Finite Group Databasebeta

← #20286398

Subgroups — #20286398

Order 6784.

526 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×#13133237C211yesmaximal, characteristic, complemented
2C2×D8×D53C211yesmaximal, characteristic, complemented
3#16811270C211yesmaximal, characteristic, complemented
4#16811261C211yesmaximal, characteristic, complemented
5#16811004C211yesmaximal, characteristic, complemented
6#16810863C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810841C211yesmaximal, characteristic, complemented
9#16810836C211yesmaximal, characteristic, complemented
10#16810775C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810770C211yesmaximal, characteristic, complemented
13#16810703C211yesmaximal, characteristic, complemented
14#16810686C211yesmaximal, characteristic, complemented
15C53×#449C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C2×D4×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133237—24no—
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719913C2211yescharacteristic, complemented
22C2×#11719923C2211yescharacteristic, complemented
23C2×#11719924C2211yescharacteristic, complemented
24C2×D424C2211yescharacteristic
25C2×#11719922C2211yescharacteristic
26C2×#11719922C2211yescharacteristic
27D8×D53—24no—
28C22×#88322C2211yescharacteristic, complemented
29C2×C8×D53C2211yescharacteristic
30#13133206C2211yescharacteristic, complemented
31#13133204C2211yescharacteristic, complemented
32#13133200—21no—
33#13133199—21no—
34#13133197C2211yescharacteristic
35#13133193C2211yescharacteristic
36#13133191C2211yescharacteristic, complemented
37#13133189—21no—
38#13133188—21no—
39#13133185—21no—
40#13133185—21no—
41D4×Dic53C2211yescharacteristic, complemented
42#13133179C2211yescharacteristic, complemented
43C4×C22⋊D53C2211yescharacteristic
44C2×#11719921—21no—
45C22⋊C4×D53—21no—
46#13133150C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133139C2211yescharacteristic, complemented
50#13133138C2211yescharacteristic, complemented
51#13133137C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic
53C8×Dic53C2211yescharacteristic, complemented
54C2×#41×C53C2211yescharacteristic, complemented
55C2×SD16×C53C2211yescharacteristic, complemented
56C2×D8×C53C2211yescharacteristic
57#121×C53C2211yescharacteristic, complemented
58#103×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic, complemented
60C22⋊C8×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C23×D53—21no—
63#41×C53—22no—
64#41×C53—41no—
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53C2311yescharacteristic
69C2×D4×C53C2311yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53C2311yescharacteristic
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53D411yescharacteristic
77C22×Dic53—21no—
78D4×D53—22no—
79D4×D53—22no—
80D4×D53—41no—
81D4×D53—41no—
82#11719945—22no—
83#11719945—41no—
84C2×D212C2311yescharacteristic
85C2×D212C2311yescharacteristic
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53C2311yescharacteristic
89C2×Dic106C2311yescharacteristic
90SD16×C53—41no—
91D8×C53—22no—
92C2×C8×C53C2311yescharacteristic
93C2×C8×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53D411yescharacteristic
100#11719925—22no—
101#11719924—22no—
102#11719923—22no—
103#11719922—22no—
104#11719922D414yescomplemented
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921—21no—
111#11719921D411yescharacteristic
112C4⋊Dic53—21no—
113C4⋊Dic53C2311yescharacteristic
114#11719919—21no—
115#11719919D411yescharacteristic
116C4×Dic53C2311yescharacteristic
117C2×#88322—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120D424—22no—
121#11719913—41no—
122C8×D53—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53C2×D411yescharacteristic
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—21no—
140D4×C53—42no—
141D4×C53—81no—
142D4×C53C2×D412yes—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—42no—
148C2×C4×C53C2×D411yescharacteristic
149C2×C4×C53C2411yescharacteristic
150C22⋊D53—22no—
151C22⋊D53—42no—
152C22⋊D53—42no—
153C22⋊D53—42no—
154C22⋊D53—81no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—81no—
158C2×Dic53C2×D411yescharacteristic
159C2×Dic53C2×D411yescharacteristic
160C2×Dic53C2×D411yescharacteristic
161D212—21no—
162D212—21no—
163D212—21no—
164D212C2×D412yes—
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—21no—
176C22×C53—22no—
177C22×C53—21no—
178C22×C53—21no—
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—41no—
189C2×D53—41no—
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—84no—
194C2×D53—84no—
195C2×D53#13512yescomplemented
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53C22×D411yescharacteristic
200C4×C53H(5,𝔽2)11yescharacteristic
201Dic53—21no—
202Dic53—21no—
203Dic53—21no—
204Dic53—41no—
205Dic53—41no—
206#3280—531nomaximal, Sylow
207C2×C53—42no—
208C2×C53—41no—
209C2×C53—81no—
210C2×C53D4211yescharacteristic
211C2×C53C2×#13511yescharacteristic
212C2×C53#57611yescharacteristic
213D53—22no—
214D53—42no—
215D53—81no—
216C2×#135—531no—
217C22×D8—531no—
218#546—531no—
219#546—531no—
220#495—531no—
221#467—531no—
222#466—531no—
223#463—531no—
224#463—531no—
225#450—531no—
226#450—531no—
227#449—531no—
228#449D5311yescharacteristic, complemented
229C4×D8—531no—
230C8×D4—531no—
231C53#328011yescharacteristic, complemented, Sylow
232C2×#41—531no—
233C2×#41C2×D5311yescharacteristic, complemented
234C22×D4—531no—
235C22×D4—531no—
236#135—1064no—
237C2×Q16—531no—
238C2×SD16—531no—
239C2×SD16—531no—
240C2×SD16—531no—
241C2×SD16C2×D5311yescharacteristic, complemented
242C2×D8—531no—
243C2×D8—531no—
244C2×D8—531no—
245C2×D8—1064no—
246C2×D8C2×D5311yescharacteristic
247C22×C8—531no—
248C22×C8—531no—
249#124—531no—
250#124—531no—
251#123—1061no—
252#123—1061no—
253C22⋊Q8—531no—
254C22⋊Q8—531no—
255#121—531no—
256#121—1061no—
257#121—1061no—
258#121C2×D5311yescharacteristic, complemented
259#120—1061no—
260#120—1061no—
261C4×D4—531no—
262C4×D4—531no—
263C4×D4—531no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266#107—531no—
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⋊C8C2×D5311yescharacteristic, complemented
276C4×C8—531no—
277C24—1061no—
278C24—1061no—
279#41—22no—
280#41—41no—
281#41—1062no—
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—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—1061no—
303C22×C4—1061no—
304C22×C4—1061no—
305C22×C4C22×D5311yescharacteristic
306Q16—1062no—
307SD16—41no—
308SD16—1062no—
309SD16—1062no—
310SD16—2121no—
311D8—22no—
312D8—534no—
313D8—1062no—
314D8—1062no—
315C2×C8—531no—
316C2×C8—531no—
317C2×C8—1062no—
318C2×C8—1062no—
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—531no—
328C22⋊C4—531no—
329C22⋊C4—1061no—
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—21no—
341C23—531no—
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—21no—
358D4—42no—
359D4—81no—
360D4—532no—
361D4—1062no—
362D4—1061no—
363D4—1061no—
364D4—1061no—
365D4—4241no—
366D4—2122no—
367D4—2122no—
368D4—2122no—
369D4D4×D5312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—531no—
376C2×C4—531no—
377C2×C4—531no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—2122no—
385C2×C4D4×D5311yescharacteristic
386C2×C4C23×D5311yescharacteristic
387C8—21no—
388C8—21no—
389C8—532no—
390C8—1061no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—532no—
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—41no—
414C4—41no—
415C4—1061no—
416C4—1061no—
417C4—1061no—
418C4—2121no—
419C4—2121no—
420C4C2×D4×D5311yescharacteristic
421C4#1313326211yescharacteristic
422C2—41no—
423C2—42no—
424C2—81no—
425C2—1062no—
426C2—4241no—
427C2—2122no—
428C2#1681125511yescharacteristic
429C2#135×D5311yescharacteristic
430C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#449C2
socleC22×C53C22×D4
hypercenter#449D53
hypocenterC53#3280
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×#13133237C2semidirect
2. C2×D8×D53C2semidirect
3. #16811270C2semidirect
4. #16811261C2semidirect
5. #16811004C2semidirect
6. #16810863C2semidirect
7. #16810857C2semidirect
8. #16810841C2semidirect
9. #16810836C2semidirect
10. #16810775C2semidirect
11. #16810772C2semidirect
12. #16810770C2semidirect
13. #16810703C2semidirect
14. #16810686C2semidirect
15. C53×#449C2semidirect
16. C2×#11719945C22semidirect
20. C2×#11719925C22semidirect
21. C2×#11719913C22semidirect
22. C2×#11719923C22semidirect
23. C2×#11719924C22semidirect
28. C22×#88322C22semidirect
30. #13133206C22semidirect
31. #13133204C22semidirect
36. #13133191C22semidirect
41. D4×Dic53C22semidirect
42. #13133179C22semidirect
46. #13133150C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
49. #13133139C22semidirect
50. #13133138C22semidirect
51. #13133137C22semidirect
53. C8×Dic53C22semidirect
54. C2×#41×C53C22semidirect
55. C2×SD16×C53C22semidirect
57. #121×C53C22semidirect
58. #103×C53C22semidirect
59. #102×C53C22semidirect
60. C22⋊C8×C53C22semidirect
104. #11719922D4semidirect
195. C2×D53#135semidirect
228. #449D53semidirect
231. C53#3280semidirect
233. C2×#41C2×D53semidirect
241. C2×SD16C2×D53semidirect
258. #121C2×D53semidirect
269. #103C2×D53semidirect
273. #102C2×D53semidirect
275. C22⋊C8C2×D53semidirect