Finite Group Databasebeta

← #31783607

Subgroups — #31783607

Order 17536.

1252 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D137C211yesmaximal, characteristic
2C2×C4×C22⋊D137C211yesmaximal, characteristic, complemented
3C2×#17984048C211yesmaximal, characteristic, complemented
4C22×#14555487C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D137C211yesmaximal, characteristic, complemented
6#22577303C211yesmaximal, characteristic, complemented
7#22577303C211yesmaximal, characteristic, complemented
8#22577283C211yesmaximal, characteristic, complemented
9#22577281C211yesmaximal, characteristic, complemented
10#22577281C211yesmaximal, characteristic, complemented
11#22577262C211yesmaximal, characteristic, complemented
12#22577258C211yesmaximal, characteristic, complemented
13#22577258C211yesmaximal, characteristic, complemented
14#22577250C211yesmaximal, characteristic, complemented
15#386×C137C211yesmaximal, characteristic, complemented
16C24×D137C2211yescharacteristic
17C24×D137C2211yescharacteristic
18C22×D548C2211yescharacteristic
19C22×C22⋊D137C2211yescharacteristic
20C22×C22⋊D137C2211yescharacteristic
21C2×D4×D137—24no—
22C2×D4×D137—24no—
23C2×D4×D137C418yescomplemented
24C23×Dic137C2211yescharacteristic, complemented
25C22×C4×D137C2211yescharacteristic
26C22×C4×D137C2211yescharacteristic, complemented
27C22×C4×D137C2211yescharacteristic, complemented
28C4×C22⋊D137—24no—
29#17984048—24no—
30C2×#14555489C2211yescharacteristic, complemented
31C2×#14555489C2211yescharacteristic, complemented
32C2×C22⋊Dic137C2211yescharacteristic, complemented
33C2×C22⋊Dic137C2211yescharacteristic, complemented
34C2×C22⋊Dic137C2211yescharacteristic, complemented
35C2×#14555487—24no—
36C2×#14555487C2211yescharacteristic
37C2×#14555487C2211yescharacteristic
38C2×#14555487C2211yescharacteristic
39C2×#14555487C2211yescharacteristic
40C2×#14555487C2211yescharacteristic, complemented
41C2×#14555487C2211yescharacteristic, complemented
42C2×#14555487C2211yescharacteristic, complemented
43C2×#14555487C2211yescharacteristic, complemented
44C2×#14555487C2211yescharacteristic, complemented
45C2×#14555487C2211yescharacteristic, complemented
46C22⋊C4×D137—24no—
47C2×C4×Dic137C2211yescharacteristic, complemented
48C2×C4×Dic137C2211yescharacteristic, complemented
49#17983995C2211yescharacteristic, complemented
50#17983995C2211yescharacteristic, complemented
51C22×D4×C137C2211yescharacteristic
52C23×C4×C137C2211yescharacteristic, complemented
53C2×C22⋊C4×C137C2211yescharacteristic, complemented
54C2×C22⋊C4×C137C2211yescharacteristic, complemented
55C2×C22⋊C4×C137C2211yescharacteristic, complemented
56#95×C137C2211yescharacteristic, complemented
57#95×C137C2211yescharacteristic, complemented
58C23×D137—22no—
59C23×D137—22no—
60C23×D137—22no—
61C23×D137—22no—
62C23×D137—22no—
63C23×D137—22no—
64C23×D137—22no—
65C23×D137—22no—
66C23×D137—41no—
67C23×D137—41no—
68C23×D137—41no—
69C23×D137—41no—
70C23×D137C2×C414yescomplemented
71C23×D137C2×C414yescomplemented
72C23×D137C2311yescharacteristic
73C23×D137C2311yescharacteristic
74C23×D137C2311yescharacteristic
75C2×D548—22no—
76C2×D548—22no—
77C2×D548C2×C414yescomplemented
78C2×C22⋊D137—22no—
79C2×C22⋊D137—22no—
80C2×C22⋊D137C2×C414yes—
81C2×C22⋊D137C2×C414yescomplemented
82C2×C22⋊D137D414yes—
83C2×C22⋊D137D414yes—
84D4×D137—232no—
85C22×Dic137—22no—
86C22×Dic137—22no—
87C22×Dic137—41no—
88C22×Dic137C2311yescharacteristic
89C22×Dic137C2311yescharacteristic
90C22×Dic137C2311yescharacteristic
91C22×Dic137C2311yescharacteristic
92C22×Dic137C2311yescharacteristic
93C2×C4×D137—22no—
94C2×C4×D137—22no—
95C2×C4×D137—22no—
96C2×C4×D137—22no—
97C2×C4×D137—22no—
98C2×C4×D137—22no—
99C2×C4×D137—41no—
100C2×C4×D137—41no—
101C2×C4×D137C2×C414yescomplemented
102#14555489—22no—
103#14555489—22no—
104C22⋊Dic137—22no—
105C22⋊Dic137—22no—
106C22⋊Dic137—41no—
107#14555487—22no—
108#14555487—22no—
109#14555487—22no—
110#14555487—22no—
111#14555487—22no—
112#14555487—22no—
113#14555487—22no—
114#14555487—41no—
115#14555487—41no—
116#14555487D414yescomplemented
117C4×Dic137—22no—
118C4×Dic137—22no—
119C24×C137C2311yescharacteristic
120C24×C137C2311yescharacteristic
121C2×D4×C137—22no—
122C2×D4×C137—22no—
123C2×D4×C137C2×C414yescomplemented
124C22×C4×C137—22no—
125C22×C4×C137—22no—
126C22×C4×C137—41no—
127C22×C4×C137C2311yescharacteristic
128C22×C4×C137C2311yescharacteristic
129C22×C4×C137C2311yescharacteristic
130C22×C4×C137C2311yescharacteristic
131C22×C4×C137C2311yescharacteristic
132C22⋊C4×C137—22no—
133C22⋊C4×C137—22no—
134C22⋊C4×C137—41no—
135C22×D137—21no—
136C22×D137—21no—
137C22×D137—21no—
138C22×D137—21no—
139C22×D137—21no—
140C22×D137—21no—
141C22×D137—21no—
142C22×D137—21no—
143C22×D137—28no—
144C22×D137—21no—
145C22×D137—21no—
146C22×D137—21no—
147C22×D137—21no—
148C22×D137—21no—
149C22×D137—21no—
150C22×D137—28no—
151C22×D137—28no—
152C22×D137—28no—
153C22×D137—44no—
154C22×D137—44no—
155C22×D137—44no—
156C22×D137—44no—
157C22×D137—44no—
158C22×D137—44no—
159C22×D137—44no—
160C22×D137—44no—
161C22×D137—44no—
162C22×D137—44no—
163C22×D137C22×C412yes—
164C22×D137C22×C412yes—
165C22×D137C22×C412yes—
166C22×D137C2×D412yes—
167C22×D137C2×D412yes—
168C22×D137C2×D412yes—
169C22×D137#4112yes—
170D548—28no—
171C22⋊D137—28no—
172C22⋊D137C22⋊C4116yescomplemented
173C2×Dic137—21no—
174C2×Dic137—21no—
175C2×Dic137—21no—
176C2×Dic137—21no—
177C2×Dic137—21no—
178C2×Dic137—21no—
179C2×Dic137—21no—
180C2×Dic137—21no—
181C2×Dic137—21no—
182C2×Dic137—21no—
183C2×Dic137—21no—
184C2×Dic137—44no—
185C2×Dic137C22×C412yes—
186C2×Dic137C2×D412yes—
187C2×Dic137C2×D412yes—
188C2×Dic137C2×D412yescomplemented
189C4×D137—28no—
190C4×D137—44no—
191C4×D137—44no—
192C23×C137—21no—
193C23×C137—21no—
194C23×C137—21no—
195C23×C137—21no—
196C23×C137—21no—
197C23×C137—21no—
198C23×C137—21no—
199C23×C137—21no—
200C23×C137—21no—
201C23×C137—21no—
202C23×C137C22×C412yes—
203C23×C137C22×C412yes—
204C23×C137C2×D412yes—
205C23×C137C2×D412yes—
206C23×C137C2411yescharacteristic
207D4×C137—28no—
208C2×C4×C137—21no—
209C2×C4×C137—21no—
210C2×C4×C137—21no—
211C2×C4×C137—21no—
212C2×C4×C137—21no—
213C2×C4×C137—21no—
214C2×C4×C137—21no—
215C2×C4×C137—21no—
216C2×C4×C137—21no—
217C2×C4×C137—21no—
218C2×C4×C137—21no—
219C2×C4×C137—21no—
220C2×C4×C137—21no—
221C2×C4×C137—44no—
222C2×C4×C137C22×C412yes—
223C2×C4×C137C2×D412yescomplemented
224C22×C137—22no—
225C22×C137—22no—
226C22×C137—22no—
227C22×C137—22no—
228C22×C137—22no—
229C22×C137—22no—
230C22×C137—22no—
231C22×C137—22no—
232C22×C137—22no—
233C22×C137—41no—
234C22×C137—41no—
235C22×C137—41no—
236C22×C137—41no—
237C22×C137C2×C22⋊C414yescomplemented
238C22×C137C23×C411yescharacteristic
239C22×C137C22×D411yescharacteristic
240C22×C137C22×D411yescharacteristic
241C22×C137C22×D411yescharacteristic
242C22×C137C2×#4111yescharacteristic
243C22×C137H(5,𝔽2)11yescharacteristic
244C22×C137H(5,𝔽2)11yescharacteristic
245C2×D137—22no—
246C2×D137—22no—
247C2×D137—22no—
248C2×D137—22no—
249C2×D137—22no—
250C2×D137—22no—
251C2×D137—22no—
252C2×D137—22no—
253C2×D137—22no—
254C2×D137—22no—
255C2×D137—22no—
256C2×D137—22no—
257C2×D137—22no—
258C2×D137—22no—
259C2×D137—22no—
260C2×D137—416no—
261C2×D137—416no—
262C2×D137—41no—
263C2×D137—41no—
264C2×D137—41no—
265C2×D137—41no—
266C2×D137C2×C22⋊C414yescomplemented
267C2×D137C4×D414yescomplemented
268C4×C137—22no—
269C4×C137—22no—
270C4×C137—41no—
271C4×C137—41no—
272C4×C137—41no—
273Dic137—22no—
274Dic137—41no—
275Dic137—41no—
276Dic137—41no—
277Dic137C2×C22⋊C414yescomplemented
278C2×C137—24no—
279C2×C137—42no—
280C2×C137C22×C22⋊C411yescharacteristic
281C2×C137C2×C4×D411yescharacteristic
282C2×C137#51811yescharacteristic
283C2×C137D4211yescharacteristic
284C2×C137D4211yescharacteristic
285C2×C137#54611yescharacteristic
286C2×C137#54611yescharacteristic
287D137—24no—
288D137—24no—
289D137—42no—
290C137D4×C22⋊C411yescharacteristic, complemented, Sylow
291D4×C22⋊C4—1371nomaximal, Sylow
292C23×D4—1371no—
293C2×C4×D4—1371no—
294C2×C4×D4—1371no—
295C22×C22⋊C4—1371no—
296C22×C22⋊C4—1371no—
297#390—1371no—
298#390—1371no—
299#386—1371no—
300#386—1371no—
301#386—1371no—
302#386D13711yescharacteristic, complemented
303#380—1371no—
304#379—1371no—
305#379—1371no—
306C4×C22⋊C4—1371no—
307C25—1371no—
308C25—1371no—
309C22×D4—2744no—
310C22×D4—2744no—
311C22×D4—1371no—
312C22×D4—1378no—
313C22×D4—1371no—
314C22×D4—1371no—
315C22×D4C2×D13711yescharacteristic
316C23×C4—1371no—
317C23×C4—1371no—
318C23×C4—1371no—
319C23×C4—1371no—
320C23×C4C2×D13711yescharacteristic, complemented
321C4×D4—2744no—
322C4×D4—2744no—
323C2×C4⋊C4—1371no—
324C2×C4⋊C4—1371no—
325C2×C22⋊C4—2744no—
326C2×C22⋊C4—2744no—
327C2×C22⋊C4—1371no—
328C2×C22⋊C4—1371no—
329C2×C22⋊C4—1371no—
330C2×C22⋊C4—1371no—
331C2×C22⋊C4—1371no—
332C2×C22⋊C4—1371no—
333C2×C22⋊C4—1371no—
334C2×C22⋊C4—1371no—
335C2×C22⋊C4—1371no—
336C2×C22⋊C4—1371no—
337C2×C22⋊C4—1371no—
338C2×C22⋊C4—1371no—
339C2×C22⋊C4—1371no—
340C2×C22⋊C4C2×D13711yescharacteristic, complemented
341C2×C22⋊C4C2×D13711yescharacteristic, complemented
342C2×C22⋊C4C2×D13711yescharacteristic, complemented
343C2×C42—1371no—
344C2×C42—1371no—
345#95—1371no—
346#95—1371no—
347#95C2×D13711yescharacteristic, complemented
348#95C2×D13711yescharacteristic, complemented
349C24—2742no—
350C24—2742no—
351C24—2742no—
352C24—2742no—
353C24—2742no—
354C24—2742no—
355C24—2742no—
356C24—2742no—
357C24—5481no—
358C24—5481no—
359C24—5481no—
360C24—5481no—
361C24—1371no—
362C24—1374no—
363C24—1374no—
364C24—1371no—
365C24—1371no—
366C24C22×D13711yescharacteristic
367C24C22×D13711yescharacteristic
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—2742no—
371C2×D4—2742no—
372C2×D4—27432no—
373C2×D4—2742no—
374C2×D4—2742no—
375C2×D4—1374no—
376C2×D4—1374no—
377C2×D4—1374no—
378C2×D4—1374no—
379C2×D4—1374no—
380C2×D4C4×D13714yescomplemented
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—2742no—
385C22×C4—2742no—
386C22×C4—2742no—
387C22×C4—2742no—
388C22×C4—2742no—
389C22×C4—2742no—
390C22×C4—2742no—
391C22×C4—2742no—
392C22×C4—5481no—
393C22×C4—5481no—
394C22×C4—5481no—
395C22×C4—1371no—
396C22×C4—1371no—
397C22×C4—1371no—
398C22×C4—1371no—
399C22×C4—1374no—
400C22×C4—1371no—
401C22×C4C22×D13711yescharacteristic
402C22×C4C22×D13711yescharacteristic
403C22×C4C22×D13711yescharacteristic
404C22×C4C22×D13711yescharacteristic
405C22×C4C22×D13711yescharacteristic
406C4⋊C4—2742no—
407C4⋊C4—2742no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—41no—
411C22⋊C4—2742no—
412C22⋊C4—2742no—
413C22⋊C4—2742no—
414C22⋊C4—2742no—
415C22⋊C4—2742no—
416C22⋊C4—2742no—
417C22⋊C4—2742no—
418C22⋊C4—2742no—
419C22⋊C4—2742no—
420C22⋊C4—5481no—
421C22⋊C4—5481no—
422C22⋊C4—5481no—
423C22⋊C4—1374no—
424C42—2742no—
425C42—2742no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—2748no—
437C23—2748no—
438C23—2748no—
439C23—2741no—
440C23—2741no—
441C23—2741no—
442C23—2741no—
443C23—2741no—
444C23—2741no—
445C23—2741no—
446C23—2741no—
447C23—2748no—
448C23—2741no—
449C23—2741no—
450C23—2741no—
451C23—2741no—
452C23—2741no—
453C23—2741no—
454C23—5484no—
455C23—5484no—
456C23—5484no—
457C23—5484no—
458C23—5484no—
459C23—5484no—
460C23—5484no—
461C23—5484no—
462C23—5484no—
463C23—5484no—
464C23—1372no—
465C23—1372no—
466C23—1372no—
467C23—1372no—
468C23—1372no—
469C23—1372no—
470C23—1372no—
471C23C2×C4×D13712yes—
472C23C2×C4×D13712yes—
473C23D4×D13712yes—
474C23D4×D13712yes—
475C23C23×D13711yescharacteristic
476D4—28no—
477D4—2748no—
478D4—2748no—
479D4—13716no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—44no—
494C2×C4—2748no—
495C2×C4—2741no—
496C2×C4—2741no—
497C2×C4—2741no—
498C2×C4—2741no—
499C2×C4—2741no—
500C2×C4—2741no—
501C2×C4—2741no—
502C2×C4—2741no—
503C2×C4—2741no—
504C2×C4—2741no—
505C2×C4—2741no—
506C2×C4—5484no—
507C2×C4—5484no—
508C2×C4—5484no—
509C2×C4—1372no—
510C2×C4—1372no—
511C2×C4—1372no—
512C2×C4—1372no—
513C2×C4C2×C4×D13712yes—
514C2×C4D4×D13712yescomplemented
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—2742no—
529C22—2742no—
530C22—2742no—
531C22—2742no—
532C22—2742no—
533C22—2742no—
534C22—2742no—
535C22—2742no—
536C22—2742no—
537C22—2742no—
538C22—2742no—
539C22—2742no—
540C22—2742no—
541C22—2742no—
542C22—2742no—
543C22—54816no—
544C22—54816no—
545C22—5481no—
546C22—5481no—
547C22—5481no—
548C22—5481no—
549C22—1374no—
550C22—1374no—
551C22C22⋊C4×D13714yescomplemented
552C22C22×C4×D13711yescharacteristic
553C22C2×D4×D13711yescharacteristic
554C22C2×D4×D13711yescharacteristic
555C22C2×D4×D13711yescharacteristic
556C22#41×D13711yescharacteristic
557C22#1798413311yescharacteristic
558C22#1798413411yescharacteristic
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—2742no—
565C4—5481no—
566C4—5481no—
567C4—5481no—
568C4—1374no—
569C2—24no—
570C2—42no—
571C2—2744no—
572C2—2744no—
573C2—5482no—
574C2C2×C22⋊C4×D13711yescharacteristic
575C2C4×D4×D13711yescharacteristic
576C2#2257787511yescharacteristic
577C2#2257803511yescharacteristic
578C2#2257803711yescharacteristic
579C2#2257804211yescharacteristic
580C2#2257804511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#386×C137C2
socle, cosocleC23×C137C24
prime radicalC22×D4×D137C2
hypercenter#386D137
hypocenterC137D4×C22⋊C4
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D137C2semidirect
3. C2×#17984048C2semidirect
4. C22×#14555487C2semidirect
5. C2×C22⋊C4×D137C2semidirect
6. #22577303C2semidirect
7. #22577303C2semidirect
8. #22577283C2semidirect
9. #22577281C2semidirect
10. #22577281C2semidirect
11. #22577262C2semidirect
12. #22577258C2semidirect
13. #22577258C2semidirect
14. #22577250C2semidirect
15. #386×C137C2semidirect
23. C2×D4×D137C4semidirect
24. C23×Dic137C22semidirect
26. C22×C4×D137C22semidirect
27. C22×C4×D137C22semidirect
30. C2×#14555489C22semidirect
31. C2×#14555489C22semidirect
32. C2×C22⋊Dic137C22semidirect
33. C2×C22⋊Dic137C22semidirect
34. C2×C22⋊Dic137C22semidirect
40. C2×#14555487C22semidirect
41. C2×#14555487C22semidirect
42. C2×#14555487C22semidirect
43. C2×#14555487C22semidirect
44. C2×#14555487C22semidirect
45. C2×#14555487C22semidirect
47. C2×C4×Dic137C22semidirect
48. C2×C4×Dic137C22semidirect
49. #17983995C22semidirect
50. #17983995C22semidirect
52. C23×C4×C137C22semidirect
53. C2×C22⋊C4×C137C22semidirect
54. C2×C22⋊C4×C137C22semidirect
55. C2×C22⋊C4×C137C22semidirect
56. #95×C137C22semidirect
57. #95×C137C22semidirect
70. C23×D137C2×C4semidirect
71. C23×D137C2×C4semidirect
77. C2×D548C2×C4semidirect
81. C2×C22⋊D137C2×C4semidirect
101. C2×C4×D137C2×C4semidirect
116. #14555487D4semidirect
123. C2×D4×C137C2×C4semidirect
172. C22⋊D137C22⋊C4semidirect
188. C2×Dic137C2×D4semidirect
223. C2×C4×C137C2×D4semidirect
237. C22×C137C2×C22⋊C4semidirect
266. C2×D137C2×C22⋊C4semidirect
267. C2×D137C4×D4semidirect
277. Dic137C2×C22⋊C4semidirect
290. C137D4×C22⋊C4semidirect
302. #386D137semidirect
320. C23×C4C2×D137semidirect
340. C2×C22⋊C4C2×D137semidirect
341. C2×C22⋊C4C2×D137semidirect
342. C2×C22⋊C4C2×D137semidirect
347. #95C2×D137semidirect
348. #95C2×D137semidirect
380. C2×D4C4×D137semidirect
514. C2×C4D4×D137semidirect
551. C22C22⋊C4×D137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×D4×D1372
C2×C4×C22⋊D1372
C2×#179840482
C22×#145554872
C2×C22⋊C4×D1372
#225773032
#225772832
#225772812
#225772622
#225772582
#225772502
#386×C1372
D4×C22⋊C4137