Finite Group Databasebeta

← #20288813

Subgroups — #20288813

Order 6784.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811359C213yesmaximal, complemented
2#16811349C211yesmaximal, characteristic, complemented
3#16811337C213yesmaximal, complemented
4#16811325C213yesmaximal, complemented
5#16811313C213yesmaximal, complemented
6#16811280C211yesmaximal, characteristic, complemented
7#16811280C211yesmaximal, characteristic, complemented
8#16811240C213yesmaximal, complemented
9#16811227C213yesmaximal, complemented
10#16811154C213yesmaximal, complemented
11#16811153C213yesmaximal, complemented
12#16811106C211yesmaximal, characteristic, complemented
13#16811099C211yesmaximal, characteristic, complemented
14#41×Dic53C211yesmaximal, characteristic, complemented
15C53×#550C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17#13133216C2213yes—
18#13133212C2211yescharacteristic
19#13133210C2216yes—
20#13133210C2213yes—
21#13133209C2213yes—
22#13133208C2213yes—
23#13133207C2213yes—
24#13133204C2213yes—
25#13133204C2213yes—
26#13133204C2216yes—
27#13133203C2213yes—
28#13133201C2213yes—
29#13133198C2216yes—
30#13133198C2213yes—
31#13133198C2216yes—
32#13133196C2213yes—
33#13133196C2213yes—
34#13133194C2213yes—
35#13133189C2216yes—
36#13133189C2213yes—
37#13133188C2213yes—
38#13133188C2213yes—
39Q8×Dic53C2211yescharacteristic
40Q8×Dic53C2211yescharacteristic
41Q8×Dic53C2211yescharacteristic
42C4×Dic106C2213yes—
43C4×Dic106C2213yes—
44#13133182C2213yes—
45D4×Dic53C2213yes—
46D4×Dic53C2216yes—
47D4×Dic53C2213yes—
48D4×Dic53C2213yes—
49D4×Dic53C2213yes—
50C4×C22⋊D53C2213yes—
51C4×C22⋊D53C2213yes—
52#13133176C2213yes—
53#13133176C2213yes—
54#13133175C2213yes—
55#13133174C2213yes—
56#13133173C2216yes—
57C2×C4⋊Dic53C2213yes—
58C2×C4×Dic53C2213yes—
59C2×#41×C53C2211yescharacteristic
60C2×#41×C53C2211yescharacteristic
61#127×C53C2213yes—
62#121×C53C2213yes—
63#121×C53C2213yes—
64C4×Q8×C53C2211yescharacteristic
65C4×D4×C53C2213yes—
66#41×C53—24no—
67#41×C53—24no—
68C2×Q8×C53C2311yescharacteristic
69C2×D4×C53C2313yes—
70C2×D4×C53C2313yes—
71C2×D4×C53C2313yes—
72C2×D4×C53C2316yes—
73C22×C4×C53C2313yes—
74C22×C4×C53C2313yes—
75C2×C22⋊D53C2313yes—
76C2×C22⋊D53C2313yes—
77C22×Dic53C2313yes—
78C22×Dic53C2313yes—
79C22×Dic53C2316yes—
80Q8×D53—24no—
81C2×C4×D53C2313yes—
82C2×Dic106C2313yes—
83C2×Dic106C2313yes—
84C4⋊C4×C53C2311yescharacteristic
85C4⋊C4×C53C2313yes—
86C22⋊C4×C53C2313yes—
87C22⋊C4×C53C2313yes—
88C42×C53C2313yes—
89C22⋊Dic53C2316yes—
90C22⋊Dic53C2313yes—
91C22⋊Dic53C2316yes—
92C22⋊Dic53C2313yes—
93C22⋊Dic53C2313yes—
94C22⋊Dic53C2313yes—
95C22⋊Dic53C2316yes—
96#11719921C2311yescharacteristic
97#11719921C2313yes—
98#11719921C2311yescharacteristic
99#11719921C2311yescharacteristic
100C4⋊Dic53C2313yes—
101C4⋊Dic53C2313yes—
102C4⋊Dic53C2313yes—
103C4⋊Dic53C2311yescharacteristic
104C4⋊Dic53C2313yes—
105C4⋊Dic53C2313yes—
106#11719919C2313yes—
107#11719919C2316yes—
108#11719919C2313yes—
109#11719919C2313yes—
110#11719919C2313yes—
111#11719919C2311yescharacteristic
112#11719919C2313yes—
113C4×Dic53C2311yescharacteristic
114C4×Dic53C2313yes—
115C4×Dic53C2313yes—
116C4×Dic53C2313yes—
117C4×Dic53C2316yes—
118C4×Dic53C2313yes—
119C23×C53C2413yes—
120C23×C53C2413yes—
121C22×D53C2411yescharacteristic
122Q8×C53#4114yes—
123D4×C53—26no—
124D4×C53—26no—
125D4×C53—46no—
126D4×C53—43no—
127C2×C4×C53—26no—
128C2×C4×C53—26no—
129C2×C4×C53C2413yes—
130C2×C4×C53C2411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C2×C4×C53C2413yes—
134C22⋊D53—43no—
135C22⋊D53—43no—
136C2×Dic53—43no—
137C2×Dic53—43no—
138C2×Dic53—46no—
139C2×Dic53C2411yescharacteristic
140C2×Dic53C2413yes—
141C2×Dic53C2413yes—
142C2×Dic53C2411yescharacteristic
143C2×Dic53C2411yescharacteristic
144C2×Dic53C2413yes—
145C2×Dic53C2413yes—
146C4×D53—26no—
147Dic106—26no—
148Dic106—43no—
149C22×C53—23no—
150C22×C53—23no—
151C22×C53—23no—
152C22×C53—23no—
153C22×C53—23no—
154C22×C53—23no—
155C22×C53C2511yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—23no—
162C4×C53—21no—
163C4×C53C2×#4116yes—
164Dic53—23no—
165Dic53—21no—
166Dic53—23no—
167Dic53—21no—
168Dic53—21no—
169Dic53—23no—
170Dic53—23no—
171#3524—531nomaximal, Sylow
172C2×C53—43no—
173C2×C53—43no—
174C2×C53C22×#4111yescharacteristic
175C2×C53#58511yescharacteristic
176C2×C53#58511yescharacteristic
177D53—41no—
178#559—533no—
179#557—531no—
180#555—533no—
181#553—533no—
182#552—533no—
183#550D5311yescharacteristic, complemented
184#548—531no—
185#548—531no—
186#542—533no—
187#540—533no—
188#529—533no—
189#529—533no—
190#520—531no—
191#519—531no—
192C4×#41—531no—
193C53#352411yescharacteristic, complemented, Sylow
194C2×#41C2×D5311yescharacteristic
195C2×#41C2×D5311yescharacteristic
196C22×Q8—531no—
197C4⋊Q8—533no—
198#127C2×D5313yes—
199#126—531no—
200#126—533no—
201#126—536no—
202#125—533no—
203#125—533no—
204#125—533no—
205#124—533no—
206#124—533no—
207#124—536no—
208#124—533no—
209#123—536no—
210#123—533no—
211#123—536no—
212#123—533no—
213C22⋊Q8—533no—
214C22⋊Q8—533no—
215C22⋊Q8—533no—
216#121—533no—
217#121—533no—
218#121—533no—
219#121—536no—
220#121C2×D5313yes—
221#121C2×D5313yes—
222C4×Q8—531no—
223C4×Q8—531no—
224C4×Q8—531no—
225C4×Q8—533no—
226C4×Q8—533no—
227C4×Q8—533no—
228C4×Q8C2×D5311yescharacteristic
229C4×D4—533no—
230C4×D4—533no—
231C4×D4—533no—
232C4×D4—533no—
233C4×D4—536no—
234C4×D4—533no—
235C4×D4—533no—
236C4×D4C2×D5313yes—
237#117—533no—
238#117—533no—
239#117—533no—
240#117—536no—
241#117—533no—
242C2×C4⋊C4—533no—
243C2×C42—533no—
244#41—24no—
245#41—24no—
246C2×Q8—533no—
247C2×Q8—533no—
248C2×Q8—1064no—
249C2×Q8C22×D5311yescharacteristic
250C2×D4—533no—
251C2×D4—533no—
252C2×D4C22×D5313yes—
253C2×D4C22×D5313yes—
254C2×D4C22×D5316yes—
255C2×D4C22×D5313yes—
256C22×C4—533no—
257C22×C4—533no—
258C22×C4—536no—
259C22×C4—533no—
260C22×C4C22×D5313yes—
261C22×C4C22×D5313yes—
262C4⋊C4—533no—
263C4⋊C4—533no—
264C4⋊C4—533no—
265C4⋊C4—531no—
266C4⋊C4—533no—
267C4⋊C4—533no—
268C4⋊C4—533no—
269C4⋊C4—533no—
270C4⋊C4—536no—
271C4⋊C4—533no—
272C4⋊C4—533no—
273C4⋊C4—531no—
274C4⋊C4—533no—
275C4⋊C4C22×D5313yes—
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—533no—
278C22⋊C4—536no—
279C22⋊C4—533no—
280C22⋊C4—536no—
281C22⋊C4—533no—
282C22⋊C4—536no—
283C22⋊C4—533no—
284C22⋊C4—531no—
285C22⋊C4—533no—
286C22⋊C4—531no—
287C22⋊C4—531no—
288C22⋊C4C22×D5313yes—
289C22⋊C4C22×D5313yes—
290C42—531no—
291C42—533no—
292C42—533no—
293C42—533no—
294C42—536no—
295C42—533no—
296C42C22×D5313yes—
297C23—531no—
298C23C23×D5313yes—
299C23C23×D5313yes—
300Q8—1066no—
301Q8—2123no—
302Q8#1171994714yes—
303D4—26no—
304D4—26no—
305D4—46no—
306D4—43no—
307D4—2123no—
308D4—2123no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—533no—
314C2×C4—531no—
315C2×C4—533no—
316C2×C4—533no—
317C2×C4—533no—
318C2×C4—1066no—
319C2×C4—2123no—
320C2×C4—2123no—
321C2×C4—2126no—
322C2×C4C23×D5311yescharacteristic
323C2×C4C23×D5313yes—
324C2×C4C23×D5311yescharacteristic
325C2×C4C23×D5311yescharacteristic
326C2×C4C23×D5313yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—23no—
331C22—23no—
332C22—23no—
333C22—1061no—
334C22—1061no—
335C22—1061no—
336C22C24×D5311yescharacteristic
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—1063no—
342C4—1061no—
343C4—1063no—
344C4—1061no—
345C4—1061no—
346C4—1063no—
347C4—1063no—
348C4C2×#1171994716yes—
349C2—43no—
350C2—43no—
351C2—2121no—
352C2C22×#1171994711yescharacteristic
353C2#1681145811yescharacteristic
354C2#1681145911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#550C2
hypercenter#550D53
hypocenterC53#3524
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #16811359C2semidirect
2. #16811349C2semidirect
3. #16811337C2semidirect
4. #16811325C2semidirect
5. #16811313C2semidirect
6. #16811280C2semidirect
7. #16811280C2semidirect
8. #16811240C2semidirect
9. #16811227C2semidirect
10. #16811154C2semidirect
11. #16811153C2semidirect
12. #16811106C2semidirect
13. #16811099C2semidirect
14. #41×Dic53C2semidirect
15. C53×#550C2semidirect
183. #550D53semidirect
193. C53#3524semidirect