Finite Group Databasebeta

← #20286721

Subgroups — #20286721

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#16811319C211yesmaximal, characteristic, complemented
2#16810987C211yesmaximal, characteristic, complemented
3#16810944C211yesmaximal, characteristic, complemented
4#16810934C211yesmaximal, characteristic, complemented
5#16810930C211yesmaximal, characteristic, complemented
6#16810874C211yesmaximal, characteristic, complemented
7#16810857C211yesmaximal, characteristic, complemented
8#16810815C211yesmaximal, characteristic, complemented
9#16810768C211yesmaximal, characteristic, complemented
10#16810768C211yesmaximal, characteristic, complemented
11#16810721C211yesmaximal, characteristic, complemented
12#16810695C211yesmaximal, characteristic, complemented
13#16810624C211yesmaximal, characteristic, complemented
14#16810619C211yesmaximal, characteristic, complemented
15C53×#550C211yesmaximal, characteristic, complemented
16C2×#11719924C2211yescharacteristic
17C2×#11719922C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20C22×#88322C2211yescharacteristic
21#13133212—21no—
22#13133209C2211yescharacteristic
23#13133199—21no—
24#13133191C2211yescharacteristic
25#13133191C2211yescharacteristic
26#13133189—21no—
27C4×D212C2211yescharacteristic
28#13133177—21no—
29#13133176C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic
31#13133150C2211yescharacteristic
32#13133150C2211yescharacteristic
33#13133148C2211yescharacteristic
34C4⋊#88322C2211yescharacteristic
35#13133141C2211yescharacteristic
36#13133141C2211yescharacteristic
37#13133141C2211yescharacteristic
38#13133138C2211yescharacteristic
39#13133138C2211yescharacteristic
40#13133138C2211yescharacteristic
41#13133138C2211yescharacteristic
42#13133138C2211yescharacteristic
43#13133138C2211yescharacteristic
44#13133137C2211yescharacteristic
45C22⋊#88322C2211yescharacteristic
46C22⋊#88322C2211yescharacteristic
47#13133128C2211yescharacteristic
48C2×#41×C53C2211yescharacteristic
49C2×#41×C53C2211yescharacteristic
50#127×C53—21no—
51#127×C53C2211yescharacteristic
52#121×C53—21no—
53#121×C53—21no—
54#121×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56C4×Q8×C53C2211yescharacteristic
57C4×D4×C53—21no—
58C4×D4×C53C2211yescharacteristic
59#41×C53—22no—
60#41×C53—22no—
61#41×C53—41no—
62#41×C53—41no—
63C2×Q8×C53C2311yescharacteristic
64C2×D4×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C2×D4×C53C2311yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C2311yescharacteristic
76C22×C4×C53C2311yescharacteristic
77C2×C22⋊D53—21no—
78C2×C22⋊D53—21no—
79C22×Dic53—21no—
80C2×D212C2311yescharacteristic
81C2×C4×D53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53D411yescharacteristic
84C4⋊C4×C53C2311yescharacteristic
85C22⋊C4×C53—21no—
86C22⋊C4×C53—21no—
87C22⋊C4×C53D411yescharacteristic
88C22⋊C4×C53D411yescharacteristic
89C42×C53—21no—
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92#11719924—22no—
93#11719922—41no—
94#11719922—41no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105#11719917—41no—
106C2×#88322—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C2×#88322C2311yescharacteristic
111C23×C53—21no—
112C23×C53—21no—
113C23×C53C2×D411yescharacteristic
114C23×C53C2×D411yescharacteristic
115C22×D53—21no—
116Q8×C53—21no—
117Q8×C53#4112yes—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122D4×C53—41no—
123D4×C53—42no—
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—81no—
127D4×C53—81no—
128D4×C53—81no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53C2×D411yescharacteristic
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2×D411yescharacteristic
141C2×C4×C53C2411yescharacteristic
142C22⋊D53—42no—
143C22⋊D53—81no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—81no—
148D212—21no—
149D212—21no—
150C4×D53—81no—
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×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53C22×D411yescharacteristic
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—22no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—41no—
177C4×C53C2×#4111yescharacteristic
178C4×C53H(5,𝔽2)11yescharacteristic
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182#3309—531nomaximal, Sylow
183C2×C53—41no—
184C2×C53—41no—
185C2×C53—81no—
186C2×C53—81no—
187C2×C53#54611yescharacteristic
188C2×C53#57511yescharacteristic
189C2×C53#57611yescharacteristic
190D53—81no—
191#553—531no—
192#550D5311yescharacteristic, complemented
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×#41C2×D5311yescharacteristic
208C2×#41C2×D5311yescharacteristic
209C2×SD16—531no—
210C2×D8—531no—
211C2×D8—531no—
212C2×M16—531no—
213C22×C8—531no—
214#127—21no—
215#127C2×D5311yescharacteristic
216#126—1061no—
217#125—531no—
218#123—1061no—
219#121—21no—
220#121—21no—
221#121—531no—
222#121—531no—
223#121—1061no—
224#121C2×D5311yescharacteristic
225#121C2×D5311yescharacteristic
226C4×Q8C2×D5311yescharacteristic
227C4×D4—21no—
228C4×D4—531no—
229C4×D4—1061no—
230C4×D4C2×D5311yescharacteristic
231#117—531no—
232C2×C4⋊C4—531no—
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—22no—
251#41—22no—
252#41—41no—
253#41—41no—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—531no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4C22⋊D5311yescharacteristic
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—531no—
284C4⋊C4—531no—
285C4⋊C4—531no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4C22⋊D5311yescharacteristic
290C4⋊C4C22×D5311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4C22⋊D5311yescharacteristic
298C22⋊C4C22⋊D5311yescharacteristic
299C42—21no—
300C42—1061no—
301C42C22×D5311yescharacteristic
302C23—21no—
303C23—21no—
304C23—1061no—
305C23C2×C22⋊D5311yescharacteristic
306C23C2×C22⋊D5311yescharacteristic
307Q8—21no—
308Q8#1171994712yes—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—1061no—
321D4—1061no—
322D4—4241no—
323D4—2122no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—4241no—
336C2×C4—4241no—
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—41no—
356C22—41no—
357C22—41no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C22×C22⋊D5311yescharacteristic
362C4—21no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—2121no—
369C4—2121no—
370C4—2121no—
371C4C2×#1171994711yescharacteristic
372C4#1313326211yescharacteristic
373C2—41no—
374C2—41no—
375C2—81no—
376C2—81no—
377C2—4241no—
378C2#1681126711yescharacteristic
379C2#1681142111yescharacteristic
380C2#1681142611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#550C2
socleC22×C53C22×D4
hypercenter#550D53
hypocenterC53#3309
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #16811319C2semidirect
2. #16810987C2semidirect
3. #16810944C2semidirect
4. #16810934C2semidirect
5. #16810930C2semidirect
6. #16810874C2semidirect
7. #16810857C2semidirect
8. #16810815C2semidirect
9. #16810768C2semidirect
10. #16810768C2semidirect
11. #16810721C2semidirect
12. #16810695C2semidirect
13. #16810624C2semidirect
14. #16810619C2semidirect
15. C53×#550C2semidirect
192. #550D53semidirect
206. C53#3309semidirect