Finite Group Databasebeta

← #33086712

Subgroups — #33086712

Order 18688.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D73C211yesmaximal, characteristic
2C2×#17988625C211yesmaximal, characteristic, complemented
3C2×#17988623C211yesmaximal, characteristic, complemented
4C22×#14556503C211yesmaximal, characteristic
5#22612310C211yesmaximal, characteristic
6#22610133C212yesmaximal, complemented
7#22610131C212yesmaximal, complemented
8#22610125C212yesmaximal, complemented
9#22610120C212yesmaximal, complemented
10#22609647C211yesmaximal, characteristic, complemented
11C73×#1735C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C23×#10596831C2211yescharacteristic
14C2×#14556550C2211yescharacteristic
15C2×#14556550C2211yescharacteristic
16C2×#14556549C2212yes—
17C2×#14556548—21no—
18C2×#14556547—21no—
19#117×D73—24no—
20#117×D73—24no—
21C2×C4⋊C4×D73C2212yes—
22C2×C22⋊C4×D73—21no—
23C2×C42×D73—21no—
24#17988625—24no—
25#17988623—24no—
26C2×#14556525C2212yescomplemented
27C2×#14556524C2212yescomplemented
28C2×#14556523C2212yescomplemented
29C2×#14556522C2212yescomplemented
30C2×#14556503C418yescomplemented
31C2×#14556503C2211yescharacteristic
32C2×#14556503C2211yescharacteristic
33C2×#14556503C2211yescharacteristic
34C2×#14556503C2211yescharacteristic
35#17988448—21no—
36#17988444C2212yes—
37#17988444C2212yes—
38#17988442—21no—
39C4×#11924005—21no—
40C4×#11924005—21no—
41#17988349C2212yescomplemented
42#17988349C2212yescomplemented
43C23×C8×C73C2211yescharacteristic
44C2×#117×C73C2211yescharacteristic
45C2×#117×C73C2211yescharacteristic
46#340×C73C2212yescomplemented
47#340×C73C2212yescomplemented
48C24×D73D411yescharacteristic
49C23×Dic73D411yescharacteristic
50C22×C4×D73—22no—
51C22×C4×D73—22no—
52C22×C4×D73—22no—
53C22×C4×D73—22no—
54C22×C4×D73—22no—
55C22×C4×D73C2×C414yes—
56C22×C4×D73D411yescharacteristic
57C22×C4×D73D411yescharacteristic
58C22×C4×D73Q814yes—
59C22×C4×D73C2311yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C22×#10596831—22no—
62C22×#10596831—22no—
63C22×#10596831C2×C414yescomplemented
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556550—22no—
67#14556550—22no—
68#14556550—22no—
69#14556550D414yes—
70#14556549—24no—
71#14556549—24no—
72#14556548—28no—
73#14556547—28no—
74C2×C4⋊Dic73C2312yes—
75C2×C4⋊Dic73C2312yes—
76C2×#11924007—22no—
77C4⋊C4×D73—24no—
78C4⋊C4×D73—24no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×#11924005—22no—
82C2×#11924005—22no—
83C2×#11924005—22no—
84C22⋊C4×D73—28no—
85C2×C4×Dic73—22no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C42×D73—28no—
89#14556525—24no—
90#14556524—24no—
91#14556523—24no—
92#14556522—24no—
93#14556503C2×C414yes—
94#14556503C2×C414yes—
95#14556503C2×C414yescomplemented
96#14556503C2×C414yescomplemented
97#14556498—22no—
98#14556498—22no—
99C23×C4×C73C2311yescharacteristic
100C22×C8×C73—22no—
101C22×C8×C73—22no—
102C22×C8×C73C2×C414yescomplemented
103C22×C8×C73C2311yescharacteristic
104C22×C8×C73C2311yescharacteristic
105#117×C73—22no—
106#117×C73—22no—
107#117×C73—22no—
108#117×C73D414yes—
109C2×C4⋊C4×C73C2312yes—
110C2×C4⋊C4×C73C2312yes—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C42×C73—21no—
114C2×C42×C73—21no—
115C23×D73—22no—
116C23×D73—22no—
117C23×D73—24no—
118C23×D73—42no—
119C23×D73C4⋊C414yes—
120C23×D73C2×D411yescharacteristic
121C23×D73C2×D411yescharacteristic
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73C4⋊C414yes—
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C2×C4×D73—28no—
130C2×C4×D73—28no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—28no—
134C2×C4×D73—28no—
135C2×C4×D73—22no—
136C2×C4×D73—22no—
137C2×C4×D73—21no—
138C2×C4×D73—21no—
139C2×C4×D73—21no—
140C2×C4×D73—21no—
141C2×C4×D73—21no—
142C2×C4×D73—21no—
143C2×C4×D73—21no—
144C2×C4×D73—21no—
145C2×C4×D73—28no—
146C2×C4×D73—44no—
147C2×C4×D73—44no—
148C2×C4×D73C4⋊C414yes—
149C2×C4×D73C4⋊C414yes—
150C2×C4×D73C22×C412yes—
151C2×C4×D73C22×C412yes—
152C2×C4×D73C2×Q812yes—
153C2×C4×D73C2×Q812yes—
154C2×#10596831—28no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831C22×C412yes—
160C2×#10596831C22×C412yes—
161C4⋊Dic73—22no—
162C4⋊Dic73—22no—
163C4⋊Dic73—22no—
164C4⋊Dic73C2×D414yes—
165#11924007—28no—
166C22⋊Dic73—24no—
167C22⋊Dic73—42no—
168#11924005—28no—
169#11924005—44no—
170#11924005—44no—
171C4×Dic73—28no—
172C4×Dic73—24no—
173C4×Dic73—42no—
174C24×C73C2×D411yescharacteristic
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73C22×C412yes—
178C22×C4×C73C2×D412yes—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D412yes—
181C22×C4×C73C2×Q812yes—
182C22×C4×C73#4112yes—
183C22×C4×C73#4112yes—
184C22×C4×C73C2411yescharacteristic
185C2×C8×C73—28no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73C22×C412yes—
191C2×C8×C73C22×C412yes—
192C4⋊C4×C73—22no—
193C4⋊C4×C73—22no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73C2×D414yes—
196C22⋊C4×C73—24no—
197C22⋊C4×C73—42no—
198C42×C73—24no—
199C42×C73—42no—
200C23×C73—22no—
201C23×C73—22no—
202C23×C73C2×C4⋊C412yes—
203C23×C73#12012yes—
204C23×C73#12312yes—
205C23×C73C22×D411yescharacteristic
206C22×D73—21no—
207C22×D73—21no—
208C22×D73—22no—
209C22×D73—21no—
210C22×D73—21no—
211C22×D73—22no—
212C22×D73—22no—
213C22×D73—28no—
214C22×D73—28no—
215C22×D73—44no—
216C22×D73—48no—
217C22×D73—48no—
218C22×D73C2×C4⋊C412yes—
219C22×D73C2×C4⋊C412yes—
220C22×D73#13514yes—
221C2×C4×C73—24no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—42no—
225C2×C4×C73—42no—
226C2×C4×C73—42no—
227C2×C4×C73C2×C4⋊C412yes—
228C2×C4×C73C4×D414yes—
229C2×C4×C73C4×D414yes—
230C2×C4×C73#12012yes—
231C2×C4×C73C22⋊Q814yes—
232C2×C4×C73C22⋊Q814yes—
233C2×C4×C73#12312yes—
234C2×C4×C73C23×C411yescharacteristic
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C22×Q811yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—28no—
245C2×Dic73—24no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73C2×C4⋊C412yes—
252C2×Dic73C2×C4⋊C412yes—
253C4×D73—22no—
254C4×D73—22no—
255C4×D73—22no—
256C4×D73—22no—
257C4×D73—22no—
258C4×D73—22no—
259C4×D73—22no—
260C4×D73—22no—
261C4×D73—28no—
262C4×D73—28no—
263C4×D73—416no—
264C8×C73—22no—
265C8×C73—22no—
266#10596831—22no—
267#10596831—22no—
268C22×C73—24no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—24no—
272C22×C73#38514yes—
273C22×C73C22×C4⋊C411yescharacteristic
274C22×C73C2×#12011yescharacteristic
275C22×C73C2×#12311yescharacteristic
276C22×C73C2×#13512yes—
277C22×C73#57512yes—
278C2×D73—24no—
279C2×D73—22no—
280C2×D73—24no—
281C2×D73—22no—
282C2×D73—22no—
283C2×D73—22no—
284C2×D73—22no—
285C2×D73—416no—
286C2×D73—42no—
287C2×D73—42no—
288C2×D73#42714yescomplemented
289C4×C73—44no—
290C4×C73—82no—
291C4×C73#38514yes—
292C4×C73C2×C4×D411yescharacteristic
293C4×C73C2×C4×D411yescharacteristic
294C4×C73C2×C22⋊Q811yescharacteristic
295C4×C73C2×C22⋊Q811yescharacteristic
296Dic73—22no—
297Dic73—22no—
298Dic73—44no—
299Dic73—82no—
300#20843—731nomaximal, Sylow
301C2×C73—24no—
302C2×C73C2×#38511yescharacteristic
303C2×C73C2×#42711yescharacteristic
304C2×C73#291111yescharacteristic
305C2×C73#300212yes—
306C2×C73#309112yes—
307D73—24no—
308D73—42no—
309C22×#117—731no—
310C2×#427—731no—
311C2×#427—731no—
312C22×C22⋊C8—731no—
313#2280—731no—
314#1854—732no—
315#1854—732no—
316#1853—732no—
317#1852—732no—
318#1735—731no—
319#1735D7311yescharacteristic, complemented
320C73#2084311yescharacteristic, complemented, Sylow
321C24×C4—731no—
322C23×C8—731no—
323C23×C8C2×D7311yescharacteristic
324C2×#117—731no—
325C2×#117—731no—
326C2×#117—732no—
327C2×#117—1461no—
328C2×#117—1461no—
329C2×#117—1464no—
330C2×#117—1464no—
331C2×#117C2×D7311yescharacteristic
332C2×#117C2×D7311yescharacteristic
333C22×C4⋊C4—732no—
334C22×C22⋊C4—1461no—
335C22×C42—1461no—
336#427—1464no—
337#427—1464no—
338C2×#107—732no—
339C2×#107—732no—
340C2×#106—732no—
341C2×#106—732no—
342C2×C22⋊C8—738no—
343C2×C22⋊C8—731no—
344C2×C22⋊C8—731no—
345C2×C22⋊C8—731no—
346C2×C22⋊C8—731no—
347#381—1461no—
348#380—732no—
349#380—732no—
350#379—1461no—
351C4×C22⋊C4—1461no—
352C4×C22⋊C4—1461no—
353#340—732no—
354#340—732no—
355#340C2×D7312yescomplemented
356#340C2×D7312yescomplemented
357C25—731no—
358C23×C4—731no—
359C23×C4—734no—
360C23×C4—731no—
361C23×C4—731no—
362C23×C4—734no—
363C23×C4—731no—
364C23×C4—731no—
365C23×C4—1462no—
366C23×C4—1462no—
367C23×C4—1462no—
368C23×C4—1462no—
369C23×C4—1462no—
370C23×C4C22×D7311yescharacteristic
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—734no—
374C22×C8—731no—
375C22×C8—731no—
376C22×C8—1462no—
377C22×C8—1462no—
378C22×C8C4×D7314yescomplemented
379C22×C8C22×D7311yescharacteristic
380C22×C8C22×D7311yescharacteristic
381#117—22no—
382#117—22no—
383#117—22no—
384#117—734no—
385#117—1468no—
386#117—1462no—
387#117—1462no—
388#117—1468no—
389#117—1464no—
390#117—1462no—
391#117—1464no—
392#117D29214yes—
393C2×C4⋊C4—732no—
394C2×C4⋊C4—732no—
395C2×C4⋊C4—1464no—
396C2×C4⋊C4—1462no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4C22×D7312yes—
399C2×C4⋊C4C22×D7312yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1468no—
403C2×C22⋊C4—1462no—
404C2×C22⋊C4—1462no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1462no—
407C2×C22⋊C4—1461no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1468no—
411C2×C42—1461no—
412C2×C42—1462no—
413C2×C42—1461no—
414#107—1464no—
415#107—1464no—
416#106—1464no—
417#106—1464no—
418C22⋊C8—734no—
419C22⋊C8—734no—
420C22⋊C8—734no—
421C22⋊C8—734no—
422#95—1462no—
423#95—1462no—
424C24—2922no—
425C24—734no—
426C24—731no—
427C24—731no—
428C24—1464no—
429C24—1462no—
430C24—1462no—
431C24D4×D7311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—2924no—
435C22×C4—2924no—
436C22×C4—734no—
437C22×C4—731no—
438C22×C4—731no—
439C22×C4—734no—
440C22×C4—734no—
441C22×C4—732no—
442C22×C4—732no—
443C22×C4—732no—
444C22×C4—732no—
445C22×C4—1468no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1468no—
451C22×C4—1468no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1468no—
455C22×C4—1468no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4C2×C4×D7312yes—
467C22×C4D4×D7312yes—
468C22×C4D4×D7311yescharacteristic
469C22×C4C2×D29212yes—
470C22×C4Q8×D7312yes—
471C22×C4#1192402712yes—
472C22×C4#1192402812yes—
473C22×C4C23×D7311yescharacteristic
474C2×C8—28no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—732no—
480C2×C8—732no—
481C2×C8—1468no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8C2×C4×D7312yes—
487C2×C8C2×C4×D7312yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—734no—
492C4⋊C4—1468no—
493C4⋊C4—1462no—
494C4⋊C4—1462no—
495C4⋊C4—1462no—
496C4⋊C4C2×D29214yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—2924no—
500C22⋊C4—2924no—
501C22⋊C4—2922no—
502C22⋊C4—1468no—
503C22⋊C4—1464no—
504C42—24no—
505C42—42no—
506C42—2922no—
507C42—1468no—
508C42—1464no—
509C23—22no—
510C23—22no—
511C23—2928no—
512C23—2928no—
513C23—2924no—
514C23—732no—
515C23—734no—
516C23—732no—
517C23—1468no—
518C23—1462no—
519C23—1468no—
520C23—1461no—
521C23—1461no—
522C23—1462no—
523C23—1462no—
524C23—1461no—
525C23—1461no—
526C23C4⋊C4×D7312yes—
527C23#1455656112yes—
528C23#1455657312yes—
529C23C2×D4×D7311yescharacteristic
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—29216no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—2922no—
540C2×C4—2922no—
541C2×C4—2922no—
542C2×C4—732no—
543C2×C4—732no—
544C2×C4—1464no—
545C2×C4—1468no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1468no—
551C2×C4—1468no—
552C2×C4—1462no—
553C2×C4—1462no—
554C2×C4—1462no—
555C2×C4—1462no—
556C2×C4—1462no—
557C2×C4—1462no—
558C2×C4—1462no—
559C2×C4—1462no—
560C2×C4C4⋊C4×D7312yes—
561C2×C4#1455655114yes—
562C2×C4C4×D29214yes—
563C2×C4#1455656112yes—
564C2×C4#1455656914yes—
565C2×C4#1455657114yes—
566C2×C4#1455657312yes—
567C2×C4C22×C4×D7311yescharacteristic
568C2×C4C2×D4×D7311yescharacteristic
569C2×C4C22×D29211yescharacteristic
570C2×C4C2×Q8×D7311yescharacteristic
571C2×C4C2×#1192402711yescharacteristic
572C2×C4C2×#1192402811yescharacteristic
573C8—22no—
574C8—22no—
575C8—1462no—
576C8—1462no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—24no—
581C22—29216no—
582C22—2922no—
583C22—2922no—
584C22—734no—
585C22—1462no—
586C22—1462no—
587C22—1462no—
588C22—1464no—
589C22—1462no—
590C22—1462no—
591C22—1464no—
592C22#1798846114yes—
593C22C2×C4⋊C4×D7311yescharacteristic
594C22C2×#1455656111yescharacteristic
595C22C2×#1455657311yescharacteristic
596C22#135×D7312yes—
597C22#1798938412yes—
598C4—44no—
599C4—82no—
600C4—2924no—
601C4—5842no—
602C4—1462no—
603C4—1462no—
604C4#1798846114yes—
605C4C2×#1455655111yescharacteristic
606C4C2×C4×D29211yescharacteristic
607C4C2×#1455656911yescharacteristic
608C4C2×#1455657111yescharacteristic
609C2—24no—
610C2—2922no—
611C2—1464no—
612C2C2×#1798846111yescharacteristic
613C2#427×D7311yescharacteristic
614C2#2261851911yescharacteristic
615C2#2261915112yes—
616C2#2261984212yes—
subgroupG/H
centerC22×C4D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1735C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenter#1735D73
hypocenterC73#20843
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. C2×#17988625C2semidirect
3. C2×#17988623C2semidirect
6. #22610133C2semidirect
7. #22610131C2semidirect
8. #22610125C2semidirect
9. #22610120C2semidirect
10. #22609647C2semidirect
11. C73×#1735C2semidirect
26. C2×#14556525C22semidirect
27. C2×#14556524C22semidirect
28. C2×#14556523C22semidirect
29. C2×#14556522C22semidirect
30. C2×#14556503C4semidirect
41. #17988349C22semidirect
42. #17988349C22semidirect
46. #340×C73C22semidirect
47. #340×C73C22semidirect
63. C22×#10596831C2×C4semidirect
95. #14556503C2×C4semidirect
96. #14556503C2×C4semidirect
102. C22×C8×C73C2×C4semidirect
288. C2×D73#427semidirect
319. #1735D73semidirect
320. C73#20843semidirect
355. #340C2×D73semidirect
356. #340C2×D73semidirect
378. C22×C8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D732
C2×#179886252
C2×#179886232
C22×#145565032
#226123102
#226101332
#226101312
#226101252
#226101202
#226096472
C73×#17352
#2084373