Finite Group Databasebeta

← #20286227

Subgroups — #20286227

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
2C2×SD16×D53C211yesmaximal, characteristic, complemented
3Q8×D212C211yesmaximal, characteristic, complemented
4#16811268C211yesmaximal, characteristic
5#16811017C211yesmaximal, characteristic, complemented
6#16810870C211yesmaximal, characteristic, complemented
7#16810865C211yesmaximal, characteristic, complemented
8#16810845C211yesmaximal, characteristic, complemented
9#16810828C211yesmaximal, characteristic, complemented
10#16810786C211yesmaximal, characteristic, complemented
11#16810780C211yesmaximal, characteristic, complemented
12#16810775C211yesmaximal, characteristic, complemented
13#16810763C211yesmaximal, characteristic, complemented
14#16810696C211yesmaximal, characteristic, complemented
15C53×#440C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic, complemented
19C2×D4×D53C2211yescharacteristic
20#13133246—24no—
21C2×Dic212C2211yescharacteristic, complemented
22C2×Dic212C2211yescharacteristic, complemented
23C2×#11719925C2211yescharacteristic, complemented
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic
26C2×#11719923C2211yescharacteristic
27C2×#11719923C2211yescharacteristic
28C2×#11719924C2211yescharacteristic
29SD16×D53—24no—
30C2×#11719912C2211yescharacteristic, complemented
31C2×C8×D53C2211yescharacteristic
32#13133217—21no—
33#13133217C2211yescharacteristic, complemented
34#13133205C2211yescharacteristic
35#13133201—21no—
36#13133197—21no—
37#13133197—21no—
38#13133197C2211yescharacteristic
39#13133197C2211yescharacteristic
40#13133197C2211yescharacteristic, complemented
41#13133192C2211yescharacteristic
42#13133191—21no—
43#13133187—21no—
44C4×D212—21no—
45C4×D212C2211yescharacteristic
46C2×#11719921—21no—
47C4⋊#88322C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133140C2211yescharacteristic, complemented
51#13133137C2211yescharacteristic
52#13133129C2211yescharacteristic
53#13133129C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic, complemented
56C4×D4×C53C2211yescharacteristic
57#107×C53C2211yescharacteristic, complemented
58#103×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic
60#97×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68Q8×D53—22no—
69Q8×D53—22no—
70Q8×D53—41no—
71Q8×D53—41no—
72#11719947—22no—
73#11719947—41no—
74D4×D53—22no—
75D4×D53—41no—
76C2×D212—21no—
77C2×D212D411yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53C2311yescharacteristic
82C2×C4×D53C2311yescharacteristic
83C2×Dic106—21no—
84C2×Dic106—21no—
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
88SD16×C53D414yescomplemented
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C4⋊C4×C53—21no—
92C4⋊C4×C53C2311yescharacteristic
93C4⋊C4×C53C2311yescharacteristic
94C22⋊C4×C53—21no—
95C42×C53—21no—
96C42×C53C2311yescharacteristic
97#11719925—22no—
98#11719924—22no—
99#11719923—22no—
100#11719923—22no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921D411yescharacteristic
108#11719921D411yescharacteristic
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53D411yescharacteristic
115C2×#88322C2311yescharacteristic
116C2×#88322C2311yescharacteristic
117Dic212—22no—
118Dic212—41no—
119#11719913—22no—
120#11719913—41no—
121#11719912—22no—
122C8×D53—22no—
123C23×C53—21no—
124C22×D53—21no—
125C22×D53—42no—
126C22×D53—42no—
127C22×D53—81no—
128C22×D53C2×D411yescharacteristic
129C22×D53C2×D411yescharacteristic
130Q8×C53—21no—
131Q8×C53C2×D412yes—
132D4×C53—21no—
133D4×C53C2×D412yes—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—81no—
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C22⋊D53—42no—
142C22⋊D53—42no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—42no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149D212—22no—
150D212—21no—
151D212—21no—
152D212—81no—
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—42no—
158C4×D53—42no—
159Dic106—21no—
160Dic106—21no—
161Dic106—21no—
162Dic106—21no—
163Dic106—21no—
164Dic106—21no—
165Dic106—42no—
166Dic106—42no—
167C8×C53—21no—
168C8×C53C2×D412yes—
169#88322—21no—
170#88322—21no—
171C22×C53—22no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53C22×D411yescharacteristic
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C2×D53#13712yescomplemented
185C4×C53—22no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—41no—
189C4×C53—41no—
190C4×C53C22×D411yescharacteristic
191C4×C53H(5,𝔽2)11yescharacteristic
192Dic53—21no—
193Dic53—21no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#3266—531nomaximal, Sylow
198C2×C53—42no—
199C2×C53D4211yescharacteristic
200C2×C53C2×#13711yescharacteristic
201C2×C53#57711yescharacteristic
202D53—22no—
203D53—41no—
204D53—81no—
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#450—531no—
217#448—531no—
218#440D5311yescharacteristic, complemented
219#435—531no—
220C53#326611yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C22×Q8—531no—
223C22×Q8—531no—
224C22×D4—531no—
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—531no—
234C2×SD16—1064no—
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—531no—
245C22⋊Q8—1061no—
246C22⋊Q8—1061no—
247#121—531no—
248#121—1061no—
249#120—1061no—
250C4×Q8C2×D5311yescharacteristic, complemented
251C4×D4—531no—
252C4×D4—1061no—
253C4×D4C2×D5311yescharacteristic
254C2×C22⋊C4—1061no—
255#107C2×D5311yescharacteristic, complemented
256C4⋊C8—531no—
257#103—531no—
258#103—531no—
259#103—531no—
260#103C2×D5311yescharacteristic, complemented
261#102—531no—
262#102C2×D5311yescharacteristic
263C22⋊C8—531no—
264C22⋊C8—531no—
265#97C2×D5311yescharacteristic, complemented
266C24—1061no—
267#41—1062no—
268#41—2121no—
269C2×Q8—531no—
270C2×Q8—531no—
271C2×Q8—531no—
272C2×Q8—1061no—
273C2×Q8—1061no—
274C2×Q8—1062no—
275C2×Q8—1062no—
276C2×Q8—2121no—
277C2×Q8—2121no—
278C2×Q8C22×D5311yescharacteristic
279C2×D4—531no—
280C2×D4—531no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1062no—
284C2×D4—1061no—
285C2×D4—2121no—
286C2×D4C22×D5311yescharacteristic
287C22×C4—21no—
288C22×C4—531no—
289C22×C4—531no—
290C22×C4—1061no—
291C22×C4—1061no—
292C22×C4—1061no—
293Q16—1062no—
294Q16—1062no—
295Q16—2121no—
296SD16—1062no—
297SD16—1062no—
298SD16—1062no—
299SD16—1062no—
300SD16—2121no—
301SD16D21214yescomplemented
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—1061no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4C22×D5311yescharacteristic
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—21no—
327C42C22×D5311yescharacteristic
328C23—21no—
329C23—531no—
330C23—531no—
331C23—1061no—
332C23—4241no—
333C23—2122no—
334C23—2122no—
335Q8—21no—
336Q8—1061no—
337Q8—1061no—
338Q8—1061no—
339Q8—1061no—
340Q8—1061no—
341Q8—1061no—
342Q8—2122no—
343Q8—2122no—
344Q8C2×D21212yes—
345D4—21no—
346D4—1062no—
347D4—1061no—
348D4—1061no—
349D4—4241no—
350D4—2122no—
351D4—2122no—
352D4C2×D21212yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—531no—
358C2×C4—531no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4D4×D5311yescharacteristic
370C2×C4D4×D5311yescharacteristic
371C2×C4C23×D5311yescharacteristic
372C8—21no—
373C8—1061no—
374C8—1061no—
375C8C2×D21212yes—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—532no—
380C22—1061no—
381C22—1061no—
382C22—1061no—
383C22—1061no—
384C22—1061no—
385C22—4244no—
386C22—2121no—
387C22—2121no—
388C22—2121no—
389C22C2×D4×D5311yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—1061no—
396C4—1061no—
397C4—2121no—
398C4—2121no—
399C4—2121no—
400C4C22×D21211yescharacteristic
401C4#1313326311yescharacteristic
402C2—42no—
403C2—1062no—
404C2—4241no—
405C2—2121no—
406C2D4×D21211yescharacteristic
407C2#137×D5311yescharacteristic
408C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#440C2
socleC22×C53C22×D4
hypercenter#440D53
hypocenterC53#3266
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. C2×SD16×D53C2semidirect
3. Q8×D212C2semidirect
5. #16811017C2semidirect
6. #16810870C2semidirect
7. #16810865C2semidirect
8. #16810845C2semidirect
9. #16810828C2semidirect
10. #16810786C2semidirect
11. #16810780C2semidirect
12. #16810775C2semidirect
13. #16810763C2semidirect
14. #16810696C2semidirect
15. C53×#440C2semidirect
18. C2×Q8×D53C22semidirect
21. C2×Dic212C22semidirect
22. C2×Dic212C22semidirect
23. C2×#11719925C22semidirect
30. C2×#11719912C22semidirect
33. #13133217C22semidirect
40. #13133197C22semidirect
47. C4⋊#88322C22semidirect
48. #13133140C22semidirect
49. #13133140C22semidirect
50. #13133140C22semidirect
53. #13133129C22semidirect
55. C4×Q8×C53C22semidirect
57. #107×C53C22semidirect
58. #103×C53C22semidirect
60. #97×C53C22semidirect
88. SD16×C53D4semidirect
184. C2×D53#137semidirect
218. #440D53semidirect
220. C53#3266semidirect
250. C4×Q8C2×D53semidirect
255. #107C2×D53semidirect
260. #103C2×D53semidirect
265. #97C2×D53semidirect
301. SD16D212semidirect