirect
6. #65502C2semidirect
7. #65502C2semidirect
8. C4×C22⋊Dic5C2semidirect
9. #65250C2semidirect
10. #65248C2semidirect
11. #65246C2semidirect
12. #65241C2semidirect
13. #65240C2semidirect
14. #65236C2semidirect
15. #65233C2semidirect
22. C22×Dic10C22semidirect
38. C2×#685C22semidirect
39. C2×#685C22semidirect
40. C2×#685C22semidirect
41. C2×#685C22semidirect
42. C2×C4⋊Dic5C22semidirect
44. C2×#683C22semidirect
45. C2×#683C22semidirect
46. C2×C4×Dic5C22semidirect
47. C2×C4×Dic5C22semidirect
49. #3963C22semidirect
50. #3963C22semidirect
51. #3963C22semidirect
52. #3963C22semidirect
53. #3963C22semidirect
54. #3963C22semidirect
99. C5×C22⋊C4D4semidirect
121. C22×C22⋊C4D5semidirect
268. C22×C5#124semidirect
353. C22⋊C4C22⋊D5semidirect
442. C5#2472semidirect
471. C22#4031semidirect

Maximal subgroups

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

subgroupindex
C22×C5×C22⋊C42
C2×#40992
C2×#40702
C2×#40312
#655362
#655022
C4×C22⋊Dic52
#652502
#652482
#652462
#652412
#652402
#652362
#652332
#24725