Finite Group Databasebeta

← #20278249

Subgroups — #20278249

Order 6784.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133192C211yesmaximal, characteristic, complemented
2C2×#13133191C211yesmaximal, characteristic, complemented
3C2×#13133187C211yesmaximal, characteristic, complemented
4C2×#13133187C211yesmaximal, characteristic, complemented
5C2×#13133177C211yesmaximal, characteristic, complemented
6C22×#11719921C211yesmaximal, characteristic, complemented
7#16810545C211yesmaximal, characteristic, complemented
8#16810541C211yesmaximal, characteristic, complemented
9#16810540C211yesmaximal, characteristic, complemented
10#16810533C211yesmaximal, characteristic, complemented
11#16810524C211yesmaximal, characteristic, complemented
12#16810515C211yesmaximal, characteristic, complemented
13#16810501C211yesmaximal, characteristic, complemented
14#16810500C211yesmaximal, characteristic, complemented
15C53×#385C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×D212C2211yescharacteristic
21C22×D212C2211yescharacteristic, complemented
22C22×C22⋊D53C2211yescharacteristic
23C22×C22⋊D53C2211yescharacteristic
24C23×Dic53C2211yescharacteristic
25C22×C4×D53C2211yescharacteristic, complemented
26#13133192—24no—
27#13133191—24no—
28#13133187—24no—
29#13133187—24no—
30#13133177—24no—
31C2×C4⋊Dic53C2211yescharacteristic, complemented
32C2×#11719919C2211yescharacteristic, complemented
33C2×#11719921—24no—
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic, complemented
43C2×#11719921C2211yescharacteristic, complemented
44C2×#11719921C2211yescharacteristic, complemented
45C2×#11719921C2211yescharacteristic, complemented
46C2×#11719921C2211yescharacteristic, complemented
47C2×C4×Dic53C2211yescharacteristic, complemented
48#13133124C2211yescharacteristic, complemented
49#13133124C2211yescharacteristic, complemented
50C23×C4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic, complemented
52C2×C4⋊C4×C53C2211yescharacteristic, complemented
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55#95×C53C2211yescharacteristic, complemented
56#95×C53C2211yescharacteristic, complemented
57C24×C53C2311yescharacteristic
58C23×D53—22no—
59C23×D53—22no—
60C23×D53—22no—
61C23×D53—22no—
62C23×D53—22no—
63C23×D53—41no—
64C23×D53—41no—
65C23×D53C2311yescharacteristic
66C23×D53C2311yescharacteristic
67C23×D53C2311yescharacteristic
68C23×D53C2311yescharacteristic
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—41no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C22×C4×C53C2311yescharacteristic
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
81C2×C22⋊D53—22no—
82C2×C22⋊D53—22no—
83C2×C22⋊D53D414yescomplemented
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—41no—
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—22no—
98C2×D212—22no—
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—41no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—41no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53D414yescomplemented
113#11719921—22no—
114#11719921—22no—
115#11719921—22no—
116#11719921—22no—
117#11719921—22no—
118#11719921—22no—
119#11719921—22no—
120#11719921—22no—
121#11719921—22no—
122#11719921—22no—
123#11719921—22no—
124#11719921—41no—
125#11719921D414yescomplemented
126C4⋊Dic53—22no—
127#11719919—22no—
128C4×Dic53—22no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53C2×D412yes—
134C23×C53C2×D412yes—
135C23×C53#4112yes—
136C23×C53C2411yescharacteristic
137C22×D53—28no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53—21no—
163C22×D53—44no—
164C22×D53—44no—
165C22×D53—44no—
166C22×D53—44no—
167C22×D53—44no—
168C22×D53—44no—
169C22×D53C2×D412yes—
170C22×D53C2×D412yes—
171C22×D53#4112yes—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—44no—
188C2×C4×C53C2×D412yes—
189C2×C4×C53C2×D412yes—
190C2×C4×C53C2×D412yes—
191C22⋊D53—28no—
192C22⋊D53—44no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—44no—
204C2×Dic53C2×D412yes—
205C2×Dic53C2×D412yes—
206D212—44no—
207D212—44no—
208D212—44no—
209D212—44no—
210D212—44no—
211C4×D53—44no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53#12114yescomplemented
219C22×C53C22×D411yescharacteristic
220C22×C53C22×D411yescharacteristic
221C22×C53C22×D411yescharacteristic
222C22×C53C2×#4111yescharacteristic
223C22×C53C2×#4111yescharacteristic
224C22×C53H(5,𝔽2)11yescharacteristic
225C22×C53H(5,𝔽2)11yescharacteristic
226C2×D53—22no—
227C2×D53—22no—
228C2×D53—22no—
229C2×D53—22no—
230C2×D53—22no—
231C2×D53—22no—
232C2×D53—22no—
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—416no—
236C2×D53—41no—
237C2×D53—41no—
238C2×D53—41no—
239C2×D53—41no—
240C2×D53—41no—
241C2×D53—41no—
242C2×D53—41no—
243C2×D53—41no—
244C2×D53—41no—
245C2×D53—41no—
246C2×D53—41no—
247C2×D53—41no—
248C2×D53—41no—
249C2×D53—41no—
250C2×D53—41no—
251C2×D53—41no—
252C2×D53—41no—
253C2×D53—41no—
254C2×D53—41no—
255C4×C53—22no—
256C4×C53—22no—
257C4×C53—22no—
258C4×C53—41no—
259C4×C53—41no—
260C4×C53—41no—
261Dic53—22no—
262Dic53—22no—
263Dic53—41no—
264Dic53—41no—
265#2424—531nomaximal, Sylow
266C2×C53—24no—
267C2×C53C2×#12111yescharacteristic
268C2×C53#52511yescharacteristic
269C2×C53#53811yescharacteristic
270C2×C53D4211yescharacteristic
271C2×C53D4211yescharacteristic
272C2×C53#54611yescharacteristic
273C2×C53#54611yescharacteristic
274D53—42no—
275D53—42no—
276D53—42no—
277D53—81no—
278C2×#121—531no—
279C2×#121—531no—
280C2×#120—531no—
281C2×#120—531no—
282C2×C4×D4—531no—
283C22×C22⋊C4—531no—
284#396—531no—
285#394—531no—
286#394—531no—
287#392—531no—
288#390—531no—
289#388—531no—
290#386—531no—
291#386—531no—
292#385D5311yescharacteristic, complemented
293C53#242411yescharacteristic, complemented, Sylow
294C25—531no—
295C22×D4—531no—
296C22×D4—531no—
297C22×D4—531no—
298C22×D4—531no—
299C22×D4—531no—
300C22×D4—531no—
301C22×D4—531no—
302C23×C4—531no—
303C23×C4—531no—
304C23×C4C2×D5311yescharacteristic
305#121—1064no—
306#121—1064no—
307#120—1064no—
308#120—1064no—
309C4×D4—1064no—
310C2×C4⋊C4—531no—
311C2×C4⋊C4—531no—
312C2×C4⋊C4C2×D5311yescharacteristic, complemented
313C2×C4⋊C4C2×D5311yescharacteristic, complemented
314C2×C22⋊C4—531no—
315C2×C22⋊C4—531no—
316C2×C22⋊C4—531no—
317C2×C22⋊C4—531no—
318C2×C22⋊C4—531no—
319C2×C22⋊C4—531no—
320C2×C22⋊C4—531no—
321C2×C22⋊C4—531no—
322C2×C22⋊C4—531no—
323C2×C22⋊C4—531no—
324C2×C22⋊C4—531no—
325C2×C22⋊C4—531no—
326C2×C22⋊C4—531no—
327C2×C22⋊C4—1064no—
328C2×C22⋊C4C2×D5311yescharacteristic
329C2×C22⋊C4C2×D5311yescharacteristic
330C2×C42—531no—
331#95—531no—
332#95—531no—
333#95C2×D5311yescharacteristic, complemented
334#95C2×D5311yescharacteristic, complemented
335C24—531no—
336C24—531no—
337C24—531no—
338C24—531no—
339C24—1062no—
340C24—1062no—
341C24—1062no—
342C24—1062no—
343C24—1062no—
344C24—2121no—
345C24—2121no—
346C24C22×D5311yescharacteristic
347C2×D4—534no—
348C2×D4—1062no—
349C2×D4—1062no—
350C2×D4—1062no—
351C2×D4—1062no—
352C2×D4—1062no—
353C2×D4—1062no—
354C2×D4—1062no—
355C2×D4—1062no—
356C2×D4—1062no—
357C2×D4—1062no—
358C2×D4—1062no—
359C2×D4—1062no—
360C2×D4—1062no—
361C2×D4—1062no—
362C2×D4—1062no—
363C2×D4—2121no—
364C2×D4—2121no—
365C2×D4—2121no—
366C2×D4—2121no—
367C2×D4—2121no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—531no—
374C22×C4—531no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—2121no—
380C22×C4—2121no—
381C22×C4C22×D5311yescharacteristic
382C22×C4C22×D5311yescharacteristic
383C22×C4C22×D5311yescharacteristic
384C22×C4C22×D5311yescharacteristic
385C22×C4C22×D5311yescharacteristic
386C22×C4C22×D5311yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—1062no—
390C4⋊C4—1062no—
391C22⋊C4—22no—
392C22⋊C4—534no—
393C22⋊C4—1062no—
394C22⋊C4—1062no—
395C22⋊C4—1062no—
396C22⋊C4—1062no—
397C22⋊C4—1062no—
398C22⋊C4—1062no—
399C22⋊C4—1062no—
400C22⋊C4—1062no—
401C22⋊C4—1062no—
402C22⋊C4—1062no—
403C22⋊C4—1062no—
404C22⋊C4—2121no—
405C22⋊C4D21214yescomplemented
406C42—1062no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—532no—
412C23—532no—
413C23—532no—
414C23—1068no—
415C23—1061no—
416C23—1061no—
417C23—1061no—
418C23—1061no—
419C23—1061no—
420C23—1061no—
421C23—1061no—
422C23—1061no—
423C23—1061no—
424C23—1061no—
425C23—1061no—
426C23—1061no—
427C23—1061no—
428C23—1061no—
429C23—1061no—
430C23—1061no—
431C23—1061no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23—1061no—
436C23—1061no—
437C23—1061no—
438C23—1061no—
439C23—1061no—
440C23—2124no—
441C23—2124no—
442C23—2124no—
443C23—2124no—
444C23—2124no—
445C23—2124no—
446C23C2×D21212yes—
447C23D4×D5312yes—
448C23#1171994912yes—
449C23C23×D5311yescharacteristic
450D4—1068no—
451D4—2124no—
452D4—2124no—
453D4—2124no—
454D4—2124no—
455D4—2124no—
456D4—2124no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—44no—
473C2×C4—532no—
474C2×C4—532no—
475C2×C4—1061no—
476C2×C4—1061no—
477C2×C4—1061no—
478C2×C4—1061no—
479C2×C4—1061no—
480C2×C4—1061no—
481C2×C4—1061no—
482C2×C4—1061no—
483C2×C4—1061no—
484C2×C4—1061no—
485C2×C4—2124no—
486C2×C4—2124no—
487C2×C4C2×D21212yes—
488C2×C4C2×D21212yes—
489C2×C4D4×D5312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—1062no—
497C22—1062no—
498C22—1062no—
499C22—1062no—
500C22—1062no—
501C22—1062no—
502C22—1062no—
503C22—1062no—
504C22—1062no—
505C22—21216no—
506C22—2121no—
507C22—2121no—
508C22—2121no—
509C22—2121no—
510C22—2121no—
511C22—2121no—
512C22—2121no—
513C22—2121no—
514C22—2121no—
515C22—2121no—
516C22—2121no—
517C22—2121no—
518C22—2121no—
519C22—2121no—
520C22—2121no—
521C22—2121no—
522C22—2121no—
523C22—2121no—
524C22—2121no—
525C22#1313319214yescomplemented
526C22C2×D4×D5311yescharacteristic
527C22C2×D4×D5311yescharacteristic
528C22C22×D21211yescharacteristic
529C22#41×D5311yescharacteristic
530C22C2×#1171994911yescharacteristic
531C22#1313326211yescharacteristic
532C22#1313326311yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—1062no—
540C4—1062no—
541C4—2121no—
542C4—2121no—
543C2—24no—
544C2—4241no—
545C2—2122no—
546C2—2122no—
547C2—2122no—
548C2C2×#1313319211yescharacteristic
549C2#1681113611yescharacteristic
550C2#1681121311yescharacteristic
551C2#1681125511yescharacteristic
552C2D4×D21211yescharacteristic
553C2#1681126511yescharacteristic
554C2#1681126811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2424
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#13133192C2semidirect
2. C2×#13133191C2semidirect
3. C2×#13133187C2semidirect
4. C2×#13133187C2semidirect
5. C2×#13133177C2semidirect
6. C22×#11719921C2semidirect
7. #16810545C2semidirect
8. #16810541C2semidirect
9. #16810540C2semidirect
10. #16810533C2semidirect
11. #16810524C2semidirect
12. #16810515C2semidirect
13. #16810501C2semidirect
14. #16810500C2semidirect
15. C53×#385C2semidirect
21. C22×D212C22semidirect
25. C22×C4×D53C22semidirect
31. C2×C4⋊Dic53C22semidirect
32. C2×#11719919C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. C2×#11719921C22semidirect
46. C2×#11719921C22semidirect
47. C2×C4×Dic53C22semidirect
48. #13133124C22semidirect
49. #13133124C22semidirect
51. C2×C4⋊C4×C53C22semidirect
52. C2×C4⋊C4×C53C22semidirect
55. #95×C53C22semidirect
56. #95×C53C22semidirect
83. C2×C22⋊D53D4semidirect
112. C22⋊C4×C53D4semidirect
125. #11719921D4semidirect
218. C22×C53#121semidirect
292. #385D53semidirect
293. C53#2424semidirect
312. C2×C4⋊C4C2×D53semidirect
313. C2×C4⋊C4C2×D53semidirect
333. #95C2×D53semidirect
334. #95C2×D53semidirect
405. C22⋊C4D212semidirect
525. C22#13133192semidirect