Finite Group Databasebeta

← #20278606

Subgroups — #20278606

Order 6784.

718 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133199C211yesmaximal, characteristic, complemented
2C2×#13133190C211yesmaximal, characteristic, complemented
3C2×C4×D212C211yesmaximal, characteristic, complemented
4C2×#13133179C211yesmaximal, characteristic, complemented
5#16810547C211yesmaximal, characteristic, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810532C211yesmaximal, characteristic, complemented
9#16810523C211yesmaximal, characteristic, complemented
10#16810514C211yesmaximal, characteristic, complemented
11#16810506C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic, complemented
13#16810499C211yesmaximal, characteristic, complemented
14#16810494C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×D212C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C22×C22⋊D53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23C22×C4×D53C2211yescharacteristic
24C22×C4×D53C2211yescharacteristic
25#13133199—24no—
26#13133190—24no—
27C4×D212—24no—
28#13133179—24no—
29C2×C4⋊Dic53C2211yescharacteristic
30C2×#11719919C2211yescharacteristic
31C2×#11719919C2211yescharacteristic
32C2×#11719919C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×#11719921C2211yescharacteristic
40C2×#11719921C2211yescharacteristic
41C2×#11719921C2211yescharacteristic
42C2×#11719921C2211yescharacteristic
43C2×C4×Dic53C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47#13133124C2211yescharacteristic
48C2×C4⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C42×C53C2211yescharacteristic
53#95×C53C2211yescharacteristic
54#95×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C23×D53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×C4×C53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C2×C22⋊D53—22no—
66C2×C22⋊D53—22no—
67C2×C22⋊D53—22no—
68C2×C22⋊D53—22no—
69C2×C22⋊D53—22no—
70C2×C22⋊D53—41no—
71C2×C22⋊D53—41no—
72C2×C22⋊D53—41no—
73C2×C22⋊D53—41no—
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C22×Dic53C2311yescharacteristic
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2311yescharacteristic
79C2×D212—22no—
80C2×D212—22no—
81C2×D212—22no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—22no—
86C2×D212—22no—
87C2×D212—41no—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—22no—
91C2×C4×D53—22no—
92C2×C4×D53—22no—
93C2×C4×D53—22no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97C4⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53—22no—
101C42×C53—22no—
102C22⋊Dic53—22no—
103#11719921—22no—
104#11719921—22no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—41no—
109#11719921—41no—
110#11719921—41no—
111#11719921D414yescomplemented
112C4⋊Dic53D414yescomplemented
113#11719919—22no—
114#11719919—22no—
115#11719919—41no—
116C4×Dic53—22no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53C2411yescharacteristic
125C22×D53—21no—
126C22×D53—21no—
127C22×D53—21no—
128C22×D53—21no—
129C22×D53—21no—
130C22×D53—21no—
131C22×D53—21no—
132C22×D53—21no—
133C22×D53—21no—
134C22×D53—21no—
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53C2×D412yes—
144C22×D53#4112yes—
145C22×D53#4112yes—
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×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×C53C2×D412yes—
162C2×C4×C53C2×D412yes—
163C2×C4×C53#4112yes—
164C22⋊D53—44no—
165C22⋊D53—44no—
166C22⋊D53—44no—
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×Dic53C2×D412yes—
180C2×Dic53C2×D412yes—
181C2×Dic53C2×D412yes—
182D212—28no—
183D212—28no—
184D212—44no—
185C4×D53—44no—
186C4×D53—44no—
187C4×D53—44no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53C22×D411yescharacteristic
196C22×C53C22×D411yescharacteristic
197C22×C53C2×#4111yescharacteristic
198C22×C53C2×#4111yescharacteristic
199C22×C53C2×#4111yescharacteristic
200C22×C53H(5,𝔽2)11yescharacteristic
201C22×C53H(5,𝔽2)11yescharacteristic
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—22no—
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—22no—
209C2×D53—22no—
210C2×D53—22no—
211C2×D53—41no—
212C2×D53—41no—
213C2×D53—41no—
214C2×D53—41no—
215C2×D53—41no—
216C2×D53—41no—
217C2×D53—41no—
218C2×D53—41no—
219C2×D53—41no—
220C2×D53—41no—
221C2×D53—41no—
222C2×D53—41no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—22no—
226C4×C53—41no—
227C4×C53—41no—
228C4×C53—41no—
229Dic53—22no—
230Dic53—22no—
231Dic53—22no—
232Dic53—41no—
233Dic53—41no—
234#2456—531nomaximal, Sylow
235C2×C53—81no—
236C2×C53#52511yescharacteristic
237C2×C53#52511yescharacteristic
238C2×C53#53811yescharacteristic
239C2×C53D4211yescharacteristic
240C2×C53#54611yescharacteristic
241C2×C53#55011yescharacteristic
242C2×C53#55311yescharacteristic
243D53—42no—
244D53—42no—
245D53—42no—
246C2×#123—531no—
247C2×#121—531no—
248C2×C4×D4—531no—
249C2×C4×D4—531no—
250#396—531no—
251#392—531no—
252#392—531no—
253#392—531no—
254#390—531no—
255#388—531no—
256#388D5311yescharacteristic, complemented
257#387—531no—
258#386—531no—
259#386—531no—
260#385—531no—
261C53#245611yescharacteristic, complemented, Sylow
262C22×D4—531no—
263C22×D4—531no—
264C22×D4—531no—
265C22×D4—531no—
266C22×D4—531no—
267C22×D4—531no—
268C23×C4—531no—
269C23×C4—531no—
270C23×C4—531no—
271#123—1064no—
272#121—1064no—
273C4×D4—1064no—
274C4×D4—1064no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4—531no—
279C2×C4⋊C4C2×D5311yescharacteristic
280C2×C22⋊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⋊C4C2×D5311yescharacteristic
291C2×C22⋊C4C2×D5311yescharacteristic
292C2×C22⋊C4C2×D5311yescharacteristic
293C2×C42—531no—
294C2×C42C2×D5311yescharacteristic
295#95—531no—
296#95—531no—
297#95—531no—
298#95—531no—
299#95C2×D5311yescharacteristic
300#95C2×D5311yescharacteristic
301C24—531no—
302C24—531no—
303C24—531no—
304C24C22×D5311yescharacteristic
305C2×D4—1062no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—1062no—
312C2×D4—1062no—
313C2×D4—1062no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—1062no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C22×C4—531no—
324C22×C4—531no—
325C22×C4—531no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—1062no—
329C22×C4—1062no—
330C22×C4—1062no—
331C22×C4—1062no—
332C22×C4—1062no—
333C22×C4—1062no—
334C22×C4—2121no—
335C22×C4—2121no—
336C22×C4—2121no—
337C22×C4C22×D5311yescharacteristic
338C22×C4C22×D5311yescharacteristic
339C22×C4C22×D5311yescharacteristic
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—534no—
345C4⋊C4—1062no—
346C4⋊C4—1062no—
347C4⋊C4—2121no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—534no—
352C22⋊C4—1062no—
353C22⋊C4—1062no—
354C22⋊C4—1062no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C22⋊C4—2121no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C42—22no—
362C42—1062no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—532no—
371C23—532no—
372C23—532no—
373C23—1061no—
374C23—1061no—
375C23—1061no—
376C23—1061no—
377C23—1061no—
378C23—1061no—
379C23—1061no—
380C23—1061no—
381C23—1061no—
382C23—1061no—
383C23—1061no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23—1061no—
391C23C23×D5311yescharacteristic
392D4—1068no—
393D4—1068no—
394D4—2124no—
395D4—2124no—
396D4—2124no—
397D4—2124no—
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—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—532no—
414C2×C4—532no—
415C2×C4—532no—
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×C4—2124no—
431C2×C4#1171994512yes—
432C2×C4D4×D5312yes—
433C2×C4D4×D5312yes—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—1062no—
442C22—1062no—
443C22—1062no—
444C22—1062no—
445C22—1062no—
446C22—1062no—
447C22—1062no—
448C22—1062no—
449C22—1062no—
450C22—2121no—
451C22—2121no—
452C22—2121no—
453C22—2121no—
454C22—2121no—
455C22—2121no—
456C22—2121no—
457C22—2121no—
458C22—2121no—
459C22—2121no—
460C22—2121no—
461C22—2121no—
462C22C2×D4×D5311yescharacteristic
463C22C2×D4×D5311yescharacteristic
464C22#41×D5311yescharacteristic
465C22#41×D5311yescharacteristic
466C22C2×#1171994511yescharacteristic
467C22#1313326311yescharacteristic
468C22#1313326311yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—1062no—
476C4—1062no—
477C4—1062no—
478C4—2121no—
479C4—2121no—
480C2—81no—
481C2—2122no—
482C2—2122no—
483C2—2122no—
484C2#1681113111yescharacteristic
485C2#1681113211yescharacteristic
486C2#1681121211yescharacteristic
487C2#1681125711yescharacteristic
488C2#1681126611yescharacteristic
489C2#1681129411yescharacteristic
490C2#1681131711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2456
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133199C2semidirect
2. C2×#13133190C2semidirect
3. C2×C4×D212C2semidirect
4. C2×#13133179C2semidirect
5. #16810547C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810532C2semidirect
9. #16810523C2semidirect
10. #16810514C2semidirect
11. #16810506C2semidirect
12. #16810501C2semidirect
13. #16810499C2semidirect
14. #16810494C2semidirect
15. C53×#388C2semidirect
111. #11719921D4semidirect
112. C4⋊Dic53D4semidirect
256. #388D53semidirect
261. C53#2456semidirect