Finite Group Databasebeta

← C73×#18817 #32928221

Subgroups — C73×#18817 #32928221

Order 18688.

884 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3503C211yesmaximal, characteristic
2C73×#3237C211yesmaximal, characteristic, complemented
3C73×#3209C211yesmaximal, characteristic, complemented
4C73×#3116C211yesmaximal, characteristic
5C73×#1645C212yesmaximal, complemented
6C73×#1643C212yesmaximal, complemented
7C73×#1633C212yesmaximal, complemented
8C73×#1632C212yesmaximal, complemented
9C73×#1558C211yesmaximal, characteristic, complemented
10C73×#1497C211yesmaximal, characteristic, complemented
11C73×#1488C211yesmaximal, characteristic
12Q82×C73C2212yes—
13#557×C73—21no—
14#554×C73—22no—
15#554×C73C2212yes—
16D4×Q8×C73—22no—
17D4×Q8×C73—21no—
18#544×C73—22no—
19#544×C73—21no—
20#533×C73—21no—
21#533×C73C2211yescharacteristic
22#533×C73C2211yescharacteristic
23C2×C4⋊Q8×C73—21no—
24C2×C4⋊Q8×C73C2211yescharacteristic
25#520×C73—21no—
26C4×#41×C73C2211yescharacteristic
27#501×C73—21no—
28#494×C73C2212yes—
29#491×C73—21no—
30#487×C73C2212yes—
31#485×C73—22no—
32#485×C73—21no—
33#477×C73C2212yes—
34#477×C73C2211yescharacteristic
35#476×C73C2211yescharacteristic
36#475×C73C2212yes—
37#462×C73—22no—
38#462×C73C2211yescharacteristic
39#461×C73C2211yescharacteristic
40#452×C73—21no—
41#451×C73—22no—
42#441×C73—21no—
43#440×C73—21no—
44#432×C73C2211yescharacteristic
45#432×C73C2211yescharacteristic
46#431×C73C2211yescharacteristic
47#333×C73C2212yes—
48#333×C73C2212yes—
49Q8⋊C8×C73C2212yes—
50Q8⋊C8×C73C2212yes—
51Q8⋊C8×C73C2212yes—
52D4⋊C8×C73C2212yes—
53C2×#41×C73D411yescharacteristic
54C22×Q8×C73—22no—
55C22×Q8×C73D412yes—
56C2×Q16×C73—21no—
57C2×Q16×C73—24no—
58C2×SD16×C73—21no—
59C4⋊Q8×C73—22no—
60C4⋊Q8×C73—22no—
61C4⋊Q8×C73—24no—
62C4⋊Q8×C73—22no—
63C4⋊Q8×C73—22no—
64C4⋊Q8×C73—21no—
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73D412yes—
67C4⋊Q8×C73D411yescharacteristic
68C4⋊Q8×C73D411yescharacteristic
69C4⋊Q8×C73C2311yescharacteristic
70C4⋊Q8×C73C2311yescharacteristic
71C4⋊Q8×C73C2312yes—
72#125×C73—22no—
73#125×C73—24no—
74#125×C73—22no—
75#125×C73—21no—
76#125×C73—21no—
77C22⋊Q8×C73—22no—
78C22⋊Q8×C73—24no—
79C22⋊Q8×C73—24no—
80C22⋊Q8×C73—22no—
81C22⋊Q8×C73—24no—
82C22⋊Q8×C73—21no—
83C22⋊Q8×C73—21no—
84C22⋊Q8×C73—21no—
85C22⋊Q8×C73D412yes—
86C4×Q8×C73—22no—
87C4×Q8×C73—22no—
88C4×Q8×C73—22no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73D411yescharacteristic
93C4×Q8×C73C2311yescharacteristic
94C4×Q8×C73C2312yes—
95C4×D4×C73—24no—
96C4×D4×C73—22no—
97C4×D4×C73—21no—
98C4×D4×C73—21no—
99C4×D4×C73D411yescharacteristic
100C4×D4×C73C2311yescharacteristic
101#117×C73—22no—
102#117×C73—21no—
103#117×C73—21no—
104#117×C73—21no—
105#117×C73D411yescharacteristic
106C2×C4⋊C4×C73—22no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73—22no—
110C2×C42×C73—21no—
111C2×C42×C73C2311yescharacteristic
112#107×C73—21no—
113#107×C73—22no—
114#106×C73—21no—
115#106×C73—22no—
116C4⋊C8×C73—21no—
117C4⋊C8×C73C2311yescharacteristic
118C4⋊C8×C73C2311yescharacteristic
119C4⋊C8×C73C2312yes—
120#103×C73—21no—
121#103×C73—24no—
122#103×C73—22no—
123#103×C73—22no—
124#103×C73—21no—
125#103×C73—21no—
126#102×C73—21no—
127C22⋊C8×C73—21no—
128C22⋊C8×C73—21no—
129C22⋊C8×C73—21no—
130#97×C73—21no—
131#97×C73—21no—
132C4×C8×C73C2312yes—
133#41×C73—22no—
134#41×C73—22no—
135C2×Q8×C73—24no—
136C2×Q8×C73—22no—
137C2×Q8×C73—21no—
138C2×Q8×C73—21no—
139C2×Q8×C73—21no—
140C2×Q8×C73—84no—
141C2×Q8×C73—82no—
142C2×Q8×C73C2×D412yes—
143C2×Q8×C73C2×D411yescharacteristic
144C2×Q8×C73C2×D414yes—
145C2×D4×C73—21no—
146C2×D4×C73C2×D411yescharacteristic
147C22×C4×C73—21no—
148C22×C4×C73—24no—
149C22×C4×C73—22no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155Q16×C73—42no—
156Q16×C73—84no—
157SD16×C73—42no—
158C2×C8×C73—22no—
159C2×C8×C73—21no—
160C2×C8×C73—22no—
161C2×C8×C73—21no—
162C4⋊C4×C73—24no—
163C4⋊C4×C73—24no—
164C4⋊C4×C73—24no—
165C4⋊C4×C73—24no—
166C4⋊C4×C73—22no—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—22no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—22no—
173C4⋊C4×C73—22no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—22no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—41no—
182C4⋊C4×C73—41no—
183C4⋊C4×C73C2×D411yescharacteristic
184C4⋊C4×C73C2×D411yescharacteristic
185C4⋊C4×C73C2×D412yes—
186C22⋊C4×C73—24no—
187C22⋊C4×C73—22no—
188C22⋊C4×C73—21no—
189C22⋊C4×C73—21no—
190C22⋊C4×C73—21no—
191C22⋊C4×C73—41no—
192C42×C73—24no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—22no—
197C42×C73—41no—
198C42×C73C2×D411yescharacteristic
199C42×C73C2×D411yescharacteristic
200C42×C73C2×D411yescharacteristic
201C42×C73C2411yescharacteristic
202C23×C73—21no—
203C23×C73#12011yescharacteristic
204Q8×C73—21no—
205Q8×C73—24no—
206Q8×C73—44no—
207Q8×C73—44no—
208Q8×C73—42no—
209Q8×C73—42no—
210Q8×C73—41no—
211Q8×C73—41no—
212Q8×C73—84no—
213Q8×C73#13712yes—
214D4×C73—24no—
215D4×C73—21no—
216D4×C73#13712yes—
217C2×C4×C73—24no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
227C2×C4×C73—41no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—82no—
233C2×C4×C73—84no—
234C2×C4×C73#12011yescharacteristic
235C2×C4×C73#12011yescharacteristic
236C2×C4×C73#12011yescharacteristic
237C2×C4×C73C22×D411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C22×D411yescharacteristic
240C8×C73—42no—
241C8×C73—41no—
242C8×C73—41no—
243C8×C73—42no—
244C22×C73—21no—
245C22×C73—22no—
246C22×C73—21no—
247C22×C73—21no—
248C22×C73—41no—
249C22×C73—41no—
250C22×C73C2×#12011yescharacteristic
251C4×C73—21no—
252C4×C73—22no—
253C4×C73—21no—
254C4×C73—21no—
255C4×C73—21no—
256C4×C73—41no—
257C4×C73—42no—
258C4×C73—41no—
259C4×C73—44no—
260C4×C73—42no—
261C4×C73—81no—
262C4×C73—81no—
263C4×C73C2×#13711yescharacteristic
264C4×C73C2×#13711yescharacteristic
265C4×C73#57812yes—
266#18817C7311yesmaximal, characteristic, direct product factor, Sylow
267C2×C73—41no—
268C2×C73—42no—
269C2×C73#300411yescharacteristic
270C2×C73#301111yescharacteristic
271C2×C73#301911yescharacteristic
272#3503C2×C7311yescharacteristic
273#3237C2×C7311yescharacteristic, complemented
274#3209C2×C7311yescharacteristic, complemented
275#3116C2×C7311yescharacteristic
276#1645C2×C7312yescomplemented
277#1643C2×C7312yescomplemented
278#1633C2×C7312yescomplemented
279#1632C2×C7312yescomplemented
280#1558C2×C7311yescharacteristic, complemented
281#1497C2×C7311yescharacteristic, complemented
282#1488C2×C7311yescharacteristic
283C73#1881711yescharacteristic, direct product factor, Sylow
284Q82C22×C7312yes—
285#557—21no—
286#554—22no—
287#554C22×C7312yes—
288D4×Q8—22no—
289D4×Q8—21no—
290#544—22no—
291#544—21no—
292#533—21no—
293#533C22×C7311yescharacteristic
294#533C22×C7311yescharacteristic
295C2×C4⋊Q8—21no—
296C2×C4⋊Q8C22×C7311yescharacteristic
297#520—21no—
298C4×#41C22×C7311yescharacteristic
299#501—21no—
300#494C22×C7312yes—
301#491—21no—
302#487C22×C7312yes—
303#485—22no—
304#485—21no—
305#477C22×C7311yescharacteristic
306#477C22×C7312yes—
307#476C22×C7311yescharacteristic
308#475C22×C7312yes—
309#462—22no—
310#462C22×C7311yescharacteristic
311#461C22×C7311yescharacteristic
312#452—21no—
313#451—22no—
314#441—21no—
315#440—21no—
316#432C22×C7311yescharacteristic
317#432C22×C7311yescharacteristic
318#431C22×C7311yescharacteristic
319#333C22×C7312yes—
320#333C22×C7312yes—
321Q8⋊C8C22×C7312yes—
322Q8⋊C8C22×C7312yes—
323Q8⋊C8C22×C7312yes—
324D4⋊C8C22×C7312yes—
325C2×#41D4×C7311yescharacteristic
326C22×Q8—22no—
327C22×Q8D4×C7312yes—
328C2×Q16—21no—
329C2×Q16—24no—
330C2×SD16—21no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—24no—
334C4⋊Q8—22no—
335C4⋊Q8—22no—
336C4⋊Q8—21no—
337C4⋊Q8—21no—
338C4⋊Q8D4×C7312yes—
339C4⋊Q8D4×C7311yescharacteristic
340C4⋊Q8D4×C7311yescharacteristic
341C4⋊Q8C23×C7311yescharacteristic
342C4⋊Q8C23×C7311yescharacteristic
343C4⋊Q8C23×C7312yes—
344#125—22no—
345#125—24no—
346#125—22no—
347#125—21no—
348#125—21no—
349C22⋊Q8—22no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8D4×C7312yes—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—22no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8D4×C7311yescharacteristic
365C4×Q8C23×C7311yescharacteristic
366C4×Q8C23×C7312yes—
367C4×D4—24no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4D4×C7311yescharacteristic
372C4×D4C23×C7311yescharacteristic
373#117—22no—
374#117—21no—
375#117—21no—
376#117—21no—
377#117D4×C7311yescharacteristic
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—22no—
382C2×C42—21no—
383C2×C42C23×C7311yescharacteristic
384#107—21no—
385#107—22no—
386#106—21no—
387#106—22no—
388C4⋊C8—21no—
389C4⋊C8C23×C7312yes—
390C4⋊C8C23×C7311yescharacteristic
391C4⋊C8C23×C7311yescharacteristic
392#103—21no—
393#103—24no—
394#103—22no—
395#103—22no—
396#103—21no—
397#103—21no—
398#102—21no—
399C22⋊C8—21no—
400C22⋊C8—21no—
401C22⋊C8—21no—
402#97—21no—
403#97—21no—
404C4×C8C23×C7312yes—
405#41—22no—
406#41—22no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—84no—
413C2×Q8—82no—
414C2×Q8C2×D4×C7312yes—
415C2×Q8C2×D4×C7311yescharacteristic
416C2×Q8C2×D4×C7314yes—
417C2×D4—21no—
418C2×D4C2×D4×C7311yescharacteristic
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4C2×D4×C7311yescharacteristic
425C22×C4C2×D4×C7311yescharacteristic
426C22×C4C2×D4×C7311yescharacteristic
427Q16—42no—
428Q16—84no—
429SD16—42no—
430C2×C8—22no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—21no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4C2×D4×C7311yescharacteristic
456C4⋊C4C2×D4×C7311yescharacteristic
457C4⋊C4C2×D4×C7312yes—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—41no—
464C42—24no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—22no—
469C42—41no—
470C42C2×D4×C7311yescharacteristic
471C42C2×D4×C7311yescharacteristic
472C42C2×D4×C7311yescharacteristic
473C42C24×C7311yescharacteristic
474C23—21no—
475C23#120×C7311yescharacteristic
476Q8—21no—
477Q8—24no—
478Q8—44no—
479Q8—44no—
480Q8—42no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8—84no—
485Q8#137×C7312yes—
486D4—24no—
487D4—21no—
488D4#137×C7312yes—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4#120×C7311yescharacteristic
507C2×C4#120×C7311yescharacteristic
508C2×C4#120×C7311yescharacteristic
509C2×C4C22×D4×C7311yescharacteristic
510C2×C4C22×D4×C7311yescharacteristic
511C2×C4C22×D4×C7311yescharacteristic
512C8—42no—
513C8—41no—
514C8—41no—
515C8—42no—
516C22—21no—
517C22—22no—
518C22—21no—
519C22—21no—
520C22—41no—
521C22—41no—
522C22C2×#120×C7311yescharacteristic
523C4—21no—
524C4—22no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—44no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4C2×#137×C7311yescharacteristic
536C4C2×#137×C7311yescharacteristic
537C4#578×C7312yes—
538C2—41no—
539C2—42no—
540C2C73×#300411yescharacteristic
541C2C73×#301111yescharacteristic
542C2C73×#301911yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC2×#41×C73D4
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
2. C73×#3237C2semidirect
3. C73×#3209C2semidirect
5. C73×#1645C2semidirect
6. C73×#1643C2semidirect
7. C73×#1633C2semidirect
8. C73×#1632C2semidirect
9. C73×#1558C2semidirect
10. C73×#1497C2semidirect
266. #18817C73direct (N × H)
273. #3237C2×C73semidirect
274. #3209C2×C73semidirect
276. #1645C2×C73semidirect
277. #1643C2×C73semidirect
278. #1633C2×C73semidirect
279. #1632C2×C73semidirect
280. #1558C2×C73semidirect
281. #1497C2×C73semidirect
283. C73#18817direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35032
C73×#32372
C73×#32092
C73×#31162
C73×#16452
C73×#16432
C73×#16332
C73×#16322
C73×#15582
C73×#14972
C73×#14882
#1881773