Finite Group Databasebeta

← C73×#18588 #32927992

Subgroups — C73×#18588 #32927992

Order 18688.

898 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3523C211yesmaximal, characteristic
2C73×#3239C211yesmaximal, characteristic
3C73×#3165C211yesmaximal, characteristic, complemented
4C2×#462×C73C211yesmaximal, characteristic, complemented
5C73×#1653C212yesmaximal, complemented
6C73×#1652C212yesmaximal, complemented
7C73×#1649C212yesmaximal, complemented
8C73×#1648C212yesmaximal, complemented
9C73×#1552C211yesmaximal, characteristic, complemented
10C73×#1491C211yesmaximal, characteristic, complemented
11C73×#1483C211yesmaximal, characteristic
12C22×Q16×C73—21no—
13#557×C73—21no—
14#557×C73C2211yescharacteristic
15#557×C73C2211yescharacteristic
16#556×C73—21no—
17#556×C73C2211yescharacteristic
18#552×C73—22no—
19#552×C73—21no—
20D4×Q8×C73—21no—
21D4×Q8×C73C2211yescharacteristic
22#541×C73—22no—
23#541×C73—21no—
24#533×C73—21no—
25#533×C73C2211yescharacteristic
26C2×C4⋊Q8×C73C2211yescharacteristic
27#529×C73—21no—
28#529×C73C2211yescharacteristic
29#519×C73—21no—
30C2×C4×Q8×C73C2211yescharacteristic
31#501×C73—21no—
32#501×C73—22no—
33#500×C73C2212yes—
34#498×C73C2212yes—
35#489×C73—21no—
36#484×C73—21no—
37#481×C73—21no—
38#475×C73C2212yes—
39#474×C73C2212yes—
40#462×C73—21no—
41#462×C73—21no—
42#462×C73C2212yes—
43#462×C73C2212yes—
44#432×C73C2211yescharacteristic
45#428×C73—22no—
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×Q16×C73—22no—
60C2×Q16×C73—44no—
61C2×M16×C73—21no—
62C22×C8×C73—21no—
63C4⋊Q8×C73—22no—
64C4⋊Q8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73—21no—
68C4⋊Q8×C73—21no—
69C4⋊Q8×C73—21no—
70C4⋊Q8×C73C2312yes—
71C4⋊Q8×C73C2311yescharacteristic
72C4⋊Q8×C73C2311yescharacteristic
73#126×C73—22no—
74#126×C73—22no—
75#126×C73—22no—
76#126×C73—22no—
77#126×C73—21no—
78#125×C73—22no—
79#125×C73—22no—
80#125×C73—22no—
81#125×C73—21no—
82#125×C73—22no—
83#125×C73—22no—
84#125×C73—21no—
85#125×C73D411yescharacteristic
86#125×C73D412yes—
87#124×C73—21no—
88#124×C73D411yescharacteristic
89#123×C73—22no—
90#123×C73—21no—
91C22⋊Q8×C73—22no—
92C22⋊Q8×C73—22no—
93C22⋊Q8×C73—21no—
94C22⋊Q8×C73—21no—
95C22⋊Q8×C73—21no—
96C22⋊Q8×C73—21no—
97C22⋊Q8×C73—22no—
98C22⋊Q8×C73D411yescharacteristic
99#121×C73D411yescharacteristic
100C4×Q8×C73—22no—
101C4×Q8×C73—22no—
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×C73—21no—
111C4×Q8×C73D411yescharacteristic
112C4×Q8×C73D411yescharacteristic
113C4×Q8×C73C2311yescharacteristic
114C4×Q8×C73C2311yescharacteristic
115C4×Q8×C73C2311yescharacteristic
116C4×D4×C73—22no—
117C4×D4×C73—21no—
118C4×D4×C73—21no—
119C4×D4×C73C2311yescharacteristic
120#117×C73—22no—
121#117×C73—22no—
122#117×C73—22no—
123#117×C73—21no—
124#117×C73—21no—
125#117×C73D412yes—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73—21no—
128C2×C4⋊C4×C73—21no—
129C2×C4⋊C4×C73D411yescharacteristic
130C2×C42×C73—21no—
131C2×C42×C73C2311yescharacteristic
132#107×C73—24no—
133#107×C73—22no—
134#106×C73—24no—
135#106×C73—22no—
136C4⋊C8×C73—21no—
137C4⋊C8×C73C2312yes—
138C4⋊C8×C73C2312yes—
139#103×C73—22no—
140#103×C73—22no—
141#103×C73—22no—
142#102×C73—22no—
143C22⋊C8×C73—21no—
144#97×C73—21no—
145#97×C73—21no—
146C4×C8×C73C2312yes—
147C2×Q8×C73—22no—
148C2×Q8×C73—21no—
149C2×Q8×C73—21no—
150C2×Q8×C73—21no—
151C2×Q8×C73—22no—
152C2×Q8×C73—22no—
153C2×Q8×C73—41no—
154C2×Q8×C73—42no—
155C2×Q8×C73—42no—
156C2×Q8×C73—42no—
157C2×Q8×C73—42no—
158C2×Q8×C73—41no—
159C2×Q8×C73C2×D411yescharacteristic
160C2×Q8×C73C2×D411yescharacteristic
161C2×Q8×C73C2×D411yescharacteristic
162C2×D4×C73—21no—
163C2×D4×C73C2×D411yescharacteristic
164C22×C4×C73—22no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—41no—
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172Q16×C73—44no—
173Q16×C73—44no—
174M16×C73—24no—
175C2×C8×C73—22no—
176C2×C8×C73—22no—
177C2×C8×C73—22no—
178C2×C8×C73—42no—
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—22no—
186C4⋊C4×C73—22no—
187C4⋊C4×C73—21no—
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—22no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—22no—
201C4⋊C4×C73—22no—
202C4⋊C4×C73—41no—
203C4⋊C4×C73—41no—
204C4⋊C4×C73C2×D411yescharacteristic
205C4⋊C4×C73C2×D411yescharacteristic
206C4⋊C4×C73C2×D412yes—
207C4⋊C4×C73C2×D411yescharacteristic
208C4⋊C4×C73C2×D412yes—
209C4⋊C4×C73C2×D411yescharacteristic
210C22⋊C4×C73—22no—
211C22⋊C4×C73—22no—
212C22⋊C4×C73—22no—
213C22⋊C4×C73—21no—
214C22⋊C4×C73—22no—
215C22⋊C4×C73—21no—
216C22⋊C4×C73—21no—
217C22⋊C4×C73—21no—
218C22⋊C4×C73—21no—
219C42×C73—22no—
220C42×C73—22no—
221C42×C73—22no—
222C42×C73—21no—
223C42×C73—21no—
224C42×C73—22no—
225C42×C73—21no—
226C42×C73—21no—
227C42×C73—21no—
228C42×C73C2×D411yescharacteristic
229C42×C73C2×D411yescharacteristic
230C42×C73C2×D411yescharacteristic
231C42×C73C2411yescharacteristic
232C23×C73—21no—
233C23×C73#12011yescharacteristic
234Q8×C73—21no—
235Q8×C73—21no—
236Q8×C73—21no—
237Q8×C73—22no—
238Q8×C73—42no—
239Q8×C73—44no—
240Q8×C73—41no—
241Q8×C73—41no—
242Q8×C73—42no—
243Q8×C73—42no—
244Q8×C73—42no—
245Q8×C73—82no—
246Q8×C73AGL(1,ℤ/8)12yes—
247D4×C73—22no—
248D4×C73—42no—
249C2×C4×C73—21no—
250C2×C4×C73—22no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—22no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—22no—
257C2×C4×C73—21no—
258C2×C4×C73—21no—
259C2×C4×C73—21no—
260C2×C4×C73—21no—
261C2×C4×C73—22no—
262C2×C4×C73—42no—
263C2×C4×C73—44no—
264C2×C4×C73—42no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—42no—
268C2×C4×C73—82no—
269C2×C4×C73#12011yescharacteristic
270C2×C4×C73#12011yescharacteristic
271C2×C4×C73#12011yescharacteristic
272C2×C4×C73C22×D411yescharacteristic
273C2×C4×C73C22×D411yescharacteristic
274C2×C4×C73C22×D411yescharacteristic
275C8×C73—24no—
276C8×C73—42no—
277C8×C73—42no—
278C22×C73—21no—
279C22×C73—21no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73#45612yes—
284C22×C73C2×#12011yescharacteristic
285C4×C73—21no—
286C4×C73—22no—
287C4×C73—21no—
288C4×C73—21no—
289C4×C73—21no—
290C4×C73—41no—
291C4×C73—42no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—42no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—41no—
298C4×C73—82no—
299C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
300C4×C73#57511yescharacteristic
301C4×C73#57812yes—
302#18588C7311yesmaximal, characteristic, direct product factor, Sylow
303C2×C73—22no—
304C2×C73—81no—
305C2×C73#300611yescharacteristic
306C2×C73#301111yescharacteristic
307C2×C73C2×#45611yescharacteristic
308#3523C2×C7311yescharacteristic
309#3239C2×C7311yescharacteristic
310#3165C2×C7311yescharacteristic, complemented
311C2×#462C2×C7311yescharacteristic, complemented
312#1653C2×C7312yescomplemented
313#1652C2×C7312yescomplemented
314#1649C2×C7312yescomplemented
315#1648C2×C7312yescomplemented
316#1552C2×C7311yescharacteristic, complemented
317#1491C2×C7311yescharacteristic, complemented
318#1483C2×C7311yescharacteristic
319C73#1858811yescharacteristic, direct product factor, Sylow
320C22×Q16—21no—
321#557—21no—
322#557C22×C7311yescharacteristic
323#557C22×C7311yescharacteristic
324#556—21no—
325#556C22×C7311yescharacteristic
326#552—22no—
327#552—21no—
328D4×Q8—21no—
329D4×Q8C22×C7311yescharacteristic
330#541—22no—
331#541—21no—
332#533—21no—
333#533C22×C7311yescharacteristic
334C2×C4⋊Q8C22×C7311yescharacteristic
335#529—21no—
336#529C22×C7311yescharacteristic
337#519—21no—
338C2×C4×Q8C22×C7311yescharacteristic
339#501—22no—
340#501—21no—
341#500C22×C7312yes—
342#498C22×C7312yes—
343#489—21no—
344#484—21no—
345#481—21no—
346#475C22×C7312yes—
347#474C22×C7312yes—
348#462—21no—
349#462—21no—
350#462C22×C7312yes—
351#462C22×C7312yes—
352#432C22×C7311yescharacteristic
353#428—22no—
354C2×C4⋊C8C22×C7311yescharacteristic
355C2×#103—21no—
356C4×M16C22×C7311yescharacteristic
357#332C22×C7312yes—
358#332C22×C7312yes—
359Q8⋊C8C22×C7312yes—
360Q8⋊C8C22×C7312yes—
361Q8⋊C8C22×C7312yes—
362D4⋊C8C22×C7312yes—
363C22×Q8—21no—
364C22×Q8—21no—
365C22×Q8D4×C7311yescharacteristic
366C2×Q16—22no—
367C2×Q16—22no—
368C2×Q16—44no—
369C2×M16—21no—
370C22×C8—21no—
371C4⋊Q8—22no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—21no—
376C4⋊Q8—21no—
377C4⋊Q8—21no—
378C4⋊Q8C23×C7312yes—
379C4⋊Q8C23×C7311yescharacteristic
380C4⋊Q8C23×C7311yescharacteristic
381#126—22no—
382#126—22no—
383#126—22no—
384#126—22no—
385#126—21no—
386#125—22no—
387#125—22no—
388#125—22no—
389#125—21no—
390#125—22no—
391#125—22no—
392#125—21no—
393#125D4×C7311yescharacteristic
394#125D4×C7312yes—
395#124—21no—
396#124D4×C7311yescharacteristic
397#123—22no—
398#123—21no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—21no—
404C22⋊Q8—21no—
405C22⋊Q8—22no—
406C22⋊Q8D4×C7311yescharacteristic
407#121D4×C7311yescharacteristic
408C4×Q8—22no—
409C4×Q8—22no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8D4×C7311yescharacteristic
420C4×Q8D4×C7311yescharacteristic
421C4×Q8C23×C7311yescharacteristic
422C4×Q8C23×C7311yescharacteristic
423C4×Q8C23×C7311yescharacteristic
424C4×D4—22no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4C23×C7311yescharacteristic
428#117—22no—
429#117—22no—
430#117—22no—
431#117—21no—
432#117—21no—
433#117D4×C7312yes—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4D4×C7311yescharacteristic
438C2×C42—21no—
439C2×C42C23×C7311yescharacteristic
440#107—24no—
441#107—22no—
442#106—24no—
443#106—22no—
444C4⋊C8—21no—
445C4⋊C8C23×C7312yes—
446C4⋊C8C23×C7312yes—
447#103—22no—
448#103—22no—
449#103—22no—
450#102—22no—
451C22⋊C8—21no—
452#97—21no—
453#97—21no—
454C4×C8C23×C7312yes—
455C2×Q8—22no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—41no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—41no—
467C2×Q8C2×D4×C7311yescharacteristic
468C2×Q8C2×D4×C7311yescharacteristic
469C2×Q8C2×D4×C7311yescharacteristic
470C2×D4—21no—
471C2×D4C2×D4×C7311yescharacteristic
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—41no—
477C22×C4C2×D4×C7311yescharacteristic
478C22×C4C2×D4×C7311yescharacteristic
479C22×C4C2×D4×C7311yescharacteristic
480Q16—44no—
481Q16—44no—
482M16—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—42no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
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—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4C2×D4×C7311yescharacteristic
513C4⋊C4C2×D4×C7311yescharacteristic
514C4⋊C4C2×D4×C7312yes—
515C4⋊C4C2×D4×C7311yescharacteristic
516C4⋊C4C2×D4×C7312yes—
517C4⋊C4C2×D4×C7311yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42C2×D4×C7311yescharacteristic
537C42C2×D4×C7311yescharacteristic
538C42C2×D4×C7311yescharacteristic
539C42C24×C7311yescharacteristic
540C23—21no—
541C23#120×C7311yescharacteristic
542Q8—21no—
543Q8—21no—
544Q8—21no—
545Q8—22no—
546Q8—42no—
547Q8—44no—
548Q8—41no—
549Q8—41no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—82no—
554Q8AGL(1,ℤ/8)×C7312yes—
555D4—22no—
556D4—42no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—82no—
577C2×C4#120×C7311yescharacteristic
578C2×C4#120×C7311yescharacteristic
579C2×C4#120×C7311yescharacteristic
580C2×C4C22×D4×C7311yescharacteristic
581C2×C4C22×D4×C7311yescharacteristic
582C2×C4C22×D4×C7311yescharacteristic
583C8—24no—
584C8—42no—
585C8—42no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22#456×C7312yes—
592C22C2×#120×C7311yescharacteristic
593C4—21no—
594C4—22no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—82no—
607C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
608C4#575×C7311yescharacteristic
609C4#578×C7312yes—
610C2—22no—
611C2—81no—
612C2C73×#300611yescharacteristic
613C2C73×#301111yescharacteristic
614C2C2×#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×#462×C73C2semidirect
5. C73×#1653C2semidirect
6. C73×#1652C2semidirect
7. C73×#1649C2semidirect
8. C73×#1648C2semidirect
9. C73×#1552C2semidirect
10. C73×#1491C2semidirect
302. #18588C73direct (N × H)
310. #3165C2×C73semidirect
311. C2×#462C2×C73semidirect
312. #1653C2×C73semidirect
313. #1652C2×C73semidirect
314. #1649C2×C73semidirect
315. #1648C2×C73semidirect
316. #1552C2×C73semidirect
317. #1491C2×C73semidirect
319. C73#18588direct (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×#32392
C73×#31652
C2×#462×C732
C73×#16532
C73×#16522
C73×#16492
C73×#16482
C73×#15522
C73×#14912
C73×#14832
#1858873