Finite Group Databasebeta

← #20277902

Subgroups — #20277902

Order 6784.

456 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133210C211yesmaximal, characteristic
2#16810521C211yesmaximal, characteristic, complemented
3#16810520C211yesmaximal, characteristic, complemented
4#16810518C211yesmaximal, characteristic, complemented
5#16810517C211yesmaximal, characteristic
6#16810511C211yesmaximal, characteristic
7#16810510C211yesmaximal, characteristic, complemented
8#16810508C211yesmaximal, characteristic, complemented
9#16810507C211yesmaximal, characteristic, complemented
10#16810498C211yesmaximal, characteristic
11#16810495C211yesmaximal, characteristic
12#16810495C211yesmaximal, characteristic
13#16810492C211yesmaximal, characteristic, complemented
14#16810490C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17#13133210C418yescomplemented
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33C2×C4×Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C2×C4⋊C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C42×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52C24×C53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×Dic53—41no—
60C22×Dic53—41no—
61C22×Dic53—41no—
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C4⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53C2×C414yes—
74C42×C53—41no—
75C22⋊Dic53—41no—
76C22⋊Dic53C2×C414yes—
77C22⋊Dic53C2×C414yes—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80C4⋊Dic53C2×C414yescomplemented
81#11719919—41no—
82#11719919—41no—
83#11719919—41no—
84#11719919—41no—
85#11719919—41no—
86#11719919—41no—
87#11719919C2×C414yescomplemented
88#11719919C2×C414yescomplemented
89C4×Dic53—41no—
90C4×Dic53—41no—
91C4×Dic53C2×C414yescomplemented
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53C22×C412yes—
99C23×C53C2411yescharacteristic
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53C22×C412yes—
117C2×C4×C53C22×C412yes—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—44no—
139C2×Dic53C22×C412yes—
140C2×Dic53C22×C412yes—
141C2×Dic53C22×C412yes—
142C2×Dic53C22×C412yes—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53C23×C411yescharacteristic
151C22×C53H(5,𝔽2)11yescharacteristic
152C22×C53H(5,𝔽2)11yescharacteristic
153C22×C53H(5,𝔽2)11yescharacteristic
154C22×C532−1+411yescharacteristic
155C22×C532−1+411yescharacteristic
156C22×C532−1+411yescharacteristic
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163Dic53—22no—
164Dic53—22no—
165Dic53—22no—
166Dic53—22no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53—41no—
171#2383—531nomaximal, Sylow
172C2×C53—42no—
173C2×C53#52011yescharacteristic
174C2×C53#52011yescharacteristic
175C2×C53#52011yescharacteristic
176C2×C53#56011yescharacteristic
177C2×C53#56311yescharacteristic
178C2×C53#56311yescharacteristic
179C2×C53#56311yescharacteristic
180C2×#126—531no—
181#389—531no—
182#389—531no—
183#389—531no—
184#388—531no—
185#388—531no—
186#388D5311yescharacteristic
187#387—531no—
188#387—531no—
189#387—531no—
190#385—531no—
191#385—531no—
192#385—531no—
193#384—531no—
194#383—531no—
195C53#238311yescharacteristic, complemented, Sylow
196C23×C4—531no—
197#126—538no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4C2×D5311yescharacteristic
210C2×C22⋊C4—531no—
211C2×C22⋊C4—531no—
212C2×C22⋊C4—531no—
213C2×C22⋊C4C2×D5311yescharacteristic
214C2×C22⋊C4C2×D5311yescharacteristic
215C2×C22⋊C4C2×D5311yescharacteristic
216C2×C42—531no—
217C2×C42—531no—
218C2×C42—531no—
219C2×C42C2×D5311yescharacteristic
220#95—531no—
221#95—531no—
222#95—531no—
223#95—531no—
224#95—531no—
225#95—531no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95C2×D5311yescharacteristic
231#95C2×D5311yescharacteristic
232C24C22×D5311yescharacteristic
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—2121no—
242C22×C4—2121no—
243C22×C4—2121no—
244C22×C4C22×D5311yescharacteristic
245C22×C4C22×D5311yescharacteristic
246C22×C4C22×D5311yescharacteristic
247C22×C4C22×D5311yescharacteristic
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—534no—
252C4⋊C4—534no—
253C4⋊C4—534no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—534no—
265C22⋊C4—534no—
266C22⋊C4—2121no—
267C22⋊C4C4×D5314yes—
268C42—41no—
269C42—534no—
270C42—2121no—
271C42—2121no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D5312yes—
279C23C23×D5311yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—532no—
297C2×C4—532no—
298C2×C4—532no—
299C2×C4—532no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—2124no—
321C2×C4C2×C4×D5312yes—
322C2×C4C2×C4×D5312yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C22×C4×D5311yescharacteristic
331C22#1313326211yescharacteristic
332C22#1313326211yescharacteristic
333C22#1313326311yescharacteristic
334C22#1313326411yescharacteristic
335C22#1313326411yescharacteristic
336C22#1313326411yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—1062no—
344C4—1062no—
345C4—1062no—
346C4—1062no—
347C4—2121no—
348C4—2121no—
349C4—2121no—
350C4—2121no—
351C2—42no—
352C2#1681110211yescharacteristic
353C2#1681110211yescharacteristic
354C2#1681110311yescharacteristic
355C2#1681136111yescharacteristic
356C2#1681137311yescharacteristic
357C2#1681137511yescharacteristic
358C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#388C2
socleC23×C53C24
hypercenter#388D53
hypocenterC53#2383
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #16810521C2semidirect
3. #16810520C2semidirect
4. #16810518C2semidirect
7. #16810510C2semidirect
8. #16810508C2semidirect
9. #16810507C2semidirect
13. #16810492C2semidirect
14. #16810490C2semidirect
17. #13133210C4semidirect
80. C4⋊Dic53C2×C4semidirect
87. #11719919C2×C4semidirect
88. #11719919C2×C4semidirect
91. C4×Dic53C2×C4semidirect
195. C53#2383semidirect