Finite Group Databasebeta

← #20286685

Subgroups — #20286685

Order 6784.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811278C211yesmaximal, characteristic, complemented
2#16810993C211yesmaximal, characteristic, complemented
3#16810964C211yesmaximal, characteristic, complemented
4#16810953C211yesmaximal, characteristic, complemented
5#16810920C211yesmaximal, characteristic, complemented
6#16810867C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810828C211yesmaximal, characteristic, complemented
9#16810766C211yesmaximal, characteristic, complemented
10#16810766C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810622C211yesmaximal, characteristic, complemented
14#16810617C211yesmaximal, characteristic, complemented
15C53×#553C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23#13133217C2211yescharacteristic
24#13133199—21no—
25#13133199—21no—
26#13133196C2211yescharacteristic
27#13133196C2211yescharacteristic
28#13133188—21no—
29C4×D212C2211yescharacteristic
30C4⋊C4×D53—21no—
31#13133150C2211yescharacteristic
32#13133148C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133140C2211yescharacteristic
36#13133140C2211yescharacteristic
37#13133140C2211yescharacteristic
38#13133140C2211yescharacteristic
39#13133140C2211yescharacteristic
40#13133140C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48#126×C53—21no—
49#125×C53C2211yescharacteristic
50#123×C53—21no—
51#121×C53—21no—
52#121×C53C2211yescharacteristic
53#121×C53C2211yescharacteristic
54C4×D4×C53—21no—
55C4×D4×C53C2211yescharacteristic
56C4×D4×C53C2211yescharacteristic
57#117×C53C2211yescharacteristic
58C2×C4⋊C4×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53D411yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69#11719945—22no—
70#11719945—22no—
71#11719945—41no—
72#11719945—41no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79C4⋊C4×C53—21no—
80C4⋊C4×C53—21no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53D411yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53—21no—
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C22⋊C4×C53D411yescharacteristic
91C22⋊C4×C53D411yescharacteristic
92C42×C53—21no—
93C42×C53C2311yescharacteristic
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96#11719923—41no—
97#11719923—41no—
98#11719922—22no—
99#11719921—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—41no—
118D4×C53—42no—
119D4×C53—81no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—42no—
135C22⋊D53—42no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139D212—21no—
140D212#4112yes—
141C4×D53—42no—
142C4×D53—42no—
143C4×D53—42no—
144Dic106—21no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148#88322—21no—
149#88322—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—22no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177#3307—531nomaximal, Sylow
178C2×C53—41no—
179C2×C53—41no—
180C2×C53—81no—
181C2×C53#54611yescharacteristic
182C2×C53#57511yescharacteristic
183C2×C53#57711yescharacteristic
184D53—42no—
185#553D5311yescharacteristic, complemented
186#547—531no—
187#491—531no—
188#485—531no—
189#483—531no—
190#478—531no—
191#468—531no—
192#465—531no—
193#461—531no—
194#449—531no—
195#449—531no—
196#442—531no—
197#435—531no—
198#417—531no—
199#416—531no—
200C53#330711yescharacteristic, complemented, Sylow
201C2×#41—531no—
202C2×#41—531no—
203C2×SD16—531no—
204C2×SD16—531no—
205C2×D8—531no—
206C2×M16—531no—
207C22×C8—531no—
208C4⋊Q8—531no—
209#126—21no—
210#125C2×D5311yescharacteristic
211#123—21no—
212#123—1061no—
213#123—1061no—
214C22⋊Q8—531no—
215C22⋊Q8—531no—
216#121—21no—
217#121—1061no—
218#121C2×D5311yescharacteristic
219#121C2×D5311yescharacteristic
220C4×D4—21no—
221C4×D4—531no—
222C4×D4C2×D5311yescharacteristic
223C4×D4C2×D5311yescharacteristic
224#117C2×D5311yescharacteristic
225C2×C4⋊C4—1061no—
226C2×C4⋊C4C2×D5311yescharacteristic
227#107—531no—
228#106—531no—
229#106—531no—
230C4⋊C8—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241C22⋊C8—531no—
242C22⋊C8—531no—
243#97—531no—
244#41—1062no—
245#41—1062no—
246#41—2121no—
247#41—2121no—
248C2×Q8—531no—
249C2×Q8—531no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—531no—
253C2×D4—1061no—
254C2×D4—1061no—
255C2×D4C22⋊D5311yescharacteristic
256C2×D4C22×D5311yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4C22×D5311yescharacteristic
263C22×C4C22×D5311yescharacteristic
264SD16—2121no—
265SD16—2121no—
266D8—1062no—
267M16—2121no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8—531no—
271C2×C8—531no—
272C2×C8—2121no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—531no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4C22⋊D5311yescharacteristic
282C4⋊C4C22×D5311yescharacteristic
283C4⋊C4C22×D5311yescharacteristic
284C4⋊C4C22×D5311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4C22⋊D5311yescharacteristic
293C22⋊C4C22⋊D5311yescharacteristic
294C42—21no—
295C42C22×D5311yescharacteristic
296C23—21no—
297C23—1061no—
298C23C2×C22⋊D5311yescharacteristic
299C23C2×C22⋊D5311yescharacteristic
300Q8—1061no—
301Q8—1061no—
302Q8—1061no—
303Q8—1061no—
304D4—21no—
305D4—21no—
306D4—41no—
307D4—42no—
308D4—81no—
309D4—532no—
310D4—1061no—
311D4—2122no—
312D4—2122no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4C2×C22⋊D5311yescharacteristic
329C2×C4C2×C22⋊D5311yescharacteristic
330C2×C4C2×C22⋊D5311yescharacteristic
331C2×C4C2×C22⋊D5311yescharacteristic
332C2×C4C23×D5311yescharacteristic
333C8—1061no—
334C8—1061no—
335C8—1061no—
336C8—1061no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—1062no—
347C22—2121no—
348C22—2121no—
349C22C22×C22⋊D5311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2121no—
358C4—2121no—
359C4—2121no—
360C4#41×D5311yescharacteristic
361C4#1313326311yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—2122no—
366C2#1681126111yescharacteristic
367C2#1681142111yescharacteristic
368C2#1681143411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#553C2
socleC22×C53C22×D4
hypercenter#553D53
hypocenterC53#3307
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #16811278C2semidirect
2. #16810993C2semidirect
3. #16810964C2semidirect
4. #16810953C2semidirect
5. #16810920C2semidirect
6. #16810867C2semidirect
7. #16810850C2semidirect
8. #16810828C2semidirect
9. #16810766C2semidirect
10. #16810766C2semidirect
11. #16810733C2semidirect
12. #16810695C2semidirect
13. #16810622C2semidirect
14. #16810617C2semidirect
15. C53×#553C2semidirect
185. #553D53semidirect
200. C53#3307semidirect