Finite Group Databasebeta

← #20286906

Subgroups — #20286906

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811338C211yesmaximal, characteristic, complemented
2#16811285C211yesmaximal, characteristic, complemented
3#16810983C211yesmaximal, characteristic, complemented
4#16810966C211yesmaximal, characteristic, complemented
5#16810950C211yesmaximal, characteristic, complemented
6#16810908C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810851C211yesmaximal, characteristic, complemented
9#16810838C211yesmaximal, characteristic, complemented
10#16810767C211yesmaximal, characteristic, complemented
11#16810760C211yesmaximal, characteristic, complemented
12#16810696C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810618C211yesmaximal, characteristic, complemented
15C53×#440C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133217C2211yescharacteristic
23#13133205—21no—
24#13133205C2211yescharacteristic
25#13133203—21no—
26#13133203—21no—
27#13133197—21no—
28#13133197C2211yescharacteristic
29#13133192—21no—
30#13133192C2211yescharacteristic
31#13133191—21no—
32#13133188C2211yescharacteristic
33#13133188C2211yescharacteristic
34C4×D212—21no—
35C4×D212C2211yescharacteristic
36#13133176—21no—
37#13133175C2211yescharacteristic
38#13133175C2211yescharacteristic
39#13133148C2211yescharacteristic
40#13133148C2211yescharacteristic
41C4⋊#88322C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C4×Q8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64#11719949—22no—
65#11719949—41no—
66Q8×D53—22no—
67Q8×D53—41no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212D411yescharacteristic
71C2×D212C2311yescharacteristic
72C2×C4×D53—21no—
73C2×C4×D53—21no—
74C2×C4×D53C2311yescharacteristic
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106—21no—
78C2×Dic106C2311yescharacteristic
79SD16×C53—22no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719923—41no—
91#11719922—41no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99#11719921D411yescharacteristic
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53D411yescharacteristic
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4×Dic53—21no—
106C4×Dic53—21no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719912—41no—
110C8×D53—41no—
111C23×C53—21no—
112C22×D53—21no—
113C22×D53C2×D411yescharacteristic
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53#4112yes—
117D4×C53—21no—
118D4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—42no—
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—81no—
127C22⋊D53—81no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53C2×D411yescharacteristic
132C2×Dic53C2×D411yescharacteristic
133D212—21no—
134D212—21no—
135D212—41no—
136D212—42no—
137D212—42no—
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—42no—
143C4×D53—42no—
144Dic106—21no—
145Dic106—21no—
146Dic106—42no—
147Dic106—81no—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53C22×D411yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53C2×#4111yescharacteristic
171C4×C53H(5,𝔽2)11yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177#3321—531nomaximal, Sylow
178C2×C53—81no—
179C2×C53#54611yescharacteristic
180C2×C53#57511yescharacteristic
181C2×C53#57711yescharacteristic
182D53—41no—
183D53—41no—
184D53—81no—
185#555—531no—
186#548—531no—
187#489—531no—
188#485—531no—
189#482—531no—
190#475—531no—
191#469—531no—
192#465—531no—
193#463—531no—
194#449—531no—
195#448—531no—
196#440D5311yescharacteristic, complemented
197#435—531no—
198#419—531no—
199#416—531no—
200C53#332111yescharacteristic, complemented, Sylow
201C2×#41—531no—
202C22×Q8—531no—
203C2×SD16—531no—
204C2×SD16C2×D5311yescharacteristic
205C2×D8—531no—
206C2×M16—531no—
207C22×C8—531no—
208C4⋊Q8—531no—
209#124—531no—
210#124—1061no—
211#124—1061no—
212#124—1061no—
213C22⋊Q8—531no—
214C22⋊Q8—1061no—
215#121—531no—
216#121—531no—
217#121—531no—
218#121—1061no—
219#121—1061no—
220C4×Q8C2×D5311yescharacteristic
221C4×D4—531no—
222C4×D4—1061no—
223C4×D4C2×D5311yescharacteristic
224#117—531no—
225#117—531no—
226#117—1061no—
227#107C2×D5311yescharacteristic
228#106—531no—
229#106—531no—
230C4⋊C8—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103C2×D5311yescharacteristic
237#102—531no—
238#102—531no—
239#102—531no—
240#102C2×D5311yescharacteristic
241C22⋊C8—531no—
242C22⋊C8—531no—
243#97C2×D5311yescharacteristic
244#41—1062no—
245#41—2121no—
246C2×Q8—531no—
247C2×Q8—1061no—
248C2×Q8—1061no—
249C2×Q8—1062no—
250C2×Q8—2121no—
251C2×Q8C22×D5311yescharacteristic
252C2×D4—531no—
253C2×D4—531no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4C22×D5311yescharacteristic
259C22×C4—21no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—1061no—
263C22×C4—1061no—
264SD16—22no—
265SD16—2121no—
266D8—2121no—
267M16—2121no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—2121no—
271C2×C8C22×D5311yescharacteristic
272C2×C8C22×D5311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4C22×D5311yescharacteristic
280C4⋊C4C22×D5311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—531no—
283C22⋊C4—531no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C42—21no—
293C42—1061no—
294C42—1061no—
295C42C22×D5311yescharacteristic
296C23—21no—
297C23—531no—
298C23—531no—
299C23—1061no—
300Q8—21no—
301Q8—1061no—
302Q8—1061no—
303Q8—4241no—
304Q8—2122no—
305Q8#1171994512yes—
306D4—21no—
307D4—21no—
308D4—1061no—
309D4—1061no—
310D4—4241no—
311D4—4241no—
312D4—2121no—
313D4—2122no—
314D4—2122no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4D4×D5311yescharacteristic
331C2×C4D4×D5311yescharacteristic
332C2×C4C23×D5311yescharacteristic
333C8—21no—
334C8—21no—
335C8—1061no—
336C8—1061no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—1061no—
341C22—1061no—
342C22—1061no—
343C22—1061no—
344C22—1061no—
345C22—1061no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C2×D4×D5311yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—41no—
354C4—41no—
355C4—1061no—
356C4—1061no—
357C4—2121no—
358C4—2121no—
359C4—2121no—
360C4C2×#1171994511yescharacteristic
361C4#1313326311yescharacteristic
362C2—81no—
363C2—4241no—
364C2—2121no—
365C2—2121no—
366C2#1681126911yescharacteristic
367C2#1681141911yescharacteristic
368C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#440C2
socleC22×C53C22×D4
hypercenter#440D53
hypocenterC53#3321
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. #16811338C2semidirect
2. #16811285C2semidirect
3. #16810983C2semidirect
4. #16810966C2semidirect
5. #16810950C2semidirect
6. #16810908C2semidirect
7. #16810875C2semidirect
8. #16810851C2semidirect
9. #16810838C2semidirect
10. #16810767C2semidirect
11. #16810760C2semidirect
12. #16810696C2semidirect
13. #16810630C2semidirect
14. #16810618C2semidirect
15. C53×#440C2semidirect
196. #440D53semidirect
200. C53#3321semidirect