Finite Group Databasebeta

← #20285569

Subgroups — #20285569

Order 6784.

422 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
1#16811201C211yesmaximal, characteristic, complemented
2#16811187C211yesmaximal, characteristic, complemented
3#16810965C211yesmaximal, characteristic, complemented
4#16810948C211yesmaximal, characteristic, complemented
5#16810942C211yesmaximal, characteristic, complemented
6#16810937C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810869C211yesmaximal, characteristic, complemented
9#16810859C211yesmaximal, characteristic, complemented
10#16810852C211yesmaximal, characteristic, complemented
11#16810776C211yesmaximal, characteristic, complemented
12#16810770C211yesmaximal, characteristic, complemented
13#16810770C211yesmaximal, characteristic, complemented
14#16810759C211yesmaximal, characteristic, complemented
15C53×#408C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic, complemented
17C22×D212C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×D424C2211yescharacteristic
21C2×D424C2211yescharacteristic
22#13133213—21no—
23#13133206—21no—
24#13133196C2211yescharacteristic
25#13133196C2211yescharacteristic, complemented
26#13133193—21no—
27#13133191C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic
30#13133191C2211yescharacteristic
31#13133191C2211yescharacteristic
32#13133191C2211yescharacteristic, complemented
33#13133189—21no—
34#13133189—21no—
35#13133188—21no—
36#13133185—21no—
37#13133185—21no—
38#13133176C2211yescharacteristic
39C2×C4⋊Dic53C2211yescharacteristic, complemented
40#13133151C2211yescharacteristic, complemented
41#13133151C2211yescharacteristic, complemented
42#13133149C2211yescharacteristic, complemented
43#13133149C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133142C2211yescharacteristic, complemented
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic, complemented
51#13133139C2211yescharacteristic, complemented
52C2×#41×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic, complemented
54C22×C8×C53C2211yescharacteristic, complemented
55C22⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60#41×C53—41no—
61#41×C53—41no—
62C2×Q8×C53D411yescharacteristic
63C2×D4×C53D411yescharacteristic
64C2×D4×C53D411yescharacteristic
65C2×D4×C53D411yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C2×C22⋊D53—21no—
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C22×Dic53—21no—
77#11719945—41no—
78#11719945—41no—
79C2×D212—21no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212C2311yescharacteristic
83C2×D212C2311yescharacteristic
84C2×D212C2311yescharacteristic
85C2×C4×D53—21no—
86C2×Dic106C2311yescharacteristic
87M16×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C2×C8×C53C2311yescharacteristic
92C2×C8×C53C2311yescharacteristic
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102C4⋊Dic53—21no—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110D424—41no—
111D424—41no—
112#11719913—41no—
113#11719913—41no—
114C23×C53C2×D411yescharacteristic
115C23×C53C2×D411yescharacteristic
116C23×C53C2×D411yescharacteristic
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—42no—
121C22×D53—81no—
122C22×D53—81no—
123Q8×C53—41no—
124D4×C53—41no—
125D4×C53—41no—
126D4×C53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2×D411yescharacteristic
135C2×C4×C53C2×D411yescharacteristic
136C2×C4×C53C2411yescharacteristic
137C22⋊D53—42no—
138C22⋊D53—81no—
139C22⋊D53—81no—
140C22⋊D53—81no—
141C22⋊D53—81no—
142C22⋊D53—81no—
143C22⋊D53—81no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—81no—
150D212—21no—
151D212—21no—
152D212—21no—
153D212—21no—
154D212—21no—
155D212—21no—
156D212—42no—
157C4×D53—42no—
158Dic106—21no—
159Dic106—21no—
160C8×C53—21no—
161C8×C53—21no—
162C8×C53—21no—
163C8×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53C22×D411yescharacteristic
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—41no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53H(5,𝔽2)11yescharacteristic
188C4×C53H(5,𝔽2)11yescharacteristic
189Dic53—41no—
190Dic53—41no—
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194#3198—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—41no—
197C2×C53—41no—
198C2×C53#53411yescharacteristic
199C2×C53#57611yescharacteristic
200C2×C53#57711yescharacteristic
201D53—81no—
202D53—81no—
203D53—81no—
204#537—531no—
205#535—531no—
206#485—531no—
207#482—531no—
208#481—531no—
209#480—531no—
210#469—531no—
211#468—531no—
212#466—531no—
213#465—531no—
214#450—531no—
215#449—531no—
216#449—531no—
217#447—531no—
218#408D5311yescharacteristic, complemented
219C53#319811yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C2×#41C2×D5311yescharacteristic
222C22×D4—531no—
223C2×SD16—531no—
224C2×SD16—531no—
225C2×D8—531no—
226C2×D8—531no—
227C2×M16C2×D5311yescharacteristic, complemented
228C22×C8C2×D5311yescharacteristic, complemented
229#127—1061no—
230#124—1061no—
231C22⋊Q8—531no—
232C22⋊Q8—531no—
233C22⋊Q8—1061no—
234#121—531no—
235#121—531no—
236#121—531no—
237#121—531no—
238#121—531no—
239#121—531no—
240#121—1061no—
241#121—1061no—
242#121—1061no—
243#120—1061no—
244#120—1061no—
245#117—531no—
246C2×C4⋊C4—531no—
247#107—531no—
248#107—531no—
249#106—531no—
250#106—531no—
251#103—531no—
252#103—531no—
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102—531no—
259C22⋊C8C2×D5311yescharacteristic
260C22⋊C8C2×D5311yescharacteristic
261C22⋊C8C2×D5311yescharacteristic, complemented
262C22⋊C8C2×D5311yescharacteristic, complemented
263C24—1061no—
264#41—41no—
265#41—41no—
266#41—2121no—
267#41—2121no—
268C2×Q8—531no—
269C2×Q8D21211yescharacteristic
270C2×D4—531no—
271C2×D4—531no—
272C2×D4—531no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—2121no—
282C2×D4—2121no—
283C2×D4D21211yescharacteristic
284C2×D4D21211yescharacteristic
285C2×D4D21211yescharacteristic
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4C22×D5311yescharacteristic
289C22×C4C22×D5311yescharacteristic
290C22×C4C22×D5311yescharacteristic
291SD16—2121no—
292SD16—2121no—
293D8—2121no—
294D8—2121no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D5311yescharacteristic
298C2×C8C22×D5311yescharacteristic
299C2×C8C22×D5311yescharacteristic
300C2×C8C22×D5311yescharacteristic
301C4⋊C4—531no—
302C4⋊C4—531no—
303C4⋊C4—531no—
304C4⋊C4—531no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C22⋊C4—1061no—
317C42—1061no—
318C23—1061no—
319C23—1061no—
320C23—1061no—
321C23—4241no—
322C23—4241no—
323C23—2122no—
324C23C2×D21211yescharacteristic
325C23C2×D21211yescharacteristic
326C23C2×D21211yescharacteristic
327Q8—41no—
328Q8—1061no—
329Q8—1061no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—1061no—
334D4—1061no—
335D4—1061no—
336D4—1061no—
337D4—1061no—
338D4—1061no—
339D4—4241no—
340D4—4241no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—4241no—
345D4—2122no—
346D4—2122no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—4241no—
359C2×C4—2122no—
360C2×C4C2×D21211yescharacteristic
361C2×C4C2×D21211yescharacteristic
362C2×C4C2×D21211yescharacteristic
363C2×C4C23×D5311yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—4244no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22—2121no—
383C22—2121no—
384C22—2121no—
385C22—2121no—
386C22—2121no—
387C22C22×D21211yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—2121no—
392C4—2121no—
393C4—2121no—
394C4—2121no—
395C4—2121no—
396C4#1313326211yescharacteristic
397C4#1313326211yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—4241no—
402C2—4241no—
403C2—4241no—
404C2#1681118511yescharacteristic
405C2#1681142811yescharacteristic
406C2#1681143511yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3198
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #16811201C2semidirect
2. #16811187C2semidirect
3. #16810965C2semidirect
4. #16810948C2semidirect
5. #16810942C2semidirect
6. #16810937C2semidirect
7. #16810876C2semidirect
8. #16810869C2semidirect
9. #16810859C2semidirect
10. #16810852C2semidirect
11. #16810776C2semidirect
12. #16810770C2semidirect
13. #16810770C2semidirect
14. #16810759C2semidirect
15. C53×#408C2semidirect
16. C2×#11719945C22semidirect
25. #13133196C22semidirect
32. #13133191C22semidirect
39. C2×C4⋊Dic53C22semidirect
40. #13133151C22semidirect
41. #13133151C22semidirect
42. #13133149C22semidirect
43. #13133149C22semidirect
44. #13133142C22semidirect
45. #13133142C22semidirect
50. #13133139C22semidirect
51. #13133139C22semidirect
53. C2×M16×C53C22semidirect
54. C22×C8×C53C22semidirect
57. C22⋊C8×C53C22semidirect
58. C22⋊C8×C53C22semidirect
218. #408D53semidirect
219. C53#3198semidirect
227. C2×M16C2×D53semidirect
228. C22×C8C2×D53semidirect
261. C22⋊C8C2×D53semidirect
262. C22⋊C8C2×D53semidirect