Finite Group Databasebeta

← C73×#18589 #32927993

Subgroups — C73×#18589 #32927993

Order 18688.

878 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3523C211yesmaximal, characteristic
2C73×#3218C211yesmaximal, characteristic
3C73×#3165C211yesmaximal, characteristic, complemented
4C2×#475×C73C211yesmaximal, characteristic, complemented
5C73×#1653C212yesmaximal, complemented
6C73×#1651C212yesmaximal, complemented
7C73×#1650C212yesmaximal, complemented
8C73×#1648C212yesmaximal, complemented
9C73×#1552C211yesmaximal, characteristic, complemented
10C73×#1491C211yesmaximal, characteristic, complemented
11C73×#1483C211yesmaximal, characteristic
12#557×C73—21no—
13#557×C73C2211yescharacteristic
14#557×C73C2211yescharacteristic
15#556×C73—21no—
16#556×C73C2211yescharacteristic
17#552×C73—22no—
18#552×C73—21no—
19D4×Q8×C73—21no—
20D4×Q8×C73C2211yescharacteristic
21#541×C73—22no—
22#541×C73—21no—
23#533×C73—21no—
24#533×C73C2211yescharacteristic
25C2×C4⋊Q8×C73C2211yescharacteristic
26#529×C73—21no—
27#529×C73C2211yescharacteristic
28#519×C73—21no—
29C2×C4×Q8×C73C2211yescharacteristic
30#501×C73—22no—
31#500×C73C2212yes—
32#498×C73C2212yes—
33#497×C73—21no—
34#491×C73—21no—
35#475×C73—21no—
36#475×C73—21no—
37#475×C73C2212yes—
38#475×C73C2212yes—
39#462×C73C2212yes—
40#461×C73C2212yes—
41#451×C73—21no—
42#448×C73—21no—
43#432×C73C2211yescharacteristic
44#428×C73—22no—
45C2×#106×C73—21no—
46C2×C4⋊C8×C73C2211yescharacteristic
47C2×#103×C73—21no—
48C4×M16×C73C2211yescharacteristic
49#332×C73C2212yes—
50#332×C73C2212yes—
51Q8⋊C8×C73C2212yes—
52Q8⋊C8×C73C2212yes—
53Q8⋊C8×C73C2212yes—
54D4⋊C8×C73C2212yes—
55C22×Q8×C73—21no—
56C22×Q8×C73—21no—
57C22×Q8×C73D411yescharacteristic
58C2×Q16×C73—22no—
59C2×SD16×C73—22no—
60C2×M16×C73—21no—
61C22×C8×C73—21no—
62C4⋊Q8×C73—22no—
63C4⋊Q8×C73—21no—
64C4⋊Q8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73—21no—
68C4⋊Q8×C73—21no—
69C4⋊Q8×C73C2312yes—
70C4⋊Q8×C73C2311yescharacteristic
71C4⋊Q8×C73C2311yescharacteristic
72#126×C73—22no—
73#126×C73—22no—
74#126×C73—22no—
75#126×C73—22no—
76#126×C73—21no—
77#125×C73—22no—
78#125×C73—22no—
79#125×C73—22no—
80#125×C73—21no—
81#125×C73—22no—
82#125×C73—22no—
83#125×C73—21no—
84#125×C73D411yescharacteristic
85#125×C73D412yes—
86#124×C73—21no—
87#124×C73D411yescharacteristic
88#123×C73—22no—
89#123×C73—21no—
90C22⋊Q8×C73—22no—
91C22⋊Q8×C73—22no—
92C22⋊Q8×C73—21no—
93C22⋊Q8×C73—21no—
94C22⋊Q8×C73—21no—
95C22⋊Q8×C73—21no—
96C22⋊Q8×C73—22no—
97C22⋊Q8×C73D411yescharacteristic
98#121×C73D411yescharacteristic
99C4×Q8×C73—22no—
100C4×Q8×C73—22no—
101C4×Q8×C73—21no—
102C4×Q8×C73—21no—
103C4×Q8×C73—21no—
104C4×Q8×C73—21no—
105C4×Q8×C73—21no—
106C4×Q8×C73—21no—
107C4×Q8×C73—21no—
108C4×Q8×C73—21no—
109C4×Q8×C73—21no—
110C4×Q8×C73D411yescharacteristic
111C4×Q8×C73D411yescharacteristic
112C4×Q8×C73C2311yescharacteristic
113C4×Q8×C73C2311yescharacteristic
114C4×Q8×C73C2311yescharacteristic
115C4×D4×C73—22no—
116C4×D4×C73—21no—
117C4×D4×C73—21no—
118C4×D4×C73C2311yescharacteristic
119#117×C73—22no—
120#117×C73—22no—
121#117×C73—22no—
122#117×C73—21no—
123#117×C73—21no—
124#117×C73D412yes—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73—21no—
128C2×C4⋊C4×C73D411yescharacteristic
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131#107×C73—24no—
132#106×C73—22no—
133#106×C73—22no—
134#106×C73—24no—
135C4⋊C8×C73—21no—
136C4⋊C8×C73C2312yes—
137C4⋊C8×C73C2312yes—
138#103×C73—22no—
139#103×C73—22no—
140#103×C73—22no—
141#103×C73—22no—
142C22⋊C8×C73—21no—
143#97×C73—21no—
144#97×C73—21no—
145C4×C8×C73C2312yes—
146C2×Q8×C73—22no—
147C2×Q8×C73—21no—
148C2×Q8×C73—21no—
149C2×Q8×C73—21no—
150C2×Q8×C73—22no—
151C2×Q8×C73—22no—
152C2×Q8×C73—41no—
153C2×Q8×C73—42no—
154C2×Q8×C73—42no—
155C2×Q8×C73—41no—
156C2×Q8×C73—82no—
157C2×Q8×C73C2×D411yescharacteristic
158C2×Q8×C73C2×D411yescharacteristic
159C2×Q8×C73C2×D411yescharacteristic
160C2×D4×C73—21no—
161C2×D4×C73C2×D411yescharacteristic
162C22×C4×C73—22no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—41no—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170Q16×C73—82no—
171SD16×C73—82no—
172M16×C73—24no—
173C2×C8×C73—22no—
174C2×C8×C73—22no—
175C2×C8×C73—22no—
176C2×C8×C73—42no—
177C4⋊C4×C73—22no—
178C4⋊C4×C73—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—22no—
181C4⋊C4×C73—22no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73—22no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—22no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—22no—
199C4⋊C4×C73—22no—
200C4⋊C4×C73—41no—
201C4⋊C4×C73—41no—
202C4⋊C4×C73C2×D411yescharacteristic
203C4⋊C4×C73C2×D411yescharacteristic
204C4⋊C4×C73C2×D411yescharacteristic
205C4⋊C4×C73C2×D411yescharacteristic
206C4⋊C4×C73C2×D412yes—
207C4⋊C4×C73C2×D412yes—
208C22⋊C4×C73—22no—
209C22⋊C4×C73—22no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—21no—
212C22⋊C4×C73—22no—
213C22⋊C4×C73—21no—
214C22⋊C4×C73—21no—
215C22⋊C4×C73—21no—
216C22⋊C4×C73—21no—
217C42×C73—22no—
218C42×C73—22no—
219C42×C73—22no—
220C42×C73—21no—
221C42×C73—21no—
222C42×C73—22no—
223C42×C73—21no—
224C42×C73—21no—
225C42×C73—21no—
226C42×C73C2×D411yescharacteristic
227C42×C73C2×D411yescharacteristic
228C42×C73C2×D411yescharacteristic
229C42×C73C2411yescharacteristic
230C23×C73—21no—
231C23×C73#12011yescharacteristic
232Q8×C73—21no—
233Q8×C73—21no—
234Q8×C73—21no—
235Q8×C73—22no—
236Q8×C73—42no—
237Q8×C73—44no—
238Q8×C73—41no—
239Q8×C73—41no—
240Q8×C73—42no—
241Q8×C73—42no—
242Q8×C73—42no—
243Q8×C73—82no—
244Q8×C73#13712yes—
245D4×C73—22no—
246D4×C73—42no—
247C2×C4×C73—21no—
248C2×C4×C73—22no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—22no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—22no—
255C2×C4×C73—21no—
256C2×C4×C73—21no—
257C2×C4×C73—21no—
258C2×C4×C73—21no—
259C2×C4×C73—22no—
260C2×C4×C73—42no—
261C2×C4×C73—44no—
262C2×C4×C73—42no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—42no—
266C2×C4×C73—82no—
267C2×C4×C73#12011yescharacteristic
268C2×C4×C73#12011yescharacteristic
269C2×C4×C73#12011yescharacteristic
270C2×C4×C73C22×D411yescharacteristic
271C2×C4×C73C22×D411yescharacteristic
272C2×C4×C73C22×D411yescharacteristic
273C8×C73—24no—
274C8×C73—42no—
275C8×C73—42no—
276C22×C73—21no—
277C22×C73—21no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73#45612yes—
282C22×C73C2×#12011yescharacteristic
283C4×C73—21no—
284C4×C73—22no—
285C4×C73—21no—
286C4×C73—21no—
287C4×C73—21no—
288C4×C73—41no—
289C4×C73—42no—
290C4×C73—41no—
291C4×C73—41no—
292C4×C73—42no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73—82no—
297C4×C73C2×#13711yescharacteristic
298C4×C73#57511yescharacteristic
299C4×C73#57712yes—
300#18589C7311yesmaximal, characteristic, direct product factor, Sylow
301C2×C73—22no—
302C2×C73—81no—
303C2×C73#300711yescharacteristic
304C2×C73#301011yescharacteristic
305C2×C73C2×#45611yescharacteristic
306#3523C2×C7311yescharacteristic
307#3218C2×C7311yescharacteristic
308#3165C2×C7311yescharacteristic, complemented
309C2×#475C2×C7311yescharacteristic, complemented
310#1653C2×C7312yescomplemented
311#1651C2×C7312yescomplemented
312#1650C2×C7312yescomplemented
313#1648C2×C7312yescomplemented
314#1552C2×C7311yescharacteristic, complemented
315#1491C2×C7311yescharacteristic, complemented
316#1483C2×C7311yescharacteristic
317C73#1858911yescharacteristic, direct product factor, Sylow
318#557—21no—
319#557C22×C7311yescharacteristic
320#557C22×C7311yescharacteristic
321#556—21no—
322#556C22×C7311yescharacteristic
323#552—22no—
324#552—21no—
325D4×Q8—21no—
326D4×Q8C22×C7311yescharacteristic
327#541—22no—
328#541—21no—
329#533—21no—
330#533C22×C7311yescharacteristic
331C2×C4⋊Q8C22×C7311yescharacteristic
332#529—21no—
333#529C22×C7311yescharacteristic
334#519—21no—
335C2×C4×Q8C22×C7311yescharacteristic
336#501—22no—
337#500C22×C7312yes—
338#498C22×C7312yes—
339#497—21no—
340#491—21no—
341#475—21no—
342#475—21no—
343#475C22×C7312yes—
344#475C22×C7312yes—
345#462C22×C7312yes—
346#461C22×C7312yes—
347#451—21no—
348#448—21no—
349#432C22×C7311yescharacteristic
350#428—22no—
351C2×#106—21no—
352C2×C4⋊C8C22×C7311yescharacteristic
353C2×#103—21no—
354C4×M16C22×C7311yescharacteristic
355#332C22×C7312yes—
356#332C22×C7312yes—
357Q8⋊C8C22×C7312yes—
358Q8⋊C8C22×C7312yes—
359Q8⋊C8C22×C7312yes—
360D4⋊C8C22×C7312yes—
361C22×Q8—21no—
362C22×Q8—21no—
363C22×Q8D4×C7311yescharacteristic
364C2×Q16—22no—
365C2×SD16—22no—
366C2×M16—21no—
367C22×C8—21no—
368C4⋊Q8—22no—
369C4⋊Q8—21no—
370C4⋊Q8—21no—
371C4⋊Q8—21no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8C23×C7312yes—
376C4⋊Q8C23×C7311yescharacteristic
377C4⋊Q8C23×C7311yescharacteristic
378#126—22no—
379#126—22no—
380#126—22no—
381#126—22no—
382#126—21no—
383#125—22no—
384#125—22no—
385#125—22no—
386#125—21no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125D4×C7311yescharacteristic
391#125D4×C7312yes—
392#124—21no—
393#124D4×C7311yescharacteristic
394#123—22no—
395#123—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—21no—
400C22⋊Q8—21no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8D4×C7311yescharacteristic
404#121D4×C7311yescharacteristic
405C4×Q8—22no—
406C4×Q8—22no—
407C4×Q8—21no—
408C4×Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8D4×C7311yescharacteristic
417C4×Q8D4×C7311yescharacteristic
418C4×Q8C23×C7311yescharacteristic
419C4×Q8C23×C7311yescharacteristic
420C4×Q8C23×C7311yescharacteristic
421C4×D4—22no—
422C4×D4—21no—
423C4×D4—21no—
424C4×D4C23×C7311yescharacteristic
425#117—22no—
426#117—22no—
427#117—22no—
428#117—21no—
429#117—21no—
430#117D4×C7312yes—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4D4×C7311yescharacteristic
435C2×C42—21no—
436C2×C42C23×C7311yescharacteristic
437#107—24no—
438#106—22no—
439#106—22no—
440#106—24no—
441C4⋊C8—21no—
442C4⋊C8C23×C7312yes—
443C4⋊C8C23×C7312yes—
444#103—22no—
445#103—22no—
446#103—22no—
447#103—22no—
448C22⋊C8—21no—
449#97—21no—
450#97—21no—
451C4×C8C23×C7312yes—
452C2×Q8—22no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—41no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—41no—
462C2×Q8—82no—
463C2×Q8C2×D4×C7311yescharacteristic
464C2×Q8C2×D4×C7311yescharacteristic
465C2×Q8C2×D4×C7311yescharacteristic
466C2×D4—21no—
467C2×D4C2×D4×C7311yescharacteristic
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4C2×D4×C7311yescharacteristic
474C22×C4C2×D4×C7311yescharacteristic
475C22×C4C2×D4×C7311yescharacteristic
476Q16—82no—
477SD16—82no—
478M16—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—42no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4C2×D4×C7311yescharacteristic
509C4⋊C4C2×D4×C7311yescharacteristic
510C4⋊C4C2×D4×C7311yescharacteristic
511C4⋊C4C2×D4×C7311yescharacteristic
512C4⋊C4C2×D4×C7312yes—
513C4⋊C4C2×D4×C7312yes—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—21no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—21no—
528C42—22no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42C2×D4×C7311yescharacteristic
533C42C2×D4×C7311yescharacteristic
534C42C2×D4×C7311yescharacteristic
535C42C24×C7311yescharacteristic
536C23—21no—
537C23#120×C7311yescharacteristic
538Q8—21no—
539Q8—21no—
540Q8—21no—
541Q8—22no—
542Q8—42no—
543Q8—44no—
544Q8—41no—
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—82no—
550Q8#137×C7312yes—
551D4—22no—
552D4—42no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4#120×C7311yescharacteristic
574C2×C4#120×C7311yescharacteristic
575C2×C4#120×C7311yescharacteristic
576C2×C4C22×D4×C7311yescharacteristic
577C2×C4C22×D4×C7311yescharacteristic
578C2×C4C22×D4×C7311yescharacteristic
579C8—24no—
580C8—42no—
581C8—42no—
582C22—21no—
583C22—21no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22#456×C7312yes—
588C22C2×#120×C7311yescharacteristic
589C4—21no—
590C4—22no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—82no—
603C4C2×#137×C7311yescharacteristic
604C4#575×C7311yescharacteristic
605C4#577×C7312yes—
606C2—22no—
607C2—81no—
608C2C73×#300711yescharacteristic
609C2C73×#301011yescharacteristic
610C2C2×#456×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radical#121×C73D4
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
3. C73×#3165C2semidirect
4. C2×#475×C73C2semidirect
5. C73×#1653C2semidirect
6. C73×#1651C2semidirect
7. C73×#1650C2semidirect
8. C73×#1648C2semidirect
9. C73×#1552C2semidirect
10. C73×#1491C2semidirect
300. #18589C73direct (N × H)
308. #3165C2×C73semidirect
309. C2×#475C2×C73semidirect
310. #1653C2×C73semidirect
311. #1651C2×C73semidirect
312. #1650C2×C73semidirect
313. #1648C2×C73semidirect
314. #1552C2×C73semidirect
315. #1491C2×C73semidirect
317. C73#18589direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35232
C73×#32182
C73×#31652
C2×#475×C732
C73×#16532
C73×#16512
C73×#16502
C73×#16482
C73×#15522
C73×#14912
C73×#14832
#1858973