Finite Group Databasebeta

← C73×#18667 #32928071

Subgroups — C73×#18667 #32928071

Order 18688.

882 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3507C211yesmaximal, characteristic
2C73×#3249C211yesmaximal, characteristic, complemented
3C73×#3167C211yesmaximal, characteristic
4C73×#3078C211yesmaximal, characteristic, complemented
5C73×#1653C212yesmaximal, complemented
6C73×#1640C212yesmaximal, complemented
7C73×#1636C212yesmaximal, complemented
8C73×#1630C212yesmaximal, complemented
9C73×#1547C211yesmaximal, characteristic
10C73×#1493C211yesmaximal, characteristic, complemented
11C2×Q8⋊C8×C73C211yesmaximal, characteristic, complemented
12#559×C73C2212yes—
13#557×C73C2212yes—
14#556×C73—21no—
15#556×C73—22no—
16#554×C73—21no—
17#552×C73—22no—
18#539×C73—22no—
19#536×C73—21no—
20#533×C73C2211yescharacteristic
21#533×C73C2211yescharacteristic
22#532×C73C2211yescharacteristic
23#529×C73—22no—
24#520×C73—21no—
25C2×C4×Q8×C73C2211yescharacteristic
26#498×C73C2211yescharacteristic
27#492×C73C2211yescharacteristic
28#487×C73C2211yescharacteristic
29#486×C73C2211yescharacteristic
30#479×C73—22no—
31#477×C73—21no—
32#477×C73C2212yes—
33#477×C73C2212yes—
34#475×C73C2212yes—
35#467×C73—22no—
36#465×C73—22no—
37#462×C73—21no—
38#462×C73C2212yes—
39C4×SD16×C73—22no—
40#428×C73—21no—
41#423×C73C2211yescharacteristic
42C2×C4⋊C8×C73C2211yescharacteristic
43#419×C73—21no—
44C2×C4×C8×C73C2211yescharacteristic
45#332×C73C2212yes—
46#331×C73C2212yes—
47Q8⋊C8×C73C2212yes—
48Q8⋊C8×C73C2212yes—
49Q8⋊C8×C73C2212yes—
50Q8⋊C8×C73C2212yes—
51C2×#41×C73—21no—
52C22×Q8×C73D411yescharacteristic
53C2×Q16×C73—22no—
54C2×SD16×C73—22no—
55C2×M16×C73—21no—
56C22×C8×C73—21no—
57C22×C8×C73—21no—
58C4⋊Q8×C73—22no—
59C4⋊Q8×C73D411yescharacteristic
60C4⋊Q8×C73D411yescharacteristic
61C4⋊Q8×C73C2311yescharacteristic
62C4⋊Q8×C73C2311yescharacteristic
63C4⋊Q8×C73C2311yescharacteristic
64#127×C73C2311yescharacteristic
65#126×C73—24no—
66#126×C73—24no—
67#125×C73—22no—
68#125×C73—24no—
69#125×C73—21no—
70#125×C73—21no—
71#124×C73—22no—
72#124×C73—22no—
73#124×C73—21no—
74#123×C73—24no—
75#123×C73—24no—
76C22⋊Q8×C73—22no—
77C22⋊Q8×C73—22no—
78C22⋊Q8×C73—24no—
79C22⋊Q8×C73—21no—
80C22⋊Q8×C73—21no—
81C22⋊Q8×C73—21no—
82C22⋊Q8×C73D412yes—
83#121×C73—21no—
84#121×C73D412yes—
85C4×Q8×C73—22no—
86C4×Q8×C73—22no—
87C4×Q8×C73—21no—
88C4×Q8×C73—21no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73—21no—
92C4×Q8×C73D412yes—
93C4×Q8×C73C2312yes—
94C4×Q8×C73C2312yes—
95C4×D4×C73—24no—
96C4×D4×C73—22no—
97C4×D4×C73—22no—
98C4×D4×C73—21no—
99C4×D4×C73D412yes—
100#117×C73—22no—
101#117×C73—24no—
102#117×C73—21no—
103#117×C73—21no—
104C2×C4⋊C4×C73—22no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73D411yescharacteristic
107C2×C42×C73—21no—
108C2×C42×C73C2311yescharacteristic
109#107×C73—22no—
110#107×C73—22no—
111#106×C73—22no—
112#106×C73—22no—
113C4⋊C8×C73—21no—
114C4⋊C8×C73—21no—
115C4⋊C8×C73C2312yes—
116C4⋊C8×C73C2312yes—
117#103×C73—22no—
118#103×C73—22no—
119#103×C73—22no—
120#103×C73—22no—
121#103×C73—22no—
122#102×C73—22no—
123C4×C8×C73—21no—
124C4×C8×C73C2311yescharacteristic
125C4×C8×C73C2311yescharacteristic
126#41×C73—42no—
127#41×C73—42no—
128C2×Q8×C73—21no—
129C2×Q8×C73—21no—
130C2×Q8×C73—21no—
131C2×Q8×C73—22no—
132C2×Q8×C73—22no—
133C2×Q8×C73—21no—
134C2×Q8×C73—21no—
135C2×Q8×C73C2×D412yes—
136C2×Q8×C73C2×D412yes—
137C2×D4×C73—22no—
138C2×D4×C73—21no—
139C2×D4×C73—21no—
140C2×D4×C73C2×D412yes—
141C22×C4×C73—21no—
142C22×C4×C73—24no—
143C22×C4×C73—21no—
144C22×C4×C73—22no—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D411yescharacteristic
148Q16×C73—82no—
149SD16×C73—44no—
150M16×C73—42no—
151C2×C8×C73—21no—
152C2×C8×C73—21no—
153C2×C8×C73—22no—
154C2×C8×C73—22no—
155C2×C8×C73—42no—
156C2×C8×C73—81no—
157C4⋊C4×C73—24no—
158C4⋊C4×C73—24no—
159C4⋊C4×C73—24no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—22no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73—21no—
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—21no—
173C4⋊C4×C73—41no—
174C4⋊C4×C73—41no—
175C4⋊C4×C73C2×D412yes—
176C4⋊C4×C73C2×D412yes—
177C4⋊C4×C73C2×D412yes—
178C22⋊C4×C73—24no—
179C22⋊C4×C73—24no—
180C22⋊C4×C73—24no—
181C22⋊C4×C73—22no—
182C22⋊C4×C73—22no—
183C22⋊C4×C73—22no—
184C22⋊C4×C73—41no—
185C42×C73—24no—
186C42×C73—22no—
187C42×C73—22no—
188C42×C73—22no—
189C42×C73—41no—
190C42×C73C2×D411yescharacteristic
191C42×C73C2×D411yescharacteristic
192C42×C73C2×D411yescharacteristic
193C42×C73C2411yescharacteristic
194C23×C73—22no—
195C23×C73#12011yescharacteristic
196Q8×C73—22no—
197Q8×C73—22no—
198Q8×C73—22no—
199Q8×C73—22no—
200Q8×C73—42no—
201Q8×C73—41no—
202Q8×C73—41no—
203Q8×C73—42no—
204Q8×C73—41no—
205Q8×C73—41no—
206Q8×C73—81no—
207D4×C73—42no—
208D4×C73—44no—
209D4×C73—41no—
210D4×C73—41no—
211D4×C73—41no—
212D4×C73—41no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—41no—
221C2×C4×C73—42no—
222C2×C4×C73—44no—
223C2×C4×C73—41no—
224C2×C4×C73—41no—
225C2×C4×C73—81no—
226C2×C4×C73—84no—
227C2×C4×C73#12011yescharacteristic
228C2×C4×C73#12011yescharacteristic
229C2×C4×C73#12011yescharacteristic
230C2×C4×C73#13512yes—
231C2×C4×C73#13712yes—
232C2×C4×C73C22×D411yescharacteristic
233C2×C4×C73C22×D411yescharacteristic
234C2×C4×C73C22×D411yescharacteristic
235C8×C73—22no—
236C8×C73—41no—
237C8×C73—42no—
238C8×C73—42no—
239C22×C73—21no—
240C22×C73—21no—
241C22×C73—42no—
242C22×C73—42no—
243C22×C73—42no—
244C22×C73#45212yes—
245C22×C73C2×#12011yescharacteristic
246C4×C73—21no—
247C4×C73—21no—
248C4×C73—21no—
249C4×C73—42no—
250C4×C73—42no—
251C4×C73—42no—
252C4×C73—42no—
253C4×C73—42no—
254C4×C73—81no—
255C4×C73—81no—
256C4×C73#45212yes—
257C4×C73C2×#13511yescharacteristic
258C4×C73C2×#13711yescharacteristic
259C4×C73#57711yescharacteristic
260C4×C73#57811yescharacteristic
261#18667C7311yesmaximal, characteristic, direct product factor, Sylow
262C2×C73—22no—
263C2×C73—82no—
264C2×C73C2×#45211yescharacteristic
265C2×C73#301311yescharacteristic
266C2×C73#301811yescharacteristic
267#3507C2×C7311yescharacteristic
268#3249C2×C7311yescharacteristic, complemented
269#3167C2×C7311yescharacteristic
270#3078C2×C7311yescharacteristic, complemented
271#1653C2×C7312yescomplemented
272#1640C2×C7312yescomplemented
273#1636C2×C7312yescomplemented
274#1630C2×C7312yescomplemented
275#1547C2×C7311yescharacteristic
276#1493C2×C7311yescharacteristic, complemented
277C2×Q8⋊C8C2×C7311yescharacteristic, complemented
278C73#1866711yescharacteristic, direct product factor, Sylow
279#559C22×C7312yes—
280#557C22×C7312yes—
281#556—21no—
282#556—22no—
283#554—21no—
284#552—22no—
285#539—22no—
286#536—21no—
287#533C22×C7311yescharacteristic
288#533C22×C7311yescharacteristic
289#532C22×C7311yescharacteristic
290#529—22no—
291#520—21no—
292C2×C4×Q8C22×C7311yescharacteristic
293#498C22×C7311yescharacteristic
294#492C22×C7311yescharacteristic
295#487C22×C7311yescharacteristic
296#486C22×C7311yescharacteristic
297#479—22no—
298#477—21no—
299#477C22×C7312yes—
300#477C22×C7312yes—
301#475C22×C7312yes—
302#467—22no—
303#465—22no—
304#462—21no—
305#462C22×C7312yes—
306C4×SD16—22no—
307#428—21no—
308#423C22×C7311yescharacteristic
309C2×C4⋊C8C22×C7311yescharacteristic
310#419—21no—
311C2×C4×C8C22×C7311yescharacteristic
312#332C22×C7312yes—
313#331C22×C7312yes—
314Q8⋊C8C22×C7312yes—
315Q8⋊C8C22×C7312yes—
316Q8⋊C8C22×C7312yes—
317Q8⋊C8C22×C7312yes—
318C2×#41—21no—
319C22×Q8D4×C7311yescharacteristic
320C2×Q16—22no—
321C2×SD16—22no—
322C2×M16—21no—
323C22×C8—21no—
324C22×C8—21no—
325C4⋊Q8—22no—
326C4⋊Q8D4×C7311yescharacteristic
327C4⋊Q8D4×C7311yescharacteristic
328C4⋊Q8C23×C7311yescharacteristic
329C4⋊Q8C23×C7311yescharacteristic
330C4⋊Q8C23×C7311yescharacteristic
331#127C23×C7311yescharacteristic
332#126—24no—
333#126—24no—
334#125—22no—
335#125—24no—
336#125—21no—
337#125—21no—
338#124—22no—
339#124—22no—
340#124—21no—
341#123—24no—
342#123—24no—
343C22⋊Q8—22no—
344C22⋊Q8—22no—
345C22⋊Q8—24no—
346C22⋊Q8—21no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8D4×C7312yes—
350#121—21no—
351#121D4×C7312yes—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8—21no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8D4×C7312yes—
360C4×Q8C23×C7312yes—
361C4×Q8C23×C7312yes—
362C4×D4—24no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—21no—
366C4×D4D4×C7312yes—
367#117—22no—
368#117—24no—
369#117—21no—
370#117—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4D4×C7311yescharacteristic
374C2×C42—21no—
375C2×C42C23×C7311yescharacteristic
376#107—22no—
377#107—22no—
378#106—22no—
379#106—22no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8C23×C7312yes—
383C4⋊C8C23×C7312yes—
384#103—22no—
385#103—22no—
386#103—22no—
387#103—22no—
388#103—22no—
389#102—22no—
390C4×C8—21no—
391C4×C8C23×C7311yescharacteristic
392C4×C8C23×C7311yescharacteristic
393#41—42no—
394#41—42no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C2×D4×C7312yes—
403C2×Q8C2×D4×C7312yes—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×D4×C7312yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4C2×D4×C7311yescharacteristic
413C22×C4C2×D4×C7311yescharacteristic
414C22×C4C2×D4×C7311yescharacteristic
415Q16—82no—
416SD16—44no—
417M16—42no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—42no—
423C2×C8—81no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×D4×C7312yes—
443C4⋊C4C2×D4×C7312yes—
444C4⋊C4C2×D4×C7312yes—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—41no—
452C42—24no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42C2×D4×C7311yescharacteristic
458C42C2×D4×C7311yescharacteristic
459C42C2×D4×C7311yescharacteristic
460C42C24×C7311yescharacteristic
461C23—22no—
462C23#120×C7311yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—22no—
466Q8—22no—
467Q8—42no—
468Q8—42no—
469Q8—41no—
470Q8—41no—
471Q8—41no—
472Q8—41no—
473Q8—81no—
474D4—42no—
475D4—44no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—81no—
493C2×C4—84no—
494C2×C4#120×C7311yescharacteristic
495C2×C4#120×C7311yescharacteristic
496C2×C4#120×C7311yescharacteristic
497C2×C4#135×C7312yes—
498C2×C4#137×C7312yes—
499C2×C4C22×D4×C7311yescharacteristic
500C2×C4C22×D4×C7311yescharacteristic
501C2×C4C22×D4×C7311yescharacteristic
502C8—22no—
503C8—41no—
504C8—42no—
505C8—42no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22#452×C7312yes—
512C22C2×#120×C7311yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4#452×C7312yes—
524C4C2×#135×C7311yescharacteristic
525C4C2×#137×C7311yescharacteristic
526C4#577×C7311yescharacteristic
527C4#578×C7311yescharacteristic
528C2—22no—
529C2—82no—
530C2C2×#452×C7311yescharacteristic
531C2C73×#301311yescharacteristic
532C2C73×#301811yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radical#532×C73C22
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×#3249C2semidirect
4. C73×#3078C2semidirect
5. C73×#1653C2semidirect
6. C73×#1640C2semidirect
7. C73×#1636C2semidirect
8. C73×#1630C2semidirect
10. C73×#1493C2semidirect
11. C2×Q8⋊C8×C73C2semidirect
261. #18667C73direct (N × H)
268. #3249C2×C73semidirect
270. #3078C2×C73semidirect
271. #1653C2×C73semidirect
272. #1640C2×C73semidirect
273. #1636C2×C73semidirect
274. #1630C2×C73semidirect
276. #1493C2×C73semidirect
277. C2×Q8⋊C8C2×C73semidirect
278. C73#18667direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35072
C73×#32492
C73×#31672
C73×#30782
C73×#16532
C73×#16402
C73×#16362
C73×#16302
C73×#15472
C73×#14932
C2×Q8⋊C8×C732
#1866773