Finite Group Databasebeta

← #20280937

Subgroups — #20280937

Order 6784.

796 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133204C211yesmaximal, characteristic, complemented
2C2×#13133203C211yesmaximal, characteristic, complemented
3C2×#13133186C211yesmaximal, characteristic, complemented
4#16810539C211yesmaximal, characteristic, complemented
5#16810537C211yesmaximal, characteristic, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810531C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810526C211yesmaximal, characteristic, complemented
11#16810523C211yesmaximal, characteristic, complemented
12#16810516C211yesmaximal, characteristic, complemented
13#16810511C211yesmaximal, characteristic, complemented
14#16810479C211yesmaximal, characteristic, complemented
15C2×#120×C53C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic, complemented
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic, complemented
21#13133204—24no—
22#13133203—24no—
23#13133186—24no—
24C2×C4⋊Dic53C2211yescharacteristic, complemented
25C2×#11719919C2211yescharacteristic, complemented
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic, complemented
36C2×C22⋊Dic53C2211yescharacteristic, complemented
37C2×C22⋊Dic53C2211yescharacteristic, complemented
38C2×#11719921C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
40C2×#11719921C2211yescharacteristic, complemented
41C2×C4×Dic53C2211yescharacteristic, complemented
42C2×C4×Dic53C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47C25×C53C2211yescharacteristic
48C22×D4×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51#120×C53—24no—
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53—22no—
56C24×C53—22no—
57C24×C53—22no—
58C24×C53—41no—
59C24×C53—41no—
60C24×C53—41no—
61C24×C53—41no—
62C24×C53C2311yescharacteristic
63C24×C53C2311yescharacteristic
64C24×C53C2311yescharacteristic
65C24×C53C2311yescharacteristic
66C23×D53C2311yescharacteristic
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C22×C4×C53C2311yescharacteristic
77C22×C4×C53C2311yescharacteristic
78C22×C4×C53C2311yescharacteristic
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
81C2×C22⋊D53—22no—
82C2×C22⋊D53—22no—
83C2×C22⋊D53—22no—
84C2×C22⋊D53—41no—
85C2×C22⋊D53—41no—
86C2×C22⋊D53—41no—
87C2×C22⋊D53—41no—
88C2×C22⋊D53—41no—
89C2×C22⋊D53—41no—
90C2×C22⋊D53—41no—
91C22×Dic53C2311yescharacteristic
92C22×Dic53C2311yescharacteristic
93C22×Dic53C2311yescharacteristic
94C22×Dic53C2311yescharacteristic
95C22×Dic53C2311yescharacteristic
96C22×Dic53C2311yescharacteristic
97C22×Dic53C2311yescharacteristic
98C2×Dic106—22no—
99C2×Dic106—22no—
100C2×Dic106—41no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53D414yescomplemented
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—22no—
107C22⋊Dic53—22no—
108C22⋊Dic53—22no—
109C22⋊Dic53—22no—
110C22⋊Dic53—41no—
111C22⋊Dic53—41no—
112C22⋊Dic53—41no—
113C22⋊Dic53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53D414yescomplemented
116#11719921—22no—
117#11719921—22no—
118#11719921—41no—
119C4⋊Dic53—41no—
120#11719919—41no—
121C4×Dic53—22no—
122C4×Dic53—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
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×C53—44no—
149C23×C53—44no—
150C23×C53—44no—
151C23×C53—44no—
152C23×C53—44no—
153C23×C53—44no—
154C23×C53—44no—
155C23×C53C2×D412yes—
156C23×C53C2×D412yes—
157C23×C53#4112yes—
158C23×C53C2411yescharacteristic
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—21no—
165C22×D53—21no—
166D4×C53—44no—
167D4×C53—44no—
168D4×C53—44no—
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×C53C2×D412yes—
177C2×C4×C53C2×D412yes—
178C22⋊D53—44no—
179C22⋊D53—44no—
180C22⋊D53—44no—
181C22⋊D53—44no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
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—21no—
201C2×Dic53C2×D412yes—
202C2×Dic53C2×D412yes—
203Dic106—44no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53—22no—
212C22×C53—22no—
213C22×C53—416no—
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×C53—41no—
230C22×C53—41no—
231C22×C53—41no—
232C22×C53—41no—
233C22×C53C22×D411yescharacteristic
234C22×C53C22×D411yescharacteristic
235C22×C53C2×#4111yescharacteristic
236C22×C53H(5,𝔽2)11yescharacteristic
237C22×C53H(5,𝔽2)11yescharacteristic
238C22×C53H(5,𝔽2)11yescharacteristic
239C22×C53H(5,𝔽2)11yescharacteristic
240C2×D53—41no—
241C2×D53—41no—
242C2×D53—41no—
243C2×D53—41no—
244C2×D53—41no—
245C2×D53—41no—
246C2×D53—41no—
247C4×C53—22no—
248C4×C53—22no—
249C4×C53—41no—
250Dic53—22no—
251Dic53—22no—
252Dic53—41no—
253Dic53—41no—
254Dic53—41no—
255Dic53—41no—
256Dic53—41no—
257#2670—531nomaximal, Sylow
258C2×C53—42no—
259C2×C53—42no—
260C2×C53—42no—
261C2×C53—81no—
262C2×C53#53511yescharacteristic
263C2×C53#53511yescharacteristic
264C2×C53#53811yescharacteristic
265C2×C53D4211yescharacteristic
266C2×C53#54611yescharacteristic
267C2×C53#54611yescharacteristic
268C2×C53H(3,𝔽4)11yescharacteristic
269D53—81no—
270C2×#124—531no—
271C2×#124—531no—
272C2×#120—531no—
273C2×#120D5311yescharacteristic, complemented
274#394—531no—
275#394—531no—
276#393—531no—
277#392—531no—
278#392—531no—
279#392—531no—
280#390—531no—
281#390—531no—
282#388—531no—
283#388—531no—
284#379—531no—
285C53#267011yescharacteristic, complemented, Sylow
286C25C2×D5311yescharacteristic
287C22×Q8—531no—
288C22×D4—531no—
289C22×D4—531no—
290C22×D4—531no—
291C22×D4—531no—
292C22×D4C2×D5311yescharacteristic
293C22×D4C2×D5311yescharacteristic
294C22×D4C2×D5311yescharacteristic
295#124—1064no—
296#124—1064no—
297#120—24no—
298#120—1064no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—531no—
301C2×C22⋊C4—531no—
302C2×C22⋊C4—531no—
303C2×C22⋊C4—531no—
304C2×C22⋊C4—531no—
305C2×C22⋊C4—531no—
306C2×C22⋊C4—531no—
307C2×C22⋊C4—531no—
308C2×C22⋊C4—531no—
309C2×C22⋊C4—531no—
310C2×C22⋊C4—531no—
311C2×C22⋊C4—531no—
312C2×C22⋊C4—531no—
313C2×C22⋊C4—531no—
314C2×C22⋊C4—531no—
315C2×C22⋊C4—531no—
316C2×C22⋊C4C2×D5311yescharacteristic
317C2×C22⋊C4C2×D5311yescharacteristic
318C2×C22⋊C4C2×D5311yescharacteristic
319C2×C42—531no—
320C2×C42—531no—
321#95—531no—
322#95—531no—
323#95—531no—
324#95—531no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—41no—
329C24—41no—
330C24—41no—
331C24—41no—
332C24—531no—
333C24C22×D5311yescharacteristic
334C24C22×D5311yescharacteristic
335C24C22×D5311yescharacteristic
336C24C22×D5311yescharacteristic
337C2×Q8—1062no—
338C2×Q8—1062no—
339C2×Q8—2121no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—1062no—
350C2×D4—1062no—
351C2×D4—1062no—
352C2×D4—1062no—
353C2×D4—1062no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2121no—
358C2×D4—2121no—
359C2×D4—2121no—
360C2×D4—2121no—
361C22×C4—531no—
362C22×C4—531no—
363C22×C4—531no—
364C22×C4—531no—
365C22×C4—531no—
366C22×C4—531no—
367C22×C4—531no—
368C22×C4C22×D5311yescharacteristic
369C22×C4C22×D5311yescharacteristic
370C22×C4C22×D5311yescharacteristic
371C4⋊C4—2121no—
372C4⋊C4—2121no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—534no—
376C22⋊C4—1062no—
377C22⋊C4—1062no—
378C22⋊C4—1062no—
379C22⋊C4—1062no—
380C22⋊C4—1062no—
381C22⋊C4—1062no—
382C22⋊C4—1062no—
383C22⋊C4—1062no—
384C22⋊C4—2121no—
385C22⋊C4—2121no—
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4C22⋊D5314yescomplemented
391C42—1062no—
392C42—1062no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
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—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—1061no—
426C23—1061no—
427C23—1061no—
428C23—1061no—
429C23—1061no—
430C23—1061no—
431C23—1061no—
432C23D4×D5312yes—
433C23#1171994712yes—
434C23C2×C22⋊D5312yes—
435C23C23×D5311yescharacteristic
436Q8—2124no—
437D4—44no—
438D4—44no—
439D4—44no—
440D4—2124no—
441D4—2124no—
442D4—2124no—
443D4—2124no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—532no—
452C2×C4—532no—
453C2×C4—1061no—
454C2×C4—1061no—
455C2×C4—1061no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—1061no—
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×C4C2×C22⋊D5312yes—
473C2×C4C2×C22⋊D5312yes—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—416no—
484C22—41no—
485C22—41no—
486C22—41no—
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—41no—
503C22—2121no—
504C22—2121no—
505C22—2121no—
506C22—2121no—
507C22—2121no—
508C22—2121no—
509C22—2121no—
510C22C2×D4×D5311yescharacteristic
511C22C22×C22⋊D5311yescharacteristic
512C22C2×#1171994711yescharacteristic
513C22#1313326211yescharacteristic
514C22#1313326211yescharacteristic
515C22#1313326211yescharacteristic
516C22#1313326211yescharacteristic
517C4—22no—
518C4—22no—
519C4—41no—
520C4—1062no—
521C4—1062no—
522C4—2121no—
523C4—2121no—
524C4—2121no—
525C4—2121no—
526C4—2121no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—4241no—
532C2#1681118811yescharacteristic
533C2#1681119011yescharacteristic
534C2#1681120811yescharacteristic
535C2D4×C22⋊D5311yescharacteristic
536C2#1681126011yescharacteristic
537C2#1681126711yescharacteristic
538C2#1681136611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#120×C53C2
hypercenterC2×#120D53
hypocenterC53#2670
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#13133204C2semidirect
2. C2×#13133203C2semidirect
3. C2×#13133186C2semidirect
4. #16810539C2semidirect
5. #16810537C2semidirect
6. #16810535C2semidirect
7. #16810531C2semidirect
8. #16810531C2semidirect
9. #16810531C2semidirect
10. #16810526C2semidirect
11. #16810523C2semidirect
12. #16810516C2semidirect
13. #16810511C2semidirect
14. #16810479C2semidirect
15. C2×#120×C53C2semidirect
16. C22×Dic106C22semidirect
20. C22×C22⋊D53C22semidirect
24. C2×C4⋊Dic53C22semidirect
25. C2×#11719919C22semidirect
35. C2×C22⋊Dic53C22semidirect
36. C2×C22⋊Dic53C22semidirect
37. C2×C22⋊Dic53C22semidirect
38. C2×#11719921C22semidirect
39. C2×#11719921C22semidirect
40. C2×#11719921C22semidirect
41. C2×C4×Dic53C22semidirect
42. C2×C4×Dic53C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
103. C22⋊C4×C53D4semidirect
115. C22⋊Dic53D4semidirect
273. C2×#120D53semidirect
285. C53#2670semidirect
390. C22⋊C4C22⋊D53semidirect