Finite Group Databasebeta

← #20285564

Subgroups — #20285564

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#16811189C211yesmaximal, characteristic, complemented
3#16810963C211yesmaximal, characteristic, complemented
4#16810945C211yesmaximal, characteristic, complemented
5#16810939C211yesmaximal, characteristic, complemented
6#16810937C211yesmaximal, characteristic, complemented
7#16810875C211yesmaximal, characteristic, complemented
8#16810866C211yesmaximal, characteristic, complemented
9#16810856C211yesmaximal, characteristic, complemented
10#16810855C211yesmaximal, characteristic, complemented
11#16810772C211yesmaximal, characteristic, complemented
12#16810768C211yesmaximal, characteristic, complemented
13#16810755C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#449C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719912C2211yescharacteristic, complemented
21C2×C8×D53C2211yescharacteristic, complemented
22#13133213—21no—
23#13133203—21no—
24#13133196—21no—
25#13133193C2211yescharacteristic
26#13133193C2211yescharacteristic, complemented
27#13133192C2211yescharacteristic
28#13133191—21no—
29#13133191C2211yescharacteristic
30#13133189—21no—
31#13133189—21no—
32#13133188C2211yescharacteristic
33#13133188C2211yescharacteristic
34#13133188C2211yescharacteristic, complemented
35#13133187—21no—
36#13133185—21no—
37#13133175C2211yescharacteristic
38C2×C4⋊Dic53C2211yescharacteristic, complemented
39#13133151C2211yescharacteristic, complemented
40#13133151C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133139C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic
51#13133129C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic, complemented
53C2×SD16×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60#41×C53—41no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53—21no—
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⋊D53D411yescharacteristic
75C2×C22⋊D53D411yescharacteristic
76C22×Dic53—21no—
77D4×D53—41no—
78D4×D53—41no—
79#11719945—41no—
80#11719945—41no—
81C2×D212—21no—
82C2×D212D411yescharacteristic
83C2×D212C2311yescharacteristic
84C2×C4×D53C2311yescharacteristic
85C2×C4×D53C2311yescharacteristic
86C2×Dic106D411yescharacteristic
87SD16×C53—41no—
88D8×C53—41no—
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊Dic53—21no—
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96#11719924—41no—
97#11719922—41no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719921—21no—
102#11719921—21no—
103C4⋊Dic53—21no—
104C4⋊Dic53C2311yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2311yescharacteristic
107#11719919—21no—
108#11719919—21no—
109C4×Dic53—21no—
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112#11719912—41no—
113C8×D53—41no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53C2×D411yescharacteristic
117C22×D53—21no—
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53C2×D411yescharacteristic
122C22×D53C2×D411yescharacteristic
123Q8×C53—21no—
124Q8×C53—21no—
125D4×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—21no—
129D4×C53—42no—
130D4×C53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—42no—
136C2×C4×C53C2×D411yescharacteristic
137C2×C4×C53C2411yescharacteristic
138C22⋊D53—41no—
139C22⋊D53—41no—
140C22⋊D53—42no—
141C22⋊D53—81no—
142C22⋊D53—81no—
143C22⋊D53—81no—
144C22⋊D53—81no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—81no—
149C2×Dic53C2×D411yescharacteristic
150C2×Dic53C2×D411yescharacteristic
151D212—21no—
152D212—21no—
153D212—41no—
154D212—81no—
155C4×D53—21no—
156C4×D53—21no—
157C4×D53—21no—
158C4×D53—21no—
159Dic106—41no—
160C8×C53—21no—
161C8×C53—21no—
162#88322—21no—
163#88322—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
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—21no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C4×C53—21no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53H(5,𝔽2)11yescharacteristic
188C4×C53H(5,𝔽2)11yescharacteristic
189Dic53—21no—
190Dic53—21no—
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194#3198—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—81no—
197C2×C53—81no—
198C2×C53#53411yescharacteristic
199C2×C53#57611yescharacteristic
200C2×C53#57711yescharacteristic
201D53—41no—
202D53—41no—
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#449D5311yescharacteristic, complemented
217#447—531no—
218#408—531no—
219C53#319811yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C2×#41C2×D5311yescharacteristic, complemented
222C22×D4—531no—
223C2×SD16—531no—
224C2×SD16C2×D5311yescharacteristic
225C2×D8—531no—
226C2×D8C2×D5311yescharacteristic
227C2×M16—531no—
228C22×C8—531no—
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—1061no—
240#121—1061no—
241#121—1061no—
242#121C2×D5311yescharacteristic
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#103C2×D5311yescharacteristic, complemented
253#102—531no—
254#102—531no—
255#102—531no—
256#102—531no—
257#102—531no—
258#102C2×D5311yescharacteristic, complemented
259C22⋊C8—531no—
260C22⋊C8—531no—
261C22⋊C8—531no—
262C22⋊C8C2×D5311yescharacteristic, complemented
263C24—1061no—
264#41—41no—
265#41—41no—
266#41—2121no—
267#41—2121no—
268C2×Q8—531no—
269C2×Q8C22×D5311yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—531no—
273C2×D4—531no—
274C2×D4—531no—
275C2×D4—531no—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—2121no—
283C2×D4—2121no—
284C2×D4C22×D5311yescharacteristic
285C2×D4C22×D5311yescharacteristic
286C22×C4—21no—
287C22×C4—531no—
288C22×C4—531no—
289C22×C4—1061no—
290C22×C4C22×D5311yescharacteristic
291SD16—41no—
292SD16—2121no—
293D8—41no—
294D8—2121no—
295M16—2121no—
296C2×C8—531no—
297C2×C8—531no—
298C2×C8—2121no—
299C2×C8C22×D5311yescharacteristic
300C2×C8C22×D5311yescharacteristic
301C4⋊C4—531no—
302C4⋊C4—531no—
303C4⋊C4—531no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4C22×D5311yescharacteristic
308C22⋊C4—21no—
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—21no—
319C23—21no—
320C23—531no—
321C23—531no—
322C23—1061no—
323C23—4241no—
324C23—4241no—
325C23—2122no—
326C23D4×D5311yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—2121no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—81no—
336D4—1061no—
337D4—1061no—
338D4—4241no—
339D4—4241no—
340D4—4241no—
341D4—4241no—
342D4—4241no—
343D4—2121no—
344D4—2121no—
345D4—2121no—
346D4—2122no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—4241no—
362C2×C4D4×D5311yescharacteristic
363C2×C4C23×D5311yescharacteristic
364C8—21no—
365C8—21no—
366C8—1061no—
367C8—1061no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—1061no—
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—1061no—
392C4—1061no—
393C4—2121no—
394C4—2121no—
395C4—2121no—
396C4#1313326211yescharacteristic
397C4#1313326211yescharacteristic
398C2—41no—
399C2—81no—
400C2—81no—
401C2—4241no—
402C2—2121no—
403C2—2121no—
404C2#1681118311yescharacteristic
405C2#1681142511yescharacteristic
406C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#449C2
socleC22×C53C22×D4
hypercenter#449D53
hypocenterC53#3198
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #16811201C2semidirect
2. #16811189C2semidirect
3. #16810963C2semidirect
4. #16810945C2semidirect
5. #16810939C2semidirect
6. #16810937C2semidirect
7. #16810875C2semidirect
8. #16810866C2semidirect
9. #16810856C2semidirect
10. #16810855C2semidirect
11. #16810772C2semidirect
12. #16810768C2semidirect
13. #16810755C2semidirect
14. #16810588C2semidirect
15. C53×#449C2semidirect
20. C2×#11719912C22semidirect
21. C2×C8×D53C22semidirect
26. #13133193C22semidirect
34. #13133188C22semidirect
38. C2×C4⋊Dic53C22semidirect
39. #13133151C22semidirect
40. #13133151C22semidirect
41. #13133148C22semidirect
42. #13133148C22semidirect
43. #13133141C22semidirect
48. #13133138C22semidirect
49. C22⋊#88322C22semidirect
52. C2×#41×C53C22semidirect
56. #103×C53C22semidirect
57. #102×C53C22semidirect
58. C22⋊C8×C53C22semidirect
216. #449D53semidirect
219. C53#3198semidirect
221. C2×#41C2×D53semidirect
252. #103C2×D53semidirect
258. #102C2×D53semidirect
262. C22⋊C8C2×D53semidirect