Finite Group Databasebeta

← C73×#18821 #32928225

Subgroups — C73×#18821 #32928225

Order 18688.

906 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3511C211yesmaximal, characteristic
2C73×#3235C211yesmaximal, characteristic, complemented
3C73×#3209C211yesmaximal, characteristic, complemented
4C73×#3116C211yesmaximal, characteristic, complemented
5C73×#1646C212yesmaximal, complemented
6C73×#1644C212yesmaximal, complemented
7C73×#1634C212yesmaximal, complemented
8C73×#1632C212yesmaximal, complemented
9C73×#1558C211yesmaximal, characteristic, complemented
10C73×#1496C211yesmaximal, characteristic, complemented
11C73×#1488C211yesmaximal, characteristic, complemented
12#556×C73—21no—
13#555×C73C2212yes—
14#554×C73C2212yes—
15#553×C73—22no—
16#548×C73—22no—
17#547×C73—21no—
18#542×C73—22no—
19#537×C73—21no—
20#533×C73—21no—
21#533×C73C2211yescharacteristic
22#532×C73—21no—
23#532×C73C2211yescharacteristic
24C2×C4⋊Q8×C73C2211yescharacteristic
25#520×C73—21no—
26C4×#41×C73C2211yescharacteristic
27#501×C73—21no—
28#494×C73C2212yes—
29#490×C73—21no—
30#487×C73C2212yes—
31#485×C73—21no—
32#482×C73—22no—
33#477×C73C2211yescharacteristic
34#476×C73C2212yes—
35#476×C73C2211yescharacteristic
36#474×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—
50D4⋊C8×C73C2212yes—
51D4⋊C8×C73C2212yes—
52D4⋊C8×C73C2212yes—
53C2×#41×C73—22no—
54C2×#41×C73—21no—
55C2×#41×C73D411yescharacteristic
56C22×Q8×C73D412yes—
57C2×Q16×C73—21no—
58C2×Q16×C73—24no—
59C2×SD16×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73—21no—
62C4⋊Q8×C73D412yes—
63C4⋊Q8×C73D411yescharacteristic
64C4⋊Q8×C73C2311yescharacteristic
65C4⋊Q8×C73C2311yescharacteristic
66C4⋊Q8×C73C2312yes—
67#127×C73D411yescharacteristic
68#126×C73—24no—
69#125×C73—22no—
70#125×C73—21no—
71#124×C73—22no—
72#124×C73—24no—
73#124×C73—21no—
74#123×C73—24no—
75C22⋊Q8×C73—22no—
76C22⋊Q8×C73—22no—
77C22⋊Q8×C73—22no—
78C22⋊Q8×C73—24no—
79C22⋊Q8×C73—21no—
80C22⋊Q8×C73—21no—
81#121×C73—22no—
82#121×C73—24no—
83#121×C73—22no—
84#121×C73—22no—
85#121×C73—21no—
86#121×C73D412yes—
87C4×Q8×C73—22no—
88C4×Q8×C73—22no—
89C4×Q8×C73—21no—
90C4×Q8×C73—21no—
91C4×Q8×C73D411yescharacteristic
92C4×Q8×C73C2311yescharacteristic
93C4×D4×C73—22no—
94C4×D4×C73—24no—
95C4×D4×C73—22no—
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73—21no—
99C4×D4×C73D411yescharacteristic
100C4×D4×C73C2311yescharacteristic
101C4×D4×C73C2312yes—
102#117×C73—22no—
103#117×C73—22no—
104#117×C73—21no—
105#117×C73—21no—
106#117×C73D411yescharacteristic
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—21no—
123#103×C73—21no—
124#102×C73—21no—
125#102×C73—22no—
126#102×C73—22no—
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—
135#41×C73—42no—
136#41×C73—42no—
137#41×C73—84no—
138C2×Q8×C73—21no—
139C2×Q8×C73—22no—
140C2×Q8×C73—21no—
141C2×Q8×C73—21no—
142C2×Q8×C73—82no—
143C2×Q8×C73C2×D411yescharacteristic
144C2×Q8×C73C2×D414yes—
145C2×D4×C73—21no—
146C2×D4×C73—22no—
147C2×D4×C73—24no—
148C2×D4×C73—21no—
149C2×D4×C73—21no—
150C2×D4×C73C2×D412yes—
151C2×D4×C73C2×D411yescharacteristic
152C22×C4×C73—24no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73C2×D411yescharacteristic
159C22×C4×C73C2×D411yescharacteristic
160C22×C4×C73C2×D411yescharacteristic
161Q16×C73—42no—
162Q16×C73—84no—
163SD16×C73—42no—
164C2×C8×C73—22no—
165C2×C8×C73—22no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C4⋊C4×C73—24no—
169C4⋊C4×C73—24no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—22no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—22no—
175C4⋊C4×C73—22no—
176C4⋊C4×C73—21no—
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—24no—
188C22⋊C4×C73—24no—
189C22⋊C4×C73—21no—
190C22⋊C4×C73—21no—
191C22⋊C4×C73—22no—
192C22⋊C4×C73—21no—
193C22⋊C4×C73—22no—
194C22⋊C4×C73—41no—
195C42×C73—24no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—41no—
200C42×C73C2×D411yescharacteristic
201C42×C73C2×D411yescharacteristic
202C42×C73C2×D411yescharacteristic
203C42×C73C2411yescharacteristic
204C23×C73—21no—
205C23×C73—22no—
206C23×C73#12011yescharacteristic
207Q8×C73—21no—
208Q8×C73—21no—
209Q8×C73—44no—
210Q8×C73—44no—
211Q8×C73—42no—
212Q8×C73—42no—
213Q8×C73—41no—
214Q8×C73—41no—
215D4×C73—21no—
216D4×C73—21no—
217D4×C73—24no—
218D4×C73—42no—
219D4×C73—42no—
220D4×C73—42no—
221D4×C73—44no—
222D4×C73—84no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—24no—
231C2×C4×C73—41no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—82no—
240C2×C4×C73—84no—
241C2×C4×C73#12011yescharacteristic
242C2×C4×C73#12011yescharacteristic
243C2×C4×C73#12011yescharacteristic
244C2×C4×C73C22×D411yescharacteristic
245C2×C4×C73C22×D411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C8×C73—42no—
248C8×C73—41no—
249C8×C73—41no—
250C8×C73—42no—
251C22×C73—21no—
252C22×C73—21no—
253C22×C73—21no—
254C22×C73—41no—
255C22×C73—42no—
256C22×C73—41no—
257C22×C73—42no—
258C22×C73—41no—
259C22×C73—42no—
260C22×C73C2×#12011yescharacteristic
261C4×C73—21no—
262C4×C73—21no—
263C4×C73—21no—
264C4×C73—21no—
265C4×C73—41no—
266C4×C73—42no—
267C4×C73—41no—
268C4×C73—41no—
269C4×C73—44no—
270C4×C73—81no—
271C4×C73—81no—
272C4×C73#57511yescharacteristic
273C4×C73#57511yescharacteristic
274C4×C73#57812yes—
275#18821C7311yesmaximal, characteristic, direct product factor, Sylow
276C2×C73—41no—
277C2×C73—81no—
278C2×C73—82no—
279C2×C73#300511yescharacteristic
280C2×C73#301111yescharacteristic
281C2×C73#301911yescharacteristic
282#3511C2×C7311yescharacteristic
283#3235C2×C7311yescharacteristic, complemented
284#3209C2×C7311yescharacteristic, complemented
285#3116C2×C7311yescharacteristic, complemented
286#1646C2×C7312yescomplemented
287#1644C2×C7312yescomplemented
288#1634C2×C7312yescomplemented
289#1632C2×C7312yescomplemented
290#1558C2×C7311yescharacteristic, complemented
291#1496C2×C7311yescharacteristic, complemented
292#1488C2×C7311yescharacteristic, complemented
293C73#1882111yescharacteristic, direct product factor, Sylow
294#556—21no—
295#555C22×C7312yes—
296#554C22×C7312yes—
297#553—22no—
298#548—22no—
299#547—21no—
300#542—22no—
301#537—21no—
302#533—21no—
303#533C22×C7311yescharacteristic
304#532—21no—
305#532C22×C7311yescharacteristic
306C2×C4⋊Q8C22×C7311yescharacteristic
307#520—21no—
308C4×#41C22×C7311yescharacteristic
309#501—21no—
310#494C22×C7312yes—
311#490—21no—
312#487C22×C7312yes—
313#485—21no—
314#482—22no—
315#477C22×C7311yescharacteristic
316#476C22×C7311yescharacteristic
317#476C22×C7312yes—
318#474C22×C7312yes—
319#462—22no—
320#462C22×C7311yescharacteristic
321#461C22×C7311yescharacteristic
322#452—21no—
323#451—22no—
324#441—21no—
325#440—21no—
326#432C22×C7311yescharacteristic
327#432C22×C7311yescharacteristic
328#431C22×C7311yescharacteristic
329#333C22×C7312yes—
330#333C22×C7312yes—
331Q8⋊C8C22×C7312yes—
332D4⋊C8C22×C7312yes—
333D4⋊C8C22×C7312yes—
334D4⋊C8C22×C7312yes—
335C2×#41—22no—
336C2×#41—21no—
337C2×#41D4×C7311yescharacteristic
338C22×Q8D4×C7312yes—
339C2×Q16—21no—
340C2×Q16—24no—
341C2×SD16—21no—
342C4⋊Q8—21no—
343C4⋊Q8—21no—
344C4⋊Q8D4×C7312yes—
345C4⋊Q8D4×C7311yescharacteristic
346C4⋊Q8C23×C7311yescharacteristic
347C4⋊Q8C23×C7311yescharacteristic
348C4⋊Q8C23×C7312yes—
349#127D4×C7311yescharacteristic
350#126—24no—
351#125—22no—
352#125—21no—
353#124—22no—
354#124—24no—
355#124—21no—
356#123—24no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—24no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363#121—22no—
364#121—24no—
365#121—22no—
366#121—22no—
367#121—21no—
368#121D4×C7312yes—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—21no—
372C4×Q8—21no—
373C4×Q8D4×C7311yescharacteristic
374C4×Q8C23×C7311yescharacteristic
375C4×D4—22no—
376C4×D4—24no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4—21no—
381C4×D4D4×C7311yescharacteristic
382C4×D4C23×C7311yescharacteristic
383C4×D4C23×C7312yes—
384#117—22no—
385#117—22no—
386#117—21no—
387#117—21no—
388#117D4×C7311yescharacteristic
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—22no—
392C2×C42—21no—
393C2×C42C23×C7311yescharacteristic
394#107—21no—
395#107—22no—
396#106—21no—
397#106—22no—
398C4⋊C8—21no—
399C4⋊C8C23×C7311yescharacteristic
400C4⋊C8C23×C7311yescharacteristic
401C4⋊C8C23×C7312yes—
402#103—21no—
403#103—24no—
404#103—21no—
405#103—21no—
406#102—21no—
407#102—22no—
408#102—22no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412#97—21no—
413#97—21no—
414C4×C8C23×C7312yes—
415#41—41no—
416#41—41no—
417#41—42no—
418#41—42no—
419#41—84no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—82no—
425C2×Q8C2×D4×C7311yescharacteristic
426C2×Q8C2×D4×C7314yes—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C2×D4×C7312yes—
433C2×D4C2×D4×C7311yescharacteristic
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4C2×D4×C7311yescharacteristic
441C22×C4C2×D4×C7311yescharacteristic
442C22×C4C2×D4×C7311yescharacteristic
443Q16—42no—
444Q16—84no—
445SD16—42no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—21no—
449C2×C8—21no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4C2×D4×C7311yescharacteristic
466C4⋊C4C2×D4×C7311yescharacteristic
467C4⋊C4C2×D4×C7312yes—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—21no—
475C22⋊C4—22no—
476C22⋊C4—41no—
477C42—24no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42C2×D4×C7311yescharacteristic
483C42C2×D4×C7311yescharacteristic
484C42C2×D4×C7311yescharacteristic
485C42C24×C7311yescharacteristic
486C23—21no—
487C23—22no—
488C23#120×C7311yescharacteristic
489Q8—21no—
490Q8—21no—
491Q8—44no—
492Q8—44no—
493Q8—42no—
494Q8—42no—
495Q8—41no—
496Q8—41no—
497D4—21no—
498D4—21no—
499D4—24no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—84no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4#120×C7311yescharacteristic
524C2×C4#120×C7311yescharacteristic
525C2×C4#120×C7311yescharacteristic
526C2×C4C22×D4×C7311yescharacteristic
527C2×C4C22×D4×C7311yescharacteristic
528C2×C4C22×D4×C7311yescharacteristic
529C8—42no—
530C8—41no—
531C8—41no—
532C8—42no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—42no—
542C22C2×#120×C7311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—41no—
548C4—42no—
549C4—41no—
550C4—41no—
551C4—44no—
552C4—81no—
553C4—81no—
554C4#575×C7311yescharacteristic
555C4#575×C7311yescharacteristic
556C4#578×C7312yes—
557C2—41no—
558C2—81no—
559C2—82no—
560C2C73×#300511yescharacteristic
561C2C73×#301111yescharacteristic
562C2C73×#301911yescharacteristic
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC42C24×C73
prime radicalC73×#3511C2
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×#3235C2semidirect
3. C73×#3209C2semidirect
4. C73×#3116C2semidirect
5. C73×#1646C2semidirect
6. C73×#1644C2semidirect
7. C73×#1634C2semidirect
8. C73×#1632C2semidirect
9. C73×#1558C2semidirect
10. C73×#1496C2semidirect
11. C73×#1488C2semidirect
275. #18821C73direct (N × H)
283. #3235C2×C73semidirect
284. #3209C2×C73semidirect
285. #3116C2×C73semidirect
286. #1646C2×C73semidirect
287. #1644C2×C73semidirect
288. #1634C2×C73semidirect
289. #1632C2×C73semidirect
290. #1558C2×C73semidirect
291. #1496C2×C73semidirect
292. #1488C2×C73semidirect
293. C73#18821direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#35112
C73×#32352
C73×#32092
C73×#31162
C73×#16462
C73×#16442
C73×#16342
C73×#16322
C73×#15582
C73×#14962
C73×#14882
#1882173