Finite Group Databasebeta

← #20278106

Subgroups — #20278106

Order 6784.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133199C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
5#16810551C211yesmaximal, characteristic, complemented
6#16810548C211yesmaximal, characteristic, complemented
7#16810537C211yesmaximal, characteristic, complemented
8#16810499C211yesmaximal, characteristic, complemented
9#16810499C211yesmaximal, characteristic, complemented
10#16810499C211yesmaximal, characteristic, complemented
11#16810494C211yesmaximal, characteristic, complemented
12#16810485C211yesmaximal, characteristic, complemented
13#16810482C211yesmaximal, characteristic, complemented
14#16810480C211yesmaximal, characteristic, complemented
15C53×#394C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23#13133199—24no—
24#13133185—24no—
25C2×C4⋊Dic53C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×#11719919C2211yescharacteristic, complemented
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic, complemented
32C2×C22⋊Dic53C2211yescharacteristic, complemented
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C22⋊C4×D53—24no—
40C22⋊C4×D53—24no—
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48C22×D4×C53C2211yescharacteristic
49C2×C4⋊C4×C53C2211yescharacteristic, complemented
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic, complemented
53C2×C22⋊C4×C53C2211yescharacteristic, complemented
54#95×C53C2211yescharacteristic, complemented
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—22no—
60C23×D53—22no—
61C23×D53—41no—
62C23×D53—41no—
63C23×D53D414yes—
64C23×D53C2311yescharacteristic
65C23×D53C2311yescharacteristic
66C2×D4×C53—22no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—22no—
75C2×C22⋊D53—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22×Dic53C2311yescharacteristic
84C22×Dic53C2311yescharacteristic
85C22×Dic53C2311yescharacteristic
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—22no—
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—22no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C2×C4×D53—41no—
98C4⋊C4×C53—41no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—41no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—22no—
107C22⋊Dic53—41no—
108#11719921—22no—
109#11719921—22no—
110#11719921—22no—
111#11719921—22no—
112#11719921—22no—
113#11719921D414yescomplemented
114C4⋊Dic53—22no—
115#11719919—22no—
116#11719919—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
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×C53C2×D412yes—
131C23×C53C2411yescharacteristic
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—28no—
142C22×D53—44no—
143C22×D53—44no—
144C22×D53—44no—
145C22×D53—44no—
146C22×D53—44no—
147C22×D53C2×D412yes—
148C22×D53C2×D412yes—
149C22×D53C2×D412yes—
150C22×D53#4112yes—
151C22×D53#4112yes—
152C22×D53#4112yes—
153D4×C53—44no—
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×C53—21no—
168C2×C4×C53C2×D412yes—
169C22⋊D53—44no—
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×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53C2×D412yes—
189C4×D53—44no—
190C4×D53—44no—
191C4×D53—44no—
192C4×D53—44no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53C22×D411yescharacteristic
208C22×C53C22×D411yescharacteristic
209C22×C53C2×#4111yescharacteristic
210C22×C53C2×#4111yescharacteristic
211C22×C53C2×#4111yescharacteristic
212C22×C53H(5,𝔽2)11yescharacteristic
213C22×C53H(5,𝔽2)11yescharacteristic
214C2×D53—22no—
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—22no—
225C2×D53—22no—
226C2×D53—416no—
227C2×D53#12114yes—
228C2×D53#12314yescomplemented
229C4×C53—22no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—41no—
234Dic53—22no—
235Dic53—41no—
236Dic53—41no—
237Dic53—41no—
238Dic53—41no—
239Dic53—41no—
240#2412—531nomaximal, Sylow
241C2×C53—42no—
242C2×C53—81no—
243C2×C53C2×#12111yescharacteristic
244C2×C53C2×#12311yescharacteristic
245C2×C53#53411yescharacteristic
246C2×C53#54611yescharacteristic
247C2×C53#54611yescharacteristic
248C2×C53#55111yescharacteristic
249C2×C53#55111yescharacteristic
250D53—24no—
251D53—24no—
252C2×#123—531no—
253C2×#120—531no—
254C22×C22⋊C4—531no—
255C22×C22⋊C4—531no—
256#397—531no—
257#397—531no—
258#394—531no—
259#394D5311yescharacteristic, complemented
260#386—531no—
261#386—531no—
262#386—531no—
263#385—531no—
264#381—531no—
265#380—531no—
266#380—531no—
267C53#241211yescharacteristic, complemented, Sylow
268C25—531no—
269C22×D4—531no—
270C22×D4—531no—
271C22×D4C2×D5311yescharacteristic
272C23×C4—531no—
273C23×C4—531no—
274C23×C4—531no—
275C23×C4—531no—
276#123—1064no—
277#120—1064no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4—531no—
281C2×C4⋊C4C2×D5311yescharacteristic, complemented
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—531no—
292C2×C22⋊C4—531no—
293C2×C22⋊C4—1064no—
294C2×C22⋊C4—1064no—
295C2×C22⋊C4C2×D5311yescharacteristic
296C2×C22⋊C4C2×D5311yescharacteristic
297C2×C22⋊C4C2×D5311yescharacteristic, complemented
298C2×C22⋊C4C2×D5311yescharacteristic, complemented
299#95—531no—
300#95—531no—
301#95—531no—
302#95—531no—
303#95—531no—
304#95—531no—
305#95—531no—
306#95C2×D5311yescharacteristic, complemented
307C24—531no—
308C24—531no—
309C24—534no—
310C24—1062no—
311C24—1062no—
312C24—1062no—
313C24—1062no—
314C24—2121no—
315C24—2121no—
316C24C22×D5311yescharacteristic
317C24C22×D5311yescharacteristic
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—1062no—
322C2×D4—1062no—
323C2×D4—1062no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—1062no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—1062no—
334C22×C4—1062no—
335C22×C4—1062no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—1062no—
339C22×C4—1062no—
340C22×C4—1062no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2121no—
345C22×C4C22×D5311yescharacteristic
346C22×C4C22×D5311yescharacteristic
347C22×C4C22×D5311yescharacteristic
348C22×C4C22×D5311yescharacteristic
349C22×C4C22×D5311yescharacteristic
350C4⋊C4—41no—
351C4⋊C4—1062no—
352C4⋊C4—1062no—
353C4⋊C4—2121no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—41no—
358C22⋊C4—534no—
359C22⋊C4—1062no—
360C22⋊C4—1062no—
361C22⋊C4—1062no—
362C22⋊C4—1062no—
363C22⋊C4—1062no—
364C22⋊C4—1062no—
365C22⋊C4—1062no—
366C22⋊C4—1062no—
367C22⋊C4—1062no—
368C22⋊C4—2121no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—532no—
383C23—532no—
384C23—532no—
385C23—532no—
386C23—532no—
387C23—532no—
388C23—1068no—
389C23—1068no—
390C23—1061no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—1061no—
397C23—1061no—
398C23—2124no—
399C23—2124no—
400C23—2124no—
401C23—2124no—
402C23—2124no—
403C23D4×D5312yes—
404C23C23×D5311yescharacteristic
405D4—44no—
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—21no—
421C2×C4—21no—
422C2×C4—532no—
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—1061no—
436C2×C4—1061no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—2124no—
441C2×C4—2124no—
442C2×C4—2124no—
443C2×C4—2124no—
444C2×C4D4×D5312yes—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—534no—
460C22—534no—
461C22—1062no—
462C22—1062no—
463C22—1062no—
464C22—1062no—
465C22—1062no—
466C22—1062no—
467C22—1062no—
468C22—1062no—
469C22—1062no—
470C22—1062no—
471C22—1062no—
472C22—1062no—
473C22—21216no—
474C22C2×D4×D5311yescharacteristic
475C22C2×D4×D5311yescharacteristic
476C22#41×D5311yescharacteristic
477C22#41×D5311yescharacteristic
478C22#41×D5311yescharacteristic
479C22#1313326211yescharacteristic
480C22#1313326211yescharacteristic
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—1062no—
487C4—2121no—
488C4—2121no—
489C4—2121no—
490C4—2121no—
491C4—2121no—
492C2—42no—
493C2—81no—
494C2—1064no—
495C2—1064no—
496C2#121×D5311yescharacteristic
497C2#123×D5311yescharacteristic
498C2#1681118311yescharacteristic
499C2#1681126211yescharacteristic
500C2#1681127011yescharacteristic
501C2#1681130111yescharacteristic
502C2#1681130311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2412
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. C2×C22⋊C4×D53C2semidirect
5. #16810551C2semidirect
6. #16810548C2semidirect
7. #16810537C2semidirect
8. #16810499C2semidirect
9. #16810499C2semidirect
10. #16810499C2semidirect
11. #16810494C2semidirect
12. #16810485C2semidirect
13. #16810482C2semidirect
14. #16810480C2semidirect
15. C53×#394C2semidirect
25. C2×C4⋊Dic53C22semidirect
26. C2×#11719919C22semidirect
27. C2×#11719919C22semidirect
31. C2×C22⋊Dic53C22semidirect
32. C2×C22⋊Dic53C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
49. C2×C4⋊C4×C53C22semidirect
52. C2×C22⋊C4×C53C22semidirect
53. C2×C22⋊C4×C53C22semidirect
54. #95×C53C22semidirect
113. #11719921D4semidirect
228. C2×D53#123semidirect
259. #394D53semidirect
267. C53#2412semidirect
281. C2×C4⋊C4C2×D53semidirect
297. C2×C22⋊C4C2×D53semidirect
298. C2×C22⋊C4C2×D53semidirect
306. #95C2×D53semidirect