Finite Group Databasebeta

← #20286792

Subgroups — #20286792

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#16811328C211yesmaximal, characteristic, complemented
2#16811267C211yesmaximal, characteristic
3#16810981C211yesmaximal, characteristic, complemented
4#16810958C211yesmaximal, characteristic, complemented
5#16810935C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810868C211yesmaximal, characteristic, complemented
8#16810864C211yesmaximal, characteristic, complemented
9#16810844C211yesmaximal, characteristic, complemented
10#16810786C211yesmaximal, characteristic, complemented
11#16810774C211yesmaximal, characteristic, complemented
12#16810720C211yesmaximal, characteristic, complemented
13#16810694C211yesmaximal, characteristic, complemented
14#16810621C211yesmaximal, characteristic, complemented
15C2×#102×C53C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Dic212C2211yescharacteristic, complemented
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719912C2211yescharacteristic, complemented
21#13133216C2211yescharacteristic, complemented
22#13133207—21no—
23#13133204C2211yescharacteristic
24#13133198—21no—
25#13133197C2211yescharacteristic, complemented
26#13133196—21no—
27#13133196C2211yescharacteristic
28#13133196C2211yescharacteristic
29#13133195—21no—
30#13133189—21no—
31#13133188C2211yescharacteristic
32#13133186—21no—
33#13133182C2211yescharacteristic, complemented
34D4×Dic53C2211yescharacteristic
35C4×C22⋊D53—21no—
36C4×C22⋊D53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic, complemented
38C2×C22⋊Dic53—21no—
39#13133151C2211yescharacteristic, complemented
40#13133150C2211yescharacteristic, complemented
41#13133150C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133140C2211yescharacteristic, complemented
46#13133140C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic, complemented
51#13133127C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C2×C4⋊C4×C53C2211yescharacteristic
55#102×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C24×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53D411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72#11719947—22no—
73#11719947—41no—
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79C2×C8×C53—41no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53D411yescharacteristic
92#11719923—22no—
93#11719923—41no—
94#11719921—21no—
95#11719921D411yescharacteristic
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919D411yescharacteristic
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109Dic212—41no—
110#11719912—41no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—42no—
114C23×C53—42no—
115C23×C53—81no—
116C23×C53C2×D411yescharacteristic
117C22×D53C2×D411yescharacteristic
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—42no—
122D4×C53#4112yes—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—81no—
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2411yescharacteristic
130C22⋊D53—22no—
131C22⋊D53—42no—
132C22⋊D53—81no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—42no—
138C2×Dic53—81no—
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—81no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148Dic106—21no—
149Dic106—81no—
150C8×C53—21no—
151C8×C53—21no—
152#88322—21no—
153#88322—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—22no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—84no—
163C22×C53#13712yes—
164C22×C53C22×D411yescharacteristic
165C2×D53—21no—
166C2×D53—21no—
167C2×D53—21no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53C2×#4111yescharacteristic
172C4×C53H(5,𝔽2)11yescharacteristic
173Dic53—21no—
174Dic53—21no—
175Dic53—21no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3314—531nomaximal, Sylow
181C2×C53—22no—
182C2×C53—42no—
183C2×C53—81no—
184C2×C53#54611yescharacteristic
185C2×C53C2×#13711yescharacteristic
186C2×C53#57611yescharacteristic
187D53—41no—
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#452—531no—
198#450—531no—
199#440—531no—
200#435—531no—
201#417—531no—
202C2×#102D5311yescharacteristic, complemented
203C53#331411yescharacteristic, complemented, Sylow
204C2×#41—531no—
205C22×D4C2×D5311yescharacteristic
206C2×Q16—531no—
207C2×SD16—531no—
208C2×SD16—531no—
209C2×M16—531no—
210C22×C8C2×D5311yescharacteristic
211C4⋊Q8—531no—
212#125—1061no—
213#124—531no—
214#123—1061no—
215C22⋊Q8—531no—
216C22⋊Q8—531no—
217C22⋊Q8—531no—
218C22⋊Q8—1061no—
219C22⋊Q8—1061no—
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⋊C4C2×D5311yescharacteristic
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#103—531no—
238#102—531no—
239#102—531no—
240#102C2×D5311yescharacteristic
241#102C2×D5311yescharacteristic
242#102C2×D5311yescharacteristic
243#102C2×D5311yescharacteristic
244C22⋊C8—531no—
245C22⋊C8—531no—
246#97—531no—
247C24—21no—
248#41—1062no—
249#41—2121no—
250C2×Q8—531no—
251C2×Q8—531no—
252C2×Q8—1061no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—531no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4C22×D5311yescharacteristic
260C2×D4C22×D5311yescharacteristic
261C22×C4—21no—
262C22×C4—531no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4C22×D5311yescharacteristic
267Q16—2121no—
268SD16—1062no—
269SD16—2121no—
270M16—2121no—
271C2×C8—41no—
272C2×C8—531no—
273C2×C8—531no—
274C2×C8C22×D5311yescharacteristic
275C2×C8C22×D5311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4C22×D5311yescharacteristic
287C4⋊C4C22×D5311yescharacteristic
288C22⋊C4—531no—
289C22⋊C4—531no—
290C22⋊C4—1061no—
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—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—531no—
305C23D4×D5311yescharacteristic
306Q8—1061no—
307Q8—1061no—
308Q8—1061no—
309Q8—1061no—
310Q8—4241no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—1062no—
316D4—4241no—
317D4—2122no—
318D4#1171994512yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—2122no—
336C2×C4D4×D5311yescharacteristic
337C2×C4C23×D5311yescharacteristic
338C8—21no—
339C8—21no—
340C8—1061no—
341C8—1061no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—1061no—
352C22—1061no—
353C22—1061no—
354C22#1313324612yes—
355C22C2×D4×D5311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—1061no—
360C4—1061no—
361C4—1061no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4C2×#1171994511yescharacteristic
367C4#1313326211yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—2121no—
372C2#1681126311yescharacteristic
373C2C2×#1313324611yescharacteristic
374C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#102×C53C2
socleC22×C53C22×D4
hypercenterC2×#102D53
hypocenterC53#3314
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #16811328C2semidirect
3. #16810981C2semidirect
4. #16810958C2semidirect
5. #16810935C2semidirect
6. #16810901C2semidirect
7. #16810868C2semidirect
8. #16810864C2semidirect
9. #16810844C2semidirect
10. #16810786C2semidirect
11. #16810774C2semidirect
12. #16810720C2semidirect
13. #16810694C2semidirect
14. #16810621C2semidirect
15. C2×#102×C53C2semidirect
17. C2×Dic212C22semidirect
20. C2×#11719912C22semidirect
21. #13133216C22semidirect
25. #13133197C22semidirect
33. #13133182C22semidirect
37. C4⋊C4×D53C22semidirect
39. #13133151C22semidirect
40. #13133150C22semidirect
41. #13133150C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
45. #13133140C22semidirect
46. #13133140C22semidirect
50. #13133129C22semidirect
51. #13133127C22semidirect
202. C2×#102D53semidirect
203. C53#3314semidirect