Finite Group Databasebeta

← #20283972

Subgroups — #20283972

Order 6784.

754 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D53C211yesmaximal, characteristic
2C2×#13133249C211yesmaximal, characteristic, complemented
3C2×#13133246C211yesmaximal, characteristic, complemented
4C22⋊Q8×D53C211yesmaximal, characteristic, complemented
5#16810787C211yesmaximal, characteristic, complemented
6#16810785C211yesmaximal, characteristic, complemented
7#16810783C211yesmaximal, characteristic, complemented
8#16810780C211yesmaximal, characteristic, complemented
9#16810777C211yesmaximal, characteristic, complemented
10#16810775C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810623C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14#16810585C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945—21no—
18C2×#11719945C2211yescharacteristic
19C2×#11719947—21no—
20C2×#11719947C2211yescharacteristic
21#41×D53—24no—
22#41×D53—24no—
23C2×Q8×D53C2211yescharacteristic
24C2×Q8×D53C2211yescharacteristic, complemented
25C2×D4×D53—21no—
26C2×D4×D53C2211yescharacteristic
27C22×C4×D53—21no—
28C22×C4×D53C2211yescharacteristic
29#13133249—24no—
30#13133246—24no—
31C2×Dic212C2211yescharacteristic, complemented
32C2×#11719925C2211yescharacteristic, complemented
33C2×#11719925C2211yescharacteristic, complemented
34C2×#11719913C2211yescharacteristic, complemented
35C2×#11719923C2211yescharacteristic, complemented
36C2×#11719924C2211yescharacteristic, complemented
37C2×#11719912C2211yescharacteristic, complemented
38C2×#11719912C2211yescharacteristic, complemented
39#13133197C2211yescharacteristic
40#13133196C2211yescharacteristic, complemented
41#13133195—21no—
42#13133194—21no—
43#13133193C2211yescharacteristic
44C4⋊C4×D53—21no—
45C4⋊C4×D53C2211yescharacteristic, complemented
46C22⋊C4×D53—21no—
47#13133142C2211yescharacteristic, complemented
48#13133141C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic, complemented
51#13133138C2211yescharacteristic, complemented
52#13133137C2211yescharacteristic, complemented
53C22⋊#88322C2211yescharacteristic, complemented
54#13133129C2211yescharacteristic
55#13133129C2211yescharacteristic
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×D53D411yescharacteristic
65#41×C53—22no—
66#41×C53—22no—
67C2×Q8×C53C2311yescharacteristic
68C2×Q8×C53C2311yescharacteristic
69C2×D4×C53—21no—
70C2×D4×C53C2311yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C22×Dic53—21no—
77C22×Dic53D411yescharacteristic
78#11719949—22no—
79#11719949D414yes—
80Q8×D53—22no—
81Q8×D53—22no—
82Q8×D53—22no—
83Q8×D53—22no—
84#11719947—28no—
85#11719947—22no—
86#11719947D414yes—
87D4×D53—28no—
88D4×D53—22no—
89D4×D53—22no—
90#11719945—28no—
91#11719945—22no—
92#11719945—22no—
93C2×D212—21no—
94C2×D212C2311yescharacteristic
95C2×C4×D53—28no—
96C2×C4×D53—21no—
97C2×C4×D53—21no—
98C2×C4×D53—21no—
99C2×C4×D53—22no—
100C2×C4×D53—22no—
101C2×C4×D53—21no—
102C2×C4×D53—21no—
103C2×C4×D53D411yescharacteristic
104C2×C4×D53D411yescharacteristic
105C2×C4×D53C2311yescharacteristic
106C2×C4×D53C2311yescharacteristic
107C2×Dic106—21no—
108C2×Dic106—21no—
109C2×Dic106C2311yescharacteristic
110C2×Dic106C2311yescharacteristic
111Q16×C53—22no—
112SD16×C53—22no—
113C2×C8×C53C2311yescharacteristic
114C2×C8×C53C2311yescharacteristic
115C4⋊C4×C53—21no—
116C4⋊C4×C53C2311yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊Dic53—21no—
119#11719925—22no—
120#11719925—22no—
121#11719924—22no—
122#11719923—22no—
123#11719921—21no—
124#11719921—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53C2311yescharacteristic
127#11719919—21no—
128#11719919—21no—
129#11719919—21no—
130C2×#88322C2311yescharacteristic
131C2×#88322C2311yescharacteristic
132Dic212—22no—
133#11719913—22no—
134#11719912—22no—
135#11719912—22no—
136C23×C53—21no—
137C23×C53C2×D411yescharacteristic
138C22×D53—24no—
139C22×D53—21no—
140C22×D53—22no—
141C22×D53—22no—
142C22×D53—41no—
143C22×D53—42no—
144C22×D53—42no—
145C22×D53C2×D411yescharacteristic
146C22×D53C2×D411yescharacteristic
147Q8×C53—21no—
148Q8×C53—21no—
149Q8×C53—21no—
150Q8×C53C2×D412yes—
151D4×C53—24no—
152D4×C53—21no—
153D4×C53C2×D412yes—
154C2×C4×C53—24no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53C2×D411yescharacteristic
161C2×C4×C53C2411yescharacteristic
162C22⋊D53—24no—
163C22⋊D53—24no—
164C22⋊D53—24no—
165C2×Dic53—21no—
166C2×Dic53—22no—
167C2×Dic53—24no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53C2×D411yescharacteristic
171C2×Dic53C2×D411yescharacteristic
172D212—24no—
173D212—21no—
174D212C2×D412yes—
175C4×D53—24no—
176C4×D53—24no—
177C4×D53—22no—
178C4×D53—21no—
179C4×D53—21no—
180C4×D53—22no—
181C4×D53—24no—
182C4×D53—42no—
183C4×D53—42no—
184C4×D53C2×D412yes—
185C4×D53C2×D412yes—
186Dic106—24no—
187Dic106—21no—
188Dic106—21no—
189Dic106—21no—
190Dic106—42no—
191Dic106C2×D412yes—
192C8×C53—21no—
193C8×C53—21no—
194#88322—21no—
195#88322—21no—
196C22×C53—21no—
197C22×C53—22no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53C22×D411yescharacteristic
203C2×D53—21no—
204C2×D53—21no—
205C2×D53—21no—
206C2×D53—21no—
207C2×D53—22no—
208C2×D53—44no—
209C2×D53—48no—
210C2×D53—41no—
211C2×D53—41no—
212C2×D53#12012yes—
213C2×D53#13712yescomplemented
214C4×C53—22no—
215C4×C53—21no—
216C4×C53—41no—
217C4×C53—41no—
218C4×C53C22×D411yescharacteristic
219C4×C53C22×D411yescharacteristic
220Dic53—21no—
221Dic53—22no—
222Dic53—41no—
223Dic53—41no—
224Dic53#12012yes—
225#3007—531nomaximal, Sylow
226C2×C53—41no—
227C2×C53—42no—
228C2×C53C2×#12011yescharacteristic
229C2×C53C2×#13711yescharacteristic
230C2×C53#57511yescharacteristic
231D53—22no—
232D53—22no—
233D53—42no—
234C22×#41—531no—
235C2×#137—531no—
236C2×#137—531no—
237C2×C22⋊Q8—531no—
238#452—531no—
239#452—531no—
240#452—531no—
241#452D5311yescharacteristic, complemented
242#451—531no—
243#451—531no—
244#450—531no—
245#450—531no—
246#417—531no—
247#417—531no—
248#407—531no—
249C53#300711yescharacteristic, complemented, Sylow
250C2×#41—531no—
251C2×#41—531no—
252C2×#41—531no—
253C2×#41—1061no—
254C2×#41—1061no—
255C2×#41—1064no—
256C2×#41—1064no—
257C2×#41C2×D5311yescharacteristic
258C22×Q8—531no—
259C22×Q8—531no—
260C22×D4—531no—
261C22×D4—1061no—
262C23×C4—531no—
263C23×C4—1061no—
264#137—1064no—
265#137—1064no—
266C2×Q16—531no—
267C2×Q16—531no—
268C2×Q16—531no—
269C2×Q16C2×D5311yescharacteristic, complemented
270C2×SD16—531no—
271C2×SD16—531no—
272C2×SD16—531no—
273C2×SD16C2×D5311yescharacteristic, complemented
274C2×M16—531no—
275C2×M16—531no—
276C22⋊Q8—531no—
277C22⋊Q8—531no—
278C22⋊Q8—531no—
279C22⋊Q8—1061no—
280C22⋊Q8—1061no—
281C22⋊Q8C2×D5311yescharacteristic, complemented
282C2×C4⋊C4—531no—
283C2×C4⋊C4—1061no—
284C2×C22⋊C4—1061no—
285#103—531no—
286#103—531no—
287#103—531no—
288#103C2×D5311yescharacteristic, complemented
289#102—531no—
290#102—531no—
291#102—531no—
292#102C2×D5311yescharacteristic, complemented
293C22⋊C8—531no—
294C22⋊C8—531no—
295C22⋊C8—531no—
296C22⋊C8C2×D5311yescharacteristic, complemented
297C24—531no—
298C24—1061no—
299#41—22no—
300#41—22no—
301#41—534no—
302#41—534no—
303#41—1062no—
304#41—1062no—
305#41—1068no—
306#41—1062no—
307#41—1062no—
308#41—1068no—
309C2×Q8—531no—
310C2×Q8—531no—
311C2×Q8—1061no—
312C2×Q8—1061no—
313C2×Q8—1062no—
314C2×Q8—1062no—
315C2×Q8—1062no—
316C2×Q8—1062no—
317C2×Q8C22×D5311yescharacteristic
318C2×Q8C22×D5311yescharacteristic
319C2×D4—21no—
320C2×D4—531no—
321C2×D4—1061no—
322C2×D4—1061no—
323C2×D4—1062no—
324C2×D4—1068no—
325C2×D4—1061no—
326C2×D4—1061no—
327C2×D4—1062no—
328C2×D4C22×D5311yescharacteristic
329C22×C4—21no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—531no—
335C22×C4—1062no—
336C22×C4—1062no—
337C22×C4—1061no—
338C22×C4—1061no—
339C22×C4—1068no—
340C22×C4—1061no—
341C22×C4—1061no—
342C22×C4—1061no—
343C22×C4—1061no—
344C22×C4C22×D5311yescharacteristic
345Q16—22no—
346Q16—1062no—
347Q16—1062no—
348Q16—1062no—
349SD16—22no—
350SD16—1062no—
351SD16—1062no—
352SD16—1062no—
353M16—1062no—
354M16—1062no—
355C2×C8—531no—
356C2×C8—531no—
357C2×C8C22×D5311yescharacteristic
358C2×C8C22×D5311yescharacteristic
359C4⋊C4—21no—
360C4⋊C4—531no—
361C4⋊C4—1061no—
362C4⋊C4—1061no—
363C4⋊C4—1061no—
364C4⋊C4—1061no—
365C4⋊C4C22×D5311yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—1061no—
368C22⋊C4—1061no—
369C22⋊C4—1061no—
370C23—21no—
371C23—531no—
372C23—531no—
373C23—1062no—
374C23—1062no—
375C23—1064no—
376C23—1061no—
377C23—2121no—
378C23—2122no—
379C23—2122no—
380C23D4×D5311yescharacteristic
381Q8—21no—
382Q8—21no—
383Q8—21no—
384Q8—532no—
385Q8—1064no—
386Q8—1061no—
387Q8—1061no—
388Q8—1061no—
389Q8—2122no—
390Q8D4×D5312yes—
391D4—24no—
392D4—21no—
393D4—532no—
394D4—1064no—
395D4—1064no—
396D4—1064no—
397D4—1064no—
398D4—1061no—
399D4D4×D5312yes—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—531no—
407C2×C4—532no—
408C2×C4—532no—
409C2×C4—531no—
410C2×C4—1062no—
411C2×C4—1064no—
412C2×C4—1062no—
413C2×C4—1064no—
414C2×C4—1064no—
415C2×C4—1062no—
416C2×C4—1064no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1061no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—2122no—
423C2×C4—2122no—
424C2×C4D4×D5311yescharacteristic
425C2×C4C23×D5311yescharacteristic
426C8—21no—
427C8—21no—
428C8—1061no—
429C8—1061no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—532no—
437C22—532no—
438C22—1061no—
439C22—1061no—
440C22—1061no—
441C22—1061no—
442C22—1062no—
443C22—2124no—
444C22—2128no—
445C22—2121no—
446C22—2121no—
447C22C2×D4×D5311yescharacteristic
448C4—21no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—532no—
453C4—1061no—
454C4—1062no—
455C4—2121no—
456C4—2121no—
457C4C2×D4×D5311yescharacteristic
458C4C2×D4×D5311yescharacteristic
459C2—41no—
460C2—42no—
461C2—1062no—
462C2—1062no—
463C2—2122no—
464C2#120×D5311yescharacteristic
465C2#137×D5311yescharacteristic
466C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3007
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
2. C2×#13133249C2semidirect
3. C2×#13133246C2semidirect
4. C22⋊Q8×D53C2semidirect
5. #16810787C2semidirect
6. #16810785C2semidirect
7. #16810783C2semidirect
8. #16810780C2semidirect
9. #16810777C2semidirect
10. #16810775C2semidirect
11. #16810772C2semidirect
12. #16810623C2semidirect
13. #16810621C2semidirect
14. #16810585C2semidirect
15. C53×#452C2semidirect
24. C2×Q8×D53C22semidirect
31. C2×Dic212C22semidirect
32. C2×#11719925C22semidirect
33. C2×#11719925C22semidirect
34. C2×#11719913C22semidirect
35. C2×#11719923C22semidirect
36. C2×#11719924C22semidirect
37. C2×#11719912C22semidirect
38. C2×#11719912C22semidirect
40. #13133196C22semidirect
45. C4⋊C4×D53C22semidirect
47. #13133142C22semidirect
48. #13133141C22semidirect
49. #13133140C22semidirect
50. #13133139C22semidirect
51. #13133138C22semidirect
52. #13133137C22semidirect
53. C22⋊#88322C22semidirect
57. C2×Q16×C53C22semidirect
58. C2×SD16×C53C22semidirect
59. C22⋊Q8×C53C22semidirect
60. #103×C53C22semidirect
61. #102×C53C22semidirect
62. C22⋊C8×C53C22semidirect
213. C2×D53#137semidirect
241. #452D53semidirect
249. C53#3007semidirect
269. C2×Q16C2×D53semidirect
273. C2×SD16C2×D53semidirect
281. C22⋊Q8C2×D53semidirect
288. #103C2×D53semidirect
292. #102C2×D53semidirect
296. C22⋊C8C2×D53semidirect