Finite Group Databasebeta

← C73×#19825 #32929229

Subgroups — C73×#19825 #32929229

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3394C211yesmaximal, characteristic, complemented
2C73×#3361C211yesmaximal, characteristic, complemented
3C73×#3324C211yesmaximal, characteristic, complemented
4C73×#3295C211yesmaximal, characteristic, complemented
5C73×#1717C211yesmaximal, characteristic, complemented
6C73×#1697C211yesmaximal, characteristic, complemented
7C73×#1682C211yesmaximal, characteristic, complemented
8C73×#1678C211yesmaximal, characteristic, complemented
9C73×#1665C211yesmaximal, characteristic, complemented
10C73×#1644C211yesmaximal, characteristic, complemented
11C73×#1638C211yesmaximal, characteristic, complemented
12C73×#1620C211yesmaximal, characteristic, complemented
13C73×#1619C211yesmaximal, characteristic
14C73×#1583C211yesmaximal, characteristic, complemented
15C73×SD16⋊C8C211yesmaximal, characteristic, complemented
16#559×C73C2211yescharacteristic, complemented
17#556×C73C2211yescharacteristic
18#554×C73C2211yescharacteristic
19D42×C73C2211yescharacteristic
20#495×C73—21no—
21#493×C73C2211yescharacteristic
22#492×C73C2211yescharacteristic, complemented
23#488×C73—21no—
24#487×C73C2211yescharacteristic, complemented
25#486×C73C2211yescharacteristic
26#486×C73C2211yescharacteristic
27#486×C73C2211yescharacteristic, complemented
28#482×C73—21no—
29#481×C73—21no—
30#478×C73—21no—
31#477×C73C2211yescharacteristic, complemented
32#476×C73C2211yescharacteristic
33#474×C73C2211yescharacteristic
34#468×C73—21no—
35#465×C73—21no—
36#464×C73—21no—
37#463×C73—21no—
38#463×C73—21no—
39#462×C73C2211yescharacteristic, complemented
40#460×C73C2211yescharacteristic
41#460×C73C2211yescharacteristic
42#459×C73C2211yescharacteristic
43#459×C73C2211yescharacteristic
44#450×C73—21no—
45#450×C73—21no—
46C8×Q8×C73C2211yescharacteristic, complemented
47C8×Q8×C73C2211yescharacteristic, complemented
48C4×SD16×C73—21no—
49C4×SD16×C73C2211yescharacteristic
50C4×SD16×C73C2211yescharacteristic
51C4×D8×C73—21no—
52#436×C73C2211yescharacteristic
53C8×D4×C73C2211yescharacteristic
54#418×C73—21no—
55C2×#102×C73—21no—
56#335×C73C2211yescharacteristic, complemented
57#334×C73C2211yescharacteristic, complemented
58#333×C73C2211yescharacteristic, complemented
59#332×C73C2211yescharacteristic, complemented
60#331×C73C2211yescharacteristic
61#331×C73C2211yescharacteristic, complemented
62Q8⋊C8×C73C2211yescharacteristic, complemented
63Q8⋊C8×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic, complemented
67C22×D4×C73—21no—
68C22×D4×C73—21no—
69C2×Q16×C73—21no—
70C2×SD16×C73—21no—
71C2×SD16×C73—21no—
72C2×SD16×C73—21no—
73C2×SD16×C73—21no—
74C2×SD16×C73D411yescharacteristic
75C2×SD16×C73D411yescharacteristic
76C2×D8×C73—21no—
77C2×D8×C73—21no—
78C2×D8×C73—41no—
79C2×M16×C73—21no—
80C22×C8×C73—21no—
81C4⋊Q8×C73C2311yescharacteristic
82C4⋊Q8×C73C2311yescharacteristic
83#127×C73C2311yescharacteristic
84#127×C73C2311yescharacteristic
85#126×C73—41no—
86#125×C73—21no—
87#124×C73—21no—
88#124×C73—21no—
89#124×C73—21no—
90#123×C73—41no—
91C22⋊Q8×C73—21no—
92C22⋊Q8×C73—21no—
93C22⋊Q8×C73—41no—
94#121×C73—21no—
95#121×C73—21no—
96#120×C73—41no—
97C4×Q8×C73—21no—
98C4×Q8×C73C2311yescharacteristic
99C4×Q8×C73C2311yescharacteristic
100C4×D4×C73—21no—
101C4×D4×C73—21no—
102C4×D4×C73—21no—
103C4×D4×C73C2311yescharacteristic
104C4×D4×C73C2311yescharacteristic
105#117×C73—21no—
106C2×C4⋊C4×C73—21no—
107#107×C73—21no—
108#107×C73—21no—
109#106×C73—21no—
110#106×C73D411yescharacteristic
111#106×C73D411yescharacteristic
112C4⋊C8×C73—21no—
113C4⋊C8×C73—21no—
114C4⋊C8×C73C2311yescharacteristic
115C4⋊C8×C73C2311yescharacteristic
116C4⋊C8×C73C2311yescharacteristic
117C4⋊C8×C73C2311yescharacteristic
118#103×C73—21no—
119#103×C73—21no—
120#103×C73—41no—
121#103×C73D411yescharacteristic
122#103×C73D411yescharacteristic
123#102×C73—21no—
124#102×C73—21no—
125#102×C73—21no—
126#102×C73—21no—
127#102×C73—21no—
128#102×C73—21no—
129#102×C73—21no—
130#102×C73—21no—
131#102×C73—21no—
132#102×C73—21no—
133#102×C73—21no—
134#102×C73—21no—
135#102×C73D411yescharacteristic
136#102×C73D411yescharacteristic
137C22⋊C8×C73—21no—
138C22⋊C8×C73—21no—
139C4×C8×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C24×C73—41no—
145C2×Q8×C73—21no—
146C2×Q8×C73—21no—
147C2×Q8×C73C2×D411yescharacteristic
148C2×Q8×C73C2×D411yescharacteristic
149C2×D4×C73—21no—
150C2×D4×C73—21no—
151C2×D4×C73—21no—
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—41no—
156C2×D4×C73—41no—
157C2×D4×C73—42no—
158C2×D4×C73—42no—
159C2×D4×C73—81no—
160C2×D4×C73—81no—
161C2×D4×C73C2×D411yescharacteristic
162C2×D4×C73C2×D411yescharacteristic
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—41no—
166C22×C4×C73—41no—
167Q16×C73—81no—
168SD16×C73—22no—
169SD16×C73—41no—
170SD16×C73—42no—
171SD16×C73—81no—
172SD16×C73—81no—
173SD16×C73—81no—
174D8×C73—42no—
175D8×C73—81no—
176D8×C73—82no—
177M16×C73—81no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C2×C8×C73—21no—
183C2×C8×C73—81no—
184C2×C8×C73C2×D411yescharacteristic
185C2×C8×C73C2×D411yescharacteristic
186C2×C8×C73C2×D411yescharacteristic
187C2×C8×C73C2×D411yescharacteristic
188C4⋊C4×C73—21no—
189C4⋊C4×C73—21no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—41no—
195C4⋊C4×C73—41no—
196C4⋊C4×C73—41no—
197C4⋊C4×C73C2×D411yescharacteristic
198C4⋊C4×C73C2×D411yescharacteristic
199C4⋊C4×C73C2×D411yescharacteristic
200C4⋊C4×C73C2×D411yescharacteristic
201C22⋊C4×C73—21no—
202C22⋊C4×C73—21no—
203C22⋊C4×C73—41no—
204C22⋊C4×C73—41no—
205C22⋊C4×C73—41no—
206C22⋊C4×C73—41no—
207C22⋊C4×C73—41no—
208C22⋊C4×C73—41no—
209C22⋊C4×C73—41no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—41no—
213C42×C73C2411yescharacteristic
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—82no—
219C23×C73—82no—
220C23×C73—161no—
221Q8×C73—21no—
222Q8×C73—21no—
223Q8×C73—21no—
224Q8×C73—21no—
225Q8×C73—41no—
226Q8×C73—41no—
227Q8×C73—81no—
228D4×C73—21no—
229D4×C73—21no—
230D4×C73—21no—
231D4×C73—41no—
232D4×C73—41no—
233D4×C73—41no—
234D4×C73—41no—
235D4×C73—41no—
236D4×C73—41no—
237D4×C73—41no—
238D4×C73—41no—
239D4×C73—81no—
240D4×C73—81no—
241D4×C73—82no—
242D4×C73—82no—
243D4×C73#13512yes—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—161no—
257C2×C4×C73—161no—
258C2×C4×C73C22×D411yescharacteristic
259C2×C4×C73C22×D411yescharacteristic
260C2×C4×C73H(5,𝔽2)11yescharacteristic
261C8×C73—21no—
262C8×C73—21no—
263C8×C73—21no—
264C8×C73—21no—
265C8×C73—41no—
266C8×C73—41no—
267C8×C73—41no—
268C8×C73—41no—
269C8×C73—41no—
270C22×C73—22no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—81no—
277C22×C73—81no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73—81no—
282C22×C73—164no—
283C22×C73D4211yescharacteristic
284C4×C73—21no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—81no—
292C4×C73—81no—
293C4×C73C2×#13511yescharacteristic
294C4×C73#57511yescharacteristic
295C4×C73#57611yescharacteristic
296C4×C73#57711yescharacteristic
297#19825C7311yesmaximal, characteristic, direct product factor, Sylow
298C2×C73—42no—
299C2×C73—81no—
300C2×C73—161no—
301C2×C73—161no—
302C2×C73#326811yescharacteristic
303C2×C73#328211yescharacteristic
304C2×C73#328811yescharacteristic
305#3394C2×C7311yescharacteristic, complemented
306#3361C2×C7311yescharacteristic, complemented
307#3324C2×C7311yescharacteristic, complemented
308#3295C2×C7311yescharacteristic, complemented
309#1717C2×C7311yescharacteristic, complemented
310#1697C2×C7311yescharacteristic, complemented
311#1682C2×C7311yescharacteristic, complemented
312#1678C2×C7311yescharacteristic, complemented
313#1665C2×C7311yescharacteristic, complemented
314#1644C2×C7311yescharacteristic, complemented
315#1638C2×C7311yescharacteristic, complemented
316#1620C2×C7311yescharacteristic, complemented
317#1619C2×C7311yescharacteristic
318#1583C2×C7311yescharacteristic, complemented
319SD16⋊C8C2×C7311yescharacteristic, complemented
320C73#1982511yescharacteristic, direct product factor, Sylow
321#559C22×C7311yescharacteristic, complemented
322#556C22×C7311yescharacteristic
323#554C22×C7311yescharacteristic
324D42C22×C7311yescharacteristic
325#495—21no—
326#493C22×C7311yescharacteristic
327#492C22×C7311yescharacteristic, complemented
328#488—21no—
329#487C22×C7311yescharacteristic, complemented
330#486C22×C7311yescharacteristic
331#486C22×C7311yescharacteristic
332#486C22×C7311yescharacteristic, complemented
333#482—21no—
334#481—21no—
335#478—21no—
336#477C22×C7311yescharacteristic, complemented
337#476C22×C7311yescharacteristic
338#474C22×C7311yescharacteristic
339#468—21no—
340#465—21no—
341#464—21no—
342#463—21no—
343#463—21no—
344#462C22×C7311yescharacteristic, complemented
345#460C22×C7311yescharacteristic
346#460C22×C7311yescharacteristic
347#459C22×C7311yescharacteristic
348#459C22×C7311yescharacteristic
349#450—21no—
350#450—21no—
351C8×Q8C22×C7311yescharacteristic, complemented
352C8×Q8C22×C7311yescharacteristic, complemented
353C4×SD16—21no—
354C4×SD16C22×C7311yescharacteristic
355C4×SD16C22×C7311yescharacteristic
356C4×D8—21no—
357#436C22×C7311yescharacteristic
358C8×D4C22×C7311yescharacteristic
359#418—21no—
360C2×#102—21no—
361#335C22×C7311yescharacteristic, complemented
362#334C22×C7311yescharacteristic, complemented
363#333C22×C7311yescharacteristic, complemented
364#332C22×C7311yescharacteristic, complemented
365#331C22×C7311yescharacteristic
366#331C22×C7311yescharacteristic, complemented
367Q8⋊C8C22×C7311yescharacteristic, complemented
368Q8⋊C8C22×C7311yescharacteristic, complemented
369D4⋊C8C22×C7311yescharacteristic
370D4⋊C8C22×C7311yescharacteristic
371#322C22×C7311yescharacteristic, complemented
372C22×D4—21no—
373C22×D4—21no—
374C2×Q16—21no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×SD16—21no—
378C2×SD16—21no—
379C2×SD16D4×C7311yescharacteristic
380C2×SD16D4×C7311yescharacteristic
381C2×D8—21no—
382C2×D8—21no—
383C2×D8—41no—
384C2×M16—21no—
385C22×C8—21no—
386C4⋊Q8C23×C7311yescharacteristic
387C4⋊Q8C23×C7311yescharacteristic
388#127C23×C7311yescharacteristic
389#127C23×C7311yescharacteristic
390#126—41no—
391#125—21no—
392#124—21no—
393#124—21no—
394#124—21no—
395#123—41no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398C22⋊Q8—41no—
399#121—21no—
400#121—21no—
401#120—41no—
402C4×Q8—21no—
403C4×Q8C23×C7311yescharacteristic
404C4×Q8C23×C7311yescharacteristic
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—21no—
408C4×D4C23×C7311yescharacteristic
409C4×D4C23×C7311yescharacteristic
410#117—21no—
411C2×C4⋊C4—21no—
412#107—21no—
413#107—21no—
414#106—21no—
415#106D4×C7311yescharacteristic
416#106D4×C7311yescharacteristic
417C4⋊C8—21no—
418C4⋊C8—21no—
419C4⋊C8C23×C7311yescharacteristic
420C4⋊C8C23×C7311yescharacteristic
421C4⋊C8C23×C7311yescharacteristic
422C4⋊C8C23×C7311yescharacteristic
423#103—21no—
424#103—21no—
425#103—41no—
426#103D4×C7311yescharacteristic
427#103D4×C7311yescharacteristic
428#102—21no—
429#102—21no—
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—21no—
435#102—21no—
436#102—21no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102D4×C7311yescharacteristic
441#102D4×C7311yescharacteristic
442C22⋊C8—21no—
443C22⋊C8—21no—
444C4×C8—21no—
445C4×C8—21no—
446C4×C8C23×C7311yescharacteristic
447C4×C8C23×C7311yescharacteristic
448C4×C8C23×C7311yescharacteristic
449C24—41no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×D4×C7311yescharacteristic
453C2×Q8C2×D4×C7311yescharacteristic
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—81no—
465C2×D4—81no—
466C2×D4C2×D4×C7311yescharacteristic
467C2×D4C2×D4×C7311yescharacteristic
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—41no—
471C22×C4—41no—
472Q16—81no—
473SD16—22no—
474SD16—41no—
475SD16—42no—
476SD16—81no—
477SD16—81no—
478SD16—81no—
479D8—42no—
480D8—81no—
481D8—82no—
482M16—81no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—81no—
489C2×C8C2×D4×C7311yescharacteristic
490C2×C8C2×D4×C7311yescharacteristic
491C2×C8C2×D4×C7311yescharacteristic
492C2×C8C2×D4×C7311yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4C2×D4×C7311yescharacteristic
503C4⋊C4C2×D4×C7311yescharacteristic
504C4⋊C4C2×D4×C7311yescharacteristic
505C4⋊C4C2×D4×C7311yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C42—21no—
516C42—21no—
517C42—41no—
518C42C24×C7311yescharacteristic
519C23—21no—
520C23—21no—
521C23—41no—
522C23—41no—
523C23—82no—
524C23—82no—
525C23—161no—
526Q8—21no—
527Q8—21no—
528Q8—21no—
529Q8—21no—
530Q8—41no—
531Q8—41no—
532Q8—81no—
533D4—21no—
534D4—21no—
535D4—21no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—81no—
545D4—81no—
546D4—82no—
547D4—82no—
548D4#135×C7312yes—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—161no—
562C2×C4—161no—
563C2×C4C22×D4×C7311yescharacteristic
564C2×C4C22×D4×C7311yescharacteristic
565C2×C4H(5,𝔽2)×C7311yescharacteristic
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—41no—
573C8—41no—
574C8—41no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—164no—
588C22D42×C7311yescharacteristic
589C4—21no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4C2×#135×C7311yescharacteristic
599C4#575×C7311yescharacteristic
600C4#576×C7311yescharacteristic
601C4#577×C7311yescharacteristic
602C2—42no—
603C2—81no—
604C2—161no—
605C2—161no—
606C2C73×#326811yescharacteristic
607C2C73×#328211yescharacteristic
608C2C73×#328811yescharacteristic
subgroupG/H
center, socleC22×C73D42
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#1619C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C73×#3394C2semidirect
2. C73×#3361C2semidirect
3. C73×#3324C2semidirect
4. C73×#3295C2semidirect
5. C73×#1717C2semidirect
6. C73×#1697C2semidirect
7. C73×#1682C2semidirect
8. C73×#1678C2semidirect
9. C73×#1665C2semidirect
10. C73×#1644C2semidirect
11. C73×#1638C2semidirect
12. C73×#1620C2semidirect
14. C73×#1583C2semidirect
15. C73×SD16⋊C8C2semidirect
16. #559×C73C22semidirect
22. #492×C73C22semidirect
24. #487×C73C22semidirect
27. #486×C73C22semidirect
31. #477×C73C22semidirect
39. #462×C73C22semidirect
46. C8×Q8×C73C22semidirect
47. C8×Q8×C73C22semidirect
56. #335×C73C22semidirect
57. #334×C73C22semidirect
58. #333×C73C22semidirect
59. #332×C73C22semidirect
61. #331×C73C22semidirect
62. Q8⋊C8×C73C22semidirect
63. Q8⋊C8×C73C22semidirect
66. #322×C73C22semidirect
297. #19825C73direct (N × H)
305. #3394C2×C73semidirect
306. #3361C2×C73semidirect
307. #3324C2×C73semidirect
308. #3295C2×C73semidirect
309. #1717C2×C73semidirect
310. #1697C2×C73semidirect
311. #1682C2×C73semidirect
312. #1678C2×C73semidirect
313. #1665C2×C73semidirect
314. #1644C2×C73semidirect
315. #1638C2×C73semidirect
316. #1620C2×C73semidirect
318. #1583C2×C73semidirect
319. SD16⋊C8C2×C73semidirect
320. C73#19825direct (N × H)
321. #559C22×C73semidirect
327. #492C22×C73semidirect
329. #487C22×C73semidirect
332. #486C22×C73semidirect
336. #477C22×C73semidirect
344. #462C22×C73semidirect
351. C8×Q8C22×C73semidirect
352. C8×Q8C22×C73semidirect
361. #335C22×C73semidirect
362. #334C22×C73semidirect
363. #333C22×C73semidirect
364. #332C22×C73semidirect
366. #331C22×C73semidirect
367. Q8⋊C8C22×C73semidirect
368. Q8⋊C8C22×C73semidirect
371. #322C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#33942
C73×#33612
C73×#33242
C73×#32952
C73×#17172
C73×#16972
C73×#16822
C73×#16782
C73×#16652
C73×#16442
C73×#16382
C73×#16202
C73×#16192
C73×#15832
C73×SD16⋊C82
#1982573