Finite Group Databasebeta

← #20286321

Subgroups — #20286321

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×#13133244C211yesmaximal, characteristic, complemented
2C2×#13133241C211yesmaximal, characteristic, complemented
3#16811268C211yesmaximal, characteristic, complemented
4#16811266C211yesmaximal, characteristic, complemented
5#16811005C211yesmaximal, characteristic, complemented
6#16810872C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810839C211yesmaximal, characteristic, complemented
9#16810788C211yesmaximal, characteristic, complemented
10#16810782C211yesmaximal, characteristic, complemented
11#16810758C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810700C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic, complemented
20#13133244—24no—
21#13133241—24no—
22C2×#11719925C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic, complemented
24C2×#11719923C2211yescharacteristic, complemented
25C2×#11719924C2211yescharacteristic, complemented
26C2×D424C2211yescharacteristic
27C2×#11719922C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719912C2211yescharacteristic, complemented
30C2×#11719912C2211yescharacteristic, complemented
31#13133205C2211yescharacteristic, complemented
32#13133201—21no—
33#13133199—21no—
34#13133197C2211yescharacteristic, complemented
35#13133193C2211yescharacteristic, complemented
36#13133192C2211yescharacteristic
37#13133191—21no—
38#13133190—21no—
39#13133188C2211yescharacteristic
40#13133187—21no—
41#13133187—21no—
42C4×D212C2211yescharacteristic
43C4×D212C2211yescharacteristic, complemented
44C2×#11719921—21no—
45C22⋊C4×D53—21no—
46#13133150C2211yescharacteristic, complemented
47#13133142C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic, complemented
49#13133138C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53C4×#88322C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×D8×C53C2211yescharacteristic
56#124×C53C2211yescharacteristic, complemented
57C4×D4×C53C2211yescharacteristic, complemented
58C4⋊C8×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72#11719949—22no—
73#11719949—41no—
74#11719947—22no—
75#11719947—41no—
76D4×D53—22no—
77D4×D53—22no—
78D4×D53—41no—
79D4×D53—41no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212D411yescharacteristic
83C2×D212C2311yescharacteristic
84C2×D212C2311yescharacteristic
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53C2311yescharacteristic
88C2×C4×D53C2311yescharacteristic
89C2×Dic106C2311yescharacteristic
90SD16×C53—22no—
91D8×C53—22no—
92C2×C8×C53C2311yescharacteristic
93C2×C8×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C42×C53C2311yescharacteristic
99C22⋊Dic53—21no—
100#11719925—41no—
101#11719924—22no—
102#11719923—22no—
103#11719922—41no—
104#11719922D414yescomplemented
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921D411yescharacteristic
112#11719921D411yescharacteristic
113C4⋊Dic53—21no—
114C4⋊Dic53D411yescharacteristic
115C4⋊Dic53C2311yescharacteristic
116#11719919—21no—
117C2×#88322C2311yescharacteristic
118C2×#88322C2311yescharacteristic
119D424—22no—
120#11719913—22no—
121#11719912—22no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53—21no—
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—42no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53C2×D411yescharacteristic
134C22×D53C2×D411yescharacteristic
135Q8×C53—21no—
136Q8×C53—21no—
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—21no—
140D4×C53C2×D412yes—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—81no—
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2411yescharacteristic
147C22⋊D53—42no—
148C22⋊D53—42no—
149C22⋊D53—42no—
150C22⋊D53—81no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—42no—
154C2×Dic53C2×D411yescharacteristic
155C2×Dic53C2×D411yescharacteristic
156D212—21no—
157D212—21no—
158D212—21no—
159D212—41no—
160D212—42no—
161D212—81no—
162D212C2×D412yes—
163C4×D53—21no—
164C4×D53—21no—
165C4×D53—21no—
166C4×D53—21no—
167C4×D53—42no—
168C4×D53—81no—
169Dic106—21no—
170Dic106—21no—
171C8×C53—21no—
172C8×C53—21no—
173#88322—21no—
174#88322C2×D412yes—
175C22×C53—22no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53C22×D411yescharacteristic
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—22no—
185C2×D53—21no—
186C2×D53—21no—
187C2×D53—21no—
188C2×D53—21no—
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—21no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53C22×D411yescharacteristic
201C4×C53H(5,𝔽2)11yescharacteristic
202Dic53—21no—
203Dic53—21no—
204Dic53—41no—
205Dic53—41no—
206#3273—531nomaximal, Sylow
207C2×C53—42no—
208C2×C53—81no—
209C2×C53D4211yescharacteristic
210C2×C53#57511yescharacteristic
211C2×C53#57611yescharacteristic
212D53—41no—
213D53—41no—
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#468—531no—
223#463—531no—
224#463D5311yescharacteristic, complemented
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×D8C2×D5311yescharacteristic
247C2×M16—531no—
248C2×M16—531no—
249#124—531no—
250#124C2×D5311yescharacteristic, complemented
251#123—1061no—
252#123—1061no—
253C22⋊Q8—531no—
254C22⋊Q8—531no—
255#121—531no—
256#121—531no—
257#121—1061no—
258#121—1061no—
259#120—1061no—
260#120—1061no—
261C4×D4—531no—
262C4×D4—531no—
263C4×D4C2×D5311yescharacteristic, complemented
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266#107—531no—
267C4⋊C8C2×D5311yescharacteristic, complemented
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—531no—
286C2×D4—531no—
287C2×D4—531no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—1062no—
293C2×D4—1062no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4C22×D5311yescharacteristic
299C2×D4C22×D5311yescharacteristic
300C22×C4—21no—
301C22×C4—531no—
302C22×C4—531no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4—1061no—
306Q16—2121no—
307SD16—22no—
308SD16—1062no—
309SD16—1062no—
310SD16—1062no—
311D8—22no—
312D8—534no—
313D8—1062no—
314D8—2121no—
315M16—1062no—
316M16—1062no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8C22×D5311yescharacteristic
320C2×C8C22×D5311yescharacteristic
321C4⋊C4—531no—
322C4⋊C4—531no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4C22×D5311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—531no—
330C22⋊C4—531no—
331C22⋊C4—1061no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C22⋊C4—1061no—
336C22⋊C4—1061no—
337C22⋊C4—1061no—
338C42C22×D5311yescharacteristic
339C23—21no—
340C23—21no—
341C23—531no—
342C23—531no—
343C23—1061no—
344C23—1061no—
345C23—4241no—
346C23—4241no—
347C23—2122no—
348C23—2122no—
349C23—2122no—
350C23—2122no—
351Q8—21no—
352Q8—21no—
353Q8—1061no—
354Q8—1061no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—532no—
359D4—1061no—
360D4—1061no—
361D4—1061no—
362D4—4241no—
363D4—4241no—
364D4—2121no—
365D4—2122no—
366D4—2122no—
367D4—2122no—
368D4—2122no—
369D4D4×D5312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—531no—
374C2×C4—531no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—4241no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4D4×D5311yescharacteristic
385C2×C4D4×D5311yescharacteristic
386C2×C4C23×D5311yescharacteristic
387C8—21no—
388C8—21no—
389C8—532no—
390C8—1061no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—1061no—
398C22—1061no—
399C22—1061no—
400C22—1061no—
401C22—1061no—
402C22—1061no—
403C22—1062no—
404C22—4244no—
405C22—4244no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22—2121no—
410C22—2121no—
411C22C2×D4×D5311yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—1061no—
417C4—1061no—
418C4—2121no—
419C4—2121no—
420C4C2×D4×D5311yescharacteristic
421C4#1313326311yescharacteristic
422C2—42no—
423C2—81no—
424C2—4241no—
425C2—2121no—
426C2—2121no—
427C2—2122no—
428C2#1681125711yescharacteristic
429C2#1681142011yescharacteristic
430C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3273
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. C2×#13133241C2semidirect
3. #16811268C2semidirect
4. #16811266C2semidirect
5. #16811005C2semidirect
6. #16810872C2semidirect
7. #16810868C2semidirect
8. #16810839C2semidirect
9. #16810788C2semidirect
10. #16810782C2semidirect
11. #16810758C2semidirect
12. #16810755C2semidirect
13. C4×#11719922C2semidirect
14. #16810700C2semidirect
15. C53×#463C2semidirect
18. C2×D4×D53C22semidirect
19. C2×D4×D53C22semidirect
22. C2×#11719925C22semidirect
23. C2×#11719913C22semidirect
24. C2×#11719923C22semidirect
25. C2×#11719924C22semidirect
29. C2×#11719912C22semidirect
30. C2×#11719912C22semidirect
31. #13133205C22semidirect
34. #13133197C22semidirect
35. #13133193C22semidirect
43. C4×D212C22semidirect
46. #13133150C22semidirect
47. #13133142C22semidirect
48. #13133139C22semidirect
49. #13133138C22semidirect
50. #13133137C22semidirect
51. #13133129C22semidirect
52. #13133129C22semidirect
53. C4×#88322C22semidirect
54. C2×SD16×C53C22semidirect
56. #124×C53C22semidirect
57. C4×D4×C53C22semidirect
58. C4⋊C8×C53C22semidirect
59. #103×C53C22semidirect
60. #102×C53C22semidirect
104. #11719922D4semidirect
224. #463D53semidirect
231. C53#3273semidirect
242. C2×SD16C2×D53semidirect
250. #124C2×D53semidirect
263. C4×D4C2×D53semidirect
267. C4⋊C8C2×D53semidirect
269. #103C2×D53semidirect
273. #102C2×D53semidirect