Finite Group Databasebeta

← #20278377

Subgroups — #20278377

Order 6784.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133197C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic
4C2×C22⋊C4×D53C211yesmaximal, characteristic
5#16810559C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810535C211yesmaximal, characteristic, complemented
8#16810497C211yesmaximal, characteristic, complemented
9#16810494C211yesmaximal, characteristic, complemented
10#16810494C211yesmaximal, characteristic, complemented
11#16810494C211yesmaximal, characteristic, complemented
12#16810482C211yesmaximal, characteristic, complemented
13#16810480C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic
15C53×#396C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×Dic106C2211yescharacteristic, complemented
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22#13133197—24no—
23#13133195—24no—
24C2×C4⋊Dic53C2211yescharacteristic, complemented
25C2×C4⋊Dic53C2211yescharacteristic, complemented
26C2×#11719919C2211yescharacteristic, complemented
27C2×#11719919C2211yescharacteristic, complemented
28C2×#11719919C2211yescharacteristic, complemented
29C2×#11719919C2211yescharacteristic, complemented
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C22⋊C4×D53—24no—
42C22⋊C4×D53—24no—
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48C2×C4⋊C4×C53C2211yescharacteristic, complemented
49C2×C4⋊C4×C53C2211yescharacteristic, complemented
50C2×C4⋊C4×C53C2211yescharacteristic, complemented
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic, complemented
55C24×C53C2311yescharacteristic
56C23×D53—22no—
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—22no—
60C23×D53—41no—
61C23×D53—41no—
62C23×D53Q814yescomplemented
63C23×D53C2311yescharacteristic
64C23×D53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—22no—
81C2×C4×D53—22no—
82C2×C4×D53—22no—
83C2×C4×D53—22no—
84C2×C4×D53—22no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×Dic106—22no—
90C2×Dic106—22no—
91C2×Dic106—41no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—41no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—41no—
98C22⋊Dic53—22no—
99C22⋊Dic53—22no—
100C22⋊Dic53—41no—
101#11719921—22no—
102#11719921—22no—
103#11719921—22no—
104#11719921—22no—
105#11719921—22no—
106#11719921—22no—
107#11719921D414yes—
108#11719921D414yes—
109C4⋊Dic53—22no—
110C4⋊Dic53—22no—
111#11719919—22no—
112#11719919—22no—
113#11719919—41no—
114#11719919—41no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53C2×Q812yes—
122C23×C53C2411yescharacteristic
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53—28no—
132C22×D53—28no—
133C22×D53—44no—
134C22×D53—44no—
135C22×D53—44no—
136C22×D53—44no—
137C22×D53—44no—
138C22×D53C2×D412yes—
139C22×D53C2×D412yes—
140C22×D53C2×Q812yes—
141C22×D53C2×Q812yes—
142C22×D53#4112yes—
143C22×D53#4112yes—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
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×C53C2×D412yes—
161C2×C4×C53C2×D412yes—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
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×Dic53C2×D412yes—
179C2×Dic53C2×D412yes—
180C4×D53—44no—
181C4×D53—44no—
182C4×D53—44no—
183C4×D53—44no—
184Dic106—44no—
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53C22×D411yescharacteristic
193C22×C53C22×D411yescharacteristic
194C22×C53C22×Q811yescharacteristic
195C22×C53C2×#4111yescharacteristic
196C22×C53C2×#4111yescharacteristic
197C22×C53H(5,𝔽2)11yescharacteristic
198C22×C53H(5,𝔽2)11yescharacteristic
199C2×D53—22no—
200C2×D53—22no—
201C2×D53—22no—
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—22no—
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—22no—
209C2×D53—22no—
210C2×D53—22no—
211C2×D53—416no—
212C2×D53C22⋊Q814yescomplemented
213C2×D53C22⋊Q814yescomplemented
214C4×C53—22no—
215C4×C53—22no—
216C4×C53—41no—
217C4×C53—41no—
218C4×C53—41no—
219C4×C53—41no—
220Dic53—22no—
221Dic53—22no—
222Dic53—41no—
223Dic53—41no—
224Dic53—41no—
225Dic53—41no—
226#2437—531nomaximal, Sylow
227C2×C53—42no—
228C2×C53C2×C22⋊Q811yescharacteristic
229C2×C53C2×C22⋊Q811yescharacteristic
230C2×C53#54311yescharacteristic
231C2×C53D4211yescharacteristic
232C2×C53#54611yescharacteristic
233C2×C53#54611yescharacteristic
234C2×C53#55111yescharacteristic
235D53—24no—
236D53—24no—
237C2×C22⋊Q8—531no—
238C2×C22⋊Q8—531no—
239C22×C22⋊C4—531no—
240C22×C22⋊C4—531no—
241#399—531no—
242#396—531no—
243#396D5311yescharacteristic, complemented
244#393—531no—
245#385—531no—
246#385—531no—
247#385—531no—
248#385—531no—
249#380—531no—
250#380—531no—
251#379—531no—
252C53#243711yescharacteristic, complemented, Sylow
253C25—531no—
254C22×Q8—531no—
255C23×C4—531no—
256C23×C4—531no—
257C23×C4—531no—
258C23×C4—531no—
259C22⋊Q8—1064no—
260C22⋊Q8—1064no—
261C2×C4⋊C4—531no—
262C2×C4⋊C4—531no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4C2×D5311yescharacteristic, complemented
268C2×C4⋊C4C2×D5311yescharacteristic, complemented
269C2×C4⋊C4C2×D5311yescharacteristic, complemented
270C2×C22⋊C4—531no—
271C2×C22⋊C4—531no—
272C2×C22⋊C4—531no—
273C2×C22⋊C4—531no—
274C2×C22⋊C4—531no—
275C2×C22⋊C4—531no—
276C2×C22⋊C4—531no—
277C2×C22⋊C4—531no—
278C2×C22⋊C4—531no—
279C2×C22⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—1064no—
282C2×C22⋊C4—1064no—
283C2×C22⋊C4C2×D5311yescharacteristic
284C2×C22⋊C4C2×D5311yescharacteristic
285C2×C22⋊C4C2×D5311yescharacteristic
286#95—531no—
287#95—531no—
288#95—531no—
289#95—531no—
290#95—531no—
291#95C2×D5311yescharacteristic, complemented
292C24—531no—
293C24—531no—
294C24—534no—
295C24—1062no—
296C24—1062no—
297C24—1062no—
298C24—1062no—
299C24—2121no—
300C24—2121no—
301C24C22×D5311yescharacteristic
302C2×Q8—1062no—
303C2×Q8—1062no—
304C2×Q8—2121no—
305C22×C4—531no—
306C22×C4—531no—
307C22×C4—531no—
308C22×C4—531no—
309C22×C4—531no—
310C22×C4—531no—
311C22×C4—1062no—
312C22×C4—1062no—
313C22×C4—1062no—
314C22×C4—1062no—
315C22×C4—1062no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4—1062no—
319C22×C4—2121no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4—2121no—
323C22×C4C22×D5311yescharacteristic
324C22×C4C22×D5311yescharacteristic
325C22×C4C22×D5311yescharacteristic
326C22×C4C22×D5311yescharacteristic
327C22×C4C22×D5311yescharacteristic
328C22×C4C22×D5311yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—1062no—
335C4⋊C4—1062no—
336C4⋊C4—2121no—
337C4⋊C4—2121no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—41no—
341C22⋊C4—534no—
342C22⋊C4—534no—
343C22⋊C4—1062no—
344C22⋊C4—1062no—
345C22⋊C4—1062no—
346C22⋊C4—1062no—
347C22⋊C4—1062no—
348C22⋊C4—1062no—
349C22⋊C4—1062no—
350C22⋊C4—1062no—
351C22⋊C4—2121no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—532no—
359C23—532no—
360C23—532no—
361C23—532no—
362C23—532no—
363C23—532no—
364C23—1068no—
365C23—1068no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23—1061no—
370C23—1061no—
371C23—1061no—
372C23—1061no—
373C23—1061no—
374C23—2124no—
375C23—2124no—
376C23—2124no—
377C23—2124no—
378C23—2124no—
379C23Q8×D5312yes—
380C23C23×D5311yescharacteristic
381Q8—2124no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—532no—
399C2×C4—532no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—1061no—
408C2×C4—1061no—
409C2×C4—1061no—
410C2×C4—1061no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—2124no—
417C2×C4—2124no—
418C2×C4—2124no—
419C2×C4—2124no—
420C2×C4D4×D5312yes—
421C2×C4D4×D5312yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—534no—
430C22—534no—
431C22—1062no—
432C22—1062no—
433C22—1062no—
434C22—1062no—
435C22—1062no—
436C22—1062no—
437C22—1062no—
438C22—1062no—
439C22—1062no—
440C22—1062no—
441C22—1062no—
442C22—1062no—
443C22—21216no—
444C22C2×D4×D5311yescharacteristic
445C22C2×D4×D5311yescharacteristic
446C22C2×Q8×D5311yescharacteristic
447C22#41×D5311yescharacteristic
448C22#41×D5311yescharacteristic
449C22#1313326211yescharacteristic
450C22#1313326311yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—1062no—
458C4—1062no—
459C4—2121no—
460C4—2121no—
461C4—2121no—
462C4—2121no—
463C2—42no—
464C2—1064no—
465C2—1064no—
466C2C22⋊Q8×D5311yescharacteristic
467C2C22⋊Q8×D5311yescharacteristic
468C2#1681124811yescharacteristic
469C2#1681125711yescharacteristic
470C2#1681126211yescharacteristic
471C2#1681127011yescharacteristic
472C2#1681129811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#396C2
hypercenter#396D53
hypocenterC53#2437
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#13133197C2semidirect
2. C2×#13133195C2semidirect
5. #16810559C2semidirect
6. #16810546C2semidirect
7. #16810535C2semidirect
8. #16810497C2semidirect
9. #16810494C2semidirect
10. #16810494C2semidirect
11. #16810494C2semidirect
12. #16810482C2semidirect
13. #16810480C2semidirect
15. C53×#396C2semidirect
17. C22×Dic106C22semidirect
24. C2×C4⋊Dic53C22semidirect
25. C2×C4⋊Dic53C22semidirect
26. C2×#11719919C22semidirect
27. C2×#11719919C22semidirect
28. C2×#11719919C22semidirect
29. C2×#11719919C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. C2×C4⋊C4×C53C22semidirect
49. C2×C4⋊C4×C53C22semidirect
50. C2×C4⋊C4×C53C22semidirect
54. #95×C53C22semidirect
62. C23×D53Q8semidirect
212. C2×D53C22⋊Q8semidirect
213. C2×D53C22⋊Q8semidirect
243. #396D53semidirect
252. C53#2437semidirect
267. C2×C4⋊C4C2×D53semidirect
268. C2×C4⋊C4C2×D53semidirect
269. C2×C4⋊C4C2×D53semidirect
291. #95C2×D53semidirect