Finite Group Databasebeta

← #20279788

Subgroups — #20279788

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×#13133204C211yesmaximal, characteristic, complemented
2C2×#13133203C211yesmaximal, characteristic, complemented
3C2×#13133189C211yesmaximal, characteristic, complemented
4#16810543C211yesmaximal, characteristic, complemented
5#16810537C211yesmaximal, characteristic, complemented
6#16810531C211yesmaximal, characteristic, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810523C211yesmaximal, characteristic, complemented
9#16810513C211yesmaximal, characteristic, complemented
10#16810511C211yesmaximal, characteristic, complemented
11#16810503C211yesmaximal, characteristic, complemented
12#16810502C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
15C2×#121×C53C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2211yescharacteristic
20C23×Dic53C2211yescharacteristic
21#13133204—24no—
22#13133203—24no—
23#13133189—24no—
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C2×#11719921C2211yescharacteristic
37C2×#11719921C2211yescharacteristic
38C2×C4×Dic53C2211yescharacteristic
39C2×C4×Dic53C2211yescharacteristic
40C2×C4×Dic53C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45#13133124C2211yescharacteristic
46#13133124C2211yescharacteristic
47C22×D4×C53C2211yescharacteristic
48C22×D4×C53C2211yescharacteristic
49C22×D4×C53C2211yescharacteristic
50C23×C4×C53C2211yescharacteristic
51#121×C53—24no—
52C2×C4⋊C4×C53C2211yescharacteristic
53C2×C22⋊C4×C53C2211yescharacteristic
54C2×C22⋊C4×C53C2211yescharacteristic
55C24×C53C2311yescharacteristic
56C24×C53C2311yescharacteristic
57C24×C53C2311yescharacteristic
58C23×D53C2311yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—22no—
63C2×D4×C53—22no—
64C2×D4×C53—22no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—41no—
71C22×C4×C53C2311yescharacteristic
72C22×C4×C53C2311yescharacteristic
73C22×C4×C53C2311yescharacteristic
74C22×C4×C53C2311yescharacteristic
75C2×C22⋊D53—22no—
76C2×C22⋊D53—22no—
77C2×C22⋊D53—22no—
78C2×C22⋊D53—22no—
79C2×C22⋊D53—22no—
80C2×C22⋊D53—41no—
81C2×C22⋊D53—41no—
82C2×C22⋊D53—41no—
83C2×C22⋊D53—41no—
84C22×Dic53—22no—
85C22×Dic53—22no—
86C22×Dic53—41no—
87C22×Dic53C2311yescharacteristic
88C22×Dic53C2311yescharacteristic
89C22×Dic53C2311yescharacteristic
90C22×Dic53C2311yescharacteristic
91C22×Dic53C2311yescharacteristic
92C22×Dic53C2311yescharacteristic
93C22×Dic53C2311yescharacteristic
94C2×Dic106—22no—
95C2×Dic106—22no—
96C2×Dic106—41no—
97C4⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53D414yescomplemented
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C22⋊Dic53—22no—
105C22⋊Dic53—22no—
106C22⋊Dic53—22no—
107C22⋊Dic53D414yescomplemented
108#11719921—22no—
109#11719921—22no—
110#11719921—22no—
111#11719921—22no—
112#11719919—22no—
113#11719919—41no—
114C4×Dic53—22no—
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×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—
146D4×C53—44no—
147D4×C53—44no—
148D4×C53—44no—
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—44no—
159C2×C4×C53C2×D412yes—
160C2×C4×C53C2×D412yes—
161C2×C4×C53#4112yes—
162C22⋊D53—44no—
163C22⋊D53—44no—
164C22⋊D53—44no—
165C2×Dic53—21no—
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×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—44no—
183C2×Dic53C2×D412yes—
184C2×Dic53C2×D412yes—
185C2×Dic53#4112yes—
186C2×Dic53#4112yes—
187Dic106—44no—
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—22no—
193C22×C53—22no—
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×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53C22×D411yescharacteristic
210C22×C53C22×D411yescharacteristic
211C22×C53C2×#4111yescharacteristic
212C22×C53C2×#4111yescharacteristic
213C22×C53C2×#4111yescharacteristic
214C22×C53H(5,𝔽2)11yescharacteristic
215C22×C53H(5,𝔽2)11yescharacteristic
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—
227Dic53—22no—
228Dic53—22no—
229Dic53—22no—
230Dic53—22no—
231Dic53—41no—
232Dic53—41no—
233Dic53—41no—
234#2555—531nomaximal, Sylow
235C2×C53—42no—
236C2×C53—42no—
237C2×C53—81no—
238C2×C53#53211yescharacteristic
239C2×C53#53211yescharacteristic
240C2×C53#53811yescharacteristic
241C2×C53D4211yescharacteristic
242C2×C53#54611yescharacteristic
243C2×C53#54611yescharacteristic
244C2×C53#55511yescharacteristic
245D53—81no—
246C2×#124—531no—
247C2×#124—531no—
248C2×#121—531no—
249C2×#121D5311yescharacteristic, complemented
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×Q8—531no—
263C22×D4—531no—
264C22×D4—531no—
265C22×D4—531no—
266C22×D4C2×D5311yescharacteristic
267C22×D4C2×D5311yescharacteristic
268C22×D4C2×D5311yescharacteristic
269C23×C4—531no—
270C23×C4C2×D5311yescharacteristic
271#124—1064no—
272#124—1064no—
273#121—24no—
274#121—1064no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4—531no—
289C2×C22⋊C4—531no—
290C2×C22⋊C4C2×D5311yescharacteristic
291C2×C22⋊C4C2×D5311yescharacteristic
292C2×C42—531no—
293C2×C42—531no—
294C2×C42—531no—
295#95—531no—
296#95—531no—
297#95—531no—
298#95—531no—
299#95—531no—
300#95—531no—
301C24—531no—
302C24C22×D5311yescharacteristic
303C24C22×D5311yescharacteristic
304C24C22×D5311yescharacteristic
305C2×Q8—1062no—
306C2×Q8—1062no—
307C2×Q8—2121no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—1062no—
318C2×D4—1062no—
319C2×D4—1062no—
320C2×D4—1062no—
321C2×D4—1062no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—531no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—531no—
335C22×C4—531no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—2121no—
339C22×C4C22×D5311yescharacteristic
340C22×C4C22×D5311yescharacteristic
341C22×C4C22×D5311yescharacteristic
342C22×C4C22×D5311yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—1062no—
345C4⋊C4—2121no—
346C22⋊C4—22no—
347C22⋊C4—534no—
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⋊C4—1062no—
357C22⋊C4—1062no—
358C22⋊C4—1062no—
359C22⋊C4C22⋊D5314yescomplemented
360C42—1062no—
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—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—1061no—
383C23—1061no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23D4×D5312yes—
390C23C2×C22⋊D5312yes—
391C23C23×D5311yescharacteristic
392Q8—2124no—
393D4—44no—
394D4—44no—
395D4—44no—
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—44no—
409C2×C4—532no—
410C2×C4—532no—
411C2×C4—532no—
412C2×C4—532no—
413C2×C4—1061no—
414C2×C4—1061no—
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—1061no—
430C2×C4—2124no—
431C2×C4#1171994712yes—
432C2×C4C2×C22⋊D5312yes—
433C2×C4C2×C22⋊D5312yes—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
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
466C22C2×#1171994711yescharacteristic
467C22#1313326211yescharacteristic
468C22#1313326211yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—1062no—
474C4—1062no—
475C4—1062no—
476C4—1062no—
477C4—2121no—
478C4—2121no—
479C4—2121no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—4241no—
484C2#1681116811yescharacteristic
485C2#1681116911yescharacteristic
486C2#1681120511yescharacteristic
487C2D4×C22⋊D5311yescharacteristic
488C2#1681126211yescharacteristic
489C2#1681126711yescharacteristic
490C2#1681133711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#121×C53C2
hypercenterC2×#121D53
hypocenterC53#2555
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#13133204C2semidirect
2. C2×#13133203C2semidirect
3. C2×#13133189C2semidirect
4. #16810543C2semidirect
5. #16810537C2semidirect
6. #16810531C2semidirect
7. #16810531C2semidirect
8. #16810523C2semidirect
9. #16810513C2semidirect
10. #16810511C2semidirect
11. #16810503C2semidirect
12. #16810502C2semidirect
13. C22⋊C4×Dic53C2semidirect
14. C4×C22⋊Dic53C2semidirect
15. C2×#121×C53C2semidirect
99. C22⋊C4×C53D4semidirect
107. C22⋊Dic53D4semidirect
249. C2×#121D53semidirect
261. C53#2555semidirect
359. C22⋊C4C22⋊D53semidirect