Finite Group Databasebeta

← #20286229

Subgroups — #20286229

Order 6784.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic, complemented
2C22×#11719923C211yesmaximal, characteristic, complemented
3#16811286C211yesmaximal, characteristic, complemented
4#16811267C211yesmaximal, characteristic
5#16811014C211yesmaximal, characteristic, complemented
6#16810868C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810844C211yesmaximal, characteristic, complemented
9#16810826C211yesmaximal, characteristic, complemented
10#16810786C211yesmaximal, characteristic, complemented
11#16810781C211yesmaximal, characteristic, complemented
12#16810760C211yesmaximal, characteristic, complemented
13#16810720C211yesmaximal, characteristic, complemented
14#16810691C211yesmaximal, characteristic, complemented
15C53×#450C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic, complemented
19#13133246—24no—
20C2×Dic212C2211yescharacteristic, complemented
21C2×Dic212C2211yescharacteristic, complemented
22C2×#11719925C2211yescharacteristic, complemented
23C2×#11719923—24no—
24C2×#11719923C2211yescharacteristic
25C2×#11719923C2211yescharacteristic
26C2×#11719923C2211yescharacteristic
27C2×#11719923C2211yescharacteristic
28C2×#11719912C2211yescharacteristic, complemented
29C22×#88322C2211yescharacteristic
30#13133216—21no—
31#13133215C2211yescharacteristic, complemented
32#13133204C2211yescharacteristic
33#13133198—21no—
34#13133197C2211yescharacteristic, complemented
35#13133196C2211yescharacteristic
36#13133195—21no—
37#13133194—21no—
38#13133189—21no—
39#13133188C2211yescharacteristic
40#13133186—21no—
41#13133182C2211yescharacteristic, complemented
42D4×Dic53C2211yescharacteristic
43C4×C22⋊D53C2211yescharacteristic
44#13133177—21no—
45C2×C22⋊Dic53—21no—
46#13133150C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133142C2211yescharacteristic, complemented
50#13133141C2211yescharacteristic, complemented
51#13133140C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic, complemented
53#13133127C2211yescharacteristic, complemented
54C22×D4×C53C2211yescharacteristic
55C2×SD16×C53C2211yescharacteristic
56C2×SD16×C53C2211yescharacteristic
57C22⋊Q8×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59#102×C53C2211yescharacteristic
60C22⋊C8×C53C2211yescharacteristic
61C24×C53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—22no—
64C2×D4×C53—21no—
65C2×D4×C53—41no—
66C2×D4×C53C2311yescharacteristic
67C2×D4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75Q8×D53—22no—
76Q8×D53—41no—
77#11719947—22no—
78#11719947—41no—
79C2×C4×D53—21no—
80C2×C4×D53C2311yescharacteristic
81C2×Dic106—22no—
82C2×Dic106—21no—
83C2×Dic106—21no—
84C2×Dic106—41no—
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
88SD16×C53—22no—
89SD16×C53—41no—
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53—21no—
93C4⋊C4×C53C2311yescharacteristic
94C22⋊C4×C53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53D411yescharacteristic
101#11719925—22no—
102#11719923—22no—
103#11719923—22no—
104#11719923—22no—
105#11719923D414yescomplemented
106#11719921—21no—
107#11719921D411yescharacteristic
108C4⋊Dic53—21no—
109C4⋊Dic53C2311yescharacteristic
110#11719919—21no—
111#11719919—21no—
112#11719919—21no—
113#11719919—21no—
114#11719919D411yescharacteristic
115C4×Dic53—21no—
116C4×Dic53C2311yescharacteristic
117C2×#88322—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120Dic212—22no—
121Dic212—41no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—42no—
126C23×C53—42no—
127C23×C53—81no—
128C23×C53C2×D411yescharacteristic
129C22×D53C2×D411yescharacteristic
130Q8×C53—21no—
131Q8×C53—21no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—42no—
136D4×C53C2×D412yes—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53C2×D411yescharacteristic
142C2×C4×C53C2411yescharacteristic
143C22⋊D53—22no—
144C22⋊D53—42no—
145C22⋊D53—81no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—81no—
153C2×Dic53C2×D411yescharacteristic
154C2×Dic53C2×D411yescharacteristic
155C2×Dic53C2×D411yescharacteristic
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—42no—
159Dic106—21no—
160Dic106—21no—
161Dic106—21no—
162Dic106—21no—
163Dic106—21no—
164Dic106—42no—
165Dic106—42no—
166Dic106C2×D412yes—
167C8×C53—21no—
168C8×C53—21no—
169#88322—21no—
170#88322C2×D412yes—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—22no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—84no—
180C22×C53#13712yescomplemented
181C22×C53C22×D411yescharacteristic
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53C22×D411yescharacteristic
189C4×C53H(5,𝔽2)11yescharacteristic
190Dic53—22no—
191Dic53—21no—
192Dic53—21no—
193Dic53—21no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#3266—531nomaximal, Sylow
198C2×C53—22no—
199C2×C53—42no—
200C2×C53—81no—
201C2×C53D4211yescharacteristic
202C2×C53C2×#13711yescharacteristic
203C2×C53#57711yescharacteristic
204D53—41no—
205C2×#137—531no—
206C22×SD16—531no—
207D4×Q8—531no—
208#546—531no—
209#497—531no—
210#468—531no—
211#467—531no—
212#464—531no—
213#461—531no—
214#452—531no—
215#451—531no—
216#450D5311yescharacteristic, complemented
217#448—531no—
218#440—531no—
219#435—531no—
220C53#326611yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C22×Q8—531no—
223C22×Q8—531no—
224C22×D4C2×D5311yescharacteristic
225#137—1064no—
226C2×Q16—531no—
227C2×Q16—531no—
228C2×Q16—531no—
229C2×SD16—531no—
230C2×SD16—531no—
231C2×SD16—531no—
232C2×SD16—531no—
233C2×SD16—1064no—
234C2×SD16C2×D5311yescharacteristic
235C2×SD16C2×D5311yescharacteristic
236C2×M16—531no—
237C22×C8—531no—
238C4⋊Q8—531no—
239C4⋊Q8—1061no—
240#124—531no—
241#123—1061no—
242C22⋊Q8—531no—
243C22⋊Q8—531no—
244C22⋊Q8—1061no—
245C22⋊Q8—1061no—
246C22⋊Q8C2×D5311yescharacteristic
247#121—531no—
248#121—1061no—
249#120—1061no—
250C4×Q8—531no—
251C4×D4—531no—
252C4×D4—531no—
253C4×D4—1061no—
254C2×C22⋊C4—1061no—
255#107—531no—
256C4⋊C8—531no—
257#103—531no—
258#103—531no—
259#103—531no—
260#103—531no—
261#102C2×D5311yescharacteristic
262#102C2×D5311yescharacteristic
263C22⋊C8—531no—
264C22⋊C8C2×D5311yescharacteristic
265#97—531no—
266C24—21no—
267#41—1062no—
268#41—2121no—
269C2×Q8—531no—
270C2×Q8—531no—
271C2×Q8—531no—
272C2×Q8—1061no—
273C2×Q8—1062no—
274C2×Q8—1061no—
275C2×Q8—1062no—
276C2×Q8—2121no—
277C2×Q8—2121no—
278C2×Q8C22×D5311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—531no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4C22×D5311yescharacteristic
286C2×D4C22×D5311yescharacteristic
287C22×C4—531no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4C22×D5311yescharacteristic
293Q16—1062no—
294Q16—1062no—
295Q16—2121no—
296SD16—22no—
297SD16—41no—
298SD16—534no—
299SD16—1062no—
300SD16—1062no—
301SD16—1062no—
302M16—1062no—
303C2×C8—531no—
304C2×C8—531no—
305C2×C8—1062no—
306C2×C8C22×D5311yescharacteristic
307C2×C8C22×D5311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—531no—
310C4⋊C4—531no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C4⋊C4C22×D5311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—531no—
319C22⋊C4—531no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C42—531no—
327C42—1061no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—531no—
334C23D4×D5311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—532no—
338Q8—1061no—
339Q8—1061no—
340Q8—1061no—
341Q8—1061no—
342Q8—1061no—
343Q8—2122no—
344Q8—2122no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—1062no—
350D4—4241no—
351D4—2122no—
352D4D4×D5312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—531no—
358C2×C4—531no—
359C2×C4—531no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—4241no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4—2122no—
370C2×C4D4×D5311yescharacteristic
371C2×C4C23×D5311yescharacteristic
372C8—21no—
373C8—21no—
374C8—532no—
375C8—1061no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—1061no—
386C22—1061no—
387C22—1061no—
388C22#1313324612yescomplemented
389C22C2×D4×D5311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—1061no—
394C4—1061no—
395C4—1061no—
396C4—1062no—
397C4—2121no—
398C4—2121no—
399C4—2121no—
400C4C2×D4×D5311yescharacteristic
401C4#1313326211yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—2121no—
406C2#1681125511yescharacteristic
407C2C2×#1313324611yescharacteristic
408C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3266
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. C22×#11719923C2semidirect
3. #16811286C2semidirect
5. #16811014C2semidirect
6. #16810868C2semidirect
7. #16810862C2semidirect
8. #16810844C2semidirect
9. #16810826C2semidirect
10. #16810786C2semidirect
11. #16810781C2semidirect
12. #16810760C2semidirect
13. #16810720C2semidirect
14. #16810691C2semidirect
15. C53×#450C2semidirect
18. C2×Q8×D53C22semidirect
20. C2×Dic212C22semidirect
21. C2×Dic212C22semidirect
22. C2×#11719925C22semidirect
28. C2×#11719912C22semidirect
31. #13133215C22semidirect
34. #13133197C22semidirect
41. #13133182C22semidirect
46. #13133150C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
49. #13133142C22semidirect
50. #13133141C22semidirect
51. #13133140C22semidirect
52. #13133129C22semidirect
53. #13133127C22semidirect
105. #11719923D4semidirect
180. C22×C53#137semidirect
216. #450D53semidirect
220. C53#3266semidirect
388. C22#13133246semidirect