Finite Group Databasebeta

← #20284977

Subgroups — #20284977

Order 6784.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811244C211yesmaximal, characteristic, complemented
2#16810958C211yesmaximal, characteristic, complemented
3#16810936C211yesmaximal, characteristic, complemented
4#16810918C211yesmaximal, characteristic, complemented
5#16810903C211yesmaximal, characteristic, complemented
6#16810845C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810785C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic
10#16810723C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12C4×#11719925C211yesmaximal, characteristic, complemented
13C4×#11719922C211yesmaximal, characteristic, complemented
14#16810684C211yesmaximal, characteristic, complemented
15C53×#520C211yesmaximal, characteristic, complemented
16C2×#11719925C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18C2×#11719924C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20#13133217C2211yescharacteristic
21#13133212—21no—
22#13133205C2211yescharacteristic
23#13133203—21no—
24#13133199—21no—
25#13133196C2211yescharacteristic
26#13133195—21no—
27#13133191C2211yescharacteristic
28C4×Dic106C2211yescharacteristic
29C4×D212C2211yescharacteristic
30#13133150C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133150C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35C4⋊#88322C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133140C2211yescharacteristic
39#13133140C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44C22⋊#88322C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46#13133128C2211yescharacteristic
47C4×#88322C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C4×Q8×C53C2211yescharacteristic
50C4×Q8×C53C2211yescharacteristic
51C4×D4×C53—21no—
52C4×D4×C53—21no—
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#117×C53—21no—
56#117×C53C2211yescharacteristic
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—22no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—21no—
70C2×D212C2311yescharacteristic
71C2×C4×D53—21no—
72C2×Dic106—21no—
73C2×Dic106C2311yescharacteristic
74C4⋊C4×C53—21no—
75C4⋊C4×C53—21no—
76C4⋊C4×C53—21no—
77C4⋊C4×C53D411yescharacteristic
78C4⋊C4×C53D411yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C4⋊C4×C53C2311yescharacteristic
81C22⋊C4×C53—21no—
82C22⋊C4×C53—21no—
83C22⋊C4×C53D411yescharacteristic
84C22⋊C4×C53D411yescharacteristic
85C42×C53—21no—
86C42×C53—21no—
87C42×C53C2311yescharacteristic
88C42×C53C2311yescharacteristic
89C22⋊Dic53—21no—
90#11719925—22no—
91#11719924—22no—
92#11719923—22no—
93#11719922—22no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—21no—
104C2×#88322C2311yescharacteristic
105C2×#88322C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—21no—
109C23×C53C2×D411yescharacteristic
110C22×D53—21no—
111Q8×C53—21no—
112Q8×C53#4112yes—
113D4×C53—24no—
114D4×C53—21no—
115D4×C53#4112yes—
116C2×C4×C53—24no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2×D411yescharacteristic
129C2×C4×C53C2×D411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135D212—21no—
136D212—21no—
137C4×D53—81no—
138Dic106—21no—
139Dic106—21no—
140Dic106—81no—
141#88322—21no—
142#88322—21no—
143#88322—21no—
144#88322—21no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53C22×D411yescharacteristic
152C2×D53—41no—
153C2×D53—41no—
154C2×D53—41no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53C2×#4111yescharacteristic
164C4×C53C2×#4111yescharacteristic
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168#3124—531nomaximal, Sylow
169C2×C53—41no—
170C2×C53—42no—
171C2×C53#52511yescharacteristic
172C2×C53#57611yescharacteristic
173C2×C53#57811yescharacteristic
174D53—81no—
175#542—531no—
176#520D5311yescharacteristic, complemented
177#484—531no—
178#480—531no—
179#477—531no—
180#474—531no—
181#464—531no—
182#463—531no—
183#452—531no—
184#449—531no—
185#440—531no—
186#440—531no—
187C4×Q16—531no—
188C4×D8—531no—
189#433—531no—
190C53#312411yescharacteristic, complemented, Sylow
191C2×#41C2×D5311yescharacteristic
192C2×Q16—531no—
193C2×SD16—531no—
194C2×SD16—531no—
195C2×D8—531no—
196C4⋊Q8—531no—
197#126—1061no—
198#124—531no—
199#124—1061no—
200#123—1061no—
201C22⋊Q8—531no—
202C22⋊Q8—1061no—
203#121—531no—
204C4×Q8—531no—
205C4×Q8C2×D5311yescharacteristic
206C4×Q8C2×D5311yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—531no—
210C4×D4C2×D5311yescharacteristic
211C4×D4C2×D5311yescharacteristic
212#117—21no—
213#117C2×D5311yescharacteristic
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×D5311yescharacteristic
216#107—531no—
217#107—531no—
218#107—531no—
219#107—531no—
220C4⋊C8—531no—
221C4⋊C8—531no—
222#103—531no—
223#103—531no—
224#103—531no—
225#103—531no—
226#102—531no—
227#102—531no—
228#102—531no—
229#102—531no—
230C22⋊C8—531no—
231C22⋊C8—531no—
232#97—531no—
233C4×C8—531no—
234#41—22no—
235#41—22no—
236C2×Q8—531no—
237C2×Q8—1061no—
238C2×Q8C22×D5311yescharacteristic
239C2×D4—21no—
240C2×D4—531no—
241C2×D4—1061no—
242C2×D4C22×D5311yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1061no—
248C22×C4C22×D5311yescharacteristic
249Q16—1062no—
250SD16—1062no—
251SD16—1062no—
252D8—1062no—
253C2×C8—531no—
254C2×C8—531no—
255C2×C8—531no—
256C2×C8—531no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—531no—
261C4⋊C4—531no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4C22⋊D5311yescharacteristic
266C4⋊C4C22⋊D5311yescharacteristic
267C4⋊C4C22×D5311yescharacteristic
268C4⋊C4C22×D5311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—1061no—
272C22⋊C4—1061no—
273C22⋊C4—1061no—
274C22⋊C4—1061no—
275C22⋊C4—1061no—
276C22⋊C4C22⋊D5311yescharacteristic
277C22⋊C4C22⋊D5311yescharacteristic
278C42—21no—
279C42—21no—
280C42—1061no—
281C42C22×D5311yescharacteristic
282C42C22×D5311yescharacteristic
283C23—21no—
284C23—1061no—
285C23C2×C22⋊D5311yescharacteristic
286Q8—21no—
287Q8—1061no—
288Q8—1061no—
289Q8—4241no—
290Q8#1171994512yes—
291D4—24no—
292D4—21no—
293D4—1061no—
294D4—1061no—
295D4—4241no—
296D4#1171994512yes—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—4241no—
310C2×C4C2×C22⋊D5311yescharacteristic
311C2×C4C2×C22⋊D5311yescharacteristic
312C2×C4C2×C22⋊D5311yescharacteristic
313C2×C4C2×C22⋊D5311yescharacteristic
314C2×C4C2×C22⋊D5311yescharacteristic
315C2×C4C23×D5311yescharacteristic
316C8—1061no—
317C8—1061no—
318C8—1061no—
319C8—1061no—
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—2121no—
327C22—2121no—
328C22—2121no—
329C22C22×C22⋊D5311yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—2121no—
339C4—2121no—
340C4—2121no—
341C4C2×#1171994511yescharacteristic
342C4C2×#1171994511yescharacteristic
343C2—41no—
344C2—42no—
345C2—4241no—
346C2#1681113411yescharacteristic
347C2#1681142611yescharacteristic
348C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#520C2
socleC22×C53C22×D4
hypercenter#520D53
hypocenterC53#3124
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #16811244C2semidirect
2. #16810958C2semidirect
3. #16810936C2semidirect
4. #16810918C2semidirect
5. #16810903C2semidirect
6. #16810845C2semidirect
7. #16810838C2semidirect
8. #16810785C2semidirect
10. #16810723C2semidirect
11. #16810721C2semidirect
12. C4×#11719925C2semidirect
13. C4×#11719922C2semidirect
14. #16810684C2semidirect
15. C53×#520C2semidirect
176. #520D53semidirect
190. C53#3124semidirect