Finite Group Databasebeta

← C73×#24860 #32934264

Subgroups — C73×#24860 #32934264

Order 18688.

704 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3080C211yesmaximal, characteristic
2C2×#479×C73C211yesmaximal, characteristic, complemented
3C4×#137×C73C211yesmaximal, characteristic
4C73×#2906C211yesmaximal, characteristic
5C73×#2090C211yesmaximal, characteristic, complemented
6C73×#2089C211yesmaximal, characteristic, complemented
7C73×#2064C211yesmaximal, characteristic, complemented
8C73×#2064C211yesmaximal, characteristic, complemented
9C73×#1998C211yesmaximal, characteristic, complemented
10C73×#1950C211yesmaximal, characteristic, complemented
11C73×#1908C211yesmaximal, characteristic
12C73×#1908C211yesmaximal, characteristic
13C73×#1816C211yesmaximal, characteristic
14C73×#1815C211yesmaximal, characteristic
15C73×#1801C211yesmaximal, characteristic, complemented
16C2×#137×C73C2211yescharacteristic
17C2×#131×C73C2211yescharacteristic
18C2×#131×C73C2211yescharacteristic
19C2×#125×C73C2211yescharacteristic
20C4×#41×C73C2211yescharacteristic
21C2×C4×Q8×C73C2211yescharacteristic
22#479×C73—21no—
23#479×C73—21no—
24#479×C73—21no—
25#479×C73—21no—
26#479×C73—21no—
27#479×C73—21no—
28#478×C73—21no—
29#478×C73—21no—
30#441×C73—21no—
31#440×C73—21no—
32C4×Q16×C73—21no—
33C4×SD16×C73—21no—
34C2×#108×C73C2211yescharacteristic
35C2×#108×C73C2211yescharacteristic
36#428×C73C2211yescharacteristic
37#428×C73C2211yescharacteristic
38#428×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40C2×#106×C73C2211yescharacteristic
41#424×C73—21no—
42#424×C73—21no—
43#424×C73C2211yescharacteristic
44#424×C73C2211yescharacteristic
45#423×C73—21no—
46#423×C73C2211yescharacteristic
47#423×C73C2211yescharacteristic
48C2×C4⋊C8×C73—21no—
49C2×C4⋊C8×C73C2211yescharacteristic
50#419×C73C2211yescharacteristic
51#417×C73C2211yescharacteristic
52#417×C73C2211yescharacteristic
53#417×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#101×C73C2211yescharacteristic
57C2×#101×C73C2211yescharacteristic
58C4×M16×C73C2211yescharacteristic
59#343×C73C2211yescharacteristic
60#343×C73C2211yescharacteristic
61#343×C73C2211yescharacteristic
62#343×C73C2211yescharacteristic
63#343×C73C2211yescharacteristic
64#343×C73C2211yescharacteristic
65#341×C73C2211yescharacteristic
66#341×C73C2211yescharacteristic
67C2×#41×C73C2311yescharacteristic
68C22×Q8×C73C2311yescharacteristic
69#137×C73—22no—
70#137×C73Q814yes—
71C2×Q16×C73—21no—
72C2×SD16×C73—21no—
73#131×C73—22no—
74#131×C73—22no—
75#131×C73—41no—
76#131×C73—41no—
77C2×M16×C73—21no—
78C2×M16×C73—21no—
79C2×M16×C73C2311yescharacteristic
80C2×M16×C73C2311yescharacteristic
81C2×M16×C73C2311yescharacteristic
82C2×M16×C73C2311yescharacteristic
83C2×M16×C73C2311yescharacteristic
84C2×M16×C73C2311yescharacteristic
85C22×C8×C73—21no—
86C22×C8×C73—21no—
87C22×C8×C73C2311yescharacteristic
88C22×C8×C73C2311yescharacteristic
89#125×C73—21no—
90#125×C73—21no—
91#125×C73—21no—
92#125×C73—21no—
93C4×Q8×C73—21no—
94C4×Q8×C73—21no—
95C4×Q8×C73—21no—
96C4×Q8×C73—21no—
97C4×Q8×C73D411yescharacteristic
98C4×Q8×C73D411yescharacteristic
99C4×D4×C73—21no—
100C4×D4×C73—21no—
101C4×D4×C73D411yescharacteristic
102C4×D4×C73D411yescharacteristic
103#117×C73—21no—
104#117×C73C2311yescharacteristic
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C42×C73—21no—
112C2×C42×C73—21no—
113C2×C42×C73C2311yescharacteristic
114#108×C73—41no—
115#108×C73—41no—
116#107×C73—21no—
117#107×C73—21no—
118#107×C73—21no—
119#107×C73—21no—
120#107×C73—21no—
121#106×C73—21no—
122#106×C73—21no—
123#106×C73—21no—
124#106×C73—21no—
125#106×C73—21no—
126C4⋊C8×C73—21no—
127C4⋊C8×C73—21no—
128C4⋊C8×C73—21no—
129C4⋊C8×C73—21no—
130C4⋊C8×C73—21no—
131C4⋊C8×C73—21no—
132C4⋊C8×C73—21no—
133C4⋊C8×C73—21no—
134C4⋊C8×C73D411yescharacteristic
135C4⋊C8×C73D411yescharacteristic
136C4⋊C8×C73D411yescharacteristic
137C4⋊C8×C73D411yescharacteristic
138#103×C73—21no—
139#103×C73—21no—
140#103×C73—21no—
141#103×C73—21no—
142#103×C73—21no—
143#103×C73—21no—
144#103×C73—21no—
145#103×C73—21no—
146#103×C73—21no—
147#102×C73—21no—
148#102×C73—21no—
149#102×C73—21no—
150#101×C73—41no—
151#101×C73—41no—
152#97×C73—21no—
153C4×C8×C73—21no—
154#41×C73—21no—
155#41×C73—21no—
156#41×C73C2×Q812yes—
157#41×C73#4112yes—
158C2×Q8×C73—21no—
159C2×Q8×C73—21no—
160C2×Q8×C73—21no—
161C2×Q8×C73—21no—
162C2×Q8×C73—21no—
163C2×Q8×C73C2×D411yescharacteristic
164C2×Q8×C73C2×Q812yes—
165C2×D4×C73—21no—
166C2×D4×C73C2×D411yescharacteristic
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—21no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73#4111yescharacteristic
175C22×C4×C73#4111yescharacteristic
176C22×C4×C73C2411yescharacteristic
177Q16×C73—24no—
178SD16×C73—24no—
179M16×C73—21no—
180M16×C73—21no—
181M16×C73—21no—
182M16×C73—21no—
183M16×C73—21no—
184M16×C73—21no—
185M16×C73—21no—
186M16×C73—21no—
187M16×C73—21no—
188M16×C73—21no—
189M16×C73—21no—
190M16×C73—42no—
191M16×C73—42no—
192M16×C73C2×Q812yes—
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×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—42no—
204C2×C8×C73—42no—
205C2×C8×C73C2×D411yescharacteristic
206C2×C8×C73C2×D411yescharacteristic
207C2×C8×C73C2×D411yescharacteristic
208C2×C8×C73C2×D411yescharacteristic
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—41no—
218C4⋊C4×C73—41no—
219C4⋊C4×C73—41no—
220C4⋊C4×C73—41no—
221C4⋊C4×C73—41no—
222C4⋊C4×C73—41no—
223C4⋊C4×C73C2×D411yescharacteristic
224C4⋊C4×C73C2×D411yescharacteristic
225C22⋊C4×C73—21no—
226C22⋊C4×C73—21no—
227C22⋊C4×C73—21no—
228C42×C73—21no—
229C42×C73—21no—
230C42×C73—21no—
231C42×C73—41no—
232C42×C73—41no—
233C42×C73C2×D411yescharacteristic
234C42×C73C2×D411yescharacteristic
235C42×C73C2×D411yescharacteristic
236C42×C73C2×D411yescharacteristic
237C23×C73—21no—
238C23×C73C2×#4111yescharacteristic
239Q8×C73—21no—
240Q8×C73—22no—
241Q8×C73—22no—
242Q8×C73—41no—
243Q8×C73—41no—
244Q8×C73C22⋊Q812yes—
245D4×C73—22no—
246D4×C73—22no—
247D4×C73—21no—
248D4×C73C22⋊Q812yes—
249C2×C4×C73—21no—
250C2×C4×C73—22no—
251C2×C4×C73—22no—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—22no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—42no—
258C2×C4×C73—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—81no—
264C2×C4×C73—81no—
265C2×C4×C73—81no—
266C2×C4×C73—81no—
267C2×C4×C73#12111yescharacteristic
268C2×C4×C73#12111yescharacteristic
269C2×C4×C73#12111yescharacteristic
270C2×C4×C73#12111yescharacteristic
271C2×C4×C73C22×D411yescharacteristic
272C2×C4×C73C22×D411yescharacteristic
273C2×C4×C73C22×Q811yescharacteristic
274C2×C4×C73C2×#4111yescharacteristic
275C2×C4×C73H(5,𝔽2)11yescharacteristic
276C2×C4×C732−1+411yescharacteristic
277C8×C73—22no—
278C8×C73—21no—
279C8×C73—21no—
280C8×C73—21no—
281C8×C73—21no—
282C8×C73—41no—
283C8×C73—41no—
284C8×C73—41no—
285C8×C73—41no—
286C8×C73—41no—
287C22×C73—21no—
288C22×C73—21no—
289C22×C73—22no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73C2×#12111yescharacteristic
293C22×C73#55211yescharacteristic
294C22×C73#55411yescharacteristic
295C4×C73—22no—
296C4×C73—21no—
297C4×C73—21no—
298C4×C73—21no—
299C4×C73—21no—
300C4×C73—42no—
301C4×C73—41no—
302C4×C73—41no—
303C4×C73—81no—
304C4×C73—81no—
305C4×C73—81no—
306C4×C73C2×C22⋊Q811yescharacteristic
307C4×C73#53711yescharacteristic
308C4×C73#54611yescharacteristic
309C4×C73D4×Q811yescharacteristic
310#24860C7311yesmaximal, characteristic, direct product factor, Sylow
311C2×C73—21no—
312C2×C73—21no—
313C2×C73—42no—
314C2×C73#241711yescharacteristic
315C2×C73#306711yescharacteristic
316C2×C73#306911yescharacteristic
317#3080C2×C7311yescharacteristic
318C2×#479C2×C7311yescharacteristic, complemented
319C4×#137C2×C7311yescharacteristic
320#2906C2×C7311yescharacteristic
321#2090C2×C7311yescharacteristic, complemented
322#2089C2×C7311yescharacteristic, complemented
323#2064C2×C7311yescharacteristic, complemented
324#2064C2×C7311yescharacteristic, complemented
325#1998C2×C7311yescharacteristic, complemented
326#1950C2×C7311yescharacteristic, complemented
327#1908C2×C7311yescharacteristic
328#1908C2×C7311yescharacteristic
329#1816C2×C7311yescharacteristic
330#1815C2×C7311yescharacteristic
331#1801C2×C7311yescharacteristic, complemented
332C73#2486011yescharacteristic, direct product factor, Sylow
333C2×#137C22×C7311yescharacteristic
334C2×#131C22×C7311yescharacteristic
335C2×#131C22×C7311yescharacteristic
336C2×#125C22×C7311yescharacteristic
337C4×#41C22×C7311yescharacteristic
338C2×C4×Q8C22×C7311yescharacteristic
339#479—21no—
340#479—21no—
341#479—21no—
342#479—21no—
343#479—21no—
344#479—21no—
345#478—21no—
346#478—21no—
347#441—21no—
348#440—21no—
349C4×Q16—21no—
350C4×SD16—21no—
351C2×#108C22×C7311yescharacteristic
352C2×#108C22×C7311yescharacteristic
353#428C22×C7311yescharacteristic
354#428C22×C7311yescharacteristic
355#428C22×C7311yescharacteristic
356C2×#107C22×C7311yescharacteristic
357C2×#106C22×C7311yescharacteristic
358#424—21no—
359#424—21no—
360#424C22×C7311yescharacteristic
361#424C22×C7311yescharacteristic
362#423—21no—
363#423C22×C7311yescharacteristic
364#423C22×C7311yescharacteristic
365C2×C4⋊C8—21no—
366C2×C4⋊C8C22×C7311yescharacteristic
367#419C22×C7311yescharacteristic
368#417C22×C7311yescharacteristic
369#417C22×C7311yescharacteristic
370#417C22×C7311yescharacteristic
371C2×#103C22×C7311yescharacteristic
372C2×#103C22×C7311yescharacteristic
373C2×#101C22×C7311yescharacteristic
374C2×#101C22×C7311yescharacteristic
375C4×M16C22×C7311yescharacteristic
376#343C22×C7311yescharacteristic
377#343C22×C7311yescharacteristic
378#343C22×C7311yescharacteristic
379#343C22×C7311yescharacteristic
380#343C22×C7311yescharacteristic
381#343C22×C7311yescharacteristic
382#341C22×C7311yescharacteristic
383#341C22×C7311yescharacteristic
384C2×#41C23×C7311yescharacteristic
385C22×Q8C23×C7311yescharacteristic
386#137—22no—
387#137Q8×C7314yes—
388C2×Q16—21no—
389C2×SD16—21no—
390#131—22no—
391#131—22no—
392#131—41no—
393#131—41no—
394C2×M16—21no—
395C2×M16—21no—
396C2×M16C23×C7311yescharacteristic
397C2×M16C23×C7311yescharacteristic
398C2×M16C23×C7311yescharacteristic
399C2×M16C23×C7311yescharacteristic
400C2×M16C23×C7311yescharacteristic
401C2×M16C23×C7311yescharacteristic
402C22×C8—21no—
403C22×C8—21no—
404C22×C8C23×C7311yescharacteristic
405C22×C8C23×C7311yescharacteristic
406#125—21no—
407#125—21no—
408#125—21no—
409#125—21no—
410C4×Q8—21no—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8D4×C7311yescharacteristic
415C4×Q8D4×C7311yescharacteristic
416C4×D4—21no—
417C4×D4—21no—
418C4×D4D4×C7311yescharacteristic
419C4×D4D4×C7311yescharacteristic
420#117—21no—
421#117C23×C7311yescharacteristic
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4C23×C7311yescharacteristic
426C2×C4⋊C4C23×C7311yescharacteristic
427C2×C4⋊C4C23×C7311yescharacteristic
428C2×C42—21no—
429C2×C42—21no—
430C2×C42C23×C7311yescharacteristic
431#108—41no—
432#108—41no—
433#107—21no—
434#107—21no—
435#107—21no—
436#107—21no—
437#107—21no—
438#106—21no—
439#106—21no—
440#106—21no—
441#106—21no—
442#106—21no—
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8—21no—
446C4⋊C8—21no—
447C4⋊C8—21no—
448C4⋊C8—21no—
449C4⋊C8—21no—
450C4⋊C8—21no—
451C4⋊C8D4×C7311yescharacteristic
452C4⋊C8D4×C7311yescharacteristic
453C4⋊C8D4×C7311yescharacteristic
454C4⋊C8D4×C7311yescharacteristic
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#101—41no—
468#101—41no—
469#97—21no—
470C4×C8—21no—
471#41—21no—
472#41—21no—
473#41C2×Q8×C7312yes—
474#41#41×C7312yes—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8C2×D4×C7311yescharacteristic
481C2×Q8C2×Q8×C7312yes—
482C2×D4—21no—
483C2×D4C2×D4×C7311yescharacteristic
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4#41×C7311yescharacteristic
492C22×C4#41×C7311yescharacteristic
493C22×C4C24×C7311yescharacteristic
494Q16—24no—
495SD16—24no—
496M16—21no—
497M16—21no—
498M16—21no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—21no—
503M16—21no—
504M16—21no—
505M16—21no—
506M16—21no—
507M16—42no—
508M16—42no—
509M16C2×Q8×C7312yes—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—42no—
521C2×C8—42no—
522C2×C8C2×D4×C7311yescharacteristic
523C2×C8C2×D4×C7311yescharacteristic
524C2×C8C2×D4×C7311yescharacteristic
525C2×C8C2×D4×C7311yescharacteristic
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4C2×D4×C7311yescharacteristic
541C4⋊C4C2×D4×C7311yescharacteristic
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—41no—
549C42—41no—
550C42C2×D4×C7311yescharacteristic
551C42C2×D4×C7311yescharacteristic
552C42C2×D4×C7311yescharacteristic
553C42C2×D4×C7311yescharacteristic
554C23—21no—
555C23C2×#41×C7311yescharacteristic
556Q8—21no—
557Q8—22no—
558Q8—22no—
559Q8—41no—
560Q8—41no—
561Q8C22⋊Q8×C7312yes—
562D4—22no—
563D4—22no—
564D4—21no—
565D4C22⋊Q8×C7312yes—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4#121×C7311yescharacteristic
585C2×C4#121×C7311yescharacteristic
586C2×C4#121×C7311yescharacteristic
587C2×C4#121×C7311yescharacteristic
588C2×C4C22×D4×C7311yescharacteristic
589C2×C4C22×D4×C7311yescharacteristic
590C2×C4C22×Q8×C7311yescharacteristic
591C2×C4C2×#41×C7311yescharacteristic
592C2×C4H(5,𝔽2)×C7311yescharacteristic
593C2×C42−1+4×C7311yescharacteristic
594C8—22no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—41no—
600C8—41no—
601C8—41no—
602C8—41no—
603C8—41no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22C2×#121×C7311yescharacteristic
610C22#552×C7311yescharacteristic
611C22#554×C7311yescharacteristic
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4C2×C22⋊Q8×C7311yescharacteristic
624C4#537×C7311yescharacteristic
625C4#546×C7311yescharacteristic
626C4D4×Q8×C7311yescharacteristic
627C2—21no—
628C2—21no—
629C2—42no—
630C2C73×#241711yescharacteristic
631C2C73×#306711yescharacteristic
632C2C73×#306911yescharacteristic
subgroupG/H
center, socleC22×C73C2×#121
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC2×#41×C73C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#479×C73C2semidirect
5. C73×#2090C2semidirect
6. C73×#2089C2semidirect
7. C73×#2064C2semidirect
8. C73×#2064C2semidirect
9. C73×#1998C2semidirect
10. C73×#1950C2semidirect
15. C73×#1801C2semidirect
310. #24860C73direct (N × H)
318. C2×#479C2×C73semidirect
321. #2090C2×C73semidirect
322. #2089C2×C73semidirect
323. #2064C2×C73semidirect
324. #2064C2×C73semidirect
325. #1998C2×C73semidirect
326. #1950C2×C73semidirect
331. #1801C2×C73semidirect
332. C73#24860direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#30802
C2×#479×C732
C4×#137×C732
C73×#29062
C73×#20902
C73×#20892
C73×#20642
C73×#19982
C73×#19502
C73×#19082
C73×#18162
C73×#18152
C73×#18012
#2486073