Finite Group Databasebeta

← #20285723

Subgroups — #20285723

Order 6784.

390 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811175C211yesmaximal, characteristic, complemented
2#16811169C211yesmaximal, characteristic
3#16811016C211yesmaximal, characteristic, complemented
4#16811014C211yesmaximal, characteristic, complemented
5#16810984C211yesmaximal, characteristic, complemented
6#16810981C211yesmaximal, characteristic, complemented
7#16810832C211yesmaximal, characteristic, complemented
8#16810831C211yesmaximal, characteristic, complemented
9#16810822C211yesmaximal, characteristic, complemented
10#16810821C211yesmaximal, characteristic, complemented
11#16810787C211yesmaximal, characteristic, complemented
12#16810785C211yesmaximal, characteristic, complemented
13#16810783C211yesmaximal, characteristic, complemented
14#16810677C211yesmaximal, characteristic, complemented
15C53×#452C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×Dic212C2211yescharacteristic
18C2×#11719925C2211yescharacteristic
19C2×#11719925C2211yescharacteristic
20C2×#11719913C2211yescharacteristic
21C2×#11719923C2211yescharacteristic
22C2×#11719924C2211yescharacteristic
23#13133216C2211yescharacteristic
24#13133215C2211yescharacteristic
25#13133215C2211yescharacteristic
26#13133213C2211yescharacteristic
27#13133207—21no—
28#13133206C2211yescharacteristic
29#13133204C2211yescharacteristic
30#13133196C2211yescharacteristic
31#13133194—21no—
32#13133189—21no—
33#13133189—21no—
34Q8×Dic53C2211yescharacteristic
35#13133182—21no—
36#13133182C2211yescharacteristic
37C4×C22⋊D53—21no—
38C4×C22⋊D53—21no—
39#13133173—21no—
40C2×C4×Dic53C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic
51#13133127C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×SD16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59#41×C53—41no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×Q8×C53C2311yescharacteristic
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C22×Dic53D411yescharacteristic
71C22×Dic53D411yescharacteristic
72#11719945—41no—
73#11719945—41no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79Q16×C53—41no—
80SD16×C53—41no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89#11719925—41no—
90#11719925—41no—
91#11719924—41no—
92#11719923—41no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53D411yescharacteristic
107C4×Dic53D411yescharacteristic
108C4×Dic53C2311yescharacteristic
109C4×Dic53C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C2×#88322C2311yescharacteristic
112Dic212—41no—
113#11719913—41no—
114C23×C53—21no—
115C23×C53C2×D411yescharacteristic
116C22×D53—21no—
117Q8×C53—21no—
118Q8×C53—21no—
119Q8×C53—21no—
120Q8×C53—21no—
121D4×C53—21no—
122D4×C53—21no—
123D4×C53—42no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—42no—
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C22⋊D53—42no—
133C22⋊D53—42no—
134C22⋊D53—42no—
135C2×Dic53—22no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—41no—
140C2×Dic53C2×D411yescharacteristic
141C2×Dic53C2×D411yescharacteristic
142C2×Dic53C2×D411yescharacteristic
143C2×Dic53C2×D411yescharacteristic
144D212—21no—
145D212—21no—
146C4×D53—42no—
147Dic106—21no—
148Dic106—21no—
149Dic106—21no—
150Dic106—21no—
151Dic106—42no—
152C8×C53—21no—
153C8×C53—21no—
154#88322—21no—
155#88322—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53C22×D411yescharacteristic
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53H(5,𝔽2)11yescharacteristic
171C4×C53H(5,𝔽2)11yescharacteristic
172Dic53—21no—
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178Dic53#13712yes—
179#3216—531nomaximal, Sylow
180C2×C53—41no—
181C2×C53—81no—
182C2×C53#53511yescharacteristic
183C2×C53C2×#13711yescharacteristic
184C2×C53#57511yescharacteristic
185D53—81no—
186#533—531no—
187#532—531no—
188#497—531no—
189#497—531no—
190#489—531no—
191#489—531no—
192#462—531no—
193#462—531no—
194#460—531no—
195#460—531no—
196#452—531no—
197#452—531no—
198#452—531no—
199#452D5311yescharacteristic, complemented
200#432—531no—
201C53#321611yescharacteristic, complemented, Sylow
202C2×#41—531no—
203C2×#41C2×D5311yescharacteristic
204C2×Q16—531no—
205C2×Q16—531no—
206C2×Q16—531no—
207C2×Q16C2×D5311yescharacteristic
208C2×SD16—531no—
209C2×SD16—531no—
210C2×SD16—531no—
211C2×SD16C2×D5311yescharacteristic
212C4⋊Q8—531no—
213C4⋊Q8—531no—
214C4⋊Q8—531no—
215#127—531no—
216#125—1061no—
217#124—531no—
218#124—531no—
219C22⋊Q8—531no—
220C22⋊Q8—1061no—
221C22⋊Q8C2×D5311yescharacteristic
222#121—1061no—
223#121—1061no—
224C4×Q8—531no—
225C4×Q8—531no—
226C4×Q8—1061no—
227C4×D4—1061no—
228C4×D4—1061no—
229#117—1061no—
230C2×C42—531no—
231C4⋊C8—531no—
232C4⋊C8—531no—
233#103—531no—
234#103—531no—
235#103—531no—
236#103C2×D5311yescharacteristic
237#102—531no—
238#102—531no—
239#102—531no—
240#102C2×D5311yescharacteristic
241C22⋊C8—531no—
242C22⋊C8C2×D5311yescharacteristic
243#97—531no—
244#97—531no—
245#41—41no—
246#41—41no—
247#41—2121no—
248#41—2121no—
249C2×Q8—531no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8C22×D5311yescharacteristic
253C2×Q8C22×D5311yescharacteristic
254C2×D4—21no—
255C2×D4—531no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4C22×D5311yescharacteristic
260C22×C4—21no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—1061no—
264C22×C4C22×D5311yescharacteristic
265Q16—41no—
266Q16—2121no—
267Q16—2121no—
268Q16—2121no—
269SD16—41no—
270SD16—2121no—
271SD16—2121no—
272SD16—2121no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8C22×D5311yescharacteristic
276C2×C8C22×D5311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—531no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
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—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C42—531no—
295C42—531no—
296C42—531no—
297C42—531no—
298C42—1061no—
299C42—1061no—
300C23—21no—
301C23—1061no—
302C23D4×D5311yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—1061no—
308Q8—1061no—
309Q8—1061no—
310Q8—1061no—
311Q8—2122no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—1061no—
316D4—1061no—
317D4—2122no—
318D4—2122no—
319D4—2122no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—531no—
329C2×C4—531no—
330C2×C4—1062no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—2122no—
335C2×C4—2121no—
336C2×C4D4×D5311yescharacteristic
337C2×C4C23×D5311yescharacteristic
338C8—21no—
339C8—21no—
340C8—1061no—
341C8—1061no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22C2×D4×D5311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—532no—
357C4—1061no—
358C4—1061no—
359C4—1061no—
360C4—2121no—
361C4—2121no—
362C4—2121no—
363C4#1313326211yescharacteristic
364C4#1313326211yescharacteristic
365C2—41no—
366C2—81no—
367C2—4241no—
368C2#1681118811yescharacteristic
369C2#137×D5311yescharacteristic
370C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#452C2
socleC22×C53C22×D4
hypercenter#452D53
hypocenterC53#3216
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. #16811175C2semidirect
3. #16811016C2semidirect
4. #16811014C2semidirect
5. #16810984C2semidirect
6. #16810981C2semidirect
7. #16810832C2semidirect
8. #16810831C2semidirect
9. #16810822C2semidirect
10. #16810821C2semidirect
11. #16810787C2semidirect
12. #16810785C2semidirect
13. #16810783C2semidirect
14. #16810677C2semidirect
15. C53×#452C2semidirect
199. #452D53semidirect
201. C53#3216semidirect