Finite Group Databasebeta

← #20278838

Subgroups — #20278838

Order 6784.

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
1C2×#13133203C211yesmaximal, characteristic, complemented
2C2×#13133194C211yesmaximal, characteristic, complemented
3C2×#13133186C211yesmaximal, characteristic
4C22×C22⋊Dic53C211yesmaximal, characteristic, complemented
5#16810551C211yesmaximal, characteristic, complemented
6#16810539C211yesmaximal, characteristic, complemented
7#16810538C211yesmaximal, characteristic, complemented
8#16810535C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic, complemented
10#16810516C211yesmaximal, characteristic, complemented
11#16810511C211yesmaximal, characteristic, complemented
12#16810500C211yesmaximal, characteristic, complemented
13#16810500C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
15C53×#379C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic, complemented
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C23×Dic53C2211yescharacteristic
21C23×Dic53C2211yescharacteristic
22#13133203—24no—
23#13133194—24no—
24#13133186—24no—
25C2×C4⋊Dic53C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×#11719919C2211yescharacteristic, complemented
28C2×C22⋊Dic53—24no—
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×#11719921C2211yescharacteristic, complemented
37C2×#11719921C2211yescharacteristic, complemented
38C2×#11719921C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
40C2×C4×Dic53C2211yescharacteristic, complemented
41C2×C4×Dic53C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48C25×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53—22no—
56C24×C53—22no—
57C24×C53—22no—
58C24×C53—22no—
59C24×C53—22no—
60C24×C53—41no—
61C24×C53—41no—
62C24×C53C2311yescharacteristic
63C24×C53C2311yescharacteristic
64C24×C53C2311yescharacteristic
65C23×D53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C2×C22⋊D53—22no—
71C2×C22⋊D53—22no—
72C2×C22⋊D53—22no—
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—41no—
79C22×Dic53—22no—
80C22×Dic53—22no—
81C22×Dic53—22no—
82C22×Dic53—22no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C22×Dic53C2311yescharacteristic
92C2×Dic106—22no—
93C2×Dic106—22no—
94C2×Dic106—41no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊Dic53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53D414yes—
107C22⋊Dic53D414yescomplemented
108#11719921—22no—
109#11719921—22no—
110#11719921—41no—
111#11719921—41no—
112C4⋊Dic53—22no—
113#11719919—22no—
114#11719919—22no—
115C4×Dic53—22no—
116C4×Dic53—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—28no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—44no—
135C23×C53—44no—
136C23×C53—44no—
137C23×C53—44no—
138C23×C53—44no—
139C23×C53—44no—
140C23×C53C2×D412yes—
141C23×C53C2×D412yes—
142C23×C53#4112yes—
143C23×C53#4112yes—
144C23×C53#4112yes—
145C23×C53C2411yescharacteristic
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—21no—
152C22×D53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C22⋊D53—44no—
166C22⋊D53—44no—
167C22⋊D53—44no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—44no—
186C2×Dic53—44no—
187C2×Dic53C2×D412yes—
188C2×Dic53C2×D412yes—
189C2×Dic53C2×D412yes—
190C2×Dic53C2×D412yes—
191Dic106—44no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—22no—
202C22×C53—22no—
203C22×C53—22no—
204C22×C53—416no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53#12414yescomplemented
214C22×C53C22×D411yescharacteristic
215C22×C53C22×D411yescharacteristic
216C22×C53C2×#4111yescharacteristic
217C22×C53C2×#4111yescharacteristic
218C22×C53C2×#4111yescharacteristic
219C22×C53H(5,𝔽2)11yescharacteristic
220C22×C53H(5,𝔽2)11yescharacteristic
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C4×C53—41no—
229C4×C53—41no—
230C4×C53—41no—
231C4×C53—41no—
232Dic53—22no—
233Dic53—22no—
234Dic53—22no—
235Dic53—22no—
236Dic53—41no—
237Dic53—41no—
238Dic53—41no—
239#2472—531nomaximal, Sylow
240C2×C53—24no—
241C2×C53—42no—
242C2×C53—42no—
243C2×C53#52511yescharacteristic
244C2×C53C2×#12411yescharacteristic
245C2×C53#53811yescharacteristic
246C2×C53D4211yescharacteristic
247C2×C53#54611yescharacteristic
248C2×C53#54611yescharacteristic
249C2×C53#55111yescharacteristic
250D53—81no—
251C2×#124—531no—
252C2×C22⋊Q8—531no—
253C2×#120—531no—
254C22×C22⋊C4—531no—
255#397—531no—
256#394—531no—
257#394—531no—
258#393—531no—
259#390—531no—
260#388—531no—
261#388—531no—
262#386—531no—
263#386—531no—
264#379D5311yescharacteristic, complemented
265C4×C22⋊C4—531no—
266C53#247211yescharacteristic, complemented, Sylow
267C25C2×D5311yescharacteristic
268C22×Q8—531no—
269C22×D4—531no—
270C22×D4—531no—
271C22×D4—531no—
272C23×C4—531no—
273C23×C4—531no—
274#124—1064no—
275C22⋊Q8—1064no—
276#120—1064no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—531no—
282C2×C22⋊C4—531no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—531no—
286C2×C22⋊C4—531no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—531no—
290C2×C22⋊C4—531no—
291C2×C22⋊C4—1064no—
292C2×C22⋊C4C2×D5311yescharacteristic
293C2×C22⋊C4C2×D5311yescharacteristic
294C2×C22⋊C4C2×D5311yescharacteristic
295C2×C22⋊C4C2×D5311yescharacteristic
296C2×C22⋊C4C2×D5311yescharacteristic
297C2×C22⋊C4C2×D5311yescharacteristic
298C2×C42—531no—
299C2×C42—531no—
300#95—531no—
301#95—531no—
302#95—531no—
303#95—531no—
304#95—531no—
305#95—531no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24—531no—
314C24C22×D5311yescharacteristic
315C24C22×D5311yescharacteristic
316C24C22×D5311yescharacteristic
317C2×Q8—1062no—
318C2×Q8—1062no—
319C2×Q8—2121no—
320C2×D4—1062no—
321C2×D4—1062no—
322C2×D4—1062no—
323C2×D4—1062no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—1062no—
327C2×D4—2121no—
328C2×D4—2121no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—531no—
335C22×C4—531no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—1062no—
339C22×C4—1062no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4C22×D5311yescharacteristic
343C22×C4C22×D5311yescharacteristic
344C22×C4C22×D5311yescharacteristic
345C22×C4C22×D5311yescharacteristic
346C4⋊C4—1062no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—534no—
356C22⋊C4—534no—
357C22⋊C4—1062no—
358C22⋊C4—1062no—
359C22⋊C4—1062no—
360C22⋊C4—1062no—
361C22⋊C4—1062no—
362C22⋊C4—1062no—
363C22⋊C4—1062no—
364C22⋊C4—2121no—
365C22⋊C4—2121no—
366C42—1062no—
367C42—2121no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—1061no—
397C23—1061no—
398C23#1171994512yes—
399C23D4×D5312yes—
400C23D4×D5312yes—
401C23#1171994712yes—
402C23#1171994712yes—
403C23C23×D5311yescharacteristic
404Q8—2124no—
405D4—2124no—
406D4—2124no—
407D4—2124no—
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—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—532no—
421C2×C4—532no—
422C2×C4—532no—
423C2×C4—532no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—1061no—
434C2×C4—1061no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—1061no—
441C2×C4—2124no—
442C2×C4—2124no—
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—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22—2121no—
470C22—2121no—
471C22#1313320314yescomplemented
472C22C2×D4×D5311yescharacteristic
473C22C2×D4×D5311yescharacteristic
474C22C2×#1171994711yescharacteristic
475C22C2×#1171994711yescharacteristic
476C22C2×#1171994511yescharacteristic
477C22#1313326211yescharacteristic
478C22#1313326211yescharacteristic
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—1062no—
484C4—1062no—
485C4—1062no—
486C4—1062no—
487C4—2121no—
488C4—2121no—
489C4—2121no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—4241no—
494C2#1681112911yescharacteristic
495C2C2×#1313320311yescharacteristic
496C2#1681120811yescharacteristic
497C2#1681125511yescharacteristic
498C2#1681126011yescharacteristic
499C2#1681126311yescharacteristic
500C2#1681129911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#379C2
hypercenter#379D53
hypocenterC53#2472
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133194C2semidirect
4. C22×C22⋊Dic53C2semidirect
5. #16810551C2semidirect
6. #16810539C2semidirect
7. #16810538C2semidirect
8. #16810535C2semidirect
9. #16810523C2semidirect
10. #16810516C2semidirect
11. #16810511C2semidirect
12. #16810500C2semidirect
13. #16810500C2semidirect
14. C22⋊C4×Dic53C2semidirect
15. C53×#379C2semidirect
16. C22×Dic106C22semidirect
25. C2×C4⋊Dic53C22semidirect
26. C2×#11719919C22semidirect
27. C2×#11719919C22semidirect
36. C2×#11719921C22semidirect
37. C2×#11719921C22semidirect
38. C2×#11719921C22semidirect
39. C2×#11719921C22semidirect
40. C2×C4×Dic53C22semidirect
41. C2×C4×Dic53C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
107. C22⋊Dic53D4semidirect
213. C22×C53#124semidirect
264. #379D53semidirect
266. C53#2472semidirect
471. C22#13133203semidirect