Finite Group Databasebeta

← #20278055

Subgroups — #20278055

Order 6784.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719947C211yesmaximal, characteristic, complemented
2C2×#13133195C211yesmaximal, characteristic, complemented
3C2×#13133190C211yesmaximal, characteristic, complemented
4C2×#13133177C211yesmaximal, characteristic, complemented
5#16810538C211yesmaximal, characteristic, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810527C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic, complemented
10#16810501C211yesmaximal, characteristic, complemented
11#16810500C211yesmaximal, characteristic, complemented
12#16810495C211yesmaximal, characteristic, complemented
13#16810494C211yesmaximal, characteristic, complemented
14#16810481C211yesmaximal, characteristic, complemented
15C53×#394C211yesmaximal, characteristic, complemented
16C2×#11719947—24no—
17C2×#11719947—24no—
18C2×#11719947—24no—
19C22×Dic106C2211yescharacteristic
20C22×D212C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic
23C22×C22⋊D53C2211yescharacteristic
24C22×C22⋊D53C2211yescharacteristic
25C23×Dic53C2211yescharacteristic
26C23×Dic53C2211yescharacteristic
27C22×C4×D53C2211yescharacteristic
28C22×C4×D53C2211yescharacteristic
29#13133195—24no—
30#13133190—24no—
31#13133177—24no—
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C2×#11719919C2211yescharacteristic
35C2×#11719919C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic
44C2×C4×Dic53C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
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
56#95×C53C2211yescharacteristic
57C24×C53C2311yescharacteristic
58C24×C53C2311yescharacteristic
59C23×D53C2311yescharacteristic
60C23×D53C2311yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—22no—
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⋊D53D414yescomplemented
81C22×Dic53—22no—
82C22×Dic53—22no—
83C22×Dic53—22no—
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53D414yescomplemented
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C22×Dic53C2311yescharacteristic
92C22×Dic53C2311yescharacteristic
93#11719947—232no—
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—41no—
97C2×C4×D53—22no—
98C2×C4×D53—22no—
99C2×C4×D53—22no—
100C2×C4×D53—22no—
101C2×C4×D53—22no—
102C2×C4×D53—22no—
103C2×Dic106—22no—
104C2×Dic106—22no—
105C2×Dic106D414yescomplemented
106C4⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊Dic53—22no—
112C22⋊Dic53—41no—
113#11719921—22no—
114#11719921—22no—
115#11719921—22no—
116#11719921—22no—
117#11719921—22no—
118#11719921—41no—
119#11719919—22no—
120#11719919—22no—
121#11719919—22no—
122#11719919—22no—
123C4×Dic53—22no—
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×C53C2×D412yes—
137C23×C53C2×D412yes—
138C23×C53C2411yescharacteristic
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×D53C2×D412yes—
152C22×D53#4112yescomplemented
153D4×C53—28no—
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×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53C2×D412yes—
169C22⋊D53—28no—
170C22⋊D53—28no—
171C22⋊D53—28no—
172C22⋊D53—44no—
173C2×Dic53—28no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—28no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53C2×D412yes—
188C2×Dic53C2×D412yes—
189C2×Dic53C2×D412yes—
190C2×Dic53C2×D412yes—
191C2×Dic53C2×D412yes—
192C2×Dic53#4112yescomplemented
193D212—44no—
194C4×D53—28no—
195C4×D53—44no—
196Dic106—28no—
197C22×C53—22no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—22no—
202C22×C53—22no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53C22×D411yescharacteristic
212C22×C53C22×D411yescharacteristic
213C22×C53C22×D411yescharacteristic
214C22×C53C2×#4111yescharacteristic
215C22×C53C2×#4111yescharacteristic
216C22×C53H(5,𝔽2)11yescharacteristic
217C22×C532−1+411yescharacteristic
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—41no—
229C2×D53—41no—
230C2×D53—41no—
231C2×D53—41no—
232C4×C53—22no—
233C4×C53—41no—
234C4×C53—41no—
235C4×C53—41no—
236C4×C53—41no—
237Dic53—22no—
238Dic53—22no—
239Dic53—22no—
240Dic53—41no—
241Dic53—41no—
242Dic53#12014yes—
243#2407—531nomaximal, Sylow
244C2×C53—42no—
245C2×C53—42no—
246C2×C53C2×#12011yescharacteristic
247C2×C53#52511yescharacteristic
248C2×C53#53711yescharacteristic
249C2×C53#54611yescharacteristic
250C2×C53#54711yescharacteristic
251C2×C53#54811yescharacteristic
252C2×C53#55011yescharacteristic
253D53—42no—
254D53—42no—
255C22×#41—531no—
256C2×C22⋊Q8—531no—
257C2×#121—531no—
258C2×C4×D4—531no—
259#394—531no—
260#394D5311yescharacteristic, complemented
261#392—531no—
262#392—531no—
263#391—531no—
264#390—531no—
265#386—531no—
266#386—531no—
267#385—531no—
268#385—531no—
269#380—531no—
270C53#240711yescharacteristic, complemented, Sylow
271C2×#41—1064no—
272C2×#41—1064no—
273C2×#41—1064no—
274C22×Q8—531no—
275C22×D4—531no—
276C22×D4—531no—
277C22×D4—531no—
278C22×D4—531no—
279C22×D4—531no—
280C22×D4C2×D5311yescharacteristic
281C23×C4—531no—
282C23×C4—531no—
283C23×C4—531no—
284C23×C4—531no—
285C22⋊Q8—1064no—
286#121—1064no—
287C4×D4—1064no—
288C2×C4⋊C4—531no—
289C2×C4⋊C4—531no—
290C2×C4⋊C4—531no—
291C2×C4⋊C4—531no—
292C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4—531no—
301C2×C22⋊C4C2×D5311yescharacteristic
302C2×C22⋊C4C2×D5311yescharacteristic
303C2×C22⋊C4C2×D5311yescharacteristic
304C2×C22⋊C4C2×D5311yescharacteristic
305C2×C42—531no—
306#95—531no—
307#95—531no—
308#95—531no—
309#95—531no—
310#95—531no—
311#95C2×D5311yescharacteristic
312C24—531no—
313C24—531no—
314C24C22×D5311yescharacteristic
315C24C22×D5311yescharacteristic
316#41—10632no—
317C2×Q8—534no—
318C2×Q8—1062no—
319C2×Q8—1062no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—534no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—1062no—
327C2×D4—1062no—
328C2×D4—1062no—
329C2×D4—1062no—
330C2×D4—1062no—
331C2×D4—1062no—
332C2×D4—1062no—
333C2×D4—1062no—
334C2×D4—1062no—
335C2×D4—1062no—
336C2×D4—2121no—
337C2×D4—2121no—
338C22×C4—531no—
339C22×C4—534no—
340C22×C4—531no—
341C22×C4—531no—
342C22×C4—531no—
343C22×C4—531no—
344C22×C4—531no—
345C22×C4—1062no—
346C22×C4—1062no—
347C22×C4—1062no—
348C22×C4—1062no—
349C22×C4—1062no—
350C22×C4—1062no—
351C22×C4—1062no—
352C22×C4—1062no—
353C22×C4—1062no—
354C22×C4—1062no—
355C22×C4—1062no—
356C22×C4C22×D5311yescharacteristic
357C22×C4C22×D5311yescharacteristic
358C22×C4C22×D5311yescharacteristic
359C22×C4C22×D5311yescharacteristic
360C22×C4C22×D5311yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—1062no—
363C4⋊C4—1062no—
364C4⋊C4—1062no—
365C4⋊C4—1062no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—1062no—
371C22⋊C4—1062no—
372C22⋊C4—1062no—
373C22⋊C4—1062no—
374C22⋊C4—1062no—
375C22⋊C4—1062no—
376C22⋊C4—2121no—
377C22⋊C4—2121no—
378C42—1062no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—532no—
392C23—532no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—1061no—
397C23—1061no—
398C23—1061no—
399C23—1061no—
400C23—1061no—
401C23—1061no—
402C23—1061no—
403C23—1061no—
404C23—1061no—
405C23D4×D5312yes—
406C23D4×D5312yes—
407C23C23×D5311yescharacteristic
408Q8—1068no—
409D4—28no—
410D4—1068no—
411D4—1068no—
412D4—1068no—
413D4—2124no—
414D4—2124no—
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—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—532no—
430C2×C4—532no—
431C2×C4—532no—
432C2×C4—532no—
433C2×C4—532no—
434C2×C4—532no—
435C2×C4—1068no—
436C2×C4—1068no—
437C2×C4—1068no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—1061no—
441C2×C4—1061no—
442C2×C4—1061no—
443C2×C4—1061no—
444C2×C4—1061no—
445C2×C4—1061no—
446C2×C4—1061no—
447C2×C4—1061no—
448C2×C4—1061no—
449C2×C4—1061no—
450C2×C4—2124no—
451C2×C4D4×D5312yes—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—1062no—
467C22—1062no—
468C22—1062no—
469C22—1062no—
470C22—1062no—
471C22—1062no—
472C22—2121no—
473C22—2121no—
474C22—2121no—
475C22—2121no—
476C22—2121no—
477C22—2121no—
478C22—2121no—
479C22—2121no—
480C22C2×D4×D5311yescharacteristic
481C22C2×D4×D5311yescharacteristic
482C22C2×D4×D5311yescharacteristic
483C22#41×D5311yescharacteristic
484C22#41×D5311yescharacteristic
485C22#1313326211yescharacteristic
486C22#1313326411yescharacteristic
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—534no—
493C4—1062no—
494C4—1062no—
495C4—1062no—
496C4—2121no—
497C4—2121no—
498C2—42no—
499C2—42no—
500C2—2122no—
501C2—2122no—
502C2#120×D5311yescharacteristic
503C2#1681113111yescharacteristic
504C2#1681120011yescharacteristic
505C2#1681126211yescharacteristic
506C2#1681127111yescharacteristic
507C2#1681128111yescharacteristic
508C2#1681129111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2407
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#11719947C2semidirect
2. C2×#13133195C2semidirect
3. C2×#13133190C2semidirect
4. C2×#13133177C2semidirect
5. #16810538C2semidirect
6. #16810532C2semidirect
7. #16810531C2semidirect
8. #16810527C2semidirect
9. #16810523C2semidirect
10. #16810501C2semidirect
11. #16810500C2semidirect
12. #16810495C2semidirect
13. #16810494C2semidirect
14. #16810481C2semidirect
15. C53×#394C2semidirect
80. C2×C22⋊D53D4semidirect
86. C22×Dic53D4semidirect
105. C2×Dic106D4semidirect
152. C22×D53#41semidirect
192. C2×Dic53#41semidirect
260. #394D53semidirect
270. C53#2407semidirect