Finite Group Databasebeta

← #20278323

Subgroups — #20278323

Order 6784.

914 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719947C211yesmaximal, characteristic, complemented
2C2×#13133201C211yesmaximal, characteristic, complemented
3C2×#13133191C211yesmaximal, characteristic, complemented
4C2×#13133189C211yesmaximal, characteristic, complemented
5#16810544C211yesmaximal, characteristic, complemented
6#16810539C211yesmaximal, characteristic, complemented
7#16810536C211yesmaximal, characteristic, complemented
8#16810532C211yesmaximal, characteristic, complemented
9#16810501C211yesmaximal, characteristic, complemented
10#16810500C211yesmaximal, characteristic, complemented
11#16810500C211yesmaximal, characteristic, complemented
12#16810498C211yesmaximal, characteristic, complemented
13#16810497C211yesmaximal, characteristic, complemented
14C2×#13133124C211yesmaximal, characteristic, complemented
15C53×#386C211yesmaximal, 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
27C23×Dic53C2211yescharacteristic
28C22×C4×D53C2211yescharacteristic
29#13133201—24no—
30#13133191—24no—
31#13133189—24no—
32C2×C4⋊Dic53C2211yescharacteristic
33C2×C4⋊Dic53C2211yescharacteristic
34C2×C4⋊Dic53C2211yescharacteristic
35C2×#11719919C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×#11719921C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48#13133124C2211yescharacteristic
49#13133124C2211yescharacteristic
50C22×D4×C53C2211yescharacteristic
51C23×C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55#95×C53C2211yescharacteristic
56#95×C53C2211yescharacteristic
57C24×C53C2311yescharacteristic
58C24×C53C2311yescharacteristic
59C23×D53C2311yescharacteristic
60C23×D53C2311yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53D414yescomplemented
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—41no—
67C22×C4×C53C2311yescharacteristic
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—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C2×C22⋊D53—22no—
81C2×C22⋊D53—22no—
82C2×C22⋊D53—41no—
83C2×C22⋊D53D414yescomplemented
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—22no—
87C22×Dic53—22no—
88C22×Dic53—22no—
89C22×Dic53—22no—
90C22×Dic53—22no—
91C22×Dic53—41no—
92C22×Dic53D414yescomplemented
93C22×Dic53C2311yescharacteristic
94C22×Dic53C2311yescharacteristic
95C22×Dic53C2311yescharacteristic
96C22×Dic53C2311yescharacteristic
97C22×Dic53C2311yescharacteristic
98C22×Dic53C2311yescharacteristic
99#11719947—232no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—41no—
103C2×C4×D53—22no—
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×Dic106—22no—
107C2×Dic106—22no—
108C2×Dic106—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—41no—
112C22⋊Dic53—22no—
113#11719921—22no—
114#11719921—22no—
115#11719921—22no—
116#11719921—22no—
117#11719921—22no—
118#11719921—22no—
119#11719921—41no—
120C4⋊Dic53—22no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123#11719919—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×C53C2×D412yes—
135C23×C53C2×D412yes—
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×D53—21no—
152C22×D53C2×D412yes—
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—44no—
168C2×C4×C53C2×D412yes—
169C2×C4×C53#4112yescomplemented
170C22⋊D53—28no—
171C22⋊D53—28no—
172C22⋊D53—44no—
173C22⋊D53—44no—
174C2×Dic53—28no—
175C2×Dic53—28no—
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—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—44no—
191C2×Dic53C2×D412yes—
192C2×Dic53C2×D412yes—
193C2×Dic53C2×D412yes—
194C2×Dic53#4112yescomplemented
195D212—44no—
196C4×D53—28no—
197Dic106—28no—
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—22no—
202C22×C53—22no—
203C22×C53—22no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—41no—
211C22×C53#12014yes—
212C22×C53C22×D411yescharacteristic
213C22×C53C22×D411yescharacteristic
214C22×C53C22×D411yescharacteristic
215C22×C53C2×#4111yescharacteristic
216C22×C53C2×#4111yescharacteristic
217C22×C53H(5,𝔽2)11yescharacteristic
218C22×C532−1+411yescharacteristic
219C2×D53—22no—
220C2×D53—22no—
221C2×D53—22no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—41no—
229C2×D53—41no—
230C2×D53—41no—
231C2×D53—41no—
232C2×D53—41no—
233C4×C53—22no—
234C4×C53—22no—
235C4×C53—41no—
236C4×C53—41no—
237C4×C53—41no—
238Dic53—22no—
239Dic53—22no—
240Dic53—22no—
241Dic53—22no—
242Dic53—41no—
243Dic53—41no—
244#2430—531nomaximal, Sylow
245C2×C53—24no—
246C2×C53—42no—
247C2×C53C2×#12011yescharacteristic
248C2×C53#53211yescharacteristic
249C2×C53#53711yescharacteristic
250C2×C53#54611yescharacteristic
251C2×C53#54611yescharacteristic
252C2×C53#54711yescharacteristic
253C2×C53#54711yescharacteristic
254D53—42no—
255D53—81no—
256C22×#41—531no—
257C2×#123—531no—
258C2×#121—531no—
259C2×#121—531no—
260#395—531no—
261#394—531no—
262#393—531no—
263#392—531no—
264#386—531no—
265#386—531no—
266#386—531no—
267#386D5311yescharacteristic, complemented
268#385—531no—
269#385—531no—
270C2×#95—531no—
271C53#243011yescharacteristic, complemented, Sylow
272C2×#41—1064no—
273C2×#41—1064no—
274C2×#41—1064no—
275C22×Q8—531no—
276C22×D4—531no—
277C22×D4—531no—
278C22×D4—531no—
279C22×D4—531no—
280C22×D4—531no—
281C22×D4C2×D5311yescharacteristic
282C23×C4—531no—
283C23×C4—531no—
284C23×C4—531no—
285C23×C4—531no—
286C23×C4C2×D5311yescharacteristic
287#123—1064no—
288#121—1064no—
289#121—1064no—
290C2×C4⋊C4—531no—
291C2×C4⋊C4—531no—
292C2×C4⋊C4—531no—
293C2×C4⋊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⋊C4—531no—
302C2×C22⋊C4C2×D5311yescharacteristic
303C2×C22⋊C4C2×D5311yescharacteristic
304C2×C22⋊C4C2×D5311yescharacteristic
305#95—531no—
306#95—531no—
307#95—531no—
308#95—531no—
309#95—531no—
310#95—531no—
311#95C2×D5311yescharacteristic
312#95C2×D5311yescharacteristic
313C24—531no—
314C24—531no—
315C24C22×D5311yescharacteristic
316C24C22×D5311yescharacteristic
317#41—10632no—
318C2×Q8—1062no—
319C2×Q8—1062no—
320C2×Q8—1062no—
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—
338C2×D4D21214yescomplemented
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—534no—
343C22×C4—531no—
344C22×C4—531no—
345C22×C4—531no—
346C22×C4—531no—
347C22×C4—531no—
348C22×C4—531no—
349C22×C4—1062no—
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—2121no—
360C22×C4C22×D5311yescharacteristic
361C22×C4C22×D5311yescharacteristic
362C22×C4C22×D5311yescharacteristic
363C22×C4C22×D5311yescharacteristic
364C22×C4C22×D5311yescharacteristic
365C4⋊C4—1062no—
366C4⋊C4—1062no—
367C4⋊C4—1062no—
368C4⋊C4—1062no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—41no—
372C22⋊C4—1062no—
373C22⋊C4—1062no—
374C22⋊C4—1062no—
375C22⋊C4—1062no—
376C22⋊C4—1062no—
377C22⋊C4—1062no—
378C22⋊C4—1062no—
379C22⋊C4—2121no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—532no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—1061no—
397C23—1061no—
398C23—1061no—
399C23—1061no—
400C23—1061no—
401C23—1061no—
402C23—1061no—
403C23—1061no—
404C23C2×D21212yes—
405C23C2×D21212yes—
406C23D4×D5312yes—
407C23D4×D5312yes—
408C23C23×D5311yescharacteristic
409Q8—1068no—
410D4—28no—
411D4—1068no—
412D4—1068no—
413D4—2124no—
414D4—2124no—
415D4—2124no—
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—44no—
430C2×C4—532no—
431C2×C4—532no—
432C2×C4—532no—
433C2×C4—532no—
434C2×C4—1068no—
435C2×C4—1068no—
436C2×C4—1068no—
437C2×C4—1061no—
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—1061no—
451C2×C4—2124no—
452C2×C4C2×D21212yes—
453C2×C4#1171994712yescomplemented
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—1062no—
468C22—1062no—
469C22—1062no—
470C22—2121no—
471C22—2121no—
472C22—2121no—
473C22—2121no—
474C22—2121no—
475C22—2121no—
476C22—2121no—
477C22—2121no—
478C22—2121no—
479C22—2121no—
480C22—2121no—
481C22#1313318714yes—
482C22C2×D4×D5311yescharacteristic
483C22C2×D4×D5311yescharacteristic
484C22C22×D21211yescharacteristic
485C22#41×D5311yescharacteristic
486C22C2×#1171994711yescharacteristic
487C22#1313326211yescharacteristic
488C22#1313326411yescharacteristic
489C4—22no—
490C4—22no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—1062no—
495C4—1062no—
496C4—1062no—
497C4—1062no—
498C4—2121no—
499C4—2121no—
500C2—24no—
501C2—42no—
502C2—4241no—
503C2—2122no—
504C2C2×#1313318711yescharacteristic
505C2#1681116811yescharacteristic
506C2#1681120011yescharacteristic
507C2#1681126011yescharacteristic
508C2#1681126811yescharacteristic
509C2#1681127111yescharacteristic
510C2#1681127711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2430
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#11719947C2semidirect
2. C2×#13133201C2semidirect
3. C2×#13133191C2semidirect
4. C2×#13133189C2semidirect
5. #16810544C2semidirect
6. #16810539C2semidirect
7. #16810536C2semidirect
8. #16810532C2semidirect
9. #16810501C2semidirect
10. #16810500C2semidirect
11. #16810500C2semidirect
12. #16810498C2semidirect
13. #16810497C2semidirect
14. C2×#13133124C2semidirect
15. C53×#386C2semidirect
63. C2×D4×C53D4semidirect
83. C2×C22⋊D53D4semidirect
92. C22×Dic53D4semidirect
169. C2×C4×C53#41semidirect
194. C2×Dic53#41semidirect
267. #386D53semidirect
271. C53#2430semidirect
338. C2×D4D212semidirect
453. C2×C4#11719947semidirect