Finite Group Databasebeta

← #20285774

Subgroups — #20285774

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811226C211yesmaximal, characteristic, complemented
2#16811194C211yesmaximal, characteristic, complemented
3#16811011C211yesmaximal, characteristic, complemented
4#16810994C211yesmaximal, characteristic, complemented
5#16810990C211yesmaximal, characteristic, complemented
6#16810976C211yesmaximal, characteristic, complemented
7#16810837C211yesmaximal, characteristic, complemented
8#16810836C211yesmaximal, characteristic, complemented
9#16810827C211yesmaximal, characteristic, complemented
10#16810826C211yesmaximal, characteristic, complemented
11#16810768C211yesmaximal, characteristic, complemented
12#16810764C211yesmaximal, characteristic, complemented
13#16810761C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#449C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C2×#11719913C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C2×#11719924C2211yescharacteristic
22C2×#11719922C2211yescharacteristic
23#13133215C2211yescharacteristic
24#13133215C2211yescharacteristic
25#13133213C2211yescharacteristic
26#13133208C2211yescharacteristic
27#13133204C2211yescharacteristic
28#13133204C2211yescharacteristic
29#13133201—21no—
30#13133199—21no—
31#13133198—21no—
32#13133198—21no—
33#13133196—21no—
34#13133196—21no—
35#13133191C2211yescharacteristic
36#13133189—21no—
37#13133188—21no—
38D4×Dic53C2211yescharacteristic
39#13133179C2211yescharacteristic
40#13133176C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133140C2211yescharacteristic
47#13133140C2211yescharacteristic
48#13133138C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×SD16×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×Dic106—21no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×Dic106C2311yescharacteristic
79C2×Dic106C2311yescharacteristic
80SD16×C53—41no—
81D8×C53—41no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53D411yescharacteristic
92C22⋊Dic53D411yescharacteristic
93#11719924—41no—
94#11719923—41no—
95#11719923—41no—
96#11719922—41no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53—21no—
100C4⋊Dic53D411yescharacteristic
101C4⋊Dic53D411yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107#11719919—21no—
108C4×Dic53C2311yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112#11719913—41no—
113#11719913—41no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53C2×D411yescharacteristic
117C22×D53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—21no—
124D4×C53—42no—
125D4×C53—81no—
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—81no—
134C22⋊D53—81no—
135C22⋊D53—81no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—42no—
140C2×Dic53—81no—
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144C2×Dic53C2×D411yescharacteristic
145D212—21no—
146D212—21no—
147C4×D53—81no—
148Dic106—21no—
149Dic106—21no—
150Dic106—21no—
151Dic106—21no—
152Dic106—42no—
153C8×C53—21no—
154C8×C53—21no—
155#88322—21no—
156#88322—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53C22×D411yescharacteristic
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C4×C53—21no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53H(5,𝔽2)11yescharacteristic
174C4×C53H(5,𝔽2)11yescharacteristic
175Dic53—21no—
176Dic53—21no—
177Dic53—21no—
178Dic53—21no—
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182#3221—531nomaximal, Sylow
183C2×C53—41no—
184C2×C53—81no—
185C2×C53—81no—
186C2×C53#53511yescharacteristic
187C2×C53#57711yescharacteristic
188C2×C53#57711yescharacteristic
189D53—81no—
190#540—531no—
191#536—531no—
192#496—531no—
193#492—531no—
194#491—531no—
195#488—531no—
196#463—531no—
197#463—531no—
198#461—531no—
199#461—531no—
200#449—531no—
201#449D5311yescharacteristic, complemented
202#448—531no—
203#448—531no—
204#433—531no—
205C53#322111yescharacteristic, complemented, Sylow
206C2×#41C2×D5311yescharacteristic
207C22×Q8—531no—
208C2×SD16—531no—
209C2×SD16—531no—
210C2×SD16—531no—
211C2×SD16—531no—
212C2×SD16—531no—
213C2×SD16C2×D5311yescharacteristic
214C2×D8—531no—
215C2×D8C2×D5311yescharacteristic
216C4⋊Q8—531no—
217C4⋊Q8—531no—
218#127—531no—
219#125—531no—
220#124—531no—
221#124—531no—
222#123—1061no—
223#123—1061no—
224#123—1061no—
225#123—1061no—
226C22⋊Q8—1061no—
227C22⋊Q8—1061no—
228#121—531no—
229#121—1061no—
230#121—1061no—
231#121C2×D5311yescharacteristic
232C4×D4—531no—
233C4×D4—531no—
234#117—531no—
235C4⋊C8—531no—
236C4⋊C8—531no—
237#103—531no—
238#103—531no—
239#103—531no—
240#103—531no—
241#103—531no—
242#103C2×D5311yescharacteristic
243#102—531no—
244#102C2×D5311yescharacteristic
245C22⋊C8—531no—
246C22⋊C8C2×D5311yescharacteristic
247#97—531no—
248C4×C8—531no—
249#41—41no—
250#41—41no—
251C2×Q8—531no—
252C2×Q8—531no—
253C2×Q8—1061no—
254C2×Q8—2121no—
255C2×Q8—2121no—
256C2×Q8C22×D5311yescharacteristic
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—531no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4C22×D5311yescharacteristic
264C2×D4C22×D5311yescharacteristic
265C22×C4—21no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4C22×D5311yescharacteristic
270SD16—41no—
271SD16—2121no—
272SD16—2121no—
273SD16—2121no—
274SD16—2121no—
275SD16—2121no—
276D8—41no—
277D8—2121no—
278C2×C8—531no—
279C2×C8—531no—
280C2×C8C22×D5311yescharacteristic
281C2×C8C22×D5311yescharacteristic
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C4⋊C4C22×D5311yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—531no—
294C22⋊C4—531no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C22⋊C4—1061no—
299C22⋊C4—1061no—
300C22⋊C4—1061no—
301C22⋊C4—1061no—
302C42—531no—
303C42—531no—
304C23—21no—
305C23—21no—
306C23—1061no—
307C23D4×D5311yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—1061no—
311Q8—1061no—
312Q8—1061no—
313Q8—1061no—
314Q8—2122no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—81no—
321D4—1061no—
322D4—1061no—
323D4—4241no—
324D4—4241no—
325D4—4241no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—4241no—
339C2×C4—4241no—
340C2×C4—2122no—
341C2×C4D4×D5311yescharacteristic
342C2×C4C23×D5311yescharacteristic
343C8—21no—
344C8—21no—
345C8—1061no—
346C8—1061no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22C2×D4×D5311yescharacteristic
360C4—21no—
361C4—41no—
362C4—41no—
363C4—1061no—
364C4—1061no—
365C4—1061no—
366C4—1061no—
367C4—2121no—
368C4—2121no—
369C4—2121no—
370C4#1313326211yescharacteristic
371C4#1313326211yescharacteristic
372C2—41no—
373C2—81no—
374C2—81no—
375C2—4241no—
376C2#1681118811yescharacteristic
377C2#1681143011yescharacteristic
378C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#449C2
socleC22×C53C22×D4
hypercenter#449D53
hypocenterC53#3221
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. #16811226C2semidirect
2. #16811194C2semidirect
3. #16811011C2semidirect
4. #16810994C2semidirect
5. #16810990C2semidirect
6. #16810976C2semidirect
7. #16810837C2semidirect
8. #16810836C2semidirect
9. #16810827C2semidirect
10. #16810826C2semidirect
11. #16810768C2semidirect
12. #16810764C2semidirect
13. #16810761C2semidirect
14. #16810681C2semidirect
15. C53×#449C2semidirect
201. #449D53semidirect
205. C53#3221semidirect