Finite Group Databasebeta

← #20288365

Subgroups — #20288365

Order 6784.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133265C211yesmaximal, characteristic, complemented
2C22×Q8×D53C211yesmaximal, characteristic, complemented
3#16811288C213yesmaximal, complemented
4Q8×C22⋊D53C211yesmaximal, characteristic, complemented
5#16811286C213yesmaximal, complemented
6#16811285C211yesmaximal, characteristic, complemented
7#16811284C211yesmaximal, characteristic, complemented
8#16811283C213yesmaximal, complemented
9#16811282C213yesmaximal, complemented
10#16811198C213yesmaximal, complemented
11#16811196C213yesmaximal, complemented
12#16811136C213yesmaximal, complemented
13#16811132C213yesmaximal, complemented
14#16811099C211yesmaximal, characteristic, complemented
15D4×Q8×C53C211yesmaximal, characteristic, complemented
16#13133265—28no—
17C2×#11719949C2213yes—
18C2×#11719949C2211yescharacteristic, complemented
19C2×#11719945C2213yes—
20C2×#11719945C2213yescomplemented
21C22×Dic106C2213yes—
22C2×Q8×D53—28no—
23C2×Q8×D53C2213yes—
24C2×Q8×D53C2211yescharacteristic
25C2×Q8×D53C2213yes—
26C2×Q8×D53C2211yescharacteristic
27C2×Q8×D53C2213yescomplemented
28C22×C4×D53C2213yes—
29#13133216C2216yescomplemented
30#13133215C2213yescomplemented
31#13133206C2213yescomplemented
32#13133205C2213yescomplemented
33#13133203C2216yescomplemented
34#13133202C2213yes—
35#13133202C2216yes—
36#13133199C2213yes—
37#13133197C2213yescomplemented
38#13133196C2213yescomplemented
39#13133195C2213yes—
40#13133195C2216yescomplemented
41#13133195C2216yescomplemented
42#13133194C2213yes—
43#13133193C2213yes—
44#13133193C2213yescomplemented
45#13133192C2213yes—
46#13133191C2213yes—
47#13133190C2213yes—
48#13133189C2213yes—
49#13133187C2213yes—
50#13133185C2211yescharacteristic
51Q8×Dic53C2211yescharacteristic, complemented
52C4×Dic106C2213yescomplemented
53#13133182C2213yescomplemented
54C4×D212C2213yes—
55#13133179C2216yes—
56C4×C22⋊D53C2213yes—
57C4×C22⋊D53C2213yes—
58#13133177C2216yes—
59#13133175C2213yes—
60#13133174C2213yes—
61C22×Q8×C53C2211yescharacteristic
62C22×Q8×C53C2211yescharacteristic, complemented
63C4⋊Q8×C53C2213yescomplemented
64C22⋊Q8×C53C2213yes—
65C22⋊Q8×C53C2213yescomplemented
66C4×Q8×C53C2211yescharacteristic, complemented
67C4×D4×C53C2213yes—
68C23×D53C2311yescharacteristic
69C2×Q8×C53—24no—
70C2×Q8×C53—24no—
71C2×Q8×C53C2313yes—
72C2×Q8×C53C2313yes—
73C2×Q8×C53C2311yescharacteristic
74C2×D4×C53C2311yescharacteristic
75C22×C4×C53C2313yes—
76C22×C4×C53C2313yes—
77C2×C22⋊D53C2313yes—
78C2×C22⋊D53C2311yescharacteristic
79C2×C22⋊D53C2311yescharacteristic
80C2×C22⋊D53C2313yes—
81C22×Dic53C2313yes—
82#11719949—24no—
83#11719949—212no—
84Q8×D53—24no—
85Q8×D53—212no—
86Q8×D53—212no—
87Q8×D53—212no—
88Q8×D53D418yescomplemented
89#11719945—212no—
90#11719945—212no—
91C2×D212C2313yes—
92C2×D212C2313yes—
93C2×D212C2313yes—
94C2×C4×D53—212no—
95C2×C4×D53C2316yes—
96C2×C4×D53C2313yes—
97C2×C4×D53C2313yes—
98C2×C4×D53C2313yes—
99C2×C4×D53C2313yes—
100C2×C4×D53C2313yes—
101C2×Dic106—212no—
102C2×Dic106C2313yes—
103C2×Dic106C2316yes—
104C2×Dic106C2313yes—
105C2×Dic106C2313yes—
106C4⋊C4×C53C2313yes—
107C4⋊C4×C53C2313yes—
108C4⋊C4×C53C2316yes—
109C22⋊C4×C53C2313yes—
110C22⋊C4×C53C2313yes—
111C42×C53C2313yes—
112C22⋊Dic53C2311yescharacteristic
113C22⋊Dic53C2313yes—
114#11719921C2311yescharacteristic
115#11719921C2313yes—
116#11719921C2316yes—
117#11719921C2313yes—
118#11719921C2313yes—
119#11719921C2316yes—
120#11719921C2311yescharacteristic
121#11719921C2313yes—
122C4⋊Dic53C2313yes—
123C4⋊Dic53C2313yes—
124#11719919C2313yes—
125#11719919C2313yes—
126#11719919C2316yes—
127#11719919C2313yes—
128#11719919C2313yes—
129C4×Dic53C2313yes—
130C4×Dic53C2316yes—
131C23×C53C2411yescharacteristic
132C23×C53C2411yescharacteristic
133C22×D53—22no—
134C22×D53—22no—
135C22×D53—22no—
136C22×D53C2413yes—
137C22×D53C2411yescharacteristic
138C22×D53C2411yescharacteristic
139Q8×C53—26no—
140Q8×C53—26no—
141Q8×C53C2×D414yes—
142D4×C53—22no—
143C2×C4×C53—26no—
144C2×C4×C53—26no—
145C2×C4×C53C2413yes—
146C2×C4×C53C2413yes—
147C2×C4×C53C2413yes—
148C2×C4×C53C2411yescharacteristic
149C2×C4×C53C2413yes—
150C22⋊D53—26no—
151C22⋊D53—22no—
152C22⋊D53—22no—
153C22⋊D53—26no—
154C2×Dic53—26no—
155C2×Dic53C2413yes—
156C2×Dic53C2411yescharacteristic
157C2×Dic53C2411yescharacteristic
158C2×Dic53C2413yes—
159C2×Dic53C2413yes—
160D212—26no—
161D212—26no—
162D212—26no—
163C4×D53—26no—
164C4×D53—26no—
165C4×D53—26no—
166C4×D53—26no—
167C4×D53—212no—
168C4×D53C2×D4112yes—
169Dic106—26no—
170Dic106—212no—
171Dic106—26no—
172Dic106C2×D4112yes—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C532−1+412yescomplemented
179C22×C53C2511yescharacteristic
180C2×D53—28no—
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—21no—
184C2×D53—21no—
185C2×D53—23no—
186C2×D53—23no—
187C2×D53—23no—
188C2×D53C22×D412yes—
189C2×D532−1+412yescomplemented
190C4×C53—23no—
191C4×C53—23no—
192C4×C53—23no—
193C4×C53—21no—
194C4×C53C22×D416yes—
195Dic53—23no—
196Dic53—21no—
197Dic53—21no—
198Dic53—23no—
199Dic53C22×D416yes—
200#3486—531nomaximal, Sylow
201C2×C53—22no—
202C2×C53—41no—
203C2×C53C23×D411yescharacteristic
204C2×C53C2×2−1+411yescharacteristic
205C2×C53C2×2−1+411yescharacteristic
206D53—22no—
207D53—22no—
208D53—43no—
209C2×2−1+4—531no—
210C23×Q8—531no—
211D4×Q8—531no—
212D4×Q8—533no—
213D4×Q8—533no—
214D4×Q8D5311yescharacteristic, complemented
215#548—531no—
216#548—533no—
217#548—531no—
218#548—533no—
219#536—533no—
220#536—533no—
221#525—533no—
222#525—533no—
223#519—531no—
224C53#348611yescharacteristic, complemented, Sylow
2252−1+4—1068no—
226C2×#41—533no—
227C2×#41—531no—
228C2×#41—533no—
229C2×#41—533no—
230C22×Q8—531no—
231C22×Q8—533no—
232C22×Q8—533no—
233C22×Q8—531no—
234C22×Q8—533no—
235C22×Q8—533no—
236C22×Q8—1068no—
237C22×Q8C2×D5311yescharacteristic
238C22×Q8C2×D5311yescharacteristic, complemented
239C23×C4—533no—
240C4⋊Q8—533no—
241C4⋊Q8—536no—
242C4⋊Q8C2×D5313yescomplemented
243#124—533no—
244#124—536no—
245#124—533no—
246#123—536no—
247#123—533no—
248#123—533no—
249C22⋊Q8—533no—
250C22⋊Q8—533no—
251C22⋊Q8—533no—
252C22⋊Q8—533no—
253C22⋊Q8—536no—
254C22⋊Q8—536no—
255C22⋊Q8—533no—
256C22⋊Q8—533no—
257C22⋊Q8C2×D5313yes—
258C22⋊Q8C2×D5313yescomplemented
259#121—533no—
260#121—533no—
261#121—533no—
262#121—533no—
263#120—531no—
264#120—533no—
265C4×Q8—531no—
266C4×Q8—533no—
267C4×Q8—533no—
268C4×Q8C2×D5311yescharacteristic, complemented
269C4×D4—533no—
270C4×D4—533no—
271C4×D4—536no—
272C4×D4—536no—
273C4×D4—533no—
274C4×D4C2×D5313yes—
275#117—533no—
276#117—533no—
277C24—531no—
278#41—10612no—
279#41—10612no—
280#41—10612no—
281#41—1064no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—538no—
285C2×Q8—533no—
286C2×Q8—536no—
287C2×Q8—533no—
288C2×Q8—533no—
289C2×Q8—10612no—
290C2×Q8—10612no—
291C2×Q8—10612no—
292C2×Q8—10612no—
293C2×Q8—1064no—
294C2×Q8C22×D5313yes—
295C2×Q8C22×D5313yes—
296C2×Q8C22×D5311yescharacteristic
297C2×D4—531no—
298C2×D4—533no—
299C2×D4—531no—
300C2×D4—533no—
301C2×D4—533no—
302C2×D4—533no—
303C2×D4—533no—
304C2×D4C22×D5311yescharacteristic
305C22×C4—533no—
306C22×C4—533no—
307C22×C4—533no—
308C22×C4—533no—
309C22×C4—536no—
310C22×C4—533no—
311C22×C4—533no—
312C22×C4—10612no—
313C22×C4C22×D5313yes—
314C22×C4C22×D5313yes—
315C4⋊C4—533no—
316C4⋊C4—533no—
317C4⋊C4—533no—
318C4⋊C4—533no—
319C4⋊C4—536no—
320C4⋊C4—533no—
321C4⋊C4—533no—
322C4⋊C4C22×D5313yes—
323C4⋊C4C22×D5316yes—
324C4⋊C4C22×D5313yes—
325C22⋊C4—531no—
326C22⋊C4—533no—
327C22⋊C4—533no—
328C22⋊C4—531no—
329C22⋊C4—533no—
330C22⋊C4—531no—
331C22⋊C4—533no—
332C22⋊C4—536no—
333C22⋊C4—536no—
334C22⋊C4—533no—
335C22⋊C4C22×D5313yes—
336C22⋊C4C22×D5313yes—
337C42—536no—
338C42—533no—
339C42C22×D5313yes—
340C23—531no—
341C23—531no—
342C23—533no—
343C23—1062no—
344C23—1062no—
345C23—1062no—
346C23C23×D5311yescharacteristic
347C23C23×D5311yescharacteristic
348Q8—26no—
349Q8—26no—
350Q8—5312no—
351Q8—10612no—
352Q8—1066no—
353Q8—1066no—
354Q8D4×D5314yes—
355D4—22no—
356D4—1066no—
357D4—1066no—
358D4—1066no—
359D4—1062no—
360D4—1062no—
361D4—1066no—
362D4—1066no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—533no—
366C2×C4—5312no—
367C2×C4—531no—
368C2×C4—533no—
369C2×C4—533no—
370C2×C4—531no—
371C2×C4—1066no—
372C2×C4—10612no—
373C2×C4—1066no—
374C2×C4—1066no—
375C2×C4—1066no—
376C2×C4—1066no—
377C2×C4C23×D5313yes—
378C2×C4C23×D5311yescharacteristic
379C2×C4C23×D5313yes—
380C2×C4C23×D5313yes—
381C2×C4C23×D5313yes—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—532no—
388C22—532no—
389C22—1068no—
390C22—1061no—
391C22—1061no—
392C22—1061no—
393C22—1061no—
394C22—1063no—
395C22—1063no—
396C22—1063no—
397C22#1313326512yescomplemented
398C22C24×D5311yescharacteristic
399C4—23no—
400C4—23no—
401C4—23no—
402C4—21no—
403C4—536no—
404C4—1061no—
405C4—1063no—
406C4—1061no—
407C4—1063no—
408C4C2×D4×D5316yes—
409C2—22no—
410C2—41no—
411C2—1062no—
412C2—1062no—
413C2—2123no—
414C2C22×D4×D5311yescharacteristic
415C22−1+4×D5311yescharacteristic
416C2C2×#1313326511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingD4×Q8×C53C2
hypercenterD4×Q8D53
hypocenterC53#3486
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#13133265C2semidirect
2. C22×Q8×D53C2semidirect
3. #16811288C2semidirect
4. Q8×C22⋊D53C2semidirect
5. #16811286C2semidirect
6. #16811285C2semidirect
7. #16811284C2semidirect
8. #16811283C2semidirect
9. #16811282C2semidirect
10. #16811198C2semidirect
11. #16811196C2semidirect
12. #16811136C2semidirect
13. #16811132C2semidirect
14. #16811099C2semidirect
15. D4×Q8×C53C2semidirect
18. C2×#11719949C22semidirect
20. C2×#11719945C22semidirect
27. C2×Q8×D53C22semidirect
29. #13133216C22semidirect
30. #13133215C22semidirect
31. #13133206C22semidirect
32. #13133205C22semidirect
33. #13133203C22semidirect
37. #13133197C22semidirect
38. #13133196C22semidirect
40. #13133195C22semidirect
41. #13133195C22semidirect
44. #13133193C22semidirect
51. Q8×Dic53C22semidirect
52. C4×Dic106C22semidirect
53. #13133182C22semidirect
62. C22×Q8×C53C22semidirect
63. C4⋊Q8×C53C22semidirect
65. C22⋊Q8×C53C22semidirect
66. C4×Q8×C53C22semidirect
88. Q8×D53D4semidirect
178. C22×C532−1+4semidirect
189. C2×D532−1+4semidirect
214. D4×Q8D53semidirect
224. C53#3486semidirect
238. C22×Q8C2×D53semidirect
242. C4⋊Q8C2×D53semidirect
258. C22⋊Q8C2×D53semidirect
268. C4×Q8C2×D53semidirect
397. C22#13133265semidirect