Finite Group Databasebeta

← #20284670

Subgroups — #20284670

Order 6784.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811189C211yesmaximal, characteristic
2#16811103C211yesmaximal, characteristic, complemented
3#16810950C211yesmaximal, characteristic, complemented
4#16810947C211yesmaximal, characteristic, complemented
5#16810945C211yesmaximal, characteristic, complemented
6#16810943C211yesmaximal, characteristic, complemented
7#16810939C211yesmaximal, characteristic, complemented
8#16810937C211yesmaximal, characteristic, complemented
9#16810935C211yesmaximal, characteristic, complemented
10#16810662C211yesmaximal, characteristic, complemented
11#16810660C211yesmaximal, characteristic, complemented
12#16810626C211yesmaximal, characteristic, complemented
13#102×D53C211yesmaximal, characteristic, complemented
14#16810588C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×D4×D53C2211yescharacteristic
18C2×#11719912C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133213—21no—
21#13133203—21no—
22#13133192C2211yescharacteristic
23#13133191C2211yescharacteristic
24#13133188C2211yescharacteristic
25#13133187—21no—
26#13133185—21no—
27C4×Dic106—21no—
28C4×D212—21no—
29#13133177—21no—
30#13133177—21no—
31#13133175C2211yescharacteristic
32#13133175C2211yescharacteristic
33#13133175C2211yescharacteristic, complemented
34C2×C4⋊Dic53C2211yescharacteristic, complemented
35C4⋊C4×D53C2211yescharacteristic
36C4⋊C4×D53C2211yescharacteristic, complemented
37#13133151C2211yescharacteristic, complemented
38#13133151C2211yescharacteristic, complemented
39#13133150C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133148C2211yescharacteristic, complemented
42#13133148C2211yescharacteristic, complemented
43#13133139C2211yescharacteristic
44#13133139C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic, complemented
50#13133129C2211yescharacteristic, complemented
51#13133129C2211yescharacteristic, complemented
52#121×C53C2211yescharacteristic
53#117×C53C2211yescharacteristic, complemented
54#107×C53C2211yescharacteristic, complemented
55#106×C53C2211yescharacteristic, complemented
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53D411yescharacteristic
67C2×C22⋊D53D411yescharacteristic
68C22×Dic53—21no—
69D4×D53—22no—
70D4×D53—22no—
71#11719945—22no—
72#11719945—22no—
73C2×D212—21no—
74C2×D212D411yescharacteristic
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×C4×D53C2311yescharacteristic
80C2×Dic106D411yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106#11719912—22no—
107C8×D53—22no—
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—21no—
111C22×D53—42no—
112C22×D53—81no—
113C22×D53—81no—
114C22×D53C2×D411yescharacteristic
115C22×D53C2×D411yescharacteristic
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—81no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—22no—
127C22⋊D53—22no—
128C22⋊D53—42no—
129C22⋊D53—81no—
130C22⋊D53—81no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—81no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137D212—22no—
138D212—21no—
139D212—21no—
140D212—81no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—81no—
144C4×D53—81no—
145C4×D53#4112yes—
146C4×D53#4112yes—
147Dic106—22no—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—84no—
168C2×D53#12312yes—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53C2×#4111yescharacteristic
174C4×C53C2×#4111yescharacteristic
175Dic53—21no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179Dic53#12312yes—
180#3096—531nomaximal, Sylow
181C2×C53—41no—
182C2×C53—81no—
183C2×C53C2×#12311yescharacteristic
184C2×C53#57611yescharacteristic
185C2×C53#57711yescharacteristic
186D53—22no—
187D53—41no—
188D53—81no—
189#535—531no—
190#520—531no—
191#482—531no—
192#482—531no—
193#482—531no—
194#482D5311yescharacteristic, complemented
195#481—531no—
196#481—531no—
197#480—531no—
198#480—531no—
199#428—531no—
200#427—531no—
201#418—531no—
202C2×#102—531no—
203#408—531no—
204C53#309611yescharacteristic, complemented, Sylow
205C2×#41—531no—
206C22×D4—531no—
207C2×M16—531no—
208C22×C8—531no—
209#127—1061no—
210#124—1061no—
211#121—531no—
212#121—531no—
213#121—531no—
214#121C2×D5311yescharacteristic
215#120—1061no—
216#120—1061no—
217C4×Q8—1061no—
218C4×D4—1061no—
219C4×D4—1061no—
220C4×D4—1061no—
221#117—531no—
222#117—531no—
223#117—531no—
224#117C2×D5311yescharacteristic, complemented
225C2×C4⋊C4—531no—
226C2×C4⋊C4—531no—
227C2×C4⋊C4—531no—
228#107—531no—
229#107—531no—
230#107—531no—
231#107C2×D5311yescharacteristic, complemented
232#106—531no—
233#106—531no—
234#106—531no—
235#106C2×D5311yescharacteristic, complemented
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242#102C2×D5311yescharacteristic
243#102C2×D5311yescharacteristic
244C22⋊C8—531no—
245C22⋊C8—531no—
246C22⋊C8—531no—
247C22⋊C8C2×D5311yescharacteristic, complemented
248C24—1061no—
249#41—1062no—
250#41—1062no—
251C2×Q8—531no—
252C2×D4—21no—
253C2×D4—531no—
254C2×D4—531no—
255C2×D4—531no—
256C2×D4—531no—
257C2×D4—1061no—
258C2×D4—1062no—
259C2×D4—1062no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4C22×D5311yescharacteristic
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4C22×D5311yescharacteristic
270M16—1062no—
271C2×C8—531no—
272C2×C8—531no—
273C2×C8—1062no—
274C2×C8C22×D5311yescharacteristic
275C2×C8C22×D5311yescharacteristic
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4C22×D5311yescharacteristic
283C4⋊C4C22×D5311yescharacteristic
284C4⋊C4C22×D5311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—21no—
294C42—1061no—
295C42—1061no—
296C42—1061no—
297C23—21no—
298C23—531no—
299C23—531no—
300C23—1061no—
301C23—4241no—
302C23—4241no—
303C23—2122no—
304C23D4×D5311yescharacteristic
305Q8—1062no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—1062no—
310D4—1062no—
311D4—1062no—
312D4—1061no—
313D4—1061no—
314D4—4241no—
315D4—4241no—
316D4—4241no—
317D4—2122no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—531no—
324C2×C4—532no—
325C2×C4—532no—
326C2×C4—531no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4D4×D5311yescharacteristic
336C2×C4C23×D5311yescharacteristic
337C8—21no—
338C8—21no—
339C8—1061no—
340C8—1061no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—532no—
348C22—1061no—
349C22—1061no—
350C22—1061no—
351C22—1061no—
352C22—1061no—
353C22—4244no—
354C22—2121no—
355C22—2121no—
356C22—2121no—
357C22C2×D4×D5311yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—532no—
363C4—1061no—
364C4—2121no—
365C4—2121no—
366C4—2121no—
367C4#41×D5311yescharacteristic
368C4#41×D5311yescharacteristic
369C2—41no—
370C2—81no—
371C2—1062no—
372C2—4241no—
373C2—2121no—
374C2#123×D5311yescharacteristic
375C2#1681142511yescharacteristic
376C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3096
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #16811103C2semidirect
3. #16810950C2semidirect
4. #16810947C2semidirect
5. #16810945C2semidirect
6. #16810943C2semidirect
7. #16810939C2semidirect
8. #16810937C2semidirect
9. #16810935C2semidirect
10. #16810662C2semidirect
11. #16810660C2semidirect
12. #16810626C2semidirect
13. #102×D53C2semidirect
14. #16810588C2semidirect
15. C53×#482C2semidirect
33. #13133175C22semidirect
34. C2×C4⋊Dic53C22semidirect
36. C4⋊C4×D53C22semidirect
37. #13133151C22semidirect
38. #13133151C22semidirect
39. #13133150C22semidirect
40. #13133148C22semidirect
41. #13133148C22semidirect
42. #13133148C22semidirect
49. C22⋊#88322C22semidirect
50. #13133129C22semidirect
51. #13133129C22semidirect
53. #117×C53C22semidirect
54. #107×C53C22semidirect
55. #106×C53C22semidirect
58. C22⋊C8×C53C22semidirect
194. #482D53semidirect
204. C53#3096semidirect
224. #117C2×D53semidirect
231. #107C2×D53semidirect
235. #106C2×D53semidirect
247. C22⋊C8C2×D53semidirect