Finite Group Databasebeta

← #20284023

Subgroups — #20284023

Order 6784.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133263C211yesmaximal, characteristic, complemented
2C2×#13133244C211yesmaximal, characteristic, complemented
3C2×#13133240C211yesmaximal, characteristic, complemented
4#16811203C211yesmaximal, characteristic, complemented
5#16810784C211yesmaximal, characteristic, complemented
6#16810773C211yesmaximal, characteristic, complemented
7#16810772C211yesmaximal, characteristic, complemented
8#16810771C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810759C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810623C211yesmaximal, characteristic, complemented
13#102×D53C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16#13133263—24no—
17#13133263—24no—
18C2×#11719949C2211yescharacteristic
19C2×#11719949C2211yescharacteristic
20C2×#11719949C2211yescharacteristic, complemented
21C2×#11719945—21no—
22C2×#11719945C2211yescharacteristic
23C22×D212—21no—
24C22×D212C2211yescharacteristic
25C2×D4×D53—21no—
26C2×D4×D53—21no—
27C2×D4×D53C2211yescharacteristic
28C2×D4×D53C2211yescharacteristic
29#13133244—24no—
30#13133240—24no—
31C2×#11719924C2211yescharacteristic, complemented
32C2×#11719924C2211yescharacteristic, complemented
33C2×#11719924C2211yescharacteristic, complemented
34C2×D424C2211yescharacteristic, complemented
35C2×D424C2211yescharacteristic, complemented
36C2×#11719922C2211yescharacteristic, complemented
37C2×#11719912C2211yescharacteristic, complemented
38C2×C8×D53C2211yescharacteristic, complemented
39#13133195—21no—
40#13133193C2211yescharacteristic, complemented
41#13133192—21no—
42#13133192C2211yescharacteristic, complemented
43#13133191C2211yescharacteristic, complemented
44#13133190—21no—
45#13133190—21no—
46C4⋊C4×D53C2211yescharacteristic, complemented
47#13133141C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133138C2211yescharacteristic, complemented
51#13133137C2211yescharacteristic, complemented
52#13133137C2211yescharacteristic, complemented
53C22⋊#88322C2211yescharacteristic, complemented
54#13133129C2211yescharacteristic, complemented
55#13133129C2211yescharacteristic, complemented
56C2×#41×C53C2211yescharacteristic
57C2×Q16×C53C2211yescharacteristic, complemented
58C2×SD16×C53C2211yescharacteristic, complemented
59C22⋊Q8×C53C2211yescharacteristic, complemented
60#103×C53C2211yescharacteristic, complemented
61#102×C53C2211yescharacteristic, complemented
62C22⋊C8×C53C2211yescharacteristic, complemented
63C23×D53—21no—
64C23×D53—21no—
65C23×D53—21no—
66#41×C53—22no—
67#41×C53—22no—
68C2×Q8×C53C2311yescharacteristic
69C2×Q8×C53C2311yescharacteristic
70C2×D4×C53—21no—
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53D411yescharacteristic
78C2×C22⋊D53D411yescharacteristic
79#11719949—22no—
80#11719949—22no—
81#11719949—22no—
82#11719949—22no—
83#11719949—22no—
84#11719949D414yescomplemented
85D4×D53—28no—
86D4×D53—28no—
87D4×D53—22no—
88D4×D53—22no—
89D4×D53—22no—
90D4×D53D414yescomplemented
91#11719945—28no—
92#11719945—22no—
93#11719945—22no—
94C2×D212—28no—
95C2×D212—21no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212—22no—
99C2×D212—22no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212D411yescharacteristic
103C2×D212C2311yescharacteristic
104C2×D212C2311yescharacteristic
105C2×D212C2311yescharacteristic
106C2×C4×D53—21no—
107C2×C4×D53—21no—
108C2×C4×D53—21no—
109C2×C4×D53—21no—
110C2×C4×D53C2311yescharacteristic
111C2×C4×D53C2311yescharacteristic
112C2×Dic106D411yescharacteristic
113Q16×C53—22no—
114SD16×C53—22no—
115C2×C8×C53C2311yescharacteristic
116C2×C8×C53C2311yescharacteristic
117C4⋊C4×C53—21no—
118C4⋊C4×C53C2311yescharacteristic
119C22⋊C4×C53—21no—
120#11719924—22no—
121#11719924—22no—
122#11719924—22no—
123#11719922—22no—
124#11719921—21no—
125#11719921—21no—
126#11719921—21no—
127C4⋊Dic53C2311yescharacteristic
128#11719919—21no—
129#11719919—21no—
130C2×#88322C2311yescharacteristic
131C2×#88322C2311yescharacteristic
132D424—22no—
133D424—22no—
134#11719912—22no—
135C8×D53—22no—
136C23×C53—21no—
137C23×C53C2×D411yescharacteristic
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—24no—
141C22×D53—24no—
142C22×D53—24no—
143C22×D53—21no—
144C22×D53—81no—
145C22×D53—81no—
146C22×D53—81no—
147C22×D53—81no—
148C22×D53—81no—
149C22×D53—81no—
150C22×D53C2×D411yescharacteristic
151C22×D53C2×D411yescharacteristic
152Q8×C53—21no—
153Q8×C53—21no—
154Q8×C53—21no—
155Q8×C53C2×D412yes—
156D4×C53—24no—
157D4×C53—21no—
158D4×C53C2×D412yes—
159C2×C4×C53—24no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53C2×D411yescharacteristic
166C2×C4×C53C2411yescharacteristic
167C22⋊D53—24no—
168C22⋊D53—22no—
169C22⋊D53—22no—
170C22⋊D53—24no—
171C22⋊D53—81no—
172C2×Dic53—21no—
173C2×Dic53C2×D411yescharacteristic
174C2×Dic53C2×D411yescharacteristic
175D212—24no—
176D212—24no—
177D212—22no—
178D212—24no—
179D212—21no—
180D212—21no—
181D212—21no—
182D212—21no—
183D212—42no—
184D212—81no—
185D212C2×D412yes—
186D212C2×D412yes—
187C4×D53—24no—
188C4×D53—21no—
189C4×D53—21no—
190C4×D53—24no—
191C4×D53—42no—
192C4×D53—81no—
193C4×D53C2×D412yes—
194C4×D53C2×D412yes—
195Dic106—22no—
196C8×C53—21no—
197C8×C53—21no—
198#88322—21no—
199#88322—21no—
200C22×C53—21no—
201C22×C53—22no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53C22×D411yescharacteristic
207C2×D53—21no—
208C2×D53—21no—
209C2×D53—21no—
210C2×D53—22no—
211C2×D53—22no—
212C2×D53—21no—
213C2×D53—21no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53—41no—
217C2×D53—41no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—84no—
222C2×D53—84no—
223C2×D53—84no—
224C2×D53#12012yes—
225C4×C53—21no—
226C4×C53—22no—
227C4×C53—41no—
228C4×C53—41no—
229C4×C53C22×D411yescharacteristic
230C4×C53C22×D411yescharacteristic
231Dic53—21no—
232Dic53—41no—
233Dic53#12012yes—
234#3012—531nomaximal, Sylow
235C2×C53—41no—
236C2×C53—42no—
237C2×C53C2×#12011yescharacteristic
238C2×C53#57611yescharacteristic
239C2×C53#57711yescharacteristic
240D53—22no—
241D53—41no—
242D53—42no—
243D53—42no—
244D53—81no—
245C2×H(5,𝔽2)—531no—
246C2×AGL(1,ℤ/8)—531no—
247C2×#135—531no—
248#537—531no—
249#452—531no—
250#452D5311yescharacteristic, complemented
251#450—531no—
252#450—531no—
253#449—531no—
254#449—531no—
255#447—531no—
256#447—531no—
257#417—531no—
258C2×#102—531no—
259#408—531no—
260C53#301211yescharacteristic, complemented, Sylow
261H(5,𝔽2)—1064no—
262H(5,𝔽2)—1064no—
263C2×#41—531no—
264C2×#41—531no—
265C2×#41—531no—
266C2×#41—531no—
267C2×#41—1061no—
268C2×#41C2×D5311yescharacteristic
269C22×D4—531no—
270C22×D4—531no—
271C22×D4—531no—
272C22×D4—1061no—
273C22×D4—1061no—
274C22×D4—1061no—
275AGL(1,ℤ/8)—1064no—
276#135—1064no—
277C2×Q16C2×D5311yescharacteristic, complemented
278C2×SD16—531no—
279C2×SD16—531no—
280C2×SD16—531no—
281C2×SD16C2×D5311yescharacteristic, complemented
282C2×D8—531no—
283C2×D8—531no—
284C2×D8—531no—
285C2×M16—531no—
286C22×C8—531no—
287C22⋊Q8—531no—
288C22⋊Q8—1061no—
289C22⋊Q8C2×D5311yescharacteristic, complemented
290#121—531no—
291#121—531no—
292#121—1061no—
293#121—1061no—
294#121—1061no—
295C2×C4⋊C4—531no—
296#103—531no—
297#103C2×D5311yescharacteristic, complemented
298#102—531no—
299#102—531no—
300#102—531no—
301#102—531no—
302#102—531no—
303#102C2×D5311yescharacteristic, complemented
304C22⋊C8—531no—
305C22⋊C8—531no—
306C22⋊C8—531no—
307C22⋊C8C2×D5311yescharacteristic, complemented
308C24—1061no—
309C24—1061no—
310C24—1061no—
311#41—22no—
312#41—22no—
313#41—534no—
314#41—1062no—
315#41—1062no—
316#41—1068no—
317#41—1062no—
318#41—1062no—
319#41—1062no—
320#41—1062no—
321#41—1062no—
322C2×Q8—531no—
323C2×Q8C22×D5311yescharacteristic
324C2×Q8C22×D5311yescharacteristic
325C2×D4—21no—
326C2×D4—534no—
327C2×D4—531no—
328C2×D4—531no—
329C2×D4—531no—
330C2×D4—531no—
331C2×D4—531no—
332C2×D4—531no—
333C2×D4—1062no—
334C2×D4—1062no—
335C2×D4—1061no—
336C2×D4—1061no—
337C2×D4—1068no—
338C2×D4—1061no—
339C2×D4—1061no—
340C2×D4—1062no—
341C2×D4—1061no—
342C2×D4—1061no—
343C2×D4—1062no—
344C2×D4—1068no—
345C2×D4—1068no—
346C2×D4—1062no—
347C2×D4—1061no—
348C2×D4—1061no—
349C2×D4C22×D5311yescharacteristic
350C22×C4—21no—
351C22×C4—531no—
352C22×C4—531no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4—1061no—
356C22×C4—1061no—
357C22×C4C22×D5311yescharacteristic
358Q16—22no—
359SD16—22no—
360SD16—1062no—
361SD16—1062no—
362SD16—1062no—
363D8—1062no—
364D8—1062no—
365D8—1062no—
366M16—1062no—
367C2×C8—531no—
368C2×C8—531no—
369C2×C8—1062no—
370C2×C8C22×D5311yescharacteristic
371C2×C8C22×D5311yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—531no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4C22×D5311yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—1061no—
379C22⋊C4—1061no—
380C22⋊C4—1061no—
381C23—21no—
382C23—531no—
383C23—531no—
384C23—1064no—
385C23—1064no—
386C23—1064no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23—4241no—
391C23—4241no—
392C23—4241no—
393C23—4241no—
394C23—4241no—
395C23—4241no—
396C23D4×D5311yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—21no—
400Q8—1062no—
401Q8D4×D5312yes—
402D4—24no—
403D4—21no—
404D4—532no—
405D4—532no—
406D4—1062no—
407D4—1064no—
408D4—1064no—
409D4—1064no—
410D4—1062no—
411D4—1064no—
412D4—1062no—
413D4—1064no—
414D4—1061no—
415D4—1061no—
416D4—1061no—
417D4—1061no—
418D4—4241no—
419D4—4241no—
420D4—2122no—
421D4D4×D5312yes—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—531no—
429C2×C4—532no—
430C2×C4—532no—
431C2×C4—531no—
432C2×C4—1064no—
433C2×C4—1064no—
434C2×C4—1061no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—4241no—
438C2×C4—2122no—
439C2×C4D4×D5311yescharacteristic
440C2×C4C23×D5311yescharacteristic
441C8—21no—
442C8—21no—
443C8—1061no—
444C8—1061no—
445C22—21no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—41no—
450C22—41no—
451C22—532no—
452C22—1061no—
453C22—1061no—
454C22—1062no—
455C22—1062no—
456C22—1061no—
457C22—1061no—
458C22—1061no—
459C22—4244no—
460C22—4244no—
461C22—4244no—
462C22—2121no—
463C22—2121no—
464C22—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22C2×D4×D5311yescharacteristic
470C4—21no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—532no—
475C4—1061no—
476C4—2121no—
477C4C2×D4×D5311yescharacteristic
478C4C2×D4×D5311yescharacteristic
479C2—41no—
480C2—42no—
481C2—1062no—
482C2—4241no—
483C2—2121no—
484C2—2122no—
485C2—2122no—
486C2#120×D5311yescharacteristic
487C2#1681142511yescharacteristic
488C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3012
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. C2×#13133244C2semidirect
3. C2×#13133240C2semidirect
4. #16811203C2semidirect
5. #16810784C2semidirect
6. #16810773C2semidirect
7. #16810772C2semidirect
8. #16810771C2semidirect
9. #16810768C2semidirect
10. #16810759C2semidirect
11. #16810758C2semidirect
12. #16810623C2semidirect
13. #102×D53C2semidirect
14. #16810588C2semidirect
15. C53×#452C2semidirect
20. C2×#11719949C22semidirect
31. C2×#11719924C22semidirect
32. C2×#11719924C22semidirect
33. C2×#11719924C22semidirect
34. C2×D424C22semidirect
35. C2×D424C22semidirect
36. C2×#11719922C22semidirect
37. C2×#11719912C22semidirect
38. C2×C8×D53C22semidirect
40. #13133193C22semidirect
42. #13133192C22semidirect
43. #13133191C22semidirect
46. C4⋊C4×D53C22semidirect
47. #13133141C22semidirect
48. #13133139C22semidirect
49. #13133139C22semidirect
50. #13133138C22semidirect
51. #13133137C22semidirect
52. #13133137C22semidirect
53. C22⋊#88322C22semidirect
54. #13133129C22semidirect
55. #13133129C22semidirect
57. C2×Q16×C53C22semidirect
58. C2×SD16×C53C22semidirect
59. C22⋊Q8×C53C22semidirect
60. #103×C53C22semidirect
61. #102×C53C22semidirect
62. C22⋊C8×C53C22semidirect
84. #11719949D4semidirect
90. D4×D53D4semidirect
250. #452D53semidirect
260. C53#3012semidirect
277. C2×Q16C2×D53semidirect
281. C2×SD16C2×D53semidirect
289. C22⋊Q8C2×D53semidirect
297. #103C2×D53semidirect
303. #102C2×D53semidirect
307. C22⋊C8C2×D53semidirect