Finite Group Databasebeta

← #20288305

Subgroups — #20288305

Order 6784.

1112 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#16811355C213yesmaximal, complemented
2#16811349C211yesmaximal, characteristic, complemented
3#16811344C213yesmaximal, complemented
4#16811325C213yesmaximal, complemented
5#16811319C213yesmaximal, complemented
6#16811292C211yesmaximal, characteristic, complemented
7#16811284C211yesmaximal, characteristic, complemented
8#16811168C213yesmaximal, complemented
9#16811159C213yesmaximal, complemented
10#16811152C213yesmaximal, complemented
11#41×Dic53C211yesmaximal, characteristic, complemented
12C4×#11719949C211yesmaximal, characteristic, complemented
13C4×#11719947C213yesmaximal, complemented
14C2×Q8×Dic53C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2213yes—
18#13133216C2213yes—
19#13133213C2213yes—
20#13133212C2216yes—
21#13133210C2216yes—
22#13133209C2213yes—
23#13133208C2213yes—
24#13133206C2213yes—
25#13133203C2216yes—
26#13133201C2216yes—
27#13133199C2216yes—
28#13133193C2213yes—
29#13133191C2213yes—
30#13133191C2213yes—
31#13133189C2216yes—
32#13133188C2213yes—
33Q8×Dic53C2211yescharacteristic
34Q8×Dic53C2211yescharacteristic
35Q8×Dic53C2213yes—
36Q8×Dic53C2211yescharacteristic
37Q8×Dic53C2213yes—
38C4×Dic106C2213yes—
39#13133182C2213yes—
40C4×D212C2213yes—
41D4×Dic53C2213yes—
42D4×Dic53C2213yes—
43D4×Dic53C2213yes—
44#13133179C2213yes—
45C4×C22⋊D53C2213yes—
46C4×C22⋊D53C2213yes—
47#13133177C2213yes—
48#13133177C2213yes—
49#13133177C2216yes—
50#13133176C2213yes—
51#13133175C2213yes—
52#13133175C2213yes—
53#13133173C2211yescharacteristic
54#13133173C2213yes—
55C2×C4⋊Dic53C2213yes—
56C2×C4×Dic53C2213yes—
57C2×C4×Dic53C2213yes—
58C42×D53C2213yes—
59C2×#41×C53C2211yescharacteristic
60C22×Q8×C53C2211yescharacteristic
61#124×C53C2213yes—
62C22⋊Q8×C53C2213yes—
63#121×C53C2213yes—
64C4×Q8×C53C2211yescharacteristic
65C4×D4×C53C2213yes—
66#41×C53—24no—
67C2×Q8×C53—24no—
68C2×Q8×C53C2313yes—
69C2×Q8×C53C2311yescharacteristic
70C2×D4×C53C2313yes—
71C2×D4×C53C2313yes—
72C22×C4×C53C2313yes—
73C22×C4×C53C2313yes—
74C2×C22⋊D53C2313yes—
75C2×C22⋊D53C2313yes—
76C2×C22⋊D53C2316yes—
77C22×Dic53C2313yes—
78C22×Dic53C2313yes—
79C22×Dic53C2313yes—
80C22×Dic53C2311yescharacteristic
81#11719949—24no—
82#11719947—212no—
83C2×D212C2313yes—
84C2×C4×D53C2313yes—
85C2×C4×D53C2313yes—
86C2×C4×D53C2313yes—
87C2×Dic106C2313yes—
88C4⋊C4×C53C2313yes—
89C4⋊C4×C53C2313yes—
90C22⋊C4×C53C2311yescharacteristic
91C22⋊C4×C53C2313yes—
92C22⋊C4×C53C2316yes—
93C42×C53C2313yes—
94C22⋊Dic53C2311yescharacteristic
95C22⋊Dic53C2313yes—
96C22⋊Dic53C2316yes—
97#11719921C2313yes—
98#11719921C2313yes—
99#11719921C2316yes—
100#11719921C2316yes—
101C4⋊Dic53C2313yes—
102C4⋊Dic53C2313yes—
103C4⋊Dic53C2316yes—
104C4⋊Dic53C2313yes—
105C4⋊Dic53C2313yes—
106#11719919C2311yescharacteristic
107#11719919C2313yes—
108#11719919C2311yescharacteristic
109#11719919C2313yes—
110#11719919C2316yes—
111C4×Dic53C2311yescharacteristic
112C4×Dic53C2313yes—
113C4×Dic53C2313yes—
114C4×Dic53C2311yescharacteristic
115C4×Dic53C2313yes—
116C4×Dic53C2313yes—
117C4×Dic53C2313yes—
118C4×Dic53C2313yes—
119C4×Dic53C2313yes—
120C23×C53C2413yes—
121C23×C53C2411yescharacteristic
122C22×D53C2413yes—
123Q8×C53—26no—
124Q8×C53#4114yescomplemented
125D4×C53—26no—
126D4×C53—26no—
127C2×C4×C53—26no—
128C2×C4×C53—26no—
129C2×C4×C53C2413yes—
130C2×C4×C53C2411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C2×C4×C53C2413yes—
133C2×C4×C53C2413yes—
134C22⋊D53—26no—
135C22⋊D53—26no—
136C22⋊D53—46no—
137C2×Dic53—26no—
138C2×Dic53—43no—
139C2×Dic53—43no—
140C2×Dic53#4114yescomplemented
141C2×Dic53C2413yes—
142C2×Dic53C2411yescharacteristic
143C2×Dic53C2413yes—
144C2×Dic53C2411yescharacteristic
145C2×Dic53C2413yes—
146C2×Dic53C2411yescharacteristic
147C2×Dic53C2411yescharacteristic
148D212—26no—
149C4×D53—26no—
150C4×D53—26no—
151C4×D53—43no—
152Dic106—26no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—23no—
156C22×C53—23no—
157C22×C53—23no—
158C22×C53C2×#4112yes—
159C22×C53C2511yescharacteristic
160C2×D53—23no—
161C2×D53—23no—
162C2×D53—23no—
163C4×C53—23no—
164C4×C53—21no—
165C4×C53—23no—
166C4×C53—21no—
167C4×C53C2×#4116yes—
168Dic53—23no—
169Dic53—21no—
170Dic53—23no—
171Dic53—21no—
172Dic53—23no—
173Dic53C2×#4112yes—
174Dic53C2×#4112yes—
175#3480—531nomaximal, Sylow
176C2×C53—22no—
177C2×C53—43no—
178C2×C53C22×#4111yescharacteristic
179C2×C53C22×#4111yescharacteristic
180C2×C53#58511yescharacteristic
181D53—43no—
182#559—533no—
183#557—531no—
184#556—533no—
185#553—533no—
186#553—533no—
187#550—531no—
188#548—531no—
189#548D5311yescharacteristic, complemented
190#532—533no—
191#529—533no—
192#529—533no—
193C4×#41—531no—
194C4×#41—531no—
195C4×#41—533no—
196C2×C4×Q8—531no—
197C53#348011yescharacteristic, complemented, Sylow
198C2×#41—531no—
199C2×#41—533no—
200C2×#41C2×D5311yescharacteristic
201C22×Q8C2×D5311yescharacteristic
202C4⋊Q8—533no—
203#127—533no—
204#126—536no—
205#126—536no—
206#125—533no—
207#125—533no—
208#124—533no—
209#124—536no—
210#124C2×D5313yes—
211#123—536no—
212#123—536no—
213C22⋊Q8—533no—
214C22⋊Q8C2×D5313yes—
215#121—533no—
216#121—536no—
217#121—533no—
218#121—533no—
219#121C2×D5313yes—
220C4×Q8—531no—
221C4×Q8—533no—
222C4×Q8—531no—
223C4×Q8—533no—
224C4×Q8—531no—
225C4×Q8—533no—
226C4×Q8—533no—
227C4×Q8C2×D5311yescharacteristic
228C4×D4—533no—
229C4×D4—533no—
230C4×D4—533no—
231C4×D4—533no—
232C4×D4—533no—
233C4×D4—533no—
234C4×D4—533no—
235C4×D4—533no—
236C4×D4—536no—
237C4×D4—533no—
238C4×D4C2×D5313yes—
239#117—533no—
240#117—533no—
241#117—533no—
242#117—531no—
243#117—533no—
244C2×C4⋊C4—533no—
245C2×C42—533no—
246C2×C42—533no—
247C2×C42—533no—
248#41—24no—
249#41—10612no—
250#41—1064no—
251C2×Q8—24no—
252C2×Q8—533no—
253C2×Q8C22×D5313yes—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—533no—
256C2×D4—533no—
257C2×D4—536no—
258C2×D4—533no—
259C2×D4C22×D5313yes—
260C2×D4C22×D5313yes—
261C22×C4—531no—
262C22×C4—533no—
263C22×C4—533no—
264C22×C4—533no—
265C22×C4—533no—
266C22×C4—533no—
267C22×C4—533no—
268C22×C4C22×D5313yes—
269C22×C4C22×D5313yes—
270C4⋊C4—533no—
271C4⋊C4—533no—
272C4⋊C4—536no—
273C4⋊C4—531no—
274C4⋊C4—531no—
275C4⋊C4—533no—
276C4⋊C4—533no—
277C4⋊C4—533no—
278C4⋊C4—533no—
279C4⋊C4—536no—
280C4⋊C4C22×D5313yes—
281C4⋊C4C22×D5313yes—
282C22⋊C4—531no—
283C22⋊C4—533no—
284C22⋊C4—536no—
285C22⋊C4—533no—
286C22⋊C4—536no—
287C22⋊C4—533no—
288C22⋊C4—536no—
289C22⋊C4C22×D5311yescharacteristic
290C22⋊C4C22×D5313yes—
291C22⋊C4C22×D5316yes—
292C42—531no—
293C42—531no—
294C42—533no—
295C42—533no—
296C42—533no—
297C42—533no—
298C42—533no—
299C42—533no—
300C42—533no—
301C42C22×D5313yes—
302C23—533no—
303C23C23×D5313yes—
304C23C23×D5311yescharacteristic
305Q8—26no—
306Q8—1066no—
307Q8#1171994714yescomplemented
308D4—26no—
309D4—26no—
310D4—1066no—
311D4—1066no—
312D4—1066no—
313D4—2126no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—534no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—533no—
321C2×C4—531no—
322C2×C4—533no—
323C2×C4—533no—
324C2×C4—1066no—
325C2×C4—1066no—
326C2×C4—1066no—
327C2×C4—2123no—
328C2×C4—2123no—
329C2×C4—2123no—
330C2×C4C23×D5311yescharacteristic
331C2×C4C23×D5313yes—
332C2×C4C23×D5311yescharacteristic
333C2×C4C23×D5313yes—
334C2×C4C23×D5313yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—1063no—
341C22—1063no—
342C22—1063no—
343C22#41×D5312yes—
344C22C24×D5311yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—532no—
350C4—532no—
351C4—1063no—
352C4—1061no—
353C4—1063no—
354C4—1061no—
355C4—1063no—
356C4C2×#1171994716yes—
357C2—22no—
358C2—43no—
359C2—2123no—
360C2C2×#41×D5311yescharacteristic
361C2C22×#1171994711yescharacteristic
362C2#1681146011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#548C2
hypercenter#548D53
hypocenterC53#3480
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16811355C2semidirect
2. #16811349C2semidirect
3. #16811344C2semidirect
4. #16811325C2semidirect
5. #16811319C2semidirect
6. #16811292C2semidirect
7. #16811284C2semidirect
8. #16811168C2semidirect
9. #16811159C2semidirect
10. #16811152C2semidirect
11. #41×Dic53C2semidirect
12. C4×#11719949C2semidirect
13. C4×#11719947C2semidirect
14. C2×Q8×Dic53C2semidirect
15. C53×#548C2semidirect
124. Q8×C53#41semidirect
140. C2×Dic53#41semidirect
189. #548D53semidirect
197. C53#3480semidirect
307. Q8#11719947semidirect