Finite Group Databasebeta

← #20286254

Subgroups — #20286254

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×#13133241C211yesmaximal, characteristic, complemented
2C2×#13133238C211yesmaximal, characteristic, complemented
3#16811293C211yesmaximal, characteristic, complemented
4#16811266C211yesmaximal, characteristic, complemented
5#16810866C211yesmaximal, characteristic, complemented
6#16810856C211yesmaximal, characteristic, complemented
7#16810842C211yesmaximal, characteristic, complemented
8#16810815C211yesmaximal, characteristic, complemented
9#16810782C211yesmaximal, characteristic, complemented
10#16810767C211yesmaximal, characteristic, complemented
11#16810765C211yesmaximal, characteristic, complemented
12#16810755C211yesmaximal, characteristic, complemented
13#16810725C211yesmaximal, characteristic, complemented
14#16810696C211yesmaximal, characteristic, complemented
15C53×#496C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×#11719947C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic
20#13133241—24no—
21#13133238—24no—
22C2×#11719925C2211yescharacteristic, complemented
23C2×#11719913C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719922C2211yescharacteristic
26C2×#11719922C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic, complemented
29#13133213—21no—
30#13133199—21no—
31#13133193C2211yescharacteristic
32#13133190—21no—
33#13133189—21no—
34#13133189—21no—
35#13133188C2211yescharacteristic
36#13133188C2211yescharacteristic
37#13133188C2211yescharacteristic, complemented
38#13133187—21no—
39C4×Dic106C2211yescharacteristic, complemented
40C4×D212C2211yescharacteristic
41C4×D212C2211yescharacteristic
42D4×Dic53—21no—
43C22⋊C4×D53—21no—
44#13133151C2211yescharacteristic, complemented
45C4⋊#88322C2211yescharacteristic, complemented
46#13133142C2211yescharacteristic, complemented
47#13133141C2211yescharacteristic, complemented
48#13133139C2211yescharacteristic
49#13133138C2211yescharacteristic
50#13133138C2211yescharacteristic, complemented
51#13133138C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic
53#13133129C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic, complemented
56C2×D8×C53C2211yescharacteristic
57C2×D8×C53C2211yescharacteristic
58#127×C53C2211yescharacteristic, complemented
59#124×C53C2211yescharacteristic
60#97×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C2×D4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76#11719949—22no—
77#11719949—41no—
78#11719947—22no—
79#11719947—22no—
80#11719947—41no—
81#11719947—41no—
82D4×D53—22no—
83D4×D53—41no—
84C2×D212—21no—
85C2×D212D411yescharacteristic
86C2×D212C2311yescharacteristic
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53C2311yescharacteristic
90C2×C4×D53C2311yescharacteristic
91C2×Dic106C2311yescharacteristic
92SD16×C53—22no—
93SD16×C53—41no—
94D8×C53—22no—
95D8×C53—41no—
96C2×C8×C53C2311yescharacteristic
97C2×C8×C53C2311yescharacteristic
98C22⋊C4×C53—21no—
99C22⋊C4×C53—21no—
100C42×C53C2311yescharacteristic
101C22⋊Dic53—21no—
102C22⋊Dic53—21no—
103C22⋊Dic53—21no—
104#11719925—22no—
105#11719923—22no—
106#11719922—22no—
107#11719922—22no—
108#11719921—21no—
109#11719921—21no—
110#11719921D411yescharacteristic
111#11719921D411yescharacteristic
112C4⋊Dic53D411yescharacteristic
113C4⋊Dic53C2311yescharacteristic
114C4⋊Dic53C2311yescharacteristic
115#11719919—21no—
116#11719919—21no—
117C4×Dic53—21no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120#11719913D414yescomplemented
121#11719912—22no—
122C8×D53—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C22×D53—21no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—81no—
130C22×D53C2×D411yescharacteristic
131C22×D53C2×D411yescharacteristic
132Q8×C53—21no—
133Q8×C53—21no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—21no—
140D4×C53—42no—
141D4×C53—81no—
142C2×C4×C53—21no—
143C2×C4×C53C2×D411yescharacteristic
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2411yescharacteristic
146C22⋊D53—42no—
147C22⋊D53—42no—
148C22⋊D53—42no—
149C22⋊D53—81no—
150C22⋊D53—81no—
151C22⋊D53—81no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53C2×D411yescharacteristic
158C2×Dic53C2×D411yescharacteristic
159D212—21no—
160D212—41no—
161D212—42no—
162D212C2×D412yes—
163C4×D53—21no—
164C4×D53—21no—
165C4×D53—21no—
166C4×D53—21no—
167C4×D53—42no—
168C4×D53—81no—
169Dic106—21no—
170Dic106C2×D412yes—
171C8×C53—21no—
172C8×C53C2×D412yes—
173#88322—21no—
174#88322—21no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
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—21no—
187C2×D53—22no—
188C2×D53—21no—
189C2×D53—21no—
190C2×D53—21no—
191C2×D53—21no—
192C2×D53—41no—
193C2×D53—41no—
194C2×D53—84no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53C22×D411yescharacteristic
199C4×C53H(5,𝔽2)11yescharacteristic
200Dic53—22no—
201Dic53—21no—
202Dic53—21no—
203Dic53—41no—
204Dic53—41no—
205#3268—531nomaximal, Sylow
206C2×C53—81no—
207C2×C53—81no—
208C2×C53—81no—
209C2×C53D4211yescharacteristic
210C2×C53#57511yescharacteristic
211C2×C53#57611yescharacteristic
212D53—41no—
213D53—41no—
214D53—42no—
215C2×AGL(1,ℤ/8)—531no—
216C2×#135—531no—
217#550—531no—
218#546—531no—
219#496D5311yescharacteristic, complemented
220#468—531no—
221#466—531no—
222#464—531no—
223#459—531no—
224#452—531no—
225#449—531no—
226#449—531no—
227#447—531no—
228#440—531no—
229#435—531no—
230C53#326811yescharacteristic, complemented, Sylow
231C2×#41—531no—
232C2×#41—531no—
233C2×#41—531no—
234C22×D4—531no—
235AGL(1,ℤ/8)—1064no—
236#135—1064no—
237C2×Q16—531no—
238C2×SD16—531no—
239C2×SD16—531no—
240C2×SD16C2×D5311yescharacteristic, complemented
241C2×SD16C2×D5311yescharacteristic, complemented
242C2×D8—531no—
243C2×D8—531no—
244C2×D8C2×D5311yescharacteristic
245C2×D8C2×D5311yescharacteristic
246C2×M16—531no—
247C22×C8—531no—
248#127—1061no—
249#127C2×D5311yescharacteristic, complemented
250#124C2×D5311yescharacteristic
251#123—1061no—
252C22⋊Q8—531no—
253#121—531no—
254#121—531no—
255#121—531no—
256#121—1061no—
257#121—1061no—
258#121—1061no—
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#103—531no—
269#102—531no—
270#102—531no—
271#102—531no—
272#102—531no—
273C22⋊C8—531no—
274C22⋊C8—531no—
275#97C2×D5311yescharacteristic, complemented
276C24—1061no—
277#41—1062no—
278#41—1062no—
279#41—1062no—
280#41—2121no—
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—1061no—
290C2×D4—1061no—
291C2×D4—1062no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—2121no—
298C2×D4C22×D5311yescharacteristic
299C2×D4C22×D5311yescharacteristic
300C2×D4C22×D5311yescharacteristic
301C22×C4—531no—
302C22×C4—531no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—1061no—
307Q16—1062no—
308SD16—22no—
309SD16—41no—
310SD16—534no—
311SD16—1062no—
312D8—22no—
313D8—41no—
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—531no—
325C4⋊C4—1061no—
326C4⋊C4—1061no—
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—
336C42—1061no—
337C42C22×D5311yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—531no—
342C23—531no—
343C23—1061no—
344C23—4241no—
345C23—2122no—
346C23—2122no—
347Q8—21no—
348Q8—21no—
349Q8—532no—
350Q8—1061no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—532no—
360D4—1061no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—2121no—
365D4—2122no—
366D4—2122no—
367D4—2122no—
368D4—2122no—
369C2×C4—21no—
370C2×C4—531no—
371C2×C4—531no—
372C2×C4—1061no—
373C2×C4—1061no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—1061no—
379C2×C4—4241no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4D4×D5311yescharacteristic
384C2×C4D4×D5311yescharacteristic
385C2×C4C23×D5311yescharacteristic
386C8—21no—
387C8—1061no—
388C8—1061no—
389C8D4×D5312yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—1061no—
400C22—1061no—
401C22—1061no—
402C22—1061no—
403C22—1061no—
404C22—1061no—
405C22—1062no—
406C22—4244no—
407C22—2121no—
408C22—2121no—
409C22C2×D4×D5311yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—1061no—
414C4—1061no—
415C4—1062no—
416C4—2121no—
417C4—2121no—
418C4C2×D4×D5311yescharacteristic
419C4#1313326211yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—2121no—
424C2—2121no—
425C2—2122no—
426C2#1681125811yescharacteristic
427C2#1681141911yescharacteristic
428C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#496C2
socleC22×C53C22×D4
hypercenter#496D53
hypocenterC53#3268
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#13133241C2semidirect
2. C2×#13133238C2semidirect
3. #16811293C2semidirect
4. #16811266C2semidirect
5. #16810866C2semidirect
6. #16810856C2semidirect
7. #16810842C2semidirect
8. #16810815C2semidirect
9. #16810782C2semidirect
10. #16810767C2semidirect
11. #16810765C2semidirect
12. #16810755C2semidirect
13. #16810725C2semidirect
14. #16810696C2semidirect
15. C53×#496C2semidirect
18. C2×#11719947C22semidirect
22. C2×#11719925C22semidirect
28. C2×C8×D53C22semidirect
37. #13133188C22semidirect
39. C4×Dic106C22semidirect
44. #13133151C22semidirect
45. C4⋊#88322C22semidirect
46. #13133142C22semidirect
47. #13133141C22semidirect
50. #13133138C22semidirect
51. #13133138C22semidirect
53. #13133129C22semidirect
54. C2×SD16×C53C22semidirect
55. C2×SD16×C53C22semidirect
58. #127×C53C22semidirect
60. #97×C53C22semidirect
120. #11719913D4semidirect
219. #496D53semidirect
230. C53#3268semidirect
240. C2×SD16C2×D53semidirect
241. C2×SD16C2×D53semidirect
249. #127C2×D53semidirect
275. #97C2×D53semidirect