Finite Group Databasebeta

← #20278338

Subgroups — #20278338

Order 6784.

836 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11719947C211yesmaximal, characteristic
2C2×#13133216C211yesmaximal, characteristic, complemented
3C2×#13133203C211yesmaximal, characteristic, complemented
4C2×#13133194C211yesmaximal, characteristic, complemented
5#16810539C211yesmaximal, characteristic, complemented
6#16810538C211yesmaximal, characteristic, complemented
7#16810537C211yesmaximal, characteristic, complemented
8#16810527C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic, complemented
10#16810498C211yesmaximal, characteristic, complemented
11#16810497C211yesmaximal, characteristic, complemented
12#16810495C211yesmaximal, characteristic, complemented
13#16810494C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
15C53×#394C211yesmaximal, characteristic, complemented
16C2×#11719947—24no—
17C2×#11719947—24no—
18C2×#11719947—24no—
19C22×Dic106C2211yescharacteristic
20C22×Dic106C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C22⋊D53C2211yescharacteristic
23C23×Dic53C2211yescharacteristic
24C23×Dic53C2211yescharacteristic
25C22×C4×D53C2211yescharacteristic
26#13133216—24no—
27#13133203—24no—
28#13133194—24no—
29C2×C4⋊Dic53C2211yescharacteristic
30C2×C4⋊Dic53C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×#11719919C2211yescharacteristic
34C2×#11719919C2211yescharacteristic
35C2×C22⋊Dic53C2211yescharacteristic
36C2×C22⋊Dic53C2211yescharacteristic
37C2×C22⋊Dic53C2211yescharacteristic
38C2×C22⋊Dic53C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×C4×Dic53C2211yescharacteristic
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
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C2×C22⋊D53—22no—
69C2×C22⋊D53—22no—
70C2×C22⋊D53—22no—
71C2×C22⋊D53—22no—
72C2×C22⋊D53—22no—
73C2×C22⋊D53D414yes—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—22no—
77C22×Dic53—22no—
78C22×Dic53—22no—
79C22×Dic53D414yes—
80C22×Dic53C2311yescharacteristic
81C22×Dic53C2311yescharacteristic
82C22×Dic53C2311yescharacteristic
83C22×Dic53C2311yescharacteristic
84C22×Dic53C2311yescharacteristic
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87#11719947—232no—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—22no—
91C2×Dic106—22no—
92C2×Dic106—22no—
93C2×Dic106—22no—
94C2×Dic106—22no—
95C2×Dic106—22no—
96C2×Dic106D414yes—
97C4⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—22no—
109#11719921—22no—
110C4⋊Dic53—22no—
111C4⋊Dic53—22no—
112#11719919—22no—
113#11719919—22no—
114#11719919—22no—
115#11719919—22no—
116C4×Dic53—22no—
117C4×Dic53—22no—
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×C53C2×D412yes—
131C23×C53C2×D412yes—
132C23×C53C2411yescharacteristic
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53C2×D412yes—
140D4×C53—28no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53C2×D412yes—
155C2×C4×C53#4112yescomplemented
156C22⋊D53—28no—
157C22⋊D53—28no—
158C2×Dic53—28no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—28no—
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×Dic53C2×D412yes—
176C2×Dic53C2×D412yes—
177C2×Dic53C2×D412yes—
178C2×Dic53C2×D412yes—
179C2×Dic53C2×D412yes—
180C2×Dic53#4112yescomplemented
181C4×D53—28no—
182Dic106—28no—
183Dic106—44no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—22no—
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×C53C22×D411yescharacteristic
199C22×C53C22×D411yescharacteristic
200C22×C53C22×D411yescharacteristic
201C22×C53C2×#4111yescharacteristic
202C22×C53C2×#4111yescharacteristic
203C22×C532−1+411yescharacteristic
204C22×C532−1+411yescharacteristic
205C2×D53—22no—
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—41no—
209C2×D53—41no—
210C2×D53—41no—
211C2×D53—41no—
212C4×C53—22no—
213C4×C53—22no—
214C4×C53—41no—
215C4×C53—41no—
216C4×C53—41no—
217Dic53—22no—
218Dic53—22no—
219Dic53—22no—
220Dic53—41no—
221Dic53—41no—
222Dic53—41no—
223Dic53#12014yes—
224#2432—531nomaximal, Sylow
225C2×C53—42no—
226C2×C53—42no—
227C2×C53C2×#12011yescharacteristic
228C2×C53#53211yescharacteristic
229C2×C53#53611yescharacteristic
230C2×C53#54711yescharacteristic
231C2×C53#54711yescharacteristic
232C2×C53#54811yescharacteristic
233C2×C53#54811yescharacteristic
234D53—42no—
235C22×#41—531no—
236C2×C4⋊Q8—531no—
237C2×#124—531no—
238C2×C22⋊Q8—531no—
239#394—531no—
240#394—531no—
241#394—531no—
242#394D5311yescharacteristic, complemented
243#391—531no—
244#390—531no—
245#385—531no—
246#385—531no—
247#385—531no—
248#385—531no—
249C4×C22⋊C4—531no—
250C53#243211yescharacteristic, complemented, Sylow
251C2×#41—1064no—
252C2×#41—1064no—
253C2×#41—1064no—
254C22×Q8—531no—
255C22×Q8—531no—
256C22×D4—531no—
257C22×D4—531no—
258C22×D4C2×D5311yescharacteristic
259C23×C4—531no—
260C23×C4—531no—
261C23×C4—531no—
262C4⋊Q8—1064no—
263#124—1064no—
264C22⋊Q8—1064no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—531no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4—531no—
271C2×C4⋊C4C2×D5311yescharacteristic
272C2×C22⋊C4—531no—
273C2×C22⋊C4—531no—
274C2×C22⋊C4—531no—
275C2×C22⋊C4—531no—
276C2×C22⋊C4—531no—
277C2×C22⋊C4—531no—
278C2×C22⋊C4—531no—
279C2×C22⋊C4—531no—
280C2×C22⋊C4C2×D5311yescharacteristic
281C2×C22⋊C4C2×D5311yescharacteristic
282C2×C22⋊C4C2×D5311yescharacteristic
283C2×C22⋊C4C2×D5311yescharacteristic
284C2×C42—531no—
285C2×C42—531no—
286#95—531no—
287#95—531no—
288#95—531no—
289#95—531no—
290#95—531no—
291#95C2×D5311yescharacteristic
292C24—531no—
293C24C22×D5311yescharacteristic
294C24C22×D5311yescharacteristic
295#41—10632no—
296C2×Q8—534no—
297C2×Q8—1062no—
298C2×Q8—1062no—
299C2×Q8—1062no—
300C2×Q8—1062no—
301C2×Q8—1062no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—534no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—1062no—
311C22×C4—531no—
312C22×C4—534no—
313C22×C4—531no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—531no—
318C22×C4—531no—
319C22×C4—1062no—
320C22×C4—1062no—
321C22×C4—1062no—
322C22×C4—1062no—
323C22×C4—1062no—
324C22×C4—1062no—
325C22×C4—1062no—
326C22×C4—1062no—
327C22×C4C22×D5311yescharacteristic
328C22×C4C22×D5311yescharacteristic
329C22×C4C22×D5311yescharacteristic
330C22×C4C22×D5311yescharacteristic
331C22×C4C22×D5311yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—1062no—
334C4⋊C4—1062no—
335C4⋊C4—1062no—
336C4⋊C4—1062no—
337C4⋊C4—1062no—
338C4⋊C4—1062no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—1062no—
344C22⋊C4—1062no—
345C22⋊C4—1062no—
346C22⋊C4—1062no—
347C22⋊C4—1062no—
348C22⋊C4—1062no—
349C22⋊C4—1062no—
350C22⋊C4—1062no—
351C42—1062no—
352C42—1062no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—532no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23—1061no—
370C23—1061no—
371C23—1061no—
372C23D4×D5312yes—
373C23D4×D5312yes—
374C23C23×D5311yescharacteristic
375Q8—1068no—
376Q8—2124no—
377D4—28no—
378D4—1068no—
379D4—1068no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—532no—
394C2×C4—532no—
395C2×C4—532no—
396C2×C4—532no—
397C2×C4—532no—
398C2×C4—532no—
399C2×C4—1068no—
400C2×C4—1068no—
401C2×C4—1068no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—1061no—
408C2×C4—1061no—
409C2×C4—1061no—
410C2×C4—1061no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—1061no—
416C2×C4—1061no—
417C2×C4D4×D5312yes—
418C2×C4#1171994712yescomplemented
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—1062no—
434C22—1062no—
435C22—1062no—
436C22—2121no—
437C22—2121no—
438C22—2121no—
439C22—2121no—
440C22C2×D4×D5311yescharacteristic
441C22C2×D4×D5311yescharacteristic
442C22C2×D4×D5311yescharacteristic
443C22#41×D5311yescharacteristic
444C22C2×#1171994711yescharacteristic
445C22#1313326411yescharacteristic
446C22#1313326411yescharacteristic
447C4—22no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—534no—
453C4—1062no—
454C4—1062no—
455C4—1062no—
456C4—2121no—
457C4—2121no—
458C4—2121no—
459C2—42no—
460C2—42no—
461C2—2122no—
462C2#120×D5311yescharacteristic
463C2#1681116811yescharacteristic
464C2#1681119511yescharacteristic
465C2#1681127111yescharacteristic
466C2#1681127311yescharacteristic
467C2#1681127911yescharacteristic
468C2#1681128111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2432
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133216C2semidirect
3. C2×#13133203C2semidirect
4. C2×#13133194C2semidirect
5. #16810539C2semidirect
6. #16810538C2semidirect
7. #16810537C2semidirect
8. #16810527C2semidirect
9. #16810523C2semidirect
10. #16810498C2semidirect
11. #16810497C2semidirect
12. #16810495C2semidirect
13. #16810494C2semidirect
14. C22⋊C4×Dic53C2semidirect
15. C53×#394C2semidirect
155. C2×C4×C53#41semidirect
180. C2×Dic53#41semidirect
242. #394D53semidirect
250. C53#2432semidirect
418. C2×C4#11719947semidirect