Finite Group Databasebeta

← C73×#18602 #32928006

Subgroups — C73×#18602 #32928006

Order 18688.

900 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3507C211yesmaximal, characteristic, complemented
2C73×#3216C211yesmaximal, characteristic, complemented
3C73×#3184C211yesmaximal, characteristic
4C73×#3103C211yesmaximal, characteristic, complemented
5C73×#1653C212yesmaximal, complemented
6C73×#1640C212yesmaximal, complemented
7C73×#1626C212yesmaximal, complemented
8C73×#1620C212yesmaximal, complemented
9C73×#1553C211yesmaximal, characteristic, complemented
10C73×#1491C211yesmaximal, characteristic, complemented
11C73×#1485C211yesmaximal, 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#532×C73C2211yescharacteristic
22#532×C73C2211yescharacteristic
23#529×C73—22no—
24#520×C73—21no—
25C2×C4×Q8×C73C2211yescharacteristic
26#498×C73C2211yescharacteristic
27#497×C73—21no—
28#493×C73C2211yescharacteristic
29#489×C73—21no—
30#486×C73C2211yescharacteristic
31#486×C73C2211yescharacteristic
32#481×C73—21no—
33#475×C73C2212yes—
34#469×C73—22no—
35#462×C73C2212yes—
36#460×C73C2212yes—
37#460×C73C2212yes—
38#452×C73—22no—
39#448×C73—21no—
40#442×C73—22no—
41C4×SD16×C73—22no—
42#432×C73C2211yescharacteristic
43#431×C73C2211yescharacteristic
44C4×M16×C73C2211yescharacteristic
45#332×C73C2212yes—
46#331×C73C2212yes—
47Q8⋊C8×C73C2212yes—
48Q8⋊C8×C73C2212yes—
49Q8⋊C8×C73C2212yes—
50Q8⋊C8×C73C2212yes—
51C2×#41×C73—21no—
52C2×#41×C73—21no—
53C22×Q8×C73D411yescharacteristic
54C2×Q16×C73—22no—
55C2×SD16×C73—22no—
56C2×SD16×C73—22no—
57C2×D8×C73—22no—
58C2×M16×C73—21no—
59C4⋊Q8×C73—22no—
60C4⋊Q8×C73D411yescharacteristic
61C4⋊Q8×C73D411yescharacteristic
62C4⋊Q8×C73C2311yescharacteristic
63C4⋊Q8×C73C2311yescharacteristic
64#127×C73C2311yescharacteristic
65#127×C73C2311yescharacteristic
66#126×C73—24no—
67#126×C73—24no—
68#125×C73—22no—
69#125×C73—24no—
70#125×C73—21no—
71#124×C73—22no—
72#124×C73—22no—
73#124×C73—21no—
74#124×C73—21no—
75#123×C73—24no—
76#123×C73—24no—
77C22⋊Q8×C73—22no—
78C22⋊Q8×C73—22no—
79C22⋊Q8×C73—24no—
80C22⋊Q8×C73—21no—
81C22⋊Q8×C73D412yes—
82#121×C73—21no—
83#121×C73—21no—
84#121×C73—21no—
85#121×C73D412yes—
86C4×Q8×C73—22no—
87C4×Q8×C73—22no—
88C4×Q8×C73—21no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73D412yes—
92C4×Q8×C73C2312yes—
93C4×Q8×C73C2312yes—
94C4×D4×C73—24no—
95C4×D4×C73—22no—
96C4×D4×C73—22no—
97C4×D4×C73—21no—
98C4×D4×C73—21no—
99C4×D4×C73—21no—
100C4×D4×C73D412yes—
101#117×C73—22no—
102#117×C73—24no—
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#106×C73—22no—
110#106×C73—22no—
111C4⋊C8×C73C2312yes—
112C4⋊C8×C73C2312yes—
113#103×C73—22no—
114#103×C73—22no—
115#102×C73—22no—
116#102×C73—22no—
117#102×C73—22no—
118#102×C73—22no—
119C22⋊C8×C73—21no—
120C22⋊C8×C73—21no—
121#97×C73—21no—
122#97×C73—21no—
123C4×C8×C73—21no—
124C4×C8×C73C2311yescharacteristic
125C4×C8×C73C2311yescharacteristic
126#41×C73—44no—
127#41×C73—82no—
128C2×Q8×C73—21no—
129C2×Q8×C73—21no—
130C2×Q8×C73—21no—
131C2×Q8×C73—21no—
132C2×Q8×C73—42no—
133C2×Q8×C73C2×D412yes—
134C2×Q8×C73C2×D412yes—
135C2×D4×C73—22no—
136C2×D4×C73—21no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73—21no—
140C2×D4×C73—21no—
141C2×D4×C73—41no—
142C2×D4×C73C2×D412yes—
143C22×C4×C73—21no—
144C22×C4×C73—24no—
145C22×C4×C73—21no—
146C22×C4×C73—22no—
147C22×C4×C73—41no—
148C22×C4×C73C2×D411yescharacteristic
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73C2×D411yescharacteristic
151Q16×C73—82no—
152SD16×C73—44no—
153SD16×C73—82no—
154D8×C73—44no—
155M16×C73—42no—
156C2×C8×C73—21no—
157C2×C8×C73—22no—
158C2×C8×C73—21no—
159C2×C8×C73—22no—
160C4⋊C4×C73—24no—
161C4⋊C4×C73—24no—
162C4⋊C4×C73—24no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73—21no—
165C4⋊C4×C73—22no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—22no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—22no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73—22no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73C2×D412yes—
175C4⋊C4×C73C2×D412yes—
176C4⋊C4×C73C2×D412yes—
177C22⋊C4×C73—24no—
178C22⋊C4×C73—24no—
179C22⋊C4×C73—24no—
180C22⋊C4×C73—22no—
181C22⋊C4×C73—22no—
182C22⋊C4×C73—22no—
183C22⋊C4×C73—41no—
184C22⋊C4×C73—41no—
185C42×C73—24no—
186C42×C73—22no—
187C42×C73—22no—
188C42×C73—22no—
189C42×C73C2×D411yescharacteristic
190C42×C73C2×D411yescharacteristic
191C42×C73C2×D411yescharacteristic
192C42×C73C2411yescharacteristic
193C23×C73—22no—
194C23×C73—41no—
195C23×C73#12011yescharacteristic
196Q8×C73—22no—
197Q8×C73—22no—
198Q8×C73—22no—
199Q8×C73—22no—
200Q8×C73—42no—
201Q8×C73—42no—
202Q8×C73—42no—
203Q8×C73—81no—
204D4×C73—42no—
205D4×C73—42no—
206D4×C73—44no—
207D4×C73—42no—
208D4×C73—42no—
209D4×C73—42no—
210D4×C73—42no—
211D4×C73—82no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—41no—
220C2×C4×C73—44no—
221C2×C4×C73—42no—
222C2×C4×C73—41no—
223C2×C4×C73—82no—
224C2×C4×C73—81no—
225C2×C4×C73—84no—
226C2×C4×C73#12011yescharacteristic
227C2×C4×C73#12011yescharacteristic
228C2×C4×C73#12011yescharacteristic
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C22×D411yescharacteristic
232C8×C73—22no—
233C8×C73—41no—
234C8×C73—42no—
235C8×C73—42no—
236C22×C73—21no—
237C22×C73—21no—
238C22×C73—21no—
239C22×C73—42no—
240C22×C73—42no—
241C22×C73—42no—
242C22×C73—81no—
243C22×C73—81no—
244C22×C73—81no—
245C22×C73C2×#12011yescharacteristic
246C4×C73—21no—
247C4×C73—21no—
248C4×C73—21no—
249C4×C73—21no—
250C4×C73—42no—
251C4×C73—42no—
252C4×C73—42no—
253C4×C73—42no—
254C4×C73—42no—
255C4×C73—81no—
256C4×C73#57512yes—
257C4×C73#57712yes—
258#18602C7311yesmaximal, characteristic, direct product factor, Sylow
259C2×C73—41no—
260C2×C73—82no—
261C2×C73—161no—
262C2×C73#300511yescharacteristic
263C2×C73#301011yescharacteristic
264C2×C73#301811yescharacteristic
265#3507C2×C7311yescharacteristic, complemented
266#3216C2×C7311yescharacteristic, complemented
267#3184C2×C7311yescharacteristic
268#3103C2×C7311yescharacteristic, complemented
269#1653C2×C7312yescomplemented
270#1640C2×C7312yescomplemented
271#1626C2×C7312yescomplemented
272#1620C2×C7312yescomplemented
273#1553C2×C7311yescharacteristic, complemented
274#1491C2×C7311yescharacteristic, complemented
275#1485C2×C7311yescharacteristic, complemented
276C73#1860211yescharacteristic, direct product factor, Sylow
277#559C22×C7312yes—
278#557C22×C7312yes—
279#556—21no—
280#556—22no—
281#554—21no—
282#552—22no—
283#539—22no—
284#536—21no—
285#533C22×C7311yescharacteristic
286#532C22×C7311yescharacteristic
287#532C22×C7311yescharacteristic
288#529—22no—
289#520—21no—
290C2×C4×Q8C22×C7311yescharacteristic
291#498C22×C7311yescharacteristic
292#497—21no—
293#493C22×C7311yescharacteristic
294#489—21no—
295#486C22×C7311yescharacteristic
296#486C22×C7311yescharacteristic
297#481—21no—
298#475C22×C7312yes—
299#469—22no—
300#462C22×C7312yes—
301#460C22×C7312yes—
302#460C22×C7312yes—
303#452—22no—
304#448—21no—
305#442—22no—
306C4×SD16—22no—
307#432C22×C7311yescharacteristic
308#431C22×C7311yescharacteristic
309C4×M16C22×C7311yescharacteristic
310#332C22×C7312yes—
311#331C22×C7312yes—
312Q8⋊C8C22×C7312yes—
313Q8⋊C8C22×C7312yes—
314Q8⋊C8C22×C7312yes—
315Q8⋊C8C22×C7312yes—
316C2×#41—21no—
317C2×#41—21no—
318C22×Q8D4×C7311yescharacteristic
319C2×Q16—22no—
320C2×SD16—22no—
321C2×SD16—22no—
322C2×D8—22no—
323C2×M16—21no—
324C4⋊Q8—22no—
325C4⋊Q8D4×C7311yescharacteristic
326C4⋊Q8D4×C7311yescharacteristic
327C4⋊Q8C23×C7311yescharacteristic
328C4⋊Q8C23×C7311yescharacteristic
329#127C23×C7311yescharacteristic
330#127C23×C7311yescharacteristic
331#126—24no—
332#126—24no—
333#125—22no—
334#125—24no—
335#125—21no—
336#124—22no—
337#124—22no—
338#124—21no—
339#124—21no—
340#123—24no—
341#123—24no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—24no—
345C22⋊Q8—21no—
346C22⋊Q8D4×C7312yes—
347#121—21no—
348#121—21no—
349#121—21no—
350#121D4×C7312yes—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—21no—
354C4×Q8—21no—
355C4×Q8—21no—
356C4×Q8D4×C7312yes—
357C4×Q8C23×C7312yes—
358C4×Q8C23×C7312yes—
359C4×D4—24no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4—21no—
365C4×D4D4×C7312yes—
366#117—22no—
367#117—24no—
368#117—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4D4×C7311yescharacteristic
372C2×C42—21no—
373C2×C42C23×C7311yescharacteristic
374#106—22no—
375#106—22no—
376C4⋊C8C23×C7312yes—
377C4⋊C8C23×C7312yes—
378#103—22no—
379#103—22no—
380#102—22no—
381#102—22no—
382#102—22no—
383#102—22no—
384C22⋊C8—21no—
385C22⋊C8—21no—
386#97—21no—
387#97—21no—
388C4×C8—21no—
389C4×C8C23×C7311yescharacteristic
390C4×C8C23×C7311yescharacteristic
391#41—44no—
392#41—82no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—42no—
398C2×Q8C2×D4×C7312yes—
399C2×Q8C2×D4×C7312yes—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—41no—
407C2×D4C2×D4×C7312yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4C2×D4×C7311yescharacteristic
414C22×C4C2×D4×C7311yescharacteristic
415C22×C4C2×D4×C7311yescharacteristic
416Q16—82no—
417SD16—44no—
418SD16—82no—
419D8—44no—
420M16—42no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—21no—
424C2×C8—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4C2×D4×C7312yes—
440C4⋊C4C2×D4×C7312yes—
441C4⋊C4C2×D4×C7312yes—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C42—24no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42C2×D4×C7311yescharacteristic
455C42C2×D4×C7311yescharacteristic
456C42C2×D4×C7311yescharacteristic
457C42C24×C7311yescharacteristic
458C23—22no—
459C23—41no—
460C23#120×C7311yescharacteristic
461Q8—22no—
462Q8—22no—
463Q8—22no—
464Q8—22no—
465Q8—42no—
466Q8—42no—
467Q8—42no—
468Q8—81no—
469D4—42no—
470D4—42no—
471D4—44no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—82no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—84no—
491C2×C4#120×C7311yescharacteristic
492C2×C4#120×C7311yescharacteristic
493C2×C4#120×C7311yescharacteristic
494C2×C4C22×D4×C7311yescharacteristic
495C2×C4C22×D4×C7311yescharacteristic
496C2×C4C22×D4×C7311yescharacteristic
497C8—22no—
498C8—41no—
499C8—42no—
500C8—42no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22C2×#120×C7311yescharacteristic
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—81no—
521C4#575×C7312yes—
522C4#577×C7312yes—
523C2—41no—
524C2—82no—
525C2—161no—
526C2C73×#300511yescharacteristic
527C2C73×#301011yescharacteristic
528C2C73×#301811yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3184C2
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. C73×#3507C2semidirect
2. C73×#3216C2semidirect
4. C73×#3103C2semidirect
5. C73×#1653C2semidirect
6. C73×#1640C2semidirect
7. C73×#1626C2semidirect
8. C73×#1620C2semidirect
9. C73×#1553C2semidirect
10. C73×#1491C2semidirect
11. C73×#1485C2semidirect
258. #18602C73direct (N × H)
265. #3507C2×C73semidirect
266. #3216C2×C73semidirect
268. #3103C2×C73semidirect
269. #1653C2×C73semidirect
270. #1640C2×C73semidirect
271. #1626C2×C73semidirect
272. #1620C2×C73semidirect
273. #1553C2×C73semidirect
274. #1491C2×C73semidirect
275. #1485C2×C73semidirect
276. C73#18602direct (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×#32162
C73×#31842
C73×#31032
C73×#16532
C73×#16402
C73×#16262
C73×#16202
C73×#15532
C73×#14912
C73×#14852
#1860273