Finite Group Databasebeta

← #33087208

Subgroups — #33087208

Order 18688.

2258 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989094C211yesmaximal, characteristic
2C22×C4⋊C4×D73C211yesmaximal, characteristic, complemented
3C2×#17988588C211yesmaximal, characteristic, complemented
4C2×#17988550C211yesmaximal, characteristic, complemented
5#22610245C214yesmaximal, complemented
6#22610235C214yesmaximal, complemented
7#22609651C211yesmaximal, characteristic, complemented
8#22609650C211yesmaximal, characteristic, complemented
9C2×#417×C73C211yesmaximal, characteristic, complemented
10C23×C4×D73—21no—
11C23×C4×D73C2211yescharacteristic
12C22×#11924012C2211yescharacteristic
13#17989094—24no—
14#17989094C418yescomplemented
15C2×#14556572—21no—
16C2×#14556567C2212yes—
17C2×#14556562C2212yes—
18C2×#14556559—21no—
19C2×C4×C22⋊D73—22no—
20C2×#14556550C2211yescharacteristic
21C22×C4⋊Dic73C2211yescharacteristic
22C22×#11924007—21no—
23C2×C4⋊C4×D73—22no—
24C2×C4⋊C4×D73—24no—
25C2×C4⋊C4×D73—24no—
26C2×C4⋊C4×D73C2214yes—
27#17988588—24no—
28C2×#14556515C2212yescomplemented
29C2×#14556512C2212yescomplemented
30#17988550—24no—
31C2×#14556503C2214yes—
32#17988350C2214yescomplemented
33#17988349C2214yescomplemented
34C22×#41×C73C2211yescharacteristic
35C22×M16×C73C2211yescharacteristic
36C22×C4⋊C4×C73C2211yescharacteristic
37#417×C73—24no—
38C2×#103×C73C2212yescomplemented
39C2×#102×C73C2212yescomplemented
40C24×D73D411yescharacteristic
41C22×C22⋊D73—22no—
42C23×Dic73—21no—
43C23×Dic73D411yescharacteristic
44C22×C4×D73—28no—
45C22×C4×D73—24no—
46C22×C4×D73—22no—
47C22×C4×D73—22no—
48C22×C4×D73—22no—
49C22×C4×D73—22no—
50C22×C4×D73—48no—
51C22×C4×D73C2×C414yes—
52C22×C4×D73D412yes—
53C22×C4×D73D414yes—
54C22×C4×D73C2312yes—
55C2×#11924012—22no—
56C2×#11924012—22no—
57C2×#11924012—22no—
58C22×#10596831C2312yes—
59#14556572—28no—
60#14556567—24no—
61#14556567C2×C418yescomplemented
62#14556562—24no—
63#14556562C2×C418yescomplemented
64#14556559—28no—
65C4×C22⋊D73—216no—
66#14556550—22no—
67#14556550C2×C414yescomplemented
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—22no—
70C2×C4⋊Dic73D414yescomplemented
71C2×C4⋊Dic73C2312yes—
72C2×C4⋊Dic73C2312yes—
73C2×#11924007—28no—
74C2×#11924007—24no—
75C2×#11924007—22no—
76C4⋊C4×D73—216no—
77C4⋊C4×D73—28no—
78C4⋊C4×D73—28no—
79C2×C22⋊Dic73—22no—
80C2×C22⋊Dic73—21no—
81C2×#11924005—22no—
82C2×C4×Dic73—21no—
83#14556515—24no—
84#14556512—24no—
85#14556503—28no—
86C2×#41×C73—22no—
87C2×#41×C73—22no—
88C2×#41×C73—22no—
89C2×#41×C73—22no—
90C2×#41×C73—22no—
91C2×#41×C73C2×C414yescomplemented
92C22×Q8×C73C2311yescharacteristic
93C22×D4×C73—21no—
94C22×D4×C73C2311yescharacteristic
95C23×C4×C73—21no—
96C23×C4×C73—21no—
97C23×C4×C73C2311yescharacteristic
98C2×M16×C73—22no—
99C2×M16×C73—22no—
100C2×M16×C73—22no—
101C22×C8×C73C2312yes—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—22no—
104C2×C4⋊C4×C73D414yescomplemented
105C2×C4⋊C4×C73C2312yes—
106#103×C73—24no—
107#102×C73—24no—
108C23×D73—22no—
109C23×D73—22no—
110C23×D73—24no—
111C23×D73—24no—
112C23×D73C22⋊C414yes—
113C23×D73C2×D412yes—
114C2×C22⋊D73—28no—
115C2×C22⋊D73—44no—
116C2×C22⋊D73—44no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—24no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—44no—
123C22×Dic73C22⋊C414yes—
124C22×Dic73C2×D412yes—
125C2×C4×D73—28no—
126C2×C4×D73—28no—
127C2×C4×D73—24no—
128C2×C4×D73—28no—
129C2×C4×D73—28no—
130C2×C4×D73—24no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—22no—
135C2×C4×D73—216no—
136C2×C4×D73—432no—
137C2×C4×D73—48no—
138C2×C4×D73—48no—
139C2×C4×D73C22⋊C418yes—
140C2×C4×D73C22×C414yes—
141C2×C4×D73C2×D414yes—
142#11924012—44no—
143C2×#10596831—22no—
144C2×#10596831—22no—
145C2×#10596831—22no—
146C4⋊Dic73—24no—
147C4⋊Dic73—22no—
148C4⋊Dic73—22no—
149C4⋊Dic73C22×C414yescomplemented
150C4⋊Dic73C2×D414yes—
151#11924007—216no—
152#11924007—28no—
153C22⋊Dic73—24no—
154C22⋊Dic73—28no—
155#11924005—28no—
156C4×Dic73—24no—
157C24×C73—21no—
158C24×C73C2×D411yescharacteristic
159#41×C73—28no—
160#41×C73—28no—
161#41×C73—44no—
162#41×C73—44no—
163C2×Q8×C73—21no—
164C2×Q8×C73—21no—
165C2×Q8×C73—21no—
166C2×Q8×C73—21no—
167C2×Q8×C73—21no—
168C2×Q8×C73C22×C412yescomplemented
169C2×D4×C73—24no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73—21no—
173C2×D4×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—42no—
176C2×D4×C73—42no—
177C2×D4×C73C22×C412yescomplemented
178C22×C4×C73—24no—
179C22×C4×C73—22no—
180C22×C4×C73—21no—
181C22×C4×C73—24no—
182C22×C4×C73—44no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73C22×C412yes—
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D412yes—
188C22×C4×C73C2×D412yes—
189C22×C4×C73C2×D412yes—
190C22×C4×C73#4112yes—
191C22×C4×C73#4112yes—
192C22×C4×C73C2411yescharacteristic
193M16×C73—44no—
194C2×C8×C73—22no—
195C2×C8×C73—22no—
196C2×C8×C73—22no—
197C4⋊C4×C73—24no—
198C4⋊C4×C73—22no—
199C4⋊C4×C73C2×D414yes—
200C23×C73—22no—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73—42no—
206C23×C73—42no—
207C23×C73C2×C22⋊C412yes—
208C23×C73#12012yes—
209C23×C73#12312yes—
210C23×C73C22×D411yescharacteristic
211C22×D73—28no—
212C22×D73—216no—
213C22×D73—28no—
214C22×D73—22no—
215C22×D73—22no—
216C22×D73—22no—
217C22×D73—22no—
218C22×D73—28no—
219C22×D73—216no—
220C22×D73C2×C22⋊C414yes—
221C22×D73AGL(1,ℤ/8)14yes—
222C22×D73#13714yes—
223Q8×C73—22no—
224Q8×C73—22no—
225Q8×C73—41no—
226Q8×C73—41no—
227D4×C73—22no—
228D4×C73—22no—
229D4×C73—48no—
230D4×C73—41no—
231D4×C73—41no—
232C2×C4×C73—22no—
233C2×C4×C73—24no—
234C2×C4×C73—48no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—44no—
238C2×C4×C73—48no—
239C2×C4×C73C2×C22⋊C412yes—
240C2×C4×C73C4×D414yes—
241C2×C4×C73C4×D414yes—
242C2×C4×C73#12012yes—
243C2×C4×C73#12114yes—
244C2×C4×C73#12114yes—
245C2×C4×C73#12312yes—
246C2×C4×C73C23×C411yescharacteristic
247C2×C4×C73C22×D411yescharacteristic
248C2×C4×C73C22×D411yescharacteristic
249C2×C4×C73C22×D411yescharacteristic
250C2×C4×C73C2×#4111yescharacteristic
251C2×C4×C73C2×#4111yescharacteristic
252C22⋊D73—416no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—28no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—48no—
259C2×Dic73—44no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73C2×C22⋊C414yes—
263C4×D73—24no—
264C4×D73—24no—
265C4×D73—24no—
266C4×D73—24no—
267C4×D73—216no—
268C4×D73—432no—
269C8×C73—42no—
270#10596831—42no—
271C22×C73—24no—
272C22×C73—24no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—44no—
276C22×C73—42no—
277C22×C73—82no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73#38614yes—
281C22×C73C22×C22⋊C411yescharacteristic
282C22×C73C2×#12011yescharacteristic
283C22×C73C2×#12311yescharacteristic
284C22×C73C2×AGL(1,ℤ/8)12yes—
285C22×C73C2×#13712yes—
286C2×D73—232no—
287C2×D73—24no—
288C2×D73—24no—
289C2×D73—24no—
290C2×D73—24no—
291C2×D73—24no—
292C2×D73—24no—
293C2×D73#41718yescomplemented
294C4×C73—44no—
295C4×C73—42no—
296C4×C73#38614yes—
297C4×C73C2×C4×D411yescharacteristic
298C4×C73C2×C4×D411yescharacteristic
299C4×C73C2×#12111yescharacteristic
300C4×C73C2×#12111yescharacteristic
301Dic73—24no—
302Dic73—44no—
303Dic73—44no—
304#20909—731nomaximal, Sylow
305C2×C73—24no—
306C2×C73—82no—
307C2×C73C2×#38611yescharacteristic
308C2×C73C2×#41712yes—
309C2×C73#300311yescharacteristic
310C2×C73#300411yescharacteristic
311C2×C73#309212yes—
312D73—28no—
313C2×#525—731no—
314C23×C4⋊C4—731no—
315C2×#417—731no—
316C2×#417D7311yescharacteristic, complemented
317C2×#407—731no—
318#1874—734no—
319#1872—734no—
320#1736—731no—
321#1736—731no—
322C73#2090911yescharacteristic, complemented, Sylow
323C22×#41C2×D7311yescharacteristic
324C24×C4—731no—
325C24×C4—1461no—
326C22×M16—731no—
327C22×M16C2×D7311yescharacteristic
328#525—738no—
329#525—1464no—
330C2×#123—1461no—
331C2×C22⋊Q8—732no—
332C2×#121—732no—
333C2×#120—1461no—
334C2×C4×D4—1462no—
335C2×#117—731no—
336C22×C4⋊C4—731no—
337C22×C4⋊C4—734no—
338C22×C4⋊C4—1464no—
339C22×C4⋊C4—1462no—
340C22×C4⋊C4—1461no—
341C22×C4⋊C4—1464no—
342C22×C4⋊C4C2×D7311yescharacteristic
343#417—24no—
344#417—1464no—
345C2×#103—732no—
346C2×#103C2×D7312yescomplemented
347C2×#102—732no—
348C2×#102C2×D7312yescomplemented
349#407—1464no—
350C2×C22⋊C8—734no—
351#340—734no—
352#340—734no—
353C25—731no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—22no—
359C2×#41C4×D7314yescomplemented
360C22×Q8C22×D7311yescharacteristic
361C22×D4—21no—
362C22×D4—1462no—
363C22×D4C22×D7311yescharacteristic
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—2928no—
367C23×C4—731no—
368C23×C4—734no—
369C23×C4—732no—
370C23×C4—734no—
371C23×C4—732no—
372C23×C4—1464no—
373C23×C4—1468no—
374C23×C4—1461no—
375C23×C4—1462no—
376C23×C4—1462no—
377C23×C4—1462no—
378C23×C4—1462no—
379C23×C4C22×D7311yescharacteristic
380C2×M16—22no—
381C2×M16—22no—
382C2×M16—22no—
383C2×M16—1462no—
384C2×M16—1462no—
385C2×M16—1462no—
386C22×C8—732no—
387C22×C8C22×D7312yes—
388#123—1468no—
389C22⋊Q8—738no—
390C22⋊Q8—1464no—
391#121—738no—
392#121—1464no—
393#120—1468no—
394C4×D4—14616no—
395#117—734no—
396#117—1462no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—734no—
400C2×C4⋊C4—732no—
401C2×C4⋊C4—732no—
402C2×C4⋊C4—1468no—
403C2×C4⋊C4—14616no—
404C2×C4⋊C4—1464no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4—1468no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1462no—
409C2×C4⋊C4—1468no—
410C2×C4⋊C4C22⋊D7314yescomplemented
411C2×C4⋊C4C22×D7312yes—
412C2×C22⋊C4—1462no—
413C2×C22⋊C4—1462no—
414C2×C22⋊C4—1461no—
415C2×C42—1461no—
416#103—24no—
417#103—1464no—
418#102—24no—
419#102—1464no—
420C22⋊C8—1468no—
421C24—21no—
422C24—732no—
423C24—734no—
424C24—1464no—
425C24—1464no—
426C24—1462no—
427C24—1462no—
428C24D4×D7311yescharacteristic
429#41—28no—
430#41—28no—
431#41—44no—
432#41—44no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D7312yescomplemented
439C2×D4—24no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—2924no—
448C2×D4—2924no—
449C2×D4—1468no—
450C2×D4C2×C4×D7312yescomplemented
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—2928no—
459C22×C4—29232no—
460C22×C4—2928no—
461C22×C4—2924no—
462C22×C4—734no—
463C22×C4—732no—
464C22×C4—738no—
465C22×C4—734no—
466C22×C4—734no—
467C22×C4—14616no—
468C22×C4—1464no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1468no—
474C22×C4—1468no—
475C22×C4—1464no—
476C22×C4—1468no—
477C22×C4—1468no—
478C22×C4—1464no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4C2×C4×D7312yes—
484C22×C4D4×D7311yescharacteristic
485C22×C4D4×D7312yes—
486C22×C4D4×D7312yes—
487C22×C4C2×C22⋊D7312yes—
488C22×C4#1192402712yes—
489C22×C4#1192402812yes—
490C22×C4C23×D7311yescharacteristic
491M16—44no—
492M16—2924no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1462no—
497C2×C8—1462no—
498C2×C8—1462no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—734no—
502C4⋊C4—734no—
503C4⋊C4—1468no—
504C4⋊C4—14616no—
505C4⋊C4—1464no—
506C4⋊C4—1462no—
507C4⋊C4—1462no—
508C4⋊C4C2×C22⋊D7314yes—
509C22⋊C4—1468no—
510C22⋊C4—1468no—
511C22⋊C4—1464no—
512C42—1464no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—734no—
521C23—734no—
522C23—734no—
523C23—14616no—
524C23—1468no—
525C23—1468no—
526C23—14616no—
527C23—1462no—
528C23—1468no—
529C23—1462no—
530C23—1462no—
531C23—1462no—
532C23C22⋊C4×D7312yes—
533C23#1455655912yes—
534C23#1455657312yes—
535C23C2×D4×D7311yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540D4—22no—
541D4—22no—
542D4—48no—
543D4—41no—
544D4—41no—
545D4—29216no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—29232no—
554C2×C4—2928no—
555C2×C4—2922no—
556C2×C4—2924no—
557C2×C4—2922no—
558C2×C4—734no—
559C2×C4—1464no—
560C2×C4—1464no—
561C2×C4—1468no—
562C2×C4—1462no—
563C2×C4—1462no—
564C2×C4—14616no—
565C2×C4—1464no—
566C2×C4—1464no—
567C2×C4—1464no—
568C2×C4—1464no—
569C2×C4C22⋊C4×D7312yes—
570C2×C4#1455655114yes—
571C2×C4C4×C22⋊D7314yes—
572C2×C4#1455655912yes—
573C2×C4#1455656214yes—
574C2×C4#1455656414yes—
575C2×C4#1455657312yes—
576C2×C4C22×C4×D7311yescharacteristic
577C2×C4C2×D4×D7311yescharacteristic
578C2×C4C2×D4×D7311yescharacteristic
579C2×C4C22×C22⋊D7311yescharacteristic
580C2×C4C2×#1192402711yescharacteristic
581C2×C4C2×#1192402811yescharacteristic
582C8—42no—
583C8—2922no—
584C22—22no—
585C22—22no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—82no—
593C22—738no—
594C22—14632no—
595C22—1464no—
596C22—1464no—
597C22—1464no—
598C22—1464no—
599C22—1464no—
600C22—1464no—
601C22#1798846314yes—
602C22C2×C22⋊C4×D7311yescharacteristic
603C22C2×#1455655911yescharacteristic
604C22C2×#1455657311yescharacteristic
605C22AGL(1,ℤ/8)×D7312yes—
606C22#137×D7312yes—
607C4—44no—
608C4—42no—
609C4—2924no—
610C4—2924no—
611C4—1464no—
612C4#1798846314yes—
613C4C2×#1455655111yescharacteristic
614C4C2×C4×C22⋊D7311yescharacteristic
615C4C2×#1455656211yescharacteristic
616C4C2×#1455656411yescharacteristic
617C2—24no—
618C2—82no—
619C2—1468no—
620C2C2×#1798846311yescharacteristic
621C2#417×D7312yes—
622C2#2261915611yescharacteristic
623C2#2261916311yescharacteristic
624C2#2261985212yes—
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#417×C73C2
socleC23×C73C22×D4
prime radicalC2×#17989094C2
cosocleC22×C4×C73C24
hypercenterC2×#417D73
hypocenterC73#20909
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D73C2semidirect
3. C2×#17988588C2semidirect
4. C2×#17988550C2semidirect
5. #22610245C2semidirect
6. #22610235C2semidirect
7. #22609651C2semidirect
8. #22609650C2semidirect
9. C2×#417×C73C2semidirect
14. #17989094C4semidirect
28. C2×#14556515C22semidirect
29. C2×#14556512C22semidirect
32. #17988350C22semidirect
33. #17988349C22semidirect
38. C2×#103×C73C22semidirect
39. C2×#102×C73C22semidirect
61. #14556567C2×C4semidirect
63. #14556562C2×C4semidirect
67. #14556550C2×C4semidirect
70. C2×C4⋊Dic73D4semidirect
91. C2×#41×C73C2×C4semidirect
104. C2×C4⋊C4×C73D4semidirect
149. C4⋊Dic73C22×C4semidirect
168. C2×Q8×C73C22×C4semidirect
177. C2×D4×C73C22×C4semidirect
293. C2×D73#417semidirect
316. C2×#417D73semidirect
322. C73#20909semidirect
346. C2×#103C2×D73semidirect
348. C2×#102C2×D73semidirect
359. C2×#41C4×D73semidirect
410. C2×C4⋊C4C22⋊D73semidirect
438. C2×Q8C2×C4×D73semidirect
450. C2×D4C2×C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890942
C22×C4⋊C4×D732
C2×#179885882
C2×#179885502
#226102452
#226102352
#226096512
#226096502
C2×#417×C732
#2090973