Finite Group Databasebeta

← #20278300

Subgroups — #20278300

Order 6784.

832 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133213C211yesmaximal, characteristic, complemented
2C2×#13133190C211yesmaximal, characteristic, complemented
3C2×#13133189C211yesmaximal, characteristic, complemented
4C2×#13133189C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊D53C211yesmaximal, characteristic, complemented
6#16810540C211yesmaximal, characteristic, complemented
7#16810537C211yesmaximal, characteristic, complemented
8#16810531C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810523C211yesmaximal, characteristic, complemented
11#16810518C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13#16810499C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic
23C23×Dic53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic
25#13133213—24no—
26#13133190—24no—
27#13133189—24no—
28#13133189—24no—
29C4×C22⋊D53—24no—
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×C4×Dic53C2211yescharacteristic
44C2×C4×Dic53C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48#13133124C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51C22×D4×C53C2211yescharacteristic
52C23×C4×C53C2211yescharacteristic
53#121×C53—24no—
54C2×C4⋊C4×C53C2211yescharacteristic
55C2×C22⋊C4×C53C2211yescharacteristic
56C2×C22⋊C4×C53C2211yescharacteristic
57C24×C53C2311yescharacteristic
58C24×C53C2311yescharacteristic
59C24×C53C2311yescharacteristic
60C23×D53C2311yescharacteristic
61C23×D53C2311yescharacteristic
62C2×D4×C53—22no—
63C2×D4×C53—22no—
64C2×D4×C53—22no—
65C2×D4×C53—22no—
66C2×D4×C53—22no—
67C2×D4×C53—22no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C22×C4×C53—22no—
72C22×C4×C53—22no—
73C22×C4×C53—41no—
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⋊D53—22no—
84C2×C22⋊D53—22no—
85C2×C22⋊D53—22no—
86C2×C22⋊D53—22no—
87C2×C22⋊D53—22no—
88C2×C22⋊D53—22no—
89C2×C22⋊D53—22no—
90C2×C22⋊D53—22no—
91C2×C22⋊D53—41no—
92C2×C22⋊D53—41no—
93C2×C22⋊D53—41no—
94C2×C22⋊D53—41no—
95C2×C22⋊D53D414yescomplemented
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53—22no—
99C22×Dic53C2311yescharacteristic
100C22×Dic53C2311yescharacteristic
101C22×Dic53C2311yescharacteristic
102C22×Dic53C2311yescharacteristic
103C22×Dic53C2311yescharacteristic
104C22×Dic53C2311yescharacteristic
105C2×D212—22no—
106C2×D212—22no—
107C2×D212—41no—
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—41no—
111C4⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53D414yescomplemented
114C22⋊Dic53—22no—
115C22⋊Dic53—22no—
116C22⋊Dic53—22no—
117C22⋊Dic53—22no—
118C22⋊Dic53—41no—
119#11719921—22no—
120#11719921—22no—
121#11719921—22no—
122#11719921—22no—
123#11719921—41no—
124#11719919—22no—
125#11719919—22no—
126#11719919D414yescomplemented
127C4×Dic53—22no—
128C4×Dic53—22no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53C2×D412yes—
149C23×C53C2×D412yes—
150C23×C53C2411yescharacteristic
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—21no—
164C22×D53C2×D412yes—
165D4×C53—44no—
166D4×C53—44no—
167D4×C53—44no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—44no—
178C2×C4×C53C2×D412yes—
179C2×C4×C53C2×D412yes—
180C2×C4×C53C2×D412yes—
181C22⋊D53—28no—
182C22⋊D53—44no—
183C22⋊D53—44no—
184C22⋊D53—44no—
185C22⋊D53—44no—
186C22⋊D53—44no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
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—44no—
201C2×Dic53C2×D412yes—
202C2×Dic53C2×D412yes—
203C2×Dic53C2×D412yes—
204C2×Dic53#4112yes—
205C2×Dic53#4112yes—
206D212—44no—
207C4×D53—44no—
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53—22no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53—41no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53C22×D411yescharacteristic
230C22×C53C22×D411yescharacteristic
231C22×C53C22×D411yescharacteristic
232C22×C53C2×#4111yescharacteristic
233C22×C53C2×#4111yescharacteristic
234C22×C53H(5,𝔽2)11yescharacteristic
235C22×C53H(5,𝔽2)11yescharacteristic
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—22no—
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—
250C4×C53—22no—
251C4×C53—22no—
252C4×C53—22no—
253C4×C53—41no—
254Dic53—22no—
255Dic53—22no—
256Dic53—41no—
257Dic53—41no—
258Dic53—41no—
259Dic53#12114yescomplemented
260#2428—531nomaximal, Sylow
261C2×C53—42no—
262C2×C53—42no—
263C2×C53—81no—
264C2×C53C2×#12111yescharacteristic
265C2×C53#53211yescharacteristic
266C2×C53#54011yescharacteristic
267C2×C53D4211yescharacteristic
268C2×C53D4211yescharacteristic
269C2×C53#54611yescharacteristic
270C2×C53#55011yescharacteristic
271D53—42no—
272D53—81no—
273C2×#127—531no—
274C2×#121—531no—
275C2×#121—531no—
276C2×#121—531no—
277C2×#121D5311yescharacteristic, complemented
278C2×C4×D4—531no—
279#394—531no—
280#394—531no—
281#392—531no—
282#392—531no—
283#390—531no—
284#389—531no—
285#386—531no—
286#386—531no—
287C4×C22⋊C4—531no—
288C53#242811yescharacteristic, complemented, Sylow
289C22×D4—531no—
290C22×D4—531no—
291C22×D4—531no—
292C22×D4—531no—
293C22×D4—531no—
294C22×D4—531no—
295C22×D4—531no—
296C22×D4C2×D5311yescharacteristic
297C22×D4C2×D5311yescharacteristic
298C22×D4C2×D5311yescharacteristic
299C23×C4—531no—
300C23×C4—531no—
301C23×C4C2×D5311yescharacteristic
302#127—1064no—
303#121—24no—
304#121—1064no—
305#121—1064no—
306#121—1064no—
307C4×D4—1064no—
308C2×C4⋊C4—531no—
309C2×C4⋊C4—531no—
310C2×C4⋊C4—531no—
311C2×C4⋊C4C2×D5311yescharacteristic
312C2×C22⋊C4—531no—
313C2×C22⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
323C2×C22⋊C4C2×D5311yescharacteristic
324C2×C42—531no—
325C2×C42—531no—
326#95—531no—
327#95—531no—
328#95—531no—
329#95—531no—
330C24—531no—
331C24—531no—
332C24C22×D5311yescharacteristic
333C24C22×D5311yescharacteristic
334C24C22×D5311yescharacteristic
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—534no—
345C2×D4—1062no—
346C2×D4—1062no—
347C2×D4—1062no—
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—2121no—
361C2×D4—2121no—
362C2×D4—2121no—
363C2×D4—2121no—
364C2×D4—2121no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4—531no—
369C22×C4—531no—
370C22×C4—531no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—531no—
374C22×C4—1062no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—2121no—
380C22×C4C22×D5311yescharacteristic
381C22×C4C22×D5311yescharacteristic
382C22×C4C22×D5311yescharacteristic
383C22×C4C22×D5311yescharacteristic
384C4⋊C4—22no—
385C4⋊C4—534no—
386C4⋊C4—1062no—
387C4⋊C4—1062no—
388C22⋊C4—22no—
389C22⋊C4—1062no—
390C22⋊C4—1062no—
391C22⋊C4—1062no—
392C22⋊C4—1062no—
393C22⋊C4—1062no—
394C22⋊C4—1062no—
395C22⋊C4—1062no—
396C22⋊C4—1062no—
397C22⋊C4—2121no—
398C22⋊C4—2121no—
399C22⋊C4C22⋊D5314yescomplemented
400C42—1062no—
401C42—1062no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—532no—
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—
435C23D4×D5312yes—
436C23C2×C22⋊D5312yes—
437C23C23×D5311yescharacteristic
438D4—44no—
439D4—44no—
440D4—44no—
441D4—1068no—
442D4—2124no—
443D4—2124no—
444D4—2124no—
445D4—2124no—
446D4—2124no—
447D4—2124no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—44no—
458C2×C4—532no—
459C2×C4—532no—
460C2×C4—532no—
461C2×C4—532no—
462C2×C4—532no—
463C2×C4—1061no—
464C2×C4—1061no—
465C2×C4—1061no—
466C2×C4—1061no—
467C2×C4—1061no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—1061no—
473C2×C4—1061no—
474C2×C4—1061no—
475C2×C4—1061no—
476C2×C4—2124no—
477C2×C4—2124no—
478C2×C4D4×D5312yes—
479C2×C4C2×C22⋊D5312yes—
480C2×C4C2×C22⋊D5312yes—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—1062no—
503C22—1062no—
504C22—1062no—
505C22—2121no—
506C22—2121no—
507C22—2121no—
508C22—2121no—
509C22—2121no—
510C22—2121no—
511C22—2121no—
512C22—2121no—
513C22—2121no—
514C22—2121no—
515C22—2121no—
516C22C2×D4×D5311yescharacteristic
517C22C2×D4×D5311yescharacteristic
518C22C22×C22⋊D5311yescharacteristic
519C22#41×D5311yescharacteristic
520C22#41×D5311yescharacteristic
521C22#1313326211yescharacteristic
522C22#1313326211yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—41no—
527C4—534no—
528C4—1062no—
529C4—1062no—
530C4—2121no—
531C4—2121no—
532C4—2121no—
533C2—42no—
534C2—42no—
535C2—81no—
536C2—4241no—
537C2—2122no—
538C2#121×D5311yescharacteristic
539C2#1681116911yescharacteristic
540C2#1681122311yescharacteristic
541C2D4×C22⋊D5311yescharacteristic
542C2D4×C22⋊D5311yescharacteristic
543C2#1681126211yescharacteristic
544C2#1681129111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2428
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133190C2semidirect
3. C2×#13133189C2semidirect
4. C2×#13133189C2semidirect
5. C2×C4×C22⋊D53C2semidirect
6. #16810540C2semidirect
7. #16810537C2semidirect
8. #16810531C2semidirect
9. #16810531C2semidirect
10. #16810523C2semidirect
11. #16810518C2semidirect
12. #16810502C2semidirect
13. #16810499C2semidirect
14. C22⋊C4×Dic53C2semidirect
15. C2×#121×C53C2semidirect
95. C2×C22⋊D53D4semidirect
113. C22⋊C4×C53D4semidirect
126. #11719919D4semidirect
259. Dic53#121semidirect
277. C2×#121D53semidirect
288. C53#2428semidirect
399. C22⋊C4C22⋊D53semidirect