Finite Group Databasebeta

← C73×#20288 #32929692

Subgroups — C73×#20288 #32929692

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3398C211yesmaximal, characteristic, complemented
2C73×#3368C211yesmaximal, characteristic, complemented
3C73×#3324C211yesmaximal, characteristic, complemented
4C73×#3274C211yesmaximal, characteristic
5C73×#1709C211yesmaximal, characteristic, complemented
6C73×#1702C211yesmaximal, characteristic, complemented
7C73×#1701C211yesmaximal, characteristic, complemented
8C73×#1689C211yesmaximal, characteristic, complemented
9C73×#1688C211yesmaximal, characteristic, complemented
10C73×#1641C211yesmaximal, characteristic, complemented
11C73×#1639C211yesmaximal, characteristic, complemented
12C73×#1628C211yesmaximal, characteristic, complemented
13C73×#1626C211yesmaximal, characteristic, complemented
14C73×#1590C211yesmaximal, characteristic, complemented
15C73×#1590C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)×C73—21no—
17#557×C73C2211yescharacteristic, complemented
18#556×C73C2211yescharacteristic
19#550×C73C2211yescharacteristic
20D42×C73C2211yescharacteristic
21#497×C73—21no—
22#494×C73C2211yescharacteristic, complemented
23#493×C73C2211yescharacteristic
24#490×C73—21no—
25#486×C73C2211yescharacteristic
26#486×C73C2211yescharacteristic
27#486×C73C2211yescharacteristic, complemented
28#486×C73C2211yescharacteristic, complemented
29#482×C73—21no—
30#478×C73—21no—
31#474×C73C2211yescharacteristic
32#474×C73C2211yescharacteristic
33#469×C73—21no—
34#468×C73—21no—
35#464×C73—21no—
36#462×C73—21no—
37#462×C73C2211yescharacteristic, complemented
38#461×C73C2211yescharacteristic
39#460×C73—21no—
40#460×C73C2211yescharacteristic
41#460×C73C2211yescharacteristic
42#460×C73C2211yescharacteristic, complemented
43#459×C73C2211yescharacteristic
44#450×C73—21no—
45#449×C73—21no—
46#447×C73—21no—
47#446×C73C2211yescharacteristic, complemented
48#446×C73C2211yescharacteristic, complemented
49#441×C73—21no—
50#440×C73—21no—
51C4×SD16×C73C2211yescharacteristic
52C4×SD16×C73C2211yescharacteristic
53#436×C73C2211yescharacteristic
54#436×C73C2211yescharacteristic
55#418×C73—21no—
56#334×C73C2211yescharacteristic, complemented
57#334×C73C2211yescharacteristic, complemented
58#332×C73C2211yescharacteristic, complemented
59#332×C73C2211yescharacteristic, complemented
60#331×C73C2211yescharacteristic
61#331×C73C2211yescharacteristic, complemented
62Q8⋊C8×C73C2211yescharacteristic, complemented
63Q8⋊C8×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66C82×C73C2211yescharacteristic, complemented
67C2×#41×C73—21no—
68C22×D4×C73—21no—
69C22×D4×C73—21no—
70AGL(1,ℤ/8)×C73—44no—
71C2×Q16×C73—21no—
72C2×Q16×C73—21no—
73C2×Q16×C73—21no—
74C2×SD16×C73—21no—
75C2×SD16×C73—21no—
76C2×SD16×C73—21no—
77C2×SD16×C73—21no—
78C2×SD16×C73D411yescharacteristic
79C2×SD16×C73D411yescharacteristic
80C2×D8×C73—21no—
81C2×D8×C73—21no—
82C2×D8×C73—21no—
83C2×M16×C73—21no—
84C2×M16×C73—21no—
85C4⋊Q8×C73—21no—
86C4⋊Q8×C73C2311yescharacteristic
87C4⋊Q8×C73C2311yescharacteristic
88#127×C73—21no—
89#127×C73C2311yescharacteristic
90#127×C73C2311yescharacteristic
91#126×C73—41no—
92#125×C73—21no—
93#125×C73—41no—
94#124×C73—21no—
95C22⋊Q8×C73—21no—
96#121×C73—21no—
97#121×C73—21no—
98#121×C73—21no—
99#121×C73—41no—
100#120×C73—41no—
101C4×Q8×C73—21no—
102C4×Q8×C73—21no—
103C4×Q8×C73—21no—
104C4×Q8×C73C2311yescharacteristic
105C4×Q8×C73C2311yescharacteristic
106C4×D4×C73—21no—
107C4×D4×C73C2311yescharacteristic
108C4×D4×C73C2311yescharacteristic
109#117×C73—21no—
110#107×C73—21no—
111#107×C73—21no—
112#106×C73D411yescharacteristic
113#106×C73D411yescharacteristic
114C4⋊C8×C73—21no—
115C4⋊C8×C73—21no—
116C4⋊C8×C73C2311yescharacteristic
117C4⋊C8×C73C2311yescharacteristic
118C4⋊C8×C73C2311yescharacteristic
119C4⋊C8×C73C2311yescharacteristic
120#103×C73—21no—
121#103×C73—21no—
122#103×C73D411yescharacteristic
123#103×C73D411yescharacteristic
124#102×C73—21no—
125#102×C73—21no—
126#102×C73—21no—
127#102×C73—21no—
128#102×C73—21no—
129#102×C73—21no—
130#102×C73—21no—
131#102×C73—21no—
132#102×C73—21no—
133#102×C73—21no—
134#102×C73—41no—
135#102×C73—41no—
136#102×C73D411yescharacteristic
137#102×C73D411yescharacteristic
138C22⋊C8×C73—21no—
139C22⋊C8×C73—21no—
140#97×C73—21no—
141#97×C73—21no—
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C24×C73—41no—
146#41×C73—42no—
147#41×C73—81no—
148C2×Q8×C73—21no—
149C2×Q8×C73—21no—
150C2×Q8×C73C2×D411yescharacteristic
151C2×Q8×C73C2×D411yescharacteristic
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73—41no—
159C2×D4×C73—41no—
160C2×D4×C73—41no—
161C2×D4×C73—41no—
162C2×D4×C73—42no—
163C2×D4×C73—42no—
164C2×D4×C73—81no—
165C2×D4×C73—81no—
166C2×D4×C73C2×D411yescharacteristic
167C2×D4×C73C2×D411yescharacteristic
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—41no—
171Q16×C73—42no—
172Q16×C73—81no—
173Q16×C73—81no—
174SD16×C73—22no—
175SD16×C73—41no—
176SD16×C73—42no—
177SD16×C73—42no—
178SD16×C73—81no—
179SD16×C73—81no—
180D8×C73—42no—
181D8×C73—42no—
182D8×C73—81no—
183M16×C73—42no—
184M16×C73—81no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73C2×D411yescharacteristic
191C2×C8×C73C2×D411yescharacteristic
192C2×C8×C73C2×D411yescharacteristic
193C2×C8×C73C2×D411yescharacteristic
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—41no—
201C4⋊C4×C73—41no—
202C4⋊C4×C73—41no—
203C4⋊C4×C73—41no—
204C4⋊C4×C73C2×D411yescharacteristic
205C4⋊C4×C73C2×D411yescharacteristic
206C4⋊C4×C73C2×D411yescharacteristic
207C4⋊C4×C73C2×D411yescharacteristic
208C22⋊C4×C73—21no—
209C22⋊C4×C73—21no—
210C22⋊C4×C73—41no—
211C22⋊C4×C73—41no—
212C22⋊C4×C73—41no—
213C22⋊C4×C73—41no—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—41no—
217C42×C73—41no—
218C42×C73C2411yescharacteristic
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—82no—
224C23×C73—82no—
225C23×C73—161no—
226Q8×C73—21no—
227Q8×C73—21no—
228Q8×C73—21no—
229Q8×C73—21no—
230Q8×C73—41no—
231Q8×C73—41no—
232Q8×C73—41no—
233Q8×C73—41no—
234D4×C73—21no—
235D4×C73—21no—
236D4×C73—21no—
237D4×C73—21no—
238D4×C73—41no—
239D4×C73—41no—
240D4×C73—41no—
241D4×C73—41no—
242D4×C73—41no—
243D4×C73—41no—
244D4×C73—41no—
245D4×C73—41no—
246D4×C73—81no—
247D4×C73—81no—
248D4×C73—82no—
249D4×C73—82no—
250D4×C73—82no—
251D4×C73—161no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—21no—
257C2×C4×C73—21no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—82no—
265C2×C4×C73C22×D411yescharacteristic
266C2×C4×C73C22×D411yescharacteristic
267C2×C4×C73H(5,𝔽2)11yescharacteristic
268C8×C73—21no—
269C8×C73—21no—
270C8×C73—21no—
271C8×C73—41no—
272C8×C73—41no—
273C8×C73—41no—
274C8×C73—41no—
275C8×C73—41no—
276C8×C73AGL(1,ℤ/8)12yes—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—81no—
284C22×C73—81no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73—81no—
288C22×C73—81no—
289C22×C73—164no—
290C22×C73D4211yescharacteristic
291C4×C73—21no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—81no—
299C4×C73—81no—
300C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
301C4×C73#57511yescharacteristic
302C4×C73#57611yescharacteristic
303C4×C73#57711yescharacteristic
304#20288C7311yesmaximal, characteristic, direct product factor, Sylow
305C2×C73—81no—
306C2×C73—81no—
307C2×C73—161no—
308C2×C73—161no—
309C2×C73#327411yescharacteristic
310C2×C73#327511yescharacteristic
311C2×C73#329011yescharacteristic
312#3398C2×C7311yescharacteristic, complemented
313#3368C2×C7311yescharacteristic, complemented
314#3324C2×C7311yescharacteristic, complemented
315#3274C2×C7311yescharacteristic
316#1709C2×C7311yescharacteristic, complemented
317#1702C2×C7311yescharacteristic, complemented
318#1701C2×C7311yescharacteristic, complemented
319#1689C2×C7311yescharacteristic, complemented
320#1688C2×C7311yescharacteristic, complemented
321#1641C2×C7311yescharacteristic, complemented
322#1639C2×C7311yescharacteristic, complemented
323#1628C2×C7311yescharacteristic, complemented
324#1626C2×C7311yescharacteristic, complemented
325#1590C2×C7311yescharacteristic, complemented
326#1590C2×C7311yescharacteristic, complemented
327C73#2028811yescharacteristic, direct product factor, Sylow
328C2×AGL(1,ℤ/8)—21no—
329#557C22×C7311yescharacteristic, complemented
330#556C22×C7311yescharacteristic
331#550C22×C7311yescharacteristic
332D42C22×C7311yescharacteristic
333#497—21no—
334#494C22×C7311yescharacteristic, complemented
335#493C22×C7311yescharacteristic
336#490—21no—
337#486C22×C7311yescharacteristic
338#486C22×C7311yescharacteristic
339#486C22×C7311yescharacteristic, complemented
340#486C22×C7311yescharacteristic, complemented
341#482—21no—
342#478—21no—
343#474C22×C7311yescharacteristic
344#474C22×C7311yescharacteristic
345#469—21no—
346#468—21no—
347#464—21no—
348#462—21no—
349#462C22×C7311yescharacteristic, complemented
350#461C22×C7311yescharacteristic
351#460—21no—
352#460C22×C7311yescharacteristic
353#460C22×C7311yescharacteristic
354#460C22×C7311yescharacteristic, complemented
355#459C22×C7311yescharacteristic
356#450—21no—
357#449—21no—
358#447—21no—
359#446C22×C7311yescharacteristic, complemented
360#446C22×C7311yescharacteristic, complemented
361#441—21no—
362#440—21no—
363C4×SD16C22×C7311yescharacteristic
364C4×SD16C22×C7311yescharacteristic
365#436C22×C7311yescharacteristic
366#436C22×C7311yescharacteristic
367#418—21no—
368#334C22×C7311yescharacteristic, complemented
369#334C22×C7311yescharacteristic, complemented
370#332C22×C7311yescharacteristic, complemented
371#332C22×C7311yescharacteristic, complemented
372#331C22×C7311yescharacteristic
373#331C22×C7311yescharacteristic, complemented
374Q8⋊C8C22×C7311yescharacteristic, complemented
375Q8⋊C8C22×C7311yescharacteristic, complemented
376D4⋊C8C22×C7311yescharacteristic
377D4⋊C8C22×C7311yescharacteristic
378C82C22×C7311yescharacteristic, complemented
379C2×#41—21no—
380C22×D4—21no—
381C22×D4—21no—
382AGL(1,ℤ/8)—44no—
383C2×Q16—21no—
384C2×Q16—21no—
385C2×Q16—21no—
386C2×SD16—21no—
387C2×SD16—21no—
388C2×SD16—21no—
389C2×SD16—21no—
390C2×SD16D4×C7311yescharacteristic
391C2×SD16D4×C7311yescharacteristic
392C2×D8—21no—
393C2×D8—21no—
394C2×D8—21no—
395C2×M16—21no—
396C2×M16—21no—
397C4⋊Q8—21no—
398C4⋊Q8C23×C7311yescharacteristic
399C4⋊Q8C23×C7311yescharacteristic
400#127—21no—
401#127C23×C7311yescharacteristic
402#127C23×C7311yescharacteristic
403#126—41no—
404#125—21no—
405#125—41no—
406#124—21no—
407C22⋊Q8—21no—
408#121—21no—
409#121—21no—
410#121—21no—
411#121—41no—
412#120—41no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8C23×C7311yescharacteristic
417C4×Q8C23×C7311yescharacteristic
418C4×D4—21no—
419C4×D4C23×C7311yescharacteristic
420C4×D4C23×C7311yescharacteristic
421#117—21no—
422#107—21no—
423#107—21no—
424#106D4×C7311yescharacteristic
425#106D4×C7311yescharacteristic
426C4⋊C8—21no—
427C4⋊C8—21no—
428C4⋊C8C23×C7311yescharacteristic
429C4⋊C8C23×C7311yescharacteristic
430C4⋊C8C23×C7311yescharacteristic
431C4⋊C8C23×C7311yescharacteristic
432#103—21no—
433#103—21no—
434#103D4×C7311yescharacteristic
435#103D4×C7311yescharacteristic
436#102—21no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102—21no—
441#102—21no—
442#102—21no—
443#102—21no—
444#102—21no—
445#102—21no—
446#102—41no—
447#102—41no—
448#102D4×C7311yescharacteristic
449#102D4×C7311yescharacteristic
450C22⋊C8—21no—
451C22⋊C8—21no—
452#97—21no—
453#97—21no—
454C4×C8C23×C7311yescharacteristic
455C4×C8C23×C7311yescharacteristic
456C4×C8C23×C7311yescharacteristic
457C24—41no—
458#41—42no—
459#41—81no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8C2×D4×C7311yescharacteristic
463C2×Q8C2×D4×C7311yescharacteristic
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—81no—
477C2×D4—81no—
478C2×D4C2×D4×C7311yescharacteristic
479C2×D4C2×D4×C7311yescharacteristic
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—41no—
483Q16—42no—
484Q16—81no—
485Q16—81no—
486SD16—22no—
487SD16—41no—
488SD16—42no—
489SD16—42no—
490SD16—81no—
491SD16—81no—
492D8—42no—
493D8—42no—
494D8—81no—
495M16—42no—
496M16—81no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8C2×D4×C7311yescharacteristic
503C2×C8C2×D4×C7311yescharacteristic
504C2×C8C2×D4×C7311yescharacteristic
505C2×C8C2×D4×C7311yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4C2×D4×C7311yescharacteristic
517C4⋊C4C2×D4×C7311yescharacteristic
518C4⋊C4C2×D4×C7311yescharacteristic
519C4⋊C4C2×D4×C7311yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C42—21no—
527C42—21no—
528C42—41no—
529C42—41no—
530C42C24×C7311yescharacteristic
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—82no—
536C23—82no—
537C23—161no—
538Q8—21no—
539Q8—21no—
540Q8—21no—
541Q8—21no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546D4—21no—
547D4—21no—
548D4—21no—
549D4—21no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—81no—
559D4—81no—
560D4—82no—
561D4—82no—
562D4—82no—
563D4—161no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4C22×D4×C7311yescharacteristic
578C2×C4C22×D4×C7311yescharacteristic
579C2×C4H(5,𝔽2)×C7311yescharacteristic
580C8—21no—
581C8—21no—
582C8—21no—
583C8—41no—
584C8—41no—
585C8—41no—
586C8—41no—
587C8—41no—
588C8AGL(1,ℤ/8)×C7312yes—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—164no—
602C22D42×C7311yescharacteristic
603C4—21no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
613C4#575×C7311yescharacteristic
614C4#576×C7311yescharacteristic
615C4#577×C7311yescharacteristic
616C2—81no—
617C2—81no—
618C2—161no—
619C2—161no—
620C2C73×#327411yescharacteristic
621C2C73×#327511yescharacteristic
622C2C73×#329011yescharacteristic
subgroupG/H
center, socleC22×C73D42
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3274C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3398C2semidirect
2. C73×#3368C2semidirect
3. C73×#3324C2semidirect
5. C73×#1709C2semidirect
6. C73×#1702C2semidirect
7. C73×#1701C2semidirect
8. C73×#1689C2semidirect
9. C73×#1688C2semidirect
10. C73×#1641C2semidirect
11. C73×#1639C2semidirect
12. C73×#1628C2semidirect
13. C73×#1626C2semidirect
14. C73×#1590C2semidirect
15. C73×#1590C2semidirect
17. #557×C73C22semidirect
22. #494×C73C22semidirect
27. #486×C73C22semidirect
28. #486×C73C22semidirect
37. #462×C73C22semidirect
42. #460×C73C22semidirect
47. #446×C73C22semidirect
48. #446×C73C22semidirect
56. #334×C73C22semidirect
57. #334×C73C22semidirect
58. #332×C73C22semidirect
59. #332×C73C22semidirect
61. #331×C73C22semidirect
62. Q8⋊C8×C73C22semidirect
63. Q8⋊C8×C73C22semidirect
66. C82×C73C22semidirect
304. #20288C73direct (N × H)
312. #3398C2×C73semidirect
313. #3368C2×C73semidirect
314. #3324C2×C73semidirect
316. #1709C2×C73semidirect
317. #1702C2×C73semidirect
318. #1701C2×C73semidirect
319. #1689C2×C73semidirect
320. #1688C2×C73semidirect
321. #1641C2×C73semidirect
322. #1639C2×C73semidirect
323. #1628C2×C73semidirect
324. #1626C2×C73semidirect
325. #1590C2×C73semidirect
326. #1590C2×C73semidirect
327. C73#20288direct (N × H)
329. #557C22×C73semidirect
334. #494C22×C73semidirect
339. #486C22×C73semidirect
340. #486C22×C73semidirect
349. #462C22×C73semidirect
354. #460C22×C73semidirect
359. #446C22×C73semidirect
360. #446C22×C73semidirect
368. #334C22×C73semidirect
369. #334C22×C73semidirect
370. #332C22×C73semidirect
371. #332C22×C73semidirect
373. #331C22×C73semidirect
374. Q8⋊C8C22×C73semidirect
375. Q8⋊C8C22×C73semidirect
378. C82C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#33982
C73×#33682
C73×#33242
C73×#32742
C73×#17092
C73×#17022
C73×#17012
C73×#16892
C73×#16882
C73×#16412
C73×#16392
C73×#16282
C73×#16262
C73×#15902
#2028873