Finite Group Databasebeta

← #20278837

Subgroups — #20278837

Order 6784.

786 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133193C211yesmaximal, characteristic, complemented
2C2×#13133185C211yesmaximal, characteristic
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810551C211yesmaximal, characteristic, complemented
5#16810539C211yesmaximal, characteristic, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810534C211yesmaximal, characteristic, complemented
8#16810523C211yesmaximal, characteristic, complemented
9#16810516C211yesmaximal, characteristic, complemented
10#16810514C211yesmaximal, characteristic, complemented
11#16810499C211yesmaximal, characteristic, complemented
12#16810499C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C2×#124×C53C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133193—24no—
22#13133185—24no—
23C2×C4⋊Dic53C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, complemented
25C2×#11719919C2211yescharacteristic, complemented
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic, complemented
30C2×C22⋊Dic53C2211yescharacteristic, complemented
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C22⋊C4×D53—24no—
40C2×C4×Dic53C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47C22×Q8×C53C2211yescharacteristic, complemented
48C22×D4×C53C2211yescharacteristic
49#124×C53—24no—
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic, complemented
53C2×C22⋊C4×C53C2211yescharacteristic, complemented
54C2×C42×C53C2211yescharacteristic, complemented
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—22no—
60C23×D53—22no—
61C23×D53—22no—
62C23×D53—41no—
63C23×D53—41no—
64C23×D53C2311yescharacteristic
65C23×D53C2311yescharacteristic
66C2×Q8×C53—22no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—41no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
81C2×C22⋊D53—22no—
82C2×C22⋊D53—41no—
83C22×Dic53C2311yescharacteristic
84C22×Dic53C2311yescharacteristic
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C2×C4×D53—22no—
90C2×C4×D53—22no—
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53D414yescomplemented
99C42×C53—22no—
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—41no—
103C22⋊Dic53—41no—
104C22⋊Dic53—41no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—22no—
109#11719921—22no—
110#11719921—22no—
111#11719921—41no—
112#11719921D414yes—
113C4⋊Dic53—22no—
114#11719919—22no—
115#11719919—22no—
116C4×Dic53—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53C2×D412yes—
131C23×C53C2411yescharacteristic
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—28no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—44no—
144C22×D53—44no—
145C22×D53—44no—
146C22×D53—44no—
147C22×D53—44no—
148C22×D53—44no—
149C22×D53C2×D412yes—
150C22×D53#4112yes—
151C22×D53#4112yes—
152C22×D53#4112yes—
153Q8×C53—44no—
154D4×C53—44no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53C2×D412yes—
168C2×C4×C53C2×D412yes—
169C2×C4×C53C2×D412yes—
170C22⋊D53—44no—
171C22⋊D53—44no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53C2×D412yes—
190C4×D53—44no—
191C4×D53—44no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53C22×D411yescharacteristic
207C22×C53C22×D411yescharacteristic
208C22×C53C2×#4111yescharacteristic
209C22×C53C2×#4111yescharacteristic
210C22×C53C2×#4111yescharacteristic
211C22×C53H(5,𝔽2)11yescharacteristic
212C22×C53H(5,𝔽2)11yescharacteristic
213C2×D53—22no—
214C2×D53—22no—
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—416no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53#12414yescomplemented
228C4×C53—22no—
229C4×C53—22no—
230C4×C53—22no—
231C4×C53—41no—
232C4×C53—41no—
233Dic53—22no—
234Dic53—41no—
235Dic53—41no—
236Dic53—41no—
237Dic53—41no—
238Dic53—41no—
239#2472—531nomaximal, Sylow
240C2×C53—42no—
241C2×C53—81no—
242C2×C53#52511yescharacteristic
243C2×C53C2×#12411yescharacteristic
244C2×C53#53811yescharacteristic
245C2×C53D4211yescharacteristic
246C2×C53#54611yescharacteristic
247C2×C53#54611yescharacteristic
248C2×C53#55111yescharacteristic
249D53—24no—
250D53—42no—
251C2×#124D5311yescharacteristic, complemented
252C2×C22⋊Q8—531no—
253C2×#120—531no—
254C22×C22⋊C4—531no—
255#397—531no—
256#394—531no—
257#394—531no—
258#393—531no—
259#390—531no—
260#388—531no—
261#388—531no—
262#386—531no—
263#386—531no—
264#379—531no—
265C4×C22⋊C4—531no—
266C53#247211yescharacteristic, complemented, Sylow
267C25—531no—
268C22×Q8C2×D5311yescharacteristic, complemented
269C22×D4—531no—
270C22×D4—531no—
271C22×D4C2×D5311yescharacteristic
272C23×C4—531no—
273C23×C4—531no—
274#124—24no—
275C22⋊Q8—1064no—
276#120—1064no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—531no—
282C2×C22⋊C4—531no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—531no—
286C2×C22⋊C4—531no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—531no—
290C2×C22⋊C4—531no—
291C2×C22⋊C4—531no—
292C2×C22⋊C4—531no—
293C2×C22⋊C4—1064no—
294C2×C22⋊C4C2×D5311yescharacteristic
295C2×C22⋊C4C2×D5311yescharacteristic
296C2×C22⋊C4C2×D5311yescharacteristic, complemented
297C2×C22⋊C4C2×D5311yescharacteristic, complemented
298C2×C42—531no—
299C2×C42C2×D5311yescharacteristic, complemented
300#95—531no—
301#95—531no—
302#95—531no—
303#95—531no—
304#95—531no—
305#95—531no—
306C24—531no—
307C24—531no—
308C24—1062no—
309C24—1062no—
310C24—1062no—
311C24—1062no—
312C24—1062no—
313C24—2121no—
314C24—2121no—
315C24C22×D5311yescharacteristic
316C24C22×D5311yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—41no—
323C2×D4—1062no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—1062no—
327C2×D4—1062no—
328C2×D4—2121no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—531no—
335C22×C4—1062no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—1062no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C22×C4C22×D5311yescharacteristic
344C22×C4C22×D5311yescharacteristic
345C22×C4C22×D5311yescharacteristic
346C4⋊C4—1062no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—534no—
353C22⋊C4—1062no—
354C22⋊C4—1062no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C22⋊C4—1062no—
359C22⋊C4—1062no—
360C22⋊C4—1062no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C22⋊C4—2121no—
364C22⋊C4—2121no—
365C22⋊C4C22⋊D5314yescomplemented
366C42—22no—
367C42—2121no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—532no—
382C23—532no—
383C23—532no—
384C23—532no—
385C23—1068no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23—1061no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—2124no—
397C23—2124no—
398C23—2124no—
399C23—2124no—
400C23—2124no—
401C23—2124no—
402C23C2×C22⋊D5312yes—
403C23C23×D5311yescharacteristic
404Q8—44no—
405D4—44no—
406D4—2124no—
407D4—2124no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—532no—
421C2×C4—1061no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—1061no—
434C2×C4—1061no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—1061no—
438C2×C4—2124no—
439C2×C4—2124no—
440C2×C4D4×D5312yes—
441C2×C4C2×C22⋊D5312yes—
442C2×C4C2×C22⋊D5312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—534no—
458C22—1062no—
459C22—1062no—
460C22—1062no—
461C22—1062no—
462C22—1062no—
463C22—1062no—
464C22—1062no—
465C22—1062no—
466C22—1062no—
467C22—21216no—
468C22—2121no—
469C22—2121no—
470C22—2121no—
471C22—2121no—
472C22C2×D4×D5311yescharacteristic
473C22C22×C22⋊D5311yescharacteristic
474C22#41×D5311yescharacteristic
475C22#41×D5311yescharacteristic
476C22#41×D5311yescharacteristic
477C22#1313326211yescharacteristic
478C22#1313326311yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—1062no—
485C4—2121no—
486C4—2121no—
487C4—2121no—
488C4—2121no—
489C4—2121no—
490C2—42no—
491C2—81no—
492C2—1064no—
493C2—2122no—
494C2#1681113011yescharacteristic
495C2#124×D5311yescharacteristic
496C2#1681120711yescharacteristic
497C2D4×C22⋊D5311yescharacteristic
498C2#1681126111yescharacteristic
499C2#1681126611yescharacteristic
500C2#1681130311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#124×C53C2
hypercenterC2×#124D53
hypocenterC53#2472
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#13133193C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810551C2semidirect
5. #16810539C2semidirect
6. #16810537C2semidirect
7. #16810534C2semidirect
8. #16810523C2semidirect
9. #16810516C2semidirect
10. #16810514C2semidirect
11. #16810499C2semidirect
12. #16810499C2semidirect
13. #16810478C2semidirect
14. C4×#11719921C2semidirect
15. C2×#124×C53C2semidirect
23. C2×C4⋊Dic53C22semidirect
24. C2×#11719919C22semidirect
25. C2×#11719919C22semidirect
29. C2×C22⋊Dic53C22semidirect
30. C2×C22⋊Dic53C22semidirect
40. C2×C4×Dic53C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. C22×Q8×C53C22semidirect
52. C2×C22⋊C4×C53C22semidirect
53. C2×C22⋊C4×C53C22semidirect
54. C2×C42×C53C22semidirect
98. C22⋊C4×C53D4semidirect
227. C2×D53#124semidirect
251. C2×#124D53semidirect
266. C53#2472semidirect
268. C22×Q8C2×D53semidirect
296. C2×C22⋊C4C2×D53semidirect
297. C2×C22⋊C4C2×D53semidirect
299. C2×C42C2×D53semidirect
365. C22⋊C4C22⋊D53semidirect