Finite Group Databasebeta

← #20280925

Subgroups — #20280925

Order 6784.

816 conjugacy classes of subgroups.

The subgroup lattice has 542 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
5#16810558C211yesmaximal, characteristic, complemented
6#16810552C211yesmaximal, characteristic, complemented
7#16810541C211yesmaximal, characteristic, complemented
8#16810538C211yesmaximal, characteristic, complemented
9#16810533C211yesmaximal, characteristic, complemented
10#16810501C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic, complemented
13#16810499C211yesmaximal, characteristic, complemented
14#16810478C211yesmaximal, characteristic, complemented
15C53×#397C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×D212C2211yescharacteristic
21C22×D212C2211yescharacteristic, complemented
22C22×C22⋊D53C2211yescharacteristic
23C22×C22⋊D53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic, complemented
25C22×C4×D53C2211yescharacteristic, complemented
26#13133192—24no—
27#13133190—24no—
28#13133187—24no—
29#13133187—24no—
30C2×#11719919C2211yescharacteristic, complemented
31C2×#11719919C2211yescharacteristic, complemented
32C2×C22⋊Dic53C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic, complemented
42C2×#11719921C2211yescharacteristic, complemented
43C2×#11719921C2211yescharacteristic, complemented
44C2×#11719921C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47#13133124C2211yescharacteristic, complemented
48C2×C4⋊C4×C53C2211yescharacteristic, complemented
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52#95×C53C2211yescharacteristic, complemented
53#95×C53C2211yescharacteristic, complemented
54#95×C53C2211yescharacteristic, complemented
55C24×C53C2311yescharacteristic
56C23×D53—22no—
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53—41no—
62C23×D53—41no—
63C23×D53C2311yescharacteristic
64C23×D53C2311yescharacteristic
65C23×D53C2311yescharacteristic
66C23×D53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—41no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53D414yescomplemented
79C22×Dic53C2311yescharacteristic
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212D414yescomplemented
98C2×C4×D53—22no—
99C2×C4×D53—22no—
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C4×D53—41no—
104C4⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C22⋊Dic53—41no—
109#11719921—22no—
110#11719921—22no—
111#11719921—22no—
112#11719921—22no—
113#11719921—22no—
114#11719921—22no—
115#11719921—22no—
116#11719921—41no—
117#11719921—41no—
118#11719921—41no—
119#11719921—41no—
120#11719921—41no—
121#11719919—22no—
122#11719919—41no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53C2×D412yes—
130C23×C53C2411yescharacteristic
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—21no—
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—44no—
156C22×D53—44no—
157C22×D53—44no—
158C22×D53—44no—
159C22×D53—44no—
160C22×D53—44no—
161C22×D53—44no—
162C22×D53C2×D412yes—
163C22×D53C2×D412yes—
164C22×D53C2×D412yes—
165C22×D53#4112yes—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
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—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53C2×D412yes—
184C22⋊D53—44no—
185C22⋊D53—44no—
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×Dic53C2×D412yes—
198D212—44no—
199D212—44no—
200D212—44no—
201D212—44no—
202D212—44no—
203C4×D53—44no—
204C4×D53—44no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53C22×D411yescharacteristic
213C22×C53C22×D411yescharacteristic
214C22×C53C2×#4111yescharacteristic
215C22×C53H(5,𝔽2)11yescharacteristic
216C22×C53H(5,𝔽2)11yescharacteristic
217C22×C53H(5,𝔽2)11yescharacteristic
218C22×C53H(5,𝔽2)11yescharacteristic
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—22no—
225C2×D53—22no—
226C2×D53—22no—
227C2×D53—22no—
228C2×D53—22no—
229C2×D53—22no—
230C2×D53—22no—
231C2×D53—416no—
232C2×D53—41no—
233C2×D53—41no—
234C2×D53—41no—
235C2×D53—41no—
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—
248C4×C53—22no—
249C4×C53—41no—
250C4×C53—41no—
251C4×C53—41no—
252C4×C53—41no—
253C4×C53—41no—
254Dic53—22no—
255Dic53—41no—
256Dic53—41no—
257Dic53—41no—
258#2668—531nomaximal, Sylow
259C2×C53—42no—
260C2×C53#53411yescharacteristic
261C2×C53#53411yescharacteristic
262C2×C53#53811yescharacteristic
263C2×C53D4211yescharacteristic
264C2×C53#54611yescharacteristic
265C2×C53#54611yescharacteristic
266C2×C53#56011yescharacteristic
267D53—42no—
268D53—42no—
269D53—42no—
270D53—42no—
271C2×#121—531no—
272C2×#121—531no—
273C2×#120—531no—
274C2×#120—531no—
275#399—531no—
276#397—531no—
277#397D5311yescharacteristic, complemented
278#394—531no—
279#394—531no—
280#392—531no—
281#386—531no—
282#386—531no—
283#386—531no—
284#386—531no—
285#379—531no—
286C53#266811yescharacteristic, complemented, Sylow
287C25—531no—
288C22×D4—531no—
289C22×D4—531no—
290C22×D4—531no—
291C22×D4—531no—
292C22×D4—531no—
293C22×D4—531no—
294C22×D4—531no—
295C23×C4—531no—
296C23×C4—531no—
297#121—1064no—
298#121—1064no—
299#120—1064no—
300#120—1064no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4C2×D5311yescharacteristic, complemented
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⋊C4—531no—
317C2×C22⋊C4C2×D5311yescharacteristic
318C2×C22⋊C4C2×D5311yescharacteristic
319C2×C22⋊C4C2×D5311yescharacteristic
320#95—531no—
321#95—531no—
322#95—531no—
323#95C2×D5311yescharacteristic, complemented
324#95C2×D5311yescharacteristic, complemented
325#95C2×D5311yescharacteristic, complemented
326C24—531no—
327C24—531no—
328C24—531no—
329C24—531no—
330C24—1062no—
331C24—1062no—
332C24—1062no—
333C24—2121no—
334C24—2121no—
335C24—2121no—
336C24—2121no—
337C24C22×D5311yescharacteristic
338C2×D4—534no—
339C2×D4—534no—
340C2×D4—1062no—
341C2×D4—1062no—
342C2×D4—1062no—
343C2×D4—1062no—
344C2×D4—1062no—
345C2×D4—1062no—
346C2×D4—1062no—
347C2×D4—1062no—
348C2×D4—1062no—
349C2×D4—1062no—
350C2×D4—2121no—
351C2×D4—2121no—
352C2×D4—2121no—
353C2×D4—2121no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2121no—
358C2×D4—2121no—
359C22×C4—531no—
360C22×C4—531no—
361C22×C4—531no—
362C22×C4—531no—
363C22×C4—1062no—
364C22×C4—1062no—
365C22×C4—2121no—
366C22×C4—2121no—
367C22×C4—2121no—
368C22×C4—2121no—
369C22×C4C22×D5311yescharacteristic
370C22×C4C22×D5311yescharacteristic
371C22×C4C22×D5311yescharacteristic
372C22×C4C22×D5311yescharacteristic
373C22×C4C22×D5311yescharacteristic
374C22×C4C22×D5311yescharacteristic
375C4⋊C4—22no—
376C4⋊C4—1062no—
377C4⋊C4—2121no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—1062no—
382C22⋊C4—1062no—
383C22⋊C4—1062no—
384C22⋊C4—1062no—
385C22⋊C4—1062no—
386C22⋊C4—1062no—
387C22⋊C4—1062no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—532no—
401C23—532no—
402C23—532no—
403C23—532no—
404C23—1061no—
405C23—1061no—
406C23—1061no—
407C23—1061no—
408C23—1061no—
409C23—1061no—
410C23—1061no—
411C23—1061no—
412C23—1061no—
413C23—1061no—
414C23—1061no—
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—2124no—
429C23—2124no—
430C23—2124no—
431C23—2124no—
432C23—2124no—
433C23—2124no—
434C23—2124no—
435C23D4×D5312yes—
436C23C23×D5311yescharacteristic
437D4—2124no—
438D4—2124no—
439D4—2124no—
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—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—532no—
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×C4—1061no—
473C2×C4—2124no—
474C2×C4—2124no—
475C2×C4D4×D5312yes—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—1062no—
484C22—1062no—
485C22—1062no—
486C22—1062no—
487C22—1062no—
488C22—1062no—
489C22—1062no—
490C22—1062no—
491C22—1062no—
492C22—1062no—
493C22—1062no—
494C22—1062no—
495C22—21216no—
496C22—2121no—
497C22—2121no—
498C22—2121no—
499C22—2121no—
500C22—2121no—
501C22—2121no—
502C22—2121no—
503C22—2121no—
504C22—2121no—
505C22—2121no—
506C22—2121no—
507C22—2121no—
508C22—2121no—
509C22—2121no—
510C22—2121no—
511C22—2121no—
512C22C2×D4×D5311yescharacteristic
513C22C2×D4×D5311yescharacteristic
514C22#41×D5311yescharacteristic
515C22#1313326311yescharacteristic
516C22#1313326311yescharacteristic
517C22#1313326311yescharacteristic
518C22#1313326311yescharacteristic
519C4—22no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—1062no—
526C4—2121no—
527C4—2121no—
528C4—2121no—
529C2—42no—
530C2—2122no—
531C2—2122no—
532C2—2122no—
533C2—2122no—
534C2#1681118411yescharacteristic
535C2#1681118411yescharacteristic
536C2#1681121211yescharacteristic
537C2#1681125711yescharacteristic
538C2#1681126511yescharacteristic
539C2#1681126611yescharacteristic
540C2#1681136411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2668
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

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. #16810558C2semidirect
6. #16810552C2semidirect
7. #16810541C2semidirect
8. #16810538C2semidirect
9. #16810533C2semidirect
10. #16810501C2semidirect
11. #16810501C2semidirect
12. #16810501C2semidirect
13. #16810499C2semidirect
14. #16810478C2semidirect
15. C53×#397C2semidirect
21. C22×D212C22semidirect
24. C22×C4×D53C22semidirect
25. C22×C4×D53C22semidirect
30. C2×#11719919C22semidirect
31. C2×#11719919C22semidirect
41. C2×#11719921C22semidirect
42. C2×#11719921C22semidirect
43. C2×#11719921C22semidirect
44. C2×#11719921C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. C2×C4⋊C4×C53C22semidirect
52. #95×C53C22semidirect
53. #95×C53C22semidirect
54. #95×C53C22semidirect
78. C2×C22⋊D53D4semidirect
97. C2×D212D4semidirect
277. #397D53semidirect
286. C53#2668semidirect
303. C2×C4⋊C4C2×D53semidirect
323. #95C2×D53semidirect
324. #95C2×D53semidirect
325. #95C2×D53semidirect