Finite Group Databasebeta

← #20282039

Subgroups — #20282039

Order 6784.

484 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810557C211yesmaximal, characteristic, complemented
2#16810551C211yesmaximal, characteristic, complemented
3#16810537C211yesmaximal, characteristic, complemented
4#16810537C211yesmaximal, characteristic, complemented
5#16810537C211yesmaximal, characteristic, complemented
6#16810526C211yesmaximal, characteristic, complemented
7#16810517C211yesmaximal, characteristic, complemented
8#16810516C211yesmaximal, characteristic, complemented
9#16810511C211yesmaximal, characteristic, complemented
10#16810511C211yesmaximal, characteristic, complemented
11#16810503C211yesmaximal, characteristic, complemented
12#16810503C211yesmaximal, characteristic, complemented
13#16810490C211yesmaximal, characteristic, complemented
14#16810481C211yesmaximal, characteristic, complemented
15C53×#392C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊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
44C22×D4×C53C2211yescharacteristic
45C22×D4×C53C2211yescharacteristic
46C22×D4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C24×C53C2311yescharacteristic
53C24×C53C2311yescharacteristic
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×Dic53—22no—
68C22×Dic53—41no—
69C22×Dic53—41no—
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22⋊C4×C53—22no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊Dic53—22no—
82C22⋊Dic53—22no—
83C22⋊Dic53—22no—
84C22⋊Dic53—41no—
85C22⋊Dic53—41no—
86C22⋊Dic53—41no—
87C22⋊Dic53—41no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—41no—
94#11719919—41no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—22no—
98C4×Dic53—41no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53#4112yes—
120C23×C53C2411yescharacteristic
121D4×C53—44no—
122D4×C53—44no—
123D4×C53—44no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53#4112yes—
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—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—44no—
160C2×Dic53#4112yes—
161C22×C53—22no—
162C22×C53—22no—
163C22×C53—22no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53C2×#4111yescharacteristic
183C22×C53C2×#4111yescharacteristic
184C22×C53C2×#4111yescharacteristic
185C22×C53H(5,𝔽2)11yescharacteristic
186C22×C53H(5,𝔽2)11yescharacteristic
187C22×C53H(5,𝔽2)11yescharacteristic
188C22×C53H(5,𝔽2)11yescharacteristic
189C4×C53—22no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—41no—
193Dic53—22no—
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200Dic53—41no—
201#2756—531nomaximal, Sylow
202C2×C53—42no—
203C2×C53—81no—
204C2×C53—81no—
205C2×C53#53811yescharacteristic
206C2×C53#53811yescharacteristic
207C2×C53#54011yescharacteristic
208C2×C53#55111yescharacteristic
209C2×C53#55311yescharacteristic
210C2×C53#55511yescharacteristic
211C2×C53#56011yescharacteristic
212#398—531no—
213#397—531no—
214#394—531no—
215#394—531no—
216#394—531no—
217#392D5311yescharacteristic
218#390—531no—
219#388—531no—
220#388—531no—
221#388—531no—
222#388—531no—
223#386—531no—
224#386—531no—
225#383—531no—
226#380—531no—
227C53#275611yescharacteristic, complemented, Sylow
228C22×D4C2×D5311yescharacteristic
229C22×D4C2×D5311yescharacteristic
230C22×D4C2×D5311yescharacteristic
231C23×C4—531no—
232C2×C4⋊C4—531no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—531no—
237C2×C22⋊C4—531no—
238C2×C22⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4—531no—
244C2×C22⋊C4—531no—
245C2×C22⋊C4—531no—
246C2×C22⋊C4—531no—
247C2×C22⋊C4—531no—
248C2×C22⋊C4C2×D5311yescharacteristic
249C2×C22⋊C4C2×D5311yescharacteristic
250C2×C22⋊C4C2×D5311yescharacteristic
251C2×C42—531no—
252C2×C42—531no—
253#95—531no—
254#95—531no—
255#95—531no—
256#95—531no—
257#95—531no—
258#95—531no—
259#95—531no—
260#95—531no—
261#95—531no—
262#95C2×D5311yescharacteristic
263C24C22×D5311yescharacteristic
264C24C22×D5311yescharacteristic
265C24C22×D5311yescharacteristic
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—1062no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4C22×D5311yescharacteristic
287C22×C4C22×D5311yescharacteristic
288C22×C4C22×D5311yescharacteristic
289C22×C4C22×D5311yescharacteristic
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C22⋊C4—22no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C22⋊C4—1062no—
301C22⋊C4—2121no—
302C22⋊C4—2121no—
303C22⋊C4—2121no—
304C22⋊C4—2121no—
305C22⋊C4—2121no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—1062no—
310C42—2121no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23#1171994712yes—
332C23C23×D5311yescharacteristic
333D4—44no—
334D4—44no—
335D4—44no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—532no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—2124no—
372C2×C4#1171994712yes—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22#41×D5311yescharacteristic
395C22C2×#1171994711yescharacteristic
396C22C2×#1171994711yescharacteristic
397C22#1313326211yescharacteristic
398C22#1313326211yescharacteristic
399C22#1313326211yescharacteristic
400C22#1313326211yescharacteristic
401C4—22no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—1062no—
406C4—2121no—
407C4—2121no—
408C4—2121no—
409C4—2121no—
410C4—2121no—
411C4—2121no—
412C4—2121no—
413C2—42no—
414C2—81no—
415C2—81no—
416C2#1681120511yescharacteristic
417C2#1681120811yescharacteristic
418C2#1681122711yescharacteristic
419C2#1681130411yescharacteristic
420C2#1681131611yescharacteristic
421C2#1681133711yescharacteristic
422C2#1681136211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2756
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #16810557C2semidirect
2. #16810551C2semidirect
3. #16810537C2semidirect
4. #16810537C2semidirect
5. #16810537C2semidirect
6. #16810526C2semidirect
7. #16810517C2semidirect
8. #16810516C2semidirect
9. #16810511C2semidirect
10. #16810511C2semidirect
11. #16810503C2semidirect
12. #16810503C2semidirect
13. #16810490C2semidirect
14. #16810481C2semidirect
227. C53#2756semidirect