Finite Group Databasebeta

← #20288483

Subgroups — #20288483

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
2C2×#13133264C211yesmaximal, characteristic, complemented
3#16811292C211yesmaximal, characteristic, complemented
4#16811291C213yesmaximal, complemented
5Q8×C22⋊D53C211yesmaximal, characteristic, complemented
6#16811286C213yesmaximal, complemented
7#16811285C211yesmaximal, characteristic, complemented
8#16811282C213yesmaximal, complemented
9#16811281C213yesmaximal, complemented
10#16811203C213yesmaximal, complemented
11#16811202C213yesmaximal, complemented
12#16811172C213yesmaximal, complemented
13#16811170C213yesmaximal, complemented
14C4×Q8×D53C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic, complemented
16#13133265—28no—
17#13133264—28no—
18C2×#11719949C2211yescharacteristic
19C2×#11719949C2213yes—
20C2×#11719945C2213yes—
21C2×#11719945C2213yes—
22C2×#11719945C2213yes—
23C2×#11719947C2213yes—
24C2×#11719947C2213yes—
25C22×Dic106C2213yes—
26C2×Q8×D53C2213yes—
27C2×Q8×D53C2211yescharacteristic
28C2×Q8×D53C2211yescharacteristic
29#13133216C2213yes—
30#13133215C2213yes—
31#13133213C2213yes—
32#13133213C2213yes—
33#13133205C2213yes—
34#13133203C2213yes—
35#13133203C2213yes—
36#13133197C2213yes—
37#13133196C2213yes—
38#13133195C2216yes—
39#13133194C2216yes—
40#13133193C2213yes—
41#13133192C2213yes—
42#13133192C2213yes—
43#13133191C2213yes—
44#13133190C2216yes—
45#13133190C2211yescharacteristic
46#13133190C2213yes—
47#13133190C2216yes—
48#13133189C2216yes—
49#13133188C2213yes—
50#13133188C2213yes—
51Q8×Dic53C2211yescharacteristic
52C4×Dic106C2213yes—
53#13133182C2213yes—
54C4×D212C2213yes—
55C4×C22⋊D53C2213yes—
56C4×C22⋊D53C2213yes—
57#13133177C2216yes—
58#13133177C2216yes—
59C4⋊C4×D53C2213yes—
60C42×D53C2213yes—
61C2×#41×C53C2211yescharacteristic
62C22×Q8×C53C2211yescharacteristic
63#124×C53C2213yes—
64C22⋊Q8×C53C2213yes—
65#121×C53C2213yes—
66C4×Q8×C53C2211yescharacteristic
67C4×D4×C53C2213yes—
68#41×C53—24no—
69C2×Q8×C53—24no—
70C2×Q8×C53C2313yes—
71C2×Q8×C53C2311yescharacteristic
72C2×D4×C53C2313yes—
73C2×D4×C53C2313yes—
74C22×C4×C53C2313yes—
75C22×C4×C53C2313yes—
76C2×C22⋊D53C2313yes—
77C2×C22⋊D53C2311yescharacteristic
78C2×C22⋊D53C2311yescharacteristic
79C2×C22⋊D53C2313yes—
80C2×C22⋊D53C2313yes—
81C2×C22⋊D53C2313yes—
82C2×C22⋊D53C2316yes—
83C22×Dic53C2316yes—
84#11719949—24no—
85#11719949—212no—
86Q8×D53—24no—
87Q8×D53—212no—
88Q8×D53D418yescomplemented
89#11719947—212no—
90#11719947—212no—
91#11719945—212no—
92#11719945—212no—
93#11719945—212no—
94C2×D212C2313yes—
95C2×D212C2311yescharacteristic
96C2×D212C2313yes—
97C2×D212C2313yes—
98C2×C4×D53C2311yescharacteristic
99C2×C4×D53C2316yes—
100C2×C4×D53C2313yes—
101C2×C4×D53C2313yes—
102C2×C4×D53C2313yes—
103C2×C4×D53C2313yes—
104C2×Dic106—212no—
105C2×Dic106C2313yes—
106C2×Dic106C2313yes—
107C2×Dic106C2313yes—
108C2×Dic106C2313yes—
109C4⋊C4×C53C2313yes—
110C4⋊C4×C53C2313yes—
111C22⋊C4×C53C2311yescharacteristic
112C22⋊C4×C53C2313yes—
113C22⋊C4×C53C2316yes—
114C42×C53C2313yes—
115C22⋊Dic53C2313yes—
116C22⋊Dic53C2313yes—
117#11719921C2313yes—
118#11719921C2316yes—
119#11719921C2313yes—
120#11719921C2313yes—
121#11719921C2311yescharacteristic
122C4⋊Dic53C2313yes—
123C4⋊Dic53C2313yes—
124#11719919C2316yes—
125#11719919C2313yes—
126#11719919C2316yes—
127#11719919C2311yescharacteristic
128C4×Dic53C2313yes—
129C4×Dic53C2313yes—
130C4×Dic53C2313yes—
131C23×C53C2413yes—
132C23×C53C2411yescharacteristic
133C22×D53C2413yes—
134C22×D53C2411yescharacteristic
135C22×D53C2411yescharacteristic
136Q8×C53—26no—
137Q8×C53C2×D414yes—
138D4×C53—26no—
139D4×C53—43no—
140C2×C4×C53—26no—
141C2×C4×C53—26no—
142C2×C4×C53C2411yescharacteristic
143C2×C4×C53C2413yes—
144C2×C4×C53C2413yes—
145C2×C4×C53C2411yescharacteristic
146C2×C4×C53C2413yes—
147C22⋊D53—22no—
148C22⋊D53—26no—
149C22⋊D53—26no—
150C22⋊D53—26no—
151C22⋊D53—26no—
152C22⋊D53—22no—
153C22⋊D53—46no—
154C2×Dic53—212no—
155C2×Dic53C2411yescharacteristic
156C2×Dic53C2413yes—
157C2×Dic53C2413yes—
158C2×Dic53C2413yes—
159C2×Dic53C2411yescharacteristic
160D212—22no—
161D212—26no—
162D212—26no—
163D212—43no—
164C4×D53—26no—
165C4×D53—26no—
166C4×D53—212no—
167C4×D53—22no—
168C4×D53—43no—
169C4×D53C2×D4112yes—
170Dic106—26no—
171Dic106—26no—
172Dic106—26no—
173Dic106C2×D4112yes—
174C22×C53—23no—
175C22×C53—21no—
176C22×C53—23no—
177C22×C53—21no—
178C22×C53—23no—
179C22×C53—21no—
180C22×C53C2511yescharacteristic
181C2×D53—21no—
182C2×D53—21no—
183C2×D53—23no—
184C2×D53—21no—
185C2×D53—23no—
186C2×D53—21no—
187C2×D53—21no—
188C2×D53—23no—
189C2×D53C22×D412yes—
190C4×C53—21no—
191C4×C53—23no—
192C4×C53—21no—
193C4×C53—23no—
194C4×C53C22×D416yes—
195Dic53—21no—
196Dic53—23no—
197Dic53—23no—
198Dic53C22×D416yes—
199Dic532−1+412yescomplemented
200#3497—531nomaximal, Sylow
201C2×C53—43no—
202C2×C53—41no—
203C2×C53C23×D411yescharacteristic
204C2×C53C2×2−1+411yescharacteristic
205C2×C53#58511yescharacteristic
206D53—22no—
207D53—43no—
208D53—41no—
209C2×2−1+4—531no—
210C2×2−1+4—531no—
211#550—531no—
212#550—533no—
213D4×Q8—531no—
214D4×Q8—533no—
215#548—533no—
216#548—531no—
217#548—533no—
218#548D5311yescharacteristic, complemented
219#537—533no—
220#537—533no—
221#532—533no—
222#532—533no—
223C2×C4×Q8—531no—
224C53#349711yescharacteristic, complemented, Sylow
2252−1+4—1068no—
2262−1+4—1068no—
227C2×#41—533no—
228C2×#41—531no—
229C2×#41—533no—
230C2×#41—533no—
231C2×#41—533no—
232C2×#41—533no—
233C2×#41—533no—
234C2×#41C2×D5311yescharacteristic
235C22×Q8—531no—
236C22×Q8—533no—
237C22×Q8—531no—
238C22×Q8—533no—
239C22×Q8C2×D5311yescharacteristic
240C4⋊Q8—533no—
241C4⋊Q8—533no—
242#127—533no—
243#127—533no—
244#124—533no—
245#124—533no—
246#124—533no—
247#124C2×D5313yes—
248C22⋊Q8—533no—
249C22⋊Q8—533no—
250C22⋊Q8—536no—
251C22⋊Q8—533no—
252C22⋊Q8—536no—
253C22⋊Q8C2×D5313yes—
254#121—533no—
255#121—533no—
256#121—536no—
257#121—533no—
258#121—533no—
259#121—533no—
260#121—533no—
261#121—531no—
262#121—536no—
263#121—536no—
264#121C2×D5313yes—
265C4×Q8—531no—
266C4×Q8—533no—
267C4×Q8—533no—
268C4×Q8C2×D5311yescharacteristic
269C4×D4—533no—
270C4×D4—533no—
271C4×D4—536no—
272C4×D4—536no—
273C4×D4—533no—
274C4×D4C2×D5313yes—
275C2×C4⋊C4—533no—
276C2×C42—533no—
277#41—24no—
278#41—10612no—
279#41—10612no—
280#41—10612no—
281#41—10612no—
282#41—10612no—
283#41—10612no—
284#41—1064no—
285C2×Q8—24no—
286C2×Q8—538no—
287C2×Q8—533no—
288C2×Q8—533no—
289C2×Q8—533no—
290C2×Q8—533no—
291C2×Q8—10612no—
292C2×Q8—10612no—
293C2×Q8—1064no—
294C2×Q8C22×D5313yes—
295C2×Q8C22×D5311yescharacteristic
296C2×D4—531no—
297C2×D4—533no—
298C2×D4—531no—
299C2×D4—533no—
300C2×D4—533no—
301C2×D4—533no—
302C2×D4—536no—
303C2×D4—531no—
304C2×D4—533no—
305C2×D4—533no—
306C2×D4—533no—
307C2×D4C22×D5313yes—
308C2×D4C22×D5313yes—
309C22×C4—533no—
310C22×C4—536no—
311C22×C4—531no—
312C22×C4—536no—
313C22×C4—533no—
314C22×C4—533no—
315C22×C4—533no—
316C22×C4C22×D5313yes—
317C22×C4C22×D5313yes—
318C4⋊C4—536no—
319C4⋊C4—533no—
320C4⋊C4—531no—
321C4⋊C4—533no—
322C4⋊C4—536no—
323C4⋊C4—533no—
324C4⋊C4C22×D5313yes—
325C4⋊C4C22×D5313yes—
326C22⋊C4—533no—
327C22⋊C4—533no—
328C22⋊C4—533no—
329C22⋊C4—531no—
330C22⋊C4—533no—
331C22⋊C4—533no—
332C22⋊C4—536no—
333C22⋊C4C22×D5313yes—
334C22⋊C4C22×D5316yes—
335C22⋊C4C22×D5311yescharacteristic
336C42—533no—
337C42—533no—
338C42—533no—
339C42C22×D5313yes—
340C23—531no—
341C23—531no—
342C23—533no—
343C23C23×D5311yescharacteristic
344C23C23×D5313yes—
345Q8—26no—
346Q8—5312no—
347Q8—1066no—
348Q8—1066no—
349Q8—1066no—
350Q8D4×D5314yes—
351D4—26no—
352D4—43no—
353D4—1062no—
354D4—1066no—
355D4—1062no—
356D4—1066no—
357D4—1066no—
358D4—1066no—
359D4—1062no—
360D4—1066no—
361D4—1066no—
362D4—2123no—
363D4—2126no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—531no—
367C2×C4—533no—
368C2×C4—5312no—
369C2×C4—533no—
370C2×C4—531no—
371C2×C4—533no—
372C2×C4—1062no—
373C2×C4—10612no—
374C2×C4—10612no—
375C2×C4—1066no—
376C2×C4—1066no—
377C2×C4—2123no—
378C2×C4C23×D5311yescharacteristic
379C2×C4C23×D5313yes—
380C2×C4C23×D5311yescharacteristic
381C2×C4C23×D5313yes—
382C2×C4C23×D5313yes—
383C22—21no—
384C22—23no—
385C22—21no—
386C22—23no—
387C22—21no—
388C22—23no—
389C22—532no—
390C22—1061no—
391C22—1061no—
392C22—1063no—
393C22—1061no—
394C22—1061no—
395C22—1063no—
396C22—1061no—
397C22—1063no—
398C22C24×D5311yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—536no—
404C4—532no—
405C4—1063no—
406C4—1061no—
407C4—1063no—
408C4C2×D4×D5316yes—
409C2—41no—
410C2—43no—
411C2—1062no—
412C2—2123no—
413C2—2121no—
414C2C22×D4×D5311yescharacteristic
415C22−1+4×D5311yescharacteristic
416C2#1681145911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#548C2
hypercenter#548D53
hypocenterC53#3497
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#13133265C2semidirect
2. C2×#13133264C2semidirect
3. #16811292C2semidirect
4. #16811291C2semidirect
5. Q8×C22⋊D53C2semidirect
6. #16811286C2semidirect
7. #16811285C2semidirect
8. #16811282C2semidirect
9. #16811281C2semidirect
10. #16811203C2semidirect
11. #16811202C2semidirect
12. #16811172C2semidirect
13. #16811170C2semidirect
14. C4×Q8×D53C2semidirect
15. C53×#548C2semidirect
88. Q8×D53D4semidirect
199. Dic532−1+4semidirect
218. #548D53semidirect
224. C53#3497semidirect