Finite Group Databasebeta

← #20288392

Subgroups — #20288392

Order 6784.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133265C211yesmaximal, characteristic, complemented
2C22×#11719949C211yesmaximal, characteristic, complemented
3Q8×C22⋊D53C211yesmaximal, characteristic, complemented
4#16811285C211yesmaximal, characteristic, complemented
5#16811284C211yesmaximal, characteristic, complemented
6#16811276C213yesmaximal, complemented
7#16811275C213yesmaximal, complemented
8#16811269C213yesmaximal, complemented
9#16811264C213yesmaximal, complemented
10#16811203C213yesmaximal, complemented
11#16811198C213yesmaximal, complemented
12#16811132C213yesmaximal, complemented
13C22⋊Q8×D53C213yesmaximal, complemented
14#16811105C211yesmaximal, characteristic, complemented
15D4×Q8×C53C211yesmaximal, characteristic, complemented
16#13133265—28no—
17C2×#11719949—28no—
18C2×#11719949C2213yes—
19C2×#11719949C2211yescharacteristic
20C2×#11719949C2213yes—
21C2×#11719949C2211yescharacteristic
22C2×#11719949C2213yescomplemented
23C2×#11719945C2213yescomplemented
24C2×#11719945C2213yescomplemented
25C2×Q8×D53C2211yescharacteristic, complemented
26C2×Q8×D53C2213yescomplemented
27C22×D212C2213yes—
28C22×C4×D53C2213yes—
29#13133215C2213yescomplemented
30#13133206C2213yescomplemented
31#13133205C2213yescomplemented
32#13133202C2216yescomplemented
33#13133202C2216yescomplemented
34#13133202C2213yescomplemented
35#13133199C2213yescomplemented
36#13133197C2213yescomplemented
37#13133196C2213yes—
38#13133195C2216yescomplemented
39#13133195C2213yescomplemented
40#13133194C2213yes—
41#13133193C2213yescomplemented
42#13133193C2213yescomplemented
43#13133192C2216yescomplemented
44#13133192C2213yescomplemented
45#13133191C2213yescomplemented
46#13133190C2216yescomplemented
47#13133190C2213yescomplemented
48#13133189C2211yescharacteristic
49#13133187C2213yes—
50#13133185C2213yes—
51Q8×Dic53C2211yescharacteristic, complemented
52C4×D212C2213yescomplemented
53C4×D212C2213yescomplemented
54#13133179C2213yescomplemented
55C4×C22⋊D53C2213yes—
56C4×C22⋊D53C2213yescomplemented
57#13133174C2213yescomplemented
58C4⋊C4×D53C2216yescomplemented
59C4⋊C4×D53C2213yescomplemented
60C22⋊C4×D53C2216yes—
61C22×Q8×C53C2211yescharacteristic
62C22×Q8×C53C2211yescharacteristic, complemented
63C4⋊Q8×C53C2213yescomplemented
64C22⋊Q8×C53C2213yes—
65C22⋊Q8×C53C2213yescomplemented
66C4×Q8×C53C2211yescharacteristic, complemented
67C4×D4×C53C2213yes—
68C23×D53C2313yes—
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⋊D53C2311yescharacteristic
78C2×C22⋊D53C2313yes—
79C2×C22⋊D53C2311yescharacteristic
80C2×C22⋊D53C2313yes—
81C22×Dic53C2311yescharacteristic
82#11719949—24no—
83#11719949—212no—
84#11719949—212no—
85#11719949—212no—
86#11719949D418yescomplemented
87Q8×D53—24no—
88Q8×D53—212no—
89#11719945—212no—
90#11719945—212no—
91C2×D212—212no—
92C2×D212C2313yes—
93C2×D212C2316yes—
94C2×D212C2313yes—
95C2×D212C2313yes—
96C2×D212C2313yes—
97C2×C4×D53—212no—
98C2×C4×D53C2316yes—
99C2×C4×D53C2313yes—
100C2×C4×D53C2313yes—
101C2×C4×D53C2313yes—
102C2×C4×D53C2313yes—
103C2×C4×D53C2316yes—
104C2×C4×D53C2313yes—
105C2×Dic106C2313yes—
106C2×Dic106C2313yes—
107C4⋊C4×C53C2313yes—
108C4⋊C4×C53C2313yes—
109C4⋊C4×C53C2316yes—
110C22⋊C4×C53C2313yes—
111C22⋊C4×C53C2313yes—
112C42×C53C2313yes—
113C22⋊Dic53C2313yes—
114C22⋊Dic53C2311yescharacteristic
115#11719921C2316yes—
116#11719921C2316yes—
117#11719921C2313yes—
118#11719921C2313yes—
119#11719921C2313yes—
120#11719921C2313yes—
121#11719921C2311yescharacteristic
122#11719921C2313yes—
123C4⋊Dic53C2313yes—
124C4⋊Dic53C2313yes—
125#11719919C2313yes—
126#11719919C2313yes—
127#11719919C2316yes—
128#11719919C2311yescharacteristic
129#11719919C2313yes—
130C4×Dic53C2313yes—
131C23×C53C2411yescharacteristic
132C23×C53C2411yescharacteristic
133C22×D53—26no—
134C22×D53—26no—
135C22×D53—43no—
136C22×D53C2411yescharacteristic
137C22×D53C2413yes—
138C22×D53C2413yes—
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—22no—
151C22⋊D53—26no—
152C22⋊D53—22no—
153C22⋊D53—43no—
154C2×Dic53—22no—
155C2×Dic53C2413yes—
156C2×Dic53C2411yescharacteristic
157C2×Dic53C2413yes—
158C2×Dic53C2411yescharacteristic
159C2×Dic53C2411yescharacteristic
160D212—212no—
161D212—26no—
162D212—26no—
163D212—43no—
164D212C2×D4112yes—
165C4×D53—26no—
166C4×D53—212no—
167C4×D53—26no—
168C4×D53—26no—
169C4×D53—26no—
170C4×D53—46no—
171C4×D53C2×D4112yes—
172Dic106—26no—
173Dic106—26no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C532−1+412yescomplemented
180C22×C53C2511yescharacteristic
181C2×D53—23no—
182C2×D53—23no—
183C2×D53—21no—
184C2×D53—23no—
185C2×D53—21no—
186C2×D53—23no—
187C2×D53—21no—
188C2×D53—23no—
189C2×D53—412no—
190C2×D53C22×D416yes—
191C4×C53—23no—
192C4×C53—23no—
193C4×C53—21no—
194C4×C53—23no—
195C4×C53C22×D416yes—
196Dic53—23no—
197Dic53—21no—
198Dic53—21no—
199Dic53—23no—
200Dic53C22×D412yes—
201#3489—531nomaximal, Sylow
202C2×C53—22no—
203C2×C53—41no—
204C2×C53C23×D411yescharacteristic
205C2×C53C2×2−1+411yescharacteristic
206C2×C53#58511yescharacteristic
207D53—26no—
208D53—41no—
209D53—43no—
210C2×2−1+4—531no—
211C22×#41—531no—
212D4×Q8—531no—
213D4×Q8D5311yescharacteristic, complemented
214#548—531no—
215#548—531no—
216#547—533no—
217#547—533no—
218#546—533no—
219#546—533no—
220#537—533no—
221#536—533no—
222#525—533no—
223C2×C22⋊Q8—533no—
224#520—531no—
225C53#348911yescharacteristic, complemented, Sylow
2262−1+4—1068no—
227C2×#41—531no—
228C2×#41—533no—
229C2×#41—533no—
230C2×#41—533no—
231C2×#41—531no—
232C2×#41—533no—
233C2×#41—533no—
234C2×#41—1068no—
235C22×Q8—533no—
236C22×Q8—531no—
237C22×Q8C2×D5311yescharacteristic
238C22×Q8C2×D5311yescharacteristic, complemented
239C22×D4—533no—
240C23×C4—533no—
241C4⋊Q8—533no—
242C4⋊Q8C2×D5313yescomplemented
243#124—533no—
244#124—533no—
245#123—533no—
246#123—536no—
247#123—536no—
248#123—533no—
249C22⋊Q8—533no—
250C22⋊Q8—533no—
251C22⋊Q8—533no—
252C22⋊Q8—536no—
253C22⋊Q8—533no—
254C22⋊Q8—533no—
255C22⋊Q8—533no—
256C22⋊Q8C2×D5313yes—
257C22⋊Q8C2×D5313yescomplemented
258#121—531no—
259#121—533no—
260#121—536no—
261#121—533no—
262#121—533no—
263#121—536no—
264#120—533no—
265#120—533no—
266C4×Q8—531no—
267C4×Q8C2×D5311yescharacteristic, complemented
268C4×D4—533no—
269C4×D4—533no—
270C4×D4—533no—
271C4×D4—533no—
272C4×D4—533no—
273C4×D4C2×D5313yes—
274#117—533no—
275C2×C4⋊C4—533no—
276C2×C4⋊C4—536no—
277C2×C22⋊C4—536no—
278C24—533no—
279#41—538no—
280#41—10612no—
281#41—10612no—
282#41—10612no—
283#41—10612no—
284#41—10612no—
285#41—1064no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—533no—
289C2×Q8—533no—
290C2×Q8—10612no—
291C2×Q8—1064no—
292C2×Q8C22×D5313yes—
293C2×Q8C22×D5313yes—
294C2×Q8C22×D5311yescharacteristic
295C2×D4—531no—
296C2×D4—533no—
297C2×D4—531no—
298C2×D4—533no—
299C2×D4—533no—
300C2×D4—533no—
301C2×D4—536no—
302C2×D4—533no—
303C2×D4—533no—
304C2×D4—10612no—
305C2×D4C22×D5311yescharacteristic
306C22×C4—533no—
307C22×C4—531no—
308C22×C4—536no—
309C22×C4—533no—
310C22×C4—533no—
311C22×C4—533no—
312C22×C4—533no—
313C22×C4—536no—
314C22×C4—10612no—
315C22×C4C22×D5313yes—
316C22×C4C22×D5313yes—
317C4⋊C4—533no—
318C4⋊C4—533no—
319C4⋊C4—533no—
320C4⋊C4—533no—
321C4⋊C4—536no—
322C4⋊C4—531no—
323C4⋊C4—533no—
324C4⋊C4C22×D5313yes—
325C4⋊C4C22×D5316yes—
326C4⋊C4C22×D5313yes—
327C22⋊C4—533no—
328C22⋊C4—531no—
329C22⋊C4—536no—
330C22⋊C4—536no—
331C22⋊C4—533no—
332C22⋊C4—533no—
333C22⋊C4—531no—
334C22⋊C4—533no—
335C22⋊C4—533no—
336C22⋊C4—533no—
337C22⋊C4C22×D5313yes—
338C22⋊C4C22×D5313yes—
339C42—533no—
340C42C22×D5313yes—
341C23—533no—
342C23—533no—
343C23—531no—
344C23—1066no—
345C23—1066no—
346C23—2123no—
347C23C23×D5311yescharacteristic
348C23C23×D5311yescharacteristic
349Q8—26no—
350Q8—26no—
351Q8—1066no—
352Q8—1066no—
353Q8D4×D5314yes—
354D4—22no—
355D4—5312no—
356D4—10612no—
357D4—1066no—
358D4—1062no—
359D4—1066no—
360D4—1062no—
361D4—1066no—
362D4—2123no—
363D4—2123no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—531no—
367C2×C4—5312no—
368C2×C4—531no—
369C2×C4—533no—
370C2×C4—533no—
371C2×C4—531no—
372C2×C4—1066no—
373C2×C4—1066no—
374C2×C4—10612no—
375C2×C4—1062no—
376C2×C4—1066no—
377C2×C4—1066no—
378C2×C4—2126no—
379C2×C4C23×D5311yescharacteristic
380C2×C4C23×D5313yes—
381C2×C4C23×D5313yes—
382C2×C4C23×D5313yes—
383C2×C4C23×D5313yes—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—536no—
390C22—1063no—
391C22—1063no—
392C22—1061no—
393C22—1063no—
394C22—1061no—
395C22—1063no—
396C22—1061no—
397C22—1063no—
398C22—21212no—
399C22#1313326512yescomplemented
400C22C24×D5311yescharacteristic
401C4—23no—
402C4—23no—
403C4—21no—
404C4—23no—
405C4—532no—
406C4—1061no—
407C4—1063no—
408C4—1061no—
409C4—1063no—
410C4C2×D4×D5316yes—
411C2—22no—
412C2—41no—
413C2—1066no—
414C2—2123no—
415C2—2121no—
416C2C22×D4×D5311yescharacteristic
417C2C2×#1313326511yescharacteristic
418C2#1681146011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingD4×Q8×C53C2
hypercenterD4×Q8D53
hypocenterC53#3489
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×#11719949C2semidirect
3. Q8×C22⋊D53C2semidirect
4. #16811285C2semidirect
5. #16811284C2semidirect
6. #16811276C2semidirect
7. #16811275C2semidirect
8. #16811269C2semidirect
9. #16811264C2semidirect
10. #16811203C2semidirect
11. #16811198C2semidirect
12. #16811132C2semidirect
13. C22⋊Q8×D53C2semidirect
14. #16811105C2semidirect
15. D4×Q8×C53C2semidirect
22. C2×#11719949C22semidirect
23. C2×#11719945C22semidirect
24. C2×#11719945C22semidirect
25. C2×Q8×D53C22semidirect
26. C2×Q8×D53C22semidirect
29. #13133215C22semidirect
30. #13133206C22semidirect
31. #13133205C22semidirect
32. #13133202C22semidirect
33. #13133202C22semidirect
34. #13133202C22semidirect
35. #13133199C22semidirect
36. #13133197C22semidirect
38. #13133195C22semidirect
39. #13133195C22semidirect
41. #13133193C22semidirect
42. #13133193C22semidirect
43. #13133192C22semidirect
44. #13133192C22semidirect
45. #13133191C22semidirect
46. #13133190C22semidirect
47. #13133190C22semidirect
51. Q8×Dic53C22semidirect
52. C4×D212C22semidirect
53. C4×D212C22semidirect
54. #13133179C22semidirect
56. C4×C22⋊D53C22semidirect
57. #13133174C22semidirect
58. C4⋊C4×D53C22semidirect
59. C4⋊C4×D53C22semidirect
62. C22×Q8×C53C22semidirect
63. C4⋊Q8×C53C22semidirect
65. C22⋊Q8×C53C22semidirect
66. C4×Q8×C53C22semidirect
86. #11719949D4semidirect
179. C22×C532−1+4semidirect
213. D4×Q8D53semidirect
225. C53#3489semidirect
238. C22×Q8C2×D53semidirect
242. C4⋊Q8C2×D53semidirect
257. C22⋊Q8C2×D53semidirect
267. C4×Q8C2×D53semidirect
399. C22#13133265semidirect