Finite Group Databasebeta

← C73×#24350 #32933754

Subgroups — C73×#24350 #32933754

Order 18688.

1462 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3501C211yesmaximal, characteristic, complemented
2C73×#2848C212yesmaximal, complemented
3C4×#120×C73C211yesmaximal, characteristic
4C73×#2062C212yesmaximal, complemented
5C73×#2029C212yesmaximal, complemented
6C73×#1996C212yesmaximal, complemented
7C73×#1889C212yesmaximal
8C73×#1867C212yesmaximal, complemented
9C73×#1799C211yesmaximal, characteristic, complemented
10C24×C4×C73C2211yescharacteristic
11#554×C73—22no—
12#554×C73—24no—
13D4×Q8×C73—22no—
14#544×C73—21no—
15#543×C73—21no—
16#543×C73C2212yes—
17#533×C73—21no—
18C2×C22⋊Q8×C73—21no—
19C2×C22⋊Q8×C73—22no—
20C2×C22⋊Q8×C73C2212yes—
21C2×#120×C73C2211yescharacteristic
22#518×C73C2212yes—
23C2×C4×D4×C73—21no—
24C2×C4×D4×C73C2211yescharacteristic
25C2×#99×C73C2212yes—
26#386×C73—21no—
27#386×C73C2211yescharacteristic
28#385×C73—21no—
29#385×C73—24no—
30#385×C73C2211yescharacteristic
31#385×C73C2214yes—
32#380×C73—22no—
33#380×C73C2212yes—
34#379×C73C2211yescharacteristic
35#379×C73C2214yes—
36C4×C22⋊C4×C73—21no—
37C4×C22⋊C4×C73C2211yescharacteristic
38#342×C73C2214yes—
39#342×C73C2214yes—
40#342×C73C2212yes—
41C25×C73C2311yescharacteristic
42C22×Q8×C73—21no—
43C22×Q8×C73D412yescomplemented
44C22×D4×C73—21no—
45C22×D4×C73C2311yescharacteristic
46C23×C4×C73—21no—
47C23×C4×C73—22no—
48C23×C4×C73—22no—
49C23×C4×C73—21no—
50C23×C4×C73—44no—
51C23×C4×C73D411yescharacteristic
52C23×C4×C73C2311yescharacteristic
53C23×C4×C73C2311yescharacteristic
54C4⋊Q8×C73—21no—
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73—24no—
57#125×C73—22no—
58#125×C73—24no—
59C22⋊Q8×C73—22no—
60C22⋊Q8×C73—22no—
61C22⋊Q8×C73—24no—
62C22⋊Q8×C73—24no—
63C22⋊Q8×C73—22no—
64C22⋊Q8×C73—22no—
65C22⋊Q8×C73—22no—
66C22⋊Q8×C73—22no—
67C22⋊Q8×C73—22no—
68C22⋊Q8×C73—22no—
69C22⋊Q8×C73—24no—
70C22⋊Q8×C73—24no—
71C22⋊Q8×C73—48no—
72#120×C73—22no—
73#120×C73Q814yescomplemented
74C4×Q8×C73—22no—
75C4×Q8×C73—22no—
76C4×D4×C73—22no—
77C4×D4×C73—22no—
78C4×D4×C73—24no—
79C4×D4×C73—24no—
80C4×D4×C73—44no—
81#117×C73—22no—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73—22no—
84C2×C4⋊C4×C73—22no—
85C2×C4⋊C4×C73—21no—
86C2×C4⋊C4×C73—24no—
87C2×C4⋊C4×C73D411yescharacteristic
88C2×C4⋊C4×C73D412yescomplemented
89C2×C4⋊C4×C73D414yescomplemented
90C2×C22⋊C4×C73—22no—
91C2×C22⋊C4×C73—21no—
92C2×C22⋊C4×C73—21no—
93C2×C22⋊C4×C73—21no—
94C2×C22⋊C4×C73—24no—
95C2×C22⋊C4×C73—21no—
96C2×C22⋊C4×C73—24no—
97C2×C22⋊C4×C73D411yescharacteristic
98C2×C22⋊C4×C73C2311yescharacteristic
99C2×C22⋊C4×C73C2314yes—
100C2×C22⋊C4×C73C2311yescharacteristic
101C2×C22⋊C4×C73C2314yes—
102C2×C22⋊C4×C73C2311yescharacteristic
103C2×C42×C73—21no—
104C2×C42×C73D411yescharacteristic
105#99×C73—24no—
106#95×C73—21no—
107#95×C73—21no—
108#95×C73—21no—
109#95×C73—24no—
110#95×C73—22no—
111C24×C73—22no—
112C24×C73—21no—
113C24×C73—21no—
114C24×C73—22no—
115C24×C73—42no—
116C24×C73—42no—
117C24×C73C2×D411yescharacteristic
118C24×C73C2×Q812yescomplemented
119C24×C73C2411yescharacteristic
120C2×Q8×C73—22no—
121C2×Q8×C73—22no—
122C2×Q8×C73—22no—
123C2×Q8×C73—22no—
124C2×Q8×C73—44no—
125C2×Q8×C73—82no—
126C2×D4×C73—22no—
127C2×D4×C73—22no—
128C2×D4×C73—24no—
129C2×D4×C73—42no—
130C2×D4×C73—42no—
131C2×D4×C73C2×Q812yes—
132C2×D4×C73#4112yes—
133C22×C4×C73—24no—
134C22×C4×C73—28no—
135C22×C4×C73—21no—
136C22×C4×C73—21no—
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—22no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—24no—
144C22×C4×C73—42no—
145C22×C4×C73—42no—
146C22×C4×C73—42no—
147C22×C4×C73—44no—
148C22×C4×C73—42no—
149C22×C4×C73—44no—
150C22×C4×C73—42no—
151C22×C4×C73—44no—
152C22×C4×C73—42no—
153C22×C4×C73—41no—
154C22×C4×C73—42no—
155C22×C4×C73—41no—
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D412yes—
158C22×C4×C73C2×D411yescharacteristic
159C22×C4×C73C2×D411yescharacteristic
160C22×C4×C73C2×D414yes—
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2×D414yes—
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D412yes—
165C4⋊C4×C73—22no—
166C4⋊C4×C73—24no—
167C4⋊C4×C73—22no—
168C4⋊C4×C73—22no—
169C4⋊C4×C73—24no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—24no—
172C4⋊C4×C73—22no—
173C4⋊C4×C73—22no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—24no—
176C4⋊C4×C73—42no—
177C4⋊C4×C73—44no—
178C4⋊C4×C73—48no—
179C22⋊C4×C73—24no—
180C22⋊C4×C73—24no—
181C22⋊C4×C73—22no—
182C22⋊C4×C73—24no—
183C22⋊C4×C73—21no—
184C22⋊C4×C73—22no—
185C22⋊C4×C73—24no—
186C22⋊C4×C73—24no—
187C22⋊C4×C73—22no—
188C22⋊C4×C73—24no—
189C22⋊C4×C73—42no—
190C22⋊C4×C73—48no—
191C22⋊C4×C73—42no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—48no—
194C22⋊C4×C73C2×Q812yescomplemented
195C42×C73—24no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—42no—
199C23×C73—24no—
200C23×C73—24no—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73—21no—
204C23×C73—22no—
205C23×C73—24no—
206C23×C73—21no—
207C23×C73—41no—
208C23×C73—44no—
209C23×C73—41no—
210C23×C73—42no—
211C23×C73—41no—
212C23×C73—42no—
213C23×C73—41no—
214C23×C73—44no—
215C23×C73—48no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219C23×C73—41no—
220C23×C73—42no—
221C23×C73C22⋊Q812yescomplemented
222C23×C73C22×D411yescharacteristic
223C23×C73C22×D412yes—
224C23×C73C22×Q811yescharacteristic
225C23×C73C2×#4111yescharacteristic
226C23×C732−1+412yes—
227Q8×C73—44no—
228Q8×C73—82no—
229Q8×C73—82no—
230Q8×C73—82no—
231D4×C73—24no—
232D4×C73—24no—
233D4×C73—48no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—24no—
238C2×C4×C73—22no—
239C2×C4×C73—24no—
240C2×C4×C73—21no—
241C2×C4×C73—24no—
242C2×C4×C73—24no—
243C2×C4×C73—21no—
244C2×C4×C73—22no—
245C2×C4×C73—42no—
246C2×C4×C73—41no—
247C2×C4×C73—44no—
248C2×C4×C73—48no—
249C2×C4×C73—41no—
250C2×C4×C73—42no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—44no—
255C2×C4×C73—42no—
256C2×C4×C73—41no—
257C2×C4×C73—44no—
258C2×C4×C73—44no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—44no—
262C2×C4×C73—44no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—44no—
266C2×C4×C73—81no—
267C2×C4×C73—82no—
268C2×C4×C73—82no—
269C2×C4×C73#12011yescharacteristic
270C2×C4×C73#12012yes—
271C2×C4×C73#12011yescharacteristic
272C2×C4×C73C22⋊Q812yescomplemented
273C22×C73—22no—
274C22×C73—21no—
275C22×C73—22no—
276C22×C73—21no—
277C22×C73—24no—
278C22×C73—28no—
279C22×C73—22no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—42no—
285C22×C73—44no—
286C22×C73—44no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73—44no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—42no—
295C22×C73—81no—
296C22×C73C2×#12011yescharacteristic
297C22×C73C2×C22⋊Q811yescharacteristic
298C22×C73#53611yescharacteristic
299C22×C73#54812yes—
300C22×C73D4×Q812yes—
301C4×C73—22no—
302C4×C73—21no—
303C4×C73—22no—
304C4×C73—42no—
305C4×C73—42no—
306C4×C73—41no—
307C4×C73—44no—
308C4×C73—44no—
309C4×C73—41no—
310C4×C73—84no—
311C4×C73—81no—
312C4×C73B(4,𝔽2)12yes—
313#24350C7311yesmaximal, characteristic, direct product factor, Sylow
314C2×C73—21no—
315C2×C73—22no—
316C2×C73—21no—
317C2×C73—22no—
318C2×C73—44no—
319C2×C73—42no—
320C2×C73#240911yescharacteristic
321C2×C73C2×B(4,𝔽2)11yescharacteristic
322C2×C73#302511yescharacteristic
323#3501C2×C7311yescharacteristic, complemented
324#2848C2×C7312yescomplemented
325C4×#120C2×C7311yescharacteristic
326#2062C2×C7312yescomplemented
327#2029C2×C7312yescomplemented
328#1996C2×C7312yescomplemented
329#1889C2×C7312yes—
330#1867C2×C7312yescomplemented
331#1799C2×C7311yescharacteristic, complemented
332C73#2435011yescharacteristic, direct product factor, Sylow
333C24×C4C22×C7311yescharacteristic
334#554—22no—
335#554—24no—
336D4×Q8—22no—
337#544—21no—
338#543—21no—
339#543C22×C7312yes—
340#533—21no—
341C2×C22⋊Q8—21no—
342C2×C22⋊Q8—22no—
343C2×C22⋊Q8C22×C7312yes—
344C2×#120C22×C7311yescharacteristic
345#518C22×C7312yes—
346C2×C4×D4—21no—
347C2×C4×D4C22×C7311yescharacteristic
348C2×#99C22×C7312yes—
349#386—21no—
350#386C22×C7311yescharacteristic
351#385—21no—
352#385—24no—
353#385C22×C7311yescharacteristic
354#385C22×C7314yes—
355#380—22no—
356#380C22×C7312yes—
357#379C22×C7314yes—
358#379C22×C7311yescharacteristic
359C4×C22⋊C4—21no—
360C4×C22⋊C4C22×C7311yescharacteristic
361#342C22×C7312yes—
362#342C22×C7314yes—
363#342C22×C7314yes—
364C25C23×C7311yescharacteristic
365C22×Q8—21no—
366C22×Q8D4×C7312yescomplemented
367C22×D4—21no—
368C22×D4C23×C7311yescharacteristic
369C23×C4—21no—
370C23×C4—22no—
371C23×C4—22no—
372C23×C4—21no—
373C23×C4—44no—
374C23×C4D4×C7311yescharacteristic
375C23×C4C23×C7311yescharacteristic
376C23×C4C23×C7311yescharacteristic
377C4⋊Q8—21no—
378C4⋊Q8—21no—
379C4⋊Q8—24no—
380#125—22no—
381#125—24no—
382C22⋊Q8—22no—
383C22⋊Q8—22no—
384C22⋊Q8—24no—
385C22⋊Q8—24no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—22no—
389C22⋊Q8—22no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8—24no—
393C22⋊Q8—24no—
394C22⋊Q8—48no—
395#120—22no—
396#120Q8×C7314yescomplemented
397C4×Q8—22no—
398C4×Q8—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—24no—
402C4×D4—24no—
403C4×D4—44no—
404#117—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—24no—
410C2×C4⋊C4D4×C7311yescharacteristic
411C2×C4⋊C4D4×C7312yescomplemented
412C2×C4⋊C4D4×C7314yescomplemented
413C2×C22⋊C4—22no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—24no—
420C2×C22⋊C4D4×C7311yescharacteristic
421C2×C22⋊C4C23×C7311yescharacteristic
422C2×C22⋊C4C23×C7314yes—
423C2×C22⋊C4C23×C7311yescharacteristic
424C2×C22⋊C4C23×C7314yes—
425C2×C22⋊C4C23×C7311yescharacteristic
426C2×C42—21no—
427C2×C42D4×C7311yescharacteristic
428#99—24no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—24no—
433#95—22no—
434C24—22no—
435C24—21no—
436C24—21no—
437C24—22no—
438C24—42no—
439C24—42no—
440C24C2×D4×C7311yescharacteristic
441C24C2×Q8×C7312yescomplemented
442C24C24×C7311yescharacteristic
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—44no—
448C2×Q8—82no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4C2×Q8×C7312yes—
455C2×D4#41×C7312yes—
456C22×C4—24no—
457C22×C4—28no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—42no—
478C22×C4—41no—
479C22×C4C2×D4×C7311yescharacteristic
480C22×C4C2×D4×C7312yes—
481C22×C4C2×D4×C7311yescharacteristic
482C22×C4C2×D4×C7311yescharacteristic
483C22×C4C2×D4×C7314yes—
484C22×C4C2×D4×C7311yescharacteristic
485C22×C4C2×D4×C7314yes—
486C22×C4C2×D4×C7311yescharacteristic
487C22×C4C2×D4×C7312yes—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—24no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—22no—
511C22⋊C4—24no—
512C22⋊C4—42no—
513C22⋊C4—48no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—48no—
517C22⋊C4C2×Q8×C7312yescomplemented
518C42—24no—
519C42—21no—
520C42—21no—
521C42—42no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—22no—
528C23—24no—
529C23—21no—
530C23—41no—
531C23—44no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—44no—
538C23—48no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23C22⋊Q8×C7312yescomplemented
545C23C22×D4×C7311yescharacteristic
546C23C22×D4×C7312yes—
547C23C22×Q8×C7311yescharacteristic
548C23C2×#41×C7311yescharacteristic
549C232−1+4×C7312yes—
550Q8—44no—
551Q8—82no—
552Q8—82no—
553Q8—82no—
554D4—24no—
555D4—24no—
556D4—48no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#120×C7311yescharacteristic
593C2×C4#120×C7312yes—
594C2×C4#120×C7311yescharacteristic
595C2×C4C22⋊Q8×C7312yescomplemented
596C22—22no—
597C22—21no—
598C22—22no—
599C22—21no—
600C22—24no—
601C22—28no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22C2×#120×C7311yescharacteristic
620C22C2×C22⋊Q8×C7311yescharacteristic
621C22#536×C7311yescharacteristic
622C22#548×C7312yes—
623C22D4×Q8×C7312yes—
624C4—21no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—44no—
631C4—44no—
632C4—41no—
633C4—84no—
634C4—81no—
635C4B(4,𝔽2)×C7312yes—
636C2—21no—
637C2—22no—
638C2—21no—
639C2—22no—
640C2—44no—
641C2—42no—
642C2C73×#240911yescharacteristic
643C2C2×B(4,𝔽2)×C7311yescharacteristic
644C2C73×#302511yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC24C24×C73
prime radicalC2×#120×C73C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3501C2semidirect
2. C73×#2848C2semidirect
4. C73×#2062C2semidirect
5. C73×#2029C2semidirect
6. C73×#1996C2semidirect
8. C73×#1867C2semidirect
9. C73×#1799C2semidirect
43. C22×Q8×C73D4semidirect
73. #120×C73Q8semidirect
88. C2×C4⋊C4×C73D4semidirect
89. C2×C4⋊C4×C73D4semidirect
118. C24×C73C2×Q8semidirect
194. C22⋊C4×C73C2×Q8semidirect
221. C23×C73C22⋊Q8semidirect
272. C2×C4×C73C22⋊Q8semidirect
313. #24350C73direct (N × H)
323. #3501C2×C73semidirect
324. #2848C2×C73semidirect
326. #2062C2×C73semidirect
327. #2029C2×C73semidirect
328. #1996C2×C73semidirect
330. #1867C2×C73semidirect
331. #1799C2×C73semidirect
332. C73#24350direct (N × H)
366. C22×Q8D4×C73semidirect
396. #120Q8×C73semidirect
411. C2×C4⋊C4D4×C73semidirect
412. C2×C4⋊C4D4×C73semidirect
441. C24C2×Q8×C73semidirect
517. C22⋊C4C2×Q8×C73semidirect
544. C23C22⋊Q8×C73semidirect
595. C2×C4C22⋊Q8×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#35012
C73×#28482
C4×#120×C732
C73×#20622
C73×#20292
C73×#19962
C73×#18892
C73×#18672
C73×#17992
#2435073