Finite Group Databasebeta

← C73×#18323 #32927727

Subgroups — C73×#18323 #32927727

Order 18688.

1008 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3459C211yesmaximal, characteristic
2C73×#3196C212yesmaximal, complemented
3C73×#2564C211yesmaximal, characteristic, complemented
4C73×#1615C212yesmaximal, complemented
5C73×#1614C211yesmaximal, characteristic, complemented
6C73×#1613C211yesmaximal, characteristic, complemented
7C73×#1603C212yesmaximal, complemented
8C73×#1602C211yesmaximal, characteristic, complemented
9C73×#1601C211yesmaximal, characteristic, complemented
10C73×#1504C212yesmaximal, complemented
11C73×#1465C211yesmaximal, characteristic, complemented
12#543×C73C2211yescharacteristic
13#539×C73—22no—
14#539×C73C2212yes—
15#538×C73—22no—
16#538×C73—21no—
17#534×C73—21no—
18#534×C73C2211yescharacteristic
19#529×C73—22no—
20#529×C73—21no—
21#525×C73—22no—
22#525×C73C2212yes—
23C2×C22⋊Q8×C73C2211yescharacteristic, complemented
24C2×C4×D4×C73—21no—
25C2×C4×D4×C73C2211yescharacteristic
26#485×C73—22no—
27#484×C73C2212yescomplemented
28#481×C73C2212yes—
29#470×C73—22no—
30#468×C73—22no—
31#452×C73—22no—
32#451×C73C2212yescomplemented
33#448×C73C2212yes—
34#407×C73C2212yescomplemented
35#400×C73C2211yescharacteristic, complemented
36#399×C73C2211yescharacteristic
37#395×C73C2211yescharacteristic, complemented
38#393×C73C2211yescharacteristic
39#387×C73—21no—
40#386×C73—21no—
41#385×C73—21no—
42#380×C73—21no—
43C2×#95×C73C2211yescharacteristic, complemented
44#328×C73C2212yescomplemented
45#328×C73C2212yescomplemented
46#327×C73C2212yes—
47#327×C73C2212yes—
48#324×C73C2212yescomplemented
49#323×C73C2211yescharacteristic
50#323×C73C2211yescharacteristic
51C2×#41×C73D412yes—
52C22×Q8×C73C2311yescharacteristic
53C22×D4×C73—21no—
54C22×D4×C73D411yescharacteristic
55C23×C4×C73—21no—
56C23×C4×C73—21no—
57C23×C4×C73D411yescharacteristic
58C23×C4×C73C2311yescharacteristic
59C2×Q16×C73—22no—
60C2×SD16×C73—22no—
61C2×M16×C73—22no—
62#126×C73—22no—
63#126×C73—22no—
64#126×C73—22no—
65#125×C73—21no—
66#125×C73—22no—
67#124×C73—22no—
68#124×C73—21no—
69#123×C73—22no—
70#123×C73—22no—
71#123×C73—22no—
72#123×C73—22no—
73#123×C73—22no—
74#123×C73—22no—
75C22⋊Q8×C73—22no—
76C22⋊Q8×C73—22no—
77C22⋊Q8×C73—21no—
78C22⋊Q8×C73—22no—
79C22⋊Q8×C73—22no—
80C22⋊Q8×C73—22no—
81C22⋊Q8×C73D412yes—
82C22⋊Q8×C73C2312yes—
83#121×C73—22no—
84#121×C73—22no—
85#121×C73—22no—
86#121×C73—21no—
87#121×C73D412yes—
88#121×C73C2312yes—
89#120×C73—22no—
90#120×C73—22no—
91#120×C73—22no—
92C4×Q8×C73—21no—
93C4×D4×C73—22no—
94C4×D4×C73—22no—
95C4×D4×C73—21no—
96C4×D4×C73—22no—
97C4×D4×C73—22no—
98C4×D4×C73—22no—
99C4×D4×C73—22no—
100C4×D4×C73—22no—
101#117×C73—22no—
102#117×C73D412yes—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73C2311yescharacteristic
116C2×C42×C73—21no—
117C2×C42×C73D411yescharacteristic
118#107×C73—22no—
119#106×C73—22no—
120#103×C73—22no—
121#103×C73—22no—
122#103×C73—22no—
123#102×C73—22no—
124C22⋊C8×C73—22no—
125C22⋊C8×C73C2312yes—
126C22⋊C8×C73C2312yes—
127#95×C73—21no—
128#95×C73—21no—
129#95×C73—21no—
130#95×C73—21no—
131#95×C73—21no—
132#95×C73C2311yescharacteristic
133#95×C73C2311yescharacteristic
134C24×C73—21no—
135C24×C73C2×D411yescharacteristic
136C24×C73C2×D411yescharacteristic
137#41×C73—82no—
138C2×Q8×C73—21no—
139C2×Q8×C73—22no—
140C2×Q8×C73—81no—
141C2×Q8×C73C2×D412yes—
142C2×D4×C73—22no—
143C2×D4×C73—22no—
144C2×D4×C73—22no—
145C2×D4×C73—22no—
146C2×D4×C73—21no—
147C2×D4×C73—22no—
148C2×D4×C73—22no—
149C2×D4×C73—41no—
150C2×D4×C73—42no—
151C2×D4×C73—42no—
152C2×D4×C73C2×D412yes—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73—21no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
163C22×C4×C73—21no—
164C22×C4×C73—42no—
165C22×C4×C73—42no—
166C22×C4×C73—81no—
167C22×C4×C73—81no—
168C22×C4×C73—82no—
169C22×C4×C73—81no—
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2411yescharacteristic
177Q16×C73—82no—
178SD16×C73—82no—
179M16×C73—82no—
180C2×C8×C73—22no—
181C2×C8×C73—22no—
182C4⋊C4×C73—22no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73—22no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—22no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—22no—
190C4⋊C4×C73—22no—
191C4⋊C4×C73—22no—
192C4⋊C4×C73—42no—
193C4⋊C4×C73—41no—
194C4⋊C4×C73—41no—
195C4⋊C4×C73—81no—
196C4⋊C4×C73C2×D412yes—
197C4⋊C4×C73C2×D412yes—
198C22⋊C4×C73—22no—
199C22⋊C4×C73—22no—
200C22⋊C4×C73—22no—
201C22⋊C4×C73—22no—
202C22⋊C4×C73—22no—
203C22⋊C4×C73—22no—
204C22⋊C4×C73—22no—
205C22⋊C4×C73—22no—
206C22⋊C4×C73—22no—
207C22⋊C4×C73—22no—
208C22⋊C4×C73—22no—
209C22⋊C4×C73—22no—
210C22⋊C4×C73—42no—
211C22⋊C4×C73—42no—
212C22⋊C4×C73—81no—
213C42×C73—22no—
214C42×C73—22no—
215C42×C73—21no—
216C42×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—21no—
221C23×C73—22no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73—42no—
227C23×C73—81no—
228C23×C73—81no—
229C23×C73#12012yes—
230C23×C73#13712yes—
231C23×C73C22×D411yescharacteristic
232Q8×C73—42no—
233Q8×C73—42no—
234Q8×C73—42no—
235Q8×C73—81no—
236D4×C73—42no—
237D4×C73—44no—
238D4×C73—44no—
239D4×C73—42no—
240D4×C73—42no—
241D4×C73—82no—
242D4×C73—82no—
243D4×C73—81no—
244C2×C4×C73—22no—
245C2×C4×C73—21no—
246C2×C4×C73—22no—
247C2×C4×C73—21no—
248C2×C4×C73—22no—
249C2×C4×C73—21no—
250C2×C4×C73—22no—
251C2×C4×C73—21no—
252C2×C4×C73—21no—
253C2×C4×C73—42no—
254C2×C4×C73—41no—
255C2×C4×C73—42no—
256C2×C4×C73—41no—
257C2×C4×C73—42no—
258C2×C4×C73—44no—
259C2×C4×C73—41no—
260C2×C4×C73—42no—
261C2×C4×C73—41no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—41no—
268C2×C4×C73—41no—
269C2×C4×C73—41no—
270C2×C4×C73—41no—
271C2×C4×C73—41no—
272C2×C4×C73—41no—
273C2×C4×C73—81no—
274C2×C4×C73—81no—
275C2×C4×C73—82no—
276C2×C4×C73—81no—
277C2×C4×C73—84no—
278C2×C4×C73#12011yescharacteristic
279C2×C4×C73#12011yescharacteristic
280C2×C4×C73C22×D412yes—
281C8×C73—42no—
282C8×C73—42no—
283C22×C73—21no—
284C22×C73—21no—
285C22×C73—22no—
286C22×C73—22no—
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—42no—
297C22×C73—81no—
298C22×C73—84no—
299C22×C73C2×#12011yescharacteristic
300C22×C73C2×#13711yescharacteristic
301C22×C73#57511yescharacteristic
302C4×C73—22no—
303C4×C73—21no—
304C4×C73—21no—
305C4×C73—42no—
306C4×C73—41no—
307C4×C73—42no—
308C4×C73—41no—
309C4×C73—42no—
310C4×C73—41no—
311C4×C73—42no—
312C4×C73—81no—
313C4×C73—81no—
314C4×C73—81no—
315#18323C7311yesmaximal, characteristic, direct product factor, Sylow
316C2×C73—21no—
317C2×C73—21no—
318C2×C73—42no—
319C2×C73—42no—
320C2×C73—82no—
321C2×C73#300711yescharacteristic
322C2×C73#301811yescharacteristic
323C2×C73#302611yescharacteristic
324#3459C2×C7311yescharacteristic
325#3196C2×C7312yescomplemented
326#2564C2×C7311yescharacteristic, complemented
327#1615C2×C7312yescomplemented
328#1614C2×C7311yescharacteristic, complemented
329#1613C2×C7311yescharacteristic, complemented
330#1603C2×C7312yescomplemented
331#1602C2×C7311yescharacteristic, complemented
332#1601C2×C7311yescharacteristic, complemented
333#1504C2×C7312yescomplemented
334#1465C2×C7311yescharacteristic, complemented
335C73#1832311yescharacteristic, direct product factor, Sylow
336#543C22×C7311yescharacteristic
337#539—22no—
338#539C22×C7312yes—
339#538—22no—
340#538—21no—
341#534—21no—
342#534C22×C7311yescharacteristic
343#529—22no—
344#529—21no—
345#525—22no—
346#525C22×C7312yes—
347C2×C22⋊Q8C22×C7311yescharacteristic, complemented
348C2×C4×D4—21no—
349C2×C4×D4C22×C7311yescharacteristic
350#485—22no—
351#484C22×C7312yescomplemented
352#481C22×C7312yes—
353#470—22no—
354#468—22no—
355#452—22no—
356#451C22×C7312yescomplemented
357#448C22×C7312yes—
358#407C22×C7312yescomplemented
359#400C22×C7311yescharacteristic, complemented
360#399C22×C7311yescharacteristic
361#395C22×C7311yescharacteristic, complemented
362#393C22×C7311yescharacteristic
363#387—21no—
364#386—21no—
365#385—21no—
366#380—21no—
367C2×#95C22×C7311yescharacteristic, complemented
368#328C22×C7312yescomplemented
369#328C22×C7312yescomplemented
370#327C22×C7312yes—
371#327C22×C7312yes—
372#324C22×C7312yescomplemented
373#323C22×C7311yescharacteristic
374#323C22×C7311yescharacteristic
375C2×#41D4×C7312yes—
376C22×Q8C23×C7311yescharacteristic
377C22×D4—21no—
378C22×D4D4×C7311yescharacteristic
379C23×C4—21no—
380C23×C4—21no—
381C23×C4D4×C7311yescharacteristic
382C23×C4C23×C7311yescharacteristic
383C2×Q16—22no—
384C2×SD16—22no—
385C2×M16—22no—
386#126—22no—
387#126—22no—
388#126—22no—
389#125—21no—
390#125—22no—
391#124—22no—
392#124—21no—
393#123—22no—
394#123—22no—
395#123—22no—
396#123—22no—
397#123—22no—
398#123—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8—21no—
402C22⋊Q8—22no—
403C22⋊Q8—22no—
404C22⋊Q8—22no—
405C22⋊Q8D4×C7312yes—
406C22⋊Q8C23×C7312yes—
407#121—22no—
408#121—22no—
409#121—22no—
410#121—21no—
411#121D4×C7312yes—
412#121C23×C7312yes—
413#120—22no—
414#120—22no—
415#120—22no—
416C4×Q8—21no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—21no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—22no—
423C4×D4—22no—
424C4×D4—22no—
425#117—22no—
426#117D4×C7312yes—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4C23×C7311yescharacteristic
432C2×C4⋊C4C23×C7311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4D4×C7311yescharacteristic
439C2×C22⋊C4C23×C7311yescharacteristic
440C2×C42—21no—
441C2×C42D4×C7311yescharacteristic
442#107—22no—
443#106—22no—
444#103—22no—
445#103—22no—
446#103—22no—
447#102—22no—
448C22⋊C8—22no—
449C22⋊C8C23×C7312yes—
450C22⋊C8C23×C7312yes—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95C23×C7311yescharacteristic
457#95C23×C7311yescharacteristic
458C24—21no—
459C24C2×D4×C7311yescharacteristic
460C24C2×D4×C7311yescharacteristic
461#41—82no—
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—81no—
465C2×Q8C2×D4×C7312yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—41no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×D4×C7312yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—82no—
493C22×C4—81no—
494C22×C4C2×D4×C7311yescharacteristic
495C22×C4C2×D4×C7311yescharacteristic
496C22×C4C2×D4×C7311yescharacteristic
497C22×C4C2×D4×C7311yescharacteristic
498C22×C4C2×D4×C7312yes—
499C22×C4C2×D4×C7312yes—
500C22×C4C24×C7311yescharacteristic
501Q16—82no—
502SD16—82no—
503M16—82no—
504C2×C8—22no—
505C2×C8—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—81no—
520C4⋊C4C2×D4×C7312yes—
521C4⋊C4C2×D4×C7312yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—81no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23#120×C7312yes—
554C23#137×C7312yes—
555C23C22×D4×C7311yescharacteristic
556Q8—42no—
557Q8—42no—
558Q8—42no—
559Q8—81no—
560D4—42no—
561D4—44no—
562D4—44no—
563D4—42no—
564D4—42no—
565D4—82no—
566D4—82no—
567D4—81no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—84no—
602C2×C4#120×C7311yescharacteristic
603C2×C4#120×C7311yescharacteristic
604C2×C4C22×D4×C7312yes—
605C8—42no—
606C8—42no—
607C22—21no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—84no—
623C22C2×#120×C7311yescharacteristic
624C22C2×#137×C7311yescharacteristic
625C22#575×C7311yescharacteristic
626C4—22no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—81no—
639C2—21no—
640C2—21no—
641C2—42no—
642C2—42no—
643C2—82no—
644C2C73×#300711yescharacteristic
645C2C73×#301811yescharacteristic
646C2C73×#302611yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC73×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C73×#3196C2semidirect
3. C73×#2564C2semidirect
4. C73×#1615C2semidirect
5. C73×#1614C2semidirect
6. C73×#1613C2semidirect
7. C73×#1603C2semidirect
8. C73×#1602C2semidirect
9. C73×#1601C2semidirect
10. C73×#1504C2semidirect
11. C73×#1465C2semidirect
23. C2×C22⋊Q8×C73C22semidirect
27. #484×C73C22semidirect
32. #451×C73C22semidirect
34. #407×C73C22semidirect
35. #400×C73C22semidirect
37. #395×C73C22semidirect
43. C2×#95×C73C22semidirect
44. #328×C73C22semidirect
45. #328×C73C22semidirect
48. #324×C73C22semidirect
315. #18323C73direct (N × H)
325. #3196C2×C73semidirect
326. #2564C2×C73semidirect
327. #1615C2×C73semidirect
328. #1614C2×C73semidirect
329. #1613C2×C73semidirect
330. #1603C2×C73semidirect
331. #1602C2×C73semidirect
332. #1601C2×C73semidirect
333. #1504C2×C73semidirect
334. #1465C2×C73semidirect
335. C73#18323direct (N × H)
347. C2×C22⋊Q8C22×C73semidirect
351. #484C22×C73semidirect
356. #451C22×C73semidirect
358. #407C22×C73semidirect
359. #400C22×C73semidirect
361. #395C22×C73semidirect
367. C2×#95C22×C73semidirect
368. #328C22×C73semidirect
369. #328C22×C73semidirect
372. #324C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#34592
C73×#31962
C73×#25642
C73×#16152
C73×#16142
C73×#16132
C73×#16032
C73×#16022
C73×#16012
C73×#15042
C73×#14652
#1832373