Finite Group Databasebeta

← #20286789

Subgroups — #20286789

Order 6784.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811331C211yesmaximal, characteristic, complemented
2#16811261C211yesmaximal, characteristic
3#16810984C211yesmaximal, characteristic, complemented
4#16810957C211yesmaximal, characteristic, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810871C211yesmaximal, characteristic, complemented
8#16810861C211yesmaximal, characteristic, complemented
9#16810847C211yesmaximal, characteristic, complemented
10#16810772C211yesmaximal, characteristic, complemented
11#16810724C211yesmaximal, characteristic, complemented
12#16810697C211yesmaximal, characteristic, complemented
13#16810624C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C2×#11719924C2211yescharacteristic
18C2×#11719917C2211yescharacteristic, complemented
19C2×C8×D53C2211yescharacteristic
20#13133216C2211yescharacteristic, complemented
21#13133207—21no—
22#13133206C2211yescharacteristic
23#13133200—21no—
24#13133197—21no—
25#13133194—21no—
26#13133193C2211yescharacteristic
27#13133193C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133189—21no—
30#13133185—21no—
31Q8×Dic53C2211yescharacteristic, complemented
32D4×Dic53C2211yescharacteristic
33C4×C22⋊D53C2211yescharacteristic
34#13133177—21no—
35C2×C4⋊Dic53C2211yescharacteristic, complemented
36C4⋊C4×D53C2211yescharacteristic
37C2×#11719921—21no—
38#13133151C2211yescharacteristic, complemented
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133141C2211yescharacteristic, complemented
43#13133141C2211yescharacteristic, complemented
44#13133141C2211yescharacteristic, complemented
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic, complemented
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic, complemented
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic, complemented
56#103×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60#41×C53—22no—
61#41×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73D4×D53—22no—
74D4×D53—41no—
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×Dic106—21no—
79Q16×C53—41no—
80SD16×C53—22no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53D411yescharacteristic
88#11719924—41no—
89#11719921—21no—
90#11719921—21no—
91#11719921—21no—
92#11719921—21no—
93#11719921—21no—
94#11719921D411yescharacteristic
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919D411yescharacteristic
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107#11719917—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C8×D53—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—42no—
115C22×D53—42no—
116C22×D53—81no—
117C22×D53C2×D411yescharacteristic
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121Q8×C53—21no—
122D4×C53—21no—
123D4×C53—42no—
124D4×C53#4112yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C22⋊D53—22no—
134C22⋊D53—42no—
135C22⋊D53—81no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144D212—21no—
145D212—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—81no—
149Dic106—81no—
150C8×C53—21no—
151C8×C53—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—22no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53C22×D411yescharacteristic
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—84no—
167C2×D53#13712yes—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—21no—
175Dic53—21no—
176Dic53—21no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3314—531nomaximal, Sylow
181C2×C53—42no—
182C2×C53—41no—
183C2×C53#54611yescharacteristic
184C2×C53C2×#13711yescharacteristic
185C2×C53#57611yescharacteristic
186D53—22no—
187D53—81no—
188#554—531no—
189#546—531no—
190#489—531no—
191#484—531no—
192#480—531no—
193#474—531no—
194#468—531no—
195#467—531no—
196#464—531no—
197#452D5311yescharacteristic, complemented
198#450—531no—
199#440—531no—
200#435—531no—
201#417—531no—
202C2×#102—531no—
203C53#331411yescharacteristic, complemented, Sylow
204C2×#41C2×D5311yescharacteristic
205C22×D4—531no—
206C2×Q16C2×D5311yescharacteristic, complemented
207C2×SD16—531no—
208C2×SD16C2×D5311yescharacteristic
209C2×M16—531no—
210C22×C8—531no—
211C4⋊Q8—531no—
212#125—1061no—
213#124—531no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8—1061no—
218C22⋊Q8—1061no—
219C22⋊Q8C2×D5311yescharacteristic, complemented
220#121—531no—
221#121—1061no—
222#120—1061no—
223C4×Q8—531no—
224C4×D4—531no—
225C4×D4—531no—
226C4×D4—1061no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—531no—
229C2×C22⋊C4—1061no—
230#107—531no—
231#107—531no—
232#107—531no—
233C4⋊C8—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#103C2×D5311yescharacteristic, complemented
238#102—531no—
239#102—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102C2×D5311yescharacteristic
244C22⋊C8—531no—
245C22⋊C8C2×D5311yescharacteristic, complemented
246#97—531no—
247C24—1061no—
248#41—22no—
249#41—41no—
250C2×Q8—1061no—
251C2×Q8C22×D5311yescharacteristic
252C2×Q8C22×D5311yescharacteristic
253C2×D4—21no—
254C2×D4—531no—
255C2×D4—531no—
256C2×D4—1061no—
257C2×D4—1062no—
258C2×D4—1061no—
259C2×D4—2121no—
260C2×D4C22×D5311yescharacteristic
261C22×C4—21no—
262C22×C4—531no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4C22×D5311yescharacteristic
267Q16—41no—
268SD16—22no—
269SD16—2121no—
270M16—2121no—
271C2×C8—531no—
272C2×C8—531no—
273C2×C8—2121no—
274C2×C8C22×D5311yescharacteristic
275C2×C8C22×D5311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—531no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22×D5311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—531no—
290C22⋊C4—531no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C42—531no—
298C42—1061no—
299C23—21no—
300C23—531no—
301C23—1061no—
302C23—4241no—
303C23—2122no—
304C23—2122no—
305C23D4×D5311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—4241no—
311D4—21no—
312D4—42no—
313D4—1062no—
314D4—1061no—
315D4—1061no—
316D4—4241no—
317D4—2122no—
318D4#1171994712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4D4×D5311yescharacteristic
337C2×C4C23×D5311yescharacteristic
338C8—21no—
339C8—21no—
340C8—1061no—
341C8—1061no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—532no—
349C22—1061no—
350C22—1061no—
351C22—4244no—
352C22—2121no—
353C22—2121no—
354C22—2121no—
355C22C2×D4×D5311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—1061no—
361C4—1061no—
362C4—1061no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4C2×#1171994711yescharacteristic
367C4#1313326211yescharacteristic
368C2—41no—
369C2—42no—
370C2—1062no—
371C2—4241no—
372C2#1681126011yescharacteristic
373C2#137×D5311yescharacteristic
374C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3314
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #16811331C2semidirect
3. #16810984C2semidirect
4. #16810957C2semidirect
5. #16810937C2semidirect
6. #16810901C2semidirect
7. #16810871C2semidirect
8. #16810861C2semidirect
9. #16810847C2semidirect
10. #16810772C2semidirect
11. #16810724C2semidirect
12. #16810697C2semidirect
13. #16810624C2semidirect
14. #102×D53C2semidirect
15. C53×#452C2semidirect
18. C2×#11719917C22semidirect
20. #13133216C22semidirect
31. Q8×Dic53C22semidirect
35. C2×C4⋊Dic53C22semidirect
38. #13133151C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133146C22semidirect
42. #13133141C22semidirect
43. #13133141C22semidirect
44. #13133141C22semidirect
51. #13133127C22semidirect
53. C2×Q16×C53C22semidirect
55. C22⋊Q8×C53C22semidirect
56. #103×C53C22semidirect
58. C22⋊C8×C53C22semidirect
197. #452D53semidirect
203. C53#3314semidirect
206. C2×Q16C2×D53semidirect
219. C22⋊Q8C2×D53semidirect
237. #103C2×D53semidirect
245. C22⋊C8C2×D53semidirect