Finite Group Databasebeta

← #20286232

Subgroups — #20286232

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×#13133250C211yesmaximal, characteristic, complemented
2C2×SD16×D53C211yesmaximal, characteristic, complemented
3#16811286C211yesmaximal, characteristic, complemented
4#16811270C211yesmaximal, characteristic
5#16811014C211yesmaximal, characteristic, complemented
6#16810865C211yesmaximal, characteristic, complemented
7#16810844C211yesmaximal, characteristic, complemented
8#16810827C211yesmaximal, characteristic, complemented
9#16810783C211yesmaximal, characteristic, complemented
10#16810780C211yesmaximal, characteristic, complemented
11#16810775C211yesmaximal, characteristic, complemented
12#16810761C211yesmaximal, characteristic, complemented
13#16810722C211yesmaximal, characteristic, complemented
14#16810692C211yesmaximal, characteristic, complemented
15C53×#468C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic, complemented
18C2×Q8×D53C2211yescharacteristic
19C2×D4×D53C2211yescharacteristic
20#13133250—24no—
21C2×Dic212C2211yescharacteristic, complemented
22C2×Dic212C2211yescharacteristic, complemented
23C2×#11719925C2211yescharacteristic, complemented
24C2×#11719913C2211yescharacteristic
25C2×#11719913C2211yescharacteristic
26C2×#11719923C2211yescharacteristic
27C2×#11719923C2211yescharacteristic
28C2×#11719924C2211yescharacteristic
29SD16×D53—24no—
30C2×C8×D53C2211yescharacteristic
31#13133216—21no—
32#13133215C2211yescharacteristic, complemented
33#13133204C2211yescharacteristic
34#13133199—21no—
35#13133197C2211yescharacteristic
36#13133197C2211yescharacteristic
37#13133195—21no—
38#13133194—21no—
39#13133188—21no—
40#13133185—21no—
41#13133182C2211yescharacteristic, complemented
42#13133179C2211yescharacteristic
43C4×C22⋊D53C2211yescharacteristic
44#13133177—21no—
45C22⋊C4×D53—21no—
46#13133146C2211yescharacteristic, complemented
47#13133140C2211yescharacteristic, complemented
48#13133140C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133137C2211yescharacteristic
51C22⋊#88322C2211yescharacteristic, complemented
52#13133129C2211yescharacteristic
53#13133127C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C2×M16×C53C2211yescharacteristic, complemented
56C22⋊Q8×C53C2211yescharacteristic, complemented
57#121×C53C2211yescharacteristic
58#107×C53C2211yescharacteristic, complemented
59#103×C53C2211yescharacteristic, complemented
60#102×C53C2211yescharacteristic
61C23×D53—21no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53D411yescharacteristic
69C22×Dic53—21no—
70Q8×D53—22no—
71Q8×D53—41no—
72D4×D53—22no—
73D4×D53—41no—
74#11719945—22no—
75#11719945—41no—
76C2×D212C2311yescharacteristic
77C2×C4×D53—21no—
78C2×C4×D53—21no—
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—
89M16×C53—22no—
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C4⋊C4×C53—21no—
93C4⋊C4×C53C2311yescharacteristic
94C4⋊C4×C53C2311yescharacteristic
95C22⋊C4×C53—21no—
96C22⋊C4×C53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53D411yescharacteristic
100#11719925—41no—
101#11719924—22no—
102#11719923—22no—
103#11719923—41no—
104#11719921—21no—
105#11719921—21no—
106#11719921—21no—
107#11719921D411yescharacteristic
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110#11719919—21no—
111#11719919—21no—
112#11719919—21no—
113#11719919D411yescharacteristic
114C4×Dic53—21no—
115C4×Dic53C2311yescharacteristic
116C2×#88322C2311yescharacteristic
117C2×#88322C2311yescharacteristic
118Dic212—22no—
119Dic212—22no—
120#11719913—22no—
121#11719913D414yescomplemented
122C8×D53—22no—
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C22×D53—21no—
126C22×D53—42no—
127C22×D53—42no—
128C22×D53—81no—
129C22×D53C2×D411yescharacteristic
130Q8×C53—21no—
131Q8×C53—21no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—81no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—22no—
143C22⋊D53—42no—
144C22⋊D53—42no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—42no—
149C2×Dic53C2×D411yescharacteristic
150C2×Dic53C2×D411yescharacteristic
151C2×Dic53C2×D411yescharacteristic
152D212—21no—
153D212C2×D412yes—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—42no—
157C4×D53—42no—
158C4×D53—81no—
159Dic106—21no—
160Dic106—21no—
161Dic106—21no—
162Dic106—21no—
163Dic106—21no—
164Dic106—42no—
165Dic106—42no—
166Dic106C2×D412yes—
167C8×C53—21no—
168C8×C53C2×D412yes—
169#88322—21no—
170#88322—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53C22×D411yescharacteristic
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—22no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C2×D53#13712yescomplemented
185C4×C53—21no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53C22×D411yescharacteristic
190C4×C53H(5,𝔽2)11yescharacteristic
191Dic53—22no—
192Dic53—21no—
193Dic53—21no—
194Dic53—21no—
195Dic53—41no—
196Dic53—41no—
197#3266—531nomaximal, Sylow
198C2×C53—41no—
199C2×C53—81no—
200C2×C53D4211yescharacteristic
201C2×C53C2×#13711yescharacteristic
202C2×C53#57711yescharacteristic
203D53—22no—
204D53—42no—
205C2×#137—531no—
206C22×SD16—531no—
207D4×Q8—531no—
208#546—531no—
209#497—531no—
210#468D5311yescharacteristic, complemented
211#467—531no—
212#464—531no—
213#461—531no—
214#452—531no—
215#451—531no—
216#450—531no—
217#448—531no—
218#440—531no—
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×M16C2×D5311yescharacteristic, complemented
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, complemented
247#121—1061no—
248#121C2×D5311yescharacteristic
249#120—1061no—
250C4×Q8—531no—
251C4×D4—531no—
252C4×D4—531no—
253C4×D4—1061no—
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#97—531no—
266C24—1061no—
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—21no—
280C2×D4—531no—
281C2×D4—531no—
282C2×D4—1061no—
283C2×D4—1062no—
284C2×D4—1061no—
285C2×D4—2121no—
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—534no—
298SD16—1062no—
299SD16—1062no—
300SD16—1062no—
301SD16—2121no—
302M16—22no—
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—21no—
319C22⋊C4—531no—
320C22⋊C4—531no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C42—531no—
327C42—1061no—
328C23—21no—
329C23—531no—
330C23—1061no—
331C23—4241no—
332C23—2122no—
333C23—2122no—
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—81no—
348D4—532no—
349D4—1062no—
350D4—1061no—
351D4—2122no—
352D4—2122no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—531no—
359C2×C4—531no—
360C2×C4—531no—
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—1061no—
374C8—1061no—
375C8D4×D5312yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—532no—
383C22—1061no—
384C22—1061no—
385C22—1062no—
386C22—4244no—
387C22—2121no—
388C22—2121no—
389C22C2×D4×D5311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—1061no—
395C4—1061no—
396C4—1061no—
397C4—1062no—
398C4—2121no—
399C4—2121no—
400C4C2×D4×D5311yescharacteristic
401C4#1313326311yescharacteristic
402C2—41no—
403C2—81no—
404C2—1062no—
405C2—2122no—
406C2#1681125711yescharacteristic
407C2#137×D5311yescharacteristic
408C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#468C2
socleC22×C53C22×D4
hypercenter#468D53
hypocenterC53#3266
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#13133250C2semidirect
2. C2×SD16×D53C2semidirect
3. #16811286C2semidirect
5. #16811014C2semidirect
6. #16810865C2semidirect
7. #16810844C2semidirect
8. #16810827C2semidirect
9. #16810783C2semidirect
10. #16810780C2semidirect
11. #16810775C2semidirect
12. #16810761C2semidirect
13. #16810722C2semidirect
14. #16810692C2semidirect
15. C53×#468C2semidirect
17. C22×Dic106C22semidirect
21. C2×Dic212C22semidirect
22. C2×Dic212C22semidirect
23. C2×#11719925C22semidirect
32. #13133215C22semidirect
41. #13133182C22semidirect
46. #13133146C22semidirect
47. #13133140C22semidirect
48. #13133140C22semidirect
49. #13133140C22semidirect
51. C22⋊#88322C22semidirect
53. #13133127C22semidirect
55. C2×M16×C53C22semidirect
56. C22⋊Q8×C53C22semidirect
58. #107×C53C22semidirect
59. #103×C53C22semidirect
121. #11719913D4semidirect
184. C2×D53#137semidirect
210. #468D53semidirect
220. C53#3266semidirect
236. C2×M16C2×D53semidirect
246. C22⋊Q8C2×D53semidirect
255. #107C2×D53semidirect
260. #103C2×D53semidirect