Finite Group Databasebeta

← #20288816

Subgroups — #20288816

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#16811356C213yesmaximal, complemented
2#16811352C211yesmaximal, characteristic, complemented
3#16811338C213yesmaximal, complemented
4#16811321C213yesmaximal, complemented
5#16811308C213yesmaximal, complemented
6#16811292C211yesmaximal, characteristic, complemented
7#16811285C211yesmaximal, characteristic, complemented
8#16811280C211yesmaximal, characteristic, complemented
9#16811244C213yesmaximal, complemented
10#16811224C213yesmaximal, complemented
11#16811155C213yesmaximal, complemented
12#16811151C213yesmaximal, complemented
13#16811099C211yesmaximal, characteristic, complemented
14C4×#11719949C211yesmaximal, characteristic, complemented
15C53×#520C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic
18#13133217C2213yes—
19#13133213C2213yes—
20#13133212C2216yes—
21#13133212C2211yescharacteristic
22#13133210C2213yes—
23#13133208C2213yes—
24#13133208C2213yes—
25#13133207C2213yes—
26#13133205C2213yes—
27#13133204C2213yes—
28#13133203C2213yes—
29#13133203C2216yes—
30#13133202C2213yes—
31#13133200C2216yes—
32#13133199C2213yes—
33#13133199C2216yes—
34#13133197C2213yes—
35#13133196C2213yes—
36#13133195C2213yes—
37#13133192C2213yes—
38#13133191C2213yes—
39#13133191C2216yes—
40#13133190C2213yes—
41#13133188C2213yes—
42#13133188C2213yes—
43Q8×Dic53C2211yescharacteristic
44Q8×Dic53C2211yescharacteristic
45C4×Dic106C2213yes—
46C4×Dic106C2213yes—
47#13133182C2213yes—
48C4×D212C2213yes—
49C4×D212C2213yes—
50#13133179C2213yes—
51C4×C22⋊D53C2213yes—
52C4×C22⋊D53C2213yes—
53C4×C22⋊D53C2216yes—
54#13133176C2216yes—
55#13133175C2213yes—
56#13133175C2213yes—
57#13133174C2213yes—
58C42×D53C2213yes—
59C2×#41×C53C2211yescharacteristic
60C4×Q8×C53C2211yescharacteristic
61C4×Q8×C53C2211yescharacteristic
62C4×D4×C53C2213yes—
63C4×D4×C53C2213yes—
64#117×C53C2213yes—
65C2×C4⋊C4×C53C2213yes—
66#41×C53—24no—
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53C2313yes—
69C22×C4×C53C2313yes—
70C22×C4×C53C2316yes—
71C2×C22⋊D53C2313yes—
72C2×C22⋊D53C2316yes—
73C2×C22⋊D53C2313yes—
74#11719949—24no—
75Q8×D53—24no—
76C2×D212C2313yes—
77C2×D212C2313yes—
78C2×C4×D53C2313yes—
79C2×C4×D53C2313yes—
80C2×C4×D53C2313yes—
81C2×C4×D53C2313yes—
82C2×Dic106C2313yes—
83C2×Dic106C2313yes—
84C4⋊C4×C53C2313yes—
85C4⋊C4×C53C2313yes—
86C4⋊C4×C53C2313yes—
87C4⋊C4×C53C2311yescharacteristic
88C22⋊C4×C53C2313yes—
89C22⋊C4×C53C2313yes—
90C42×C53C2313yes—
91C42×C53C2313yes—
92C22⋊Dic53C2313yes—
93C22⋊Dic53C2316yes—
94C22⋊Dic53C2313yes—
95#11719921C2311yescharacteristic
96#11719921C2313yes—
97#11719921C2316yes—
98#11719921C2311yescharacteristic
99#11719921C2313yes—
100#11719921C2316yes—
101#11719921C2311yescharacteristic
102#11719921C2313yes—
103C4⋊Dic53C2313yes—
104C4⋊Dic53C2313yes—
105C4⋊Dic53C2313yes—
106C4⋊Dic53C2316yes—
107C4⋊Dic53C2311yescharacteristic
108#11719919C2313yes—
109#11719919C2313yes—
110#11719919C2311yescharacteristic
111#11719919C2313yes—
112#11719919C2313yes—
113#11719919C2313yes—
114C4×Dic53C2311yescharacteristic
115C4×Dic53C2313yes—
116C4×Dic53C2313yes—
117C4×Dic53C2313yes—
118C4×Dic53C2316yes—
119C23×C53C2413yes—
120C22×D53C2411yescharacteristic
121C22×D53C2413yes—
122Q8×C53#4114yes—
123D4×C53—26no—
124C2×C4×C53—26no—
125C2×C4×C53—46no—
126C2×C4×C53C2411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C2×C4×C53C2413yes—
129C2×C4×C53C2413yes—
130C2×C4×C53C2411yescharacteristic
131C2×C4×C53C2413yes—
132C22⋊D53—43no—
133C22⋊D53—46no—
134C22⋊D53—43no—
135C2×Dic53C2413yes—
136C2×Dic53C2413yes—
137C2×Dic53C2413yes—
138C2×Dic53C2411yescharacteristic
139C2×Dic53C2411yescharacteristic
140C2×Dic53C2411yescharacteristic
141D212—26no—
142D212—43no—
143C4×D53—26no—
144C4×D53—26no—
145C4×D53—43no—
146C4×D53—43no—
147Dic106—26no—
148Dic106—43no—
149C22×C53—23no—
150C22×C53—23no—
151C22×C53—23no—
152C22×C53C2511yescharacteristic
153C2×D53—21no—
154C2×D53—23no—
155C2×D53—23no—
156C2×D53—23no—
157C2×D53—21no—
158C2×D53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—23no—
163C4×C53—23no—
164C4×C53C2×#4116yes—
165Dic53—23no—
166Dic53—21no—
167Dic53—23no—
168Dic53—23no—
169Dic53—21no—
170Dic53—21no—
171#3524—531nomaximal, Sylow
172C2×C53—43no—
173C2×C53C22×#4111yescharacteristic
174C2×C53#58511yescharacteristic
175C2×C53#58511yescharacteristic
176D53—41no—
177D53—43no—
178#559—533no—
179#557—531no—
180#555—533no—
181#553—533no—
182#552—533no—
183#550—531no—
184#548—531no—
185#548—531no—
186#542—533no—
187#540—533no—
188#529—533no—
189#529—533no—
190#520D5311yescharacteristic, complemented
191#519—531no—
192C4×#41—531no—
193C53#352411yescharacteristic, complemented, Sylow
194C2×#41—531no—
195C2×#41C2×D5311yescharacteristic
196C22×Q8—531no—
197C4⋊Q8—533no—
198#127—533no—
199#126—531no—
200#126—536no—
201#126—533no—
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—536no—
219#121—533no—
220#121—533no—
221#121—533no—
222C4×Q8—531no—
223C4×Q8—531no—
224C4×Q8—533no—
225C4×Q8—533no—
226C4×Q8—533no—
227C4×Q8C2×D5311yescharacteristic
228C4×Q8C2×D5311yescharacteristic
229C4×D4—533no—
230C4×D4—533no—
231C4×D4—536no—
232C4×D4—533no—
233C4×D4—533no—
234C4×D4—533no—
235C4×D4C2×D5313yes—
236C4×D4C2×D5313yes—
237#117—536no—
238#117—533no—
239#117—533no—
240#117—533no—
241#117C2×D5313yes—
242C2×C4⋊C4C2×D5313yes—
243C2×C42—533no—
244#41—24no—
245#41—1064no—
246C2×Q8—533no—
247C2×Q8—533no—
248C2×Q8—1064no—
249C2×Q8C22×D5311yescharacteristic
250C2×D4—533no—
251C2×D4—536no—
252C2×D4—533no—
253C2×D4—533no—
254C2×D4—533no—
255C2×D4C22×D5313yes—
256C22×C4—533no—
257C22×C4—533no—
258C22×C4—533no—
259C22×C4—533no—
260C22×C4C22×D5313yes—
261C22×C4C22×D5316yes—
262C4⋊C4—533no—
263C4⋊C4—533no—
264C4⋊C4—533no—
265C4⋊C4—533no—
266C4⋊C4—533no—
267C4⋊C4—536no—
268C4⋊C4—531no—
269C4⋊C4—533no—
270C4⋊C4—533no—
271C4⋊C4—531no—
272C4⋊C4—533no—
273C4⋊C4C22×D5313yes—
274C4⋊C4C22×D5313yes—
275C4⋊C4C22×D5313yes—
276C4⋊C4C22×D5311yescharacteristic
277C22⋊C4—533no—
278C22⋊C4—536no—
279C22⋊C4—533no—
280C22⋊C4—531no—
281C22⋊C4—533no—
282C22⋊C4—536no—
283C22⋊C4—531no—
284C22⋊C4—531no—
285C22⋊C4—533no—
286C22⋊C4—536no—
287C22⋊C4—533no—
288C22⋊C4C22×D5313yes—
289C22⋊C4C22×D5313yes—
290C42—531no—
291C42—533no—
292C42—533no—
293C42—536no—
294C42—533no—
295C42C22×D5313yes—
296C42C22×D5313yes—
297C23—531no—
298C23—533no—
299C23C23×D5313yes—
300Q8—1066no—
301Q8—2123no—
302Q8#1171994514yes—
303D4—26no—
304D4—1066no—
305D4—2123no—
306D4—2123no—
307D4—2123no—
308D4—2126no—
309C2×C4—26no—
310C2×C4—46no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—533no—
314C2×C4—531no—
315C2×C4—533no—
316C2×C4—533no—
317C2×C4—1066no—
318C2×C4—1066no—
319C2×C4—2123no—
320C2×C4—2123no—
321C2×C4C23×D5313yes—
322C2×C4C23×D5311yescharacteristic
323C2×C4C23×D5311yescharacteristic
324C2×C4C23×D5313yes—
325C2×C4C23×D5311yescharacteristic
326C2×C4C23×D5313yes—
327C22—23no—
328C22—23no—
329C22—23no—
330C22—1061no—
331C22—1063no—
332C22—1063no—
333C22—1061no—
334C22—1063no—
335C22—1061no—
336C22C24×D5311yescharacteristic
337C4—21no—
338C4—23no—
339C4—23no—
340C4—21no—
341C4—21no—
342C4—1061no—
343C4—1063no—
344C4—1061no—
345C4—1061no—
346C4—1063no—
347C4—1063no—
348C4C2×#1171994516yes—
349C2—43no—
350C2—2121no—
351C2—2123no—
352C2C22×#1171994511yescharacteristic
353C2#1681145811yescharacteristic
354C2#1681146011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#520C2
hypercenter#520D53
hypocenterC53#3524
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #16811356C2semidirect
2. #16811352C2semidirect
3. #16811338C2semidirect
4. #16811321C2semidirect
5. #16811308C2semidirect
6. #16811292C2semidirect
7. #16811285C2semidirect
8. #16811280C2semidirect
9. #16811244C2semidirect
10. #16811224C2semidirect
11. #16811155C2semidirect
12. #16811151C2semidirect
13. #16811099C2semidirect
14. C4×#11719949C2semidirect
15. C53×#520C2semidirect
190. #520D53semidirect
193. C53#3524semidirect