Finite Group Databasebeta

← C73×#19072 #32928476

Subgroups — C73×#19072 #32928476

Order 18688.

992 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3545C211yesmaximal, characteristic, complemented
2C73×#3180C212yesmaximal, complemented
3C73×#3153C211yesmaximal, characteristic, complemented
4C73×#1661C212yesmaximal, complemented
5C73×#1660C211yesmaximal, characteristic, complemented
6C73×#1659C211yesmaximal, characteristic, complemented
7C73×#1657C211yesmaximal, characteristic, complemented
8C73×#1656C212yesmaximal, complemented
9C73×#1655C211yesmaximal, characteristic, complemented
10C73×#1524C212yesmaximal, complemented
11C2×#329×C73C211yesmaximal, characteristic, complemented
12C2×#137×C73—21no—
13C2×AGL(1,ℤ/8)×C73—21no—
14#563×C73C2211yescharacteristic
15#562×C73C2212yes—
16H(3,𝔽4)×C73C2211yescharacteristic
17#555×C73—21no—
18#551×C73—21no—
19#551×C73—22no—
20#546×C73—22no—
21#542×C73—22no—
22#538×C73—22no—
23#536×C73—21no—
24#534×C73—21no—
25#532×C73C2211yescharacteristic, complemented
26#529×C73C2212yes—
27C2×#124×C73C2211yescharacteristic
28#501×C73C2212yescomplemented
29#497×C73C2211yescharacteristic
30#497×C73C2211yescharacteristic, complemented
31#496×C73C2211yescharacteristic
32#496×C73C2211yescharacteristic, complemented
33#496×C73C2212yescomplemented
34#495×C73—21no—
35#495×C73—21no—
36#490×C73C2212yescomplemented
37#489×C73C2212yescomplemented
38#466×C73—22no—
39#465×C73—22no—
40#442×C73—22no—
41#440×C73—22no—
42C4×M16×C73C2211yescharacteristic, complemented
43C2×#97×C73C2212yescomplemented
44#330×C73C2212yescomplemented
45#330×C73C2212yescomplemented
46#329×C73C2211yescharacteristic
47#329×C73C2211yescharacteristic
48#329×C73C2212yescomplemented
49#329×C73C2212yescomplemented
50#329×C73C2212yescomplemented
51C2×#41×C73—21no—
52C2×#41×C73—21no—
53C22×Q8×C73D411yescharacteristic
54C22×D4×C73—21no—
55C22×D4×C73D411yescharacteristic
56#137×C73—44no—
57AGL(1,ℤ/8)×C73—44no—
58C2×Q16×C73—21no—
59C2×Q16×C73—21no—
60C2×SD16×C73—22no—
61C2×SD16×C73—21no—
62C2×SD16×C73—21no—
63C2×SD16×C73—21no—
64C2×SD16×C73—21no—
65C2×D8×C73—22no—
66C2×D8×C73—21no—
67C2×D8×C73—21no—
68C2×M16×C73—21no—
69C22×C8×C73—22no—
70C4⋊Q8×C73—21no—
71C4⋊Q8×C73—21no—
72C4⋊Q8×C73C2311yescharacteristic
73#127×C73C2311yescharacteristic
74#126×C73—22no—
75#126×C73—22no—
76#126×C73—22no—
77#125×C73C2312yes—
78#124×C73—21no—
79#124×C73—22no—
80#124×C73—21no—
81#124×C73—22no—
82#124×C73—22no—
83#124×C73—21no—
84#124×C73D411yescharacteristic
85#124×C73D411yescharacteristic
86#124×C73C2312yes—
87#124×C73C2311yescharacteristic
88#124×C73C2311yescharacteristic
89#123×C73—21no—
90#123×C73—22no—
91#123×C73—22no—
92#123×C73—21no—
93#123×C73—22no—
94#123×C73—22no—
95#123×C73—22no—
96C22⋊Q8×C73—22no—
97C22⋊Q8×C73—22no—
98C22⋊Q8×C73—22no—
99C22⋊Q8×C73—22no—
100C22⋊Q8×C73D412yes—
101#121×C73—22no—
102#121×C73—22no—
103#121×C73—22no—
104#121×C73—22no—
105#121×C73—22no—
106#121×C73D412yes—
107#120×C73—21no—
108#120×C73—22no—
109#120×C73—21no—
110#120×C73—22no—
111C4×Q8×C73D412yes—
112C4×D4×C73—22no—
113C4×D4×C73—22no—
114C4×D4×C73—22no—
115C4×D4×C73—22no—
116C4×D4×C73D412yes—
117#117×C73—21no—
118#117×C73—22no—
119#117×C73—22no—
120C2×C22⋊C4×C73—21no—
121C2×C22⋊C4×C73—22no—
122C2×C22⋊C4×C73—22no—
123C2×C42×C73C2311yescharacteristic
124#107×C73—22no—
125#106×C73—22no—
126#103×C73—22no—
127#102×C73—22no—
128#102×C73—22no—
129#102×C73—22no—
130#97×C73—22no—
131#97×C73C2312yes—
132#97×C73C2312yes—
133#97×C73C2311yescharacteristic
134#97×C73C2311yescharacteristic
135C4×C8×C73—21no—
136C24×C73—21no—
137C24×C73—21no—
138#41×C73—42no—
139#41×C73—42no—
140#41×C73—82no—
141C2×Q8×C73—21no—
142C2×Q8×C73—21no—
143C2×Q8×C73—21no—
144C2×Q8×C73—42no—
145C2×Q8×C73C2×D412yes—
146C2×Q8×C73C2×D411yescharacteristic
147C2×Q8×C73C2×D411yescharacteristic
148C2×D4×C73—22no—
149C2×D4×C73—22no—
150C2×D4×C73—22no—
151C2×D4×C73—21no—
152C2×D4×C73—22no—
153C2×D4×C73—21no—
154C2×D4×C73—22no—
155C2×D4×C73—21no—
156C2×D4×C73—42no—
157C2×D4×C73—41no—
158C2×D4×C73—82no—
159C2×D4×C73C2×D412yes—
160C2×D4×C73C2×D411yescharacteristic
161C2×D4×C73C2×D411yescharacteristic
162C22×C4×C73—21no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73—41no—
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D412yes—
170Q16×C73—42no—
171Q16×C73—42no—
172SD16×C73—44no—
173SD16×C73—42no—
174SD16×C73—42no—
175SD16×C73—42no—
176SD16×C73—42no—
177D8×C73—44no—
178D8×C73—42no—
179D8×C73—42no—
180M16×C73—24no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—22no—
184C2×C8×C73—22no—
185C2×C8×C73—44no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73—22no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—22no—
190C4⋊C4×C73—22no—
191C4⋊C4×C73—22no—
192C4⋊C4×C73—22no—
193C4⋊C4×C73—22no—
194C4⋊C4×C73C2×D412yes—
195C4⋊C4×C73C2×D412yes—
196C22⋊C4×C73—22no—
197C22⋊C4×C73—21no—
198C22⋊C4×C73—21no—
199C22⋊C4×C73—22no—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—22no—
202C22⋊C4×C73—22no—
203C22⋊C4×C73—22no—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—22no—
206C22⋊C4×C73—22no—
207C22⋊C4×C73—22no—
208C22⋊C4×C73—22no—
209C22⋊C4×C73—22no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—42no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73—22no—
215C42×C73C2×D411yescharacteristic
216C42×C73C2×D412yes—
217C42×C73C2411yescharacteristic
218C23×C73—21no—
219C23×C73—22no—
220C23×C73—21no—
221C23×C73—42no—
222C23×C73—41no—
223C23×C73—82no—
224C23×C73—81no—
225C23×C73—82no—
226C23×C73#12011yescharacteristic
227Q8×C73—41no—
228Q8×C73—41no—
229Q8×C73—42no—
230Q8×C73—41no—
231Q8×C73—41no—
232Q8×C73—81no—
233Q8×C73—81no—
234D4×C73—41no—
235D4×C73—41no—
236D4×C73—42no—
237D4×C73—41no—
238D4×C73—42no—
239D4×C73—41no—
240D4×C73—42no—
241D4×C73—44no—
242D4×C73—81no—
243D4×C73—82no—
244D4×C73—82no—
245D4×C73—82no—
246D4×C73—81no—
247D4×C73—82no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—21no—
251C2×C4×C73—22no—
252C2×C4×C73—22no—
253C2×C4×C73—21no—
254C2×C4×C73—42no—
255C2×C4×C73—42no—
256C2×C4×C73—41no—
257C2×C4×C73—82no—
258C2×C4×C73—82no—
259C2×C4×C73—82no—
260C2×C4×C73—81no—
261C2×C4×C73—82no—
262C2×C4×C73#12012yes—
263C2×C4×C73#12011yescharacteristic
264C2×C4×C73C22×D412yes—
265C2×C4×C73C22×D411yescharacteristic
266C8×C73—24no—
267C8×C73—22no—
268C8×C73—22no—
269C8×C73—42no—
270C22×C73—22no—
271C22×C73—42no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—42no—
277C22×C73—42no—
278C22×C73—81no—
279C22×C73—82no—
280C22×C73—162no—
281C22×C73—162no—
282C22×C73#45312yes—
283C22×C73C2×#12011yescharacteristic
284C4×C73—21no—
285C4×C73—21no—
286C4×C73—22no—
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—42no—
291C4×C73—42no—
292C4×C73—42no—
293C4×C73—81no—
294#19072C7311yesmaximal, characteristic, direct product factor, Sylow
295C2×C73—22no—
296C2×C73—82no—
297C2×C73—81no—
298C2×C73—81no—
299C2×C73—161no—
300C2×C73C2×#45311yescharacteristic
301C2×C73#301812yes—
302#3545C2×C7311yescharacteristic, complemented
303#3180C2×C7312yescomplemented
304#3153C2×C7311yescharacteristic, complemented
305#1661C2×C7312yescomplemented
306#1660C2×C7311yescharacteristic, complemented
307#1659C2×C7311yescharacteristic, complemented
308#1657C2×C7311yescharacteristic, complemented
309#1656C2×C7312yescomplemented
310#1655C2×C7311yescharacteristic, complemented
311#1524C2×C7312yescomplemented
312C2×#329C2×C7311yescharacteristic, complemented
313C73#1907211yescharacteristic, direct product factor, Sylow
314C2×#137—21no—
315C2×AGL(1,ℤ/8)—21no—
316#563C22×C7311yescharacteristic
317#562C22×C7312yes—
318H(3,𝔽4)C22×C7311yescharacteristic
319#555—21no—
320#551—21no—
321#551—22no—
322#546—22no—
323#542—22no—
324#538—22no—
325#536—21no—
326#534—21no—
327#532C22×C7311yescharacteristic, complemented
328#529C22×C7312yes—
329C2×#124C22×C7311yescharacteristic
330#501C22×C7312yescomplemented
331#497C22×C7311yescharacteristic
332#497C22×C7311yescharacteristic, complemented
333#496C22×C7311yescharacteristic
334#496C22×C7311yescharacteristic, complemented
335#496C22×C7312yescomplemented
336#495—21no—
337#495—21no—
338#490C22×C7312yescomplemented
339#489C22×C7312yescomplemented
340#466—22no—
341#465—22no—
342#442—22no—
343#440—22no—
344C4×M16C22×C7311yescharacteristic, complemented
345C2×#97C22×C7312yescomplemented
346#330C22×C7312yescomplemented
347#330C22×C7312yescomplemented
348#329C22×C7311yescharacteristic
349#329C22×C7311yescharacteristic
350#329C22×C7312yescomplemented
351#329C22×C7312yescomplemented
352#329C22×C7312yescomplemented
353C2×#41—21no—
354C2×#41—21no—
355C22×Q8D4×C7311yescharacteristic
356C22×D4—21no—
357C22×D4D4×C7311yescharacteristic
358#137—44no—
359AGL(1,ℤ/8)—44no—
360C2×Q16—21no—
361C2×Q16—21no—
362C2×SD16—21no—
363C2×SD16—21no—
364C2×SD16—22no—
365C2×SD16—21no—
366C2×SD16—21no—
367C2×D8—21no—
368C2×D8—21no—
369C2×D8—22no—
370C2×M16—21no—
371C22×C8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8C23×C7311yescharacteristic
375#127C23×C7311yescharacteristic
376#126—22no—
377#126—22no—
378#126—22no—
379#125C23×C7312yes—
380#124—21no—
381#124—22no—
382#124—21no—
383#124—22no—
384#124—22no—
385#124—21no—
386#124D4×C7311yescharacteristic
387#124D4×C7311yescharacteristic
388#124C23×C7311yescharacteristic
389#124C23×C7311yescharacteristic
390#124C23×C7312yes—
391#123—21no—
392#123—22no—
393#123—22no—
394#123—21no—
395#123—22no—
396#123—22no—
397#123—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—22no—
402C22⋊Q8D4×C7312yes—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—22no—
407#121—22no—
408#121D4×C7312yes—
409#120—21no—
410#120—22no—
411#120—21no—
412#120—22no—
413C4×Q8D4×C7312yes—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4D4×C7312yes—
419#117—21no—
420#117—22no—
421#117—22no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C42C23×C7311yescharacteristic
426#107—22no—
427#106—22no—
428#103—22no—
429#102—22no—
430#102—22no—
431#102—22no—
432#97—22no—
433#97C23×C7312yes—
434#97C23×C7312yes—
435#97C23×C7311yescharacteristic
436#97C23×C7311yescharacteristic
437C4×C8—21no—
438C24—21no—
439C24—21no—
440#41—42no—
441#41—42no—
442#41—82no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—42no—
447C2×Q8C2×D4×C7312yes—
448C2×Q8C2×D4×C7311yescharacteristic
449C2×Q8C2×D4×C7311yescharacteristic
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—82no—
461C2×D4C2×D4×C7312yes—
462C2×D4C2×D4×C7311yescharacteristic
463C2×D4C2×D4×C7311yescharacteristic
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4C2×D4×C7311yescharacteristic
471C22×C4C2×D4×C7312yes—
472Q16—42no—
473Q16—42no—
474SD16—44no—
475SD16—42no—
476SD16—42no—
477SD16—42no—
478SD16—42no—
479D8—44no—
480D8—42no—
481D8—42no—
482M16—24no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—44no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4C2×D4×C7312yes—
497C4⋊C4C2×D4×C7312yes—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—42no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42C2×D4×C7311yescharacteristic
518C42C2×D4×C7312yes—
519C42C24×C7311yescharacteristic
520C23—21no—
521C23—22no—
522C23—21no—
523C23—42no—
524C23—41no—
525C23—82no—
526C23—81no—
527C23—82no—
528C23#120×C7311yescharacteristic
529Q8—41no—
530Q8—41no—
531Q8—42no—
532Q8—41no—
533Q8—41no—
534Q8—81no—
535Q8—81no—
536D4—41no—
537D4—41no—
538D4—42no—
539D4—41no—
540D4—42no—
541D4—41no—
542D4—42no—
543D4—44no—
544D4—81no—
545D4—82no—
546D4—81no—
547D4—82no—
548D4—82no—
549D4—82no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#120×C7312yes—
565C2×C4#120×C7311yescharacteristic
566C2×C4C22×D4×C7312yes—
567C2×C4C22×D4×C7311yescharacteristic
568C8—24no—
569C8—22no—
570C8—22no—
571C8—42no—
572C22—22no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—82no—
582C22—162no—
583C22—162no—
584C22#453×C7312yes—
585C22C2×#120×C7311yescharacteristic
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—81no—
596C2—22no—
597C2—81no—
598C2—82no—
599C2—81no—
600C2—161no—
601C2C2×#453×C7311yescharacteristic
602C2C73×#301812yes—
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3545C2semidirect
2. C73×#3180C2semidirect
3. C73×#3153C2semidirect
4. C73×#1661C2semidirect
5. C73×#1660C2semidirect
6. C73×#1659C2semidirect
7. C73×#1657C2semidirect
8. C73×#1656C2semidirect
9. C73×#1655C2semidirect
10. C73×#1524C2semidirect
11. C2×#329×C73C2semidirect
25. #532×C73C22semidirect
28. #501×C73C22semidirect
30. #497×C73C22semidirect
32. #496×C73C22semidirect
33. #496×C73C22semidirect
36. #490×C73C22semidirect
37. #489×C73C22semidirect
42. C4×M16×C73C22semidirect
43. C2×#97×C73C22semidirect
44. #330×C73C22semidirect
45. #330×C73C22semidirect
48. #329×C73C22semidirect
49. #329×C73C22semidirect
50. #329×C73C22semidirect
294. #19072C73direct (N × H)
302. #3545C2×C73semidirect
303. #3180C2×C73semidirect
304. #3153C2×C73semidirect
305. #1661C2×C73semidirect
306. #1660C2×C73semidirect
307. #1659C2×C73semidirect
308. #1657C2×C73semidirect
309. #1656C2×C73semidirect
310. #1655C2×C73semidirect
311. #1524C2×C73semidirect
312. C2×#329C2×C73semidirect
313. C73#19072direct (N × H)
327. #532C22×C73semidirect
330. #501C22×C73semidirect
332. #497C22×C73semidirect
334. #496C22×C73semidirect
335. #496C22×C73semidirect
338. #490C22×C73semidirect
339. #489C22×C73semidirect
344. C4×M16C22×C73semidirect
345. C2×#97C22×C73semidirect
346. #330C22×C73semidirect
347. #330C22×C73semidirect
350. #329C22×C73semidirect
351. #329C22×C73semidirect
352. #329C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#35452
C73×#31802
C73×#31532
C73×#16612
C73×#16602
C73×#16592
C73×#16572
C73×#16562
C73×#16552
C73×#15242
C2×#329×C732
#1907273