Finite Group Databasebeta

← #20285811

Subgroups — #20285811

Order 6784.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811244C211yesmaximal, characteristic, complemented
2#16811192C211yesmaximal, characteristic, complemented
3#16811012C211yesmaximal, characteristic, complemented
4#16811007C211yesmaximal, characteristic, complemented
5#16810983C211yesmaximal, characteristic, complemented
6#16810970C211yesmaximal, characteristic, complemented
7#16810846C211yesmaximal, characteristic, complemented
8#16810839C211yesmaximal, characteristic, complemented
9#16810824C211yesmaximal, characteristic, complemented
10#16810819C211yesmaximal, characteristic, complemented
11#16810787C211yesmaximal, characteristic, complemented
12#16810776C211yesmaximal, characteristic, complemented
13#16810770C211yesmaximal, characteristic, complemented
14#16810759C211yesmaximal, characteristic, complemented
15C53×#433C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C2×Dic212C2211yescharacteristic, complemented
19C2×#11719913C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719913C2211yescharacteristic, complemented
22C2×#11719913C2211yescharacteristic, complemented
23C2×D424C2211yescharacteristic
24C2×D424C2211yescharacteristic
25C2×D424C2211yescharacteristic, complemented
26#13133217C2211yescharacteristic, complemented
27#13133214C2211yescharacteristic
28#13133214C2211yescharacteristic
29#13133212—21no—
30#13133205C2211yescharacteristic
31#13133205C2211yescharacteristic
32#13133205C2211yescharacteristic, complemented
33#13133203—21no—
34#13133199—21no—
35#13133196C2211yescharacteristic
36#13133195—21no—
37#13133192—21no—
38#13133192—21no—
39#13133191C2211yescharacteristic
40#13133187—21no—
41#13133187—21no—
42C4×Dic106C2211yescharacteristic, complemented
43C4×D212C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133142C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic, complemented
51#13133139C2211yescharacteristic, complemented
52#117×C53C2211yescharacteristic
53C4⋊C8×C53C2211yescharacteristic, complemented
54C4⋊C8×C53C2211yescharacteristic, complemented
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57#97×C53C2211yescharacteristic, complemented
58C4×C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C22×C4×C53C2311yescharacteristic
61C2×C22⋊D53—21no—
62C2×C22⋊D53—21no—
63#11719945—41no—
64#11719945—41no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212—41no—
71C2×D212—41no—
72C2×D212C2311yescharacteristic
73C2×D212C2311yescharacteristic
74C2×D212C2311yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×Dic106—21no—
79C2×Dic106C2311yescharacteristic
80C2×Dic106C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C2×C8×C53C2311yescharacteristic
85C4⋊C4×C53D411yescharacteristic
86C4⋊C4×C53D411yescharacteristic
87C22⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C42×C53C2311yescharacteristic
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106Dic212—41no—
107D424—41no—
108D424—41no—
109D424—41no—
110#11719913—41no—
111#11719913—41no—
112#11719913—41no—
113#11719913—41no—
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—42no—
120C22×D53—81no—
121C22×D53—81no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—42no—
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136D212—21no—
137D212—21no—
138D212—21no—
139D212—21no—
140D212—21no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—42no—
145D212—81no—
146D212—81no—
147D212—81no—
148D212—81no—
149C4×D53—42no—
150C4×D53—81no—
151Dic106—21no—
152Dic106—21no—
153Dic106—21no—
154Dic106—21no—
155Dic106—81no—
156C8×C53—21no—
157C8×C53—21no—
158C8×C53—21no—
159C8×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53C22×D411yescharacteristic
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C2×D53—41no—
172C2×D53—41no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—84no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53H(5,𝔽2)11yescharacteristic
183C4×C53H(5,𝔽2)11yescharacteristic
184Dic53—41no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188#3224—531nomaximal, Sylow
189C2×C53—41no—
190C2×C53#53511yescharacteristic
191C2×C53#57611yescharacteristic
192C2×C53#57711yescharacteristic
193D53—81no—
194D53—81no—
195D53—81no—
196D53—81no—
197#542—531no—
198#535—531no—
199#496—531no—
200#495—531no—
201#489—531no—
202#486—531no—
203#464—531no—
204#463—531no—
205#460—531no—
206#459—531no—
207#452—531no—
208#450—531no—
209#449—531no—
210#447—531no—
211#433D5311yescharacteristic, complemented
212C53#322411yescharacteristic, complemented, Sylow
213C2×#41—531no—
214C22×D4—531no—
215C2×Q16—531no—
216C2×SD16—531no—
217C2×SD16—531no—
218C2×SD16—531no—
219C2×SD16—531no—
220C2×D8—531no—
221C2×D8—531no—
222C2×D8—531no—
223C4⋊Q8—531no—
224#127—531no—
225#127—531no—
226#126—1061no—
227#124—531no—
228#124—531no—
229#124—531no—
230#124—1061no—
231#123—1061no—
232C22⋊Q8—531no—
233C22⋊Q8—1061no—
234#121—531no—
235#121—1061no—
236#121—1061no—
237#120—1061no—
238#120—1061no—
239C4×Q8—531no—
240C4×D4—531no—
241#117C2×D5311yescharacteristic
242C4⋊C8C2×D5311yescharacteristic, complemented
243C4⋊C8C2×D5311yescharacteristic, complemented
244#103—531no—
245#103—531no—
246#102—531no—
247#102—531no—
248#102—531no—
249#102—531no—
250#102—531no—
251#102—531no—
252C22⋊C8C2×D5311yescharacteristic
253C22⋊C8C2×D5311yescharacteristic
254#97C2×D5311yescharacteristic, complemented
255C4×C8C2×D5311yescharacteristic, complemented
256C24—1061no—
257#41—2121no—
258#41—2121no—
259C2×Q8—531no—
260C2×Q8—531no—
261C2×Q8—1061no—
262C2×D4—531no—
263C2×D4—531no—
264C2×D4—531no—
265C2×D4—531no—
266C2×D4—1061no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—2121no—
274C2×D4—2121no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4C22×D5311yescharacteristic
278Q16—2121no—
279SD16—2121no—
280SD16—2121no—
281SD16—2121no—
282SD16—2121no—
283D8—2121no—
284D8—2121no—
285D8—2121no—
286C2×C8C22×D5311yescharacteristic
287C2×C8C22×D5311yescharacteristic
288C2×C8C22×D5311yescharacteristic
289C2×C8C22×D5311yescharacteristic
290C4⋊C4—531no—
291C4⋊C4—531no—
292C4⋊C4—1061no—
293C4⋊C4—1061no—
294C4⋊C4—1061no—
295C4⋊C4D21211yescharacteristic
296C4⋊C4D21211yescharacteristic
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C22⋊C4—1061no—
306C22⋊C4D21211yescharacteristic
307C22⋊C4D21211yescharacteristic
308C42—1061no—
309C42C22×D5311yescharacteristic
310C42C22×D5311yescharacteristic
311C23—1061no—
312C23—1061no—
313C23—1061no—
314C23—1061no—
315C23—4241no—
316C23—4241no—
317C23—2122no—
318C23C2×D21211yescharacteristic
319Q8—1061no—
320Q8—1061no—
321Q8—1061no—
322Q8—1061no—
323Q8—4241no—
324D4—1061no—
325D4—1061no—
326D4—1061no—
327D4—1061no—
328D4—1061no—
329D4—1061no—
330D4—1061no—
331D4—1061no—
332D4—4241no—
333D4—4241no—
334D4—4241no—
335D4—4241no—
336D4—4241no—
337D4—2122no—
338D4—2122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—4241no—
346C2×C4—2122no—
347C2×C4C2×D21211yescharacteristic
348C2×C4C2×D21211yescharacteristic
349C2×C4C2×D21211yescharacteristic
350C2×C4C2×D21211yescharacteristic
351C2×C4C2×D21211yescharacteristic
352C2×C4C23×D5311yescharacteristic
353C8—21no—
354C8—21no—
355C8—21no—
356C8—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—4244no—
361C22—2121no—
362C22—2121no—
363C22—2121no—
364C22—2121no—
365C22—2121no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22—2121no—
373C22C22×D21211yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—2121no—
380C4—2121no—
381C4—2121no—
382C4—2121no—
383C4#1313326311yescharacteristic
384C4#1313326311yescharacteristic
385C2—41no—
386C2—4241no—
387C2—4241no—
388C2—4241no—
389C2—4241no—
390C2#1681119211yescharacteristic
391C2#1681142811yescharacteristic
392C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#433C2
socleC22×C53C22×D4
hypercenter#433D53
hypocenterC53#3224
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #16811244C2semidirect
2. #16811192C2semidirect
3. #16811012C2semidirect
4. #16811007C2semidirect
5. #16810983C2semidirect
6. #16810970C2semidirect
7. #16810846C2semidirect
8. #16810839C2semidirect
9. #16810824C2semidirect
10. #16810819C2semidirect
11. #16810787C2semidirect
12. #16810776C2semidirect
13. #16810770C2semidirect
14. #16810759C2semidirect
15. C53×#433C2semidirect
18. C2×Dic212C22semidirect
21. C2×#11719913C22semidirect
22. C2×#11719913C22semidirect
25. C2×D424C22semidirect
26. #13133217C22semidirect
32. #13133205C22semidirect
42. C4×Dic106C22semidirect
43. C4×D212C22semidirect
44. #13133142C22semidirect
45. #13133142C22semidirect
50. #13133139C22semidirect
51. #13133139C22semidirect
53. C4⋊C8×C53C22semidirect
54. C4⋊C8×C53C22semidirect
57. #97×C53C22semidirect
58. C4×C8×C53C22semidirect
211. #433D53semidirect
212. C53#3224semidirect
242. C4⋊C8C2×D53semidirect
243. C4⋊C8C2×D53semidirect
254. #97C2×D53semidirect
255. C4×C8C2×D53semidirect