Finite Group Databasebeta

← #20288715

Subgroups — #20288715

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#16811323C213yesmaximal, complemented
3#16811321C213yesmaximal, complemented
4#16811295C211yesmaximal, characteristic, complemented
5#16811294C213yesmaximal, complemented
6#16811292C211yesmaximal, characteristic, complemented
7#16811292C211yesmaximal, characteristic, complemented
8#16811274C213yesmaximal, complemented
9#16811228C213yesmaximal, complemented
10#16811224C213yesmaximal, complemented
11#16811170C213yesmaximal, complemented
12#16811151C213yesmaximal, complemented
13#16811105C211yesmaximal, characteristic, complemented
14C4×#11719949C211yesmaximal, characteristic, complemented
15C53×#520C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719949C2211yescharacteristic
18C2×#11719945C2216yes—
19#13133216C2213yes—
20#13133214C2213yes—
21#13133213C2213yes—
22#13133213C2213yes—
23#13133212C2213yes—
24#13133210C2213yes—
25#13133209C2213yes—
26#13133208C2213yes—
27#13133207C2211yescharacteristic
28#13133203C2213yes—
29#13133202C2216yes—
30#13133202C2213yes—
31#13133200C2216yes—
32#13133199C2213yes—
33#13133195C2213yes—
34#13133192C2213yes—
35#13133192C2213yes—
36#13133191C2213yes—
37#13133191C2213yes—
38#13133191C2216yes—
39#13133190C2213yes—
40#13133190C2216yes—
41#13133190C2216yes—
42#13133188C2213yes—
43#13133188C2213yes—
44Q8×Dic53C2211yescharacteristic
45Q8×Dic53C2211yescharacteristic
46C4×Dic106C2213yes—
47C4×D212C2213yes—
48C4×D212C2213yes—
49C4×D212C2213yes—
50#13133179C2213yes—
51#13133179C2213yes—
52C4×C22⋊D53C2213yes—
53C4×C22⋊D53C2213yes—
54C4×C22⋊D53C2216yes—
55#13133175C2213yes—
56#13133174C2213yes—
57C4⋊C4×D53C2213yes—
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—
74C2×C22⋊D53C2316yes—
75#11719949—24no—
76#11719949—24no—
77#11719945—412no—
78C2×D212C2313yes—
79C2×D212C2313yes—
80C2×D212C2313yes—
81C2×D212C2316yes—
82C2×C4×D53C2313yes—
83C2×C4×D53C2313yes—
84C2×C4×D53C2313yes—
85C2×C4×D53C2313yes—
86C2×C4×D53C2316yes—
87C2×Dic106C2313yes—
88C4⋊C4×C53C2313yes—
89C4⋊C4×C53C2313yes—
90C4⋊C4×C53C2313yes—
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53C2313yes—
93C22⋊C4×C53C2313yes—
94C42×C53C2313yes—
95C42×C53C2313yes—
96C22⋊Dic53C2313yes—
97C22⋊Dic53C2313yes—
98#11719921C2313yes—
99#11719921C2313yes—
100#11719921C2316yes—
101#11719921C2313yes—
102#11719921C2313yes—
103#11719921C2313yes—
104#11719921C2313yes—
105C4⋊Dic53C2313yes—
106C4⋊Dic53C2313yes—
107C4⋊Dic53C2311yescharacteristic
108C4⋊Dic53C2313yes—
109#11719919C2313yes—
110#11719919C2311yescharacteristic
111#11719919C2311yescharacteristic
112#11719919C2311yescharacteristic
113#11719919C2316yes—
114#11719919C2311yescharacteristic
115#11719919C2313yes—
116C4×Dic53C2311yescharacteristic
117C4×Dic53C2313yes—
118C4×Dic53C2313yes—
119C4×Dic53C2313yes—
120C23×C53C2413yes—
121C22×D53C2413yes—
122C22×D53C2413yes—
123Q8×C53#4114yescomplemented
124D4×C53—26no—
125C2×C4×C53—26no—
126C2×C4×C53—46no—
127C2×C4×C53C2413yes—
128C2×C4×C53C2413yes—
129C2×C4×C53C2411yescharacteristic
130C2×C4×C53C2411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C2×C4×C53C2413yes—
133C22⋊D53—26no—
134C22⋊D53—46no—
135C22⋊D53—46no—
136C22⋊D53—43no—
137C2×Dic53C2413yes—
138C2×Dic53C2411yescharacteristic
139C2×Dic53C2411yescharacteristic
140C2×Dic53C2413yes—
141C2×Dic53C2411yescharacteristic
142C2×Dic53C2411yescharacteristic
143D212—26no—
144D212—26no—
145D212—43no—
146D212—46no—
147C4×D53—26no—
148C4×D53—26no—
149C4×D53—26no—
150C4×D53—43no—
151C4×D53—46no—
152Dic106—26no—
153C22×C53—23no—
154C22×C53—23no—
155C22×C53—23no—
156C22×C53C2511yescharacteristic
157C2×D53—23no—
158C2×D53—23no—
159C2×D53—23no—
160C2×D53—23no—
161C2×D53—23no—
162C2×D53—23no—
163C4×C53—21no—
164C4×C53—23no—
165C4×C53—23no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53C2×#4116yes—
169Dic53—21no—
170Dic53—21no—
171Dic53—23no—
172Dic53—23no—
173Dic53—21no—
174Dic53H(5,𝔽2)12yes—
175#3517—531nomaximal, Sylow
176C2×C53—43no—
177C2×C53C22×#4111yescharacteristic
178C2×C53C2×H(5,𝔽2)11yescharacteristic
179C2×C53#58511yescharacteristic
180D53—43no—
181D53—43no—
182#557—531no—
183#553—533no—
184#553—533no—
185#550—531no—
186#550—531no—
187#550—531no—
188#550—533no—
189#547—533no—
190#540—533no—
191#540—533no—
192#532—533no—
193#529—533no—
194#520—531no—
195#520D5311yescharacteristic, complemented
196C4×#41—531no—
197C53#351711yescharacteristic, complemented, Sylow
198C2×#41—531no—
199C2×#41—531no—
200C2×#41—536no—
201C2×#41C2×D5311yescharacteristic
202C4⋊Q8—533no—
203#127—533no—
204#127—533no—
205#127—533no—
206#126—533no—
207#126—533no—
208#125—531no—
209#125—533no—
210#125—533no—
211#124—533no—
212#123—536no—
213#123—533no—
214#123—536no—
215#123—533no—
216C22⋊Q8—533no—
217#121—533no—
218#121—533no—
219#121—533no—
220#121—536no—
221#121—533no—
222#121—533no—
223#121—533no—
224#121—533no—
225#121—536no—
226#121—536no—
227C4×Q8—531no—
228C4×Q8—531no—
229C4×Q8—533no—
230C4×Q8C2×D5311yescharacteristic
231C4×Q8C2×D5311yescharacteristic
232C4×D4—533no—
233C4×D4—533no—
234C4×D4—536no—
235C4×D4—533no—
236C4×D4—533no—
237C4×D4—533no—
238C4×D4—533no—
239C4×D4—533no—
240C4×D4C2×D5313yes—
241C4×D4C2×D5313yes—
242#117—533no—
243#117—533no—
244#117C2×D5313yes—
245C2×C4⋊C4—533no—
246C2×C4⋊C4C2×D5313yes—
247C2×C42—533no—
248#41—24no—
249#41—1064no—
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×D4—533no—
260C2×D4—533no—
261C2×D4—536no—
262C2×D4C22×D5313yes—
263C22×C4—533no—
264C22×C4—533no—
265C22×C4—533no—
266C22×C4—533no—
267C22×C4—536no—
268C22×C4C22×D5313yes—
269C22×C4C22×D5316yes—
270C4⋊C4—533no—
271C4⋊C4—533no—
272C4⋊C4—531no—
273C4⋊C4—533no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—536no—
278C4⋊C4—531no—
279C4⋊C4—533no—
280C4⋊C4—533no—
281C4⋊C4C22×D5313yes—
282C4⋊C4C22×D5313yes—
283C4⋊C4C22×D5313yes—
284C4⋊C4C22×D5311yescharacteristic
285C22⋊C4—533no—
286C22⋊C4—533no—
287C22⋊C4—533no—
288C22⋊C4—536no—
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—
300C42C22×D5313yes—
301C42C22×D5313yes—
302C23—533no—
303C23—533no—
304C23C23×D5313yes—
305Q8—1066no—
306Q8#1171994514yescomplemented
307D4—26no—
308D4—1066no—
309D4—1066no—
310D4—1066no—
311D4—2123no—
312D4—2126no—
313D4—2126no—
314D4—2123no—
315D4—2126no—
316C2×C4—26no—
317C2×C4—46no—
318C2×C4—531no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—533no—
322C2×C4—533no—
323C2×C4—531no—
324C2×C4—1066no—
325C2×C4—1066no—
326C2×C4—1066no—
327C2×C4—2123no—
328C2×C4—2126no—
329C2×C4C23×D5311yescharacteristic
330C2×C4C23×D5313yes—
331C2×C4C23×D5311yescharacteristic
332C2×C4C23×D5313yes—
333C2×C4C23×D5311yescharacteristic
334C2×C4C23×D5313yes—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1063no—
339C22—1063no—
340C22—1063no—
341C22—1063no—
342C22—1063no—
343C22—1063no—
344C22C24×D5311yescharacteristic
345C4—21no—
346C4—23no—
347C4—23no—
348C4—21no—
349C4—21no—
350C4—532no—
351C4—1061no—
352C4—1061no—
353C4—1063no—
354C4—1063no—
355C4—1061no—
356C4C2×#1171994516yes—
357C2—43no—
358C2—2123no—
359C2—2123no—
360C2C22×#1171994511yescharacteristic
361C2H(5,𝔽2)×D5311yescharacteristic
362C2#1681146011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#520C2
hypercenter#520D53
hypocenterC53#3517
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #16811349C2semidirect
2. #16811323C2semidirect
3. #16811321C2semidirect
4. #16811295C2semidirect
5. #16811294C2semidirect
6. #16811292C2semidirect
7. #16811292C2semidirect
8. #16811274C2semidirect
9. #16811228C2semidirect
10. #16811224C2semidirect
11. #16811170C2semidirect
12. #16811151C2semidirect
13. #16811105C2semidirect
14. C4×#11719949C2semidirect
15. C53×#520C2semidirect
123. Q8×C53#41semidirect
195. #520D53semidirect
197. C53#3517semidirect
306. Q8#11719945semidirect