Finite Group Databasebeta

← C73×#24057 #32933461

Subgroups — C73×#24057 #32933461

Order 18688.

1392 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#3485C211yesmaximal, characteristic
2C73×#2853C212yesmaximal, complemented
3C73×#2297C211yesmaximal, characteristic, complemented
4C73×#2044C214yesmaximal, complemented
5C73×#1899C212yesmaximal, complemented
6C73×#1867C212yesmaximal, complemented
7C73×#1790C212yesmaximal, complemented
8C73×#1782C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C73C2211yescharacteristic
10C22×#41×C73C2211yescharacteristic
11#547×C73—24no—
12#546×C73—24no—
13#536×C73—21no—
14#534×C73—21no—
15#534×C73C2212yes—
16#525×C73—21no—
17C2×#123×C73—21no—
18C2×#123×C73C2212yes—
19#518×C73C2211yescharacteristic
20C2×#117×C73C2211yescharacteristic, complemented
21C22×C42×C73C2211yescharacteristic, complemented
22C2×#99×C73C2212yes—
23C2×#99×C73C2214yes—
24C2×#99×C73C2212yescomplemented
25#391×C73—21no—
26#390×C73—21no—
27#388×C73—24no—
28#387×C73—24no—
29#386×C73C2212yes—
30#386×C73C2214yes—
31#386×C73C2212yescomplemented
32#385×C73C2214yescomplemented
33#381×C73C2212yescomplemented
34C4×C22⋊C4×C73—21no—
35C4×C22⋊C4×C73—21no—
36#342×C73C2214yescomplemented
37H(5,𝔽2)×C73—24no—
38H(5,𝔽2)×C73—42no—
39C2×#41×C73—24no—
40C2×#41×C73—21no—
41C2×#41×C73—24no—
42C2×#41×C73—22no—
43C2×#41×C73—42no—
44C22×Q8×C73D411yescharacteristic
45C22×D4×C73—21no—
46C22×D4×C73—22no—
47C22×D4×C73D411yescharacteristic
48C22×D4×C73C2312yes—
49C22×D4×C73C2311yescharacteristic
50C22×D4×C73C2311yescharacteristic
51C23×C4×C73D411yescharacteristic
52C23×C4×C73D411yescharacteristic
53C23×C4×C73C2312yes—
54C23×C4×C73C2311yescharacteristic
55C4⋊Q8×C73—22no—
56#124×C73—22no—
57#123×C73—24no—
58#123×C73—22no—
59#123×C73—22no—
60#123×C73—22no—
61#123×C73—22no—
62#123×C73—24no—
63#123×C73—22no—
64C22⋊Q8×C73—22no—
65C22⋊Q8×C73—24no—
66#121×C73—22no—
67#121×C73—24no—
68#121×C73—24no—
69#120×C73—22no—
70#120×C73—22no—
71#120×C73—22no—
72C4×D4×C73—24no—
73C4×D4×C73—24no—
74#117×C73—21no—
75#117×C73—42no—
76C2×C4⋊C4×C73—22no—
77C2×C4⋊C4×C73—24no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73D414yes—
80C2×C22⋊C4×C73—22no—
81C2×C22⋊C4×C73—21no—
82C2×C22⋊C4×C73—22no—
83C2×C22⋊C4×C73—22no—
84C2×C22⋊C4×C73—24no—
85C2×C22⋊C4×C73D414yes—
86C2×C22⋊C4×C73C2314yes—
87C2×C22⋊C4×C73C2312yes—
88C2×C22⋊C4×C73C2312yes—
89C2×C42×C73—21no—
90C2×C42×C73—21no—
91C2×C42×C73—24no—
92C2×C42×C73—21no—
93#99×C73—24no—
94#99×C73—44no—
95#99×C73—42no—
96#95×C73—22no—
97#95×C73—22no—
98#95×C73—24no—
99#95×C73—24no—
100#95×C73—24no—
101C24×C73—21no—
102C24×C73C2×D412yes—
103C24×C73C2×D411yescharacteristic
104C24×C73C2×D411yescharacteristic
105C24×C73C2411yescharacteristic
106#41×C73—416no—
107#41×C73—48no—
108#41×C73—42no—
109#41×C73—42no—
110C2×Q8×C73—21no—
111C2×Q8×C73—22no—
112C2×Q8×C73—22no—
113C2×Q8×C73—41no—
114C2×D4×C73—22no—
115C2×D4×C73—22no—
116C2×D4×C73—22no—
117C2×D4×C73—21no—
118C2×D4×C73—22no—
119C2×D4×C73—22no—
120C2×D4×C73—22no—
121C2×D4×C73—22no—
122C2×D4×C73—22no—
123C2×D4×C73—24no—
124C2×D4×C73—22no—
125C2×D4×C73—24no—
126C2×D4×C73—24no—
127C2×D4×C73—22no—
128C2×D4×C73—41no—
129C2×D4×C73—48no—
130C2×D4×C73—42no—
131C2×D4×C73—42no—
132C2×D4×C73#4114yes—
133C22×C4×C73—22no—
134C22×C4×C73—22no—
135C22×C4×C73—21no—
136C22×C4×C73—22no—
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—24no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—24no—
145C22×C4×C73—22no—
146C22×C4×C73—44no—
147C22×C4×C73—41no—
148C22×C4×C73—41no—
149C22×C4×C73C2×D412yes—
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×D414yes—
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D414yes—
156C4⋊C4×C73—24no—
157C4⋊C4×C73—22no—
158C4⋊C4×C73—24no—
159C4⋊C4×C73—24no—
160C4⋊C4×C73—48no—
161C4⋊C4×C73—42no—
162C22⋊C4×C73—22no—
163C22⋊C4×C73—24no—
164C22⋊C4×C73—24no—
165C22⋊C4×C73—22no—
166C22⋊C4×C73—22no—
167C22⋊C4×C73—24no—
168C22⋊C4×C73—24no—
169C22⋊C4×C73—24no—
170C22⋊C4×C73—24no—
171C22⋊C4×C73—24no—
172C22⋊C4×C73—44no—
173C22⋊C4×C73—44no—
174C22⋊C4×C73—48no—
175C42×C73—22no—
176C42×C73—42no—
177C42×C73—42no—
178C42×C73—42no—
179C23×C73—21no—
180C23×C73—21no—
181C23×C73—22no—
182C23×C73—21no—
183C23×C73—21no—
184C23×C73—21no—
185C23×C73—24no—
186C23×C73—42no—
187C23×C73—42no—
188C23×C73—42no—
189C23×C73—42no—
190C23×C73—44no—
191C23×C73—82no—
192C23×C73#12312yes—
193C23×C73C22×D411yescharacteristic
194C23×C73C22×D412yes—
195C23×C73C2×#4112yes—
196C23×C73H(5,𝔽2)12yes—
197Q8×C73—44no—
198Q8×C73—42no—
199D4×C73—28no—
200D4×C73—44no—
201D4×C73—42no—
202D4×C73—44no—
203D4×C73—42no—
204D4×C73—42no—
205D4×C73—44no—
206D4×C73—44no—
207D4×C73—84no—
208C2×C4×C73—21no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—28no—
215C2×C4×C73—24no—
216C2×C4×C73—28no—
217C2×C4×C73—24no—
218C2×C4×C73—21no—
219C2×C4×C73—22no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—24no—
223C2×C4×C73—44no—
224C2×C4×C73—44no—
225C2×C4×C73—44no—
226C2×C4×C73—44no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—44no—
231C2×C4×C73—44no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—44no—
235C2×C4×C73—42no—
236C2×C4×C73—44no—
237C2×C4×C73—82no—
238C2×C4×C73#12012yes—
239C2×C4×C73#12012yes—
240C2×C4×C73#12312yes—
241C22×C73—24no—
242C22×C73—22no—
243C22×C73—22no—
244C22×C73—42no—
245C22×C73—42no—
246C22×C73—42no—
247C22×C73—42no—
248C22×C73—42no—
249C22×C73—41no—
250C22×C73—42no—
251C22×C73—42no—
252C22×C73—44no—
253C22×C73—81no—
254C22×C73—81no—
255C22×C73—84no—
256C22×C73—162no—
257C22×C73C2×#12011yescharacteristic
258C22×C73C2×#12311yescharacteristic
259C22×C73#53411yescharacteristic
260C22×C73#54614yes—
261C4×C73—22no—
262C4×C73—22no—
263C4×C73—22no—
264C4×C73—42no—
265C4×C73—41no—
266C4×C73—44no—
267C4×C73—44no—
268C4×C73—82no—
269C4×C73—84no—
270#24057C7311yesmaximal, characteristic, direct product factor, Sylow
271C2×C73—21no—
272C2×C73—24no—
273C2×C73—21no—
274C2×C73—42no—
275C2×C73—81no—
276C2×C73—82no—
277C2×C73#240511yescharacteristic
278C2×C73#302512yes—
279#3485C2×C7311yescharacteristic
280#2853C2×C7312yescomplemented
281#2297C2×C7311yescharacteristic, complemented
282#2044C2×C7314yescomplemented
283#1899C2×C7312yescomplemented
284#1867C2×C7312yescomplemented
285#1790C2×C7312yescomplemented
286#1782C2×C7311yescharacteristic, complemented
287C73#2405711yescharacteristic, direct product factor, Sylow
288C2×H(5,𝔽2)C22×C7311yescharacteristic
289C22×#41C22×C7311yescharacteristic
290#547—24no—
291#546—24no—
292#536—21no—
293#534—21no—
294#534C22×C7312yes—
295#525—21no—
296C2×#123—21no—
297C2×#123C22×C7312yes—
298#518C22×C7311yescharacteristic
299C2×#117C22×C7311yescharacteristic, complemented
300C22×C42C22×C7311yescharacteristic, complemented
301C2×#99C22×C7312yes—
302C2×#99C22×C7314yes—
303C2×#99C22×C7312yescomplemented
304#391—21no—
305#390—21no—
306#388—24no—
307#387—24no—
308#386C22×C7314yes—
309#386C22×C7312yes—
310#386C22×C7312yescomplemented
311#385C22×C7314yescomplemented
312#381C22×C7312yescomplemented
313C4×C22⋊C4—21no—
314C4×C22⋊C4—21no—
315#342C22×C7314yescomplemented
316H(5,𝔽2)—24no—
317H(5,𝔽2)—42no—
318C2×#41—24no—
319C2×#41—21no—
320C2×#41—24no—
321C2×#41—22no—
322C2×#41—42no—
323C22×Q8D4×C7311yescharacteristic
324C22×D4—21no—
325C22×D4—22no—
326C22×D4D4×C7311yescharacteristic
327C22×D4C23×C7312yes—
328C22×D4C23×C7311yescharacteristic
329C22×D4C23×C7311yescharacteristic
330C23×C4D4×C7311yescharacteristic
331C23×C4D4×C7311yescharacteristic
332C23×C4C23×C7311yescharacteristic
333C23×C4C23×C7312yes—
334C4⋊Q8—22no—
335#124—22no—
336#123—24no—
337#123—22no—
338#123—22no—
339#123—22no—
340#123—22no—
341#123—24no—
342#123—22no—
343C22⋊Q8—22no—
344C22⋊Q8—24no—
345#121—22no—
346#121—24no—
347#121—24no—
348#120—22no—
349#120—22no—
350#120—22no—
351C4×D4—24no—
352C4×D4—24no—
353#117—21no—
354#117—42no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4D4×C7314yes—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4D4×C7314yes—
365C2×C22⋊C4C23×C7312yes—
366C2×C22⋊C4C23×C7314yes—
367C2×C22⋊C4C23×C7312yes—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—24no—
371C2×C42—21no—
372#99—24no—
373#99—44no—
374#99—42no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380C24—21no—
381C24C2×D4×C7312yes—
382C24C2×D4×C7311yescharacteristic
383C24C2×D4×C7311yescharacteristic
384C24C24×C7311yescharacteristic
385#41—416no—
386#41—48no—
387#41—42no—
388#41—42no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—41no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—41no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4#41×C7314yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4C2×D4×C7312yes—
429C22×C4C2×D4×C7311yescharacteristic
430C22×C4C2×D4×C7314yes—
431C22×C4C2×D4×C7311yescharacteristic
432C22×C4C2×D4×C7311yescharacteristic
433C22×C4C2×D4×C7311yescharacteristic
434C22×C4C2×D4×C7314yes—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C42—22no—
455C42—42no—
456C42—42no—
457C42—42no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—44no—
470C23—82no—
471C23#123×C7312yes—
472C23C22×D4×C7311yescharacteristic
473C23C22×D4×C7312yes—
474C23C2×#41×C7312yes—
475C23H(5,𝔽2)×C7312yes—
476Q8—44no—
477Q8—42no—
478D4—28no—
479D4—44no—
480D4—42no—
481D4—44no—
482D4—42no—
483D4—42no—
484D4—44no—
485D4—44no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—82no—
517C2×C4#120×C7312yes—
518C2×C4#120×C7312yes—
519C2×C4#123×C7312yes—
520C22—24no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—41no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—84no—
535C22—162no—
536C22C2×#120×C7311yescharacteristic
537C22C2×#123×C7311yescharacteristic
538C22#534×C7311yescharacteristic
539C22#546×C7314yes—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—42no—
546C4—41no—
547C4—82no—
548C4—84no—
549C2—21no—
550C2—24no—
551C2—21no—
552C2—42no—
553C2—81no—
554C2—82no—
555C2C73×#240511yescharacteristic
556C2C73×#302512yes—
subgroupG/H
center, socleC22×C73C2×#120
commutator, Frattini, cosocleC24C24×C73
prime radicalC73×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C4×C73 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C73×#2853C2semidirect
3. C73×#2297C2semidirect
4. C73×#2044C2semidirect
5. C73×#1899C2semidirect
6. C73×#1867C2semidirect
7. C73×#1790C2semidirect
8. C73×#1782C2semidirect
20. C2×#117×C73C22semidirect
21. C22×C42×C73C22semidirect
24. C2×#99×C73C22semidirect
31. #386×C73C22semidirect
32. #385×C73C22semidirect
33. #381×C73C22semidirect
36. #342×C73C22semidirect
270. #24057C73direct (N × H)
280. #2853C2×C73semidirect
281. #2297C2×C73semidirect
282. #2044C2×C73semidirect
283. #1899C2×C73semidirect
284. #1867C2×C73semidirect
285. #1790C2×C73semidirect
286. #1782C2×C73semidirect
287. C73#24057direct (N × H)
299. C2×#117C22×C73semidirect
300. C22×C42C22×C73semidirect
303. C2×#99C22×C73semidirect
310. #386C22×C73semidirect
311. #385C22×C73semidirect
312. #381C22×C73semidirect
315. #342C22×C73semidirect

Maximal subgroups

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

subgroupindex
C73×#34852
C73×#28532
C73×#22972
C73×#20442
C73×#18992
C73×#18672
C73×#17902
C73×#17822
#2405773