Finite Group Databasebeta

← #20278256

Subgroups — #20278256

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×#13133190C211yesmaximal, characteristic, complemented
3C2×#13133187C211yesmaximal, characteristic, complemented
4C2×#13133187C211yesmaximal, characteristic, complemented
5C2×C4×D212C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
7#16810545C211yesmaximal, characteristic, complemented
8#16810539C211yesmaximal, characteristic, complemented
9#16810539C211yesmaximal, characteristic, complemented
10#16810533C211yesmaximal, characteristic, complemented
11#16810525C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic, complemented
13#16810501C211yesmaximal, characteristic, complemented
14#16810497C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×D212C2211yescharacteristic
21C22×D212C2211yescharacteristic, complemented
22C22×C22⋊D53C2211yescharacteristic
23C22×C22⋊D53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic
25C22×C4×D53C2211yescharacteristic
26C22×C4×D53C2211yescharacteristic, complemented
27#13133192—24no—
28#13133190—24no—
29#13133187—24no—
30#13133187—24no—
31C4×D212—24no—
32C2×C4⋊Dic53C2211yescharacteristic, complemented
33C2×C4⋊Dic53C2211yescharacteristic, complemented
34C2×#11719919C2211yescharacteristic, complemented
35C2×C22⋊Dic53C2211yescharacteristic
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
47C22⋊C4×D53—24no—
48#13133124C2211yescharacteristic, complemented
49#13133124C2211yescharacteristic, complemented
50C2×C4⋊C4×C53C2211yescharacteristic, complemented
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C42×C53C2211yescharacteristic, complemented
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×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C2×C22⋊D53—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22×Dic53C2311yescharacteristic
84C22×Dic53C2311yescharacteristic
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212D414yescomplemented
100C2×C4×D53—22no—
101C2×C4×D53—22no—
102C2×C4×D53—22no—
103C2×C4×D53—22no—
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×C4×D53—41no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C4⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53D414yescomplemented
113C42×C53—22no—
114C22⋊Dic53—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—
127C4⋊Dic53—41no—
128#11719919—22no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53C2×D412yes—
136C23×C53C2411yescharacteristic
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—28no—
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—44no—
162C22×D53—44no—
163C22×D53—44no—
164C22×D53—44no—
165C22×D53—44no—
166C22×D53—44no—
167C22×D53C2×D412yes—
168C22×D53C2×D412yes—
169C22×D53C2×D412yes—
170C22×D53#4112yes—
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×C53C2×D412yes—
187C2×C4×C53C2×D412yes—
188C2×C4×C53C2×D412yes—
189C2×C4×C53C2×D412yes—
190C22⋊D53—44no—
191C22⋊D53—44no—
192C2×Dic53—21no—
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×Dic53C2×D412yes—
204D212—28no—
205D212—44no—
206D212—44no—
207D212—44no—
208D212—44no—
209C4×D53—44no—
210C4×D53—44no—
211C4×D53—44no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
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—22no—
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—416no—
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#12114yescomplemented
255C4×C53—22no—
256C4×C53—22no—
257C4×C53—22no—
258C4×C53—22no—
259C4×C53—41no—
260C4×C53—41no—
261Dic53—22no—
262Dic53—41no—
263Dic53—41no—
264Dic53—41no—
265#2424—531nomaximal, Sylow
266C2×C53—42no—
267C2×C53C2×#12111yescharacteristic
268C2×C53#52511yescharacteristic
269C2×C53#53811yescharacteristic
270C2×C53D4211yescharacteristic
271C2×C53D4211yescharacteristic
272C2×C53#54611yescharacteristic
273C2×C53#54611yescharacteristic
274D53—24no—
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#388D5311yescharacteristic, complemented
290#386—531no—
291#386—531no—
292#385—531no—
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×C4—531no—
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⋊C4—531no—
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—1064no—
327C2×C22⋊C4C2×D5311yescharacteristic
328C2×C22⋊C4C2×D5311yescharacteristic
329C2×C22⋊C4C2×D5311yescharacteristic
330C2×C42C2×D5311yescharacteristic, complemented
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—531no—
369C22×C4—531no—
370C22×C4—531no—
371C22×C4—531no—
372C22×C4—1062no—
373C22×C4—1062no—
374C22×C4—1062no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—2121no—
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—1062no—
389C4⋊C4—1062no—
390C4⋊C4—2121no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—534no—
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—22no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—532no—
414C23—532no—
415C23—532no—
416C23—532no—
417C23—532no—
418C23—1068no—
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—1061no—
441C23—1061no—
442C23—2124no—
443C23—2124no—
444C23—2124no—
445C23—2124no—
446C23—2124no—
447C23—2124no—
448C23C2×D21212yes—
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—532no—
472C2×C4—1061no—
473C2×C4—1061no—
474C2×C4—1061no—
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—2124no—
484C2×C4—2124no—
485C2×C4—2124no—
486C2×C4C2×D21212yes—
487C2×C4C2×D21212yes—
488C2×C4D4×D5312yes—
489C2×C4D4×D5312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—534no—
498C22—1062no—
499C22—1062no—
500C22—1062no—
501C22—1062no—
502C22—1062no—
503C22—1062no—
504C22—1062no—
505C22—1062no—
506C22—1062no—
507C22—1062no—
508C22—1062no—
509C22—1062no—
510C22—21216no—
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—2121no—
526C22C2×D4×D5311yescharacteristic
527C22C2×D4×D5311yescharacteristic
528C22C22×D21211yescharacteristic
529C22#41×D5311yescharacteristic
530C22#41×D5311yescharacteristic
531C22#1313326311yescharacteristic
532C22#1313326311yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—1062no—
540C4—2121no—
541C4—2121no—
542C4—2121no—
543C2—42no—
544C2—1064no—
545C2—4241no—
546C2—2122no—
547C2—2122no—
548C2#121×D5311yescharacteristic
549C2#1681113511yescharacteristic
550C2#1681121211yescharacteristic
551C2D4×D21211yescharacteristic
552C2D4×D21211yescharacteristic
553C2#1681126511yescharacteristic
554C2#1681126611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2424
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133192C2semidirect
2. C2×#13133190C2semidirect
3. C2×#13133187C2semidirect
4. C2×#13133187C2semidirect
5. C2×C4×D212C2semidirect
6. C2×C22⋊C4×D53C2semidirect
7. #16810545C2semidirect
8. #16810539C2semidirect
9. #16810539C2semidirect
10. #16810533C2semidirect
11. #16810525C2semidirect
12. #16810501C2semidirect
13. #16810501C2semidirect
14. #16810497C2semidirect
15. C53×#388C2semidirect
21. C22×D212C22semidirect
26. C22×C4×D53C22semidirect
32. C2×C4⋊Dic53C22semidirect
33. C2×C4⋊Dic53C22semidirect
34. C2×#11719919C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. C2×#11719921C22semidirect
46. C2×#11719921C22semidirect
48. #13133124C22semidirect
49. #13133124C22semidirect
50. C2×C4⋊C4×C53C22semidirect
54. C2×C42×C53C22semidirect
55. #95×C53C22semidirect
56. #95×C53C22semidirect
99. C2×D212D4semidirect
112. C22⋊C4×C53D4semidirect
125. #11719921D4semidirect
254. C2×D53#121semidirect
289. #388D53semidirect
293. C53#2424semidirect
313. C2×C4⋊C4C2×D53semidirect
330. C2×C42C2×D53semidirect
333. #95C2×D53semidirect
334. #95C2×D53semidirect
405. C22⋊C4D212semidirect