Finite Group Databasebeta

← C73×#18802 #32928206

Subgroups — C73×#18802 #32928206

Order 18688.

958 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3480C211yesmaximal, characteristic
2C73×#3226C211yesmaximal, characteristic, complemented
3C2×#475×C73C211yesmaximal, characteristic, complemented
4C73×#3039C211yesmaximal, characteristic, complemented
5C73×#1650C212yesmaximal, complemented
6C73×#1648C212yesmaximal, complemented
7C73×#1642C212yesmaximal, complemented
8C73×#1640C212yesmaximal, complemented
9C73×#1547C211yesmaximal, characteristic, complemented
10C73×#1487C211yesmaximal, characteristic, complemented
11C73×#1477C211yesmaximal, characteristic, complemented
12C2×#137×C73—21no—
13#559×C73—21no—
14#559×C73C2211yescharacteristic
15#557×C73C2211yescharacteristic
16#556×C73—21no—
17#556×C73C2211yescharacteristic
18#553×C73—22no—
19#553×C73—21no—
20#550×C73C2211yescharacteristic
21#548×C73—21no—
22#532×C73—21no—
23#532×C73C2211yescharacteristic
24C2×C4⋊Q8×C73C2211yescharacteristic
25#529×C73—21no—
26#529×C73—22no—
27C4×#41×C73—21no—
28C4×#41×C73—21no—
29C4×#41×C73C2211yescharacteristic
30C2×C4×Q8×C73C2211yescharacteristic
31#500×C73C2212yes—
32#486×C73C2212yes—
33#480×C73—22no—
34#475×C73—21no—
35#475×C73—21no—
36#475×C73C2212yes—
37#475×C73C2212yes—
38#474×C73—22no—
39#462×C73—21no—
40#462×C73C2212yes—
41#461×C73—21no—
42#461×C73C2212yes—
43#431×C73C2211yescharacteristic
44C2×#106×C73—21no—
45#423×C73C2211yescharacteristic
46C2×C4⋊C8×C73C2211yescharacteristic
47#419×C73—21no—
48C2×#103×C73—21no—
49#332×C73C2212yes—
50#331×C73C2212yes—
51Q8⋊C8×C73C2212yes—
52Q8⋊C8×C73C2212yes—
53Q8⋊C8×C73C2212yes—
54D4⋊C8×C73C2212yes—
55C2×#41×C73—21no—
56C2×#41×C73—21no—
57C2×#41×C73D411yescharacteristic
58C22×Q8×C73—21no—
59C22×Q8×C73D411yescharacteristic
60#137×C73—44no—
61C2×Q16×C73—22no—
62C2×SD16×C73—22no—
63C2×M16×C73—21no—
64C22×C8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C4⋊Q8×C73—21no—
68C4⋊Q8×C73—21no—
69C4⋊Q8×C73C2312yes—
70C4⋊Q8×C73C2311yescharacteristic
71#127×C73—21no—
72#127×C73C2311yescharacteristic
73#126×C73—22no—
74#126×C73—22no—
75#126×C73—22no—
76#125×C73—22no—
77#125×C73—21no—
78#124×C73—22no—
79#124×C73—21no—
80#124×C73—21no—
81#124×C73—21no—
82#124×C73—21no—
83#123×C73—22no—
84#123×C73—22no—
85#123×C73—22no—
86C22⋊Q8×C73—22no—
87C22⋊Q8×C73—21no—
88#121×C73—22no—
89#121×C73—22no—
90#121×C73—22no—
91#121×C73—21no—
92#121×C73—21no—
93#121×C73D412yes—
94C4×Q8×C73—21no—
95C4×Q8×C73—21no—
96C4×Q8×C73—21no—
97C4×Q8×C73—21no—
98C4×Q8×C73—21no—
99C4×Q8×C73D411yescharacteristic
100C4×Q8×C73D411yescharacteristic
101C4×Q8×C73D411yescharacteristic
102C4×Q8×C73C2311yescharacteristic
103C4×Q8×C73C2311yescharacteristic
104C4×Q8×C73C2311yescharacteristic
105C4×D4×C73—22no—
106C4×D4×C73—22no—
107C4×D4×C73—22no—
108C4×D4×C73—22no—
109C4×D4×C73—22no—
110C4×D4×C73—21no—
111C4×D4×C73—21no—
112C4×D4×C73—21no—
113C4×D4×C73—21no—
114C4×D4×C73—21no—
115C4×D4×C73—21no—
116C4×D4×C73D411yescharacteristic
117C4×D4×C73D412yes—
118C4×D4×C73C2311yescharacteristic
119#117×C73—22no—
120#117×C73—22no—
121#117×C73—21no—
122#117×C73—21no—
123#117×C73D411yescharacteristic
124C2×C4⋊C4×C73—21no—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73D411yescharacteristic
128C2×C42×C73—22no—
129C2×C42×C73—21no—
130C2×C42×C73—21no—
131C2×C42×C73C2311yescharacteristic
132#107×C73—24no—
133#106×C73—22no—
134#106×C73—22no—
135C4⋊C8×C73—21no—
136C4⋊C8×C73—21no—
137C4⋊C8×C73—21no—
138C4⋊C8×C73C2312yes—
139C4⋊C8×C73C2312yes—
140#103×C73—22no—
141#103×C73—22no—
142#103×C73—22no—
143#103×C73—22no—
144#102×C73—24no—
145C22⋊C8×C73—21no—
146C4×C8×C73C2312yes—
147#41×C73—22no—
148#41×C73—22no—
149#41×C73—42no—
150#41×C73—42no—
151#41×C73—42no—
152#41×C73—42no—
153C2×Q8×C73—21no—
154C2×Q8×C73—21no—
155C2×Q8×C73—22no—
156C2×Q8×C73—22no—
157C2×Q8×C73—41no—
158C2×Q8×C73—41no—
159C2×Q8×C73—42no—
160C2×Q8×C73—42no—
161C2×Q8×C73C2×D411yescharacteristic
162C2×Q8×C73C2×D411yescharacteristic
163C2×Q8×C73C2×D411yescharacteristic
164C2×D4×C73—22no—
165C2×D4×C73—21no—
166C2×D4×C73—22no—
167C2×D4×C73—22no—
168C2×D4×C73—21no—
169C2×D4×C73—21no—
170C2×D4×C73C2×D411yescharacteristic
171C2×D4×C73C2×D412yes—
172C22×C4×C73—22no—
173C22×C4×C73—21no—
174C22×C4×C73—22no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—22no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—41no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184Q16×C73—44no—
185SD16×C73—44no—
186M16×C73—42no—
187C2×C8×C73—22no—
188C2×C8×C73—22no—
189C2×C8×C73—22no—
190C2×C8×C73—42no—
191C4⋊C4×C73—22no—
192C4⋊C4×C73—22no—
193C4⋊C4×C73—22no—
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—21no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—22no—
204C4⋊C4×C73—22no—
205C4⋊C4×C73—41no—
206C4⋊C4×C73—41no—
207C4⋊C4×C73C2×D411yescharacteristic
208C4⋊C4×C73C2×D411yescharacteristic
209C4⋊C4×C73C2×D411yescharacteristic
210C4⋊C4×C73C2×D411yescharacteristic
211C4⋊C4×C73C2×D412yes—
212C22⋊C4×C73—22no—
213C22⋊C4×C73—22no—
214C22⋊C4×C73—22no—
215C22⋊C4×C73—22no—
216C22⋊C4×C73—22no—
217C22⋊C4×C73—22no—
218C22⋊C4×C73—21no—
219C22⋊C4×C73—22no—
220C22⋊C4×C73—21no—
221C22⋊C4×C73—22no—
222C22⋊C4×C73—21no—
223C42×C73—22no—
224C42×C73—22no—
225C42×C73—21no—
226C42×C73—21no—
227C42×C73—21no—
228C42×C73—21no—
229C42×C73—21no—
230C42×C73—21no—
231C42×C73—21no—
232C42×C73C2×D411yescharacteristic
233C42×C73C2×D411yescharacteristic
234C42×C73C2×D411yescharacteristic
235C42×C73C2411yescharacteristic
236C23×C73—21no—
237C23×C73—22no—
238C23×C73#12011yescharacteristic
239Q8×C73—21no—
240Q8×C73—21no—
241Q8×C73—21no—
242Q8×C73—21no—
243Q8×C73—21no—
244Q8×C73—42no—
245Q8×C73—41no—
246Q8×C73—41no—
247Q8×C73—42no—
248Q8×C73—42no—
249Q8×C73—82no—
250Q8×C73#13712yescomplemented
251D4×C73—21no—
252D4×C73—21no—
253D4×C73—42no—
254D4×C73—44no—
255D4×C73—42no—
256D4×C73—44no—
257D4×C73—41no—
258D4×C73—41no—
259D4×C73—41no—
260D4×C73—41no—
261D4×C73—82no—
262C2×C4×C73—21no—
263C2×C4×C73—21no—
264C2×C4×C73—21no—
265C2×C4×C73—21no—
266C2×C4×C73—21no—
267C2×C4×C73—21no—
268C2×C4×C73—21no—
269C2×C4×C73—21no—
270C2×C4×C73—22no—
271C2×C4×C73—21no—
272C2×C4×C73—41no—
273C2×C4×C73—42no—
274C2×C4×C73—44no—
275C2×C4×C73—41no—
276C2×C4×C73—41no—
277C2×C4×C73—42no—
278C2×C4×C73—44no—
279C2×C4×C73—42no—
280C2×C4×C73—81no—
281C2×C4×C73—81no—
282C2×C4×C73—82no—
283C2×C4×C73—82no—
284C2×C4×C73#12011yescharacteristic
285C2×C4×C73#12011yescharacteristic
286C2×C4×C73#12011yescharacteristic
287C2×C4×C73#13512yes—
288C2×C4×C73AGL(1,ℤ/8)12yes—
289C2×C4×C73C22×D411yescharacteristic
290C2×C4×C73C22×D411yescharacteristic
291C2×C4×C73C22×D411yescharacteristic
292C8×C73—42no—
293C8×C73—42no—
294C8×C73—42no—
295C22×C73—21no—
296C22×C73—21no—
297C22×C73—41no—
298C22×C73—42no—
299C22×C73—42no—
300C22×C73—41no—
301C22×C73—42no—
302C22×C73—41no—
303C22×C73#44912yes—
304C22×C73C2×#12011yescharacteristic
305C4×C73—21no—
306C4×C73—22no—
307C4×C73—21no—
308C4×C73—21no—
309C4×C73—41no—
310C4×C73—42no—
311C4×C73—41no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—82no—
317C4×C73#44912yes—
318C4×C73C2×#13511yescharacteristic
319C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
320C4×C73C2×#13711yescharacteristic
321C4×C73#57511yescharacteristic
322#18802C7311yesmaximal, characteristic, direct product factor, Sylow
323C2×C73—22no—
324C2×C73—81no—
325C2×C73—82no—
326C2×C73C2×#44911yescharacteristic
327C2×C73#300711yescharacteristic
328C2×C73#302211yescharacteristic
329#3480C2×C7311yescharacteristic
330#3226C2×C7311yescharacteristic, complemented
331C2×#475C2×C7311yescharacteristic, complemented
332#3039C2×C7311yescharacteristic, complemented
333#1650C2×C7312yescomplemented
334#1648C2×C7312yescomplemented
335#1642C2×C7312yescomplemented
336#1640C2×C7312yescomplemented
337#1547C2×C7311yescharacteristic, complemented
338#1487C2×C7311yescharacteristic, complemented
339#1477C2×C7311yescharacteristic, complemented
340C73#1880211yescharacteristic, direct product factor, Sylow
341C2×#137—21no—
342#559—21no—
343#559C22×C7311yescharacteristic
344#557C22×C7311yescharacteristic
345#556—21no—
346#556C22×C7311yescharacteristic
347#553—22no—
348#553—21no—
349#550C22×C7311yescharacteristic
350#548—21no—
351#532—21no—
352#532C22×C7311yescharacteristic
353C2×C4⋊Q8C22×C7311yescharacteristic
354#529—21no—
355#529—22no—
356C4×#41—21no—
357C4×#41—21no—
358C4×#41C22×C7311yescharacteristic
359C2×C4×Q8C22×C7311yescharacteristic
360#500C22×C7312yes—
361#486C22×C7312yes—
362#480—22no—
363#475—21no—
364#475—21no—
365#475C22×C7312yes—
366#475C22×C7312yes—
367#474—22no—
368#462—21no—
369#462C22×C7312yes—
370#461—21no—
371#461C22×C7312yes—
372#431C22×C7311yescharacteristic
373C2×#106—21no—
374#423C22×C7311yescharacteristic
375C2×C4⋊C8C22×C7311yescharacteristic
376#419—21no—
377C2×#103—21no—
378#332C22×C7312yes—
379#331C22×C7312yes—
380Q8⋊C8C22×C7312yes—
381Q8⋊C8C22×C7312yes—
382Q8⋊C8C22×C7312yes—
383D4⋊C8C22×C7312yes—
384C2×#41—21no—
385C2×#41—21no—
386C2×#41D4×C7311yescharacteristic
387C22×Q8—21no—
388C22×Q8D4×C7311yescharacteristic
389#137—44no—
390C2×Q16—22no—
391C2×SD16—22no—
392C2×M16—21no—
393C22×C8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8—21no—
398C4⋊Q8C23×C7312yes—
399C4⋊Q8C23×C7311yescharacteristic
400#127—21no—
401#127C23×C7311yescharacteristic
402#126—22no—
403#126—22no—
404#126—22no—
405#125—22no—
406#125—21no—
407#124—22no—
408#124—21no—
409#124—21no—
410#124—21no—
411#124—21no—
412#123—22no—
413#123—22no—
414#123—22no—
415C22⋊Q8—22no—
416C22⋊Q8—21no—
417#121—22no—
418#121—22no—
419#121—22no—
420#121—21no—
421#121—21no—
422#121D4×C7312yes—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8D4×C7311yescharacteristic
429C4×Q8D4×C7311yescharacteristic
430C4×Q8D4×C7311yescharacteristic
431C4×Q8C23×C7311yescharacteristic
432C4×Q8C23×C7311yescharacteristic
433C4×Q8C23×C7311yescharacteristic
434C4×D4—22no—
435C4×D4—22no—
436C4×D4—22no—
437C4×D4—22no—
438C4×D4—22no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—21no—
442C4×D4—21no—
443C4×D4—21no—
444C4×D4—21no—
445C4×D4D4×C7311yescharacteristic
446C4×D4D4×C7312yes—
447C4×D4C23×C7311yescharacteristic
448#117—22no—
449#117—22no—
450#117—21no—
451#117—21no—
452#117D4×C7311yescharacteristic
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4D4×C7311yescharacteristic
457C2×C42—22no—
458C2×C42—21no—
459C2×C42—21no—
460C2×C42C23×C7311yescharacteristic
461#107—24no—
462#106—22no—
463#106—22no—
464C4⋊C8—21no—
465C4⋊C8—21no—
466C4⋊C8—21no—
467C4⋊C8C23×C7312yes—
468C4⋊C8C23×C7312yes—
469#103—22no—
470#103—22no—
471#103—22no—
472#103—22no—
473#102—24no—
474C22⋊C8—21no—
475C4×C8C23×C7312yes—
476#41—22no—
477#41—22no—
478#41—42no—
479#41—42no—
480#41—42no—
481#41—42no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—41no—
487C2×Q8—41no—
488C2×Q8—42no—
489C2×Q8—42no—
490C2×Q8C2×D4×C7311yescharacteristic
491C2×Q8C2×D4×C7311yescharacteristic
492C2×Q8C2×D4×C7311yescharacteristic
493C2×D4—22no—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4C2×D4×C7311yescharacteristic
500C2×D4C2×D4×C7312yes—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—41no—
510C22×C4C2×D4×C7311yescharacteristic
511C22×C4C2×D4×C7311yescharacteristic
512C22×C4C2×D4×C7311yescharacteristic
513Q16—44no—
514SD16—44no—
515M16—42no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—42no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4C2×D4×C7311yescharacteristic
537C4⋊C4C2×D4×C7311yescharacteristic
538C4⋊C4C2×D4×C7311yescharacteristic
539C4⋊C4C2×D4×C7311yescharacteristic
540C4⋊C4C2×D4×C7312yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—21no—
548C22⋊C4—22no—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C42—22no—
553C42—22no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42C2×D4×C7311yescharacteristic
562C42C2×D4×C7311yescharacteristic
563C42C2×D4×C7311yescharacteristic
564C42C24×C7311yescharacteristic
565C23—21no—
566C23—22no—
567C23#120×C7311yescharacteristic
568Q8—21no—
569Q8—21no—
570Q8—21no—
571Q8—21no—
572Q8—21no—
573Q8—42no—
574Q8—41no—
575Q8—41no—
576Q8—42no—
577Q8—42no—
578Q8—82no—
579Q8#137×C7312yescomplemented
580D4—21no—
581D4—21no—
582D4—42no—
583D4—44no—
584D4—42no—
585D4—44no—
586D4—41no—
587D4—41no—
588D4—41no—
589D4—41no—
590D4—82no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4#120×C7311yescharacteristic
614C2×C4#120×C7311yescharacteristic
615C2×C4#120×C7311yescharacteristic
616C2×C4#135×C7312yes—
617C2×C4AGL(1,ℤ/8)×C7312yes—
618C2×C4C22×D4×C7311yescharacteristic
619C2×C4C22×D4×C7311yescharacteristic
620C2×C4C22×D4×C7311yescharacteristic
621C8—42no—
622C8—42no—
623C8—42no—
624C22—21no—
625C22—21no—
626C22—41no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—42no—
631C22—41no—
632C22#449×C7312yes—
633C22C2×#120×C7311yescharacteristic
634C4—21no—
635C4—22no—
636C4—21no—
637C4—21no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—82no—
646C4#449×C7312yes—
647C4C2×#135×C7311yescharacteristic
648C4C2×AGL(1,ℤ/8)×C7311yescharacteristic
649C4C2×#137×C7311yescharacteristic
650C4#575×C7311yescharacteristic
651C2—22no—
652C2—81no—
653C2—82no—
654C2C2×#449×C7311yescharacteristic
655C2C73×#300711yescharacteristic
656C2C73×#302211yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3480C2
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
2. C73×#3226C2semidirect
3. C2×#475×C73C2semidirect
4. C73×#3039C2semidirect
5. C73×#1650C2semidirect
6. C73×#1648C2semidirect
7. C73×#1642C2semidirect
8. C73×#1640C2semidirect
9. C73×#1547C2semidirect
10. C73×#1487C2semidirect
11. C73×#1477C2semidirect
250. Q8×C73#137semidirect
322. #18802C73direct (N × H)
330. #3226C2×C73semidirect
331. C2×#475C2×C73semidirect
332. #3039C2×C73semidirect
333. #1650C2×C73semidirect
334. #1648C2×C73semidirect
335. #1642C2×C73semidirect
336. #1640C2×C73semidirect
337. #1547C2×C73semidirect
338. #1487C2×C73semidirect
339. #1477C2×C73semidirect
340. C73#18802direct (N × H)
579. Q8#137×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#34802
C73×#32262
C2×#475×C732
C73×#30392
C73×#16502
C73×#16482
C73×#16422
C73×#16402
C73×#15472
C73×#14872
C73×#14772
#1880273