Finite Group Databasebeta

← #20287051

Subgroups — #20287051

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#16811291C211yesmaximal, characteristic
3#16810968C211yesmaximal, characteristic, complemented
4#16810938C211yesmaximal, characteristic, complemented
5#16810917C211yesmaximal, characteristic, complemented
6#16810870C211yesmaximal, characteristic, complemented
7#16810867C211yesmaximal, characteristic, complemented
8#16810822C211yesmaximal, characteristic, complemented
9#16810787C211yesmaximal, characteristic, complemented
10#16810786C211yesmaximal, characteristic, complemented
11#16810717C211yesmaximal, characteristic, complemented
12#16810698C211yesmaximal, characteristic, complemented
13#16810622C211yesmaximal, characteristic, complemented
14#16810621C211yesmaximal, characteristic, complemented
15C53×#480C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C2×#11719912C2211yescharacteristic
23#13133216C2211yescharacteristic
24#13133213—21no—
25#13133213C2211yescharacteristic
26#13133207—21no—
27#13133197C2211yescharacteristic
28#13133196—21no—
29#13133196C2211yescharacteristic
30#13133195—21no—
31#13133192C2211yescharacteristic
32#13133190—21no—
33#13133189—21no—
34#13133182C2211yescharacteristic
35#13133182C2211yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53C2211yescharacteristic
38#13133177—21no—
39C4⋊C4×D53C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133140C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52#121×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53C2311yescharacteristic
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53D411yescharacteristic
68C22×Dic53—21no—
69#11719947—22no—
70#11719947—41no—
71#11719945—22no—
72#11719945—41no—
73C2×D212—21no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×Dic106—21no—
79C2×Dic106C2311yescharacteristic
80C2×Dic106C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53—21no—
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53D411yescharacteristic
90#11719923—41no—
91#11719921—21no—
92#11719921—21no—
93#11719921D411yescharacteristic
94C4⋊Dic53—21no—
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919D411yescharacteristic
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105#11719917—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108Dic212—22no—
109#11719913—41no—
110#11719912—41no—
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53C2×D411yescharacteristic
115D4×C53—21no—
116D4×C53—21no—
117D4×C53—81no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—81no—
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—22no—
127C22⋊D53—42no—
128C22⋊D53—42no—
129C22⋊D53—42no—
130C22⋊D53—81no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—42no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138D212—21no—
139D212—21no—
140D212—81no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—42no—
144C4×D53—81no—
145Dic106—21no—
146Dic106—21no—
147Dic106—21no—
148Dic106—81no—
149Dic106#4112yes—
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—21no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53C2×#4111yescharacteristic
172C4×C53H(5,𝔽2)11yescharacteristic
173Dic53—22no—
174Dic53—21no—
175Dic53—21no—
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#480D5311yescharacteristic, complemented
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#417—531no—
202C53#333311yescharacteristic, complemented, Sylow
203C2×#41—531no—
204C2×#41—531no—
205C2×Q16—531no—
206C2×SD16—531no—
207C2×SD16—531no—
208C2×M16—531no—
209C2×M16—531no—
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—1061no—
220#121—1061no—
221#121C2×D5311yescharacteristic
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#107C2×D5311yescharacteristic
231#107C2×D5311yescharacteristic
232C4⋊C8—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—
241#102C2×D5311yescharacteristic
242#102C2×D5311yescharacteristic
243C22⋊C8—531no—
244C22⋊C8C2×D5311yescharacteristic
245C4×C8—531no—
246#41—1062no—
247#41—1062no—
248#41—2121no—
249#41—2121no—
250C2×Q8—531no—
251C2×Q8—531no—
252C2×Q8—1061no—
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—531no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4C22×D5311yescharacteristic
268Q16—1062no—
269SD16—2121no—
270SD16—2121no—
271M16—2121no—
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—1061no—
281C4⋊C4—1061no—
282C4⋊C4—1061no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4C22×D5311yescharacteristic
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—531no—
291C22⋊C4—531no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C42—531no—
296C42—1061no—
297C42—1061no—
298C23—21no—
299C23—531no—
300C23—1061no—
301C23D4×D5311yescharacteristic
302Q8—532no—
303Q8—1061no—
304Q8—1061no—
305Q8—1061no—
306Q8—4241no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—1062no—
311D4—1061no—
312D4—1061no—
313D4—4241no—
314D4—4241no—
315D4—2122no—
316D4—2122no—
317D4—2122no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—531no—
325C2×C4—531no—
326C2×C4—531no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—4241no—
333C2×C4—2122no—
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—41no—
356C4—41no—
357C4—41no—
358C4—532no—
359C4—1061no—
360C4—1062no—
361C4—1061no—
362C4—2121no—
363C4—2121no—
364C4#41×D5311yescharacteristic
365C4#1313326311yescharacteristic
366C2—41no—
367C2—81no—
368C2—4241no—
369C2—2121no—
370C2#1681126511yescharacteristic
371C2#137×D5311yescharacteristic
372C2#1681142411yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#480C2
socleC22×C53C22×D4
hypercenter#480D53
hypocenterC53#3333
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #16811328C2semidirect
3. #16810968C2semidirect
4. #16810938C2semidirect
5. #16810917C2semidirect
6. #16810870C2semidirect
7. #16810867C2semidirect
8. #16810822C2semidirect
9. #16810787C2semidirect
10. #16810786C2semidirect
11. #16810717C2semidirect
12. #16810698C2semidirect
13. #16810622C2semidirect
14. #16810621C2semidirect
15. C53×#480C2semidirect
191. #480D53semidirect
202. C53#3333semidirect