Finite Group Databasebeta

← #20284862

Subgroups — #20284862

Order 6784.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811170C211yesmaximal, characteristic, complemented
2C4×#11719947C211yesmaximal, characteristic, complemented
3#16810949C211yesmaximal, characteristic, complemented
4#16810945C211yesmaximal, characteristic, complemented
5#16810907C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810822C211yesmaximal, characteristic, complemented
8#16810816C211yesmaximal, characteristic, complemented
9#16810769C211yesmaximal, characteristic, complemented
10#16810765C211yesmaximal, characteristic, complemented
11#16810733C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13#16810723C211yesmaximal, characteristic, complemented
14#16810678C211yesmaximal, characteristic, complemented
15C53×#442C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719947C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21#13133216C2211yescharacteristic
22#13133213C2211yescharacteristic
23#13133203—21no—
24#13133192C2211yescharacteristic
25#13133190—21no—
26#13133188C2211yescharacteristic
27C4×Dic106C2211yescharacteristic
28#13133182C2211yescharacteristic
29C4×D212C2211yescharacteristic
30D4×Dic53C2211yescharacteristic
31C4×C22⋊D53—21no—
32C4×C22⋊D53—21no—
33#13133177—21no—
34#13133175C2211yescharacteristic
35#13133173—21no—
36C2×C4×Dic53—21no—
37C42×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133150C2211yescharacteristic
40#13133148C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133146C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52C2×D8×C53C2211yescharacteristic
53C4×D4×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55#106×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69#11719947—22no—
70#11719947—22no—
71#11719945—44no—
72C2×D212—21no—
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53D411yescharacteristic
76C2×C4×D53D411yescharacteristic
77C2×C4×D53C2311yescharacteristic
78C2×Dic106—21no—
79C2×Dic106C2311yescharacteristic
80D8×C53—22no—
81C2×C8×C53C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C4⋊C4×C53C2311yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C42×C53C2311yescharacteristic
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719923—22no—
91#11719922—22no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—21no—
103C4×Dic53D411yescharacteristic
104C4×Dic53D411yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108#11719913—22no—
109C23×C53—21no—
110C23×C53—21no—
111C22×D53—21no—
112C22×D53C2×D411yescharacteristic
113D4×C53—21no—
114D4×C53—21no—
115D4×C53—21no—
116D4×C53#4112yescomplemented
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2411yescharacteristic
124C22⋊D53—24no—
125C22⋊D53—42no—
126C22⋊D53—42no—
127C22⋊D53—42no—
128C2×Dic53—24no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135C2×Dic53C2×D411yescharacteristic
136D212—21no—
137D212—21no—
138D212—42no—
139C4×D53—21no—
140C4×D53—21no—
141C4×D53—22no—
142C4×D53—41no—
143C4×D53—42no—
144Dic106—21no—
145Dic106—42no—
146Dic106#4112yescomplemented
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—22no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53C22×D411yescharacteristic
158C2×D53—21no—
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53C2×#4111yescharacteristic
169C4×C53C2×#4111yescharacteristic
170Dic53—22no—
171Dic53—21no—
172Dic53—21no—
173Dic53—41no—
174Dic53—41no—
175Dic53AGL(1,ℤ/8)12yes—
176#3115—531nomaximal, Sylow
177C2×C53—42no—
178C2×C53—81no—
179C2×C53#52511yescharacteristic
180C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C53#57511yescharacteristic
182D53—41no—
183D53—81no—
184#532—531no—
185C4×#41—531no—
186#482—531no—
187#482—531no—
188#475—531no—
189#474—531no—
190#460—531no—
191#459—531no—
192#449—531no—
193#449—531no—
194#442—531no—
195#442D5311yescharacteristic, complemented
196#440—531no—
197#440—531no—
198#432—531no—
199C53#311511yescharacteristic, complemented, Sylow
200C2×#41—531no—
201C2×#41—1061no—
202C2×SD16—531no—
203C2×SD16—531no—
204C2×D8—531no—
205C2×D8C2×D5311yescharacteristic
206C4⋊Q8—531no—
207#127—531no—
208#124—1061no—
209#121—531no—
210#121—531no—
211#121—1061no—
212C4×Q8—531no—
213C4×Q8—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—1061no—
217C4×D4—1061no—
218C4×D4—1061no—
219C4×D4C2×D5311yescharacteristic
220C4×D4C2×D5311yescharacteristic
221#117—531no—
222#117—1061no—
223C2×C42—531no—
224C2×C42—1061no—
225#107—531no—
226#107—531no—
227#106—531no—
228#106C2×D5311yescharacteristic
229C4⋊C8—531no—
230C4⋊C8—531no—
231#103—531no—
232#103—531no—
233#102—531no—
234#102—531no—
235#102—531no—
236#102—531no—
237#102C2×D5311yescharacteristic
238#102C2×D5311yescharacteristic
239C22⋊C8—531no—
240C22⋊C8—531no—
241#97—531no—
242#97C2×D5311yescharacteristic
243#41—1062no—
244#41—1062no—
245#41—2124no—
246C2×Q8—531no—
247C2×Q8—1061no—
248C2×D4—531no—
249C2×D4—1061no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4C22×D5311yescharacteristic
255C2×D4C22×D5311yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264SD16—1062no—
265SD16—1062no—
266D8—22no—
267D8—1062no—
268C2×C8—531no—
269C2×C8—531no—
270C2×C8C22×D5311yescharacteristic
271C2×C8C22×D5311yescharacteristic
272C4⋊C4—531no—
273C4⋊C4—531no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4C22×D5311yescharacteristic
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—531no—
287C42—531no—
288C42—531no—
289C42—1061no—
290C42—1061no—
291C42—1061no—
292C42C22×D5311yescharacteristic
293C23—21no—
294C23—21no—
295C23—531no—
296C23—1061no—
297Q8—532no—
298Q8—1061no—
299Q8—2122no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—1064no—
304D4—1061no—
305D4—1061no—
306D4—2122no—
307D4—2122no—
308D4—2122no—
309D4—2122no—
310D4#1171994512yescomplemented
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—1062no—
319C2×C4—1064no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—2121no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4D4×D5311yescharacteristic
329C2×C4D4×D5311yescharacteristic
330C2×C4C23×D5311yescharacteristic
331C8—21no—
332C8—21no—
333C8—1061no—
334C8—1061no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—1061no—
342C22—1061no—
343C22—1061no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22C2×D4×D5311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—532no—
353C4—1061no—
354C4—1062no—
355C4—1061no—
356C4—2121no—
357C4—2121no—
358C4#41×D5311yescharacteristic
359C4C2×#1171994511yescharacteristic
360C2—42no—
361C2—81no—
362C2—4241no—
363C2—2121no—
364C2#1681113211yescharacteristic
365C2AGL(1,ℤ/8)×D5311yescharacteristic
366C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#442C2
socleC22×C53C22×D4
hypercenter#442D53
hypocenterC53#3115
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #16811170C2semidirect
2. C4×#11719947C2semidirect
3. #16810949C2semidirect
4. #16810945C2semidirect
5. #16810907C2semidirect
6. #16810901C2semidirect
7. #16810822C2semidirect
8. #16810816C2semidirect
9. #16810769C2semidirect
10. #16810765C2semidirect
11. #16810733C2semidirect
12. #16810725C2semidirect
13. #16810723C2semidirect
14. #16810678C2semidirect
15. C53×#442C2semidirect
116. D4×C53#41semidirect
146. Dic106#41semidirect
195. #442D53semidirect
199. C53#3115semidirect
310. D4#11719945semidirect