Finite Group Databasebeta

← #20286323

Subgroups — #20286323

Order 6784.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133245C211yesmaximal, characteristic, complemented
2C2×#13133244C211yesmaximal, characteristic, complemented
3#16811270C211yesmaximal, characteristic, complemented
4#16811264C211yesmaximal, characteristic, complemented
5#16811004C211yesmaximal, characteristic, complemented
6#16810870C211yesmaximal, characteristic, complemented
7#16810841C211yesmaximal, characteristic, complemented
8#16810837C211yesmaximal, characteristic, complemented
9#16810788C211yesmaximal, characteristic, complemented
10#16810783C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810756C211yesmaximal, characteristic, complemented
13#16810705C211yesmaximal, characteristic, complemented
14#16810699C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C22×D212C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic, complemented
20#13133245—24no—
21#13133244—24no—
22C2×#11719925C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic, complemented
24C2×#11719913C2211yescharacteristic, complemented
25C2×#11719924C2211yescharacteristic, complemented
26C2×D424C2211yescharacteristic
27C2×D424C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719922C2211yescharacteristic
30C2×#11719912C2211yescharacteristic, complemented
31#13133206C2211yescharacteristic, complemented
32#13133204C2211yescharacteristic, complemented
33#13133202—21no—
34#13133199—21no—
35#13133197C2211yescharacteristic, complemented
36#13133192—21no—
37#13133192C2211yescharacteristic
38#13133188—21no—
39#13133187—21no—
40#13133185—21no—
41#13133179C2211yescharacteristic, complemented
42#13133179C2211yescharacteristic, complemented
43C4×C22⋊D53C2211yescharacteristic
44C22⋊C4×D53—21no—
45C22⋊C4×D53—21no—
46#13133146C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133137C2211yescharacteristic, complemented
49#13133137C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic, complemented
51C22⋊#88322C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53C8×Dic53C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×M16×C53C2211yescharacteristic, complemented
56C22⋊Q8×C53C2211yescharacteristic, complemented
57#121×C53C2211yescharacteristic
58#107×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70#11719949—22no—
71#11719949—41no—
72D4×D53—22no—
73D4×D53—41no—
74#11719945—22no—
75#11719945—41no—
76C2×D212—22no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—41no—
81C2×D212C2311yescharacteristic
82C2×D212C2311yescharacteristic
83C2×D212C2311yescharacteristic
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53C2311yescharacteristic
89C2×Dic106C2311yescharacteristic
90SD16×C53—22no—
91M16×C53—22no—
92C2×C8×C53C2311yescharacteristic
93C2×C8×C53C2311yescharacteristic
94C4⋊C4×C53—21no—
95C4⋊C4×C53C2311yescharacteristic
96C4⋊C4×C53C2311yescharacteristic
97C22⋊C4×C53—21no—
98C22⋊C4×C53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101C22⋊Dic53D411yescharacteristic
102#11719925—41no—
103#11719924—22no—
104#11719922—22no—
105#11719922—41no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921—21no—
112#11719921D411yescharacteristic
113C4⋊Dic53—21no—
114#11719919D411yescharacteristic
115C4×Dic53C2311yescharacteristic
116C2×#88322C2311yescharacteristic
117C2×#88322C2311yescharacteristic
118D424—22no—
119D424D414yescomplemented
120#11719913—22no—
121#11719913—22no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—42no—
132C22×D53—81no—
133C22×D53—81no—
134C22×D53C2×D411yescharacteristic
135Q8×C53—21no—
136Q8×C53—21no—
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—81no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2411yescharacteristic
147C22⋊D53—41no—
148C22⋊D53—42no—
149C22⋊D53—42no—
150C2×Dic53—21no—
151C2×Dic53C2×D411yescharacteristic
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154D212—21no—
155D212—21no—
156D212—21no—
157D212—21no—
158D212—42no—
159D212—42no—
160D212—81no—
161D212C2×D412yes—
162D212C2×D412yes—
163C4×D53—21no—
164C4×D53—21no—
165C4×D53—42no—
166C4×D53—42no—
167C4×D53—81no—
168C4×D53—81no—
169Dic106—21no—
170Dic106—21no—
171C8×C53—21no—
172C8×C53C2×D412yes—
173#88322—21no—
174#88322—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53C22×D411yescharacteristic
182C2×D53—21no—
183C2×D53—22no—
184C2×D53—22no—
185C2×D53—21no—
186C2×D53—21no—
187C2×D53—41no—
188C2×D53—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—84no—
195C2×D53—84no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53C22×D411yescharacteristic
201C4×C53H(5,𝔽2)11yescharacteristic
202Dic53—21no—
203Dic53—21no—
204Dic53—21no—
205Dic53—41no—
206#3273—531nomaximal, Sylow
207C2×C53—41no—
208C2×C53—81no—
209C2×C53D4211yescharacteristic
210C2×C53#57511yescharacteristic
211C2×C53#57611yescharacteristic
212D53—41no—
213D53—42no—
214D53—42no—
215D53—81no—
216C2×AGL(1,ℤ/8)—531no—
217C2×AGL(1,ℤ/8)—531no—
218#546—531no—
219#546—531no—
220#495—531no—
221#468—531no—
222#468D5311yescharacteristic, complemented
223#463—531no—
224#463—531no—
225#452—531no—
226#452—531no—
227#447—531no—
228#447—531no—
229C4×D8—531no—
230#436—531no—
231C53#327311yescharacteristic, complemented, Sylow
232C2×#41—531no—
233C2×#41—531no—
234C22×D4—531no—
235C22×D4—531no—
236AGL(1,ℤ/8)—1064no—
237AGL(1,ℤ/8)—1064no—
238C2×Q16—531no—
239C2×SD16—531no—
240C2×SD16—531no—
241C2×SD16—531no—
242C2×SD16C2×D5311yescharacteristic, complemented
243C2×D8—531no—
244C2×D8—531no—
245C2×D8—531no—
246C2×D8—531no—
247C2×M16—531no—
248C2×M16C2×D5311yescharacteristic, complemented
249#124—531no—
250#124—531no—
251#123—1061no—
252#123—1061no—
253C22⋊Q8—531no—
254C22⋊Q8C2×D5311yescharacteristic, complemented
255#121—531no—
256#121—1061no—
257#121—1061no—
258#121C2×D5311yescharacteristic
259#120—1061no—
260#120—1061no—
261C4×D4—531no—
262C4×D4—531no—
263C4×D4—531no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266#107C2×D5311yescharacteristic, complemented
267C4⋊C8—531no—
268#103—531no—
269#103C2×D5311yescharacteristic, complemented
270#102—531no—
271#102—531no—
272#102—531no—
273#102C2×D5311yescharacteristic, complemented
274C22⋊C8—531no—
275C22⋊C8—531no—
276C4×C8—531no—
277C24—1061no—
278C24—1061no—
279#41—1062no—
280#41—1062no—
281#41—2121no—
282#41—2121no—
283C2×Q8—531no—
284C2×Q8C22×D5311yescharacteristic
285C2×D4—21no—
286C2×D4—531no—
287C2×D4—531no—
288C2×D4—531no—
289C2×D4—531no—
290C2×D4—1062no—
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1062no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4C22×D5311yescharacteristic
300C22×C4—531no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4C22×D5311yescharacteristic
306Q16—2121no—
307SD16—22no—
308SD16—1062no—
309SD16—1062no—
310SD16—1062no—
311D8—534no—
312D8—1062no—
313D8—1062no—
314D8—2121no—
315M16—22no—
316M16—1062no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8C22×D5311yescharacteristic
320C2×C8C22×D5311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—531no—
323C4⋊C4—1061no—
324C4⋊C4C22×D5311yescharacteristic
325C4⋊C4C22×D5311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—531no—
329C22⋊C4—531no—
330C22⋊C4—1061no—
331C22⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C22⋊C4—1061no—
337C22⋊C4—1061no—
338C42—531no—
339C23—21no—
340C23—531no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23—4241no—
345C23—4241no—
346C23—2122no—
347C23—2122no—
348C23—2122no—
349C23—2122no—
350C23D4×D5311yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1061no—
354Q8—1061no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—532no—
359D4—532no—
360D4—1061no—
361D4—1061no—
362D4—1061no—
363D4—1061no—
364D4—4241no—
365D4—2122no—
366D4—2122no—
367D4—2121no—
368D4—2122no—
369D4—2122no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—531no—
376C2×C4—531no—
377C2×C4—531no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—2122no—
384C2×C4—2122no—
385C2×C4D4×D5311yescharacteristic
386C2×C4C23×D5311yescharacteristic
387C8—21no—
388C8—1061no—
389C8—1061no—
390C8D4×D5312yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—1061no—
398C22—1061no—
399C22—1061no—
400C22—1062no—
401C22—1062no—
402C22—4244no—
403C22—4244no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22—2121no—
410C22—2121no—
411C22C2×D4×D5311yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—1061no—
417C4—1061no—
418C4—1061no—
419C4—2121no—
420C4C2×D4×D5311yescharacteristic
421C4#1313326311yescharacteristic
422C2—41no—
423C2—81no—
424C2—4241no—
425C2—2121no—
426C2—2122no—
427C2—2122no—
428C2#1681125711yescharacteristic
429C2#1681142211yescharacteristic
430C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3273
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#13133245C2semidirect
2. C2×#13133244C2semidirect
3. #16811270C2semidirect
4. #16811264C2semidirect
5. #16811004C2semidirect
6. #16810870C2semidirect
7. #16810841C2semidirect
8. #16810837C2semidirect
9. #16810788C2semidirect
10. #16810783C2semidirect
11. #16810758C2semidirect
12. #16810756C2semidirect
13. #16810705C2semidirect
14. #16810699C2semidirect
15. C53×#468C2semidirect
18. C22×D212C22semidirect
19. C2×D4×D53C22semidirect
22. C2×#11719925C22semidirect
23. C2×#11719913C22semidirect
24. C2×#11719913C22semidirect
25. C2×#11719924C22semidirect
30. C2×#11719912C22semidirect
31. #13133206C22semidirect
32. #13133204C22semidirect
35. #13133197C22semidirect
41. #13133179C22semidirect
42. #13133179C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
48. #13133137C22semidirect
49. #13133137C22semidirect
50. #13133137C22semidirect
51. C22⋊#88322C22semidirect
52. #13133129C22semidirect
53. C8×Dic53C22semidirect
54. C2×SD16×C53C22semidirect
55. C2×M16×C53C22semidirect
56. C22⋊Q8×C53C22semidirect
58. #107×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
119. D424D4semidirect
222. #468D53semidirect
231. C53#3273semidirect
242. C2×SD16C2×D53semidirect
248. C2×M16C2×D53semidirect
254. C22⋊Q8C2×D53semidirect
266. #107C2×D53semidirect
269. #103C2×D53semidirect
273. #102C2×D53semidirect