Finite Group Databasebeta

← C73×#12762 #32922166

Subgroups — C73×#12762 #32922166

Order 18688.

944 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#560×C73C211yesmaximal, characteristic
2C2×#353×C73C211yesmaximal, characteristic, complemented
3C2×#352×C73C211yesmaximal, characteristic, complemented
4C2×AΣL(1,𝔽16)×C73C211yesmaximal, characteristic, complemented
5C73×#1890C211yesmaximal, characteristic, complemented
6C73×#1889C211yesmaximal, characteristic, complemented
7C73×#1887C211yesmaximal, characteristic, complemented
8#560×C73—22no—
9#560×C73—22no—
10#560×C73—22no—
11#560×C73C414yescomplemented
12C2×#127×C73C2211yescharacteristic
13C2×#126×C73—21no—
14C2×#123×C73—21no—
15C2×#123×C73C2211yescharacteristic
16C2×#121×C73—21no—
17C2×#121×C73—21no—
18C2×#121×C73—21no—
19C2×#120×C73—21no—
20C2×#120×C73C2211yescharacteristic
21#518×C73C2211yescharacteristic
22#418×C73—21no—
23C2×#100×C73C2211yescharacteristic
24C2×#99×C73C2211yescharacteristic
25C2×#99×C73C2211yescharacteristic
26#381×C73—21no—
27#379×C73—21no—
28#353×C73—22no—
29#352×C73—22no—
30AΣL(1,𝔽16)×C73—22no—
31#342×C73—21no—
32#342×C73—21no—
33#339×C73—21no—
34C25×C73D411yescharacteristic
35C22×D4×C73—21no—
36C22×D4×C73—21no—
37C22×D4×C73—21no—
38C22×D4×C73—21no—
39C22×D4×C73—21no—
40C22×D4×C73D411yescharacteristic
41C22×D4×C73C2311yescharacteristic
42C23×C4×C73—21no—
43C23×C4×C73D411yescharacteristic
44C2×M16×C73—21no—
45#127×C73—21no—
46#127×C73—21no—
47#127×C73C2×C412yes—
48#127×C73D412yes—
49#126×C73—24no—
50#126×C73—24no—
51#123×C73—24no—
52#123×C73—21no—
53#123×C73—21no—
54#123×C73—24no—
55#123×C73C2×C412yes—
56#123×C73D412yes—
57#121×C73—24no—
58#121×C73—24no—
59#121×C73—24no—
60#121×C73—24no—
61#121×C73—24no—
62#121×C73—24no—
63#120×C73—24no—
64#120×C73—24no—
65#120×C73—21no—
66#120×C73—21no—
67#120×C73C2×C412yes—
68#120×C73D412yes—
69C4×D4×C73—42no—
70#117×C73—21no—
71C2×C4⋊C4×C73—21no—
72C2×C4⋊C4×C73—21no—
73C2×C4⋊C4×C73—21no—
74C2×C22⋊C4×C73—21no—
75C2×C22⋊C4×C73—21no—
76C2×C22⋊C4×C73—21no—
77C2×C22⋊C4×C73—21no—
78C2×C22⋊C4×C73—21no—
79C2×C22⋊C4×C73—21no—
80C2×C22⋊C4×C73—21no—
81C2×C22⋊C4×C73—21no—
82C2×C22⋊C4×C73—21no—
83C2×C22⋊C4×C73—42no—
84C2×C22⋊C4×C73D411yescharacteristic
85C2×C22⋊C4×C73D411yescharacteristic
86C2×C42×C73D411yescharacteristic
87#102×C73—42no—
88#100×C73—21no—
89#100×C73—21no—
90#99×C73—21no—
91#99×C73—21no—
92#99×C73—21no—
93#99×C73—21no—
94#95×C73—42no—
95C24×C73—21no—
96C24×C73—21no—
97C24×C73—21no—
98C24×C73—41no—
99C24×C73—41no—
100C24×C73—82no—
101C24×C73C22⋊C412yes—
102C24×C73C2×D411yescharacteristic
103C24×C73C2×D411yescharacteristic
104C2×D4×C73—24no—
105C2×D4×C73—24no—
106C2×D4×C73—22no—
107C2×D4×C73—22no—
108C2×D4×C73—22no—
109C2×D4×C73—22no—
110C2×D4×C73—22no—
111C2×D4×C73—22no—
112C2×D4×C73—21no—
113C2×D4×C73—41no—
114C2×D4×C73—41no—
115C2×D4×C73—41no—
116C2×D4×C73—41no—
117C2×D4×C73—82no—
118C2×D4×C73—82no—
119C2×D4×C73—82no—
120C2×D4×C73—82no—
121C2×D4×C73—81no—
122C2×D4×C73—81no—
123C2×D4×C73C22⋊C412yescomplemented
124C2×D4×C73C22×C411yescharacteristic
125C2×D4×C73C2×D411yescharacteristic
126C2×D4×C73C2×D411yescharacteristic
127C2×D4×C73C2×D411yescharacteristic
128C22×C4×C73—21no—
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73—21no—
132C22×C4×C73—24no—
133C22×C4×C73—21no—
134C22×C4×C73—22no—
135C22×C4×C73—22no—
136C22×C4×C73—41no—
137C22×C4×C73—41no—
138C22×C4×C73—41no—
139C22×C4×C73—41no—
140C22×C4×C73—82no—
141C22×C4×C73C22⋊C412yescomplemented
142C22×C4×C73C2×D411yescharacteristic
143M16×C73—41no—
144M16×C73—41no—
145C2×C8×C73—41no—
146C4⋊C4×C73—24no—
147C4⋊C4×C73—24no—
148C4⋊C4×C73—22no—
149C4⋊C4×C73—22no—
150C4⋊C4×C73—22no—
151C22⋊C4×C73—24no—
152C22⋊C4×C73—24no—
153C22⋊C4×C73—24no—
154C22⋊C4×C73—22no—
155C22⋊C4×C73—22no—
156C22⋊C4×C73—21no—
157C22⋊C4×C73—21no—
158C22⋊C4×C73—22no—
159C22⋊C4×C73—22no—
160C22⋊C4×C73—41no—
161C22⋊C4×C73—41no—
162C22⋊C4×C73—41no—
163C22⋊C4×C73—41no—
164C22⋊C4×C73—41no—
165C22⋊C4×C73—41no—
166C22⋊C4×C73—41no—
167C22⋊C4×C73—84no—
168C22⋊C4×C73C22⋊C412yes—
169C22⋊C4×C73C22⋊C412yescomplemented
170C42×C73—21no—
171C42×C73—41no—
172C42×C73C22⋊C412yescomplemented
173C23×C73—21no—
174C23×C73—21no—
175C23×C73—21no—
176C23×C73—21no—
177C23×C73—22no—
178C23×C73—22no—
179C23×C73—42no—
180C23×C73—42no—
181C23×C73—42no—
182C23×C73—42no—
183C23×C73—42no—
184C23×C73—42no—
185C23×C73—42no—
186C23×C73—44no—
187C23×C73—41no—
188C23×C73—41no—
189C23×C73—41no—
190C23×C73—41no—
191C23×C73—81no—
192C23×C73—82no—
193C23×C73—82no—
194C23×C73—82no—
195C23×C73—84no—
196C23×C73C2×C22⋊C411yescharacteristic
197C23×C73C2×C22⋊C411yescharacteristic
198C23×C73#12011yescharacteristic
199C23×C73#12011yescharacteristic
200C23×C73#12011yescharacteristic
201D4×C73—41no—
202D4×C73—41no—
203D4×C73—41no—
204D4×C73—41no—
205D4×C73—42no—
206D4×C73—81no—
207D4×C73—82no—
208D4×C73—82no—
209D4×C73—84no—
210D4×C73—82no—
211D4×C73—82no—
212D4×C73—82no—
213D4×C73—82no—
214D4×C73—84no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—42no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—42no—
227C2×C4×C73—42no—
228C2×C4×C73—81no—
229C2×C4×C73—84no—
230C2×C4×C73—82no—
231C2×C4×C73—82no—
232C2×C4×C73—81no—
233C2×C4×C73—81no—
234C2×C4×C73—82no—
235C2×C4×C73—82no—
236C2×C4×C73C2×C22⋊C411yescharacteristic
237C2×C4×C73#12011yescharacteristic
238C8×C73—81no—
239C22×C73—22no—
240C22×C73—21no—
241C22×C73—21no—
242C22×C73—21no—
243C22×C73—21no—
244C22×C73—41no—
245C22×C73—41no—
246C22×C73—41no—
247C22×C73—41no—
248C22×C73—42no—
249C22×C73—42no—
250C22×C73—42no—
251C22×C73—42no—
252C22×C73—42no—
253C22×C73—42no—
254C22×C73—44no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—81no—
258C22×C73—81no—
259C22×C73—82no—
260C22×C73—82no—
261C22×C73—82no—
262C22×C73—84no—
263C22×C73—82no—
264C22×C73—82no—
265C22×C73#37911yescharacteristic
266C22×C73B(4,𝔽2)11yescharacteristic
267C22×C73#45811yescharacteristic
268C4×C73—21no—
269C4×C73—21no—
270C4×C73—42no—
271C4×C73—42no—
272C4×C73—42no—
273C4×C73—42no—
274C4×C73—82no—
275C4×C73—161no—
276C4×C73—161no—
277#12762C7311yesmaximal, characteristic, direct product factor, Sylow
278C2×C73—21no—
279C2×C73—21no—
280C2×C73—41no—
281C2×C73—41no—
282C2×C73—41no—
283C2×C73—41no—
284C2×C73—82no—
285C2×C73—82no—
286C2×C73#179011yescharacteristic
287C2×C73#220012yes—
288C2×#560C2×C7311yescharacteristic
289C2×#353C2×C7311yescharacteristic, complemented
290C2×#352C2×C7311yescharacteristic, complemented
291C2×AΣL(1,𝔽16)C2×C7311yescharacteristic, complemented
292#1890C2×C7311yescharacteristic, complemented
293#1889C2×C7311yescharacteristic, complemented
294#1887C2×C7311yescharacteristic, complemented
295C73#1276211yescharacteristic, direct product factor, Sylow
296#560—22no—
297#560—22no—
298#560—22no—
299#560C4×C7314yescomplemented
300C2×#127C22×C7311yescharacteristic
301C2×#126—21no—
302C2×#123—21no—
303C2×#123C22×C7311yescharacteristic
304C2×#121—21no—
305C2×#121—21no—
306C2×#121—21no—
307C2×#120—21no—
308C2×#120C22×C7311yescharacteristic
309#518C22×C7311yescharacteristic
310#418—21no—
311C2×#100C22×C7311yescharacteristic
312C2×#99C22×C7311yescharacteristic
313C2×#99C22×C7311yescharacteristic
314#381—21no—
315#379—21no—
316#353—22no—
317#352—22no—
318AΣL(1,𝔽16)—22no—
319#342—21no—
320#342—21no—
321#339—21no—
322C25D4×C7311yescharacteristic
323C22×D4—21no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4D4×C7311yescharacteristic
329C22×D4C23×C7311yescharacteristic
330C23×C4—21no—
331C23×C4D4×C7311yescharacteristic
332C2×M16—21no—
333#127—21no—
334#127—21no—
335#127C2×C4×C7312yes—
336#127D4×C7312yes—
337#126—24no—
338#126—24no—
339#123—24no—
340#123—21no—
341#123—21no—
342#123—24no—
343#123C2×C4×C7312yes—
344#123D4×C7312yes—
345#121—24no—
346#121—24no—
347#121—24no—
348#121—24no—
349#121—24no—
350#121—24no—
351#120—24no—
352#120—24no—
353#120—21no—
354#120—21no—
355#120C2×C4×C7312yes—
356#120D4×C7312yes—
357C4×D4—42no—
358#117—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—42no—
372C2×C22⋊C4D4×C7311yescharacteristic
373C2×C22⋊C4D4×C7311yescharacteristic
374C2×C42D4×C7311yescharacteristic
375#102—42no—
376#100—21no—
377#100—21no—
378#99—21no—
379#99—21no—
380#99—21no—
381#99—21no—
382#95—42no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—41no—
387C24—41no—
388C24—82no—
389C24C22⋊C4×C7312yes—
390C24C2×D4×C7311yescharacteristic
391C24C2×D4×C7311yescharacteristic
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4C22⋊C4×C7312yescomplemented
412C2×D4C22×C4×C7311yescharacteristic
413C2×D4C2×D4×C7311yescharacteristic
414C2×D4C2×D4×C7311yescharacteristic
415C2×D4C2×D4×C7311yescharacteristic
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—82no—
429C22×C4C22⋊C4×C7312yescomplemented
430C22×C4C2×D4×C7311yescharacteristic
431M16—41no—
432M16—41no—
433C2×C8—41no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—84no—
456C22⋊C4C22⋊C4×C7312yes—
457C22⋊C4C22⋊C4×C7312yescomplemented
458C42—21no—
459C42—41no—
460C42C22⋊C4×C7312yescomplemented
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—81no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—84no—
484C23C2×C22⋊C4×C7311yescharacteristic
485C23C2×C22⋊C4×C7311yescharacteristic
486C23#120×C7311yescharacteristic
487C23#120×C7311yescharacteristic
488C23#120×C7311yescharacteristic
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—42no—
494D4—81no—
495D4—82no—
496D4—82no—
497D4—84no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—84no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—81no—
517C2×C4—84no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4C2×C22⋊C4×C7311yescharacteristic
525C2×C4#120×C7311yescharacteristic
526C8—81no—
527C22—22no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—81no—
546C22—81no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—84no—
551C22—82no—
552C22—82no—
553C22#379×C7311yescharacteristic
554C22B(4,𝔽2)×C7311yescharacteristic
555C22#458×C7311yescharacteristic
556C4—21no—
557C4—21no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—161no—
564C4—161no—
565C2—21no—
566C2—21no—
567C2—41no—
568C2—41no—
569C2—41no—
570C2—41no—
571C2—82no—
572C2—82no—
573C2C73×#179011yescharacteristic
574C2C73×#220012yes—
subgroupG/H
center, socleC22×C73B(4,𝔽2)
commutatorC2×D4C22×C4×C73
Frattini, cosocleC22×D4C23×C73
prime radicalC2×#560×C73C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C73 ◁ C23×C73 ◁ C22×D4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#353×C73C2semidirect
3. C2×#352×C73C2semidirect
4. C2×AΣL(1,𝔽16)×C73C2semidirect
5. C73×#1890C2semidirect
6. C73×#1889C2semidirect
7. C73×#1887C2semidirect
11. #560×C73C4semidirect
123. C2×D4×C73C22⋊C4semidirect
141. C22×C4×C73C22⋊C4semidirect
169. C22⋊C4×C73C22⋊C4semidirect
172. C42×C73C22⋊C4semidirect
277. #12762C73direct (N × H)
289. C2×#353C2×C73semidirect
290. C2×#352C2×C73semidirect
291. C2×AΣL(1,𝔽16)C2×C73semidirect
292. #1890C2×C73semidirect
293. #1889C2×C73semidirect
294. #1887C2×C73semidirect
295. C73#12762direct (N × H)
299. #560C4×C73semidirect
411. C2×D4C22⋊C4×C73semidirect
429. C22×C4C22⋊C4×C73semidirect
457. C22⋊C4C22⋊C4×C73semidirect
460. C42C22⋊C4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#560×C732
C2×#353×C732
C2×#352×C732
C2×AΣL(1,𝔽16)×C732
C73×#18902
C73×#18892
C73×#18872
#1276273