Finite Group Databasebeta

← C73×#20154 #32929558

Subgroups — C73×#20154 #32929558

Order 18688.

628 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×#3399C211yesmaximal, characteristic, complemented
2C73×#3396C211yesmaximal, characteristic, complemented
3C73×#3361C211yesmaximal, characteristic, complemented
4C73×#3333C211yesmaximal, characteristic, complemented
5C73×#1721C211yesmaximal, characteristic, complemented
6C73×#1698C211yesmaximal, characteristic, complemented
7C73×#1687C211yesmaximal, characteristic, complemented
8C73×#1678C211yesmaximal, characteristic, complemented
9C73×#1668C211yesmaximal, characteristic, complemented
10C73×#1630C211yesmaximal, characteristic, complemented
11C73×#1622C211yesmaximal, characteristic, complemented
12C73×#1621C211yesmaximal, characteristic, complemented
13C73×#1620C211yesmaximal, characteristic, complemented
14C73×#1588C211yesmaximal, characteristic, complemented
15C73×#1587C211yesmaximal, characteristic, complemented
16#559×C73C2211yescharacteristic
17#559×C73C2211yescharacteristic
18#554×C73C2211yescharacteristic
19#550×C73C2211yescharacteristic
20#496×C73—21no—
21#495×C73—21no—
22#493×C73C2211yescharacteristic
23#493×C73C2211yescharacteristic
24#493×C73C2211yescharacteristic
25#492×C73C2211yescharacteristic
26#488×C73—21no—
27#486×C73C2211yescharacteristic
28#486×C73C2211yescharacteristic
29#480×C73—21no—
30#478×C73—21no—
31#477×C73C2211yescharacteristic
32#477×C73C2211yescharacteristic
33#468×C73—21no—
34#463×C73—21no—
35#463×C73—21no—
36#463×C73—21no—
37#463×C73—21no—
38#460×C73C2211yescharacteristic
39#460×C73C2211yescharacteristic
40#460×C73C2211yescharacteristic
41#460×C73C2211yescharacteristic
42#459×C73—21no—
43#459×C73C2211yescharacteristic
44#459×C73C2211yescharacteristic
45#452×C73—21no—
46#446×C73C2211yescharacteristic
47C8×Q8×C73C2211yescharacteristic
48C8×Q8×C73C2211yescharacteristic
49#442×C73—21no—
50C4×Q16×C73C2211yescharacteristic
51C4×SD16×C73C2211yescharacteristic
52C4×D8×C73—21no—
53C4×D8×C73—21no—
54#436×C73C2211yescharacteristic
55#417×C73—21no—
56#335×C73C2211yescharacteristic
57#334×C73C2211yescharacteristic
58#333×C73C2211yescharacteristic
59#331×C73C2211yescharacteristic
60#331×C73C2211yescharacteristic
61#331×C73C2211yescharacteristic
62Q8⋊C8×C73C2211yescharacteristic
63Q8⋊C8×C73C2211yescharacteristic
64Q8⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#41×C73—21no—
68C2×Q16×C73D411yescharacteristic
69C2×SD16×C73—21no—
70C2×SD16×C73—21no—
71C2×SD16×C73—21no—
72C2×SD16×C73—21no—
73C2×SD16×C73—41no—
74C2×SD16×C73D411yescharacteristic
75C2×D8×C73—21no—
76C2×D8×C73—21no—
77C2×D8×C73—21no—
78C2×D8×C73—21no—
79C2×D8×C73—21no—
80C2×D8×C73—21no—
81C2×D8×C73—41no—
82C2×M16×C73—21no—
83C4⋊Q8×C73C2311yescharacteristic
84#127×C73—21no—
85#127×C73C2311yescharacteristic
86#127×C73C2311yescharacteristic
87#127×C73C2311yescharacteristic
88#125×C73—21no—
89#124×C73—21no—
90#124×C73—21no—
91#124×C73—21no—
92#124×C73—21no—
93#123×C73—41no—
94#123×C73—41no—
95C22⋊Q8×C73—21no—
96C22⋊Q8×C73—41no—
97#121×C73—21no—
98#121×C73—41no—
99C4×Q8×C73C2311yescharacteristic
100C4×Q8×C73C2311yescharacteristic
101C4×Q8×C73C2311yescharacteristic
102C4×D4×C73—21no—
103C4×D4×C73—21no—
104C4×D4×C73—21no—
105C4×D4×C73—21no—
106C4×D4×C73—21no—
107C4×D4×C73—21no—
108C4×D4×C73C2311yescharacteristic
109C2×C4⋊C4×C73—21no—
110#107×C73—21no—
111#107×C73—21no—
112#107×C73D411yescharacteristic
113#106×C73D411yescharacteristic
114C4⋊C8×C73—21no—
115C4⋊C8×C73—21no—
116C4⋊C8×C73—21no—
117C4⋊C8×C73C2311yescharacteristic
118C4⋊C8×C73C2311yescharacteristic
119C4⋊C8×C73C2311yescharacteristic
120C4⋊C8×C73C2311yescharacteristic
121#103×C73—21no—
122#103×C73—21no—
123#103×C73D411yescharacteristic
124#103×C73D411yescharacteristic
125#103×C73D411yescharacteristic
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×C73—21no—
136#102×C73D411yescharacteristic
137C22⋊C8×C73—21no—
138#97×C73—21no—
139C4×C8×C73—21no—
140C4×C8×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144#41×C73—42no—
145#41×C73—81no—
146C2×Q8×C73—21no—
147C2×Q8×C73C2×D411yescharacteristic
148C2×Q8×C73C2×D411yescharacteristic
149C2×Q8×C73C2×D411yescharacteristic
150C2×D4×C73—21no—
151C2×D4×C73—21no—
152C2×D4×C73—21no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73—21no—
159C2×D4×C73—41no—
160C2×D4×C73—41no—
161C2×D4×C73C2×D411yescharacteristic
162C22×C4×C73—21no—
163C22×C4×C73—41no—
164C22×C4×C73—41no—
165C22×C4×C73—41no—
166C22×C4×C73—41no—
167Q16×C73—41no—
168SD16×C73—41no—
169SD16×C73—82no—
170SD16×C73—81no—
171SD16×C73—81no—
172SD16×C73—81no—
173SD16×C73—81no—
174D8×C73—42no—
175D8×C73—42no—
176D8×C73—42no—
177D8×C73—81no—
178D8×C73—81no—
179D8×C73—82no—
180D8×C73—81no—
181M16×C73—81no—
182C2×C8×C73—21no—
183C2×C8×C73—21no—
184C2×C8×C73—21no—
185C2×C8×C73—21no—
186C2×C8×C73—21no—
187C2×C8×C73C2×D411yescharacteristic
188C2×C8×C73C2×D411yescharacteristic
189C2×C8×C73C2×D411yescharacteristic
190C2×C8×C73C2×D411yescharacteristic
191C4⋊C4×C73—21no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—41no—
197C4⋊C4×C73—41no—
198C4⋊C4×C73C2×D411yescharacteristic
199C4⋊C4×C73C2×D411yescharacteristic
200C4⋊C4×C73C2×D411yescharacteristic
201C4⋊C4×C73C2×D411yescharacteristic
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—
210C22⋊C4×C73—41no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
214C42×C73C2411yescharacteristic
215C23×C73—21no—
216C23×C73—41no—
217C23×C73—41no—
218C23×C73—41no—
219Q8×C73—21no—
220Q8×C73—21no—
221Q8×C73—21no—
222Q8×C73—21no—
223Q8×C73—21no—
224Q8×C73—21no—
225Q8×C73—81no—
226D4×C73—21no—
227D4×C73—21no—
228D4×C73—41no—
229D4×C73—41no—
230D4×C73—41no—
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—41no—
240D4×C73—81no—
241D4×C73—81no—
242D4×C73—81no—
243D4×C73—82no—
244D4×C73—161no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—21no—
251C2×C4×C73—21no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—82no—
256C2×C4×C73—161no—
257C2×C4×C73—161no—
258C2×C4×C73—161no—
259C2×C4×C73C22×D411yescharacteristic
260C2×C4×C73C22×D411yescharacteristic
261C2×C4×C73H(5,𝔽2)11yescharacteristic
262C8×C73—22no—
263C8×C73—21no—
264C8×C73—21no—
265C8×C73—21no—
266C8×C73—21no—
267C8×C73—41no—
268C8×C73—41no—
269C8×C73—41no—
270C8×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73—81no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73—81no—
282C22×C73—81no—
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—41no—
292C4×C73—81no—
293C4×C73#57511yescharacteristic
294C4×C73#57511yescharacteristic
295C4×C73#57611yescharacteristic
296C4×C73#57711yescharacteristic
297#20154C7311yesmaximal, characteristic, direct product factor, Sylow
298C2×C73—81no—
299C2×C73—161no—
300C2×C73—161no—
301C2×C73—161no—
302C2×C73#326811yescharacteristic
303C2×C73#327111yescharacteristic
304C2×C73#329211yescharacteristic
305#3399C2×C7311yescharacteristic, complemented
306#3396C2×C7311yescharacteristic, complemented
307#3361C2×C7311yescharacteristic, complemented
308#3333C2×C7311yescharacteristic, complemented
309#1721C2×C7311yescharacteristic, complemented
310#1698C2×C7311yescharacteristic, complemented
311#1687C2×C7311yescharacteristic, complemented
312#1678C2×C7311yescharacteristic, complemented
313#1668C2×C7311yescharacteristic, complemented
314#1630C2×C7311yescharacteristic, complemented
315#1622C2×C7311yescharacteristic, complemented
316#1621C2×C7311yescharacteristic, complemented
317#1620C2×C7311yescharacteristic, complemented
318#1588C2×C7311yescharacteristic, complemented
319#1587C2×C7311yescharacteristic, complemented
320C73#2015411yescharacteristic, direct product factor, Sylow
321#559C22×C7311yescharacteristic
322#559C22×C7311yescharacteristic
323#554C22×C7311yescharacteristic
324#550C22×C7311yescharacteristic
325#496—21no—
326#495—21no—
327#493C22×C7311yescharacteristic
328#493C22×C7311yescharacteristic
329#493C22×C7311yescharacteristic
330#492C22×C7311yescharacteristic
331#488—21no—
332#486C22×C7311yescharacteristic
333#486C22×C7311yescharacteristic
334#480—21no—
335#478—21no—
336#477C22×C7311yescharacteristic
337#477C22×C7311yescharacteristic
338#468—21no—
339#463—21no—
340#463—21no—
341#463—21no—
342#463—21no—
343#460C22×C7311yescharacteristic
344#460C22×C7311yescharacteristic
345#460C22×C7311yescharacteristic
346#460C22×C7311yescharacteristic
347#459—21no—
348#459C22×C7311yescharacteristic
349#459C22×C7311yescharacteristic
350#452—21no—
351#446C22×C7311yescharacteristic
352C8×Q8C22×C7311yescharacteristic
353C8×Q8C22×C7311yescharacteristic
354#442—21no—
355C4×Q16C22×C7311yescharacteristic
356C4×SD16C22×C7311yescharacteristic
357C4×D8—21no—
358C4×D8—21no—
359#436C22×C7311yescharacteristic
360#417—21no—
361#335C22×C7311yescharacteristic
362#334C22×C7311yescharacteristic
363#333C22×C7311yescharacteristic
364#331C22×C7311yescharacteristic
365#331C22×C7311yescharacteristic
366#331C22×C7311yescharacteristic
367Q8⋊C8C22×C7311yescharacteristic
368Q8⋊C8C22×C7311yescharacteristic
369Q8⋊C8C22×C7311yescharacteristic
370D4⋊C8C22×C7311yescharacteristic
371#322C22×C7311yescharacteristic
372C2×#41—21no—
373C2×Q16D4×C7311yescharacteristic
374C2×SD16—21no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×SD16—21no—
378C2×SD16—41no—
379C2×SD16D4×C7311yescharacteristic
380C2×D8—21no—
381C2×D8—21no—
382C2×D8—21no—
383C2×D8—21no—
384C2×D8—21no—
385C2×D8—21no—
386C2×D8—41no—
387C2×M16—21no—
388C4⋊Q8C23×C7311yescharacteristic
389#127—21no—
390#127C23×C7311yescharacteristic
391#127C23×C7311yescharacteristic
392#127C23×C7311yescharacteristic
393#125—21no—
394#124—21no—
395#124—21no—
396#124—21no—
397#124—21no—
398#123—41no—
399#123—41no—
400C22⋊Q8—21no—
401C22⋊Q8—41no—
402#121—21no—
403#121—41no—
404C4×Q8C23×C7311yescharacteristic
405C4×Q8C23×C7311yescharacteristic
406C4×Q8C23×C7311yescharacteristic
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4C23×C7311yescharacteristic
414C2×C4⋊C4—21no—
415#107—21no—
416#107—21no—
417#107D4×C7311yescharacteristic
418#106D4×C7311yescharacteristic
419C4⋊C8—21no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8C23×C7311yescharacteristic
423C4⋊C8C23×C7311yescharacteristic
424C4⋊C8C23×C7311yescharacteristic
425C4⋊C8C23×C7311yescharacteristic
426#103—21no—
427#103—21no—
428#103D4×C7311yescharacteristic
429#103D4×C7311yescharacteristic
430#103D4×C7311yescharacteristic
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#102—21no—
441#102D4×C7311yescharacteristic
442C22⋊C8—21no—
443#97—21no—
444C4×C8—21no—
445C4×C8—21no—
446C4×C8C23×C7311yescharacteristic
447C4×C8C23×C7311yescharacteristic
448C4×C8C23×C7311yescharacteristic
449#41—42no—
450#41—81no—
451C2×Q8—21no—
452C2×Q8C2×D4×C7311yescharacteristic
453C2×Q8C2×D4×C7311yescharacteristic
454C2×Q8C2×D4×C7311yescharacteristic
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—41no—
465C2×D4—41no—
466C2×D4C2×D4×C7311yescharacteristic
467C22×C4—21no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472Q16—41no—
473SD16—41no—
474SD16—82no—
475SD16—81no—
476SD16—81no—
477SD16—81no—
478SD16—81no—
479D8—42no—
480D8—42no—
481D8—42no—
482D8—81no—
483D8—81no—
484D8—82no—
485D8—81no—
486M16—81no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8C2×D4×C7311yescharacteristic
493C2×C8C2×D4×C7311yescharacteristic
494C2×C8C2×D4×C7311yescharacteristic
495C2×C8C2×D4×C7311yescharacteristic
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4C2×D4×C7311yescharacteristic
504C4⋊C4C2×D4×C7311yescharacteristic
505C4⋊C4C2×D4×C7311yescharacteristic
506C4⋊C4C2×D4×C7311yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42C24×C7311yescharacteristic
520C23—21no—
521C23—41no—
522C23—41no—
523C23—41no—
524Q8—21no—
525Q8—21no—
526Q8—21no—
527Q8—21no—
528Q8—21no—
529Q8—21no—
530Q8—81no—
531D4—21no—
532D4—21no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—81no—
546D4—81no—
547D4—81no—
548D4—82no—
549D4—161no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—82no—
561C2×C4—161no—
562C2×C4—161no—
563C2×C4—161no—
564C2×C4C22×D4×C7311yescharacteristic
565C2×C4C22×D4×C7311yescharacteristic
566C2×C4H(5,𝔽2)×C7311yescharacteristic
567C8—22no—
568C8—21no—
569C8—21no—
570C8—21no—
571C8—21no—
572C8—41no—
573C8—41no—
574C8—41no—
575C8—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22D42×C7311yescharacteristic
589C4—21no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—81no—
598C4#575×C7311yescharacteristic
599C4#575×C7311yescharacteristic
600C4#576×C7311yescharacteristic
601C4#577×C7311yescharacteristic
602C2—81no—
603C2—161no—
604C2—161no—
605C2—161no—
606C2C73×#326811yescharacteristic
607C2C73×#327111yescharacteristic
608C2C73×#329211yescharacteristic
subgroupG/H
center, socleC22×C73D42
commutator, Frattini, cosocleC42C24×C73
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×#3399C2semidirect
2. C73×#3396C2semidirect
3. C73×#3361C2semidirect
4. C73×#3333C2semidirect
5. C73×#1721C2semidirect
6. C73×#1698C2semidirect
7. C73×#1687C2semidirect
8. C73×#1678C2semidirect
9. C73×#1668C2semidirect
10. C73×#1630C2semidirect
11. C73×#1622C2semidirect
12. C73×#1621C2semidirect
13. C73×#1620C2semidirect
14. C73×#1588C2semidirect
15. C73×#1587C2semidirect
297. #20154C73direct (N × H)
305. #3399C2×C73semidirect
306. #3396C2×C73semidirect
307. #3361C2×C73semidirect
308. #3333C2×C73semidirect
309. #1721C2×C73semidirect
310. #1698C2×C73semidirect
311. #1687C2×C73semidirect
312. #1678C2×C73semidirect
313. #1668C2×C73semidirect
314. #1630C2×C73semidirect
315. #1622C2×C73semidirect
316. #1621C2×C73semidirect
317. #1620C2×C73semidirect
318. #1588C2×C73semidirect
319. #1587C2×C73semidirect
320. C73#20154direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#33992
C73×#33962
C73×#33612
C73×#33332
C73×#17212
C73×#16982
C73×#16872
C73×#16782
C73×#16682
C73×#16302
C73×#16222
C73×#16212
C73×#16202
C73×#15882
C73×#15872
#2015473