Finite Group Databasebeta

← #20277377

Subgroups — #20277377

Order 6784.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133187C211yesmaximal, characteristic
2C2×C4×D212C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810524C211yesmaximal, characteristic, complemented
5#16810523C211yesmaximal, characteristic, complemented
6#16810516C211yesmaximal, characteristic, complemented
7#16810515C211yesmaximal, characteristic, complemented
8#16810514C211yesmaximal, characteristic, complemented
9#16810501C211yesmaximal, characteristic, complemented
10#16810501C211yesmaximal, characteristic, complemented
11#16810499C211yesmaximal, characteristic, complemented
12#16810494C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic, complemented
14#16810470C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic, complemented
23#13133187C418yescomplemented
24C4×D212—24no—
25C2×C4⋊Dic53C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic, complemented
38C2×#11719921C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
40C22⋊C4×D53—24no—
41C2×C4×Dic53C2211yescharacteristic, complemented
42C2×C4×Dic53C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47C2×C4⋊C4×C53C2211yescharacteristic, complemented
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C42×C53C2211yescharacteristic, complemented
52#95×C53C2211yescharacteristic, complemented
53#95×C53C2211yescharacteristic, complemented
54C24×C53C2311yescharacteristic
55C23×D53—22no—
56C23×D53—22no—
57C23×D53—41no—
58C23×D53—41no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53C2×C414yescomplemented
62C23×D53C2311yescharacteristic
63C23×D53C2311yescharacteristic
64C23×D53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C2×C22⋊D53—41no—
72C2×C22⋊D53—41no—
73C2×C22⋊D53C2×C414yescomplemented
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C2×D212—22no—
80C2×D212—22no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212C2×C414yes—
84C2×D212C2×C414yescomplemented
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53D414yescomplemented
94C4⋊C4×C53—41no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53C2×C414yescomplemented
98C42×C53—22no—
99C22⋊Dic53—22no—
100C22⋊Dic53—41no—
101#11719921—22no—
102#11719921—22no—
103#11719921—22no—
104#11719921—22no—
105#11719921—22no—
106#11719921—41no—
107#11719921—41no—
108#11719921—41no—
109#11719921—41no—
110#11719921C2×C414yescomplemented
111#11719921C2×C414yescomplemented
112C4⋊Dic53—22no—
113#11719919—41no—
114C4×Dic53—41no—
115C4×Dic53—41no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53C22×C412yes—
123C23×C53C2411yescharacteristic
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—28no—
141C22×D53—44no—
142C22×D53—44no—
143C22×D53—44no—
144C22×D53—44no—
145C22×D53—44no—
146C22×D53—44no—
147C22×D53C22×C412yes—
148C22×D53C22×C412yes—
149C22×D53C22×C412yes—
150C22×D53C2×D412yes—
151C22×D53#4112yes—
152C2×C4×C53—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—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53C22×C412yes—
168C2×C4×C53C22×C412yes—
169C2×C4×C53C2×D412yes—
170C22⋊D53—44no—
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×Dic53C22×C412yes—
185C2×Dic53C2×D412yes—
186D212—28no—
187D212—44no—
188C4×D53—44no—
189C4×D53—44no—
190C4×D53—44no—
191C22×C53—22no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53C23×C411yescharacteristic
199C22×C53C22×D411yescharacteristic
200C22×C53C2×#4111yescharacteristic
201C22×C53H(5,𝔽2)11yescharacteristic
202C22×C53H(5,𝔽2)11yescharacteristic
203C22×C53H(5,𝔽2)11yescharacteristic
204C22×C53H(5,𝔽2)11yescharacteristic
205C2×D53—22no—
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—22no—
209C2×D53—22no—
210C2×D53—22no—
211C2×D53—22no—
212C2×D53—22no—
213C2×D53—22no—
214C2×D53—22no—
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—416no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53C4×D414yes—
227C4×C53—22no—
228C4×C53—22no—
229C4×C53—22no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233Dic53—22no—
234Dic53—22no—
235Dic53—41no—
236Dic53—41no—
237Dic53—41no—
238#2321—531nomaximal, Sylow
239C2×C53—42no—
240C2×C53C2×C4×D411yescharacteristic
241C2×C53#51811yescharacteristic
242C2×C53#51811yescharacteristic
243C2×C53#53411yescharacteristic
244C2×C53#53511yescharacteristic
245C2×C53#53811yescharacteristic
246C2×C53#53811yescharacteristic
247D53—24no—
248D53—42no—
249D53—42no—
250C2×#120—531no—
251C2×C4×D4—531no—
252C22×C22⋊C4—531no—
253#390—531no—
254#390—531no—
255#388—531no—
256#388—531no—
257#388—531no—
258#388D5311yescharacteristic, complemented
259#386—531no—
260#386—531no—
261#386—531no—
262#385—531no—
263#379—531no—
264C4×C22⋊C4—531no—
265C53#232111yescharacteristic, complemented, Sylow
266C25—531no—
267C22×D4—531no—
268C22×D4—531no—
269C22×D4—531no—
270C23×C4—531no—
271C23×C4—531no—
272C23×C4—531no—
273#120—538no—
274C4×D4—1064no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4C2×D5311yescharacteristic, complemented
278C2×C22⋊C4—531no—
279C2×C22⋊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×C42—531no—
296C2×C42—531no—
297C2×C42C2×D5311yescharacteristic, complemented
298#95—531no—
299#95—531no—
300#95—531no—
301#95—531no—
302#95C2×D5311yescharacteristic, complemented
303#95C2×D5311yescharacteristic, complemented
304C24—531no—
305C24—531no—
306C24—531no—
307C24—534no—
308C24—1062no—
309C24—1062no—
310C24—2121no—
311C24—2121no—
312C24—2121no—
313C24—2121no—
314C24C22×D5311yescharacteristic
315C2×D4—534no—
316C2×D4—534no—
317C2×D4—534no—
318C2×D4—1062no—
319C2×D4—1062no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C22×C4—531no—
325C22×C4—531no—
326C22×C4—534no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—1062no—
331C22×C4—1062no—
332C22×C4—2121no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4—2121no—
336C22×C4—2121no—
337C22×C4—2121no—
338C22×C4C22×D5311yescharacteristic
339C22×C4C22×D5311yescharacteristic
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C22×C4C22×D5311yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—1062no—
346C4⋊C4—2121no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—534no—
350C22⋊C4—534no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
353C22⋊C4—1062no—
354C22⋊C4—1062no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2121no—
362C22⋊C4C4×D5314yescomplemented
363C42—22no—
364C42—2121no—
365C42—2121no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—532no—
373C23—532no—
374C23—532no—
375C23—532no—
376C23—532no—
377C23—1068no—
378C23—1061no—
379C23—1061no—
380C23—1061no—
381C23—1061no—
382C23—1061no—
383C23—1061no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23—1061no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—2124no—
395C23—2124no—
396C23—2124no—
397C23—2124no—
398C23—2124no—
399C23—2124no—
400C23C2×C4×D5312yes—
401C23C23×D5311yescharacteristic
402D4—1068no—
403D4—2124no—
404D4—2124no—
405C2×C4—21no—
406C2×C4—21no—
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—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—532no—
421C2×C4—532no—
422C2×C4—1061no—
423C2×C4—1061no—
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—2124no—
436C2×C4—2124no—
437C2×C4—2124no—
438C2×C4C2×C4×D5312yes—
439C2×C4C2×C4×D5312yes—
440C2×C4D4×D5312yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—534no—
449C22—1062no—
450C22—1062no—
451C22—1062no—
452C22—1062no—
453C22—1062no—
454C22—1062no—
455C22—1062no—
456C22—1062no—
457C22—1062no—
458C22—1062no—
459C22—1062no—
460C22—1062no—
461C22—21216no—
462C22—2121no—
463C22—2121no—
464C22—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22—2121no—
470C22C22×C4×D5311yescharacteristic
471C22C2×D4×D5311yescharacteristic
472C22#41×D5311yescharacteristic
473C22#1313326211yescharacteristic
474C22#1313326311yescharacteristic
475C22#1313326311yescharacteristic
476C22#1313326311yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—1062no—
484C4—1062no—
485C4—2121no—
486C4—2121no—
487C4—2121no—
488C2—42no—
489C2—1064no—
490C2—2122no—
491C2—2122no—
492C2C4×D4×D5311yescharacteristic
493C2#1681109511yescharacteristic
494C2#1681109711yescharacteristic
495C2#1681118411yescharacteristic
496C2#1681118911yescharacteristic
497C2#1681120711yescharacteristic
498C2#1681121211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#388C2
socleC23×C53C24
hypercenter#388D53
hypocenterC53#2321
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D212C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810524C2semidirect
5. #16810523C2semidirect
6. #16810516C2semidirect
7. #16810515C2semidirect
8. #16810514C2semidirect
9. #16810501C2semidirect
10. #16810501C2semidirect
11. #16810499C2semidirect
12. #16810494C2semidirect
13. #16810478C2semidirect
14. #16810470C2semidirect
15. C53×#388C2semidirect
22. C22×C4×D53C22semidirect
23. #13133187C4semidirect
25. C2×C4⋊Dic53C22semidirect
26. C2×#11719919C22semidirect
37. C2×#11719921C22semidirect
38. C2×#11719921C22semidirect
39. C2×#11719921C22semidirect
41. C2×C4×Dic53C22semidirect
42. C2×C4×Dic53C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. C2×C4⋊C4×C53C22semidirect
51. C2×C42×C53C22semidirect
52. #95×C53C22semidirect
53. #95×C53C22semidirect
61. C23×D53C2×C4semidirect
73. C2×C22⋊D53C2×C4semidirect
84. C2×D212C2×C4semidirect
93. C2×C4×D53D4semidirect
97. C22⋊C4×C53C2×C4semidirect
110. #11719921C2×C4semidirect
111. #11719921C2×C4semidirect
258. #388D53semidirect
265. C53#2321semidirect
277. C2×C4⋊C4C2×D53semidirect
297. C2×C42C2×D53semidirect
302. #95C2×D53semidirect
303. #95C2×D53semidirect
362. C22⋊C4C4×D53semidirect