Finite Group Databasebeta

← #20279966

Subgroups — #20279966

Order 6784.

476 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810564C211yesmaximal, characteristic, complemented
2#16810537C211yesmaximal, characteristic
3#16810535C211yesmaximal, characteristic, complemented
4#16810535C211yesmaximal, characteristic, complemented
5#16810527C211yesmaximal, characteristic, complemented
6#16810517C211yesmaximal, characteristic, complemented
7#16810511C211yesmaximal, characteristic, complemented
8#16810503C211yesmaximal, characteristic
9#16810490C211yesmaximal, characteristic, complemented
10#16810490C211yesmaximal, characteristic, complemented
11#16810486C211yesmaximal, characteristic, complemented
12#16810484C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
15C53×#386C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×C22⋊Dic53C2211yescharacteristic
21C2×C22⋊Dic53C2211yescharacteristic
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30#13133124C2211yescharacteristic
31#13133124C2211yescharacteristic
32#13133124C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44C22×D4×C53C2211yescharacteristic
45C23×C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C24×C53C2311yescharacteristic
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—41no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—41no—
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—41no—
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C2×Dic106—22no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C22⋊C4×C53—22no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊Dic53—22no—
82C22⋊Dic53—22no—
83C22⋊Dic53—22no—
84C22⋊Dic53—22no—
85C22⋊Dic53—22no—
86C22⋊Dic53—41no—
87C22⋊Dic53—41no—
88#11719919—41no—
89#11719919—41no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—41no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53#4112yes—
106C23×C53#4112yes—
107C23×C53C2411yescharacteristic
108D4×C53—44no—
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×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—44no—
124C2×C4×C53#4112yes—
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—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—44no—
148C2×Dic53#4112yes—
149C2×Dic53#4112yes—
150Dic106—44no—
151C22×C53—22no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53C2×#4111yescharacteristic
166C22×C53C2×#4111yescharacteristic
167C22×C53C2×#4111yescharacteristic
168C22×C53C2×#4111yescharacteristic
169C22×C53C2×#4111yescharacteristic
170C22×C53H(5,𝔽2)11yescharacteristic
171C22×C53H(5,𝔽2)11yescharacteristic
172C4×C53—22no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177Dic53—22no—
178Dic53—22no—
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184Dic53—41no—
185#2572—531nomaximal, Sylow
186C2×C53—42no—
187C2×C53—42no—
188C2×C53#52911yescharacteristic
189C2×C53#52911yescharacteristic
190C2×C53#53811yescharacteristic
191C2×C53#55111yescharacteristic
192C2×C53#55111yescharacteristic
193C2×C53#55111yescharacteristic
194C2×C53#55511yescharacteristic
195#400—531no—
196#394—531no—
197#393—531no—
198#393—531no—
199#391—531no—
200#388—531no—
201#388—531no—
202#386—531no—
203#386D5311yescharacteristic
204#383—531no—
205#383—531no—
206#381—531no—
207#381—531no—
208C4×C22⋊C4—531no—
209C4×C22⋊C4—531no—
210C53#257211yescharacteristic, complemented, Sylow
211C22×Q8—531no—
212C22×D4C2×D5311yescharacteristic
213C23×C4—531no—
214C23×C4C2×D5311yescharacteristic
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C22⋊C4—531no—
218C2×C22⋊C4—531no—
219C2×C22⋊C4—531no—
220C2×C22⋊C4—531no—
221C2×C22⋊C4—531no—
222C2×C22⋊C4—531no—
223C2×C22⋊C4—531no—
224C2×C22⋊C4C2×D5311yescharacteristic
225C2×C22⋊C4C2×D5311yescharacteristic
226C2×C22⋊C4C2×D5311yescharacteristic
227C2×C42—531no—
228C2×C42—531no—
229C2×C42—531no—
230#95—531no—
231#95—531no—
232#95—531no—
233#95—531no—
234#95—531no—
235#95—531no—
236#95—531no—
237#95—531no—
238#95—531no—
239#95—531no—
240#95—531no—
241#95—531no—
242#95—531no—
243#95—531no—
244#95C2×D5311yescharacteristic
245#95C2×D5311yescharacteristic
246C24C22×D5311yescharacteristic
247C24C22×D5311yescharacteristic
248C2×Q8—1062no—
249C2×Q8—2121no—
250C2×Q8—2121no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—41no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—531no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—1062no—
266C22×C4—1062no—
267C22×C4—2121no—
268C22×C4C22×D5311yescharacteristic
269C22×C4C22×D5311yescharacteristic
270C22×C4C22×D5311yescharacteristic
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—1062no—
279C22⋊C4—1062no—
280C22⋊C4—1062no—
281C22⋊C4—1062no—
282C22⋊C4—1062no—
283C22⋊C4—2121no—
284C22⋊C4—2121no—
285C42—1062no—
286C42—1062no—
287C42—2121no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23#1171994512yes—
301C23#1171994712yes—
302C23C23×D5311yescharacteristic
303Q8—2124no—
304D4—44no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—532no—
321C2×C4—532no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—1061no—
344C2×C4—2124no—
345C2×C4#1171994712yes—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22#41×D5311yescharacteristic
361C22#41×D5311yescharacteristic
362C22C2×#1171994711yescharacteristic
363C22C2×#1171994711yescharacteristic
364C22C2×#1171994511yescharacteristic
365C22#1313326211yescharacteristic
366C22#1313326211yescharacteristic
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—1062no—
373C4—1062no—
374C4—2121no—
375C4—2121no—
376C4—2121no—
377C4—2121no—
378C4—2121no—
379C4—2121no—
380C2—42no—
381C2—42no—
382C2#1681115011yescharacteristic
383C2#1681115311yescharacteristic
384C2#1681120511yescharacteristic
385C2#1681129911yescharacteristic
386C2#1681130211yescharacteristic
387C2#1681130411yescharacteristic
388C2#1681133711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2572
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. #16810564C2semidirect
3. #16810535C2semidirect
4. #16810535C2semidirect
5. #16810527C2semidirect
6. #16810517C2semidirect
7. #16810511C2semidirect
9. #16810490C2semidirect
10. #16810490C2semidirect
11. #16810486C2semidirect
12. #16810484C2semidirect
13. C22⋊C4×Dic53C2semidirect
14. C4×C22⋊Dic53C2semidirect
210. C53#2572semidirect