Finite Group Databasebeta

← #20286248

Subgroups — #20286248

Order 6784.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133244C211yesmaximal, characteristic, complemented
2C2×#13133237C211yesmaximal, characteristic, complemented
3#16811292C211yesmaximal, characteristic, complemented
4#16811270C211yesmaximal, characteristic, complemented
5#16811011C211yesmaximal, characteristic, complemented
6#16810868C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810843C211yesmaximal, characteristic, complemented
9#16810817C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810770C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810726C211yesmaximal, characteristic, complemented
14#16810691C211yesmaximal, characteristic, complemented
15C53×#449C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133244—24no—
20#13133237—24no—
21C2×#11719925C2211yescharacteristic, complemented
22C2×#11719913C2211yescharacteristic, complemented
23C2×#11719923C2211yescharacteristic, complemented
24C2×#11719924C2211yescharacteristic
25C2×D424C2211yescharacteristic
26C2×#11719922C2211yescharacteristic
27C2×#11719922C2211yescharacteristic
28C2×#11719912C2211yescharacteristic
29C22×#88322C2211yescharacteristic, complemented
30#13133213—21no—
31#13133213C2211yescharacteristic, complemented
32#13133204C2211yescharacteristic
33#13133199—21no—
34#13133197C2211yescharacteristic
35#13133191—21no—
36#13133191C2211yescharacteristic, complemented
37#13133188—21no—
38#13133188—21no—
39#13133188C2211yescharacteristic
40#13133185—21no—
41Q8×Dic53C2211yescharacteristic, complemented
42#13133179C2211yescharacteristic
43C4×C22⋊D53—21no—
44C4×C22⋊D53C2211yescharacteristic
45C22⋊C4×D53—21no—
46#13133150C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic
50#13133138C2211yescharacteristic
51#13133137C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic
53#13133127C2211yescharacteristic, complemented
54C2×#41×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic
56C2×D8×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic, complemented
60C22⋊C8×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62#41×C53—22no—
63#41×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53C2311yescharacteristic
68C2×D4×C53C2311yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53C2311yescharacteristic
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53D411yescharacteristic
78#11719949—22no—
79#11719949—41no—
80D4×D53—22no—
81D4×D53—41no—
82#11719945—22no—
83#11719945—41no—
84C2×D212—21no—
85C2×D212C2311yescharacteristic
86C2×D212C2311yescharacteristic
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53C2311yescharacteristic
91C2×Dic106C2311yescharacteristic
92SD16×C53—22no—
93D8×C53—41no—
94C2×C8×C53C2311yescharacteristic
95C2×C8×C53C2311yescharacteristic
96C4⋊C4×C53C2311yescharacteristic
97C22⋊C4×C53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101C22⋊Dic53D411yescharacteristic
102#11719925—22no—
103#11719924D414yescomplemented
104#11719923—22no—
105#11719922—22no—
106#11719922—22no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921D411yescharacteristic
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53C2311yescharacteristic
114#11719919D411yescharacteristic
115C4×Dic53—21no—
116C4×Dic53C2311yescharacteristic
117C2×#88322—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120D424—22no—
121#11719913—41no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53C2×D411yescharacteristic
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—42no—
129C22×D53—42no—
130C22×D53—81no—
131C22×D53C2×D411yescharacteristic
132Q8×C53—21no—
133Q8×C53C2×D412yes—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—21no—
138D4×C53—42no—
139D4×C53—81no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—42no—
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2411yescharacteristic
147C22⋊D53—41no—
148C22⋊D53—42no—
149C22⋊D53—42no—
150C22⋊D53—42no—
151C22⋊D53—81no—
152C22⋊D53—81no—
153C22⋊D53—81no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53C2×D411yescharacteristic
157C2×Dic53C2×D411yescharacteristic
158C2×Dic53C2×D411yescharacteristic
159D212—21no—
160D212—21no—
161D212—21no—
162D212—42no—
163D212C2×D412yes—
164C4×D53—21no—
165C4×D53—21no—
166C4×D53—42no—
167C4×D53—42no—
168C4×D53—81no—
169Dic106—21no—
170Dic106—21no—
171C8×C53—21no—
172C8×C53—21no—
173#88322—21no—
174#88322C2×D412yes—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53C22×D411yescharacteristic
185C2×D53—21no—
186C2×D53—22no—
187C2×D53—21no—
188C2×D53—21no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—84no—
195C4×C53—21no—
196C4×C53—22no—
197C4×C53—41no—
198C4×C53C22×D411yescharacteristic
199C4×C53H(5,𝔽2)11yescharacteristic
200Dic53—21no—
201Dic53—21no—
202Dic53—21no—
203Dic53—41no—
204Dic53—41no—
205#3268—531nomaximal, Sylow
206C2×C53—41no—
207C2×C53—81no—
208C2×C53—81no—
209C2×C53D4211yescharacteristic
210C2×C53#57511yescharacteristic
211C2×C53#57611yescharacteristic
212D53—41no—
213D53—42no—
214D53—81no—
215C2×AGL(1,ℤ/8)—531no—
216C2×#135—531no—
217#550—531no—
218#546—531no—
219#496—531no—
220#468—531no—
221#466—531no—
222#464—531no—
223#459—531no—
224#452—531no—
225#449—531no—
226#449D5311yescharacteristic, complemented
227#447—531no—
228#440—531no—
229#435—531no—
230C53#326811yescharacteristic, complemented, Sylow
231C2×#41—531no—
232C2×#41—531no—
233C2×#41C2×D5311yescharacteristic, complemented
234C22×D4—531no—
235AGL(1,ℤ/8)—1064no—
236#135—1064no—
237C2×Q16—531no—
238C2×SD16—531no—
239C2×SD16—531no—
240C2×SD16—531no—
241C2×SD16C2×D5311yescharacteristic
242C2×D8—531no—
243C2×D8—531no—
244C2×D8—531no—
245C2×D8C2×D5311yescharacteristic
246C2×M16—531no—
247C22×C8—531no—
248#127—531no—
249#127—1061no—
250#124—531no—
251#123—1061no—
252C22⋊Q8—531no—
253#121—531no—
254#121—531no—
255#121—1061no—
256#121—1061no—
257#121—1061no—
258#121C2×D5311yescharacteristic
259#120—1061no—
260C4×Q8—531no—
261C4×D4—531no—
262C4×D4—531no—
263C4×D4—1061no—
264C2×C22⋊C4—1061no—
265#107—531no—
266C4⋊C8—531no—
267#103—531no—
268#103C2×D5311yescharacteristic, complemented
269#102—531no—
270#102—531no—
271#102—531no—
272#102C2×D5311yescharacteristic, complemented
273C22⋊C8—531no—
274C22⋊C8C2×D5311yescharacteristic, complemented
275#97—531no—
276C24—1061no—
277#41—22no—
278#41—41no—
279#41—1062no—
280#41—1062no—
281#41—2121no—
282#41—2121no—
283C2×Q8—531no—
284C2×Q8C22×D5311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—531no—
288C2×D4—531no—
289C2×D4—531no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1062no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—2121no—
299C2×D4C22×D5311yescharacteristic
300C2×D4C22×D5311yescharacteristic
301C22×C4—21no—
302C22×C4—531no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4C22×D5311yescharacteristic
307Q16—1062no—
308SD16—22no—
309SD16—534no—
310SD16—1062no—
311SD16—2121no—
312D8—41no—
313D8—1062no—
314D8—1062no—
315D8—1062no—
316M16—1062no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—1062no—
320C2×C8C22×D5311yescharacteristic
321C2×C8C22×D5311yescharacteristic
322C4⋊C4—531no—
323C4⋊C4—531no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4C22×D5311yescharacteristic
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—
336C42—531no—
337C42—1061no—
338C23—21no—
339C23—21no—
340C23—531no—
341C23—1061no—
342C23—1061no—
343C23—4241no—
344C23—2122no—
345C23—2122no—
346C23D4×D5311yescharacteristic
347Q8—21no—
348Q8—1061no—
349Q8—1061no—
350Q8D4×D5312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—532no—
358D4—1061no—
359D4—1061no—
360D4—1061no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—2122no—
365D4—2121no—
366D4—2122no—
367D4—2122no—
368D4—2122no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—531no—
375C2×C4—531no—
376C2×C4—531no—
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×C4C23×D5311yescharacteristic
386C8—21no—
387C8—21no—
388C8—532no—
389C8—1061no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—1061no—
400C22—1061no—
401C22—1061no—
402C22—1062no—
403C22—4244no—
404C22—2121no—
405C22—2121no—
406C22—2121no—
407C22—2121no—
408C22—2121no—
409C22C2×D4×D5311yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—1061no—
414C4—1061no—
415C4—1061no—
416C4—2121no—
417C4—2121no—
418C4C2×D4×D5311yescharacteristic
419C4#1313326211yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—4241no—
424C2—2121no—
425C2—2122no—
426C2#1681125511yescharacteristic
427C2#1681142011yescharacteristic
428C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#449C2
socleC22×C53C22×D4
hypercenter#449D53
hypocenterC53#3268
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. C2×#13133237C2semidirect
3. #16811292C2semidirect
4. #16811270C2semidirect
5. #16811011C2semidirect
6. #16810868C2semidirect
7. #16810857C2semidirect
8. #16810843C2semidirect
9. #16810817C2semidirect
10. #16810788C2semidirect
11. #16810770C2semidirect
12. #16810755C2semidirect
13. #16810726C2semidirect
14. #16810691C2semidirect
15. C53×#449C2semidirect
21. C2×#11719925C22semidirect
22. C2×#11719913C22semidirect
23. C2×#11719923C22semidirect
29. C22×#88322C22semidirect
31. #13133213C22semidirect
36. #13133191C22semidirect
41. Q8×Dic53C22semidirect
46. #13133150C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
51. #13133137C22semidirect
53. #13133127C22semidirect
54. C2×#41×C53C22semidirect
58. #103×C53C22semidirect
59. #102×C53C22semidirect
60. C22⋊C8×C53C22semidirect
103. #11719924D4semidirect
226. #449D53semidirect
230. C53#3268semidirect
233. C2×#41C2×D53semidirect
268. #103C2×D53semidirect
272. #102C2×D53semidirect
274. C22⋊C8C2×D53semidirect