Finite Group Databasebeta

← #20278835

Subgroups — #20278835

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×#13133203C211yesmaximal, characteristic, complemented
2C2×#13133196C211yesmaximal, characteristic, complemented
3C2×#13133186C211yesmaximal, characteristic
4#16810551C211yesmaximal, characteristic, complemented
5#16810539C211yesmaximal, characteristic, complemented
6#16810538C211yesmaximal, characteristic, complemented
7#16810535C211yesmaximal, characteristic, complemented
8#16810523C211yesmaximal, characteristic, complemented
9#16810516C211yesmaximal, characteristic, complemented
10#16810511C211yesmaximal, characteristic, complemented
11#16810502C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13#16810479C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C53C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic, complemented
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20#13133203—24no—
21#13133196—24no—
22#13133186—24no—
23C2×C4⋊Dic53C2211yescharacteristic, complemented
24C2×#11719919C2211yescharacteristic, 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×#11719921C2211yescharacteristic, complemented
36C2×#11719921C2211yescharacteristic, complemented
37C2×#11719921C2211yescharacteristic, complemented
38C2×#11719921C2211yescharacteristic, complemented
39C2×C4×Dic53C2211yescharacteristic, complemented
40C2×C4×Dic53C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46#13133124C2211yescharacteristic, complemented
47C25×C53C2211yescharacteristic
48C23×C4×C53C2211yescharacteristic
49C23×C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53—24no—
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53—22no—
56C24×C53—22no—
57C24×C53—22no—
58C24×C53—22no—
59C24×C53—22no—
60C24×C53—41no—
61C24×C53—41no—
62C24×C53C2311yescharacteristic
63C24×C53C2311yescharacteristic
64C24×C53C2311yescharacteristic
65C23×D53C2311yescharacteristic
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C2×C22⋊D53—22no—
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
81C2×C22⋊D53—22no—
82C2×C22⋊D53—22no—
83C2×C22⋊D53—41no—
84C2×C22⋊D53—41no—
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C22×Dic53C2311yescharacteristic
92C2×Dic106—22no—
93C2×Dic106—22no—
94C2×Dic106—41no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53D414yescomplemented
99C22⋊Dic53—22no—
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—41no—
106C22⋊Dic53—41no—
107C22⋊Dic53D414yes—
108#11719921—22no—
109#11719921—22no—
110#11719921—41no—
111#11719921—41no—
112C4⋊Dic53—22no—
113#11719919—22no—
114#11719919—22no—
115C4×Dic53—22no—
116C4×Dic53—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—28no—
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×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—44no—
135C23×C53—44no—
136C23×C53—44no—
137C23×C53—44no—
138C23×C53—44no—
139C23×C53—44no—
140C23×C53C2×D412yes—
141C23×C53C2×D412yes—
142C23×C53#4112yes—
143C23×C53#4112yes—
144C23×C53#4112yes—
145C23×C53C2411yescharacteristic
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—21no—
152C22×D53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
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—44no—
164C2×C4×C53—44no—
165C2×C4×C53C2×D412yes—
166C2×C4×C53C2×D412yes—
167C22⋊D53—44no—
168C22⋊D53—44no—
169C22⋊D53—44no—
170C2×Dic53—21no—
171C2×Dic53—21no—
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—
190C2×Dic53C2×D412yes—
191Dic106—44no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—22no—
202C22×C53—22no—
203C22×C53—22no—
204C22×C53—416no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53#12414yescomplemented
214C22×C53C22×D411yescharacteristic
215C22×C53C22×D411yescharacteristic
216C22×C53C2×#4111yescharacteristic
217C22×C53C2×#4111yescharacteristic
218C22×C53C2×#4111yescharacteristic
219C22×C53H(5,𝔽2)11yescharacteristic
220C22×C53H(5,𝔽2)11yescharacteristic
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C4×C53—22no—
229C4×C53—22no—
230C4×C53—41no—
231C4×C53—41no—
232Dic53—22no—
233Dic53—22no—
234Dic53—41no—
235Dic53—41no—
236Dic53—41no—
237Dic53—41no—
238Dic53—41no—
239#2472—531nomaximal, Sylow
240C2×C53—24no—
241C2×C53—42no—
242C2×C53—42no—
243C2×C53#52511yescharacteristic
244C2×C53C2×#12411yescharacteristic
245C2×C53#53811yescharacteristic
246C2×C53D4211yescharacteristic
247C2×C53#54611yescharacteristic
248C2×C53#54611yescharacteristic
249C2×C53#55111yescharacteristic
250D53—81no—
251C2×#124—531no—
252C2×C22⋊Q8—531no—
253C2×#120—531no—
254C22×C22⋊C4D5311yescharacteristic, complemented
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
267C25C2×D5311yescharacteristic
268C22×Q8—531no—
269C22×D4—531no—
270C22×D4—531no—
271C22×D4—531no—
272C23×C4C2×D5311yescharacteristic
273C23×C4C2×D5311yescharacteristic
274#124—1064no—
275C22⋊Q8—1064no—
276#120—1064no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—531no—
280C2×C22⋊C4—24no—
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—531no—
294C2×C22⋊C4C2×D5311yescharacteristic
295C2×C22⋊C4C2×D5311yescharacteristic
296C2×C22⋊C4C2×D5311yescharacteristic
297C2×C22⋊C4C2×D5311yescharacteristic
298C2×C42—531no—
299C2×C42—531no—
300#95—531no—
301#95—531no—
302#95—531no—
303#95—531no—
304#95—531no—
305#95—531no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24—531no—
314C24C22×D5311yescharacteristic
315C24C22×D5311yescharacteristic
316C24C22×D5311yescharacteristic
317C2×Q8—1062no—
318C2×Q8—1062no—
319C2×Q8—2121no—
320C2×D4—1062no—
321C2×D4—1062no—
322C2×D4—1062no—
323C2×D4—1062no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—1062no—
327C2×D4—2121no—
328C2×D4—2121no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—531no—
336C22×C4—531no—
337C22×C4—531no—
338C22×C4—531no—
339C22×C4—531no—
340C22×C4—531no—
341C22×C4—531no—
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—1062no—
367C42—2121no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—1061no—
397C23—1061no—
398C23#1171994512yes—
399C23#1171994512yes—
400C23D4×D5312yes—
401C23#1171994712yes—
402C23C2×C22⋊D5312yes—
403C23C23×D5311yescharacteristic
404Q8—2124no—
405D4—2124no—
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—44no—
419C2×C4—44no—
420C2×C4—532no—
421C2×C4—532no—
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—1061no—
439C2×C4—1061no—
440C2×C4—1061no—
441C2×C4C2×C22⋊D5312yes—
442C2×C4C2×C22⋊D5312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—416no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22—2121no—
470C22—2121no—
471C22#1313320314yescomplemented
472C22C2×D4×D5311yescharacteristic
473C22C22×C22⋊D5311yescharacteristic
474C22C2×#1171994711yescharacteristic
475C22C2×#1171994511yescharacteristic
476C22C2×#1171994511yescharacteristic
477C22#1313326211yescharacteristic
478C22#1313326211yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—1062no—
484C4—1062no—
485C4—2121no—
486C4—2121no—
487C4—2121no—
488C4—2121no—
489C4—2121no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—4241no—
494C2#1681113411yescharacteristic
495C2C2×#1313320311yescharacteristic
496C2#1681121111yescharacteristic
497C2D4×C22⋊D5311yescharacteristic
498C2#1681126311yescharacteristic
499C2#1681126711yescharacteristic
500C2#1681129911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC22×C22⋊C4×C53C2
hypercenterC22×C22⋊C4D53
hypocenterC53#2472
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133196C2semidirect
4. #16810551C2semidirect
5. #16810539C2semidirect
6. #16810538C2semidirect
7. #16810535C2semidirect
8. #16810523C2semidirect
9. #16810516C2semidirect
10. #16810511C2semidirect
11. #16810502C2semidirect
12. #16810502C2semidirect
13. #16810479C2semidirect
14. C4×C22⋊Dic53C2semidirect
15. C22×C22⋊C4×C53C2semidirect
16. C22×Dic106C22semidirect
23. C2×C4⋊Dic53C22semidirect
24. C2×#11719919C22semidirect
25. C2×#11719919C22semidirect
35. C2×#11719921C22semidirect
36. C2×#11719921C22semidirect
37. C2×#11719921C22semidirect
38. C2×#11719921C22semidirect
39. C2×C4×Dic53C22semidirect
40. C2×C4×Dic53C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
98. C22⋊C4×C53D4semidirect
213. C22×C53#124semidirect
254. C22×C22⋊C4D53semidirect
266. C53#2472semidirect
365. C22⋊C4C22⋊D53semidirect
471. C22#13133203semidirect