Finite Group Databasebeta

← #20288712

Subgroups — #20288712

Order 6784.

1122 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811349C211yesmaximal, characteristic, complemented
2#16811320C213yesmaximal, complemented
3#16811319C213yesmaximal, complemented
4#16811293C213yesmaximal, complemented
5#16811292C211yesmaximal, characteristic, complemented
6#16811292C211yesmaximal, characteristic, complemented
7#16811273C213yesmaximal, complemented
8#16811227C213yesmaximal, complemented
9#16811226C213yesmaximal, complemented
10#16811168C213yesmaximal, complemented
11#16811153C213yesmaximal, complemented
12#16811106C211yesmaximal, characteristic, complemented
13#16811105C211yesmaximal, characteristic, complemented
14#41×Dic53C211yesmaximal, characteristic, complemented
15C53×#550C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2216yes—
18#13133216C2213yes—
19#13133213C2213yes—
20#13133213C2213yes—
21#13133212C2213yes—
22#13133210C2213yes—
23#13133209C2213yes—
24#13133208C2213yes—
25#13133207C2211yescharacteristic
26#13133203C2213yes—
27#13133201C2213yes—
28#13133199C2216yes—
29#13133198C2216yes—
30#13133198C2213yes—
31#13133194C2213yes—
32#13133191C2213yes—
33#13133191C2213yes—
34#13133189C2213yes—
35#13133189C2216yes—
36#13133189C2216yes—
37#13133188C2213yes—
38#13133188C2213yes—
39#13133188C2216yes—
40Q8×Dic53C2211yescharacteristic
41Q8×Dic53C2211yescharacteristic
42Q8×Dic53C2211yescharacteristic
43C4×Dic106C2213yes—
44C4×D212C2213yes—
45D4×Dic53C2213yes—
46D4×Dic53C2213yes—
47D4×Dic53C2213yes—
48D4×Dic53C2213yes—
49#13133179C2213yes—
50C4×C22⋊D53C2213yes—
51C4×C22⋊D53C2213yes—
52#13133177C2216yes—
53#13133176C2213yes—
54#13133176C2213yes—
55#13133174C2213yes—
56C2×C4⋊Dic53C2213yes—
57C4⋊C4×D53C2213yes—
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⋊D53C2316yes—
77C2×C22⋊D53C2313yes—
78C2×C22⋊D53C2316yes—
79C22×Dic53C2313yes—
80C22×Dic53C2313yes—
81C22×Dic53C2316yes—
82#11719949—24no—
83#11719947—412no—
84C2×D212C2313yes—
85C2×C4×D53C2313yes—
86C2×C4×D53C2316yes—
87C2×Dic106C2313yes—
88C4⋊C4×C53C2311yescharacteristic
89C4⋊C4×C53C2313yes—
90C22⋊C4×C53C2313yes—
91C22⋊C4×C53C2313yes—
92C42×C53C2313yes—
93C22⋊Dic53C2313yes—
94C22⋊Dic53C2316yes—
95C22⋊Dic53C2313yes—
96C22⋊Dic53C2313yes—
97C22⋊Dic53C2313yes—
98#11719921C2313yes—
99#11719921C2313yes—
100#11719921C2313yes—
101#11719921C2313yes—
102C4⋊Dic53C2313yes—
103C4⋊Dic53C2313yes—
104C4⋊Dic53C2313yes—
105C4⋊Dic53C2311yescharacteristic
106C4⋊Dic53C2313yes—
107C4⋊Dic53C2313yes—
108#11719919C2313yes—
109#11719919C2311yescharacteristic
110#11719919C2316yes—
111#11719919C2311yescharacteristic
112#11719919C2311yescharacteristic
113#11719919C2311yescharacteristic
114#11719919C2313yes—
115C4×Dic53C2311yescharacteristic
116C4×Dic53C2313yes—
117C4×Dic53C2313yes—
118C4×Dic53C2313yes—
119C4×Dic53C2313yes—
120C23×C53C2413yes—
121C23×C53C2413yes—
122C22×D53C2413yes—
123Q8×C53#4114yescomplemented
124D4×C53—26no—
125D4×C53—26no—
126D4×C53—46no—
127D4×C53—43no—
128C2×C4×C53—26no—
129C2×C4×C53—26no—
130C2×C4×C53C2413yes—
131C2×C4×C53C2411yescharacteristic
132C2×C4×C53C2411yescharacteristic
133C2×C4×C53C2411yescharacteristic
134C2×C4×C53C2413yes—
135C22⋊D53—26no—
136C22⋊D53—46no—
137C22⋊D53—46no—
138C22⋊D53—43no—
139C2×Dic53—26no—
140C2×Dic53—46no—
141C2×Dic53—43no—
142C2×Dic53C2413yes—
143C2×Dic53C2411yescharacteristic
144C2×Dic53C2413yes—
145C2×Dic53C2411yescharacteristic
146C2×Dic53C2413yes—
147C2×Dic53C2411yescharacteristic
148C2×Dic53C2411yescharacteristic
149D212—26no—
150C4×D53—26no—
151C4×D53—46no—
152Dic106—26no—
153C22×C53—23no—
154C22×C53—23no—
155C22×C53—23no—
156C22×C53—23no—
157C22×C53—23no—
158C22×C53—23no—
159C22×C53C2511yescharacteristic
160C2×D53—23no—
161C2×D53—23no—
162C2×D53—23no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—23no—
166C4×C53—21no—
167C4×C53C2×#4116yes—
168Dic53—21no—
169Dic53—23no—
170Dic53—23no—
171Dic53—21no—
172Dic53—23no—
173Dic53—21no—
174Dic53H(5,𝔽2)12yes—
175#3517—531nomaximal, Sylow
176C2×C53—43no—
177C2×C53—43no—
178C2×C53C22×#4111yescharacteristic
179C2×C53C2×H(5,𝔽2)11yescharacteristic
180C2×C53#58511yescharacteristic
181D53—43no—
182#557—531no—
183#553—533no—
184#553—533no—
185#550—531no—
186#550—531no—
187#550—533no—
188#550D5311yescharacteristic, complemented
189#547—533no—
190#540—533no—
191#540—533no—
192#532—533no—
193#529—533no—
194#520—531no—
195#520—531no—
196C4×#41—531no—
197C53#351711yescharacteristic, complemented, Sylow
198C2×#41—531no—
199C2×#41—536no—
200C2×#41C2×D5311yescharacteristic
201C2×#41C2×D5311yescharacteristic
202C4⋊Q8—533no—
203#127—533no—
204#127—533no—
205#127C2×D5313yes—
206#126—533no—
207#126—533no—
208#125—531no—
209#125—533no—
210#125—533no—
211#124—533no—
212#123—533no—
213#123—536no—
214#123—533no—
215#123—536no—
216C22⋊Q8—533no—
217#121—533no—
218#121—536no—
219#121—533no—
220#121—536no—
221#121—533no—
222#121—536no—
223#121—533no—
224#121—533no—
225#121C2×D5313yes—
226#121C2×D5313yes—
227C4×Q8—531no—
228C4×Q8—531no—
229C4×Q8—531no—
230C4×Q8—533no—
231C4×Q8C2×D5311yescharacteristic
232C4×D4—533no—
233C4×D4—533no—
234C4×D4—533no—
235C4×D4—533no—
236C4×D4—533no—
237C4×D4—533no—
238C4×D4—533no—
239C4×D4—536no—
240C4×D4—533no—
241C4×D4C2×D5313yes—
242#117—533no—
243#117—533no—
244#117—533no—
245C2×C4⋊C4—533no—
246C2×C4⋊C4—533no—
247C2×C42—533no—
248#41—24no—
249#41—24no—
250#41—1064no—
251#41—21212no—
252C2×Q8—533no—
253C2×Q8C22×D5311yescharacteristic
254C2×D4—533no—
255C2×D4—536no—
256C2×D4—533no—
257C2×D4—536no—
258C2×D4—533no—
259C2×D4C22×D5313yes—
260C2×D4C22×D5313yes—
261C2×D4C22×D5313yes—
262C2×D4C22×D5316yes—
263C22×C4—533no—
264C22×C4—533no—
265C22×C4—536no—
266C22×C4—533no—
267C22×C4—536no—
268C22×C4C22×D5313yes—
269C22×C4C22×D5313yes—
270C4⋊C4—533no—
271C4⋊C4—533no—
272C4⋊C4—533no—
273C4⋊C4—533no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—533no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—536no—
280C4⋊C4—531no—
281C4⋊C4—533no—
282C4⋊C4—533no—
283C4⋊C4C22×D5313yes—
284C4⋊C4C22×D5311yescharacteristic
285C22⋊C4—533no—
286C22⋊C4—536no—
287C22⋊C4—533no—
288C22⋊C4—533no—
289C22⋊C4—533no—
290C22⋊C4—533no—
291C22⋊C4—533no—
292C22⋊C4—533no—
293C22⋊C4—533no—
294C22⋊C4C22×D5313yes—
295C22⋊C4C22×D5313yes—
296C42—531no—
297C42—533no—
298C42—533no—
299C42—533no—
300C42—533no—
301C42C22×D5313yes—
302C23—533no—
303C23C23×D5313yes—
304C23C23×D5313yes—
305Q8—1066no—
306Q8#1171994714yescomplemented
307D4—26no—
308D4—26no—
309D4—43no—
310D4—46no—
311D4—1066no—
312D4—1066no—
313D4—2126no—
314D4—2123no—
315D4—2126no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—533no—
322C2×C4—533no—
323C2×C4—533no—
324C2×C4—531no—
325C2×C4—1066no—
326C2×C4—1066no—
327C2×C4—2126no—
328C2×C4—2123no—
329C2×C4—2126no—
330C2×C4C23×D5311yescharacteristic
331C2×C4C23×D5311yescharacteristic
332C2×C4C23×D5313yes—
333C2×C4C23×D5311yescharacteristic
334C2×C4C23×D5313yes—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—23no—
341C22—1063no—
342C22—1063no—
343C22—1063no—
344C22C24×D5311yescharacteristic
345C4—21no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—532no—
350C4—1061no—
351C4—1063no—
352C4—1063no—
353C4—1061no—
354C4—1063no—
355C4—1061no—
356C4C2×#1171994716yes—
357C2—43no—
358C2—43no—
359C2—2123no—
360C2C22×#1171994711yescharacteristic
361C2H(5,𝔽2)×D5311yescharacteristic
362C2#1681145911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#550C2
hypercenter#550D53
hypocenterC53#3517
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #16811349C2semidirect
2. #16811320C2semidirect
3. #16811319C2semidirect
4. #16811293C2semidirect
5. #16811292C2semidirect
6. #16811292C2semidirect
7. #16811273C2semidirect
8. #16811227C2semidirect
9. #16811226C2semidirect
10. #16811168C2semidirect
11. #16811153C2semidirect
12. #16811106C2semidirect
13. #16811105C2semidirect
14. #41×Dic53C2semidirect
15. C53×#550C2semidirect
123. Q8×C53#41semidirect
188. #550D53semidirect
197. C53#3517semidirect
306. Q8#11719947semidirect