Finite Group Databasebeta

← C73×#18993 #32928397

Subgroups — C73×#18993 #32928397

Order 18688.

1012 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#562×C73C211yesmaximal, characteristic, complemented
2C73×#3234C211yesmaximal, characteristic, complemented
3C73×#3213C211yesmaximal, characteristic, complemented
4C73×#3153C211yesmaximal, characteristic, complemented
5C73×#1661C214yesmaximal, complemented
6C73×#1656C214yesmaximal, complemented
7C73×#1533C211yesmaximal, characteristic, complemented
8C73×#1528C211yesmaximal, characteristic, complemented
9C73×#1527C211yesmaximal, characteristic, complemented
10C2×#137×C73—21no—
11C2×AGL(1,ℤ/8)×C73—21no—
12#562×C73—22no—
13#562×C73—22no—
14#562×C73—22no—
15#562×C73C2214yes—
16#532×C73C2211yescharacteristic
17C2×#125×C73C2211yescharacteristic
18C2×#124×C73C2211yescharacteristic
19C2×#124×C73C2211yescharacteristic
20C2×#123×C73—21no—
21C2×#123×C73—21no—
22C2×C22⋊Q8×C73—21no—
23C2×C22⋊Q8×C73—21no—
24C2×#121×C73—21no—
25C2×#121×C73—21no—
26#501×C73C2212yescomplemented
27#497×C73C2212yescomplemented
28#496×C73C2212yescomplemented
29#496×C73C2212yescomplemented
30#495×C73—22no—
31#490×C73C2212yescomplemented
32#489×C73C2212yescomplemented
33#481×C73—21no—
34#480×C73—21no—
35#478×C73—22no—
36#463×C73—22no—
37#448×C73—21no—
38#447×C73—21no—
39#432×C73C2211yescharacteristic
40#432×C73C2211yescharacteristic
41C4×M16×C73C2211yescharacteristic
42#330×C73C2214yescomplemented
43#329×C73C2214yescomplemented
44#329×C73C2214yescomplemented
45C2×#41×C73—21no—
46C22×Q8×C73D411yescharacteristic
47C22×Q8×C73D411yescharacteristic
48C22×D4×C73—21no—
49C22×D4×C73—21no—
50C22×D4×C73D411yescharacteristic
51C22×D4×C73D411yescharacteristic
52C23×C4×C73—21no—
53C23×C4×C73—21no—
54#137×C73—44no—
55AGL(1,ℤ/8)×C73—44no—
56C2×Q16×C73—22no—
57C2×SD16×C73—22no—
58C2×SD16×C73—22no—
59C2×SD16×C73—22no—
60C2×D8×C73—22no—
61C2×D8×C73—22no—
62C2×M16×C73—21no—
63C4⋊Q8×C73C2311yescharacteristic
64#127×C73C2311yescharacteristic
65#125×C73—21no—
66#125×C73—21no—
67#125×C73D412yes—
68#125×C73C2312yes—
69#124×C73—21no—
70#124×C73—21no—
71#124×C73—21no—
72#124×C73—21no—
73#124×C73—21no—
74#124×C73D412yes—
75#124×C73D412yes—
76#124×C73C2312yes—
77#124×C73C2312yes—
78#123×C73—24no—
79#123×C73—24no—
80#123×C73—24no—
81#123×C73—24no—
82C22⋊Q8×C73—24no—
83C22⋊Q8×C73—24no—
84C22⋊Q8×C73—24no—
85C22⋊Q8×C73—24no—
86#121×C73—24no—
87#121×C73—24no—
88#121×C73—24no—
89#121×C73—24no—
90#121×C73—21no—
91#121×C73—21no—
92C4×D4×C73—21no—
93C4×D4×C73—21no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73D411yescharacteristic
99C2×C4⋊C4×C73D411yescharacteristic
100C2×C22⋊C4×C73—21no—
101C2×C22⋊C4×C73—21no—
102C2×C22⋊C4×C73—21no—
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C22⋊C4×C73—21no—
106C2×C42×C73C2311yescharacteristic
107#107×C73—22no—
108#106×C73—22no—
109C4⋊C8×C73—21no—
110C4⋊C8×C73—21no—
111#103×C73—22no—
112#102×C73—22no—
113#102×C73—22no—
114#102×C73—22no—
115C22⋊C8×C73—21no—
116C22⋊C8×C73—21no—
117#97×C73C2312yes—
118#97×C73C2312yes—
119#97×C73C2312yes—
120C4×C8×C73—21no—
121C24×C73—21no—
122C24×C73—21no—
123#41×C73—42no—
124#41×C73—42no—
125C2×Q8×C73—21no—
126C2×Q8×C73—21no—
127C2×Q8×C73—21no—
128C2×Q8×C73—42no—
129C2×Q8×C73—81no—
130C2×Q8×C73C2×D412yes—
131C2×Q8×C73C2×D412yes—
132C2×D4×C73—24no—
133C2×D4×C73—24no—
134C2×D4×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73—42no—
140C2×D4×C73—41no—
141C2×D4×C73—81no—
142C2×D4×C73—82no—
143C2×D4×C73—82no—
144C2×D4×C73C2×D412yes—
145C2×D4×C73C2×D412yes—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—24no—
151C22×C4×C73—24no—
152C22×C4×C73—41no—
153C22×C4×C73—82no—
154C22×C4×C73—82no—
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158Q16×C73—44no—
159SD16×C73—44no—
160SD16×C73—44no—
161SD16×C73—82no—
162D8×C73—44no—
163D8×C73—82no—
164M16×C73—24no—
165C2×C8×C73—22no—
166C2×C8×C73—22no—
167C2×C8×C73—22no—
168C4⋊C4×C73—24no—
169C4⋊C4×C73—24no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—22no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73C2×D412yes—
179C4⋊C4×C73C2×D412yes—
180C22⋊C4×C73—24no—
181C22⋊C4×C73—24no—
182C22⋊C4×C73—22no—
183C22⋊C4×C73—22no—
184C22⋊C4×C73—22no—
185C22⋊C4×C73—22no—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—22no—
188C22⋊C4×C73—22no—
189C22⋊C4×C73—22no—
190C22⋊C4×C73—41no—
191C22⋊C4×C73—41no—
192C42×C73C2×D411yescharacteristic
193C42×C73C2×D411yescharacteristic
194C42×C73C2×D411yescharacteristic
195C42×C73C2411yescharacteristic
196C23×C73—22no—
197C23×C73—22no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73—42no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—41no—
205C23×C73#12011yescharacteristic
206Q8×C73—42no—
207Q8×C73—42no—
208Q8×C73—41no—
209Q8×C73—41no—
210Q8×C73—81no—
211Q8×C73—81no—
212D4×C73—42no—
213D4×C73—42no—
214D4×C73—41no—
215D4×C73—42no—
216D4×C73—41no—
217D4×C73—42no—
218D4×C73—81no—
219D4×C73—81no—
220D4×C73—84no—
221D4×C73—84no—
222D4×C73—82no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—41no—
235C2×C4×C73—82no—
236C2×C4×C73—84no—
237C2×C4×C73—84no—
238C2×C4×C73#12011yescharacteristic
239C2×C4×C73#12011yescharacteristic
240C2×C4×C73#12011yescharacteristic
241C2×C4×C73C22×D411yescharacteristic
242C2×C4×C73C22×D411yescharacteristic
243C2×C4×C73C22×D411yescharacteristic
244C8×C73—24no—
245C8×C73—42no—
246C8×C73—42no—
247C22×C73—21no—
248C22×C73—21no—
249C22×C73—42no—
250C22×C73—42no—
251C22×C73—42no—
252C22×C73—42no—
253C22×C73—42no—
254C22×C73—42no—
255C22×C73—84no—
256C22×C73—84no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73#45312yes—
261C22×C73C2×#12011yescharacteristic
262C4×C73—21no—
263C4×C73—21no—
264C4×C73—21no—
265C4×C73—21no—
266C4×C73—21no—
267C4×C73—21no—
268C4×C73—42no—
269C4×C73—42no—
270C4×C73—42no—
271C4×C73—42no—
272C4×C73—81no—
273#18993C7311yesmaximal, characteristic, direct product factor, Sylow
274C2×C73—22no—
275C2×C73—82no—
276C2×C73—82no—
277C2×C73—161no—
278C2×C73C2×#45311yescharacteristic
279C2×C73#301911yescharacteristic
280C2×C73#302011yescharacteristic
281C2×#562C2×C7311yescharacteristic, complemented
282#3234C2×C7311yescharacteristic, complemented
283#3213C2×C7311yescharacteristic, complemented
284#3153C2×C7311yescharacteristic, complemented
285#1661C2×C7314yescomplemented
286#1656C2×C7314yescomplemented
287#1533C2×C7311yescharacteristic, complemented
288#1528C2×C7311yescharacteristic, complemented
289#1527C2×C7311yescharacteristic, complemented
290C73#1899311yescharacteristic, direct product factor, Sylow
291C2×#137—21no—
292C2×AGL(1,ℤ/8)—21no—
293#562—22no—
294#562—22no—
295#562—22no—
296#562C22×C7314yes—
297#532C22×C7311yescharacteristic
298C2×#125C22×C7311yescharacteristic
299C2×#124C22×C7311yescharacteristic
300C2×#124C22×C7311yescharacteristic
301C2×#123—21no—
302C2×#123—21no—
303C2×C22⋊Q8—21no—
304C2×C22⋊Q8—21no—
305C2×#121—21no—
306C2×#121—21no—
307#501C22×C7312yescomplemented
308#497C22×C7312yescomplemented
309#496C22×C7312yescomplemented
310#496C22×C7312yescomplemented
311#495—22no—
312#490C22×C7312yescomplemented
313#489C22×C7312yescomplemented
314#481—21no—
315#480—21no—
316#478—22no—
317#463—22no—
318#448—21no—
319#447—21no—
320#432C22×C7311yescharacteristic
321#432C22×C7311yescharacteristic
322C4×M16C22×C7311yescharacteristic
323#330C22×C7314yescomplemented
324#329C22×C7314yescomplemented
325#329C22×C7314yescomplemented
326C2×#41—21no—
327C22×Q8D4×C7311yescharacteristic
328C22×Q8D4×C7311yescharacteristic
329C22×D4—21no—
330C22×D4—21no—
331C22×D4D4×C7311yescharacteristic
332C22×D4D4×C7311yescharacteristic
333C23×C4—21no—
334C23×C4—21no—
335#137—44no—
336AGL(1,ℤ/8)—44no—
337C2×Q16—22no—
338C2×SD16—22no—
339C2×SD16—22no—
340C2×SD16—22no—
341C2×D8—22no—
342C2×D8—22no—
343C2×M16—21no—
344C4⋊Q8C23×C7311yescharacteristic
345#127C23×C7311yescharacteristic
346#125—21no—
347#125—21no—
348#125D4×C7312yes—
349#125C23×C7312yes—
350#124—21no—
351#124—21no—
352#124—21no—
353#124—21no—
354#124—21no—
355#124D4×C7312yes—
356#124D4×C7312yes—
357#124C23×C7312yes—
358#124C23×C7312yes—
359#123—24no—
360#123—24no—
361#123—24no—
362#123—24no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—24no—
366C22⋊Q8—24no—
367#121—24no—
368#121—24no—
369#121—24no—
370#121—24no—
371#121—21no—
372#121—21no—
373C4×D4—21no—
374C4×D4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4D4×C7311yescharacteristic
380C2×C4⋊C4D4×C7311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C42C23×C7311yescharacteristic
388#107—22no—
389#106—22no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392#103—22no—
393#102—22no—
394#102—22no—
395#102—22no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398#97C23×C7312yes—
399#97C23×C7312yes—
400#97C23×C7312yes—
401C4×C8—21no—
402C24—21no—
403C24—21no—
404#41—42no—
405#41—42no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8C2×D4×C7312yes—
412C2×Q8C2×D4×C7312yes—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—41no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×D4×C7312yes—
426C2×D4C2×D4×C7312yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—41no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4C2×D4×C7311yescharacteristic
437C22×C4C2×D4×C7311yescharacteristic
438C22×C4C2×D4×C7311yescharacteristic
439Q16—44no—
440SD16—44no—
441SD16—44no—
442SD16—82no—
443D8—44no—
444D8—82no—
445M16—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4C2×D4×C7312yes—
460C4⋊C4C2×D4×C7312yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C42C2×D4×C7311yescharacteristic
474C42C2×D4×C7311yescharacteristic
475C42C2×D4×C7311yescharacteristic
476C42C24×C7311yescharacteristic
477C23—22no—
478C23—22no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23#120×C7311yescharacteristic
487Q8—42no—
488Q8—42no—
489Q8—41no—
490Q8—41no—
491Q8—81no—
492Q8—81no—
493D4—42no—
494D4—42no—
495D4—41no—
496D4—42no—
497D4—41no—
498D4—42no—
499D4—81no—
500D4—81no—
501D4—84no—
502D4—84no—
503D4—82no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4#120×C7311yescharacteristic
520C2×C4#120×C7311yescharacteristic
521C2×C4#120×C7311yescharacteristic
522C2×C4C22×D4×C7311yescharacteristic
523C2×C4C22×D4×C7311yescharacteristic
524C2×C4C22×D4×C7311yescharacteristic
525C8—24no—
526C8—42no—
527C8—42no—
528C22—21no—
529C22—21no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—84no—
537C22—84no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22#453×C7312yes—
542C22C2×#120×C7311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—81no—
554C2—22no—
555C2—82no—
556C2—82no—
557C2—161no—
558C2C2×#453×C7311yescharacteristic
559C2C73×#301911yescharacteristic
560C2C73×#302011yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#562×C73C2semidirect
2. C73×#3234C2semidirect
3. C73×#3213C2semidirect
4. C73×#3153C2semidirect
5. C73×#1661C2semidirect
6. C73×#1656C2semidirect
7. C73×#1533C2semidirect
8. C73×#1528C2semidirect
9. C73×#1527C2semidirect
26. #501×C73C22semidirect
27. #497×C73C22semidirect
28. #496×C73C22semidirect
29. #496×C73C22semidirect
31. #490×C73C22semidirect
32. #489×C73C22semidirect
42. #330×C73C22semidirect
43. #329×C73C22semidirect
44. #329×C73C22semidirect
273. #18993C73direct (N × H)
281. C2×#562C2×C73semidirect
282. #3234C2×C73semidirect
283. #3213C2×C73semidirect
284. #3153C2×C73semidirect
285. #1661C2×C73semidirect
286. #1656C2×C73semidirect
287. #1533C2×C73semidirect
288. #1528C2×C73semidirect
289. #1527C2×C73semidirect
290. C73#18993direct (N × H)
307. #501C22×C73semidirect
308. #497C22×C73semidirect
309. #496C22×C73semidirect
310. #496C22×C73semidirect
312. #490C22×C73semidirect
313. #489C22×C73semidirect
323. #330C22×C73semidirect
324. #329C22×C73semidirect
325. #329C22×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#562×C732
C73×#32342
C73×#32132
C73×#31532
C73×#16612
C73×#16562
C73×#15332
C73×#15282
C73×#15272
#1899373