Finite Group Databasebeta

← C73×#26922 #32936326

Subgroups — C73×#26922 #32936326

Order 18688.

732 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3082C211yesmaximal, characteristic, complemented
2C73×#3081C211yesmaximal, characteristic
3C73×#2950C211yesmaximal, characteristic, complemented
4C73×#2949C211yesmaximal, characteristic
5C73×#2047C211yesmaximal, characteristic, complemented
6C73×#2046C211yesmaximal, characteristic
7C73×#2028C211yesmaximal, characteristic, complemented
8C73×#2015C211yesmaximal, characteristic
9C73×#1931C211yesmaximal, characteristic
10C73×#1930C211yesmaximal, characteristic, complemented
11C73×#1922C211yesmaximal, characteristic, complemented
12C73×#1922C211yesmaximal, characteristic, complemented
13C73×#1888C211yesmaximal, characteristic
14C73×#1885C211yesmaximal, characteristic
15C73×#1857C211yesmaximal, characteristic, complemented
16C22×Q16×C73C2211yescharacteristic
17C22×SD16×C73C2211yescharacteristic
18#544×C73C2211yescharacteristic
19C2×#124×C73C2211yescharacteristic
20#519×C73C2211yescharacteristic
21#519×C73C2211yescharacteristic
22#518×C73C2211yescharacteristic
23#479×C73—21no—
24#479×C73—21no—
25#478×C73—21no—
26#478×C73—21no—
27#477×C73—21no—
28#476×C73—21no—
29#475×C73—21no—
30#474×C73—21no—
31#441×C73—21no—
32#441×C73—21no—
33#440×C73—21no—
34#440×C73—21no—
35C4×Q16×C73—21no—
36C4×Q16×C73—21no—
37C4×SD16×C73—21no—
38C4×SD16×C73—21no—
39#428×C73C2211yescharacteristic
40#428×C73C2211yescharacteristic
41#427×C73C2211yescharacteristic
42#427×C73C2211yescharacteristic
43#424×C73C2211yescharacteristic
44#424×C73C2211yescharacteristic
45#419×C73C2211yescharacteristic
46#419×C73C2211yescharacteristic
47#419×C73C2211yescharacteristic
48#419×C73C2211yescharacteristic
49#418×C73C2211yescharacteristic
50#418×C73C2211yescharacteristic
51C2×#103×C73C2211yescharacteristic
52C2×#102×C73C2211yescharacteristic
53C2×#101×C73C2211yescharacteristic
54C2×#101×C73C2211yescharacteristic
55C2×#100×C73C2211yescharacteristic
56#405×C73C2211yescharacteristic
57#405×C73C2211yescharacteristic
58#391×C73C2211yescharacteristic
59#343×C73C2211yescharacteristic
60#341×C73C2211yescharacteristic
61#341×C73C2211yescharacteristic
62#339×C73C2211yescharacteristic
63#339×C73C2211yescharacteristic
64#339×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67C22×Q8×C73C2311yescharacteristic
68C22×Q8×C73C2311yescharacteristic
69C22×D4×C73C2311yescharacteristic
70C2×Q16×C73—22no—
71C2×Q16×C73—22no—
72C2×Q16×C73—21no—
73C2×Q16×C73—21no—
74C2×SD16×C73—22no—
75C2×SD16×C73—22no—
76C2×SD16×C73—21no—
77C2×SD16×C73—21no—
78C2×M16×C73C2311yescharacteristic
79C2×M16×C73C2311yescharacteristic
80C2×M16×C73C2311yescharacteristic
81C2×M16×C73C2311yescharacteristic
82C22×C8×C73C2311yescharacteristic
83C22×C8×C73C2311yescharacteristic
84C4⋊Q8×C73—21no—
85C4⋊Q8×C73—21no—
86#125×C73—21no—
87#125×C73—21no—
88#124×C73—22no—
89#124×C73—22no—
90C22⋊Q8×C73—41no—
91C4×Q8×C73—21no—
92C4×Q8×C73—21no—
93C4×Q8×C73—21no—
94C4×Q8×C73—21no—
95C4×Q8×C73—21no—
96C4×Q8×C73—21no—
97C4×Q8×C73—21no—
98C4×Q8×C73—21no—
99C4×D4×C73—21no—
100C4×D4×C73—21no—
101C4×D4×C73—21no—
102C4×D4×C73—21no—
103#117×C73—41no—
104#117×C73—41no—
105#117×C73C2311yescharacteristic
106#117×C73C2311yescharacteristic
107#117×C73C2311yescharacteristic
108#117×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C22⋊C4×C73—41no—
111C2×C22⋊C4×C73—41no—
112C2×C42×C73C2311yescharacteristic
113#107×C73—21no—
114#107×C73—21no—
115#107×C73D411yescharacteristic
116#107×C73D411yescharacteristic
117#107×C73D411yescharacteristic
118#107×C73D411yescharacteristic
119#106×C73—21no—
120#106×C73—21no—
121#106×C73D411yescharacteristic
122#106×C73D411yescharacteristic
123#106×C73D411yescharacteristic
124#106×C73D411yescharacteristic
125C4⋊C8×C73—21no—
126C4⋊C8×C73—21no—
127C4⋊C8×C73—21no—
128C4⋊C8×C73—21no—
129#103×C73—21no—
130#103×C73—21no—
131#103×C73—21no—
132#103×C73—21no—
133#103×C73—21no—
134#103×C73—21no—
135#103×C73—21no—
136#103×C73—21no—
137#103×C73—21no—
138#103×C73—21no—
139#102×C73—21no—
140#102×C73—21no—
141#102×C73—21no—
142#102×C73—21no—
143#102×C73—21no—
144#102×C73—21no—
145#101×C73—22no—
146#101×C73—22no—
147#100×C73—22no—
148#97×C73—21no—
149#97×C73—21no—
150C4×C8×C73—21no—
151C4×C8×C73—21no—
152#95×C73—41no—
153C24×C73—21no—
154C2×Q8×C73—21no—
155C2×Q8×C73—21no—
156C2×Q8×C73—21no—
157C2×Q8×C73—21no—
158C2×Q8×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73—21no—
164C2×Q8×C73—41no—
165C2×Q8×C73#4112yes—
166C2×Q8×C73#4112yes—
167C2×D4×C73—21no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73—21no—
171C2×D4×C73—21no—
172C2×D4×C73#4112yes—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—41no—
178C22×C4×C73—41no—
179C22×C4×C73C2411yescharacteristic
180Q16×C73—24no—
181Q16×C73—24no—
182SD16×C73—24no—
183SD16×C73—24no—
184M16×C73—21no—
185M16×C73—21no—
186M16×C73—21no—
187M16×C73—21no—
188M16×C73—21no—
189M16×C73—21no—
190M16×C73—21no—
191M16×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73C2×D411yescharacteristic
201C2×C8×C73C2×D411yescharacteristic
202C2×C8×C73C2×D411yescharacteristic
203C2×C8×C73C2×D411yescharacteristic
204C4⋊C4×C73—21no—
205C4⋊C4×C73—21no—
206C4⋊C4×C73—41no—
207C4⋊C4×C73—41no—
208C4⋊C4×C73—41no—
209C4⋊C4×C73—41no—
210C4⋊C4×C73—41no—
211C4⋊C4×C73—41no—
212C4⋊C4×C73—41no—
213C4⋊C4×C73C2×D411yescharacteristic
214C4⋊C4×C73C2×D411yescharacteristic
215C4⋊C4×C73C2×D411yescharacteristic
216C4⋊C4×C73C2×D411yescharacteristic
217C4⋊C4×C73C2×D411yescharacteristic
218C4⋊C4×C73C2×D411yescharacteristic
219C4⋊C4×C73C2×D411yescharacteristic
220C4⋊C4×C73C2×D411yescharacteristic
221C22⋊C4×C73—21no—
222C22⋊C4×C73—21no—
223C22⋊C4×C73—21no—
224C22⋊C4×C73—21no—
225C22⋊C4×C73—41no—
226C22⋊C4×C73—41no—
227C22⋊C4×C73—44no—
228C42×C73—21no—
229C42×C73—21no—
230C42×C73—21no—
231C42×C73—21no—
232C42×C73—21no—
233C42×C73—21no—
234C42×C73—41no—
235C42×C73—41no—
236C42×C73—41no—
237C42×C73—41no—
238C23×C73—22no—
239C23×C73—41no—
240C23×C73—41no—
241C23×C73—81no—
242C23×C73—81no—
243C23×C73H(5,𝔽2)11yescharacteristic
244Q8×C73—22no—
245Q8×C73—22no—
246Q8×C73—22no—
247Q8×C73—22no—
248Q8×C73—41no—
249Q8×C73—41no—
250Q8×C73—41no—
251Q8×C73—41no—
252Q8×C73—161no—
253D4×C73—22no—
254D4×C73—22no—
255D4×C73—41no—
256D4×C73—41no—
257C2×C4×C73—22no—
258C2×C4×C73—21no—
259C2×C4×C73—21no—
260C2×C4×C73—21no—
261C2×C4×C73—21no—
262C2×C4×C73—21no—
263C2×C4×C73—21no—
264C2×C4×C73—21no—
265C2×C4×C73—21no—
266C2×C4×C73—22no—
267C2×C4×C73—41no—
268C2×C4×C73—41no—
269C2×C4×C73—41no—
270C2×C4×C73—41no—
271C2×C4×C73—41no—
272C2×C4×C73—41no—
273C2×C4×C73—41no—
274C2×C4×C73—41no—
275C2×C4×C73—81no—
276C2×C4×C73—161no—
277C2×C4×C73—161no—
278C2×C4×C73C22×D411yescharacteristic
279C2×C4×C73C22×D411yescharacteristic
280C2×C4×C73C2×#4111yescharacteristic
281C2×C4×C73C2×#4111yescharacteristic
282C2×C4×C73C2×#4111yescharacteristic
283C2×C4×C732−1+411yescharacteristic
284C8×C73—21no—
285C8×C73—21no—
286C8×C73—21no—
287C8×C73—21no—
288C8×C73—41no—
289C8×C73—41no—
290C8×C73—41no—
291C8×C73—41no—
292C22×C73—21no—
293C22×C73—21no—
294C22×C73—42no—
295C22×C73—82no—
296C22×C73—81no—
297C22×C73—81no—
298C22×C73—161no—
299C22×C73#53911yescharacteristic
300C22×C73D4211yescharacteristic
301C22×C73#55111yescharacteristic
302C4×C73—42no—
303C4×C73—42no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73—41no—
308C4×C73—41no—
309C4×C73—41no—
310C4×C73—41no—
311C4×C73—41no—
312C4×C73—81no—
313C4×C73—81no—
314C4×C73#52511yescharacteristic
315C4×C73#52511yescharacteristic
316C4×C73#54811yescharacteristic
317C4×C73#54811yescharacteristic
318#26922C7311yesmaximal, characteristic, direct product factor, Sylow
319C2×C73—21no—
320C2×C73—21no—
321C2×C73—82no—
322C2×C73#246311yescharacteristic
323C2×C73#328911yescharacteristic
324C2×C73#329011yescharacteristic
325#3082C2×C7311yescharacteristic, complemented
326#3081C2×C7311yescharacteristic
327#2950C2×C7311yescharacteristic, complemented
328#2949C2×C7311yescharacteristic
329#2047C2×C7311yescharacteristic, complemented
330#2046C2×C7311yescharacteristic
331#2028C2×C7311yescharacteristic, complemented
332#2015C2×C7311yescharacteristic
333#1931C2×C7311yescharacteristic
334#1930C2×C7311yescharacteristic, complemented
335#1922C2×C7311yescharacteristic, complemented
336#1922C2×C7311yescharacteristic, complemented
337#1888C2×C7311yescharacteristic
338#1885C2×C7311yescharacteristic
339#1857C2×C7311yescharacteristic, complemented
340C73#2692211yescharacteristic, direct product factor, Sylow
341C22×Q16C22×C7311yescharacteristic
342C22×SD16C22×C7311yescharacteristic
343#544C22×C7311yescharacteristic
344C2×#124C22×C7311yescharacteristic
345#519C22×C7311yescharacteristic
346#519C22×C7311yescharacteristic
347#518C22×C7311yescharacteristic
348#479—21no—
349#479—21no—
350#478—21no—
351#478—21no—
352#477—21no—
353#476—21no—
354#475—21no—
355#474—21no—
356#441—21no—
357#441—21no—
358#440—21no—
359#440—21no—
360C4×Q16—21no—
361C4×Q16—21no—
362C4×SD16—21no—
363C4×SD16—21no—
364#428C22×C7311yescharacteristic
365#428C22×C7311yescharacteristic
366#427C22×C7311yescharacteristic
367#427C22×C7311yescharacteristic
368#424C22×C7311yescharacteristic
369#424C22×C7311yescharacteristic
370#419C22×C7311yescharacteristic
371#419C22×C7311yescharacteristic
372#419C22×C7311yescharacteristic
373#419C22×C7311yescharacteristic
374#418C22×C7311yescharacteristic
375#418C22×C7311yescharacteristic
376C2×#103C22×C7311yescharacteristic
377C2×#102C22×C7311yescharacteristic
378C2×#101C22×C7311yescharacteristic
379C2×#101C22×C7311yescharacteristic
380C2×#100C22×C7311yescharacteristic
381#405C22×C7311yescharacteristic
382#405C22×C7311yescharacteristic
383#391C22×C7311yescharacteristic
384#343C22×C7311yescharacteristic
385#341C22×C7311yescharacteristic
386#341C22×C7311yescharacteristic
387#339C22×C7311yescharacteristic
388#339C22×C7311yescharacteristic
389#339C22×C7311yescharacteristic
390#339C22×C7311yescharacteristic
391#336C22×C7311yescharacteristic
392C22×Q8C23×C7311yescharacteristic
393C22×Q8C23×C7311yescharacteristic
394C22×D4C23×C7311yescharacteristic
395C2×Q16—22no—
396C2×Q16—22no—
397C2×Q16—21no—
398C2×Q16—21no—
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16—21no—
402C2×SD16—21no—
403C2×M16C23×C7311yescharacteristic
404C2×M16C23×C7311yescharacteristic
405C2×M16C23×C7311yescharacteristic
406C2×M16C23×C7311yescharacteristic
407C22×C8C23×C7311yescharacteristic
408C22×C8C23×C7311yescharacteristic
409C4⋊Q8—21no—
410C4⋊Q8—21no—
411#125—21no—
412#125—21no—
413#124—22no—
414#124—22no—
415C22⋊Q8—41no—
416C4×Q8—21no—
417C4×Q8—21no—
418C4×Q8—21no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—21no—
428#117—41no—
429#117—41no—
430#117C23×C7311yescharacteristic
431#117C23×C7311yescharacteristic
432#117C23×C7311yescharacteristic
433#117C23×C7311yescharacteristic
434C2×C4⋊C4C23×C7311yescharacteristic
435C2×C22⋊C4—41no—
436C2×C22⋊C4—41no—
437C2×C42C23×C7311yescharacteristic
438#107—21no—
439#107—21no—
440#107D4×C7311yescharacteristic
441#107D4×C7311yescharacteristic
442#107D4×C7311yescharacteristic
443#107D4×C7311yescharacteristic
444#106—21no—
445#106—21no—
446#106D4×C7311yescharacteristic
447#106D4×C7311yescharacteristic
448#106D4×C7311yescharacteristic
449#106D4×C7311yescharacteristic
450C4⋊C8—21no—
451C4⋊C8—21no—
452C4⋊C8—21no—
453C4⋊C8—21no—
454#103—21no—
455#103—21no—
456#103—21no—
457#103—21no—
458#103—21no—
459#103—21no—
460#103—21no—
461#103—21no—
462#103—21no—
463#103—21no—
464#102—21no—
465#102—21no—
466#102—21no—
467#102—21no—
468#102—21no—
469#102—21no—
470#101—22no—
471#101—22no—
472#100—22no—
473#97—21no—
474#97—21no—
475C4×C8—21no—
476C4×C8—21no—
477#95—41no—
478C24—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—41no—
490C2×Q8#41×C7312yes—
491C2×Q8#41×C7312yes—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4#41×C7312yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4C24×C7311yescharacteristic
505Q16—24no—
506Q16—24no—
507SD16—24no—
508SD16—24no—
509M16—21no—
510M16—21no—
511M16—21no—
512M16—21no—
513M16—21no—
514M16—21no—
515M16—21no—
516M16—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8C2×D4×C7311yescharacteristic
526C2×C8C2×D4×C7311yescharacteristic
527C2×C8C2×D4×C7311yescharacteristic
528C2×C8C2×D4×C7311yescharacteristic
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4C2×D4×C7311yescharacteristic
539C4⋊C4C2×D4×C7311yescharacteristic
540C4⋊C4C2×D4×C7311yescharacteristic
541C4⋊C4C2×D4×C7311yescharacteristic
542C4⋊C4C2×D4×C7311yescharacteristic
543C4⋊C4C2×D4×C7311yescharacteristic
544C4⋊C4C2×D4×C7311yescharacteristic
545C4⋊C4C2×D4×C7311yescharacteristic
546C22⋊C4—21no—
547C22⋊C4—21no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—44no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C23—22no—
564C23—41no—
565C23—41no—
566C23—81no—
567C23—81no—
568C23H(5,𝔽2)×C7311yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—41no—
577Q8—161no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—81no—
601C2×C4—161no—
602C2×C4—161no—
603C2×C4C22×D4×C7311yescharacteristic
604C2×C4C22×D4×C7311yescharacteristic
605C2×C4C2×#41×C7311yescharacteristic
606C2×C4C2×#41×C7311yescharacteristic
607C2×C4C2×#41×C7311yescharacteristic
608C2×C42−1+4×C7311yescharacteristic
609C8—21no—
610C8—21no—
611C8—21no—
612C8—21no—
613C8—41no—
614C8—41no—
615C8—41no—
616C8—41no—
617C22—21no—
618C22—21no—
619C22—42no—
620C22—82no—
621C22—81no—
622C22—81no—
623C22—161no—
624C22#539×C7311yescharacteristic
625C22D42×C7311yescharacteristic
626C22#551×C7311yescharacteristic
627C4—42no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4#525×C7311yescharacteristic
640C4#525×C7311yescharacteristic
641C4#548×C7311yescharacteristic
642C4#548×C7311yescharacteristic
643C2—21no—
644C2—21no—
645C2—82no—
646C2C73×#246311yescharacteristic
647C2C73×#328911yescharacteristic
648C2C73×#329011yescharacteristic
subgroupG/H
center, socleC22×C73D42
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC22×D4×C73C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C22×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3082C2semidirect
3. C73×#2950C2semidirect
5. C73×#2047C2semidirect
7. C73×#2028C2semidirect
10. C73×#1930C2semidirect
11. C73×#1922C2semidirect
12. C73×#1922C2semidirect
15. C73×#1857C2semidirect
318. #26922C73direct (N × H)
325. #3082C2×C73semidirect
327. #2950C2×C73semidirect
329. #2047C2×C73semidirect
331. #2028C2×C73semidirect
334. #1930C2×C73semidirect
335. #1922C2×C73semidirect
336. #1922C2×C73semidirect
339. #1857C2×C73semidirect
340. C73#26922direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#30822
C73×#30812
C73×#29502
C73×#29492
C73×#20472
C73×#20462
C73×#20282
C73×#20152
C73×#19312
C73×#19302
C73×#19222
C73×#18882
C73×#18852
C73×#18572
#2692273