Finite Group Databasebeta

← #20286652

Subgroups — #20286652

Order 6784.

396 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811341C211yesmaximal, characteristic, complemented
2#16811291C211yesmaximal, characteristic
3#16810968C211yesmaximal, characteristic, complemented
4#16810950C211yesmaximal, characteristic, complemented
5#16810907C211yesmaximal, characteristic, complemented
6#16810853C211yesmaximal, characteristic, complemented
7#16810850C211yesmaximal, characteristic, complemented
8#16810822C211yesmaximal, characteristic, complemented
9#16810787C211yesmaximal, characteristic, complemented
10#16810786C211yesmaximal, characteristic, complemented
11#16810717C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic, complemented
13#16810617C211yesmaximal, characteristic, complemented
14#16810616C211yesmaximal, characteristic, complemented
15C53×#482C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719923C2211yescharacteristic
21C22×#88322C2211yescharacteristic
22C2×C8×D53C2211yescharacteristic
23#13133216C2211yescharacteristic
24#13133213—21no—
25#13133213C2211yescharacteristic
26#13133211—21no—
27#13133210—21no—
28#13133203—21no—
29#13133197C2211yescharacteristic
30#13133196C2211yescharacteristic
31#13133192C2211yescharacteristic
32#13133190—21no—
33#13133189—21no—
34C4×Dic106—21no—
35#13133182C2211yescharacteristic
36#13133182C2211yescharacteristic
37C4×C22⋊D53C2211yescharacteristic
38#13133177—21no—
39#13133175C2211yescharacteristic
40#13133148C2211yescharacteristic
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
53#117×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53D411yescharacteristic
67C22×Dic53—21no—
68#11719947—22no—
69#11719947—41no—
70#11719945—22no—
71#11719945—41no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53C2311yescharacteristic
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C22⋊C4×C53—21no—
85C22⋊C4×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53D411yescharacteristic
89#11719923—41no—
90#11719921—21no—
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#11719919D411yescharacteristic
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105C2×#88322—41no—
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108Dic212—22no—
109#11719913—41no—
110C8×D53—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×C53C2×D411yescharacteristic
124C2×C4×C53C2411yescharacteristic
125C22⋊D53—22no—
126C22⋊D53—42no—
127C22⋊D53—42no—
128C22⋊D53—42no—
129C22⋊D53—81no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—42no—
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137D212—21no—
138D212—21no—
139D212—81no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—42no—
143Dic106—21no—
144Dic106—21no—
145Dic106—21no—
146Dic106—42no—
147Dic106#4112yes—
148C8×C53—21no—
149C8×C53—21no—
150#88322—21no—
151#88322—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—21no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53C2×#4111yescharacteristic
170C4×C53H(5,𝔽2)11yescharacteristic
171Dic53—22no—
172Dic53—21no—
173Dic53—21no—
174Dic53—41no—
175Dic53—41no—
176Dic53#13512yes—
177#3304—531nomaximal, Sylow
178C2×C53—41no—
179C2×C53—81no—
180C2×C53#54611yescharacteristic
181C2×C53C2×#13511yescharacteristic
182C2×C53#57711yescharacteristic
183D53—41no—
184D53—81no—
185#556—531no—
186#550—531no—
187#486—531no—
188#482—531no—
189#482D5311yescharacteristic, complemented
190#475—531no—
191#465—531no—
192#465—531no—
193#460—531no—
194#452—531no—
195#452—531no—
196C4×Q16—531no—
197C8×D4—531no—
198#416—531no—
199#416—531no—
200C53#330411yescharacteristic, complemented, Sylow
201C2×#41—531no—
202C2×#41—531no—
203C2×Q16—531no—
204C2×SD16—531no—
205C2×SD16—531no—
206C22×C8—531no—
207C22×C8—531no—
208C4⋊Q8—531no—
209#127—531no—
210#127—1061no—
211#126—1061no—
212#126—1061no—
213#124—1061no—
214C22⋊Q8—531no—
215C22⋊Q8—531no—
216#121—531no—
217#121—1061no—
218#121—1061no—
219#121C2×D5311yescharacteristic
220C4×Q8—531no—
221C4×Q8—531no—
222C4×Q8—1061no—
223C4×D4—531no—
224C4×D4—1061no—
225#117—531no—
226#117C2×D5311yescharacteristic
227#107C2×D5311yescharacteristic
228#106—531no—
229#106C2×D5311yescharacteristic
230C4⋊C8—531no—
231#103—531no—
232#103—531no—
233#103—531no—
234#103—531no—
235#102—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102C2×D5311yescharacteristic
240#102C2×D5311yescharacteristic
241C22⋊C8—531no—
242C22⋊C8C2×D5311yescharacteristic
243C4×C8—531no—
244#41—1062no—
245#41—1062no—
246#41—2121no—
247#41—2121no—
248C2×Q8—531no—
249C2×Q8—531no—
250C2×Q8—1061no—
251C2×D4—21no—
252C2×D4—531no—
253C2×D4—531no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4—1061no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4C22×D5311yescharacteristic
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4C22×D5311yescharacteristic
264Q16—1062no—
265SD16—2121no—
266SD16—2121no—
267C2×C8—531no—
268C2×C8—531no—
269C2×C8—2121no—
270C2×C8—2121no—
271C2×C8C22×D5311yescharacteristic
272C2×C8C22×D5311yescharacteristic
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4—1061no—
279C4⋊C4—1061no—
280C4⋊C4C22×D5311yescharacteristic
281C4⋊C4C22×D5311yescharacteristic
282C4⋊C4C22×D5311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—531no—
286C22⋊C4—531no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C42—21no—
292C42—531no—
293C42—1061no—
294C42—1061no—
295C42—1061no—
296C23—21no—
297C23—531no—
298C23—1061no—
299C23D4×D5311yescharacteristic
300Q8—532no—
301Q8—1061no—
302Q8—1061no—
303Q8—1061no—
304Q8—2122no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—1062no—
309D4—1061no—
310D4—1061no—
311D4—4241no—
312D4—4241no—
313D4—2122no—
314D4—2122no—
315D4—2122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—531no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4D4×D5311yescharacteristic
332C2×C4C23×D5311yescharacteristic
333C8—21no—
334C8—21no—
335C8—1061no—
336C8—1061no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—1061no—
344C22—1061no—
345C22—1061no—
346C22—2121no—
347C22—2121no—
348C22—2121no—
349C22C2×D4×D5311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—532no—
355C4—1061no—
356C4—1062no—
357C4—1061no—
358C4—2121no—
359C4—2121no—
360C4#41×D5311yescharacteristic
361C4#1313326311yescharacteristic
362C2—41no—
363C2—81no—
364C2—4241no—
365C2—2121no—
366C2#1681126511yescharacteristic
367C2#135×D5311yescharacteristic
368C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#482C2
socleC22×C53C22×D4
hypercenter#482D53
hypocenterC53#3304
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #16811341C2semidirect
3. #16810968C2semidirect
4. #16810950C2semidirect
5. #16810907C2semidirect
6. #16810853C2semidirect
7. #16810850C2semidirect
8. #16810822C2semidirect
9. #16810787C2semidirect
10. #16810786C2semidirect
11. #16810717C2semidirect
12. #16810686C2semidirect
13. #16810617C2semidirect
14. #16810616C2semidirect
15. C53×#482C2semidirect
189. #482D53semidirect
200. C53#3304semidirect