Finite Group Databasebeta

← C73×#18831 #32928235

Subgroups — C73×#18831 #32928235

Order 18688.

838 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3509C211yesmaximal, characteristic
2C73×#3237C211yesmaximal, characteristic, complemented
3C73×#3230C211yesmaximal, characteristic, complemented
4C73×#3116C211yesmaximal, characteristic
5C73×#1650C212yesmaximal, complemented
6C73×#1648C212yesmaximal, complemented
7C73×#1638C212yesmaximal, complemented
8C73×#1636C212yesmaximal, complemented
9C73×#1556C211yesmaximal, characteristic, complemented
10C73×#1497C211yesmaximal, characteristic, complemented
11C73×#1488C211yesmaximal, characteristic
12#557×C73—21no—
13#557×C73C2212yes—
14#556×C73—21no—
15#556×C73C2212yes—
16#552×C73—22no—
17#552×C73—22no—
18#541×C73—22no—
19#536×C73—21no—
20#533×C73—21no—
21#533×C73—21no—
22#533×C73C2211yescharacteristic
23#533×C73C2211yescharacteristic
24C2×C4⋊Q8×C73C2211yescharacteristic
25#519×C73—21no—
26C4×#41×C73C2211yescharacteristic
27#501×C73—21no—
28#500×C73C2212yes—
29#491×C73—21no—
30#487×C73C2212yes—
31#485×C73—22no—
32#485×C73—21no—
33#484×C73—22no—
34#477×C73—22no—
35#477×C73C2212yes—
36#477×C73C2211yescharacteristic
37#476×C73C2211yescharacteristic
38#475×C73C2212yes—
39#462×C73C2211yescharacteristic
40#461×C73C2211yescharacteristic
41#452×C73—21no—
42#441×C73—21no—
43#440×C73—21no—
44#432×C73C2211yescharacteristic
45#432×C73C2211yescharacteristic
46#431×C73C2211yescharacteristic
47#332×C73C2212yes—
48#332×C73C2212yes—
49Q8⋊C8×C73C2212yes—
50Q8⋊C8×C73C2212yes—
51Q8⋊C8×C73C2212yes—
52D4⋊C8×C73C2212yes—
53C2×#41×C73D411yescharacteristic
54C22×Q8×C73—21no—
55C2×Q16×C73—21no—
56C2×SD16×C73—21no—
57C4⋊Q8×C73—22no—
58C4⋊Q8×C73—21no—
59C4⋊Q8×C73—21no—
60C4⋊Q8×C73D411yescharacteristic
61C4⋊Q8×C73D412yes—
62C4⋊Q8×C73D411yescharacteristic
63C4⋊Q8×C73C2311yescharacteristic
64C4⋊Q8×C73C2311yescharacteristic
65C4⋊Q8×C73C2312yes—
66#126×C73—24no—
67#126×C73—24no—
68#125×C73—22no—
69#125×C73—24no—
70#125×C73—22no—
71#125×C73—24no—
72#125×C73—21no—
73#125×C73—21no—
74#124×C73—22no—
75#123×C73—24no—
76C22⋊Q8×C73—22no—
77C22⋊Q8×C73—22no—
78C22⋊Q8×C73—22no—
79C22⋊Q8×C73—22no—
80C22⋊Q8×C73—21no—
81C22⋊Q8×C73—21no—
82C22⋊Q8×C73—21no—
83C22⋊Q8×C73D412yes—
84C4×Q8×C73—22no—
85C4×Q8×C73—22no—
86C4×Q8×C73—22no—
87C4×Q8×C73—22no—
88C4×Q8×C73—22no—
89C4×Q8×C73—22no—
90C4×Q8×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73—21no—
93C4×Q8×C73D411yescharacteristic
94C4×Q8×C73C2311yescharacteristic
95C4×Q8×C73C2312yes—
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73D411yescharacteristic
99C4×D4×C73C2311yescharacteristic
100#117×C73—24no—
101#117×C73—22no—
102#117×C73—21no—
103#117×C73—22no—
104#117×C73—21no—
105#117×C73—21no—
106#117×C73D411yescharacteristic
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73D412yes—
109C2×C42×C73—21no—
110C2×C42×C73C2311yescharacteristic
111#107×C73—21no—
112#107×C73—24no—
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—22no—
122#103×C73—22no—
123#103×C73—21no—
124#103×C73—21no—
125#103×C73—24no—
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—41no—
134#41×C73—41no—
135C2×Q8×C73—21no—
136C2×Q8×C73—22no—
137C2×Q8×C73—22no—
138C2×Q8×C73—21no—
139C2×Q8×C73—21no—
140C2×Q8×C73—42no—
141C2×Q8×C73—42no—
142C2×Q8×C73C2×D412yes—
143C2×Q8×C73C2×D411yescharacteristic
144C2×D4×C73—21no—
145C2×D4×C73C2×D411yescharacteristic
146C22×C4×C73—22no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153Q16×C73—42no—
154SD16×C73—42no—
155C2×C8×C73—22no—
156C2×C8×C73—21no—
157C2×C8×C73—22no—
158C2×C8×C73—21no—
159C4⋊C4×C73—24no—
160C4⋊C4×C73—24no—
161C4⋊C4×C73—24no—
162C4⋊C4×C73—24no—
163C4⋊C4×C73—22no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73—21no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—22no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—22no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—41no—
178C4⋊C4×C73—41no—
179C4⋊C4×C73C2×D411yescharacteristic
180C4⋊C4×C73C2×D414yes—
181C4⋊C4×C73C2×D411yescharacteristic
182C4⋊C4×C73C2×D412yes—
183C22⋊C4×C73—24no—
184C22⋊C4×C73—24no—
185C22⋊C4×C73—21no—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—21no—
188C22⋊C4×C73—21no—
189C22⋊C4×C73—41no—
190C42×C73—24no—
191C42×C73—24no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—22no—
196C42×C73—41no—
197C42×C73C2×D411yescharacteristic
198C42×C73C2×D411yescharacteristic
199C42×C73C2×D411yescharacteristic
200C42×C73C2411yescharacteristic
201C23×C73—21no—
202C23×C73#12011yescharacteristic
203Q8×C73—21no—
204Q8×C73—21no—
205Q8×C73—24no—
206Q8×C73—42no—
207Q8×C73—44no—
208Q8×C73—42no—
209Q8×C73—42no—
210Q8×C73—42no—
211Q8×C73—41no—
212Q8×C73—41no—
213D4×C73—21no—
214D4×C73—21no—
215D4×C73—42no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—21no—
225C2×C4×C73—41no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
228C2×C4×C73—44no—
229C2×C4×C73—42no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73#12011yescharacteristic
233C2×C4×C73#12011yescharacteristic
234C2×C4×C73#12011yescharacteristic
235C2×C4×C73C22×D411yescharacteristic
236C2×C4×C73C22×D411yescharacteristic
237C2×C4×C73C22×D411yescharacteristic
238C8×C73—42no—
239C8×C73—41no—
240C8×C73—41no—
241C8×C73—42no—
242C22×C73—21no—
243C22×C73—21no—
244C22×C73—21no—
245C22×C73—41no—
246C22×C73—41no—
247C22×C73—41no—
248C22×C73C2×#12011yescharacteristic
249C4×C73—21no—
250C4×C73—21no—
251C4×C73—21no—
252C4×C73—21no—
253C4×C73—41no—
254C4×C73—44no—
255C4×C73—41no—
256C4×C73—42no—
257C4×C73—41no—
258C4×C73—42no—
259C4×C73—81no—
260C4×C73—81no—
261C4×C73#57511yescharacteristic
262C4×C73#57511yescharacteristic
263C4×C73#57812yes—
264#18831C7311yesmaximal, characteristic, direct product factor, Sylow
265C2×C73—41no—
266C2×C73—81no—
267C2×C73#300511yescharacteristic
268C2×C73#301111yescharacteristic
269C2×C73#302011yescharacteristic
270#3509C2×C7311yescharacteristic
271#3237C2×C7311yescharacteristic, complemented
272#3230C2×C7311yescharacteristic, complemented
273#3116C2×C7311yescharacteristic
274#1650C2×C7312yescomplemented
275#1648C2×C7312yescomplemented
276#1638C2×C7312yescomplemented
277#1636C2×C7312yescomplemented
278#1556C2×C7311yescharacteristic, complemented
279#1497C2×C7311yescharacteristic, complemented
280#1488C2×C7311yescharacteristic
281C73#1883111yescharacteristic, direct product factor, Sylow
282#557—21no—
283#557C22×C7312yes—
284#556—21no—
285#556C22×C7312yes—
286#552—22no—
287#552—22no—
288#541—22no—
289#536—21no—
290#533—21no—
291#533—21no—
292#533C22×C7311yescharacteristic
293#533C22×C7311yescharacteristic
294C2×C4⋊Q8C22×C7311yescharacteristic
295#519—21no—
296C4×#41C22×C7311yescharacteristic
297#501—21no—
298#500C22×C7312yes—
299#491—21no—
300#487C22×C7312yes—
301#485—22no—
302#485—21no—
303#484—22no—
304#477—22no—
305#477C22×C7311yescharacteristic
306#477C22×C7312yes—
307#476C22×C7311yescharacteristic
308#475C22×C7312yes—
309#462C22×C7311yescharacteristic
310#461C22×C7311yescharacteristic
311#452—21no—
312#441—21no—
313#440—21no—
314#432C22×C7311yescharacteristic
315#432C22×C7311yescharacteristic
316#431C22×C7311yescharacteristic
317#332C22×C7312yes—
318#332C22×C7312yes—
319Q8⋊C8C22×C7312yes—
320Q8⋊C8C22×C7312yes—
321Q8⋊C8C22×C7312yes—
322D4⋊C8C22×C7312yes—
323C2×#41D4×C7311yescharacteristic
324C22×Q8—21no—
325C2×Q16—21no—
326C2×SD16—21no—
327C4⋊Q8—22no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8D4×C7311yescharacteristic
331C4⋊Q8D4×C7312yes—
332C4⋊Q8D4×C7311yescharacteristic
333C4⋊Q8C23×C7311yescharacteristic
334C4⋊Q8C23×C7311yescharacteristic
335C4⋊Q8C23×C7312yes—
336#126—24no—
337#126—24no—
338#125—22no—
339#125—24no—
340#125—22no—
341#125—24no—
342#125—21no—
343#125—21no—
344#124—22no—
345#123—24no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353C22⋊Q8D4×C7312yes—
354C4×Q8—22no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8—22no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8D4×C7311yescharacteristic
364C4×Q8C23×C7311yescharacteristic
365C4×Q8C23×C7312yes—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4D4×C7311yescharacteristic
369C4×D4C23×C7311yescharacteristic
370#117—24no—
371#117—22no—
372#117—21no—
373#117—22no—
374#117—21no—
375#117—21no—
376#117D4×C7311yescharacteristic
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C7312yes—
379C2×C42—21no—
380C2×C42C23×C7311yescharacteristic
381#107—21no—
382#107—24no—
383#107—22no—
384#106—21no—
385#106—22no—
386C4⋊C8—21no—
387C4⋊C8C23×C7312yes—
388C4⋊C8C23×C7311yescharacteristic
389C4⋊C8C23×C7311yescharacteristic
390#103—21no—
391#103—22no—
392#103—22no—
393#103—21no—
394#103—21no—
395#103—24no—
396#102—21no—
397C22⋊C8—21no—
398C22⋊C8—21no—
399C22⋊C8—21no—
400#97—21no—
401#97—21no—
402C4×C8C23×C7312yes—
403#41—41no—
404#41—41no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8C2×D4×C7312yes—
413C2×Q8C2×D4×C7311yescharacteristic
414C2×D4—21no—
415C2×D4C2×D4×C7311yescharacteristic
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4C2×D4×C7311yescharacteristic
421C22×C4C2×D4×C7311yescharacteristic
422C22×C4C2×D4×C7311yescharacteristic
423Q16—42no—
424SD16—42no—
425C2×C8—22no—
426C2×C8—21no—
427C2×C8—22no—
428C2×C8—21no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4C2×D4×C7311yescharacteristic
450C4⋊C4C2×D4×C7314yes—
451C4⋊C4C2×D4×C7311yescharacteristic
452C4⋊C4C2×D4×C7312yes—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—21no—
456C22⋊C4—22no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—41no—
460C42—24no—
461C42—24no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—22no—
466C42—41no—
467C42C2×D4×C7311yescharacteristic
468C42C2×D4×C7311yescharacteristic
469C42C2×D4×C7311yescharacteristic
470C42C24×C7311yescharacteristic
471C23—21no—
472C23#120×C7311yescharacteristic
473Q8—21no—
474Q8—21no—
475Q8—24no—
476Q8—44no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—41no—
482Q8—41no—
483D4—21no—
484D4—21no—
485D4—42no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4#120×C7311yescharacteristic
503C2×C4#120×C7311yescharacteristic
504C2×C4#120×C7311yescharacteristic
505C2×C4C22×D4×C7311yescharacteristic
506C2×C4C22×D4×C7311yescharacteristic
507C2×C4C22×D4×C7311yescharacteristic
508C8—42no—
509C8—41no—
510C8—41no—
511C8—42no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22C2×#120×C7311yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—41no—
524C4—44no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—81no—
530C4—81no—
531C4#575×C7311yescharacteristic
532C4#575×C7311yescharacteristic
533C4#578×C7312yes—
534C2—41no—
535C2—81no—
536C2C73×#300511yescharacteristic
537C2C73×#301111yescharacteristic
538C2C73×#302011yescharacteristic
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×#3230C2semidirect
5. C73×#1650C2semidirect
6. C73×#1648C2semidirect
7. C73×#1638C2semidirect
8. C73×#1636C2semidirect
9. C73×#1556C2semidirect
10. C73×#1497C2semidirect
264. #18831C73direct (N × H)
271. #3237C2×C73semidirect
272. #3230C2×C73semidirect
274. #1650C2×C73semidirect
275. #1648C2×C73semidirect
276. #1638C2×C73semidirect
277. #1636C2×C73semidirect
278. #1556C2×C73semidirect
279. #1497C2×C73semidirect
281. C73#18831direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35092
C73×#32372
C73×#32302
C73×#31162
C73×#16502
C73×#16482
C73×#16382
C73×#16362
C73×#15562
C73×#14972
C73×#14882
#1883173