Finite Group Databasebeta

← C73×#18793 #32928197

Subgroups — C73×#18793 #32928197

Order 18688.

948 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3475C211yesmaximal, characteristic
2C73×#3209C211yesmaximal, characteristic, complemented
3C73×#3079C211yesmaximal, characteristic, complemented
4C2×#476×C73C211yesmaximal, characteristic, complemented
5C73×#1638C212yesmaximal, complemented
6C73×#1635C212yesmaximal, complemented
7C73×#1634C212yesmaximal, complemented
8C73×#1632C212yesmaximal, complemented
9C73×#1550C211yesmaximal, characteristic, complemented
10C73×#1486C211yesmaximal, characteristic, complemented
11C73×#1477C211yesmaximal, characteristic, complemented
12#556×C73C2211yescharacteristic
13#555×C73C2211yescharacteristic
14#554×C73C2211yescharacteristic
15#553×C73—21no—
16#552×C73—21no—
17#552×C73—21no—
18#551×C73—22no—
19#547×C73C2211yescharacteristic
20#546×C73—21no—
21#533×C73C2211yescharacteristic
22C2×C4⋊Q8×C73C2211yescharacteristic
23#529×C73—22no—
24#529×C73—21no—
25#525×C73—21no—
26C4×#41×C73—21no—
27C4×#41×C73C2211yescharacteristic
28C2×C4×D4×C73C2211yescharacteristic
29C2×#117×C73—21no—
30#494×C73C2212yescomplemented
31#487×C73C2212yescomplemented
32#477×C73C2212yescomplemented
33#476×C73—21no—
34#476×C73—21no—
35#476×C73C2212yes—
36#476×C73C2212yes—
37#476×C73C2212yescomplemented
38#475×C73—21no—
39#474×C73—21no—
40#462×C73—22no—
41#451×C73—22no—
42#431×C73C2211yescharacteristic
43#428×C73—21no—
44C2×#106×C73—21no—
45#423×C73C2211yescharacteristic
46C2×C4⋊C8×C73C2211yescharacteristic
47#417×C73—21no—
48C2×#102×C73—21no—
49#333×C73C2212yescomplemented
50#332×C73C2212yescomplemented
51Q8⋊C8×C73C2212yescomplemented
52D4⋊C8×C73C2212yes—
53D4⋊C8×C73C2212yes—
54D4⋊C8×C73C2212yescomplemented
55C2×#41×C73—21no—
56C2×#41×C73D411yescharacteristic
57C22×Q8×C73—21no—
58C22×D4×C73D411yescharacteristic
59C23×C4×C73—21no—
60C2×Q16×C73—24no—
61C2×M16×C73—21no—
62C22×C8×C73—21no—
63C4⋊Q8×C73—21no—
64C4⋊Q8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73C2312yes—
67C4⋊Q8×C73C2311yescharacteristic
68C4⋊Q8×C73C2311yescharacteristic
69#126×C73—22no—
70#126×C73—22no—
71#126×C73—22no—
72#125×C73—22no—
73#125×C73—21no—
74#125×C73—21no—
75#124×C73—22no—
76#124×C73—21no—
77#123×C73—22no—
78#123×C73—22no—
79#123×C73—22no—
80#123×C73—22no—
81C22⋊Q8×C73—21no—
82C22⋊Q8×C73—21no—
83C22⋊Q8×C73—22no—
84C22⋊Q8×C73—21no—
85C22⋊Q8×C73—21no—
86C22⋊Q8×C73D412yes—
87#121×C73—22no—
88#121×C73—21no—
89#121×C73—21no—
90#121×C73—21no—
91#120×C73—22no—
92C4×Q8×C73—21no—
93C4×Q8×C73—21no—
94C4×Q8×C73D412yes—
95C4×Q8×C73D411yescharacteristic
96C4×Q8×C73C2311yescharacteristic
97C4×D4×C73—22no—
98C4×D4×C73—22no—
99C4×D4×C73—22no—
100C4×D4×C73—21no—
101C4×D4×C73—21no—
102C4×D4×C73—21no—
103C4×D4×C73—21no—
104C4×D4×C73—21no—
105C4×D4×C73D411yescharacteristic
106C4×D4×C73D411yescharacteristic
107C4×D4×C73D411yescharacteristic
108C4×D4×C73C2311yescharacteristic
109C4×D4×C73C2311yescharacteristic
110C4×D4×C73C2311yescharacteristic
111#117×C73—22no—
112#117×C73—22no—
113#117×C73—22no—
114#117×C73—21no—
115#117×C73—21no—
116#117×C73—21no—
117#117×C73—21no—
118#117×C73—21no—
119#117×C73D411yescharacteristic
120C2×C4⋊C4×C73—21no—
121C2×C4⋊C4×C73—21no—
122C2×C4⋊C4×C73—21no—
123C2×C4⋊C4×C73—21no—
124C2×C4⋊C4×C73D411yescharacteristic
125C2×C22⋊C4×C73—22no—
126C2×C22⋊C4×C73—21no—
127C2×C42×C73—22no—
128C2×C42×C73—21no—
129C2×C42×C73C2311yescharacteristic
130#107×C73—22no—
131#106×C73—22no—
132#106×C73—22no—
133#106×C73—22no—
134C4⋊C8×C73—21no—
135C4⋊C8×C73—21no—
136C4⋊C8×C73—21no—
137C4⋊C8×C73C2312yes—
138C4⋊C8×C73C2312yes—
139#103×C73—22no—
140#103×C73—24no—
141#102×C73—22no—
142#102×C73—22no—
143#102×C73—22no—
144C22⋊C8×C73—21no—
145C4×C8×C73C2312yes—
146C24×C73—21no—
147#41×C73—22no—
148#41×C73—22no—
149#41×C73—42no—
150#41×C73—42no—
151C2×Q8×C73—21no—
152C2×Q8×C73—22no—
153C2×Q8×C73—41no—
154C2×Q8×C73—82no—
155C2×Q8×C73C2×D411yescharacteristic
156C2×Q8×C73C2×D412yes—
157C2×D4×C73—22no—
158C2×D4×C73—22no—
159C2×D4×C73—21no—
160C2×D4×C73—22no—
161C2×D4×C73—21no—
162C2×D4×C73—41no—
163C2×D4×C73C2×D411yescharacteristic
164C2×D4×C73C2×D411yescharacteristic
165C2×D4×C73C2×D411yescharacteristic
166C22×C4×C73—22no—
167C22×C4×C73—21no—
168C22×C4×C73—22no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—21no—
175C22×C4×C73—21no—
176C22×C4×C73—42no—
177C22×C4×C73—42no—
178C22×C4×C73—41no—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182Q16×C73—84no—
183M16×C73—42no—
184C2×C8×C73—22no—
185C2×C8×C73—22no—
186C2×C8×C73—22no—
187C2×C8×C73—42no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—22no—
190C4⋊C4×C73—22no—
191C4⋊C4×C73—22no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—22no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—22no—
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×D412yes—
205C4⋊C4×C73C2×D411yescharacteristic
206C4⋊C4×C73C2×D411yescharacteristic
207C4⋊C4×C73C2×D411yescharacteristic
208C4⋊C4×C73C2×D411yescharacteristic
209C22⋊C4×C73—22no—
210C22⋊C4×C73—22no—
211C22⋊C4×C73—22no—
212C22⋊C4×C73—22no—
213C22⋊C4×C73—22no—
214C22⋊C4×C73—22no—
215C22⋊C4×C73—21no—
216C22⋊C4×C73—22no—
217C22⋊C4×C73—21no—
218C22⋊C4×C73—21no—
219C22⋊C4×C73—21no—
220C22⋊C4×C73—21no—
221C22⋊C4×C73—21no—
222C22⋊C4×C73—21no—
223C42×C73—22no—
224C42×C73—22no—
225C42×C73—21no—
226C42×C73—22no—
227C42×C73—21no—
228C42×C73—21no—
229C42×C73C2×D411yescharacteristic
230C42×C73C2×D411yescharacteristic
231C42×C73C2×D411yescharacteristic
232C42×C73C2411yescharacteristic
233C23×C73—21no—
234C23×C73—21no—
235C23×C73—21no—
236C23×C73—42no—
237C23×C73—42no—
238C23×C73—81no—
239C23×C73#12011yescharacteristic
240Q8×C73—21no—
241Q8×C73—21no—
242Q8×C73—42no—
243Q8×C73—44no—
244Q8×C73—41no—
245Q8×C73—41no—
246Q8×C73—82no—
247D4×C73—21no—
248D4×C73—21no—
249D4×C73—21no—
250D4×C73—21no—
251D4×C73—21no—
252D4×C73—42no—
253D4×C73—44no—
254D4×C73—42no—
255D4×C73—82no—
256D4×C73#13712yes—
257C2×C4×C73—21no—
258C2×C4×C73—21no—
259C2×C4×C73—21no—
260C2×C4×C73—21no—
261C2×C4×C73—21no—
262C2×C4×C73—21no—
263C2×C4×C73—22no—
264C2×C4×C73—22no—
265C2×C4×C73—21no—
266C2×C4×C73—41no—
267C2×C4×C73—44no—
268C2×C4×C73—42no—
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—42no—
276C2×C4×C73—42no—
277C2×C4×C73—42no—
278C2×C4×C73—42no—
279C2×C4×C73—81no—
280C2×C4×C73—82no—
281C2×C4×C73—82no—
282C2×C4×C73#12011yescharacteristic
283C2×C4×C73#12011yescharacteristic
284C2×C4×C73#12011yescharacteristic
285C2×C4×C73#13512yes—
286C2×C4×C73#13712yes—
287C2×C4×C73C22×D411yescharacteristic
288C2×C4×C73C22×D411yescharacteristic
289C2×C4×C73C22×D411yescharacteristic
290C8×C73—42no—
291C8×C73—42no—
292C8×C73—42no—
293C22×C73—21no—
294C22×C73—21no—
295C22×C73—22no—
296C22×C73—41no—
297C22×C73—41no—
298C22×C73—41no—
299C22×C73—41no—
300C22×C73—41no—
301C22×C73—41no—
302C22×C73—41no—
303C22×C73—41no—
304C22×C73—84no—
305C22×C73#45212yes—
306C22×C73C2×#12011yescharacteristic
307C4×C73—21no—
308C4×C73—21no—
309C4×C73—21no—
310C4×C73—41no—
311C4×C73—41no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—42no—
315C4×C73—41no—
316C4×C73—42no—
317C4×C73—82no—
318C4×C73#45212yes—
319C4×C73C2×#13511yescharacteristic
320C4×C73C2×#13711yescharacteristic
321C4×C73C2×#13711yescharacteristic
322C4×C73#57511yescharacteristic
323#18793C7311yesmaximal, characteristic, direct product factor, Sylow
324C2×C73—22no—
325C2×C73—42no—
326C2×C73—81no—
327C2×C73—81no—
328C2×C73C2×#45211yescharacteristic
329C2×C73#300711yescharacteristic
330C2×C73#302111yescharacteristic
331#3475C2×C7311yescharacteristic
332#3209C2×C7311yescharacteristic, complemented
333#3079C2×C7311yescharacteristic, complemented
334C2×#476C2×C7311yescharacteristic, complemented
335#1638C2×C7312yescomplemented
336#1635C2×C7312yescomplemented
337#1634C2×C7312yescomplemented
338#1632C2×C7312yescomplemented
339#1550C2×C7311yescharacteristic, complemented
340#1486C2×C7311yescharacteristic, complemented
341#1477C2×C7311yescharacteristic, complemented
342C73#1879311yescharacteristic, direct product factor, Sylow
343#556C22×C7311yescharacteristic
344#555C22×C7311yescharacteristic
345#554C22×C7311yescharacteristic
346#553—21no—
347#552—21no—
348#552—21no—
349#551—22no—
350#547C22×C7311yescharacteristic
351#546—21no—
352#533C22×C7311yescharacteristic
353C2×C4⋊Q8C22×C7311yescharacteristic
354#529—22no—
355#529—21no—
356#525—21no—
357C4×#41—21no—
358C4×#41C22×C7311yescharacteristic
359C2×C4×D4C22×C7311yescharacteristic
360C2×#117—21no—
361#494C22×C7312yescomplemented
362#487C22×C7312yescomplemented
363#477C22×C7312yescomplemented
364#476—21no—
365#476—21no—
366#476C22×C7312yes—
367#476C22×C7312yes—
368#476C22×C7312yescomplemented
369#475—21no—
370#474—21no—
371#462—22no—
372#451—22no—
373#431C22×C7311yescharacteristic
374#428—21no—
375C2×#106—21no—
376#423C22×C7311yescharacteristic
377C2×C4⋊C8C22×C7311yescharacteristic
378#417—21no—
379C2×#102—21no—
380#333C22×C7312yescomplemented
381#332C22×C7312yescomplemented
382Q8⋊C8C22×C7312yescomplemented
383D4⋊C8C22×C7312yes—
384D4⋊C8C22×C7312yes—
385D4⋊C8C22×C7312yescomplemented
386C2×#41—21no—
387C2×#41D4×C7311yescharacteristic
388C22×Q8—21no—
389C22×D4D4×C7311yescharacteristic
390C23×C4—21no—
391C2×Q16—24no—
392C2×M16—21no—
393C22×C8—21no—
394C4⋊Q8—21no—
395C4⋊Q8—21no—
396C4⋊Q8—21no—
397C4⋊Q8C23×C7312yes—
398C4⋊Q8C23×C7311yescharacteristic
399C4⋊Q8C23×C7311yescharacteristic
400#126—22no—
401#126—22no—
402#126—22no—
403#125—22no—
404#125—21no—
405#125—21no—
406#124—22no—
407#124—21no—
408#123—22no—
409#123—22no—
410#123—22no—
411#123—22no—
412C22⋊Q8—21no—
413C22⋊Q8—21no—
414C22⋊Q8—22no—
415C22⋊Q8—21no—
416C22⋊Q8—21no—
417C22⋊Q8D4×C7312yes—
418#121—22no—
419#121—21no—
420#121—21no—
421#121—21no—
422#120—22no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8D4×C7311yescharacteristic
426C4×Q8D4×C7312yes—
427C4×Q8C23×C7311yescharacteristic
428C4×D4—22no—
429C4×D4—22no—
430C4×D4—22no—
431C4×D4—21no—
432C4×D4—21no—
433C4×D4—21no—
434C4×D4—21no—
435C4×D4—21no—
436C4×D4D4×C7311yescharacteristic
437C4×D4D4×C7311yescharacteristic
438C4×D4D4×C7311yescharacteristic
439C4×D4C23×C7311yescharacteristic
440C4×D4C23×C7311yescharacteristic
441C4×D4C23×C7311yescharacteristic
442#117—22no—
443#117—22no—
444#117—22no—
445#117—21no—
446#117—21no—
447#117—21no—
448#117—21no—
449#117—21no—
450#117D4×C7311yescharacteristic
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—21no—
454C2×C4⋊C4—21no—
455C2×C4⋊C4D4×C7311yescharacteristic
456C2×C22⋊C4—22no—
457C2×C22⋊C4—21no—
458C2×C42—22no—
459C2×C42—21no—
460C2×C42C23×C7311yescharacteristic
461#107—22no—
462#106—22no—
463#106—22no—
464#106—22no—
465C4⋊C8—21no—
466C4⋊C8—21no—
467C4⋊C8—21no—
468C4⋊C8C23×C7312yes—
469C4⋊C8C23×C7312yes—
470#103—22no—
471#103—24no—
472#102—22no—
473#102—22no—
474#102—22no—
475C22⋊C8—21no—
476C4×C8C23×C7312yes—
477C24—21no—
478#41—22no—
479#41—22no—
480#41—42no—
481#41—42no—
482C2×Q8—21no—
483C2×Q8—22no—
484C2×Q8—41no—
485C2×Q8—82no—
486C2×Q8C2×D4×C7311yescharacteristic
487C2×Q8C2×D4×C7312yes—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—21no—
493C2×D4—41no—
494C2×D4C2×D4×C7311yescharacteristic
495C2×D4C2×D4×C7311yescharacteristic
496C2×D4C2×D4×C7311yescharacteristic
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—41no—
510C22×C4C2×D4×C7311yescharacteristic
511C22×C4C2×D4×C7311yescharacteristic
512C22×C4C2×D4×C7311yescharacteristic
513Q16—84no—
514M16—42no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—42no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4C2×D4×C7312yes—
536C4⋊C4C2×D4×C7311yescharacteristic
537C4⋊C4C2×D4×C7311yescharacteristic
538C4⋊C4C2×D4×C7311yescharacteristic
539C4⋊C4C2×D4×C7311yescharacteristic
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—21no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—22no—
558C42—21no—
559C42—21no—
560C42C2×D4×C7311yescharacteristic
561C42C2×D4×C7311yescharacteristic
562C42C2×D4×C7311yescharacteristic
563C42C24×C7311yescharacteristic
564C23—21no—
565C23—21no—
566C23—21no—
567C23—42no—
568C23—42no—
569C23—81no—
570C23#120×C7311yescharacteristic
571Q8—21no—
572Q8—21no—
573Q8—42no—
574Q8—44no—
575Q8—41no—
576Q8—41no—
577Q8—82no—
578D4—21no—
579D4—21no—
580D4—21no—
581D4—21no—
582D4—21no—
583D4—42no—
584D4—44no—
585D4—42no—
586D4—82no—
587D4#137×C7312yes—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
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×C4#137×C7312yes—
618C2×C4C22×D4×C7311yescharacteristic
619C2×C4C22×D4×C7311yescharacteristic
620C2×C4C22×D4×C7311yescharacteristic
621C8—42no—
622C8—42no—
623C8—42no—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—84no—
636C22#452×C7312yes—
637C22C2×#120×C7311yescharacteristic
638C4—21no—
639C4—21no—
640C4—21no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—82no—
649C4#452×C7312yes—
650C4C2×#135×C7311yescharacteristic
651C4C2×#137×C7311yescharacteristic
652C4C2×#137×C7311yescharacteristic
653C4#575×C7311yescharacteristic
654C2—22no—
655C2—42no—
656C2—81no—
657C2—81no—
658C2C2×#452×C7311yescharacteristic
659C2C73×#300711yescharacteristic
660C2C73×#302111yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3475C2
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×#3209C2semidirect
3. C73×#3079C2semidirect
4. C2×#476×C73C2semidirect
5. C73×#1638C2semidirect
6. C73×#1635C2semidirect
7. C73×#1634C2semidirect
8. C73×#1632C2semidirect
9. C73×#1550C2semidirect
10. C73×#1486C2semidirect
11. C73×#1477C2semidirect
30. #494×C73C22semidirect
31. #487×C73C22semidirect
32. #477×C73C22semidirect
37. #476×C73C22semidirect
49. #333×C73C22semidirect
50. #332×C73C22semidirect
51. Q8⋊C8×C73C22semidirect
54. D4⋊C8×C73C22semidirect
323. #18793C73direct (N × H)
332. #3209C2×C73semidirect
333. #3079C2×C73semidirect
334. C2×#476C2×C73semidirect
335. #1638C2×C73semidirect
336. #1635C2×C73semidirect
337. #1634C2×C73semidirect
338. #1632C2×C73semidirect
339. #1550C2×C73semidirect
340. #1486C2×C73semidirect
341. #1477C2×C73semidirect
342. C73#18793direct (N × H)
361. #494C22×C73semidirect
362. #487C22×C73semidirect
363. #477C22×C73semidirect
368. #476C22×C73semidirect
380. #333C22×C73semidirect
381. #332C22×C73semidirect
382. Q8⋊C8C22×C73semidirect
385. D4⋊C8C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#34752
C73×#32092
C73×#30792
C2×#476×C732
C73×#16382
C73×#16352
C73×#16342
C73×#16322
C73×#15502
C73×#14862
C73×#14772
#1879373