Finite Group Databasebeta

← #20278839

Subgroups — #20278839

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×#13133197C211yesmaximal, characteristic, complemented
3C2×#13133187C211yesmaximal, characteristic
4C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
5#16810548C211yesmaximal, characteristic, complemented
6#16810540C211yesmaximal, characteristic, complemented
7#16810539C211yesmaximal, characteristic, complemented
8#16810536C211yesmaximal, characteristic, complemented
9#16810525C211yesmaximal, characteristic, complemented
10#16810515C211yesmaximal, characteristic, complemented
11#16810501C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic, complemented
13#16810478C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×Dic106C2211yescharacteristic, complemented
18C22×D212C2211yescharacteristic
19C22×D212C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C4×D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23#13133203—24no—
24#13133197—24no—
25#13133187—24no—
26C2×C4⋊Dic53C2211yescharacteristic, complemented
27C2×C4⋊Dic53C2211yescharacteristic, complemented
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic, complemented
39C2×#11719921C2211yescharacteristic, complemented
40C2×#11719921C2211yescharacteristic, complemented
41C2×#11719921C2211yescharacteristic, complemented
42C22⋊C4×D53—24no—
43C2×C4×Dic53C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, 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
52C2×C42×C53C2211yescharacteristic, complemented
53#95×C53C2211yescharacteristic, complemented
54#95×C53C2211yescharacteristic, complemented
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
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C2×C22⋊D53—22no—
73C2×C22⋊D53—22no—
74C2×C22⋊D53—41no—
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C22×Dic53C2311yescharacteristic
80C2×D212—22no—
81C2×D212—22no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—41no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×Dic106—22no—
93C2×Dic106—22no—
94C2×Dic106—41no—
95C4⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53D414yescomplemented
99C42×C53—41no—
100C22⋊Dic53—22no—
101C22⋊Dic53—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#11719921D414yes—
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4×Dic53—22no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53C2×D412yes—
124C23×C53C2411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—28no—
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—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—
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—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53C2×D412yes—
169C2×C4×C53C2×D412yes—
170C2×C4×C53C2×D412yes—
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×Dic53C2×D412yes—
187D212—44no—
188D212—44no—
189C4×D53—44no—
190C4×D53—44no—
191Dic106—44no—
192C22×C53—22no—
193C22×C53—22no—
194C22×C53—22no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
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—22no—
213C2×D53—22no—
214C2×D53—22no—
215C2×D53—416no—
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#12414yescomplemented
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—42no—
241C2×C53#52511yescharacteristic
242C2×C53C2×#12411yescharacteristic
243C2×C53#53811yescharacteristic
244C2×C53D4211yescharacteristic
245C2×C53#54611yescharacteristic
246C2×C53#54611yescharacteristic
247C2×C53#55111yescharacteristic
248D53—24no—
249D53—42no—
250D53—81no—
251C2×#124—531no—
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#388D5311yescharacteristic, complemented
262#386—531no—
263#386—531no—
264#379—531no—
265C4×C22⋊C4—531no—
266C53#247211yescharacteristic, complemented, Sylow
267C25—531no—
268C22×Q8—531no—
269C22×D4—531no—
270C22×D4—531no—
271C22×D4—531no—
272C23×C4—531no—
273C23×C4—531no—
274#124—1064no—
275C22⋊Q8—1064no—
276#120—1064no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
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—1064no—
295C2×C22⋊C4C2×D5311yescharacteristic
296C2×C22⋊C4C2×D5311yescharacteristic
297C2×C22⋊C4C2×D5311yescharacteristic
298C2×C42—531no—
299C2×C42C2×D5311yescharacteristic, complemented
300#95—531no—
301#95—531no—
302#95—531no—
303#95—531no—
304#95C2×D5311yescharacteristic, complemented
305#95C2×D5311yescharacteristic, complemented
306C24—531no—
307C24—531no—
308C24—531no—
309C24—1062no—
310C24—1062no—
311C24—1062no—
312C24—1062no—
313C24—1062no—
314C24—2121no—
315C24—2121no—
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—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—1062no—
335C22×C4—1062no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—2121no—
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—1062no—
348C4⋊C4—1062no—
349C22⋊C4—22no—
350C22⋊C4—41no—
351C22⋊C4—534no—
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—1062no—
362C22⋊C4—2121no—
363C22⋊C4—2121no—
364C22⋊C4—2121no—
365C22⋊C4D21214yescomplemented
366C42—41no—
367C42—1062no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—532no—
375C23—532no—
376C23—532no—
377C23—532no—
378C23—1068no—
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—1061no—
395C23—1061no—
396C23—2124no—
397C23—2124no—
398C23—2124no—
399C23—2124no—
400C23—2124no—
401C23—2124no—
402C23C2×D21212yes—
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—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—532no—
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×C4C2×D21212yes—
441C2×C4C2×D21212yes—
442C2×C4D4×D5312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—534no—
451C22—1062no—
452C22—1062no—
453C22—1062no—
454C22—1062no—
455C22—1062no—
456C22—1062no—
457C22—1062no—
458C22—1062no—
459C22—1062no—
460C22—21216no—
461C22—2121no—
462C22—2121no—
463C22—2121no—
464C22—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22—2121no—
470C22—2121no—
471C22—2121no—
472C22C2×D4×D5311yescharacteristic
473C22C22×D21211yescharacteristic
474C22#41×D5311yescharacteristic
475C22#41×D5311yescharacteristic
476C22#41×D5311yescharacteristic
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—42no—
491C2—1064no—
492C2—4241no—
493C2—2122no—
494C2#1681113511yescharacteristic
495C2#124×D5311yescharacteristic
496C2#1681120911yescharacteristic
497C2D4×D21211yescharacteristic
498C2#1681126211yescharacteristic
499C2#1681126811yescharacteristic
500C2#1681129811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2472
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133197C2semidirect
4. C2×C22⋊C4×D53C2semidirect
5. #16810548C2semidirect
6. #16810540C2semidirect
7. #16810539C2semidirect
8. #16810536C2semidirect
9. #16810525C2semidirect
10. #16810515C2semidirect
11. #16810501C2semidirect
12. #16810501C2semidirect
13. #16810478C2semidirect
14. C4×#11719921C2semidirect
15. C53×#388C2semidirect
17. C22×Dic106C22semidirect
26. C2×C4⋊Dic53C22semidirect
27. C2×C4⋊Dic53C22semidirect
38. C2×#11719921C22semidirect
39. C2×#11719921C22semidirect
40. C2×#11719921C22semidirect
41. C2×#11719921C22semidirect
43. C2×C4×Dic53C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. #13133124C22semidirect
47. #13133124C22semidirect
48. C2×C4⋊C4×C53C22semidirect
52. C2×C42×C53C22semidirect
53. #95×C53C22semidirect
54. #95×C53C22semidirect
98. C22⋊C4×C53D4semidirect
227. C2×D53#124semidirect
261. #388D53semidirect
266. C53#2472semidirect
279. C2×C4⋊C4C2×D53semidirect
299. C2×C42C2×D53semidirect
304. #95C2×D53semidirect
305. #95C2×D53semidirect
365. C22⋊C4D212semidirect