Finite Group Databasebeta

← #20284349

Subgroups — #20284349

Order 6784.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D53C211yesmaximal, characteristic, complemented
2C2×#13133223C211yesmaximal, characteristic, complemented
3#16811196C211yesmaximal, characteristic, complemented
4#16811189C211yesmaximal, characteristic
5#16810872C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810867C211yesmaximal, characteristic, complemented
9#16810853C211yesmaximal, characteristic, complemented
10#16810852C211yesmaximal, characteristic, complemented
11#16810849C211yesmaximal, characteristic, complemented
12#16810658C211yesmaximal, characteristic, complemented
13#16810630C211yesmaximal, characteristic, complemented
14#16810626C211yesmaximal, characteristic, complemented
15C53×#465C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22SD16×D53—24no—
23#13133223—24no—
24C2×#11719917C2211yescharacteristic, complemented
25C2×#11719912C2211yescharacteristic
26C2×#11719912C2211yescharacteristic, complemented
27C2×C8×D53C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic, complemented
29#13133216—21no—
30#13133213—21no—
31#13133203—21no—
32#13133203—21no—
33#13133202—21no—
34#13133199—21no—
35#13133197C2211yescharacteristic, complemented
36#13133196C2211yescharacteristic, complemented
37#13133193C2211yescharacteristic, complemented
38#13133192C2211yescharacteristic
39#13133191C2211yescharacteristic
40#13133188C2211yescharacteristic
41#13133187—21no—
42#13133185—21no—
43#13133175C2211yescharacteristic
44#13133175C2211yescharacteristic
45#13133150C2211yescharacteristic, complemented
46#13133150C2211yescharacteristic, complemented
47#13133149C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133141C2211yescharacteristic, complemented
50#13133140C2211yescharacteristic, complemented
51#13133139C2211yescharacteristic
52#13133138C2211yescharacteristic
53#13133137C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22×C8×C53C2211yescharacteristic, complemented
56C22⋊Q8×C53C2211yescharacteristic, complemented
57#121×C53C2211yescharacteristic
58#106×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⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70C2×C22⋊D53D411yescharacteristic
71Q8×D53—22no—
72Q8×D53—22no—
73D4×D53—22no—
74D4×D53—22no—
75#11719945—22no—
76#11719945—22no—
77C2×D212—21no—
78C2×D212D411yescharacteristic
79C2×D212C2311yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53C2311yescharacteristic
82C2×C4×D53C2311yescharacteristic
83C2×Dic106—21no—
84C2×Dic106D411yescharacteristic
85C2×Dic106C2311yescharacteristic
86SD16×C53—22no—
87C2×C8×C53—22no—
88C2×C8×C53C2311yescharacteristic
89C2×C8×C53C2311yescharacteristic
90C4⋊C4×C53—21no—
91C4⋊C4×C53C2311yescharacteristic
92C4⋊C4×C53C2311yescharacteristic
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719924—22no—
98#11719923—22no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103#11719921—21no—
104#11719921—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53C2311yescharacteristic
107C4⋊Dic53C2311yescharacteristic
108#11719919—21no—
109#11719919—21no—
110C4×Dic53—21no—
111C4×Dic53—21no—
112#11719917—22no—
113C2×#88322C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115#11719913—22no—
116#11719912—22no—
117#11719912—22no—
118C8×D53—22no—
119C8×D53D414yescomplemented
120C23×C53—21no—
121C23×C53C2×D411yescharacteristic
122C22×D53—21no—
123C22×D53—42no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53C2×D411yescharacteristic
127C22×D53C2×D411yescharacteristic
128Q8×C53—21no—
129Q8×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—81no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2411yescharacteristic
140C22⋊D53—22no—
141C22⋊D53—22no—
142C22⋊D53—42no—
143C22⋊D53—81no—
144C22⋊D53—81no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53C2×D411yescharacteristic
149C2×Dic53C2×D411yescharacteristic
150D212—22no—
151D212—21no—
152D212—21no—
153D212—81no—
154C4×D53—21no—
155C4×D53—21no—
156C4×D53—42no—
157C4×D53C2×D412yes—
158C4×D53#4112yes—
159Dic106—22no—
160Dic106—21no—
161Dic106—21no—
162Dic106—42no—
163C8×C53—21no—
164C8×C53C2×D412yes—
165#88322—21no—
166#88322C2×D412yes—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53C22×D411yescharacteristic
174C2×D53—21no—
175C2×D53—21no—
176C2×D53—21no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—84no—
183C2×D53#12112yes—
184C4×C53—21no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53C22×D411yescharacteristic
189C4×C53C2×#4111yescharacteristic
190Dic53—21no—
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194Dic53#12112yes—
195#3057—531nomaximal, Sylow
196C2×C53—41no—
197C2×C53—81no—
198C2×C53C2×#12111yescharacteristic
199C2×C53#57711yescharacteristic
200C2×C53#57711yescharacteristic
201D53—22no—
202D53—41no—
203D53—81no—
204C22×SD16—531no—
205C2×#131—531no—
206#536—531no—
207#535—531no—
208#468—531no—
209#468—531no—
210#468—531no—
211#468—531no—
212#465—531no—
213#465—531no—
214#465—531no—
215#465D5311yescharacteristic, complemented
216#427—531no—
217#419—531no—
218#418—531no—
219C53#305711yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C22×Q8—531no—
222C22×D4—531no—
223C2×SD16—531no—
224C2×SD16—531no—
225C2×SD16—531no—
226C2×SD16—1064no—
227C2×SD16C2×D5311yescharacteristic
228#131—1064no—
229C2×M16—531no—
230C2×M16—531no—
231C2×M16—531no—
232C22×C8—531no—
233C22×C8—531no—
234C22×C8C2×D5311yescharacteristic, complemented
235C4⋊Q8—1061no—
236#127—1061no—
237#124—1061no—
238#124—1061no—
239#123—1061no—
240#123—1061no—
241C22⋊Q8—531no—
242C22⋊Q8—531no—
243C22⋊Q8—531no—
244C22⋊Q8C2×D5311yescharacteristic, complemented
245#121—531no—
246#121—531no—
247#121—531no—
248#121C2×D5311yescharacteristic
249#120—1061no—
250#120—1061no—
251#117—531no—
252#117—531no—
253#107—531no—
254#107—531no—
255#106—531no—
256#106C2×D5311yescharacteristic, complemented
257#103—531no—
258#103—531no—
259#103—531no—
260#103C2×D5311yescharacteristic, complemented
261#102—531no—
262#102—531no—
263#102—531no—
264#102C2×D5311yescharacteristic
265C24—1061no—
266#41—1062no—
267#41—1062no—
268C2×Q8—531no—
269C2×Q8—531no—
270C2×Q8—1061no—
271C2×Q8—1062no—
272C2×Q8—1062no—
273C2×Q8C22×D5311yescharacteristic
274C2×D4—21no—
275C2×D4—531no—
276C2×D4—531no—
277C2×D4—531no—
278C2×D4—531no—
279C2×D4—1061no—
280C2×D4—1062no—
281C2×D4—1062no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4C22×D5311yescharacteristic
286C22×C4—531no—
287C22×C4—531no—
288C22×C4—1061no—
289C22×C4C22×D5311yescharacteristic
290SD16—22no—
291SD16—1062no—
292SD16—1062no—
293SD16—1062no—
294M16—1062no—
295M16—1062no—
296M16—1062no—
297C2×C8—22no—
298C2×C8—531no—
299C2×C8—534no—
300C2×C8—531no—
301C2×C8—1062no—
302C2×C8C22×D5311yescharacteristic
303C2×C8C22×D5311yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—531no—
306C4⋊C4—531no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4C22×D5311yescharacteristic
311C4⋊C4C22×D5311yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4—1061no—
322C42—1061no—
323C42—1061no—
324C23—21no—
325C23—531no—
326C23—531no—
327C23—1061no—
328C23—4241no—
329C23—4241no—
330C23—2122no—
331C23D4×D5311yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—1062no—
335Q8—1061no—
336Q8—1061no—
337Q8—2122no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—1062no—
342D4—1062no—
343D4—1062no—
344D4—1061no—
345D4—1061no—
346D4—4241no—
347D4—4241no—
348D4—4241no—
349D4—2122no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—531no—
356C2×C4—532no—
357C2×C4—532no—
358C2×C4—531no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—2122no—
365C2×C4D4×D5311yescharacteristic
366C2×C4C23×D5311yescharacteristic
367C8—21no—
368C8—532no—
369C8—1061no—
370C8D4×D5312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—532no—
378C22—1061no—
379C22—1061no—
380C22—1061no—
381C22—1061no—
382C22—1061no—
383C22—4244no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22C2×D4×D5311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—532no—
393C4—1061no—
394C4—2121no—
395C4—2121no—
396C4—2121no—
397C4C2×D4×D5311yescharacteristic
398C4#41×D5311yescharacteristic
399C2—41no—
400C2—81no—
401C2—1062no—
402C2—4241no—
403C2—2121no—
404C2#121×D5311yescharacteristic
405C2#1681142911yescharacteristic
406C2#1681143011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#465C2
socleC22×C53C22×D4
hypercenter#465D53
hypocenterC53#3057
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D53C2semidirect
2. C2×#13133223C2semidirect
3. #16811196C2semidirect
5. #16810872C2semidirect
6. #16810871C2semidirect
7. #16810868C2semidirect
8. #16810867C2semidirect
9. #16810853C2semidirect
10. #16810852C2semidirect
11. #16810849C2semidirect
12. #16810658C2semidirect
13. #16810630C2semidirect
14. #16810626C2semidirect
15. C53×#465C2semidirect
24. C2×#11719917C22semidirect
26. C2×#11719912C22semidirect
28. C2×C8×D53C22semidirect
35. #13133197C22semidirect
36. #13133196C22semidirect
37. #13133193C22semidirect
45. #13133150C22semidirect
46. #13133150C22semidirect
47. #13133149C22semidirect
48. #13133142C22semidirect
49. #13133141C22semidirect
50. #13133140C22semidirect
55. C22×C8×C53C22semidirect
56. C22⋊Q8×C53C22semidirect
58. #106×C53C22semidirect
59. #103×C53C22semidirect
119. C8×D53D4semidirect
215. #465D53semidirect
219. C53#3057semidirect
234. C22×C8C2×D53semidirect
244. C22⋊Q8C2×D53semidirect
256. #106C2×D53semidirect
260. #103C2×D53semidirect