Finite Group Databasebeta

← #20279652

Subgroups — #20279652

Order 6784.

760 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133203C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×#13133194C211yesmaximal, characteristic, complemented
4C2×#13133190C211yesmaximal, characteristic, complemented
5C2×#13133189C211yesmaximal, characteristic, complemented
6C2×#13133177C211yesmaximal, characteristic, complemented
7#16810559C211yesmaximal, characteristic, complemented
8#16810558C211yesmaximal, characteristic, complemented
9#16810532C211yesmaximal, characteristic, complemented
10#16810531C211yesmaximal, characteristic, complemented
11#16810525C211yesmaximal, characteristic, complemented
12#16810523C211yesmaximal, characteristic, complemented
13#16810523C211yesmaximal, characteristic, complemented
14C4×#11719919C211yesmaximal, characteristic, complemented
15C53×#390C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C23×Dic53C2211yescharacteristic
23C22×C4×D53C2211yescharacteristic
24#13133203—24no—
25#13133195—24no—
26#13133194—24no—
27#13133190—24no—
28#13133189—24no—
29#13133177—24no—
30C2×C4⋊Dic53C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C2×#11719919C2211yescharacteristic
35C2×#11719919C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic
44C2×#11719921C2211yescharacteristic
45C2×#11719921C2211yescharacteristic
46C2×C4×Dic53C2211yescharacteristic
47C2×C4×Dic53C2211yescharacteristic
48#13133124C2211yescharacteristic
49#13133124C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51C2×C4⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C2×C22⋊C4×C53C2211yescharacteristic
56C2×C42×C53C2211yescharacteristic
57C24×C53C2311yescharacteristic
58C24×C53C2311yescharacteristic
59C23×D53C2311yescharacteristic
60C23×D53C2311yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—41no—
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C2×C22⋊D53—22no—
70C2×C22⋊D53—22no—
71C2×C22⋊D53—22no—
72C2×C22⋊D53—22no—
73C2×C22⋊D53—22no—
74C2×C22⋊D53—22no—
75C2×C22⋊D53—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C22×Dic53—22no—
82C22×Dic53—22no—
83C22×Dic53—22no—
84C22×Dic53C2311yescharacteristic
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—41no—
93C2×C4×D53—22no—
94C2×C4×D53—22no—
95C2×C4×D53—22no—
96C2×Dic106—22no—
97C2×Dic106—22no—
98C2×Dic106—22no—
99C4⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—22no—
104C42×C53—41no—
105C22⋊Dic53—22no—
106C22⋊Dic53—22no—
107C22⋊Dic53—22no—
108#11719921—22no—
109#11719921—22no—
110#11719921—22no—
111#11719921—22no—
112#11719921—22no—
113#11719921—22no—
114#11719921—22no—
115C4⋊Dic53—22no—
116#11719919—22no—
117#11719919—22no—
118#11719919—22no—
119#11719919D414yescomplemented
120#11719919D414yescomplemented
121C4×Dic53—41no—
122C4×Dic53D414yescomplemented
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#4112yes—
137C23×C53C2411yescharacteristic
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53#4112yes—
152D4×C53—44no—
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×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53C2×D412yes—
166C2×C4×C53C2×D412yes—
167C2×C4×C53C2×D412yes—
168C22⋊D53—28no—
169C22⋊D53—44no—
170C22⋊D53—44no—
171C22⋊D53—44no—
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—21no—
183C2×Dic53—21no—
184C2×Dic53—44no—
185C2×Dic53C2×D412yes—
186C2×Dic53C2×D412yes—
187C2×Dic53C2×D412yes—
188C2×Dic53C2×D412yes—
189C2×Dic53C2×D412yes—
190C2×Dic53C2×D412yes—
191D212—44no—
192C4×D53—44no—
193Dic106—44no—
194C22×C53—22no—
195C22×C53—22no—
196C22×C53—22no—
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×C53C22×D411yescharacteristic
211C22×C53C2×#4111yescharacteristic
212C22×C53C2×#4111yescharacteristic
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—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—41no—
229C4×C53—22no—
230C4×C53—22no—
231C4×C53—22no—
232C4×C53—41no—
233C4×C53—41no—
234Dic53—22no—
235Dic53—22no—
236Dic53—22no—
237Dic53—41no—
238Dic53—41no—
239Dic53#12714yes—
240#2540—531nomaximal, Sylow
241C2×C53—42no—
242C2×C53—81no—
243C2×C53#52511yescharacteristic
244C2×C53C2×#12711yescharacteristic
245C2×C53#53711yescharacteristic
246C2×C53#54611yescharacteristic
247C2×C53#54611yescharacteristic
248C2×C53#54811yescharacteristic
249C2×C53#54811yescharacteristic
250D53—42no—
251D53—81no—
252C2×#124—531no—
253C2×C22⋊Q8—531no—
254C2×C22⋊Q8—531no—
255C2×#121—531no—
256C2×#121—531no—
257C2×C4×D4—531no—
258#399—531no—
259#399—531no—
260#392—531no—
261#392—531no—
262#390—531no—
263#390—531no—
264#390—531no—
265#390D5311yescharacteristic, complemented
266C4×C4⋊C4—531no—
267C53#254011yescharacteristic, complemented, Sylow
268C22×Q8—531no—
269C22×D4—531no—
270C22×D4—531no—
271C22×D4—531no—
272C22×D4—531no—
273C22×D4—531no—
274C22×D4C2×D5311yescharacteristic
275C23×C4—531no—
276C23×C4—531no—
277#124—1064no—
278C22⋊Q8—1064no—
279C22⋊Q8—1064no—
280#121—1064no—
281#121—1064no—
282C4×D4—1064no—
283C2×C4⋊C4—531no—
284C2×C4⋊C4—531no—
285C2×C4⋊C4—531no—
286C2×C4⋊C4—531no—
287C2×C4⋊C4—531no—
288C2×C4⋊C4—531no—
289C2×C4⋊C4C2×D5311yescharacteristic
290C2×C22⋊C4—531no—
291C2×C22⋊C4—531no—
292C2×C22⋊C4—531no—
293C2×C22⋊C4—531no—
294C2×C22⋊C4—531no—
295C2×C22⋊C4—531no—
296C2×C22⋊C4—531no—
297C2×C22⋊C4—531no—
298C2×C22⋊C4—531no—
299C2×C22⋊C4—531no—
300C2×C22⋊C4C2×D5311yescharacteristic
301C2×C22⋊C4C2×D5311yescharacteristic
302C2×C22⋊C4C2×D5311yescharacteristic
303C2×C22⋊C4C2×D5311yescharacteristic
304C2×C42—531no—
305C2×C42—531no—
306C2×C42C2×D5311yescharacteristic
307#95—531no—
308#95—531no—
309C24—531no—
310C24—531no—
311C24C22×D5311yescharacteristic
312C24C22×D5311yescharacteristic
313C2×Q8—1062no—
314C2×Q8—1062no—
315C2×Q8—1062no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—1062no—
320C2×D4—1062no—
321C2×D4—1062no—
322C2×D4—1062no—
323C2×D4—1062no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—1062no—
327C2×D4—1062no—
328C2×D4—1062no—
329C2×D4—1062no—
330C2×D4—1062no—
331C2×D4—2121no—
332C2×D4—2121no—
333C2×D4—2121no—
334C22×C4—531no—
335C22×C4—531no—
336C22×C4—531no—
337C22×C4—531no—
338C22×C4—531no—
339C22×C4—531no—
340C22×C4—1062no—
341C22×C4—1062no—
342C22×C4—1062no—
343C22×C4—1062no—
344C22×C4—1062no—
345C22×C4—1062no—
346C22×C4C22×D5311yescharacteristic
347C22×C4C22×D5311yescharacteristic
348C22×C4C22×D5311yescharacteristic
349C22×C4C22×D5311yescharacteristic
350C22×C4C22×D5311yescharacteristic
351C4⋊C4—22no—
352C4⋊C4—534no—
353C4⋊C4—534no—
354C4⋊C4—1062no—
355C4⋊C4—1062no—
356C4⋊C4—1062no—
357C4⋊C4—1062no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—1062no—
363C22⋊C4—1062no—
364C22⋊C4—1062no—
365C22⋊C4—1062no—
366C22⋊C4—1062no—
367C22⋊C4—1062no—
368C22⋊C4—1062no—
369C22⋊C4—1062no—
370C22⋊C4—1062no—
371C22⋊C4—1062no—
372C42—41no—
373C42—534no—
374C42—2121no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—532no—
389C23—1061no—
390C23—1061no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—1061no—
397C23—1061no—
398C23—1061no—
399C23—1061no—
400C23—1061no—
401C23—1061no—
402C23#1171994512yes—
403C23C23×D5311yescharacteristic
404Q8—2124no—
405D4—44no—
406D4—1068no—
407D4—2124no—
408D4—2124no—
409D4—2124no—
410D4—2124no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—532no—
424C2×C4—532no—
425C2×C4—532no—
426C2×C4—532no—
427C2×C4—532no—
428C2×C4—532no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—1061no—
434C2×C4—1061no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—1061no—
441C2×C4—2124no—
442C2×C4—2124no—
443C2×C4D4×D5312yes—
444C2×C4D4×D5312yes—
445C2×C4D4×D5312yes—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—1062no—
461C22—1062no—
462C22—1062no—
463C22—2121no—
464C22—2121no—
465C22—2121no—
466C22—2121no—
467C22—2121no—
468C22—2121no—
469C22—2121no—
470C22—2121no—
471C22—2121no—
472C22—2121no—
473C22—2121no—
474C22C2×D4×D5311yescharacteristic
475C22C2×D4×D5311yescharacteristic
476C22C2×D4×D5311yescharacteristic
477C22#41×D5311yescharacteristic
478C22C2×#1171994511yescharacteristic
479C22#1313326211yescharacteristic
480C22#1313326411yescharacteristic
481C4—22no—
482C4—22no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—534no—
487C4—1062no—
488C4—1062no—
489C4—1062no—
490C4—2121no—
491C4—2121no—
492C2—42no—
493C2—81no—
494C2—4241no—
495C2—2122no—
496C2#1681113211yescharacteristic
497C2#127×D5311yescharacteristic
498C2#1681120011yescharacteristic
499C2#1681126211yescharacteristic
500C2#1681126311yescharacteristic
501C2#1681128111yescharacteristic
502C2#1681128211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#390C2
hypercenter#390D53
hypocenterC53#2540
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#13133203C2semidirect
2. C2×#13133195C2semidirect
3. C2×#13133194C2semidirect
4. C2×#13133190C2semidirect
5. C2×#13133189C2semidirect
6. C2×#13133177C2semidirect
7. #16810559C2semidirect
8. #16810558C2semidirect
9. #16810532C2semidirect
10. #16810531C2semidirect
11. #16810525C2semidirect
12. #16810523C2semidirect
13. #16810523C2semidirect
14. C4×#11719919C2semidirect
15. C53×#390C2semidirect
119. #11719919D4semidirect
120. #11719919D4semidirect
122. C4×Dic53D4semidirect
265. #390D53semidirect
267. C53#2540semidirect