Finite Group Databasebeta

← #20279786

Subgroups — #20279786

Order 6784.

696 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×#13133206C211yesmaximal, characteristic, complemented
2C2×#13133190C211yesmaximal, characteristic, complemented
3C2×#13133189C211yesmaximal, characteristic, complemented
4#16810542C211yesmaximal, characteristic, complemented
5#16810538C211yesmaximal, characteristic, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810523C211yesmaximal, characteristic, complemented
9#16810517C211yesmaximal, characteristic, complemented
10#16810514C211yesmaximal, characteristic, complemented
11#16810500C211yesmaximal, characteristic, complemented
12#16810499C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
14C4×#11719921C211yesmaximal, characteristic, complemented
15C2×#124×C53C211yesmaximal, characteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C22×C22⋊D53C2211yescharacteristic
21C23×Dic53C2211yescharacteristic
22C22×C4×D53C2211yescharacteristic
23#13133206—24no—
24#13133190—24no—
25#13133189—24no—
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2211yescharacteristic
28C2×#11719919C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×#11719921C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C22×Q8×C53C2211yescharacteristic
48C22×D4×C53C2211yescharacteristic
49#124×C53—24no—
50C2×C22⋊C4×C53C2211yescharacteristic
51C2×C22⋊C4×C53C2211yescharacteristic
52C2×C22⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C42×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C23×D53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×D4×C53—22no—
63C2×D4×C53—22no—
64C2×D4×C53—41no—
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
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—41no—
78C2×C22⋊D53—41no—
79C2×C22⋊D53—41no—
80C2×C22⋊D53—41no—
81C2×C22⋊D53—41no—
82C22×Dic53—22no—
83C22×Dic53—22no—
84C22×Dic53—41no—
85C22×Dic53C2311yescharacteristic
86C22×Dic53C2311yescharacteristic
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—41no—
94C2×C4×D53—22no—
95C2×C4×D53—22no—
96C2×C4×D53—41no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—22no—
100C22⋊C4×C53D414yescomplemented
101C42×C53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105#11719921—22no—
106#11719921—22no—
107#11719921—22no—
108#11719921—22no—
109#11719921—22no—
110#11719921—22no—
111#11719921D414yescomplemented
112#11719919—22no—
113#11719919—22no—
114#11719919—41no—
115C4×Dic53—22no—
116C4×Dic53—41no—
117C23×C53—21no—
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×C53C2411yescharacteristic
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×D53—21no—
144C22×D53—21no—
145C22×D53C2×D412yes—
146Q8×C53—44no—
147D4×C53—44no—
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×C53C2×D412yes—
161C2×C4×C53C2×D412yes—
162C2×C4×C53C2×D412yes—
163C22⋊D53—44no—
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×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—44no—
182C2×Dic53C2×D412yes—
183C2×Dic53#4112yes—
184C2×Dic53#4112yes—
185C2×Dic53#4112yes—
186D212—44no—
187C4×D53—44no—
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53C22×D411yescharacteristic
203C22×C53C22×D411yescharacteristic
204C22×C53C2×#4111yescharacteristic
205C22×C53C2×#4111yescharacteristic
206C22×C53C2×#4111yescharacteristic
207C22×C53H(5,𝔽2)11yescharacteristic
208C22×C53H(5,𝔽2)11yescharacteristic
209C2×D53—22no—
210C2×D53—22no—
211C2×D53—22no—
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—
228Dic53—22no—
229Dic53—22no—
230Dic53—22no—
231Dic53—22no—
232Dic53—41no—
233Dic53—41no—
234#2555—531nomaximal, Sylow
235C2×C53—42no—
236C2×C53—81no—
237C2×C53#53211yescharacteristic
238C2×C53#53211yescharacteristic
239C2×C53#53811yescharacteristic
240C2×C53D4211yescharacteristic
241C2×C53#54611yescharacteristic
242C2×C53#54611yescharacteristic
243C2×C53#55511yescharacteristic
244D53—42no—
245D53—81no—
246C2×#124—531no—
247C2×#124D5311yescharacteristic, complemented
248C2×#121—531no—
249C2×#121—531no—
250#395—531no—
251#394—531no—
252#392—531no—
253#392—531no—
254#390—531no—
255#388—531no—
256#388—531no—
257#386—531no—
258#386—531no—
259C4×C22⋊C4—531no—
260C4×C22⋊C4—531no—
261C53#255511yescharacteristic, complemented, Sylow
262C22×Q8C2×D5311yescharacteristic
263C22×D4—531no—
264C22×D4—531no—
265C22×D4—531no—
266C22×D4—531no—
267C22×D4—531no—
268C22×D4C2×D5311yescharacteristic
269C23×C4—531no—
270C23×C4—531no—
271#124—24no—
272#124—1064no—
273#121—1064no—
274#121—1064no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C22⋊C4—531no—
279C2×C22⋊C4—531no—
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⋊C4C2×D5311yescharacteristic
289C2×C22⋊C4C2×D5311yescharacteristic
290C2×C22⋊C4C2×D5311yescharacteristic
291C2×C22⋊C4C2×D5311yescharacteristic
292C2×C42—531no—
293C2×C42—531no—
294C2×C42C2×D5311yescharacteristic
295#95—531no—
296#95—531no—
297#95—531no—
298#95—531no—
299#95—531no—
300#95—531no—
301C24—531no—
302C24—531no—
303C24C22×D5311yescharacteristic
304C24C22×D5311yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—41no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—1062no—
312C2×D4—1062no—
313C2×D4—1062no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—1062no—
318C2×D4—1062no—
319C2×D4—1062no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—1062no—
333C22×C4—1062no—
334C22×C4—1062no—
335C22×C4—1062no—
336C22×C4—2121no—
337C22×C4—2121no—
338C22×C4C22×D5311yescharacteristic
339C22×C4C22×D5311yescharacteristic
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C4⋊C4—1062no—
344C4⋊C4—1062no—
345C4⋊C4—2121no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—534no—
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⋊C4C22⋊D5314yescomplemented
360C42—22no—
361C42—1062no—
362C42—2121no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—532no—
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—
390C23C2×C22⋊D5312yes—
391C23C23×D5311yescharacteristic
392Q8—44no—
393D4—44no—
394D4—2124no—
395D4—2124no—
396D4—2124no—
397D4—2124no—
398D4—2124no—
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—532no—
412C2×C4—532no—
413C2×C4—532no—
414C2×C4—532no—
415C2×C4—1061no—
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—1061no—
429C2×C4—2124no—
430C2×C4—2124no—
431C2×C4D4×D5312yes—
432C2×C4C2×C22⋊D5312yes—
433C2×C4C2×C22⋊D5312yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—1062no—
449C22—1062no—
450C22—1062no—
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
463C22C22×C22⋊D5311yescharacteristic
464C22#41×D5311yescharacteristic
465C22#41×D5311yescharacteristic
466C22#41×D5311yescharacteristic
467C22#1313326211yescharacteristic
468C22#1313326311yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—1062no—
475C4—1062no—
476C4—1062no—
477C4—1062no—
478C4—2121no—
479C4—2121no—
480C2—42no—
481C2—81no—
482C2—4241no—
483C2—2122no—
484C2#1681116911yescharacteristic
485C2#1681117211yescharacteristic
486C2#1681120711yescharacteristic
487C2D4×C22⋊D5311yescharacteristic
488C2#1681126111yescharacteristic
489C2#1681126611yescharacteristic
490C2#1681133511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#124×C53C2
hypercenterC2×#124D53
hypocenterC53#2555
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#13133206C2semidirect
2. C2×#13133190C2semidirect
3. C2×#13133189C2semidirect
4. #16810542C2semidirect
5. #16810538C2semidirect
6. #16810532C2semidirect
7. #16810531C2semidirect
8. #16810523C2semidirect
9. #16810517C2semidirect
10. #16810514C2semidirect
11. #16810500C2semidirect
12. #16810499C2semidirect
13. C22⋊C4×Dic53C2semidirect
14. C4×#11719921C2semidirect
15. C2×#124×C53C2semidirect
100. C22⋊C4×C53D4semidirect
111. #11719921D4semidirect
247. C2×#124D53semidirect
261. C53#2555semidirect
359. C22⋊C4C22⋊D53semidirect