Finite Group Databasebeta

← #20286420

Subgroups — #20286420

Order 6784.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133237C211yesmaximal, characteristic, complemented
2C2×SD16×D53C211yesmaximal, characteristic, complemented
3#16811283C211yesmaximal, characteristic, complemented
4#16811261C211yesmaximal, characteristic, complemented
5#16811008C211yesmaximal, characteristic, complemented
6#16810855C211yesmaximal, characteristic, complemented
7#16810854C211yesmaximal, characteristic, complemented
8#16810848C211yesmaximal, characteristic, complemented
9#16810840C211yesmaximal, characteristic, complemented
10#16810777C211yesmaximal, characteristic, complemented
11#16810770C211yesmaximal, characteristic, complemented
12#16810758C211yesmaximal, characteristic, complemented
13#16810714C211yesmaximal, characteristic, complemented
14#16810686C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C2×Q8×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133237—24no—
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719913C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719924C2211yescharacteristic
24C2×#11719924C2211yescharacteristic, complemented
25SD16×D53—24no—
26C2×D424C2211yescharacteristic
27C2×#11719922C2211yescharacteristic
28C22×#88322C2211yescharacteristic, complemented
29C2×C8×D53C2211yescharacteristic
30#13133206C2211yescharacteristic
31#13133206C2211yescharacteristic, complemented
32#13133203—21no—
33#13133200—21no—
34#13133195—21no—
35#13133193C2211yescharacteristic
36#13133192C2211yescharacteristic
37#13133191C2211yescharacteristic
38#13133190—21no—
39#13133189—21no—
40#13133185—21no—
41#13133182C2211yescharacteristic, complemented
42D4×Dic53C2211yescharacteristic
43#13133179—21no—
44C4×C22⋊D53C2211yescharacteristic
45C2×#11719921—21no—
46#13133148C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133141C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic, complemented
51#13133137C2211yescharacteristic
52#13133129C2211yescharacteristic
53C8×Dic53C2211yescharacteristic, complemented
54C2×#41×C53C2211yescharacteristic
55C2×Q16×C53C2211yescharacteristic, complemented
56C2×SD16×C53C2211yescharacteristic
57C22⋊Q8×C53C2211yescharacteristic, complemented
58#103×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic
60C22⋊C8×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62#41×C53—22no—
63#41×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×D4×C53—21no—
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53—21no—
69C22×C4×C53C2311yescharacteristic
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53D411yescharacteristic
74C22×Dic53—21no—
75Q8×D53—22no—
76Q8×D53—41no—
77D4×D53—22no—
78D4×D53—41no—
79#11719945—22no—
80#11719945—41no—
81C2×D212—21no—
82C2×D212C2311yescharacteristic
83C2×D212C2311yescharacteristic
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53C2311yescharacteristic
87C2×Dic106—21no—
88C2×Dic106C2311yescharacteristic
89Q16×C53—41no—
90SD16×C53—22no—
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C4⋊C4×C53—21no—
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53D411yescharacteristic
98#11719925—22no—
99#11719924—22no—
100#11719924—22no—
101#11719923D414yescomplemented
102#11719922—22no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921—21no—
108#11719921—21no—
109#11719921—21no—
110#11719921D411yescharacteristic
111C4⋊Dic53C2311yescharacteristic
112#11719919—21no—
113#11719919—21no—
114#11719919D411yescharacteristic
115C4×Dic53—21no—
116C4×Dic53C2311yescharacteristic
117C2×#88322—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120D424—41no—
121#11719913—22no—
122C8×D53—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—42no—
128C22×D53—42no—
129C22×D53—81no—
130C22×D53C2×D411yescharacteristic
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53—21no—
134Q8×C53—21no—
135D4×C53—21no—
136D4×C53—42no—
137D4×C53C2×D412yes—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—42no—
144C2×C4×C53C2×D411yescharacteristic
145C2×C4×C53C2411yescharacteristic
146C22⋊D53—22no—
147C22⋊D53—42no—
148C22⋊D53—42no—
149C22⋊D53—81no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—81no—
153C2×Dic53C2×D411yescharacteristic
154C2×Dic53C2×D411yescharacteristic
155C2×Dic53C2×D411yescharacteristic
156D212—21no—
157D212—21no—
158D212—21no—
159D212—21no—
160D212—42no—
161C4×D53—21no—
162C4×D53—21no—
163C4×D53—42no—
164C4×D53—42no—
165Dic106—21no—
166Dic106—42no—
167Dic106C2×D412yes—
168C8×C53—21no—
169C8×C53—21no—
170#88322—21no—
171#88322C2×D412yes—
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53C22×D411yescharacteristic
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C2×D53—41no—
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—84no—
188C2×D53#13512yescomplemented
189C4×C53—21no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53C22×D411yescharacteristic
194C4×C53H(5,𝔽2)11yescharacteristic
195Dic53—22no—
196Dic53—21no—
197Dic53—21no—
198Dic53—21no—
199Dic53—41no—
200#3282—531nomaximal, Sylow
201C2×C53—42no—
202C2×C53—41no—
203C2×C53D4211yescharacteristic
204C2×C53C2×#13511yescharacteristic
205C2×C53#57711yescharacteristic
206D53—22no—
207D53—81no—
208D53—81no—
209C2×#135—531no—
210C22×SD16—531no—
211#548—531no—
212#546—531no—
213#495—531no—
214#465—531no—
215#465—531no—
216#464—531no—
217#463—531no—
218#452D5311yescharacteristic, complemented
219#451—531no—
220#449—531no—
221#447—531no—
222C4×SD16—531no—
223C8×D4—531no—
224C53#328211yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C2×#41C2×D5311yescharacteristic
227C22×Q8—531no—
228C22×D4—531no—
229#135—1064no—
230C2×Q16—531no—
231C2×Q16C2×D5311yescharacteristic, complemented
232C2×SD16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16—1064no—
237C2×SD16C2×D5311yescharacteristic
238C2×D8—531no—
239C2×D8—531no—
240C22×C8—531no—
241C22×C8—531no—
242#124—531no—
243#124—531no—
244#124—1061no—
245#123—1061no—
246C22⋊Q8—531no—
247C22⋊Q8—1061no—
248C22⋊Q8C2×D5311yescharacteristic, complemented
249#121—531no—
250#121—531no—
251#121—1061no—
252#121—1061no—
253#120—1061no—
254C4×Q8—531no—
255C4×D4—531no—
256C4×D4—531no—
257C4×D4—1061no—
258C2×C22⋊C4—1061no—
259#106—531no—
260C4⋊C8—531no—
261#103—531no—
262#103—531no—
263#103C2×D5311yescharacteristic, complemented
264#102—531no—
265#102—531no—
266#102C2×D5311yescharacteristic
267C22⋊C8—531no—
268C22⋊C8C2×D5311yescharacteristic, complemented
269C4×C8—531no—
270C24—1061no—
271#41—22no—
272#41—41no—
273#41—1062no—
274#41—2121no—
275C2×Q8—531no—
276C2×Q8—1061no—
277C2×Q8—1062no—
278C2×Q8—2121no—
279C2×Q8C22×D5311yescharacteristic
280C2×Q8C22×D5311yescharacteristic
281C2×D4—21no—
282C2×D4—531no—
283C2×D4—531no—
284C2×D4—531no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1062no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—2121no—
291C2×D4C22×D5311yescharacteristic
292C22×C4—21no—
293C22×C4—531no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4C22×D5311yescharacteristic
298Q16—41no—
299Q16—1062no—
300SD16—22no—
301SD16—534no—
302SD16—1062no—
303SD16—1062no—
304SD16—1062no—
305D8—1062no—
306D8—2121no—
307C2×C8—531no—
308C2×C8—531no—
309C2×C8—1062no—
310C2×C8—1062no—
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—531no—
315C4⋊C4—531no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4C22×D5311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—531no—
321C22⋊C4—531no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C22⋊C4—1061no—
330C42—531no—
331C42—1061no—
332C23—21no—
333C23—531no—
334C23—1061no—
335C23—1061no—
336C23—4241no—
337C23—2122no—
338C23—2122no—
339C23D4×D5311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—532no—
345Q8—1061no—
346Q8—2122no—
347D4—21no—
348D4—42no—
349D4—1062no—
350D4—1061no—
351D4—1061no—
352D4—1061no—
353D4—1061no—
354D4—4241no—
355D4—2122no—
356D4—2122no—
357D4—2122no—
358D4D4×D5312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—531no—
366C2×C4—531no—
367C2×C4—531no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—4241no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4D4×D5311yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—21no—
378C8—21no—
379C8—532no—
380C8—1061no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—532no—
388C22—1061no—
389C22—1061no—
390C22—4244no—
391C22—2121no—
392C22—2121no—
393C22—2121no—
394C22—2121no—
395C22—2121no—
396C22—2121no—
397C22C2×D4×D5311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—1061no—
403C4—1061no—
404C4—1061no—
405C4—1062no—
406C4—2121no—
407C4C2×D4×D5311yescharacteristic
408C4#1313326211yescharacteristic
409C2—41no—
410C2—42no—
411C2—1062no—
412C2—4241no—
413C2—4241no—
414C2#1681125511yescharacteristic
415C2#135×D5311yescharacteristic
416C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3282
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#13133237C2semidirect
2. C2×SD16×D53C2semidirect
3. #16811283C2semidirect
4. #16811261C2semidirect
5. #16811008C2semidirect
6. #16810855C2semidirect
7. #16810854C2semidirect
8. #16810848C2semidirect
9. #16810840C2semidirect
10. #16810777C2semidirect
11. #16810770C2semidirect
12. #16810758C2semidirect
13. #16810714C2semidirect
14. #16810686C2semidirect
15. C53×#452C2semidirect
16. C2×#11719945C22semidirect
20. C2×#11719925C22semidirect
24. C2×#11719924C22semidirect
28. C22×#88322C22semidirect
31. #13133206C22semidirect
41. #13133182C22semidirect
46. #13133148C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
49. #13133141C22semidirect
50. #13133139C22semidirect
53. C8×Dic53C22semidirect
55. C2×Q16×C53C22semidirect
57. C22⋊Q8×C53C22semidirect
58. #103×C53C22semidirect
60. C22⋊C8×C53C22semidirect
101. #11719923D4semidirect
188. C2×D53#135semidirect
218. #452D53semidirect
224. C53#3282semidirect
231. C2×Q16C2×D53semidirect
248. C22⋊Q8C2×D53semidirect
263. #103C2×D53semidirect
268. C22⋊C8C2×D53semidirect