Finite Group Databasebeta

← #20286288

Subgroups — #20286288

Order 6784.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133248C211yesmaximal, characteristic, complemented
2C2×Q16×D53C211yesmaximal, characteristic, complemented
3Q8×C22⋊D53C211yesmaximal, characteristic, complemented
4#16811283C211yesmaximal, characteristic
5#16811016C211yesmaximal, characteristic, complemented
6#16810879C211yesmaximal, characteristic, complemented
7#16810855C211yesmaximal, characteristic, complemented
8#16810847C211yesmaximal, characteristic, complemented
9#16810832C211yesmaximal, characteristic, complemented
10#16810787C211yesmaximal, characteristic, complemented
11#16810777C211yesmaximal, characteristic, complemented
12#16810763C211yesmaximal, characteristic, complemented
13#16810727C211yesmaximal, characteristic, complemented
14#16810697C211yesmaximal, characteristic, complemented
15C53×#451C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19#13133248—24no—
20C2×Dic212C2211yescharacteristic
21C2×#11719925C2211yescharacteristic
22C2×#11719925C2211yescharacteristic
23C2×#11719925C2211yescharacteristic
24Q16×D53—24no—
25C2×#11719913C2211yescharacteristic
26C2×#11719924C2211yescharacteristic
27C2×#11719924C2211yescharacteristic
28C2×#11719917C2211yescharacteristic
29C2×C8×D53C2211yescharacteristic
30#13133215—21no—
31#13133215C2211yescharacteristic
32#13133206C2211yescharacteristic
33#13133203—21no—
34#13133196—21no—
35#13133196C2211yescharacteristic
36#13133195—21no—
37#13133193—21no—
38#13133193C2211yescharacteristic
39#13133192C2211yescharacteristic
40#13133190—21no—
41Q8×Dic53C2211yescharacteristic
42#13133182C2211yescharacteristic
43#13133179—21no—
44C4×C22⋊D53—21no—
45C4×C22⋊D53C2211yescharacteristic
46#13133148C2211yescharacteristic
47#13133146C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133141C2211yescharacteristic
50#13133140C2211yescharacteristic
51#13133139C2211yescharacteristic
52#13133129C2211yescharacteristic
53#13133127C2211yescharacteristic
54C22×Q8×C53C2211yescharacteristic
55C2×Q16×C53C2211yescharacteristic
56C2×Q16×C53C2211yescharacteristic
57C22⋊Q8×C53C2211yescharacteristic
58#103×C53C2211yescharacteristic
59#103×C53C2211yescharacteristic
60C22⋊C8×C53C2211yescharacteristic
61C2×Q8×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×Q8×C53C2311yescharacteristic
66C2×Q8×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53D411yescharacteristic
71Q8×D53—22no—
72Q8×D53—22no—
73Q8×D53—41no—
74Q8×D53—41no—
75#11719945—22no—
76#11719945—41no—
77C2×D212—21no—
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53C2311yescharacteristic
83C2×Dic106—21no—
84C2×Dic106—21no—
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2311yescharacteristic
87Q16×C53—22no—
88Q16×C53—41no—
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C4⋊C4×C53—21no—
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53D411yescharacteristic
96#11719925—22no—
97#11719925—22no—
98#11719925D414yescomplemented
99#11719924—22no—
100#11719924—22no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105#11719921D411yescharacteristic
106C4⋊Dic53—21no—
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109#11719919—21no—
110#11719919—21no—
111#11719919—21no—
112#11719919—21no—
113#11719919D411yescharacteristic
114C4×Dic53—21no—
115C4×Dic53—21no—
116C4×Dic53C2311yescharacteristic
117#11719917—22no—
118C2×#88322C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120Dic212—22no—
121#11719913—41no—
122C8×D53—22no—
123C23×C53C2×D411yescharacteristic
124C22×D53—21no—
125C22×D53C2×D411yescharacteristic
126Q8×C53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—21no—
131Q8×C53—42no—
132Q8×C53C2×D412yes—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—42no—
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—22no—
143C22⋊D53—42no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53C2×D411yescharacteristic
148C2×Dic53C2×D411yescharacteristic
149C2×Dic53C2×D411yescharacteristic
150D212—21no—
151D212—21no—
152D212—42no—
153C4×D53—21no—
154C4×D53—21no—
155C4×D53—42no—
156C4×D53—42no—
157C4×D53—42no—
158Dic106—21no—
159Dic106—21no—
160Dic106—21no—
161Dic106—42no—
162Dic106—42no—
163Dic106C2×D412yes—
164C8×C53—21no—
165C8×C53—21no—
166#88322—21no—
167#88322C2×D412yes—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53C22×D411yescharacteristic
172C2×D53—21no—
173C2×D53—21no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53#13712yescomplemented
178C4×C53—22no—
179C4×C53—21no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53C22×D411yescharacteristic
184C4×C53H(5,𝔽2)11yescharacteristic
185Dic53—22no—
186Dic53—21no—
187Dic53—21no—
188Dic53—21no—
189Dic53—41no—
190Dic53—41no—
191#3270—531nomaximal, Sylow
192C2×C53—41no—
193C2×C53D4211yescharacteristic
194C2×C53C2×#13711yescharacteristic
195C2×C53#57811yescharacteristic
196D53—22no—
197D53—81no—
198C2×#137—531no—
199C22×Q16—531no—
200D4×Q8—531no—
201#548—531no—
202#497—531no—
203#470—531no—
204#465—531no—
205#464—531no—
206#462—531no—
207#452—531no—
208#451—531no—
209#451D5311yescharacteristic, complemented
210#448—531no—
211#441—531no—
212#435—531no—
213C53#327011yescharacteristic, complemented, Sylow
214C2×#41—531no—
215C22×Q8—531no—
216C22×Q8—531no—
217C22×Q8C2×D5311yescharacteristic
218#137—1064no—
219C2×Q16—531no—
220C2×Q16—531no—
221C2×Q16—531no—
222C2×Q16—531no—
223C2×Q16—1064no—
224C2×Q16C2×D5311yescharacteristic
225C2×Q16C2×D5311yescharacteristic
226C2×SD16—531no—
227C2×SD16—531no—
228C2×SD16—531no—
229C2×M16—531no—
230C22×C8—531no—
231C4⋊Q8—531no—
232C4⋊Q8—1061no—
233#124—531no—
234#124—1061no—
235C22⋊Q8—531no—
236C22⋊Q8—531no—
237C22⋊Q8—1061no—
238C22⋊Q8—1061no—
239C22⋊Q8—1061no—
240C22⋊Q8C2×D5311yescharacteristic
241#121—531no—
242#121—1061no—
243C4×Q8—531no—
244C4×Q8—531no—
245C4×D4—531no—
246C4×D4—1061no—
247C4×D4—1061no—
248#106—531no—
249C4⋊C8—531no—
250#103—531no—
251#103—531no—
252#103—531no—
253#103C2×D5311yescharacteristic
254#103C2×D5311yescharacteristic
255#102—531no—
256C22⋊C8—531no—
257C22⋊C8C2×D5311yescharacteristic
258#97—531no—
259#41—1062no—
260#41—2121no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—531no—
265C2×Q8—531no—
266C2×Q8—1061no—
267C2×Q8—1061no—
268C2×Q8—1062no—
269C2×Q8—1062no—
270C2×Q8—2121no—
271C2×Q8—2121no—
272C2×Q8C22×D5311yescharacteristic
273C2×Q8C22×D5311yescharacteristic
274C2×Q8C22×D5311yescharacteristic
275C2×D4—531no—
276C2×D4—531no—
277C2×D4—1061no—
278C2×D4—1061no—
279C22×C4—21no—
280C22×C4—531no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4C22×D5311yescharacteristic
285Q16—22no—
286Q16—41no—
287Q16—534no—
288Q16—1062no—
289Q16—1062no—
290Q16—1062no—
291SD16—1062no—
292SD16—1062no—
293SD16—2121no—
294M16—1062no—
295C2×C8—531no—
296C2×C8—531no—
297C2×C8—1062no—
298C2×C8C22×D5311yescharacteristic
299C2×C8C22×D5311yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—531no—
302C4⋊C4—531no—
303C4⋊C4—1061no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4C22×D5311yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—531no—
312C22⋊C4—531no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C42—531no—
319C42—1061no—
320C42—1061no—
321C23—531no—
322C23—1061no—
323C23D4×D5311yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—532no—
331Q8—1061no—
332Q8—1061no—
333Q8—1061no—
334Q8—2122no—
335Q8—2122no—
336Q8D4×D5312yes—
337D4—1062no—
338D4—1061no—
339D4—1061no—
340D4—2122no—
341D4—2122no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—531no—
350C2×C4—531no—
351C2×C4—531no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4D4×D5311yescharacteristic
361C2×C4C23×D5311yescharacteristic
362C8—21no—
363C8—21no—
364C8—532no—
365C8—1061no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—532no—
370C22—1061no—
371C22—1061no—
372C22—2121no—
373C22—2121no—
374C22—2121no—
375C22C2×D4×D5311yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—1061no—
382C4—1061no—
383C4—1061no—
384C4—1062no—
385C4—2121no—
386C4—2121no—
387C4C2×D4×D5311yescharacteristic
388C4#1313326211yescharacteristic
389C2—41no—
390C2—1062no—
391C2—4241no—
392C2#1681125511yescharacteristic
393C2#137×D5311yescharacteristic
394C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#451C2
socleC22×C53C22×D4
hypercenter#451D53
hypocenterC53#3270
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#13133248C2semidirect
2. C2×Q16×D53C2semidirect
3. Q8×C22⋊D53C2semidirect
5. #16811016C2semidirect
6. #16810879C2semidirect
7. #16810855C2semidirect
8. #16810847C2semidirect
9. #16810832C2semidirect
10. #16810787C2semidirect
11. #16810777C2semidirect
12. #16810763C2semidirect
13. #16810727C2semidirect
14. #16810697C2semidirect
15. C53×#451C2semidirect
98. #11719925D4semidirect
177. C2×D53#137semidirect
209. #451D53semidirect
213. C53#3270semidirect