Finite Group Databasebeta

← #20281082

Subgroups — #20281082

Order 6784.

668 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133199C211yesmaximal, characteristic, complemented
2C2×#13133198C211yesmaximal, characteristic, complemented
3C2×#13133189C211yesmaximal, characteristic, complemented
4C2×#13133188C211yesmaximal, characteristic, complemented
5#16810546C211yesmaximal, characteristic, complemented
6#16810546C211yesmaximal, characteristic, complemented
7#16810538C211yesmaximal, characteristic, complemented
8#16810537C211yesmaximal, characteristic, complemented
9#16810531C211yesmaximal, characteristic, complemented
10#16810526C211yesmaximal, characteristic, complemented
11#16810523C211yesmaximal, characteristic, complemented
12#16810520C211yesmaximal, characteristic, complemented
13#16810498C211yesmaximal, characteristic, complemented
14#16810497C211yesmaximal, characteristic, complemented
15C53×#392C211yesmaximal, characteristic, complemented
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133199—24no—
22#13133198—24no—
23#13133189—24no—
24#13133188—24no—
25C2×C4⋊Dic53C2211yescharacteristic
26C2×C4⋊Dic53C2211yescharacteristic
27C2×C4⋊Dic53C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×#11719919C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C22⋊Dic53C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39C2×C22⋊Dic53C2211yescharacteristic
40C2×C22⋊Dic53C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic
44C2×C4×Dic53C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C22×D4×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C24×C53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2311yescharacteristic
71C22×C4×C53C2311yescharacteristic
72C2×C22⋊D53—22no—
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—41no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—41no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53D414yescomplemented
81C22×Dic53—22no—
82C22×Dic53—22no—
83C22×Dic53—41no—
84C22×Dic53C2311yescharacteristic
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—41no—
94C22⋊C4×C53—22no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—41no—
97C22⋊Dic53—22no—
98C22⋊Dic53—22no—
99C22⋊Dic53—22no—
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—41no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108C4⋊Dic53—22no—
109C4⋊Dic53—22no—
110C4⋊Dic53—41no—
111#11719919—22no—
112#11719919—22no—
113#11719919—41no—
114#11719919—41no—
115#11719919D414yescomplemented
116C4×Dic53—41no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53C2×D412yes—
138C23×C53C2411yescharacteristic
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53C2×D412yes—
146D4×C53—44no—
147D4×C53—44no—
148D4×C53—44no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53C2×D412yes—
160C2×C4×C53#4112yes—
161C22⋊D53—44no—
162C22⋊D53—44no—
163C22⋊D53—44no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—44no—
183C2×Dic53C2×D412yes—
184C2×Dic53C2×D412yes—
185C2×Dic53C2×D412yes—
186C4×D53—44no—
187C22×C53—22no—
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53C22×D411yescharacteristic
209C22×C53C22×D411yescharacteristic
210C22×C53C2×#4111yescharacteristic
211C22×C53H(5,𝔽2)11yescharacteristic
212C22×C53H(5,𝔽2)11yescharacteristic
213C22×C53H(5,𝔽2)11yescharacteristic
214C22×C532−1+411yescharacteristic
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—41no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—41no—
225C4×C53—41no—
226Dic53—22no—
227Dic53—22no—
228Dic53—22no—
229Dic53—41no—
230Dic53—41no—
231Dic53—41no—
232Dic53—41no—
233#2682—531nomaximal, Sylow
234C2×C53—42no—
235C2×C53—81no—
236C2×C53—81no—
237C2×C53#53411yescharacteristic
238C2×C53#53711yescharacteristic
239C2×C53#53811yescharacteristic
240C2×C53D4211yescharacteristic
241C2×C53#54711yescharacteristic
242C2×C53#55011yescharacteristic
243C2×C53#56211yescharacteristic
244D53—42no—
245C2×#123—531no—
246C2×#123—531no—
247C2×#121—531no—
248C2×#121—531no—
249#396—531no—
250#396—531no—
251#394—531no—
252#394—531no—
253#392—531no—
254#392D5311yescharacteristic, complemented
255#390—531no—
256#390—531no—
257#389—531no—
258#385—531no—
259#385—531no—
260C53#268211yescharacteristic, complemented, Sylow
261C22×D4—531no—
262C22×D4—531no—
263C22×D4—531no—
264C22×D4C2×D5311yescharacteristic
265C22×D4C2×D5311yescharacteristic
266C22×D4C2×D5311yescharacteristic
267C23×C4—531no—
268C23×C4—531no—
269#123—1064no—
270#123—1064no—
271#121—1064no—
272#121—1064no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4—531no—
280C2×C4⋊C4—531no—
281C2×C22⋊C4—531no—
282C2×C22⋊C4—531no—
283C2×C22⋊C4—531no—
284C2×C22⋊C4—531no—
285C2×C22⋊C4—531no—
286C2×C22⋊C4—531no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—531no—
290C2×C22⋊C4—531no—
291C2×C22⋊C4—531no—
292C2×C22⋊C4C2×D5311yescharacteristic
293C2×C22⋊C4C2×D5311yescharacteristic
294C2×C22⋊C4C2×D5311yescharacteristic
295C2×C42—531no—
296#95—531no—
297#95—531no—
298#95—531no—
299#95C2×D5311yescharacteristic
300C24—531no—
301C24C22×D5311yescharacteristic
302C24C22×D5311yescharacteristic
303C24C22×D5311yescharacteristic
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—534no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C22×C4—531no—
323C22×C4—531no—
324C22×C4—531no—
325C22×C4—531no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—1062no—
330C22×C4—1062no—
331C22×C4—1062no—
332C22×C4—1062no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4C22×D5311yescharacteristic
336C22×C4C22×D5311yescharacteristic
337C22×C4C22×D5311yescharacteristic
338C22×C4C22×D5311yescharacteristic
339C4⋊C4—534no—
340C4⋊C4—1062no—
341C4⋊C4—1062no—
342C4⋊C4—1062no—
343C4⋊C4—1062no—
344C4⋊C4—2121no—
345C4⋊C4—2121no—
346C4⋊C4—2121no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—1062no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
353C22⋊C4—1062no—
354C22⋊C4—1062no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C22⋊C4—1062no—
359C22⋊C4—1062no—
360C22⋊C4—2121no—
361C42—2121no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—532no—
383C23—1061no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23D4×D5312yes—
390C23C23×D5311yescharacteristic
391D4—44no—
392D4—44no—
393D4—44no—
394D4—2124no—
395D4—2124no—
396D4—2124no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—532no—
408C2×C4—532no—
409C2×C4—532no—
410C2×C4—1061no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1061no—
420C2×C4—1061no—
421C2×C4—1061no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—2124no—
429C2×C4—2124no—
430C2×C4D4×D5312yes—
431C2×C4#1171994712yes—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—1062no—
454C22—1062no—
455C22—1062no—
456C22—2121no—
457C22—2121no—
458C22—2121no—
459C22—2121no—
460C22C2×D4×D5311yescharacteristic
461C22C2×D4×D5311yescharacteristic
462C22C2×#1171994711yescharacteristic
463C22#1313326211yescharacteristic
464C22#1313326211yescharacteristic
465C22#1313326211yescharacteristic
466C22#1313326411yescharacteristic
467C4—22no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—1062no—
472C4—1062no—
473C4—1062no—
474C4—2121no—
475C4—2121no—
476C4—2121no—
477C4—2121no—
478C2—42no—
479C2—81no—
480C2—81no—
481C2—2122no—
482C2#1681118311yescharacteristic
483C2#1681120211yescharacteristic
484C2#1681120811yescharacteristic
485C2#1681125511yescharacteristic
486C2#1681127311yescharacteristic
487C2#1681129311yescharacteristic
488C2#1681136811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2682
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
2. C2×#13133198C2semidirect
3. C2×#13133189C2semidirect
4. C2×#13133188C2semidirect
5. #16810546C2semidirect
6. #16810546C2semidirect
7. #16810538C2semidirect
8. #16810537C2semidirect
9. #16810531C2semidirect
10. #16810526C2semidirect
11. #16810523C2semidirect
12. #16810520C2semidirect
13. #16810498C2semidirect
14. #16810497C2semidirect
15. C53×#392C2semidirect
80. C2×C22⋊D53D4semidirect
115. #11719919D4semidirect
254. #392D53semidirect
260. C53#2682semidirect