Finite Group Databasebeta

← #20278848

Subgroups — #20278848

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×#13133206C211yesmaximal, characteristic, complemented
2C2×#13133187C211yesmaximal, characteristic
3C22×#11719921C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810540C211yesmaximal, characteristic, complemented
6#16810540C211yesmaximal, characteristic, complemented
7#16810534C211yesmaximal, characteristic, complemented
8#16810524C211yesmaximal, characteristic, complemented
9#16810513C211yesmaximal, characteristic, complemented
10#16810513C211yesmaximal, characteristic, complemented
11#16810502C211yesmaximal, characteristic, complemented
12#16810500C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C23×Dic53C2211yescharacteristic
21#13133206—24no—
22#13133187—24no—
23C2×C22⋊Dic53C2211yescharacteristic
24C2×#11719921—24no—
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
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
47C22×Q8×C53C2211yescharacteristic, complemented
48C23×C4×C53C2211yescharacteristic
49C22⋊Q8×C53—24no—
50C2×C4⋊C4×C53C2211yescharacteristic, complemented
51C2×C4⋊C4×C53C2211yescharacteristic, complemented
52C2×C4⋊C4×C53C2211yescharacteristic, complemented
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53—22no—
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—22no—
60C23×D53—22no—
61C23×D53—41no—
62C23×D53—41no—
63C23×D53C2311yescharacteristic
64C23×D53C2311yescharacteristic
65C23×D53C2311yescharacteristic
66C2×Q8×C53—22no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—41no—
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C22×C4×C53C2311yescharacteristic
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
81C22×Dic53—22no—
82C22×Dic53—22no—
83C22×Dic53—41no—
84C22×Dic53C2311yescharacteristic
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—41no—
94C2×D212—41no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53D414yes—
100C22⋊Dic53—22no—
101#11719921—22no—
102#11719921—22no—
103#11719921—22no—
104#11719921—22no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—22no—
109#11719921—22no—
110#11719921—41no—
111#11719921—41no—
112#11719921—41no—
113#11719921—41no—
114#11719921D414yescomplemented
115C4×Dic53—22no—
116C4×Dic53—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53C2×D412yes—
122C23×C53#4112yes—
123C23×C53#4112yes—
124C23×C53C2411yescharacteristic
125C22×D53—28no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
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—44no—
146C22×D53—44no—
147C22×D53—44no—
148C22×D53—44no—
149C22×D53—44no—
150C22×D53—44no—
151C22×D53C2×D412yes—
152C22×D53#4112yes—
153Q8×C53—44no—
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—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—44no—
170C2×C4×C53C2×D412yes—
171C2×C4×C53C2×D412yes—
172C2×C4×C53C2×D412yes—
173C22⋊D53—44no—
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—44no—
189C2×Dic53C2×D412yes—
190D212—44no—
191D212—44no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53—22no—
197C22×C53—22no—
198C22×C53#12414yescomplemented
199C22×C53C22×D411yescharacteristic
200C22×C53C22×D411yescharacteristic
201C22×C53C2×#4111yescharacteristic
202C22×C53C2×#4111yescharacteristic
203C22×C53C2×#4111yescharacteristic
204C22×C53H(5,𝔽2)11yescharacteristic
205C22×C53H(5,𝔽2)11yescharacteristic
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—22no—
209C2×D53—22no—
210C2×D53—22no—
211C2×D53—22no—
212C2×D53—416no—
213C2×D53—41no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53—41no—
217C2×D53—41no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
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—22no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—41no—
234Dic53—22no—
235Dic53—41no—
236Dic53—41no—
237Dic53—41no—
238Dic53—41no—
239#2472—531nomaximal, Sylow
240C2×C53—24no—
241C2×C53#52511yescharacteristic
242C2×C53C2×#12411yescharacteristic
243C2×C53#53811yescharacteristic
244C2×C53D4211yescharacteristic
245C2×C53#54611yescharacteristic
246C2×C53#54611yescharacteristic
247C2×C53#55111yescharacteristic
248D53—42no—
249D53—42no—
250D53—81no—
251C2×#124—531no—
252C2×C22⋊Q8D5311yescharacteristic, complemented
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×D4—531no—
272C23×C4—531no—
273C23×C4C2×D5311yescharacteristic
274#124—1064no—
275C22⋊Q8—24no—
276#120—1064no—
277C2×C4⋊C4C2×D5311yescharacteristic, complemented
278C2×C4⋊C4C2×D5311yescharacteristic, complemented
279C2×C4⋊C4C2×D5311yescharacteristic, complemented
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—531no—
294C2×C22⋊C4—531no—
295C2×C22⋊C4—1064no—
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—531no—
307C24—531no—
308C24—531no—
309C24—1062no—
310C24—1062no—
311C24—1062no—
312C24—1062no—
313C24—1062no—
314C24—2121no—
315C24—2121no—
316C24C22×D5311yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—41no—
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—41no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—531no—
335C22×C4—531no—
336C22×C4—531no—
337C22×C4—1062no—
338C22×C4—1062no—
339C22×C4—2121no—
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C22×C4C22×D5311yescharacteristic
344C22×C4C22×D5311yescharacteristic
345C22×C4C22×D5311yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—534no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
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⋊D5314yes—
366C42—1062no—
367C42—2121no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—532no—
373C23—532no—
374C23—1068no—
375C23—1061no—
376C23—1061no—
377C23—1061no—
378C23—1061no—
379C23—1061no—
380C23—1061no—
381C23—1061no—
382C23—1061no—
383C23—1061no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23—1061no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—2124no—
395C23—2124no—
396C23—2124no—
397C23—2124no—
398C23—2124no—
399C23—2124no—
400C23#1171994912yes—
401C23#1171994912yes—
402C23C2×C22⋊D5312yes—
403C23C23×D5311yescharacteristic
404Q8—44no—
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—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—44no—
424C2×C4—532no—
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—2124no—
440C2×C4D4×D5312yes—
441C2×C4C2×C22⋊D5312yes—
442C2×C4C2×C22⋊D5312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—1062no—
450C22—1062no—
451C22—1062no—
452C22—1062no—
453C22—1062no—
454C22—1062no—
455C22—21216no—
456C22—2121no—
457C22—2121no—
458C22—2121no—
459C22—2121no—
460C22—2121no—
461C22—2121no—
462C22—2121no—
463C22—2121no—
464C22—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22—2121no—
470C22—2121no—
471C22#1313320614yescomplemented
472C22C2×D4×D5311yescharacteristic
473C22C22×C22⋊D5311yescharacteristic
474C22#41×D5311yescharacteristic
475C22C2×#1171994911yescharacteristic
476C22C2×#1171994911yescharacteristic
477C22#1313326211yescharacteristic
478C22#1313326311yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—1062no—
486C4—2121no—
487C4—2121no—
488C4—2121no—
489C4—2121no—
490C2—24no—
491C2—4241no—
492C2—2122no—
493C2—2122no—
494C2#1681113611yescharacteristic
495C2C2×#1313320611yescharacteristic
496C2#1681121311yescharacteristic
497C2D4×C22⋊D5311yescharacteristic
498C2#1681126111yescharacteristic
499C2#1681126411yescharacteristic
500C2#1681130011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×C22⋊Q8×C53C2
hypercenterC2×C22⋊Q8D53
hypocenterC53#2472
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#13133206C2semidirect
3. C22×#11719921C2semidirect
4. #16810549C2semidirect
5. #16810540C2semidirect
6. #16810540C2semidirect
7. #16810534C2semidirect
8. #16810524C2semidirect
9. #16810513C2semidirect
10. #16810513C2semidirect
11. #16810502C2semidirect
12. #16810500C2semidirect
13. #16810478C2semidirect
14. C22⋊C4×Dic53C2semidirect
15. C2×C22⋊Q8×C53C2semidirect
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
47. C22×Q8×C53C22semidirect
50. C2×C4⋊C4×C53C22semidirect
51. C2×C4⋊C4×C53C22semidirect
52. C2×C4⋊C4×C53C22semidirect
114. #11719921D4semidirect
198. C22×C53#124semidirect
252. C2×C22⋊Q8D53semidirect
266. C53#2472semidirect
268. C22×Q8C2×D53semidirect
277. C2×C4⋊C4C2×D53semidirect
278. C2×C4⋊C4C2×D53semidirect
279. C2×C4⋊C4C2×D53semidirect
471. C22#13133206semidirect