Finite Group Databasebeta

← #33086731

Subgroups — #33086731

Order 18688.

1630 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556549C211yesmaximal, characteristic
2C2×#17988626C212yesmaximal, complemented
3C2×#17988549C211yesmaximal, characteristic
4#380×D73C211yesmaximal, characteristic
5#22610133C212yesmaximal, complemented
6#22610129C212yesmaximal, complemented
7#22610124C211yesmaximal, characteristic, complemented
8#22610121C211yesmaximal, characteristic, complemented
9#22610118C211yesmaximal, characteristic, complemented
10#22609649C212yesmaximal, complemented
11C73×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×#11924011C2212yes—
14C2×#14556549—21no—
15C2×#14556549—21no—
16C2×#14556549—28no—
17C2×#14556549C2212yes—
18C2×#14556549C2212yes—
19C22×C4⋊Dic73C2211yescharacteristic
20C2×C4⋊C4×D73C2211yescharacteristic
21C2×C4⋊C4×D73C2211yescharacteristic
22C22×#11924005—21no—
23C2×C22⋊C4×D73—21no—
24C22×C4×Dic73—21no—
25#17988626—28no—
26C2×#14556525C2212yescomplemented
27C2×#14556524C2212yescomplemented
28C2×#14556522C2212yescomplemented
29#17988549C418yescomplemented
30C2×C22⋊#10596831C2211yescharacteristic
31C2×#14556503C2212yes—
32#17988447C2211yescharacteristic
33#17988446—21no—
34#17988445—21no—
35#17988444C2211yescharacteristic
36#17988444C2211yescharacteristic
37#95×D73—21no—
38#17988350C2212yescomplemented
39#17988349C2212yescomplemented
40#17988349C2212yescomplemented
41C22×C4⋊C4×C73C2211yescharacteristic
42C2×#106×C73C2212yescomplemented
43C2×C22⋊C8×C73C2211yescharacteristic
44#380×C73C2211yescharacteristic
45#340×C73C2212yescomplemented
46C24×D73D411yescharacteristic
47C23×Dic73—21no—
48C23×Dic73D411yescharacteristic
49C22×C4×D73—24no—
50C22×C4×D73—24no—
51C22×C4×D73—21no—
52C22×C4×D73—21no—
53C22×C4×D73C2×C414yes—
54C22×C4×D73D411yescharacteristic
55C22×C4×D73D411yescharacteristic
56C22×C4×D73Q814yes—
57C22×C4×D73C2311yescharacteristic
58C22×C4×D73C2311yescharacteristic
59C2×#11924011—24no—
60C2×#11924011—24no—
61C2×#11924011C2×C418yescomplemented
62C22×#10596831C2312yes—
63#14556549—28no—
64#14556549—28no—
65#14556549—24no—
66#14556549—24no—
67#14556549—24no—
68#14556549D418yes—
69C2×C4⋊Dic73—24no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2312yes—
74C2×#11924007—21no—
75C2×#11924007—21no—
76C4⋊C4×D73—24no—
77C4⋊C4×D73—24no—
78C2×C22⋊Dic73—21no—
79C2×#11924005—28no—
80C2×#11924005—22no—
81C2×#11924005—22no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C22⋊C4×D73—44no—
85C2×C4×Dic73—28no—
86C2×C4×Dic73—22no—
87C2×C4×Dic73—22no—
88#14556525—24no—
89#14556524—24no—
90#14556522—24no—
91C22⋊#10596831C2×C414yescomplemented
92#14556503C2×C418yescomplemented
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96C23×C4×C73—21no—
97C23×C4×C73C2311yescharacteristic
98C22×C8×C73C2312yes—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—24no—
101C2×C4⋊C4×C73C2311yescharacteristic
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2312yes—
104C2×C22⋊C4×C73—21no—
105#106×C73—24no—
106C22⋊C8×C73C2×C414yescomplemented
107#95×C73—21no—
108C23×D73—22no—
109C23×D73—22no—
110C23×D73—22no—
111C23×D73—22no—
112C23×D73—41no—
113C23×D73—41no—
114C23×D73C4⋊C414yes—
115C23×D73C2×D411yescharacteristic
116C23×D73C2×D411yescharacteristic
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—24no—
120C22×Dic73—22no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—42no—
124C22×Dic73—42no—
125C22×Dic73C4⋊C414yes—
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C2×C4×D73—22no—
129C2×C4×D73—22no—
130C2×C4×D73—22no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—216no—
135C2×C4×D73—216no—
136C2×C4×D73—42no—
137C2×C4×D73—42no—
138C2×C4×D73—42no—
139C2×C4×D73—42no—
140C2×C4×D73—42no—
141C2×C4×D73—42no—
142C2×C4×D73C4⋊C414yes—
143C2×C4×D73C4⋊C414yes—
144C2×C4×D73C22×C412yes—
145C2×C4×D73C22×C412yes—
146C2×C4×D73C2×D414yes—
147C2×C4×D73C2×Q812yes—
148C2×C4×D73C2×Q812yes—
149C2×C4×D73#4114yes—
150#11924011—216no—
151C2×#10596831—22no—
152C2×#10596831—22no—
153C2×#10596831C22×C414yes—
154C4⋊Dic73—24no—
155C4⋊Dic73—22no—
156C4⋊Dic73—22no—
157C4⋊Dic73—22no—
158C4⋊Dic73C2×D414yes—
159#11924007—42no—
160#11924007—42no—
161C22⋊Dic73—42no—
162#11924005—28no—
163#11924005—28no—
164#11924005—42no—
165#11924005—42no—
166C4×Dic73—28no—
167C4×Dic73—28no—
168C24×C73C2×D411yescharacteristic
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—24no—
172C22×C4×C73—22no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—42no—
176C22×C4×C73—42no—
177C22×C4×C73C22×C412yes—
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×Q812yes—
180C22×C4×C73C2411yescharacteristic
181C2×C8×C73—22no—
182C2×C8×C73—22no—
183C2×C8×C73C22×C414yes—
184C4⋊C4×C73—24no—
185C4⋊C4×C73—22no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73—22no—
188C4⋊C4×C73C2×D414yes—
189C22⋊C4×C73—42no—
190C23×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73C2×C4⋊C412yes—
195C23×C73AGL(1,ℤ/8)12yes—
196C23×C73#13712yes—
197C23×C73C22×D411yescharacteristic
198C22×D73—21no—
199C22×D73—21no—
200C22×D73—21no—
201C22×D73—21no—
202C22×D73—21no—
203C22×D73—21no—
204C22×D73—21no—
205C22×D73—21no—
206C22×D73—21no—
207C22×D73—21no—
208C22×D73—28no—
209C22×D73—28no—
210C22×D73—44no—
211C22×D73—44no—
212C22×D73—44no—
213C22×D73—44no—
214C22×D73—44no—
215C22×D73C2×C4⋊C412yes—
216C22×D73C2×C4⋊C412yes—
217C22×D73#12012yes—
218C22×D73#12312yes—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—48no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73C2×C4⋊C412yes—
234C2×C4×C73C23×C411yescharacteristic
235C2×C4×C73C22×D412yes—
236C2×C4×C73C22×Q811yescharacteristic
237C2×C4×C73C2×#4112yes—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—28no—
241C2×Dic73—24no—
242C2×Dic73—48no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73C2×C4⋊C412yes—
252C2×Dic73C2×C4⋊C412yes—
253C2×Dic73#12012yes—
254C2×Dic73#12312yes—
255C4×D73—28no—
256C4×D73—24no—
257C4×D73—24no—
258C4×D73—28no—
259C4×D73—84no—
260C4×D73—84no—
261C4×D73C4×D418yes—
262C4×D73C22⋊Q818yes—
263C8×C73—24no—
264#10596831—24no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73#42814yescomplemented
272C22×C73C22×C4⋊C411yescharacteristic
273C22×C73C2×#12011yescharacteristic
274C22×C73C2×#12311yescharacteristic
275C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C73C2×#13711yescharacteristic
277C22×C73#57511yescharacteristic
278C22×C73#57511yescharacteristic
279C2×D73—22no—
280C2×D73—22no—
281C2×D73—22no—
282C2×D73—22no—
283C2×D73—22no—
284C2×D73—22no—
285C2×D73—22no—
286C2×D73—22no—
287C2×D73—22no—
288C2×D73—416no—
289C2×D73—41no—
290C2×D73—41no—
291C2×D73—41no—
292C2×D73—41no—
293C2×D73#38514yes—
294C4×C73—22no—
295C4×C73—44no—
296C4×C73—81no—
297C4×C73—81no—
298C4×C73C2×C4×D412yes—
299C4×C73C2×C22⋊Q812yes—
300Dic73—22no—
301Dic73—44no—
302Dic73—81no—
303Dic73—81no—
304Dic73#38514yes—
305#20846—731nomaximal, Sylow
306C2×C73—24no—
307C2×C73C2×#38511yescharacteristic
308C2×C73C2×#42811yescharacteristic
309C2×C73#291111yescharacteristic
310C2×C73#300611yescharacteristic
311C2×C73#300711yescharacteristic
312C2×C73#309411yescharacteristic
313C2×C73#309511yescharacteristic
314D73—24no—
315D73—42no—
316C22×#117—731no—
317C2×#428—732no—
318C2×#407—731no—
319C2×#380—731no—
320#1854—732no—
321#1854—732no—
322#1853—731no—
323#1853D7311yescharacteristic, complemented
324#1852—731no—
325#1852—731no—
326#1736—732no—
327C73#2084611yescharacteristic, complemented, Sylow
328C24×C4—731no—
329C22×M16—732no—
330C2×#117—732no—
331C2×#117—732no—
332C2×#117—1468no—
333C2×#117—1461no—
334C2×#117—1461no—
335C22×C4⋊C4—731no—
336C22×C4⋊C4—731no—
337C22×C4⋊C4—731no—
338C22×C4⋊C4C2×D7311yescharacteristic
339C22×C22⋊C4—1461no—
340C22×C22⋊C4—1461no—
341C22×C42—1461no—
342#428—1468no—
343C2×#107—732no—
344C2×#107—732no—
345C2×#106—732no—
346C2×#106C2×D7312yescomplemented
347#407—738no—
348C2×C22⋊C8—731no—
349C2×C22⋊C8—732no—
350C2×C22⋊C8C2×D7311yescharacteristic
351#380—731no—
352#380—731no—
353#380—731no—
354#380—1461no—
355#380—1461no—
356#380C2×D7311yescharacteristic
357C2×#95—1461no—
358#340—732no—
359#340—732no—
360#340—732no—
361#340C2×D7312yescomplemented
362C25—731no—
363C23×C4—21no—
364C23×C4—731no—
365C23×C4—731no—
366C23×C4—734no—
367C23×C4—731no—
368C23×C4—731no—
369C23×C4—734no—
370C23×C4—731no—
371C23×C4—1461no—
372C23×C4—1461no—
373C23×C4—1464no—
374C23×C4—1464no—
375C23×C4—1461no—
376C23×C4C22×D7311yescharacteristic
377C2×M16—738no—
378C2×M16—1464no—
379C2×M16—1464no—
380C22×C8—732no—
381C22×C8C22×D7312yes—
382#117—738no—
383#117—1464no—
384#117—1464no—
385#117—1468no—
386#117—1468no—
387#117—1464no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—732no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—1464no—
394C2×C4⋊C4—1464no—
395C2×C4⋊C4—1461no—
396C2×C4⋊C4—1461no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4—1461no—
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C4⋊C4C22×D7311yescharacteristic
401C2×C4⋊C4C22×D7312yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—2924no—
404C2×C22⋊C4—1461no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1462no—
408C2×C22⋊C4—1462no—
409C2×C22⋊C4—1468no—
410C2×C42—1462no—
411C2×C42—1462no—
412C2×C42—1468no—
413#107—1464no—
414#107—1464no—
415#106—24no—
416#106—1464no—
417C22⋊C8—734no—
418C22⋊C8—738no—
419C22⋊C8C4×D7314yescomplemented
420#95—21no—
421#95—1461no—
422#95—1461no—
423#95—1461no—
424C24—2921no—
425C24—2921no—
426C24—734no—
427C24—731no—
428C24—731no—
429C24—1462no—
430C24—1462no—
431C24—1462no—
432C24—1462no—
433C24D4×D7311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—2922no—
443C22×C4—2922no—
444C22×C4—2922no—
445C22×C4—2922no—
446C22×C4—2922no—
447C22×C4—2922no—
448C22×C4—2922no—
449C22×C4—2922no—
450C22×C4—731no—
451C22×C4—732no—
452C22×C4—734no—
453C22×C4—732no—
454C22×C4—734no—
455C22×C4—734no—
456C22×C4—731no—
457C22×C4—734no—
458C22×C4—734no—
459C22×C4—732no—
460C22×C4—732no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4—14616no—
464C22×C4—14616no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1464no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4C2×C4×D7312yes—
476C22×C4D4×D7311yescharacteristic
477C22×C4Q8×D7312yes—
478C22×C4C23×D7311yescharacteristic
479M16—14616no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—734no—
483C2×C8—1462no—
484C2×C8—1462no—
485C2×C8C2×C4×D7314yes—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—2922no—
491C4⋊C4—2922no—
492C4⋊C4—734no—
493C4⋊C4—1464no—
494C4⋊C4—1462no—
495C4⋊C4—1462no—
496C4⋊C4—1462no—
497C4⋊C4D4×D7314yes—
498C22⋊C4—42no—
499C22⋊C4—2922no—
500C22⋊C4—2922no—
501C22⋊C4—2922no—
502C22⋊C4—1468no—
503C22⋊C4—1468no—
504C42—1468no—
505C42—1468no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—2924no—
511C23—2924no—
512C23—2924no—
513C23—2924no—
514C23—2924no—
515C23—732no—
516C23—732no—
517C23—732no—
518C23—732no—
519C23—1468no—
520C23—1461no—
521C23—1461no—
522C23—1461no—
523C23—1468no—
524C23—1461no—
525C23—1461no—
526C23—1461no—
527C23—1461no—
528C23—1461no—
529C23—1461no—
530C23—1461no—
531C23C4⋊C4×D7312yes—
532C23#1455661812yes—
533C23#1455662012yes—
534C23C2×D4×D7311yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—2928no—
550C2×C4—2922no—
551C2×C4—2922no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—5844no—
559C2×C4—5844no—
560C2×C4—732no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—738no—
564C2×C4—738no—
565C2×C4—732no—
566C2×C4—1468no—
567C2×C4—1468no—
568C2×C4—1464no—
569C2×C4—1468no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1464no—
573C2×C4—1464no—
574C2×C4C4⋊C4×D7312yes—
575C2×C4C22×C4×D7311yescharacteristic
576C2×C4C2×D4×D7312yes—
577C2×C4C2×Q8×D7311yescharacteristic
578C2×C4#41×D7312yes—
579C8—24no—
580C8—1464no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—29216no—
588C22—2921no—
589C22—2921no—
590C22—2921no—
591C22—2921no—
592C22—734no—
593C22—1462no—
594C22—1462no—
595C22—1462no—
596C22—1462no—
597C22—1462no—
598C22—1462no—
599C22—1462no—
600C22—1462no—
601C22—1462no—
602C22#1798862614yescomplemented
603C22C2×C4⋊C4×D7311yescharacteristic
604C22#120×D7311yescharacteristic
605C22#123×D7311yescharacteristic
606C22C2×#1455661811yescharacteristic
607C22C2×#1455662011yescharacteristic
608C22#1798938311yescharacteristic
609C22#1798938611yescharacteristic
610C4—22no—
611C4—44no—
612C4—81no—
613C4—81no—
614C4—2924no—
615C4—5841no—
616C4—5841no—
617C4—734no—
618C4—1462no—
619C4C4×D4×D7312yes—
620C4C22⋊Q8×D7312yes—
621C2—24no—
622C2—2922no—
623C2—1464no—
624C2#385×D7311yescharacteristic
625C2C2×#1798862611yescharacteristic
626C2#2261851811yescharacteristic
627C2#2261918211yescharacteristic
628C2#2261918611yescharacteristic
629C2#2261986811yescharacteristic
630C2#2261987411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1853C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenter#1853D73
hypocenterC73#20846
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#17988626C2semidirect
5. #22610133C2semidirect
6. #22610129C2semidirect
7. #22610124C2semidirect
8. #22610121C2semidirect
9. #22610118C2semidirect
10. #22609649C2semidirect
11. C73×#1853C2semidirect
26. C2×#14556525C22semidirect
27. C2×#14556524C22semidirect
28. C2×#14556522C22semidirect
29. #17988549C4semidirect
38. #17988350C22semidirect
39. #17988349C22semidirect
40. #17988349C22semidirect
42. C2×#106×C73C22semidirect
45. #340×C73C22semidirect
61. C2×#11924011C2×C4semidirect
91. C22⋊#10596831C2×C4semidirect
92. #14556503C2×C4semidirect
106. C22⋊C8×C73C2×C4semidirect
271. C22×C73#428semidirect
323. #1853D73semidirect
327. C73#20846semidirect
346. C2×#106C2×D73semidirect
361. #340C2×D73semidirect
419. C22⋊C8C4×D73semidirect
602. C22#17988626semidirect

Maximal subgroups

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

subgroupindex
C22×#145565492
C2×#179886262
C2×#179885492
#380×D732
#226101332
#226101292
#226101242
#226101212
#226101182
#226096492
C73×#18532
#2084673