Finite Group Databasebeta

← #20288368

Subgroups — #20288368

Order 6784.

1536 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133264C211yesmaximal, characteristic, complemented
2C2×#41×D53C211yesmaximal, characteristic, complemented
3#16811293C213yesmaximal, complemented
4#16811292C211yesmaximal, characteristic, complemented
5#16811291C213yesmaximal, complemented
6#16811285C211yesmaximal, characteristic, complemented
7#16811282C213yesmaximal, complemented
8#16811280C211yesmaximal, characteristic, complemented
9#16811279C213yesmaximal, complemented
10#16811189C213yesmaximal, complemented
11#16811186C213yesmaximal, complemented
12#16811133C213yesmaximal, complemented
13#16811132C213yesmaximal, complemented
14#16811099C211yesmaximal, characteristic, complemented
15C53×#550C211yesmaximal, characteristic, complemented
16#13133264—28no—
17C2×#11719949C2211yescharacteristic
18C2×#11719945C2213yes—
19C2×#11719945C2213yes—
20C2×#11719947C2213yes—
21C2×#11719947C2213yes—
22C2×#11719947C2213yescomplemented
23#41×D53—28no—
24C22×Dic106C2213yescomplemented
25C2×Q8×D53C2211yescharacteristic
26C2×Q8×D53C2211yescharacteristic, complemented
27C2×D4×D53C2213yes—
28C22×C4×D53C2213yes—
29#13133213C2213yescomplemented
30#13133213C2216yescomplemented
31#13133205C2213yescomplemented
32#13133204C2213yescomplemented
33#13133203C2216yescomplemented
34#13133202C2211yescharacteristic
35#13133199C2213yes—
36#13133197C2213yescomplemented
37#13133196C2213yescomplemented
38#13133195C2213yescomplemented
39#13133194C2213yescomplemented
40#13133192C2213yescomplemented
41#13133191C2213yescomplemented
42#13133190C2213yescomplemented
43#13133189C2216yescomplemented
44#13133189C2216yescomplemented
45#13133189C2213yescomplemented
46#13133188C2213yescomplemented
47#13133188C2213yescomplemented
48#13133187C2213yes—
49#13133185C2213yes—
50#13133185C2216yes—
51Q8×Dic53C2211yescharacteristic, complemented
52C4×Dic106C2213yescomplemented
53#13133182C2213yescomplemented
54C4×D212C2213yescomplemented
55D4×Dic53C2216yescomplemented
56C4×C22⋊D53C2213yescomplemented
57C4×C22⋊D53C2213yescomplemented
58#13133177C2216yescomplemented
59#13133175C2213yes—
60#13133174C2213yes—
61C2×#41×C53C2211yescharacteristic
62C2×#41×C53C2211yescharacteristic, complemented
63#127×C53C2213yescomplemented
64#121×C53C2213yescomplemented
65#121×C53C2213yescomplemented
66C4×Q8×C53C2211yescharacteristic, complemented
67C4×D4×C53C2213yescomplemented
68C23×D53C2313yes—
69#41×C53—24no—
70#41×C53—24no—
71C2×Q8×C53C2311yescharacteristic
72C2×D4×C53C2313yes—
73C2×D4×C53C2313yes—
74C2×D4×C53C2313yes—
75C2×D4×C53C2316yes—
76C22×C4×C53C2313yes—
77C22×C4×C53C2313yes—
78C2×C22⋊D53C2313yes—
79C2×C22⋊D53C2316yes—
80C2×C22⋊D53C2313yes—
81C2×C22⋊D53C2316yes—
82C2×C22⋊D53C2313yes—
83C2×C22⋊D53C2313yes—
84C22×Dic53C2316yes—
85C22×Dic53C2313yes—
86#11719949—24no—
87Q8×D53—24no—
88Q8×D53D418yescomplemented
89#11719947—212no—
90#11719947—212no—
91#11719947—212no—
92D4×D53—212no—
93#11719945—212no—
94#11719945—212no—
95C2×D212C2311yescharacteristic
96C2×D212C2313yes—
97C2×D212C2313yes—
98C2×C4×D53—212no—
99C2×C4×D53C2313yes—
100C2×C4×D53C2313yes—
101C2×C4×D53C2313yes—
102C2×C4×D53C2311yescharacteristic
103C2×C4×D53C2313yes—
104C2×Dic106—212no—
105C2×Dic106C2313yes—
106C2×Dic106C2313yes—
107C2×Dic106C2313yes—
108C4⋊C4×C53C2311yescharacteristic
109C4⋊C4×C53C2313yes—
110C22⋊C4×C53C2313yes—
111C22⋊C4×C53C2313yes—
112C42×C53C2313yes—
113C22⋊Dic53C2313yes—
114C22⋊Dic53C2313yes—
115C22⋊Dic53C2316yes—
116#11719921C2311yescharacteristic
117#11719921C2313yes—
118#11719921C2316yes—
119#11719921C2311yescharacteristic
120#11719921C2313yes—
121#11719921C2311yescharacteristic
122#11719921C2313yes—
123C4⋊Dic53C2313yes—
124C4⋊Dic53C2313yes—
125#11719919C2313yes—
126#11719919C2313yes—
127#11719919C2313yes—
128#11719919C2311yescharacteristic
129C4×Dic53C2313yes—
130C4×Dic53C2316yes—
131C23×C53C2413yes—
132C23×C53C2413yes—
133C22×D53—26no—
134C22×D53—43no—
135C22×D53—43no—
136C22×D53C2411yescharacteristic
137C22×D53C2413yes—
138C22×D53C2411yescharacteristic
139Q8×C53C2×D414yes—
140D4×C53—26no—
141D4×C53—26no—
142D4×C53—46no—
143D4×C53—43no—
144C2×C4×C53—26no—
145C2×C4×C53—26no—
146C2×C4×C53C2411yescharacteristic
147C2×C4×C53C2411yescharacteristic
148C2×C4×C53C2413yes—
149C2×C4×C53C2411yescharacteristic
150C2×C4×C53C2413yes—
151C22⋊D53—26no—
152C22⋊D53—212no—
153C22⋊D53—26no—
154C22⋊D53—43no—
155C22⋊D53—43no—
156C22⋊D53—46no—
157C2×Dic53—212no—
158C2×Dic53—26no—
159C2×Dic53C2413yes—
160C2×Dic53C2411yescharacteristic
161C2×Dic53C2411yescharacteristic
162C2×Dic53C2413yes—
163C2×Dic53C2413yes—
164D212—22no—
165D212—26no—
166D212—43no—
167C4×D53—22no—
168C4×D53—26no—
169C4×D53—26no—
170C4×D53—26no—
171C4×D53C2×D4112yes—
172Dic106—26no—
173Dic106—26no—
174Dic106C2×D4112yes—
175C22×C53—23no—
176C22×C53—23no—
177C22×C53—23no—
178C22×C53—23no—
179C22×C53—23no—
180C22×C53—23no—
181C22×C53C2511yescharacteristic
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C2×D53—23no—
186C2×D53—21no—
187C2×D53—23no—
188C2×D53—21no—
189C2×D53—23no—
190C2×D53—412no—
191C2×D53C22×D412yes—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—23no—
195C4×C53—21no—
196C4×C53C22×D416yes—
197Dic53—23no—
198Dic53—21no—
199Dic53—21no—
200Dic53—23no—
201Dic53C22×D416yes—
202#3487—531nomaximal, Sylow
203C2×C53—43no—
204C2×C53—43no—
205C2×C53C23×D411yescharacteristic
206C2×C53#58511yescharacteristic
207C2×C53#58511yescharacteristic
208D53—22no—
209D53—41no—
210D53—43no—
211C2×2−1+4—531no—
212C22×#41—531no—
213#550—531no—
214#550—533no—
215#550—533no—
216#550D5311yescharacteristic, complemented
217#548—531no—
218#548—533no—
219#548—531no—
220#548—533no—
221#535—533no—
222#535—533no—
223#525—533no—
224#525—533no—
225#519—531no—
226C53#348711yescharacteristic, complemented, Sylow
2272−1+4—1068no—
228C2×#41—531no—
229C2×#41—533no—
230C2×#41—533no—
231C2×#41—533no—
232C2×#41—533no—
233C2×#41—533no—
234C2×#41—1068no—
235C2×#41C2×D5311yescharacteristic
236C2×#41C2×D5311yescharacteristic, complemented
237C22×Q8—531no—
238C22×Q8—531no—
239C22×Q8—533no—
240C22×D4—533no—
241C23×C4—533no—
242#127—536no—
243#127—533no—
244#127C2×D5313yescomplemented
245#124—533no—
246#124—536no—
247#124—533no—
248#123—531no—
249#123—533no—
250C22⋊Q8—533no—
251C22⋊Q8—533no—
252C22⋊Q8—533no—
253C22⋊Q8—533no—
254#121—533no—
255#121—533no—
256#121—533no—
257#121—536no—
258#121—536no—
259#121—533no—
260#121—533no—
261#121—533no—
262#121C2×D5313yescomplemented
263#121C2×D5313yescomplemented
264#120—536no—
265#120—533no—
266#120—533no—
267C4×Q8—531no—
268C4×Q8—533no—
269C4×Q8—533no—
270C4×Q8C2×D5311yescharacteristic, complemented
271C4×D4—536no—
272C4×D4—533no—
273C4×D4—533no—
274C4×D4—536no—
275C4×D4—533no—
276C4×D4C2×D5313yescomplemented
277#117—533no—
278#117—533no—
279C24—533no—
280#41—24no—
281#41—24no—
282#41—10612no—
283#41—10612no—
284#41—10612no—
285#41—10612no—
286#41—10612no—
287#41—1064no—
288C2×Q8—538no—
289C2×Q8—533no—
290C2×Q8—533no—
291C2×Q8—533no—
292C2×Q8—10612no—
293C2×Q8—1064no—
294C2×Q8C22×D5311yescharacteristic
295C2×D4—533no—
296C2×D4—536no—
297C2×D4—533no—
298C2×D4—536no—
299C2×D4—533no—
300C2×D4—533no—
301C2×D4—531no—
302C2×D4—533no—
303C2×D4—533no—
304C2×D4—10612no—
305C2×D4C22×D5313yes—
306C2×D4C22×D5313yes—
307C2×D4C22×D5316yes—
308C2×D4C22×D5313yes—
309C22×C4—533no—
310C22×C4—536no—
311C22×C4—533no—
312C22×C4—533no—
313C22×C4—533no—
314C22×C4—533no—
315C22×C4—531no—
316C22×C4—10612no—
317C22×C4C22×D5313yes—
318C22×C4C22×D5313yes—
319C4⋊C4—533no—
320C4⋊C4—533no—
321C4⋊C4—533no—
322C4⋊C4—533no—
323C4⋊C4—531no—
324C4⋊C4—533no—
325C4⋊C4C22×D5313yes—
326C4⋊C4C22×D5311yescharacteristic
327C22⋊C4—533no—
328C22⋊C4—536no—
329C22⋊C4—533no—
330C22⋊C4—533no—
331C22⋊C4—531no—
332C22⋊C4—531no—
333C22⋊C4—531no—
334C22⋊C4—533no—
335C22⋊C4—536no—
336C22⋊C4—533no—
337C22⋊C4C22×D5313yes—
338C22⋊C4C22×D5313yes—
339C42—536no—
340C42—533no—
341C42C22×D5313yes—
342C23—531no—
343C23—533no—
344C23—531no—
345C23—1066no—
346C23—2123no—
347C23—2123no—
348C23C23×D5313yes—
349C23C23×D5313yes—
350Q8—5312no—
351Q8—1066no—
352Q8—1066no—
353Q8D4×D5314yes—
354D4—26no—
355D4—26no—
356D4—46no—
357D4—43no—
358D4—1062no—
359D4—10612no—
360D4—1066no—
361D4—1066no—
362D4—1066no—
363D4—2123no—
364D4—2123no—
365D4—2126no—
366D4—2123no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—533no—
370C2×C4—5312no—
371C2×C4—531no—
372C2×C4—533no—
373C2×C4—533no—
374C2×C4—531no—
375C2×C4—1066no—
376C2×C4—1062no—
377C2×C4—1066no—
378C2×C4—10612no—
379C2×C4—1066no—
380C2×C4—1066no—
381C2×C4C23×D5313yes—
382C2×C4C23×D5311yescharacteristic
383C2×C4C23×D5311yescharacteristic
384C2×C4C23×D5311yescharacteristic
385C2×C4C23×D5313yes—
386C22—23no—
387C22—23no—
388C22—23no—
389C22—23no—
390C22—23no—
391C22—23no—
392C22—532no—
393C22—1061no—
394C22—1061no—
395C22—1061no—
396C22—1063no—
397C22—1061no—
398C22—1063no—
399C22—1061no—
400C22—1063no—
401C22—21212no—
402C22C24×D5311yescharacteristic
403C4—21no—
404C4—21no—
405C4—23no—
406C4—21no—
407C4—536no—
408C4—1061no—
409C4—1063no—
410C4—1061no—
411C4—1063no—
412C4C2×D4×D5316yes—
413C2—43no—
414C2—43no—
415C2—1062no—
416C2—2123no—
417C2—2121no—
418C2C22×D4×D5311yescharacteristic
419C2#1681145811yescharacteristic
420C2#1681145911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#550C2
hypercenter#550D53
hypocenterC53#3487
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#13133264C2semidirect
2. C2×#41×D53C2semidirect
3. #16811293C2semidirect
4. #16811292C2semidirect
5. #16811291C2semidirect
6. #16811285C2semidirect
7. #16811282C2semidirect
8. #16811280C2semidirect
9. #16811279C2semidirect
10. #16811189C2semidirect
11. #16811186C2semidirect
12. #16811133C2semidirect
13. #16811132C2semidirect
14. #16811099C2semidirect
15. C53×#550C2semidirect
22. C2×#11719947C22semidirect
24. C22×Dic106C22semidirect
26. C2×Q8×D53C22semidirect
29. #13133213C22semidirect
30. #13133213C22semidirect
31. #13133205C22semidirect
32. #13133204C22semidirect
33. #13133203C22semidirect
36. #13133197C22semidirect
37. #13133196C22semidirect
38. #13133195C22semidirect
39. #13133194C22semidirect
40. #13133192C22semidirect
41. #13133191C22semidirect
42. #13133190C22semidirect
43. #13133189C22semidirect
44. #13133189C22semidirect
45. #13133189C22semidirect
46. #13133188C22semidirect
47. #13133188C22semidirect
51. Q8×Dic53C22semidirect
52. C4×Dic106C22semidirect
53. #13133182C22semidirect
54. C4×D212C22semidirect
55. D4×Dic53C22semidirect
56. C4×C22⋊D53C22semidirect
57. C4×C22⋊D53C22semidirect
58. #13133177C22semidirect
62. C2×#41×C53C22semidirect
63. #127×C53C22semidirect
64. #121×C53C22semidirect
65. #121×C53C22semidirect
66. C4×Q8×C53C22semidirect
67. C4×D4×C53C22semidirect
88. Q8×D53D4semidirect
216. #550D53semidirect
226. C53#3487semidirect
236. C2×#41C2×D53semidirect
244. #127C2×D53semidirect
262. #121C2×D53semidirect
263. #121C2×D53semidirect
270. C4×Q8C2×D53semidirect
276. C4×D4C2×D53semidirect