Finite Group Databasebeta

← C73×#18635 #32928039

Subgroups — C73×#18635 #32928039

Order 18688.

1004 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3508C211yesmaximal, characteristic
2C73×#3209C211yesmaximal, characteristic, complemented
3C2×#486×C73C211yesmaximal, characteristic, complemented
4C73×#3079C211yesmaximal, characteristic, complemented
5C73×#1641C214yesmaximal, complemented
6C73×#1640C214yesmaximal, complemented
7C73×#1545C211yesmaximal, characteristic, complemented
8C73×#1493C211yesmaximal, characteristic, complemented
9C73×#1476C211yesmaximal, characteristic, complemented
10#559×C73—21no—
11#559×C73C2212yes—
12#550×C73—21no—
13#550×C73C2212yes—
14#546×C73—22no—
15#546×C73—22no—
16#540×C73—22no—
17#535×C73—21no—
18#533×C73C2211yescharacteristic
19#532×C73—21no—
20#532×C73—21no—
21C2×C4⋊Q8×C73C2211yescharacteristic
22C2×#127×C73C2211yescharacteristic
23#518×C73—21no—
24C4×#41×C73C2211yescharacteristic
25#494×C73—21no—
26#487×C73—21no—
27#486×C73—22no—
28#486×C73C2214yes—
29#477×C73—21no—
30#476×C73—21no—
31#475×C73C2212yescomplemented
32#474×C73C2212yescomplemented
33#462×C73C2214yescomplemented
34#451×C73—22no—
35#431×C73C2211yescharacteristic
36#428×C73—21no—
37#423×C73C2211yescharacteristic
38#417×C73—21no—
39C2×#102×C73—22no—
40C2×C4×C8×C73C2211yescharacteristic
41#331×C73C2214yes—
42Q8⋊C8×C73C2212yescomplemented
43Q8⋊C8×C73C2214yescomplemented
44D4⋊C8×C73C2212yescomplemented
45C2×#41×C73—22no—
46C2×#41×C73—22no—
47C2×#41×C73D411yescharacteristic
48C22×Q8×C73—21no—
49C22×D4×C73—21no—
50C22×D4×C73—21no—
51C22×D4×C73D412yes—
52C2×Q16×C73—24no—
53C2×M16×C73—21no—
54C22×C8×C73—21no—
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73D411yescharacteristic
59C4⋊Q8×C73D411yescharacteristic
60C4⋊Q8×C73C2312yes—
61#127×C73—22no—
62#127×C73—21no—
63#127×C73—21no—
64#127×C73D412yes—
65#127×C73C2312yes—
66#125×C73—21no—
67#124×C73—22no—
68#124×C73—22no—
69#124×C73—22no—
70#123×C73—24no—
71#123×C73—24no—
72C22⋊Q8×C73—21no—
73C22⋊Q8×C73D412yes—
74#121×C73—22no—
75#121×C73—22no—
76#121×C73—22no—
77#121×C73—24no—
78#121×C73—24no—
79#121×C73—22no—
80#120×C73—24no—
81C4×Q8×C73—21no—
82C4×Q8×C73D411yescharacteristic
83C4×Q8×C73C2312yes—
84C4×Q8×C73C2311yescharacteristic
85C4×D4×C73—22no—
86C4×D4×C73—22no—
87C4×D4×C73—22no—
88C4×D4×C73—21no—
89C4×D4×C73—21no—
90C4×D4×C73—22no—
91C4×D4×C73—22no—
92C4×D4×C73—22no—
93C4×D4×C73D411yescharacteristic
94C4×D4×C73C2311yescharacteristic
95#117×C73—21no—
96#117×C73—21no—
97#117×C73D411yescharacteristic
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C22⋊C4×C73—24no—
101C2×C42×C73—21no—
102C2×C42×C73C2311yescharacteristic
103#107×C73—22no—
104#106×C73—22no—
105C4⋊C8×C73—21no—
106C4⋊C8×C73C2312yes—
107C4⋊C8×C73C2312yes—
108#103×C73—22no—
109#103×C73—24no—
110#102×C73—28no—
111#102×C73—22no—
112C22⋊C8×C73—21no—
113C4×C8×C73—21no—
114C4×C8×C73—21no—
115C4×C8×C73C2312yes—
116C24×C73—22no—
117#41×C73—22no—
118#41×C73—22no—
119#41×C73—84no—
120#41×C73—84no—
121C2×Q8×C73—21no—
122C2×Q8×C73—22no—
123C2×Q8×C73—41no—
124C2×Q8×C73—82no—
125C2×Q8×C73C2×D411yescharacteristic
126C2×Q8×C73C2×D412yes—
127C2×D4×C73—21no—
128C2×D4×C73—24no—
129C2×D4×C73—24no—
130C2×D4×C73—22no—
131C2×D4×C73—22no—
132C2×D4×C73—22no—
133C2×D4×C73—22no—
134C2×D4×C73—22no—
135C2×D4×C73—44no—
136C2×D4×C73—42no—
137C2×D4×C73—42no—
138C2×D4×C73—82no—
139C2×D4×C73C2×D411yescharacteristic
140C2×D4×C73C2×D414yes—
141C22×C4×C73—24no—
142C22×C4×C73—24no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—41no—
147C22×C4×C73C2×D411yescharacteristic
148C22×C4×C73C2×D411yescharacteristic
149C22×C4×C73C2×D411yescharacteristic
150Q16×C73—84no—
151M16×C73—42no—
152C2×C8×C73—22no—
153C2×C8×C73—22no—
154C2×C8×C73—22no—
155C2×C8×C73—42no—
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—22no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—22no—
163C4⋊C4×C73—22no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73—41no—
166C4⋊C4×C73—41no—
167C4⋊C4×C73C2×D412yes—
168C4⋊C4×C73C2×D411yescharacteristic
169C4⋊C4×C73C2×D411yescharacteristic
170C22⋊C4×C73—24no—
171C22⋊C4×C73—24no—
172C22⋊C4×C73—24no—
173C22⋊C4×C73—24no—
174C22⋊C4×C73—21no—
175C22⋊C4×C73—22no—
176C22⋊C4×C73—21no—
177C22⋊C4×C73—21no—
178C42×C73—21no—
179C42×C73—21no—
180C42×C73—21no—
181C42×C73—22no—
182C42×C73C2×D411yescharacteristic
183C42×C73C2×D411yescharacteristic
184C42×C73C2×D411yescharacteristic
185C42×C73C2411yescharacteristic
186C23×C73—24no—
187C23×C73—21no—
188C23×C73—44no—
189C23×C73—82no—
190C23×C73—82no—
191C23×C73#12011yescharacteristic
192Q8×C73—21no—
193Q8×C73—21no—
194Q8×C73—24no—
195Q8×C73—42no—
196Q8×C73—44no—
197Q8×C73—82no—
198D4×C73—21no—
199D4×C73—21no—
200D4×C73—44no—
201D4×C73—42no—
202D4×C73—42no—
203D4×C73—42no—
204D4×C73—44no—
205D4×C73—44no—
206D4×C73—44no—
207D4×C73—82no—
208D4×C73—84no—
209D4×C73—84no—
210C2×C4×C73—22no—
211C2×C4×C73—22no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—42no—
218C2×C4×C73—42no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73—84no—
222C2×C4×C73—84no—
223C2×C4×C73—82no—
224C2×C4×C73#12011yescharacteristic
225C2×C4×C73#12011yescharacteristic
226C2×C4×C73#12011yescharacteristic
227C2×C4×C73#13512yes—
228C2×C4×C73#13712yes—
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C8×C73—42no—
233C8×C73—42no—
234C8×C73—42no—
235C22×C73—21no—
236C22×C73—21no—
237C22×C73—41no—
238C22×C73—44no—
239C22×C73—41no—
240C22×C73—44no—
241C22×C73—41no—
242C22×C73—44no—
243C22×C73—164no—
244C22×C73#45212yes—
245C22×C73C2×#12011yescharacteristic
246C4×C73—21no—
247C4×C73—21no—
248C4×C73—41no—
249C4×C73—41no—
250C4×C73—42no—
251C4×C73—41no—
252C4×C73—42no—
253C4×C73—82no—
254C4×C73#45212yes—
255C4×C73#45512yes—
256C4×C73C2×#13511yescharacteristic
257C4×C73C2×#13711yescharacteristic
258C4×C73#57612yes—
259#18635C7311yesmaximal, characteristic, direct product factor, Sylow
260C2×C73—22no—
261C2×C73—81no—
262C2×C73—84no—
263C2×C73C2×#45211yescharacteristic
264C2×C73#300811yescharacteristic
265C2×C73C2×#45511yescharacteristic
266#3508C2×C7311yescharacteristic
267#3209C2×C7311yescharacteristic, complemented
268C2×#486C2×C7311yescharacteristic, complemented
269#3079C2×C7311yescharacteristic, complemented
270#1641C2×C7314yescomplemented
271#1640C2×C7314yescomplemented
272#1545C2×C7311yescharacteristic, complemented
273#1493C2×C7311yescharacteristic, complemented
274#1476C2×C7311yescharacteristic, complemented
275C73#1863511yescharacteristic, direct product factor, Sylow
276#559—21no—
277#559C22×C7312yes—
278#550—21no—
279#550C22×C7312yes—
280#546—22no—
281#546—22no—
282#540—22no—
283#535—21no—
284#533C22×C7311yescharacteristic
285#532—21no—
286#532—21no—
287C2×C4⋊Q8C22×C7311yescharacteristic
288C2×#127C22×C7311yescharacteristic
289#518—21no—
290C4×#41C22×C7311yescharacteristic
291#494—21no—
292#487—21no—
293#486—22no—
294#486C22×C7314yes—
295#477—21no—
296#476—21no—
297#475C22×C7312yescomplemented
298#474C22×C7312yescomplemented
299#462C22×C7314yescomplemented
300#451—22no—
301#431C22×C7311yescharacteristic
302#428—21no—
303#423C22×C7311yescharacteristic
304#417—21no—
305C2×#102—22no—
306C2×C4×C8C22×C7311yescharacteristic
307#331C22×C7314yes—
308Q8⋊C8C22×C7312yescomplemented
309Q8⋊C8C22×C7314yescomplemented
310D4⋊C8C22×C7312yescomplemented
311C2×#41—22no—
312C2×#41—22no—
313C2×#41D4×C7311yescharacteristic
314C22×Q8—21no—
315C22×D4—21no—
316C22×D4—21no—
317C22×D4D4×C7312yes—
318C2×Q16—24no—
319C2×M16—21no—
320C22×C8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8D4×C7311yescharacteristic
325C4⋊Q8D4×C7311yescharacteristic
326C4⋊Q8C23×C7312yes—
327#127—22no—
328#127—21no—
329#127—21no—
330#127D4×C7312yes—
331#127C23×C7312yes—
332#125—21no—
333#124—22no—
334#124—22no—
335#124—22no—
336#123—24no—
337#123—24no—
338C22⋊Q8—21no—
339C22⋊Q8D4×C7312yes—
340#121—22no—
341#121—22no—
342#121—22no—
343#121—24no—
344#121—24no—
345#121—22no—
346#120—24no—
347C4×Q8—21no—
348C4×Q8D4×C7311yescharacteristic
349C4×Q8C23×C7311yescharacteristic
350C4×Q8C23×C7312yes—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4D4×C7311yescharacteristic
360C4×D4C23×C7311yescharacteristic
361#117—21no—
362#117—21no—
363#117D4×C7311yescharacteristic
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C22⋊C4—24no—
367C2×C42—21no—
368C2×C42C23×C7311yescharacteristic
369#107—22no—
370#106—22no—
371C4⋊C8—21no—
372C4⋊C8C23×C7312yes—
373C4⋊C8C23×C7312yes—
374#103—22no—
375#103—24no—
376#102—28no—
377#102—22no—
378C22⋊C8—21no—
379C4×C8—21no—
380C4×C8—21no—
381C4×C8C23×C7312yes—
382C24—22no—
383#41—22no—
384#41—22no—
385#41—84no—
386#41—84no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—41no—
390C2×Q8—82no—
391C2×Q8C2×D4×C7311yescharacteristic
392C2×Q8C2×D4×C7312yes—
393C2×D4—21no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—82no—
405C2×D4C2×D4×C7311yescharacteristic
406C2×D4C2×D4×C7314yes—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4C2×D4×C7311yescharacteristic
414C22×C4C2×D4×C7311yescharacteristic
415C22×C4C2×D4×C7311yescharacteristic
416Q16—84no—
417M16—42no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—42no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4C2×D4×C7312yes—
434C4⋊C4C2×D4×C7311yescharacteristic
435C4⋊C4C2×D4×C7311yescharacteristic
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—22no—
448C42C2×D4×C7311yescharacteristic
449C42C2×D4×C7311yescharacteristic
450C42C2×D4×C7311yescharacteristic
451C42C24×C7311yescharacteristic
452C23—24no—
453C23—21no—
454C23—44no—
455C23—82no—
456C23—82no—
457C23#120×C7311yescharacteristic
458Q8—21no—
459Q8—21no—
460Q8—24no—
461Q8—42no—
462Q8—44no—
463Q8—82no—
464D4—21no—
465D4—21no—
466D4—44no—
467D4—42no—
468D4—44no—
469D4—44no—
470D4—42no—
471D4—42no—
472D4—44no—
473D4—82no—
474D4—84no—
475D4—84no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—82no—
490C2×C4#120×C7311yescharacteristic
491C2×C4#120×C7311yescharacteristic
492C2×C4#120×C7311yescharacteristic
493C2×C4#135×C7312yes—
494C2×C4#137×C7312yes—
495C2×C4C22×D4×C7311yescharacteristic
496C2×C4C22×D4×C7311yescharacteristic
497C2×C4C22×D4×C7311yescharacteristic
498C8—42no—
499C8—42no—
500C8—42no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—44no—
505C22—41no—
506C22—44no—
507C22—41no—
508C22—44no—
509C22—164no—
510C22#452×C7312yes—
511C22C2×#120×C7311yescharacteristic
512C4—21no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—41no—
518C4—42no—
519C4—82no—
520C4#452×C7312yes—
521C4#455×C7312yes—
522C4C2×#135×C7311yescharacteristic
523C4C2×#137×C7311yescharacteristic
524C4#576×C7312yes—
525C2—22no—
526C2—84no—
527C2—81no—
528C2C2×#452×C7311yescharacteristic
529C2C73×#300811yescharacteristic
530C2C2×#455×C7311yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3508C2
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×#3209C2semidirect
3. C2×#486×C73C2semidirect
4. C73×#3079C2semidirect
5. C73×#1641C2semidirect
6. C73×#1640C2semidirect
7. C73×#1545C2semidirect
8. C73×#1493C2semidirect
9. C73×#1476C2semidirect
31. #475×C73C22semidirect
32. #474×C73C22semidirect
33. #462×C73C22semidirect
42. Q8⋊C8×C73C22semidirect
43. Q8⋊C8×C73C22semidirect
44. D4⋊C8×C73C22semidirect
259. #18635C73direct (N × H)
267. #3209C2×C73semidirect
268. C2×#486C2×C73semidirect
269. #3079C2×C73semidirect
270. #1641C2×C73semidirect
271. #1640C2×C73semidirect
272. #1545C2×C73semidirect
273. #1493C2×C73semidirect
274. #1476C2×C73semidirect
275. C73#18635direct (N × H)
297. #475C22×C73semidirect
298. #474C22×C73semidirect
299. #462C22×C73semidirect
308. Q8⋊C8C22×C73semidirect
309. Q8⋊C8C22×C73semidirect
310. D4⋊C8C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#35082
C73×#32092
C2×#486×C732
C73×#30792
C73×#16412
C73×#16402
C73×#15452
C73×#14932
C73×#14762
#1863573