Finite Group Databasebeta

← C73×#24690 #32934094

Subgroups — C73×#24690 #32934094

Order 18688.

808 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478×C73C211yesmaximal, characteristic
2C2×#440×C73C211yesmaximal, characteristic
3C73×#2565C211yesmaximal, characteristic, complemented
4C73×#2354C211yesmaximal, characteristic
5C73×#2073C211yesmaximal, characteristic
6C73×#2071C211yesmaximal, characteristic
7C73×#2055C211yesmaximal, characteristic
8C73×#2055C211yesmaximal, characteristic
9C73×#1992C211yesmaximal, characteristic, complemented
10C73×#1949C211yesmaximal, characteristic, complemented
11C73×#1947C211yesmaximal, characteristic, complemented
12C73×#1940C211yesmaximal, characteristic, complemented
13C73×#1928C211yesmaximal, characteristic, complemented
14C73×#1927C211yesmaximal, characteristic, complemented
15C73×#1803C211yesmaximal, characteristic, complemented
16C22×SD16×C73C2211yescharacteristic
17C2×#125×C73C2211yescharacteristic
18C2×C4×Q8×C73C2211yescharacteristic
19C2×C4×D4×C73C2211yescharacteristic
20C2×#117×C73—21no—
21#478×C73—24no—
22#440×C73—24no—
23C2×#107×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C2×#106×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27C2×C4⋊C8×C73C2211yescharacteristic
28C2×C4⋊C8×C73C2211yescharacteristic
29C2×#103×C73C2211yescharacteristic
30C2×#103×C73C2211yescharacteristic
31C2×#103×C73C2211yescharacteristic
32C2×#102×C73C2211yescharacteristic
33C2×#102×C73C2211yescharacteristic
34C2×#102×C73C2211yescharacteristic
35C2×#97×C73C2211yescharacteristic
36#398×C73—21no—
37#398×C73—21no—
38#391×C73C2211yescharacteristic
39#391×C73C2211yescharacteristic
40#390×C73C2211yescharacteristic
41#390×C73C2211yescharacteristic
42#388×C73—21no—
43#388×C73—21no—
44#387×C73—21no—
45#387×C73—21no—
46#384×C73C2211yescharacteristic
47#384×C73C2211yescharacteristic
48#382×C73C2211yescharacteristic
49#380×C73—21no—
50C4×C4⋊C4×C73C2211yescharacteristic
51C4×C4⋊C4×C73C2211yescharacteristic
52C4×C4⋊C4×C73C2211yescharacteristic
53#340×C73C2211yescharacteristic
54#340×C73C2211yescharacteristic
55#340×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Q8×C73C2311yescharacteristic
62C22×D4×C73C2311yescharacteristic
63C23×C4×C73—21no—
64C2×SD16×C73—22no—
65C2×SD16×C73—22no—
66C2×SD16×C73—22no—
67C22×C8×C73C2311yescharacteristic
68C22×C8×C73C2311yescharacteristic
69C22×C8×C73C2311yescharacteristic
70C22×C8×C73C2311yescharacteristic
71#125×C73—22no—
72#125×C73—22no—
73C4×Q8×C73—22no—
74C4×Q8×C73—22no—
75C4×D4×C73—22no—
76C4×D4×C73D414yes—
77#117×C73—44no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73—21no—
83C2×C4⋊C4×C73—21no—
84C2×C4⋊C4×C73—21no—
85C2×C4⋊C4×C73D411yescharacteristic
86C2×C4⋊C4×C73D411yescharacteristic
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73C2311yescharacteristic
89C2×C4⋊C4×C73C2311yescharacteristic
90C2×C4⋊C4×C73C2311yescharacteristic
91C2×C4⋊C4×C73C2311yescharacteristic
92C2×C4⋊C4×C73C2311yescharacteristic
93C2×C22⋊C4×C73—21no—
94C2×C22⋊C4×C73—21no—
95C2×C22⋊C4×C73—21no—
96C2×C22⋊C4×C73—21no—
97C2×C22⋊C4×C73—21no—
98C2×C42×C73—21no—
99C2×C42×C73—21no—
100C2×C42×C73—21no—
101C2×C42×C73C2311yescharacteristic
102C2×C42×C73C2311yescharacteristic
103C2×C42×C73C2311yescharacteristic
104#107×C73—22no—
105#107×C73—22no—
106#106×C73—22no—
107C4⋊C8×C73—22no—
108C4⋊C8×C73—22no—
109C4⋊C8×C73—22no—
110#103×C73—22no—
111#103×C73—22no—
112#103×C73—22no—
113#102×C73—22no—
114#102×C73—22no—
115#102×C73—22no—
116#97×C73—22no—
117#95×C73—21no—
118#95×C73—21no—
119#95×C73—21no—
120#95×C73—21no—
121#95×C73—21no—
122#95×C73—21no—
123#95×C73—21no—
124#95×C73—21no—
125#95×C73D411yescharacteristic
126#95×C73D411yescharacteristic
127C24×C73—21no—
128C2×Q8×C73—21no—
129C2×Q8×C73—21no—
130C2×Q8×C73—21no—
131C2×Q8×C73—21no—
132C2×Q8×C73—21no—
133C2×Q8×C73#4112yes—
134C2×D4×C73—21no—
135C2×D4×C73—21no—
136C2×D4×C73—21no—
137C2×D4×C73—21no—
138C2×D4×C73—21no—
139C2×D4×C73C2×D412yes—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—24no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—42no—
149C22×C4×C73—42no—
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2411yescharacteristic
157SD16×C73—28no—
158C2×C8×C73—21no—
159C2×C8×C73—21no—
160C2×C8×C73—21no—
161C2×C8×C73—21no—
162C2×C8×C73—21no—
163C2×C8×C73—21no—
164C2×C8×C73—21no—
165C2×C8×C73—21no—
166C2×C8×C73—21no—
167C2×C8×C73—21no—
168C2×C8×C73—21no—
169C2×C8×C73#4112yes—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—21no—
177C4⋊C4×C73—21no—
178C4⋊C4×C73—21no—
179C4⋊C4×C73—21no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—41no—
183C4⋊C4×C73—42no—
184C4⋊C4×C73—42no—
185C4⋊C4×C73—42no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73—42no—
188C4⋊C4×C73—42no—
189C4⋊C4×C73—81no—
190C4⋊C4×C73C2×D412yes—
191C22⋊C4×C73—24no—
192C22⋊C4×C73—42no—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—81no—
195C22⋊C4×C73—81no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—42no—
201C42×C73—42no—
202C42×C73—42no—
203C42×C73C2×D412yes—
204C42×C73#4112yes—
205C23×C73—22no—
206C23×C73—41no—
207C23×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73C22×D411yescharacteristic
213Q8×C73—22no—
214Q8×C73—22no—
215Q8×C73—41no—
216Q8×C73—41no—
217D4×C73—22no—
218D4×C73—22no—
219D4×C73—41no—
220D4×C73—41no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—22no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—48no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73#12112yes—
259C2×C4×C73#12112yes—
260C2×C4×C73AGL(1,ℤ/8)12yes—
261C2×C4×C73C22×D411yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C732−1+411yescharacteristic
266C2×C4×C732−1+411yescharacteristic
267C8×C73—22no—
268C8×C73—41no—
269C8×C73—41no—
270C8×C73—41no—
271C22×C73—42no—
272C22×C73—42no—
273C22×C73—42no—
274C22×C73—81no—
275C22×C73—81no—
276C22×C73—81no—
277C22×C73—81no—
278C22×C73C2×#12111yescharacteristic
279C22×C73#54711yescharacteristic
280C22×C73#54811yescharacteristic
281C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C73#57511yescharacteristic
283C22×C73#57711yescharacteristic
284C22×C73#57811yescharacteristic
285C4×C73—22no—
286C4×C73—22no—
287C4×C73—22no—
288C4×C73—41no—
289C4×C73—41no—
290C4×C73—41no—
291C4×C73—42no—
292C4×C73—42no—
293C4×C73—81no—
294C4×C73—81no—
295C4×C73—81no—
296C4×C73—81no—
297C4×C73—81no—
298C4×C73#52511yescharacteristic
299C4×C73#53611yescharacteristic
300C4×C73#55211yescharacteristic
301C4×C73#55611yescharacteristic
302#24690C7311yesmaximal, characteristic, direct product factor, Sylow
303C2×C73—82no—
304C2×C73#241511yescharacteristic
305C2×C73#303611yescharacteristic
306C2×C73#304611yescharacteristic
307C2×C73#334311yescharacteristic
308C2×C73#334811yescharacteristic
309C2×C73#336811yescharacteristic
310C2×C73#337711yescharacteristic
311C2×#478C2×C7311yescharacteristic
312C2×#440C2×C7311yescharacteristic
313#2565C2×C7311yescharacteristic, complemented
314#2354C2×C7311yescharacteristic
315#2073C2×C7311yescharacteristic
316#2071C2×C7311yescharacteristic
317#2055C2×C7311yescharacteristic
318#2055C2×C7311yescharacteristic
319#1992C2×C7311yescharacteristic, complemented
320#1949C2×C7311yescharacteristic, complemented
321#1947C2×C7311yescharacteristic, complemented
322#1940C2×C7311yescharacteristic, complemented
323#1928C2×C7311yescharacteristic, complemented
324#1927C2×C7311yescharacteristic, complemented
325#1803C2×C7311yescharacteristic, complemented
326C73#2469011yescharacteristic, direct product factor, Sylow
327C22×SD16C22×C7311yescharacteristic
328C2×#125C22×C7311yescharacteristic
329C2×C4×Q8C22×C7311yescharacteristic
330C2×C4×D4C22×C7311yescharacteristic
331C2×#117—21no—
332#478—24no—
333#440—24no—
334C2×#107C22×C7311yescharacteristic
335C2×#107C22×C7311yescharacteristic
336C2×#106C22×C7311yescharacteristic
337C2×C4⋊C8C22×C7311yescharacteristic
338C2×C4⋊C8C22×C7311yescharacteristic
339C2×C4⋊C8C22×C7311yescharacteristic
340C2×#103C22×C7311yescharacteristic
341C2×#103C22×C7311yescharacteristic
342C2×#103C22×C7311yescharacteristic
343C2×#102C22×C7311yescharacteristic
344C2×#102C22×C7311yescharacteristic
345C2×#102C22×C7311yescharacteristic
346C2×#97C22×C7311yescharacteristic
347#398—21no—
348#398—21no—
349#391C22×C7311yescharacteristic
350#391C22×C7311yescharacteristic
351#390C22×C7311yescharacteristic
352#390C22×C7311yescharacteristic
353#388—21no—
354#388—21no—
355#387—21no—
356#387—21no—
357#384C22×C7311yescharacteristic
358#384C22×C7311yescharacteristic
359#382C22×C7311yescharacteristic
360#380—21no—
361C4×C4⋊C4C22×C7311yescharacteristic
362C4×C4⋊C4C22×C7311yescharacteristic
363C4×C4⋊C4C22×C7311yescharacteristic
364#340C22×C7311yescharacteristic
365#340C22×C7311yescharacteristic
366#340C22×C7311yescharacteristic
367#340C22×C7311yescharacteristic
368#340C22×C7311yescharacteristic
369#340C22×C7311yescharacteristic
370#336C22×C7311yescharacteristic
371#336C22×C7311yescharacteristic
372C22×Q8C23×C7311yescharacteristic
373C22×D4C23×C7311yescharacteristic
374C23×C4—21no—
375C2×SD16—22no—
376C2×SD16—22no—
377C2×SD16—22no—
378C22×C8C23×C7311yescharacteristic
379C22×C8C23×C7311yescharacteristic
380C22×C8C23×C7311yescharacteristic
381C22×C8C23×C7311yescharacteristic
382#125—22no—
383#125—22no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×D4—22no—
387C4×D4D4×C7314yes—
388#117—44no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4D4×C7311yescharacteristic
397C2×C4⋊C4D4×C7311yescharacteristic
398C2×C4⋊C4C23×C7311yescharacteristic
399C2×C4⋊C4C23×C7311yescharacteristic
400C2×C4⋊C4C23×C7311yescharacteristic
401C2×C4⋊C4C23×C7311yescharacteristic
402C2×C4⋊C4C23×C7311yescharacteristic
403C2×C4⋊C4C23×C7311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42C23×C7311yescharacteristic
413C2×C42C23×C7311yescharacteristic
414C2×C42C23×C7311yescharacteristic
415#107—22no—
416#107—22no—
417#106—22no—
418C4⋊C8—22no—
419C4⋊C8—22no—
420C4⋊C8—22no—
421#103—22no—
422#103—22no—
423#103—22no—
424#102—22no—
425#102—22no—
426#102—22no—
427#97—22no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95D4×C7311yescharacteristic
437#95D4×C7311yescharacteristic
438C24—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8#41×C7312yes—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×D4×C7312yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4C2×D4×C7311yescharacteristic
462C22×C4C2×D4×C7311yescharacteristic
463C22×C4C2×D4×C7311yescharacteristic
464C22×C4C2×D4×C7311yescharacteristic
465C22×C4C2×D4×C7311yescharacteristic
466C22×C4C2×D4×C7311yescharacteristic
467C22×C4C24×C7311yescharacteristic
468SD16—28no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8#41×C7312yes—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4C2×D4×C7312yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42C2×D4×C7312yes—
515C42#41×C7312yes—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C22×D4×C7311yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4#121×C7312yes—
570C2×C4#121×C7312yes—
571C2×C4AGL(1,ℤ/8)×C7312yes—
572C2×C4C22×D4×C7311yescharacteristic
573C2×C4C2×#41×C7311yescharacteristic
574C2×C4C2×#41×C7311yescharacteristic
575C2×C4C2×#41×C7311yescharacteristic
576C2×C42−1+4×C7311yescharacteristic
577C2×C42−1+4×C7311yescharacteristic
578C8—22no—
579C8—41no—
580C8—41no—
581C8—41no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C2×#121×C7311yescharacteristic
590C22#547×C7311yescharacteristic
591C22#548×C7311yescharacteristic
592C22C2×AGL(1,ℤ/8)×C7311yescharacteristic
593C22#575×C7311yescharacteristic
594C22#577×C7311yescharacteristic
595C22#578×C7311yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4#525×C7311yescharacteristic
610C4#536×C7311yescharacteristic
611C4#552×C7311yescharacteristic
612C4#556×C7311yescharacteristic
613C2—82no—
614C2C73×#241511yescharacteristic
615C2C73×#303611yescharacteristic
616C2C73×#304611yescharacteristic
617C2C73×#334311yescharacteristic
618C2C73×#334811yescharacteristic
619C2C73×#336811yescharacteristic
620C2C73×#337711yescharacteristic
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC22×D4×C73C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C4×C4⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#2565C2semidirect
9. C73×#1992C2semidirect
10. C73×#1949C2semidirect
11. C73×#1947C2semidirect
12. C73×#1940C2semidirect
13. C73×#1928C2semidirect
14. C73×#1927C2semidirect
15. C73×#1803C2semidirect
302. #24690C73direct (N × H)
313. #2565C2×C73semidirect
319. #1992C2×C73semidirect
320. #1949C2×C73semidirect
321. #1947C2×C73semidirect
322. #1940C2×C73semidirect
323. #1928C2×C73semidirect
324. #1927C2×C73semidirect
325. #1803C2×C73semidirect
326. C73#24690direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#478×C732
C2×#440×C732
C73×#25652
C73×#23542
C73×#20732
C73×#20712
C73×#20552
C73×#19922
C73×#19492
C73×#19472
C73×#19402
C73×#19282
C73×#19272
C73×#18032
#2469073