Finite Group Databasebeta

← C73×#20175 #32929579

Subgroups — C73×#20175 #32929579

Order 18688.

642 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×#3384C211yesmaximal, characteristic, complemented
2C73×#3329C211yesmaximal, characteristic, complemented
3C73×#3325C211yesmaximal, characteristic
4C73×#3295C211yesmaximal, characteristic, complemented
5C73×#1720C211yesmaximal, characteristic, complemented
6C73×#1698C211yesmaximal, characteristic, complemented
7C73×#1697C211yesmaximal, characteristic, complemented
8C73×#1691C211yesmaximal, characteristic, complemented
9C73×#1683C211yesmaximal, characteristic, complemented
10C73×#1639C211yesmaximal, characteristic, complemented
11C73×#1634C211yesmaximal, characteristic, complemented
12C73×#1632C211yesmaximal, characteristic, complemented
13C73×#1627C211yesmaximal, characteristic, complemented
14C73×#1592C211yesmaximal, characteristic, complemented
15C73×#1581C211yesmaximal, characteristic, complemented
16#555×C73C2211yescharacteristic
17#554×C73C2211yescharacteristic
18D4×Q8×C73C2211yescharacteristic, complemented
19D42×C73C2211yescharacteristic
20#501×C73—21no—
21#498×C73C2211yescharacteristic, complemented
22#494×C73C2211yescharacteristic, complemented
23#487×C73C2211yescharacteristic, complemented
24#486×C73C2211yescharacteristic
25#486×C73C2211yescharacteristic
26#486×C73C2211yescharacteristic
27#485×C73—21no—
28#482×C73—21no—
29#481×C73—21no—
30#478×C73—21no—
31#477×C73C2211yescharacteristic, complemented
32#476×C73—21no—
33#476×C73C2211yescharacteristic
34#476×C73C2211yescharacteristic
35#476×C73C2211yescharacteristic, complemented
36#474×C73C2211yescharacteristic
37#469×C73—21no—
38#468×C73—21no—
39#465×C73—21no—
40#461×C73C2211yescharacteristic
41#460×C73C2211yescharacteristic
42#459×C73C2211yescharacteristic
43#450×C73—21no—
44#448×C73—21no—
45#447×C73—21no—
46#446×C73C2211yescharacteristic, complemented
47#442×C73—21no—
48C4×SD16×C73C2211yescharacteristic
49C4×D8×C73C2211yescharacteristic
50#436×C73C2211yescharacteristic
51#436×C73C2211yescharacteristic, complemented
52C8×D4×C73C2211yescharacteristic
53#419×C73—21no—
54#418×C73—21no—
55C2×#102×C73—21no—
56#335×C73C2211yescharacteristic, complemented
57#335×C73C2211yescharacteristic, complemented
58#333×C73C2211yescharacteristic, complemented
59#333×C73C2211yescharacteristic, complemented
60#332×C73C2211yescharacteristic, complemented
61#331×C73C2211yescharacteristic
62Q8⋊C8×C73C2211yescharacteristic, complemented
63D4⋊C8×C73C2211yescharacteristic
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66#322×C73C2211yescharacteristic, complemented
67C22×Q8×C73—21no—
68C22×D4×C73—21no—
69C22×D4×C73—21no—
70C2×Q16×C73—41no—
71C2×SD16×C73—21no—
72C2×SD16×C73—21no—
73C2×SD16×C73D411yescharacteristic
74C2×D8×C73—21no—
75C2×D8×C73D411yescharacteristic
76C2×M16×C73—21no—
77C2×M16×C73—21no—
78C22×C8×C73—21no—
79C4⋊Q8×C73—21no—
80C4⋊Q8×C73C2311yescharacteristic
81C4⋊Q8×C73C2311yescharacteristic
82C4⋊Q8×C73C2311yescharacteristic
83#127×C73C2311yescharacteristic
84#125×C73—21no—
85#124×C73—41no—
86C22⋊Q8×C73—21no—
87C22⋊Q8×C73—21no—
88C22⋊Q8×C73—41no—
89C22⋊Q8×C73—41no—
90#121×C73—21no—
91#121×C73—21no—
92#121×C73—21no—
93#121×C73—21no—
94#120×C73—41no—
95C4×Q8×C73C2311yescharacteristic
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73C2311yescharacteristic
99C4×D4×C73C2311yescharacteristic
100C4×D4×C73C2311yescharacteristic
101#117×C73—21no—
102#117×C73—21no—
103C2×C4⋊C4×C73—21no—
104#107×C73—21no—
105#107×C73—21no—
106#107×C73D411yescharacteristic
107#106×C73—21no—
108#106×C73—21no—
109#106×C73—21no—
110#106×C73—21no—
111#106×C73—21no—
112#106×C73D411yescharacteristic
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—21no—
121#103×C73—21no—
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—41no—
134#102×C73D411yescharacteristic
135#102×C73D411yescharacteristic
136#102×C73D411yescharacteristic
137C22⋊C8×C73—21no—
138C22⋊C8×C73—21no—
139C22⋊C8×C73—21no—
140#97×C73—21no—
141C4×C8×C73C2311yescharacteristic
142C4×C8×C73C2311yescharacteristic
143C4×C8×C73C2311yescharacteristic
144C24×C73—41no—
145C2×Q8×C73—21no—
146C2×Q8×C73—21no—
147C2×Q8×C73—21no—
148C2×Q8×C73—41no—
149C2×Q8×C73—42no—
150C2×Q8×C73—81no—
151C2×Q8×C73C2×D411yescharacteristic
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—41no—
160C2×D4×C73—81no—
161C2×D4×C73—81no—
162C2×D4×C73C2×D411yescharacteristic
163C2×D4×C73C2×D411yescharacteristic
164C2×D4×C73C2×D411yescharacteristic
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—41no—
169C22×C4×C73—41no—
170Q16×C73—82no—
171SD16×C73—41no—
172SD16×C73—81no—
173SD16×C73—81no—
174D8×C73—22no—
175D8×C73—81no—
176M16×C73—81no—
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—21no—
195C4⋊C4×C73—41no—
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—21no—
204C22⋊C4×C73—21no—
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—41no—
212C42×C73C2411yescharacteristic
213C23×C73—21no—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—41no—
217C23×C73—82no—
218C23×C73—82no—
219C23×C73—161no—
220Q8×C73—21no—
221Q8×C73—21no—
222Q8×C73—41no—
223Q8×C73—41no—
224Q8×C73—81no—
225Q8×C73—81no—
226Q8×C73—82no—
227D4×C73—21no—
228D4×C73—21no—
229D4×C73—21no—
230D4×C73—21no—
231D4×C73—21no—
232D4×C73—41no—
233D4×C73—41no—
234D4×C73—41no—
235D4×C73—41no—
236D4×C73—81no—
237D4×C73—82no—
238D4×C73—82no—
239D4×C73—161no—
240D4×C73#13712yes—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—82no—
256C2×C4×C73—161no—
257C2×C4×C73C22×D411yescharacteristic
258C2×C4×C73C22×D411yescharacteristic
259C2×C4×C73H(5,𝔽2)11yescharacteristic
260C8×C73—21no—
261C8×C73—21no—
262C8×C73—21no—
263C8×C73—21no—
264C8×C73—41no—
265C8×C73—41no—
266C8×C73—41no—
267C8×C73—41no—
268C8×C73—41no—
269C22×C73—22no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—81no—
279C22×C73—81no—
280C22×C73—81no—
281C22×C73—164no—
282C22×C73D4211yescharacteristic
283C4×C73—21no—
284C4×C73—41no—
285C4×C73—41no—
286C4×C73—41no—
287C4×C73—41no—
288C4×C73—41no—
289C4×C73—81no—
290C4×C73—81no—
291C4×C73—81no—
292C4×C73C2×#13711yescharacteristic
293C4×C73#57511yescharacteristic
294C4×C73#57611yescharacteristic
295C4×C73#57711yescharacteristic
296#20175C7311yesmaximal, characteristic, direct product factor, Sylow
297C2×C73—42no—
298C2×C73—81no—
299C2×C73—81no—
300C2×C73—161no—
301C2×C73#326611yescharacteristic
302C2×C73#327311yescharacteristic
303C2×C73#328811yescharacteristic
304#3384C2×C7311yescharacteristic, complemented
305#3329C2×C7311yescharacteristic, complemented
306#3325C2×C7311yescharacteristic
307#3295C2×C7311yescharacteristic, complemented
308#1720C2×C7311yescharacteristic, complemented
309#1698C2×C7311yescharacteristic, complemented
310#1697C2×C7311yescharacteristic, complemented
311#1691C2×C7311yescharacteristic, complemented
312#1683C2×C7311yescharacteristic, complemented
313#1639C2×C7311yescharacteristic, complemented
314#1634C2×C7311yescharacteristic, complemented
315#1632C2×C7311yescharacteristic, complemented
316#1627C2×C7311yescharacteristic, complemented
317#1592C2×C7311yescharacteristic, complemented
318#1581C2×C7311yescharacteristic, complemented
319C73#2017511yescharacteristic, direct product factor, Sylow
320#555C22×C7311yescharacteristic
321#554C22×C7311yescharacteristic
322D4×Q8C22×C7311yescharacteristic, complemented
323D42C22×C7311yescharacteristic
324#501—21no—
325#498C22×C7311yescharacteristic, complemented
326#494C22×C7311yescharacteristic, complemented
327#487C22×C7311yescharacteristic, complemented
328#486C22×C7311yescharacteristic
329#486C22×C7311yescharacteristic
330#486C22×C7311yescharacteristic
331#485—21no—
332#482—21no—
333#481—21no—
334#478—21no—
335#477C22×C7311yescharacteristic, complemented
336#476—21no—
337#476C22×C7311yescharacteristic
338#476C22×C7311yescharacteristic
339#476C22×C7311yescharacteristic, complemented
340#474C22×C7311yescharacteristic
341#469—21no—
342#468—21no—
343#465—21no—
344#461C22×C7311yescharacteristic
345#460C22×C7311yescharacteristic
346#459C22×C7311yescharacteristic
347#450—21no—
348#448—21no—
349#447—21no—
350#446C22×C7311yescharacteristic, complemented
351#442—21no—
352C4×SD16C22×C7311yescharacteristic
353C4×D8C22×C7311yescharacteristic
354#436C22×C7311yescharacteristic
355#436C22×C7311yescharacteristic, complemented
356C8×D4C22×C7311yescharacteristic
357#419—21no—
358#418—21no—
359C2×#102—21no—
360#335C22×C7311yescharacteristic, complemented
361#335C22×C7311yescharacteristic, complemented
362#333C22×C7311yescharacteristic, complemented
363#333C22×C7311yescharacteristic, complemented
364#332C22×C7311yescharacteristic, complemented
365#331C22×C7311yescharacteristic
366Q8⋊C8C22×C7311yescharacteristic, complemented
367D4⋊C8C22×C7311yescharacteristic
368D4⋊C8C22×C7311yescharacteristic
369D4⋊C8C22×C7311yescharacteristic, complemented
370#322C22×C7311yescharacteristic, complemented
371C22×Q8—21no—
372C22×D4—21no—
373C22×D4—21no—
374C2×Q16—41no—
375C2×SD16—21no—
376C2×SD16—21no—
377C2×SD16D4×C7311yescharacteristic
378C2×D8—21no—
379C2×D8D4×C7311yescharacteristic
380C2×M16—21no—
381C2×M16—21no—
382C22×C8—21no—
383C4⋊Q8—21no—
384C4⋊Q8C23×C7311yescharacteristic
385C4⋊Q8C23×C7311yescharacteristic
386C4⋊Q8C23×C7311yescharacteristic
387#127C23×C7311yescharacteristic
388#125—21no—
389#124—41no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—41no—
393C22⋊Q8—41no—
394#121—21no—
395#121—21no—
396#121—21no—
397#121—21no—
398#120—41no—
399C4×Q8C23×C7311yescharacteristic
400C4×D4—21no—
401C4×D4—21no—
402C4×D4C23×C7311yescharacteristic
403C4×D4C23×C7311yescharacteristic
404C4×D4C23×C7311yescharacteristic
405#117—21no—
406#117—21no—
407C2×C4⋊C4—21no—
408#107—21no—
409#107—21no—
410#107D4×C7311yescharacteristic
411#106—21no—
412#106—21no—
413#106—21no—
414#106—21no—
415#106—21no—
416#106D4×C7311yescharacteristic
417C4⋊C8—21no—
418C4⋊C8C23×C7311yescharacteristic
419C4⋊C8C23×C7311yescharacteristic
420C4⋊C8C23×C7311yescharacteristic
421C4⋊C8C23×C7311yescharacteristic
422#103—21no—
423#103—21no—
424#103—21no—
425#103—21no—
426#103D4×C7311yescharacteristic
427#102—21no—
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—41no—
438#102D4×C7311yescharacteristic
439#102D4×C7311yescharacteristic
440#102D4×C7311yescharacteristic
441C22⋊C8—21no—
442C22⋊C8—21no—
443C22⋊C8—21no—
444#97—21no—
445C4×C8C23×C7311yescharacteristic
446C4×C8C23×C7311yescharacteristic
447C4×C8C23×C7311yescharacteristic
448C24—41no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—41no—
453C2×Q8—42no—
454C2×Q8—81no—
455C2×Q8C2×D4×C7311yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—41no—
464C2×D4—81no—
465C2×D4—81no—
466C2×D4C2×D4×C7311yescharacteristic
467C2×D4C2×D4×C7311yescharacteristic
468C2×D4C2×D4×C7311yescharacteristic
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4—41no—
474Q16—82no—
475SD16—41no—
476SD16—81no—
477SD16—81no—
478D8—22no—
479D8—81no—
480M16—81no—
481M16—81no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—81no—
488C2×C8C2×D4×C7311yescharacteristic
489C2×C8C2×D4×C7311yescharacteristic
490C2×C8C2×D4×C7311yescharacteristic
491C2×C8C2×D4×C7311yescharacteristic
492C4⋊C4—21no—
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—21no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C42—21no—
515C42—41no—
516C42C24×C7311yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—41no—
521C23—82no—
522C23—82no—
523C23—161no—
524Q8—21no—
525Q8—21no—
526Q8—41no—
527Q8—41no—
528Q8—81no—
529Q8—81no—
530Q8—82no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—21no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—41no—
540D4—81no—
541D4—82no—
542D4—82no—
543D4—161no—
544D4#137×C7312yes—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—161no—
561C2×C4C22×D4×C7311yescharacteristic
562C2×C4C22×D4×C7311yescharacteristic
563C2×C4H(5,𝔽2)×C7311yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—41no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—164no—
586C22D42×C7311yescharacteristic
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4C2×#137×C7311yescharacteristic
597C4#575×C7311yescharacteristic
598C4#576×C7311yescharacteristic
599C4#577×C7311yescharacteristic
600C2—42no—
601C2—81no—
602C2—81no—
603C2—161no—
604C2C73×#326611yescharacteristic
605C2C73×#327311yescharacteristic
606C2C73×#328811yescharacteristic
subgroupG/H
center, socleC22×C73D42
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3325C2
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×#3384C2semidirect
2. C73×#3329C2semidirect
4. C73×#3295C2semidirect
5. C73×#1720C2semidirect
6. C73×#1698C2semidirect
7. C73×#1697C2semidirect
8. C73×#1691C2semidirect
9. C73×#1683C2semidirect
10. C73×#1639C2semidirect
11. C73×#1634C2semidirect
12. C73×#1632C2semidirect
13. C73×#1627C2semidirect
14. C73×#1592C2semidirect
15. C73×#1581C2semidirect
18. D4×Q8×C73C22semidirect
21. #498×C73C22semidirect
22. #494×C73C22semidirect
23. #487×C73C22semidirect
31. #477×C73C22semidirect
35. #476×C73C22semidirect
46. #446×C73C22semidirect
51. #436×C73C22semidirect
56. #335×C73C22semidirect
57. #335×C73C22semidirect
58. #333×C73C22semidirect
59. #333×C73C22semidirect
60. #332×C73C22semidirect
62. Q8⋊C8×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. #322×C73C22semidirect
296. #20175C73direct (N × H)
304. #3384C2×C73semidirect
305. #3329C2×C73semidirect
307. #3295C2×C73semidirect
308. #1720C2×C73semidirect
309. #1698C2×C73semidirect
310. #1697C2×C73semidirect
311. #1691C2×C73semidirect
312. #1683C2×C73semidirect
313. #1639C2×C73semidirect
314. #1634C2×C73semidirect
315. #1632C2×C73semidirect
316. #1627C2×C73semidirect
317. #1592C2×C73semidirect
318. #1581C2×C73semidirect
319. C73#20175direct (N × H)
322. D4×Q8C22×C73semidirect
325. #498C22×C73semidirect
326. #494C22×C73semidirect
327. #487C22×C73semidirect
335. #477C22×C73semidirect
339. #476C22×C73semidirect
350. #446C22×C73semidirect
355. #436C22×C73semidirect
360. #335C22×C73semidirect
361. #335C22×C73semidirect
362. #333C22×C73semidirect
363. #333C22×C73semidirect
364. #332C22×C73semidirect
366. Q8⋊C8C22×C73semidirect
369. D4⋊C8C22×C73semidirect
370. #322C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#33842
C73×#33292
C73×#33252
C73×#32952
C73×#17202
C73×#16982
C73×#16972
C73×#16912
C73×#16832
C73×#16392
C73×#16342
C73×#16322
C73×#16272
C73×#15922
C73×#15812
#2017573