Finite Group Databasebeta

← #20288701

Subgroups — #20288701

Order 6784.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811349C211yesmaximal, characteristic, complemented
2#16811342C213yesmaximal, complemented
3#16811331C213yesmaximal, complemented
4#16811320C213yesmaximal, complemented
5#16811313C213yesmaximal, complemented
6Q8×C22⋊D53C211yesmaximal, characteristic, complemented
7#16811280C211yesmaximal, characteristic, complemented
8#16811240C213yesmaximal, complemented
9#16811233C213yesmaximal, complemented
10#16811155C213yesmaximal, complemented
11#16811152C213yesmaximal, complemented
12#16811106C211yesmaximal, characteristic, complemented
13#16811101C211yesmaximal, characteristic, complemented
14C4×Q8×D53C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17#13133216C2213yes—
18#13133215C2213yes—
19#13133210C2213yes—
20#13133210C2216yes—
21#13133210C2213yes—
22#13133210C2216yes—
23#13133209C2213yes—
24#13133208C2213yes—
25#13133208C2213yes—
26#13133207C2216yes—
27#13133204C2213yes—
28#13133204C2213yes—
29#13133199C2213yes—
30#13133199C2211yescharacteristic
31#13133198C2216yes—
32#13133197C2213yes—
33#13133196C2213yes—
34#13133196C2213yes—
35#13133194C2216yes—
36#13133193C2213yes—
37#13133188C2213yes—
38#13133188C2213yes—
39Q8×Dic53C2211yescharacteristic
40Q8×Dic53C2211yescharacteristic
41Q8×Dic53C2213yes—
42Q8×Dic53C2211yescharacteristic
43Q8×Dic53C2213yes—
44C4×Dic106C2213yes—
45C4×Dic106C2213yes—
46#13133182C2213yes—
47D4×Dic53C2213yes—
48D4×Dic53C2213yes—
49C4×C22⋊D53C2213yes—
50C4×C22⋊D53C2213yes—
51#13133177C2216yes—
52#13133176C2213yes—
53#13133176C2213yes—
54#13133176C2213yes—
55#13133173C2216yes—
56C2×C4⋊Dic53C2213yes—
57C4⋊C4×D53C2213yes—
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⋊D53C2311yescharacteristic
76C22×Dic53C2316yes—
77Q8×D53—24no—
78C2×C4×D53C2311yescharacteristic
79C2×C4×D53C2313yes—
80C2×C4×D53C2313yes—
81C2×Dic106C2313yes—
82C2×Dic106C2313yes—
83C4⋊C4×C53C2313yes—
84C4⋊C4×C53C2313yes—
85C22⋊C4×C53C2311yescharacteristic
86C22⋊C4×C53C2313yes—
87C22⋊C4×C53C2316yes—
88C42×C53C2313yes—
89C22⋊Dic53C2316yes—
90C22⋊Dic53C2313yes—
91C22⋊Dic53C2311yescharacteristic
92C22⋊Dic53C2313yes—
93C22⋊Dic53C2313yes—
94C22⋊Dic53C2313yes—
95C22⋊Dic53C2313yes—
96#11719921C2313yes—
97#11719921C2311yescharacteristic
98C4⋊Dic53C2313yes—
99C4⋊Dic53C2313yes—
100C4⋊Dic53C2313yes—
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2313yes—
103C4⋊Dic53C2313yes—
104C4⋊Dic53C2313yes—
105C4⋊Dic53C2313yes—
106#11719919C2316yes—
107#11719919C2313yes—
108#11719919C2316yes—
109#11719919C2316yes—
110#11719919C2313yes—
111#11719919C2313yes—
112#11719919C2311yescharacteristic
113C4×Dic53C2313yes—
114C4×Dic53C2313yes—
115C4×Dic53C2313yes—
116C4×Dic53C2316yes—
117C4×Dic53C2313yes—
118C4×Dic53C2313yes—
119C23×C53C2413yes—
120C23×C53C2411yescharacteristic
121C22×D53C2411yescharacteristic
122Q8×C53—26no—
123Q8×C53#4114yes—
124D4×C53—26no—
125D4×C53—43no—
126C2×C4×C53—26no—
127C2×C4×C53—26no—
128C2×C4×C53C2413yes—
129C2×C4×C53C2411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C2×C4×C53C2413yes—
132C2×C4×C53C2413yes—
133C22⋊D53—26no—
134C22⋊D53—22no—
135C2×Dic53—46no—
136C2×Dic53C2411yescharacteristic
137C2×Dic53C2413yes—
138C2×Dic53C2413yes—
139C2×Dic53C2413yes—
140C2×Dic53C2411yescharacteristic
141C2×Dic53C2413yes—
142C2×Dic53C2411yescharacteristic
143C4×D53—22no—
144C4×D53—26no—
145C4×D53—43no—
146Dic106—26no—
147Dic106—43no—
148C22×C53—21no—
149C22×C53—23no—
150C22×C53—23no—
151C22×C53—21no—
152C22×C53—23no—
153C22×C53—21no—
154C22×C53C2511yescharacteristic
155C2×D53—21no—
156C2×D53—21no—
157C2×D532−1+412yes—
158C4×C53—21no—
159C4×C53—23no—
160C4×C53—21no—
161C4×C53—23no—
162C4×C53C2×#4116yes—
163Dic53—23no—
164Dic53—21no—
165Dic53—23no—
166Dic53—21no—
167Dic53—21no—
168Dic53—23no—
169Dic53—23no—
170#3516—531nomaximal, Sylow
171C2×C53—41no—
172C2×C53—43no—
173C2×C53C22×#4111yescharacteristic
174C2×C53C2×2−1+411yescharacteristic
175C2×C53#58511yescharacteristic
176D53—22no—
177#557—531no—
178#556—533no—
179#554—533no—
180#553—533no—
181#552—533no—
182D4×Q8—531no—
183#548—531no—
184#548D5311yescharacteristic, complemented
185#542—533no—
186#541—533no—
187#529—533no—
188#529—533no—
189#520—531no—
190#519—531no—
191C2×C4×Q8—531no—
192C53#351611yescharacteristic, complemented, Sylow
193C2×#41C2×D5311yescharacteristic
194C22×Q8—531no—
195C22×Q8C2×D5311yescharacteristic
196C4⋊Q8—533no—
197C4⋊Q8—533no—
198#126—536no—
199#126—533no—
200#126—533no—
201#126—536no—
202#125—533no—
203#125—536no—
204#125—533no—
205#125—533no—
206#124—533no—
207#124—533no—
208#124C2×D5313yes—
209#123—536no—
210#123—531no—
211#123—533no—
212C22⋊Q8—533no—
213C22⋊Q8—533no—
214C22⋊Q8—533no—
215C22⋊Q8—533no—
216C22⋊Q8—536no—
217C22⋊Q8C2×D5313yes—
218#121—533no—
219#121—533no—
220#121C2×D5313yes—
221C4×Q8—531no—
222C4×Q8—533no—
223C4×Q8—531no—
224C4×Q8—533no—
225C4×Q8—531no—
226C4×Q8—533no—
227C4×Q8—533no—
228C4×Q8—533no—
229C4×Q8C2×D5311yescharacteristic
230C4×D4—533no—
231C4×D4—533no—
232C4×D4—533no—
233C4×D4—533no—
234C4×D4—536no—
235C4×D4C2×D5313yes—
236#117—533no—
237#117—533no—
238#117—533no—
239#117—536no—
240C2×C4⋊C4—533no—
241C2×C4⋊C4—533no—
242C2×C42—533no—
243#41—24no—
244C2×Q8—24no—
245C2×Q8—533no—
246C2×Q8—533no—
247C2×Q8—1064no—
248C2×Q8C22×D5313yes—
249C2×Q8C22×D5311yescharacteristic
250C2×D4—531no—
251C2×D4—533no—
252C2×D4C22×D5313yes—
253C2×D4C22×D5313yes—
254C22×C4—536no—
255C22×C4—531no—
256C22×C4—533no—
257C22×C4—533no—
258C22×C4C22×D5313yes—
259C22×C4C22×D5313yes—
260C4⋊C4—533no—
261C4⋊C4—533no—
262C4⋊C4—533no—
263C4⋊C4—536no—
264C4⋊C4—531no—
265C4⋊C4—533no—
266C4⋊C4—533no—
267C4⋊C4—533no—
268C4⋊C4—533no—
269C4⋊C4—536no—
270C4⋊C4—533no—
271C4⋊C4—536no—
272C4⋊C4—533no—
273C4⋊C4—531no—
274C4⋊C4—533no—
275C4⋊C4C22×D5313yes—
276C4⋊C4C22×D5313yes—
277C22⋊C4—533no—
278C22⋊C4—531no—
279C22⋊C4—533no—
280C22⋊C4—533no—
281C22⋊C4—536no—
282C22⋊C4—533no—
283C22⋊C4—533no—
284C22⋊C4—531no—
285C22⋊C4—533no—
286C22⋊C4C22×D5313yes—
287C22⋊C4C22×D5316yes—
288C22⋊C4C22×D5311yescharacteristic
289C42—533no—
290C42—533no—
291C42—533no—
292C42—536no—
293C42—533no—
294C42—533no—
295C42C22×D5313yes—
296C23—531no—
297C23C23×D5311yescharacteristic
298C23C23×D5313yes—
299Q8—26no—
300Q8—1066no—
301Q8—2123no—
302Q8#1171994714yes—
303D4—26no—
304D4—43no—
305D4—1062no—
306D4—1066no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—531no—
310C2×C4—533no—
311C2×C4—531no—
312C2×C4—533no—
313C2×C4—531no—
314C2×C4—533no—
315C2×C4—533no—
316C2×C4—1062no—
317C2×C4—1066no—
318C2×C4—2123no—
319C2×C4—2126no—
320C2×C4C23×D5311yescharacteristic
321C2×C4C23×D5313yes—
322C2×C4C23×D5313yes—
323C2×C4C23×D5311yescharacteristic
324C2×C4C23×D5313yes—
325C22—21no—
326C22—23no—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—532no—
332C22—1061no—
333C22—1061no—
334C22C24×D5311yescharacteristic
335C4—21no—
336C4—23no—
337C4—21no—
338C4—23no—
339C4—1063no—
340C4—1061no—
341C4—1063no—
342C4—1061no—
343C4—1063no—
344C4—1063no—
345C4—1061no—
346C4C2×#1171994716yes—
347C2—41no—
348C2—43no—
349C2—1062no—
350C2C22×#1171994711yescharacteristic
351C22−1+4×D5311yescharacteristic
352C2#1681145911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#548C2
hypercenter#548D53
hypocenterC53#3516
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16811349C2semidirect
2. #16811342C2semidirect
3. #16811331C2semidirect
4. #16811320C2semidirect
5. #16811313C2semidirect
6. Q8×C22⋊D53C2semidirect
7. #16811280C2semidirect
8. #16811240C2semidirect
9. #16811233C2semidirect
10. #16811155C2semidirect
11. #16811152C2semidirect
12. #16811106C2semidirect
13. #16811101C2semidirect
14. C4×Q8×D53C2semidirect
15. C53×#548C2semidirect
184. #548D53semidirect
192. C53#3516semidirect