Finite Group Databasebeta

← #20285716

Subgroups — #20285716

Order 6784.

400 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811177C211yesmaximal, characteristic, complemented
2#124×D53C211yesmaximal, characteristic
3#16811016C211yesmaximal, characteristic, complemented
4#16811014C211yesmaximal, characteristic, complemented
5#16810984C211yesmaximal, characteristic, complemented
6#16810981C211yesmaximal, characteristic, complemented
7#16810846C211yesmaximal, characteristic, complemented
8#16810845C211yesmaximal, characteristic, complemented
9#16810842C211yesmaximal, characteristic, complemented
10#16810780C211yesmaximal, characteristic, complemented
11#16810777C211yesmaximal, characteristic, complemented
12#16810775C211yesmaximal, characteristic
13#16810772C211yesmaximal, characteristic
14#16810679C211yesmaximal, characteristic, complemented
15C53×#464C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×Dic212C2211yescharacteristic, complemented
19C2×#11719925C2211yescharacteristic, complemented
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719913C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719924C2211yescharacteristic
24#13133216C2211yescharacteristic, complemented
25#13133215C2211yescharacteristic, complemented
26#13133207—21no—
27#13133206C2211yescharacteristic
28#13133205C2211yescharacteristic
29#13133204C2211yescharacteristic
30#13133203—21no—
31#13133203—21no—
32#13133197C2211yescharacteristic
33#13133195—21no—
34#13133193C2211yescharacteristic
35C4×Dic106—21no—
36C4×Dic106C2211yescharacteristic, complemented
37#13133174—21no—
38C22⋊C4×D53—21no—
39C22⋊C4×D53—21no—
40C42×D53C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic, complemented
42#13133142C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133140C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C2×Q16×C53C2211yescharacteristic, complemented
53C2×SD16×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×Q8×C53C2211yescharacteristic, complemented
56C4⋊C8×C53C2211yescharacteristic, complemented
57#103×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic
59C23×D53—21no—
60C2×Q8×C53C2311yescharacteristic
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64Q8×D53—41no—
65Q8×D53—41no—
66D4×D53—41no—
67D4×D53—41no—
68C2×D212C2311yescharacteristic
69C2×C4×D53—21no—
70C2×C4×D53D411yescharacteristic
71C2×C4×D53D411yescharacteristic
72C2×C4×D53C2311yescharacteristic
73C2×Dic106—21no—
74C2×Dic106—21no—
75C2×Dic106C2311yescharacteristic
76C2×Dic106C2311yescharacteristic
77Q16×C53—41no—
78SD16×C53—41no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊C4×C53—21no—
85C42×C53—21no—
86C42×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719925—41no—
90#11719925—41no—
91#11719924—41no—
92#11719923—41no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53D411yescharacteristic
108C4×Dic53D411yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112Dic212—41no—
113#11719913—41no—
114C23×C53—21no—
115C22×D53—21no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53C2×D411yescharacteristic
120Q8×C53—21no—
121Q8×C53—21no—
122Q8×C53—21no—
123Q8×C53—21no—
124D4×C53—21no—
125D4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—42no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—21no—
140D212—21no—
141C4×D53—22no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—41no—
145C4×D53—42no—
146Dic106—21no—
147Dic106—21no—
148Dic106—21no—
149Dic106—21no—
150Dic106—42no—
151Dic106—42no—
152C8×C53—21no—
153C8×C53—21no—
154#88322—21no—
155#88322—21no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53C22×D411yescharacteristic
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—84no—
166C2×D53#13712yes—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53H(5,𝔽2)11yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—21no—
175Dic53—21no—
176Dic53—21no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3215—531nomaximal, Sylow
181C2×C53—81no—
182C2×C53#53511yescharacteristic
183C2×C53C2×#13711yescharacteristic
184C2×C53#57511yescharacteristic
185D53—22no—
186D53—81no—
187#533—531no—
188C2×#124—531no—
189#497—531no—
190#497—531no—
191#489—531no—
192#489—531no—
193#464—531no—
194#464—531no—
195#464—531no—
196#464D5311yescharacteristic, complemented
197#451—531no—
198#451—531no—
199#450—531no—
200#450—531no—
201#432—531no—
202C53#321511yescharacteristic, complemented, Sylow
203C22×Q8—531no—
204C22×D4—531no—
205C2×Q16—531no—
206C2×Q16—531no—
207C2×Q16—531no—
208C2×Q16C2×D5311yescharacteristic, complemented
209C2×SD16—531no—
210C2×SD16—531no—
211C2×SD16—531no—
212C2×SD16C2×D5311yescharacteristic
213C4⋊Q8—531no—
214C4⋊Q8—531no—
215#125—1061no—
216#124—531no—
217#124—531no—
218#124—531no—
219#124—1061no—
220#124—1061no—
221#124C2×D5311yescharacteristic
222C22⋊Q8—531no—
223C22⋊Q8—531no—
224C22⋊Q8—1061no—
225C4×Q8—531no—
226C4×Q8—1061no—
227C4×Q8C2×D5311yescharacteristic, complemented
228#117—1061no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1061no—
231C2×C42—531no—
232C4⋊C8—531no—
233C4⋊C8C2×D5311yescharacteristic, complemented
234#103—531no—
235#103—531no—
236#103—531no—
237#103C2×D5311yescharacteristic, complemented
238#102—531no—
239#102—531no—
240#102—531no—
241#102C2×D5311yescharacteristic
242C22⋊C8—531no—
243C22⋊C8—531no—
244#97—531no—
245#97—531no—
246C24—1061no—
247C2×Q8—531no—
248C2×Q8—531no—
249C2×Q8—1061no—
250C2×Q8—1061no—
251C2×Q8—2121no—
252C2×Q8—2121no—
253C2×Q8C22×D5311yescharacteristic
254C2×Q8C22×D5311yescharacteristic
255C2×D4—531no—
256C2×D4—1061no—
257C2×D4—2121no—
258C2×D4—2121no—
259C2×D4C22×D5311yescharacteristic
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—1061no—
264Q16—41no—
265Q16—2121no—
266Q16—2121no—
267Q16—2121no—
268SD16—41no—
269SD16—2121no—
270SD16—2121no—
271SD16—2121no—
272C2×C8—531no—
273C2×C8—531no—
274C2×C8C22×D5311yescharacteristic
275C2×C8C22×D5311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—531no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4C22×D5311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C42—21no—
296C42—531no—
297C42—531no—
298C42—531no—
299C42—1061no—
300C42C22×D5311yescharacteristic
301C23—21no—
302C23—531no—
303C23—1061no—
304C23—2122no—
305C23—2122no—
306C23—2122no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—1061no—
312Q8—1061no—
313Q8—1061no—
314Q8—1061no—
315Q8—2122no—
316Q8—2122no—
317D4—21no—
318D4—21no—
319D4—1061no—
320D4—1061no—
321D4—2122no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—1062no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—2121no—
335C2×C4—2122no—
336C2×C4D4×D5311yescharacteristic
337C2×C4D4×D5311yescharacteristic
338C2×C4C23×D5311yescharacteristic
339C8—21no—
340C8—21no—
341C8—1061no—
342C8—1061no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—532no—
347C22—1061no—
348C22—1061no—
349C22—4244no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22C2×D4×D5311yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—1061no—
360C4—1061no—
361C4—1061no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4#1313326211yescharacteristic
366C4#1313326311yescharacteristic
367C2—81no—
368C2—1062no—
369C2—4241no—
370C2#1681118911yescharacteristic
371C2#137×D5311yescharacteristic
372C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#464C2
socleC22×C53C22×D4
hypercenter#464D53
hypocenterC53#3215
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. #16811177C2semidirect
3. #16811016C2semidirect
4. #16811014C2semidirect
5. #16810984C2semidirect
6. #16810981C2semidirect
7. #16810846C2semidirect
8. #16810845C2semidirect
9. #16810842C2semidirect
10. #16810780C2semidirect
11. #16810777C2semidirect
14. #16810679C2semidirect
15. C53×#464C2semidirect
18. C2×Dic212C22semidirect
19. C2×#11719925C22semidirect
20. C2×#11719925C22semidirect
24. #13133216C22semidirect
25. #13133215C22semidirect
36. C4×Dic106C22semidirect
41. C4⋊#88322C22semidirect
42. #13133142C22semidirect
43. #13133141C22semidirect
44. #13133140C22semidirect
50. #13133127C22semidirect
51. #13133127C22semidirect
52. C2×Q16×C53C22semidirect
55. C4×Q8×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #103×C53C22semidirect
196. #464D53semidirect
202. C53#3215semidirect
208. C2×Q16C2×D53semidirect
227. C4×Q8C2×D53semidirect
233. C4⋊C8C2×D53semidirect
237. #103C2×D53semidirect