Finite Group Databasebeta

← C73×#24352 #32933756

Subgroups — C73×#24352 #32933756

Order 18688.

1198 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3501C211yesmaximal, characteristic, complemented
2C73×#2626C212yesmaximal, complemented
3C73×#2321C211yesmaximal, characteristic
4C73×#2062C212yesmaximal, complemented
5C73×#2029C212yesmaximal, complemented
6C73×#1996C212yesmaximal, complemented
7C73×#1889C212yesmaximal
8C73×#1867C212yesmaximal, complemented
9C73×#1799C211yesmaximal, characteristic, complemented
10#554×C73—24no—
11#554×C73—22no—
12D4×Q8×C73—22no—
13#544×C73—21no—
14#543×C73—21no—
15#543×C73C2212yes—
16#533×C73—21no—
17C2×C22⋊Q8×C73—21no—
18C2×C22⋊Q8×C73C2212yes—
19C2×#120×C73C2211yescharacteristic
20#518×C73C2212yes—
21C2×C4×D4×C73C2211yescharacteristic
22C22×C22⋊C4×C73C2211yescharacteristic
23C2×#99×C73C2212yes—
24#399×C73—22no—
25#396×C73—24no—
26#393×C73—22no—
27#390×C73—21no—
28#388×C73—21no—
29#388×C73—21no—
30#386×C73—21no—
31#386×C73C2211yescharacteristic
32#385×C73C2211yescharacteristic
33#385×C73C2214yes—
34#380×C73C2212yes—
35#379×C73C2211yescharacteristic
36#379×C73C2214yes—
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×C73D411yescharacteristic
47C23×C4×C73C2311yescharacteristic
48C23×C4×C73C2311yescharacteristic
49C4⋊Q8×C73—21no—
50C4⋊Q8×C73—21no—
51C4⋊Q8×C73—24no—
52#125×C73—22no—
53#125×C73—24no—
54C22⋊Q8×C73—22no—
55C22⋊Q8×C73—22no—
56C22⋊Q8×C73—24no—
57C22⋊Q8×C73—24no—
58C22⋊Q8×C73—22no—
59C22⋊Q8×C73—22no—
60C22⋊Q8×C73—22no—
61C22⋊Q8×C73—22no—
62C22⋊Q8×C73—22no—
63C22⋊Q8×C73—22no—
64C22⋊Q8×C73—24no—
65C22⋊Q8×C73—24no—
66#120×C73—22no—
67#120×C73Q814yescomplemented
68C4×Q8×C73—22no—
69C4×Q8×C73—22no—
70C4×D4×C73—22no—
71C4×D4×C73—22no—
72C4×D4×C73—24no—
73C4×D4×C73—24no—
74#117×C73—22no—
75C2×C4⋊C4×C73—21no—
76C2×C4⋊C4×C73—22no—
77C2×C4⋊C4×C73—22no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73—24no—
80C2×C4⋊C4×C73D411yescharacteristic
81C2×C4⋊C4×C73D414yescomplemented
82C2×C4⋊C4×C73D412yescomplemented
83C2×C22⋊C4×C73—22no—
84C2×C22⋊C4×C73—21no—
85C2×C22⋊C4×C73—21no—
86C2×C22⋊C4×C73—21no—
87C2×C22⋊C4×C73—22no—
88C2×C22⋊C4×C73—22no—
89C2×C22⋊C4×C73—21no—
90C2×C22⋊C4×C73—21no—
91C2×C22⋊C4×C73—21no—
92C2×C22⋊C4×C73—24no—
93C2×C22⋊C4×C73—24no—
94C2×C22⋊C4×C73D411yescharacteristic
95C2×C22⋊C4×C73C2311yescharacteristic
96C2×C22⋊C4×C73C2314yes—
97C2×C22⋊C4×C73C2311yescharacteristic
98C2×C22⋊C4×C73C2314yes—
99C2×C22⋊C4×C73C2311yescharacteristic
100C2×C42×C73—21no—
101C2×C42×C73D411yescharacteristic
102#99×C73—24no—
103#95×C73—21no—
104#95×C73—22no—
105#95×C73—21no—
106#95×C73—24no—
107#95×C73—21no—
108C24×C73—22no—
109C24×C73—21no—
110C24×C73—21no—
111C24×C73—22no—
112C24×C73—81no—
113C24×C73—81no—
114C24×C73C2×D411yescharacteristic
115C24×C73C2×Q812yescomplemented
116C24×C73C2411yescharacteristic
117C2×Q8×C73—22no—
118C2×Q8×C73—22no—
119C2×Q8×C73—22no—
120C2×Q8×C73—22no—
121C2×Q8×C73—82no—
122C2×Q8×C73—82no—
123C2×D4×C73—22no—
124C2×D4×C73—22no—
125C2×D4×C73—24no—
126C2×D4×C73—81no—
127C2×D4×C73—81no—
128C2×D4×C73C2×Q812yes—
129C2×D4×C73#4112yes—
130C22×C4×C73—21no—
131C22×C4×C73—22no—
132C22×C4×C73—22no—
133C22×C4×C73—21no—
134C22×C4×C73—21no—
135C22×C4×C73—21no—
136C22×C4×C73—21no—
137C22×C4×C73—24no—
138C22×C4×C73—22no—
139C22×C4×C73—81no—
140C22×C4×C73—81no—
141C22×C4×C73—81no—
142C22×C4×C73C2×D411yescharacteristic
143C22×C4×C73C2×D411yescharacteristic
144C22×C4×C73C2×D414yes—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D414yes—
147C22×C4×C73C2×D411yescharacteristic
148C22×C4×C73C2×D412yes—
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73C2×D412yes—
151C4⋊C4×C73—22no—
152C4⋊C4×C73—24no—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73—24no—
156C4⋊C4×C73—22no—
157C4⋊C4×C73—24no—
158C4⋊C4×C73—22no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—24no—
162C4⋊C4×C73—82no—
163C4⋊C4×C73—81no—
164C4⋊C4×C73—84no—
165C22⋊C4×C73—24no—
166C22⋊C4×C73—24no—
167C22⋊C4×C73—22no—
168C22⋊C4×C73—24no—
169C22⋊C4×C73—21no—
170C22⋊C4×C73—22no—
171C22⋊C4×C73—24no—
172C22⋊C4×C73—24no—
173C22⋊C4×C73—22no—
174C22⋊C4×C73—24no—
175C22⋊C4×C73—41no—
176C22⋊C4×C73—41no—
177C22⋊C4×C73—42no—
178C22⋊C4×C73—81no—
179C22⋊C4×C73—84no—
180C22⋊C4×C73—81no—
181C22⋊C4×C73—84no—
182C22⋊C4×C73—81no—
183C22⋊C4×C73C2×Q812yescomplemented
184C42×C73—24no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—81no—
188C23×C73—21no—
189C23×C73—21no—
190C23×C73—22no—
191C23×C73—24no—
192C23×C73—41no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—44no—
197C23×C73—41no—
198C23×C73—44no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—42no—
205C23×C73—82no—
206C23×C73—82no—
207C23×C73—84no—
208C23×C73C22⋊Q812yes—
209C23×C73C22×D411yescharacteristic
210C23×C73C22×D412yes—
211C23×C73C22×Q811yescharacteristic
212C23×C73C2×#4111yescharacteristic
213C23×C732−1+412yes—
214Q8×C73—44no—
215Q8×C73—82no—
216Q8×C73—82no—
217Q8×C73—82no—
218D4×C73—24no—
219D4×C73—24no—
220D4×C73—84no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—41no—
229C2×C4×C73—42no—
230C2×C4×C73—41no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—44no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—44no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—44no—
242C2×C4×C73—44no—
243C2×C4×C73—41no—
244C2×C4×C73—44no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—44no—
249C2×C4×C73—81no—
250C2×C4×C73—82no—
251C2×C4×C73—82no—
252C2×C4×C73#12011yescharacteristic
253C2×C4×C73#12011yescharacteristic
254C2×C4×C73#12012yes—
255C2×C4×C73C22⋊Q812yes—
256C22×C73—22no—
257C22×C73—21no—
258C22×C73—22no—
259C22×C73—21no—
260C22×C73—22no—
261C22×C73—41no—
262C22×C73—41no—
263C22×C73—42no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—48no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—42no—
271C22×C73—81no—
272C22×C73—84no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73C2×#12011yescharacteristic
278C22×C73C2×C22⋊Q811yescharacteristic
279C22×C73#53611yescharacteristic
280C22×C73#54812yes—
281C22×C73D4×Q812yes—
282C4×C73—22no—
283C4×C73—21no—
284C4×C73—21no—
285C4×C73—22no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—44no—
289C4×C73—44no—
290C4×C73—41no—
291C4×C73—84no—
292C4×C73—81no—
293C4×C73—81no—
294#24352C7311yesmaximal, characteristic, direct product factor, Sylow
295C2×C73—21no—
296C2×C73—22no—
297C2×C73—21no—
298C2×C73—22no—
299C2×C73—42no—
300C2×C73—82no—
301C2×C73#240911yescharacteristic
302C2×C73#302611yescharacteristic
303C2×C73#302711yescharacteristic
304#3501C2×C7311yescharacteristic, complemented
305#2626C2×C7312yescomplemented
306#2321C2×C7311yescharacteristic
307#2062C2×C7312yescomplemented
308#2029C2×C7312yescomplemented
309#1996C2×C7312yescomplemented
310#1889C2×C7312yes—
311#1867C2×C7312yescomplemented
312#1799C2×C7311yescharacteristic, complemented
313C73#2435211yescharacteristic, direct product factor, Sylow
314#554—24no—
315#554—22no—
316D4×Q8—22no—
317#544—21no—
318#543—21no—
319#543C22×C7312yes—
320#533—21no—
321C2×C22⋊Q8—21no—
322C2×C22⋊Q8C22×C7312yes—
323C2×#120C22×C7311yescharacteristic
324#518C22×C7312yes—
325C2×C4×D4C22×C7311yescharacteristic
326C22×C22⋊C4C22×C7311yescharacteristic
327C2×#99C22×C7312yes—
328#399—22no—
329#396—24no—
330#393—22no—
331#390—21no—
332#388—21no—
333#388—21no—
334#386—21no—
335#386C22×C7311yescharacteristic
336#385C22×C7311yescharacteristic
337#385C22×C7314yes—
338#380C22×C7312yes—
339#379C22×C7311yescharacteristic
340#379C22×C7314yes—
341C4×C22⋊C4C22×C7311yescharacteristic
342#342C22×C7312yes—
343#342C22×C7314yes—
344#342C22×C7314yes—
345C25C23×C7311yescharacteristic
346C22×Q8—21no—
347C22×Q8D4×C7312yescomplemented
348C22×D4—21no—
349C22×D4C23×C7311yescharacteristic
350C23×C4D4×C7311yescharacteristic
351C23×C4C23×C7311yescharacteristic
352C23×C4C23×C7311yescharacteristic
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—24no—
356#125—22no—
357#125—24no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—24no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—24no—
369C22⋊Q8—24no—
370#120—22no—
371#120Q8×C7314yescomplemented
372C4×Q8—22no—
373C4×Q8—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—24no—
378#117—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4D4×C7311yescharacteristic
385C2×C4⋊C4D4×C7314yescomplemented
386C2×C4⋊C4D4×C7312yescomplemented
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4D4×C7311yescharacteristic
399C2×C22⋊C4C23×C7311yescharacteristic
400C2×C22⋊C4C23×C7314yes—
401C2×C22⋊C4C23×C7311yescharacteristic
402C2×C22⋊C4C23×C7314yes—
403C2×C22⋊C4C23×C7311yescharacteristic
404C2×C42—21no—
405C2×C42D4×C7311yescharacteristic
406#99—24no—
407#95—21no—
408#95—22no—
409#95—21no—
410#95—24no—
411#95—21no—
412C24—22no—
413C24—21no—
414C24—21no—
415C24—22no—
416C24—81no—
417C24—81no—
418C24C2×D4×C7311yescharacteristic
419C24C2×Q8×C7312yescomplemented
420C24C24×C7311yescharacteristic
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4C2×Q8×C7312yes—
433C2×D4#41×C7312yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4C2×D4×C7311yescharacteristic
447C22×C4C2×D4×C7311yescharacteristic
448C22×C4C2×D4×C7314yes—
449C22×C4C2×D4×C7311yescharacteristic
450C22×C4C2×D4×C7314yes—
451C22×C4C2×D4×C7311yescharacteristic
452C22×C4C2×D4×C7312yes—
453C22×C4C2×D4×C7311yescharacteristic
454C22×C4C2×D4×C7312yes—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—24no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—22no—
478C22⋊C4—24no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—84no—
484C22⋊C4—81no—
485C22⋊C4—84no—
486C22⋊C4—81no—
487C22⋊C4C2×Q8×C7312yescomplemented
488C42—24no—
489C42—21no—
490C42—21no—
491C42—81no—
492C23—21no—
493C23—21no—
494C23—22no—
495C23—24no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—44no—
501C23—41no—
502C23—44no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—82no—
510C23—82no—
511C23—84no—
512C23C22⋊Q8×C7312yes—
513C23C22×D4×C7311yescharacteristic
514C23C22×D4×C7312yes—
515C23C22×Q8×C7311yescharacteristic
516C23C2×#41×C7311yescharacteristic
517C232−1+4×C7312yes—
518Q8—44no—
519Q8—82no—
520Q8—82no—
521Q8—82no—
522D4—24no—
523D4—24no—
524D4—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#120×C7311yescharacteristic
557C2×C4#120×C7311yescharacteristic
558C2×C4#120×C7312yes—
559C2×C4C22⋊Q8×C7312yes—
560C22—22no—
561C22—21no—
562C22—22no—
563C22—21no—
564C22—22no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—48no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—81no—
576C22—84no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#120×C7311yescharacteristic
582C22C2×C22⋊Q8×C7311yescharacteristic
583C22#536×C7311yescharacteristic
584C22#548×C7312yes—
585C22D4×Q8×C7312yes—
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—44no—
593C4—44no—
594C4—41no—
595C4—84no—
596C4—81no—
597C4—81no—
598C2—21no—
599C2—22no—
600C2—21no—
601C2—22no—
602C2—42no—
603C2—82no—
604C2C73×#240911yescharacteristic
605C2C73×#302611yescharacteristic
606C2C73×#302711yescharacteristic
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×#2626C2semidirect
4. C73×#2062C2semidirect
5. C73×#2029C2semidirect
6. C73×#1996C2semidirect
8. C73×#1867C2semidirect
9. C73×#1799C2semidirect
43. C22×Q8×C73D4semidirect
67. #120×C73Q8semidirect
81. C2×C4⋊C4×C73D4semidirect
82. C2×C4⋊C4×C73D4semidirect
115. C24×C73C2×Q8semidirect
183. C22⋊C4×C73C2×Q8semidirect
294. #24352C73direct (N × H)
304. #3501C2×C73semidirect
305. #2626C2×C73semidirect
307. #2062C2×C73semidirect
308. #2029C2×C73semidirect
309. #1996C2×C73semidirect
311. #1867C2×C73semidirect
312. #1799C2×C73semidirect
313. C73#24352direct (N × H)
347. C22×Q8D4×C73semidirect
371. #120Q8×C73semidirect
385. C2×C4⋊C4D4×C73semidirect
386. C2×C4⋊C4D4×C73semidirect
419. C24C2×Q8×C73semidirect
487. C22⋊C4C2×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×#26262
C73×#23212
C73×#20622
C73×#20292
C73×#19962
C73×#18892
C73×#18672
C73×#17992
#2435273