Finite Group Databasebeta

← #20286715

Subgroups — #20286715

Order 6784.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811294C211yesmaximal, characteristic, complemented
2#16810989C211yesmaximal, characteristic, complemented
3#16810947C211yesmaximal, characteristic, complemented
4#16810936C211yesmaximal, characteristic, complemented
5#16810927C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810818C211yesmaximal, characteristic, complemented
9#16810766C211yesmaximal, characteristic, complemented
10#16810766C211yesmaximal, characteristic, complemented
11#16810723C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810622C211yesmaximal, characteristic, complemented
14#16810617C211yesmaximal, characteristic, complemented
15C53×#553C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719923C2211yescharacteristic
19C2×#11719922C2211yescharacteristic
20C2×#11719922C2211yescharacteristic
21C2×#11719917C2211yescharacteristic
22C22×#88322C2211yescharacteristic
23#13133214C2211yescharacteristic
24#13133213—21no—
25#13133191C2211yescharacteristic
26#13133191C2211yescharacteristic
27#13133190—21no—
28#13133190—21no—
29C4×Dic106C2211yescharacteristic
30#13133179—21no—
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133140C2211yescharacteristic
36#13133140C2211yescharacteristic
37#13133140C2211yescharacteristic
38#13133138C2211yescharacteristic
39#13133137C2211yescharacteristic
40#13133137C2211yescharacteristic
41#13133137C2211yescharacteristic
42#13133137C2211yescharacteristic
43#13133137C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48#126×C53—21no—
49#125×C53C2211yescharacteristic
50#123×C53—21no—
51#121×C53—21no—
52#121×C53C2211yescharacteristic
53#121×C53C2211yescharacteristic
54C4×D4×C53—21no—
55C4×D4×C53C2211yescharacteristic
56C4×D4×C53C2211yescharacteristic
57#117×C53C2211yescharacteristic
58C2×C4⋊C4×C53C2211yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53D411yescharacteristic
62C2×D4×C53C2311yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71#11719945—22no—
72#11719945—22no—
73#11719945—41no—
74#11719945—41no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212C2311yescharacteristic
78C2×D212C2311yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×Dic106C2311yescharacteristic
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53D411yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C22⋊C4×C53D411yescharacteristic
94C22⋊C4×C53D411yescharacteristic
95C42×C53—21no—
96C42×C53C2311yescharacteristic
97#11719923—22no—
98#11719922—41no—
99#11719922—41no—
100#11719921—21no—
101#11719921—21no—
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53C2×D411yescharacteristic
113C23×C53C2×D411yescharacteristic
114C22×D53—21no—
115C22×D53—21no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—41no—
119D4×C53—42no—
120D4×C53—81no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—81no—
129C2×C4×C53—81no—
130C2×C4×C53C2×D411yescharacteristic
131C2×C4×C53C2×D411yescharacteristic
132C2×C4×C53C2×D411yescharacteristic
133C2×C4×C53C2×D411yescharacteristic
134C2×C4×C53C2411yescharacteristic
135C22⋊D53—42no—
136C22⋊D53—42no—
137C22⋊D53—81no—
138C22⋊D53—81no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—42no—
146D212—81no—
147C4×D53—42no—
148C4×D53—42no—
149Dic106—21no—
150Dic106#4112yes—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53C22×D411yescharacteristic
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178C4×C53C2×#4111yescharacteristic
179C4×C53H(5,𝔽2)11yescharacteristic
180Dic53—22no—
181Dic53—41no—
182#3309—531nomaximal, Sylow
183C2×C53—41no—
184C2×C53—41no—
185C2×C53—81no—
186C2×C53#54611yescharacteristic
187C2×C53#57511yescharacteristic
188C2×C53#57611yescharacteristic
189D53—81no—
190D53—81no—
191#553D5311yescharacteristic, complemented
192#550—531no—
193#490—531no—
194#482—531no—
195#480—531no—
196#479—531no—
197#469—531no—
198#466—531no—
199#459—531no—
200#449—531no—
201#449—531no—
202#440—531no—
203#435—531no—
204#417—531no—
205#416—531no—
206C53#330911yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C2×#41—531no—
209C2×SD16—531no—
210C2×D8—531no—
211C2×D8—531no—
212C2×M16—531no—
213C22×C8—531no—
214#127—531no—
215#127—1061no—
216#126—21no—
217#125C2×D5311yescharacteristic
218#123—21no—
219#121—21no—
220#121—531no—
221#121—531no—
222#121—1061no—
223#121—1061no—
224#121C2×D5311yescharacteristic
225#121C2×D5311yescharacteristic
226C4×Q8—531no—
227C4×D4—21no—
228C4×D4—1061no—
229C4×D4C2×D5311yescharacteristic
230C4×D4C2×D5311yescharacteristic
231#117C2×D5311yescharacteristic
232C2×C4⋊C4C2×D5311yescharacteristic
233#107—531no—
234#107—531no—
235#106—531no—
236C4⋊C8—531no—
237#103—531no—
238#103—531no—
239#103—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247C22⋊C8—531no—
248C22⋊C8—531no—
249#97—531no—
250#41—1062no—
251#41—1062no—
252#41—2121no—
253#41—2121no—
254C2×Q8—531no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4C22⋊D5311yescharacteristic
266C2×D4C22×D5311yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273SD16—1062no—
274D8—2121no—
275D8—2121no—
276M16—2121no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8—531no—
280C2×C8—531no—
281C2×C8—2121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—531no—
286C4⋊C4—1061no—
287C4⋊C4C22⋊D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C4⋊C4C22×D5311yescharacteristic
290C4⋊C4C22×D5311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4C22⋊D5311yescharacteristic
298C22⋊C4C22⋊D5311yescharacteristic
299C42—21no—
300C42—1061no—
301C42C22×D5311yescharacteristic
302C23—21no—
303C23—1061no—
304C23—1061no—
305C23C2×C22⋊D5311yescharacteristic
306C23C2×C22⋊D5311yescharacteristic
307Q8—532no—
308Q8—1061no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—81no—
314D4—1061no—
315D4—1061no—
316D4—1061no—
317D4—1061no—
318D4—4241no—
319D4—4241no—
320D4—4241no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4C2×C22⋊D5311yescharacteristic
338C2×C4C2×C22⋊D5311yescharacteristic
339C2×C4C2×C22⋊D5311yescharacteristic
340C2×C4C2×C22⋊D5311yescharacteristic
341C2×C4C23×D5311yescharacteristic
342C8—1061no—
343C8—1061no—
344C8—1061no—
345C8—1061no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C22×C22⋊D5311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—1062no—
370C4—2121no—
371C4#41×D5311yescharacteristic
372C4#1313326311yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—4241no—
377C2—4241no—
378C2#1681126111yescharacteristic
379C2#1681142111yescharacteristic
380C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#553C2
socleC22×C53C22×D4
hypercenter#553D53
hypocenterC53#3309
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #16811294C2semidirect
2. #16810989C2semidirect
3. #16810947C2semidirect
4. #16810936C2semidirect
5. #16810927C2semidirect
6. #16810874C2semidirect
7. #16810857C2semidirect
8. #16810818C2semidirect
9. #16810766C2semidirect
10. #16810766C2semidirect
11. #16810723C2semidirect
12. #16810695C2semidirect
13. #16810622C2semidirect
14. #16810617C2semidirect
15. C53×#553C2semidirect
191. #553D53semidirect
206. C53#3309semidirect