Finite Group Databasebeta

← #20287049

Subgroups — #20287049

Order 6784.

398 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811328C211yesmaximal, characteristic, complemented
2#16811292C211yesmaximal, characteristic
3#16810968C211yesmaximal, characteristic, complemented
4#16810936C211yesmaximal, characteristic, complemented
5#16810935C211yesmaximal, characteristic, complemented
6#16810916C211yesmaximal, characteristic, complemented
7#16810869C211yesmaximal, characteristic, complemented
8#16810868C211yesmaximal, characteristic, complemented
9#16810822C211yesmaximal, characteristic, complemented
10#16810788C211yesmaximal, characteristic, complemented
11#16810785C211yesmaximal, characteristic, complemented
12#16810715C211yesmaximal, characteristic, complemented
13#16810699C211yesmaximal, characteristic, complemented
14#16810623C211yesmaximal, characteristic, complemented
15C53×#417C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133213—21no—
23#13133213C2211yescharacteristic
24#13133207—21no—
25#13133197C2211yescharacteristic
26#13133196—21no—
27#13133196C2211yescharacteristic
28#13133195—21no—
29#13133191—21no—
30#13133191C2211yescharacteristic
31#13133188—21no—
32#13133188C2211yescharacteristic
33Q8×Dic53C2211yescharacteristic
34#13133182C2211yescharacteristic
35C4×C22⋊D53—21no—
36C4×C22⋊D53—21no—
37C4×C22⋊D53C2211yescharacteristic
38C4⋊C4×D53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133150C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49C22⋊#88322C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C2×#41×C53C2211yescharacteristic
53C2×M16×C53C2211yescharacteristic
54C2×C4⋊C4×C53C2211yescharacteristic
55#103×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—41no—
61C2×Q8×C53C2311yescharacteristic
62C2×D4×C53—21no—
63C2×D4×C53C2311yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53D411yescharacteristic
72#11719949—22no—
73#11719949—41no—
74C2×D212—21no—
75C2×D212C2311yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×Dic106—21no—
80C2×Dic106C2311yescharacteristic
81M16×C53—41no—
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53—21no—
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53D411yescharacteristic
90#11719925—22no—
91#11719923—41no—
92#11719921—21no—
93#11719921—21no—
94#11719921D411yescharacteristic
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109#11719913—41no—
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53C2×D411yescharacteristic
115Q8×C53—21no—
116Q8×C53#4112yes—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—42no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—42no—
126C2×C4×C53—81no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—22no—
130C22⋊D53—42no—
131C22⋊D53—81no—
132C22⋊D53—81no—
133C22⋊D53—81no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140D212—21no—
141D212—21no—
142D212—42no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—42no—
146C4×D53—81no—
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—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53C22×D411yescharacteristic
161C2×D53—21no—
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C4×C53—22no—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53C2×#4111yescharacteristic
172C4×C53H(5,𝔽2)11yescharacteristic
173Dic53—21no—
174Dic53—21no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178Dic53#13712yes—
179#3333—531nomaximal, Sylow
180C2×C53—41no—
181C2×C53—81no—
182C2×C53#54611yescharacteristic
183C2×C53C2×#13711yescharacteristic
184C2×C53#57611yescharacteristic
185D53—41no—
186D53—81no—
187#554—531no—
188#550—531no—
189#486—531no—
190#480—531no—
191#480—531no—
192#477—531no—
193#468—531no—
194#468—531no—
195#460—531no—
196#452—531no—
197#452—531no—
198C4×Q16—531no—
199#436—531no—
200#417—531no—
201#417D5311yescharacteristic, complemented
202C53#333311yescharacteristic, complemented, Sylow
203C2×#41—531no—
204C2×#41C2×D5311yescharacteristic
205C2×Q16—531no—
206C2×SD16—531no—
207C2×SD16—531no—
208C2×M16—531no—
209C2×M16C2×D5311yescharacteristic
210C4⋊Q8—531no—
211#127—531no—
212#127—1061no—
213#125—1061no—
214C22⋊Q8—531no—
215C22⋊Q8—531no—
216C22⋊Q8—1061no—
217C22⋊Q8—1061no—
218#121—531no—
219#121—531no—
220#121—1061no—
221#121—1061no—
222C4×Q8—531no—
223C4×Q8—531no—
224C4×D4—531no—
225C4×D4—1061no—
226C4×D4—1061no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4C2×D5311yescharacteristic
229#107—531no—
230#107—531no—
231#107—531no—
232C4⋊C8—531no—
233#103—531no—
234#103—531no—
235#103C2×D5311yescharacteristic
236#103C2×D5311yescharacteristic
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102C2×D5311yescharacteristic
242#102C2×D5311yescharacteristic
243C22⋊C8—531no—
244C22⋊C8—531no—
245C4×C8—531no—
246#41—22no—
247#41—41no—
248#41—1062no—
249#41—2121no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8C22×D5311yescharacteristic
253C2×D4—21no—
254C2×D4—531no—
255C2×D4—531no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4C22×D5311yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—531no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4C22×D5311yescharacteristic
268Q16—1062no—
269SD16—2121no—
270SD16—2121no—
271M16—41no—
272M16—2121no—
273C2×C8—531no—
274C2×C8—531no—
275C2×C8C22×D5311yescharacteristic
276C2×C8C22×D5311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—531no—
279C4⋊C4—531no—
280C4⋊C4—531no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C22⋊C4—531no—
290C22⋊C4—531no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C42—531no—
296C42—1061no—
297C42—1061no—
298C23—21no—
299C23—531no—
300C23—1061no—
301C23D4×D5311yescharacteristic
302Q8—21no—
303Q8—1061no—
304Q8—1061no—
305Q8—4241no—
306Q8#1171994512yes—
307D4—21no—
308D4—21no—
309D4—42no—
310D4—1062no—
311D4—1061no—
312D4—1061no—
313D4—4241no—
314D4—4241no—
315D4—4241no—
316D4—2122no—
317D4—2122no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—4241no—
334C2×C4—2122no—
335C2×C4D4×D5311yescharacteristic
336C2×C4C23×D5311yescharacteristic
337C8—21no—
338C8—21no—
339C8—1061no—
340C8—1061no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—1061no—
348C22—1061no—
349C22—1061no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22C2×D4×D5311yescharacteristic
354C4—21no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—532no—
359C4—1061no—
360C4—1061no—
361C4—2121no—
362C4—2121no—
363C4—2121no—
364C4C2×#1171994511yescharacteristic
365C4#1313326211yescharacteristic
366C2—41no—
367C2—81no—
368C2—4241no—
369C2—2121no—
370C2#1681126311yescharacteristic
371C2#137×D5311yescharacteristic
372C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#417C2
socleC22×C53C22×D4
hypercenter#417D53
hypocenterC53#3333
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #16811328C2semidirect
3. #16810968C2semidirect
4. #16810936C2semidirect
5. #16810935C2semidirect
6. #16810916C2semidirect
7. #16810869C2semidirect
8. #16810868C2semidirect
9. #16810822C2semidirect
10. #16810788C2semidirect
11. #16810785C2semidirect
12. #16810715C2semidirect
13. #16810699C2semidirect
14. #16810623C2semidirect
15. C53×#417C2semidirect
201. #417D53semidirect
202. C53#3333semidirect