Finite Group Databasebeta

← C73×#26732 #32936136

Subgroups — C73×#26732 #32936136

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C73×#2489C211yesmaximal, characteristic
2C73×#2362C211yesmaximal, characteristic
3C73×#2056C211yesmaximal, characteristic, complemented
4C73×#2036C211yesmaximal, characteristic
5C73×#2026C211yesmaximal, characteristic, complemented
6C73×#2001C211yesmaximal, characteristic
7C73×#1989C211yesmaximal, characteristic, complemented
8C73×#1933C211yesmaximal, characteristic, complemented
9C73×#1928C211yesmaximal, characteristic, complemented
10C73×#1923C211yesmaximal, characteristic
11C73×#1918C211yesmaximal, characteristic, complemented
12C73×#1878C211yesmaximal, characteristic, complemented
13C73×#1873C211yesmaximal, characteristic
14C73×#1825C211yesmaximal, characteristic
15C4×#103×C73C211yesmaximal, characteristic, complemented
16C22×Q16×C73C2211yescharacteristic
17C2×#125×C73C2211yescharacteristic
18C2×C22⋊Q8×C73C2211yescharacteristic
19C2×C22⋊Q8×C73C2211yescharacteristic
20C2×C4×Q8×C73C2211yescharacteristic
21C2×#117×C73C2211yescharacteristic
22C2×#107×C73C2211yescharacteristic
23C2×C4⋊C8×C73C2211yescharacteristic
24C2×#103×C73C2211yescharacteristic
25C2×#103×C73C2211yescharacteristic
26C2×#103×C73C2211yescharacteristic
27C2×#103×C73C2211yescharacteristic
28C2×#103×C73C2211yescharacteristic
29C2×#103×C73C2211yescharacteristic
30C2×C22⋊C8×C73C2211yescharacteristic
31C2×C22⋊C8×C73C2211yescharacteristic
32C2×C4×C8×C73C2211yescharacteristic
33#400×C73—21no—
34#397×C73—21no—
35#391×C73C2211yescharacteristic
36#391×C73C2211yescharacteristic
37#391×C73C2211yescharacteristic
38#389×C73C2211yescharacteristic
39#388×C73—21no—
40#388×C73—21no—
41#388×C73—21no—
42#387×C73—21no—
43#382×C73—21no—
44#380×C73C2211yescharacteristic
45C4×C4⋊C4×C73—21no—
46C4×C4⋊C4×C73C2211yescharacteristic
47C4×C4⋊C4×C73C2211yescharacteristic
48C4×C22⋊C4×C73C2211yescharacteristic
49#340×C73C2211yescharacteristic
50#340×C73C2211yescharacteristic
51#340×C73C2211yescharacteristic
52#340×C73C2211yescharacteristic
53#340×C73C2211yescharacteristic
54#340×C73C2211yescharacteristic
55#340×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Q8×C73C2311yescharacteristic
60C22×Q8×C73C2311yescharacteristic
61C23×C4×C73C2311yescharacteristic
62C2×Q16×C73—22no—
63C2×Q16×C73—41no—
64C2×Q16×C73—41no—
65C22×C8×C73C2311yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C22×C8×C73C2311yescharacteristic
68C22×C8×C73C2311yescharacteristic
69#125×C73—22no—
70#125×C73—41no—
71C22⋊Q8×C73—22no—
72C22⋊Q8×C73—22no—
73C22⋊Q8×C73—22no—
74C22⋊Q8×C73—41no—
75C4×Q8×C73—22no—
76C4×Q8×C73—41no—
77#117×C73—22no—
78#117×C73—41no—
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×C73C2311yescharacteristic
86C2×C4⋊C4×C73C2311yescharacteristic
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73C2311yescharacteristic
89C2×C4⋊C4×C73C2311yescharacteristic
90C2×C4⋊C4×C73C2311yescharacteristic
91C2×C22⋊C4×C73—21no—
92C2×C22⋊C4×C73—21no—
93C2×C22⋊C4×C73—21no—
94C2×C22⋊C4×C73—21no—
95C2×C22⋊C4×C73D411yescharacteristic
96C2×C22⋊C4×C73D411yescharacteristic
97C2×C42×C73—21no—
98C2×C42×C73—21no—
99C2×C42×C73—21no—
100C2×C42×C73—21no—
101C2×C42×C73C2311yescharacteristic
102C2×C42×C73C2311yescharacteristic
103#107×C73—22no—
104C4⋊C8×C73—41no—
105#103×C73—22no—
106#103×C73—22no—
107#103×C73—22no—
108#103×C73—22no—
109#103×C73—41no—
110#103×C73—41no—
111C22⋊C8×C73—41no—
112C22⋊C8×C73—41no—
113C4×C8×C73—41no—
114#95×C73—21no—
115#95×C73—21no—
116#95×C73—21no—
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×C73D411yescharacteristic
125#95×C73D411yescharacteristic
126C24×C73C2×D411yescharacteristic
127C2×Q8×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—21no—
134C2×Q8×C73—21no—
135C2×Q8×C73—21no—
136C2×Q8×C73—21no—
137C2×Q8×C73—21no—
138C2×Q8×C73#4112yes—
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73C2×D411yescharacteristic
154C22×C4×C73C2×D411yescharacteristic
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157C22×C4×C73C2×D411yescharacteristic
158C22×C4×C73C2411yescharacteristic
159Q16×C73—44no—
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—21no—
170C2×C8×C73—21no—
171C2×C8×C73—21no—
172C4⋊C4×C73—24no—
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—42no—
183C4⋊C4×C73—42no—
184C4⋊C4×C73—42no—
185C4⋊C4×C73—42no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73#4112yes—
188C4⋊C4×C73#4112yes—
189C4⋊C4×C73#4112yes—
190C22⋊C4×C73—24no—
191C22⋊C4×C73—22no—
192C22⋊C4×C73—41no—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—42no—
195C22⋊C4×C73—42no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—42no—
201C42×C73—42no—
202C42×C73—42no—
203C42×C73—42no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73C22×D411yescharacteristic
212Q8×C73—22no—
213Q8×C73—22no—
214Q8×C73—41no—
215Q8×C73—41no—
216Q8×C73—41no—
217Q8×C73—41no—
218Q8×C73—41no—
219Q8×C73—41no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—21no—
230C2×C4×C73—22no—
231C2×C4×C73—21no—
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—41no—
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—41no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73—41no—
262C2×C4×C73#13512yes—
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73H(5,𝔽2)11yescharacteristic
268C2×C4×C73H(5,𝔽2)11yescharacteristic
269C8×C73—41no—
270C8×C73—41no—
271C8×C73—41no—
272C8×C73—41no—
273C22×C73—41no—
274C22×C73—41no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73#52511yescharacteristic
281C22×C73#54611yescharacteristic
282C22×C73#54611yescharacteristic
283C22×C73C2×#13511yescharacteristic
284C22×C73#57511yescharacteristic
285C22×C73#57811yescharacteristic
286C22×C73#57811yescharacteristic
287C4×C73—22no—
288C4×C73—42no—
289C4×C73—41no—
290C4×C73—42no—
291C4×C73—41no—
292C4×C73—42no—
293C4×C73—41no—
294C4×C73—42no—
295C4×C73—41no—
296C4×C73—81no—
297C4×C73—81no—
298C4×C73—81no—
299C4×C73—81no—
300C4×C73#52911yescharacteristic
301C4×C73#53811yescharacteristic
302C4×C73#55511yescharacteristic
303C4×C73#55911yescharacteristic
304#26732C7311yesmaximal, characteristic, direct product factor, Sylow
305C2×C73—81no—
306C2×C73#245911yescharacteristic
307C2×C73#310211yescharacteristic
308C2×C73#312711yescharacteristic
309C2×C73#330011yescharacteristic
310C2×C73#330211yescharacteristic
311C2×C73#331511yescharacteristic
312C2×C73#332711yescharacteristic
313#2489C2×C7311yescharacteristic
314#2362C2×C7311yescharacteristic
315#2056C2×C7311yescharacteristic, complemented
316#2036C2×C7311yescharacteristic
317#2026C2×C7311yescharacteristic, complemented
318#2001C2×C7311yescharacteristic
319#1989C2×C7311yescharacteristic, complemented
320#1933C2×C7311yescharacteristic, complemented
321#1928C2×C7311yescharacteristic, complemented
322#1923C2×C7311yescharacteristic
323#1918C2×C7311yescharacteristic, complemented
324#1878C2×C7311yescharacteristic, complemented
325#1873C2×C7311yescharacteristic
326#1825C2×C7311yescharacteristic
327C4×#103C2×C7311yescharacteristic, complemented
328C73#2673211yescharacteristic, direct product factor, Sylow
329C22×Q16C22×C7311yescharacteristic
330C2×#125C22×C7311yescharacteristic
331C2×C22⋊Q8C22×C7311yescharacteristic
332C2×C22⋊Q8C22×C7311yescharacteristic
333C2×C4×Q8C22×C7311yescharacteristic
334C2×#117C22×C7311yescharacteristic
335C2×#107C22×C7311yescharacteristic
336C2×C4⋊C8C22×C7311yescharacteristic
337C2×#103C22×C7311yescharacteristic
338C2×#103C22×C7311yescharacteristic
339C2×#103C22×C7311yescharacteristic
340C2×#103C22×C7311yescharacteristic
341C2×#103C22×C7311yescharacteristic
342C2×#103C22×C7311yescharacteristic
343C2×C22⋊C8C22×C7311yescharacteristic
344C2×C22⋊C8C22×C7311yescharacteristic
345C2×C4×C8C22×C7311yescharacteristic
346#400—21no—
347#397—21no—
348#391C22×C7311yescharacteristic
349#391C22×C7311yescharacteristic
350#391C22×C7311yescharacteristic
351#389C22×C7311yescharacteristic
352#388—21no—
353#388—21no—
354#388—21no—
355#387—21no—
356#382—21no—
357#380C22×C7311yescharacteristic
358C4×C4⋊C4—21no—
359C4×C4⋊C4C22×C7311yescharacteristic
360C4×C4⋊C4C22×C7311yescharacteristic
361C4×C22⋊C4C22×C7311yescharacteristic
362#340C22×C7311yescharacteristic
363#340C22×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×Q8C23×C7311yescharacteristic
374C23×C4C23×C7311yescharacteristic
375C2×Q16—22no—
376C2×Q16—41no—
377C2×Q16—41no—
378C22×C8C23×C7311yescharacteristic
379C22×C8C23×C7311yescharacteristic
380C22×C8C23×C7311yescharacteristic
381C22×C8C23×C7311yescharacteristic
382#125—22no—
383#125—41no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—41no—
388C4×Q8—22no—
389C4×Q8—41no—
390#117—22no—
391#117—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
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⋊C4D4×C7311yescharacteristic
409C2×C22⋊C4D4×C7311yescharacteristic
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42C23×C7311yescharacteristic
415C2×C42C23×C7311yescharacteristic
416#107—22no—
417C4⋊C8—41no—
418#103—22no—
419#103—22no—
420#103—22no—
421#103—22no—
422#103—41no—
423#103—41no—
424C22⋊C8—41no—
425C22⋊C8—41no—
426C4×C8—41no—
427#95—21no—
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#95—21no—
437#95D4×C7311yescharacteristic
438#95D4×C7311yescharacteristic
439C24C2×D4×C7311yescharacteristic
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8#41×C7312yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4C2×D4×C7311yescharacteristic
467C22×C4C2×D4×C7311yescharacteristic
468C22×C4C2×D4×C7311yescharacteristic
469C22×C4C2×D4×C7311yescharacteristic
470C22×C4C2×D4×C7311yescharacteristic
471C22×C4C24×C7311yescharacteristic
472Q16—44no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C4⋊C4—24no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4#41×C7312yes—
501C4⋊C4#41×C7312yes—
502C4⋊C4#41×C7312yes—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23C22×D4×C7311yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
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—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4#135×C7312yes—
576C2×C4C2×#41×C7311yescharacteristic
577C2×C4C2×#41×C7311yescharacteristic
578C2×C4C2×#41×C7311yescharacteristic
579C2×C4C2×#41×C7311yescharacteristic
580C2×C4H(5,𝔽2)×C7311yescharacteristic
581C2×C4H(5,𝔽2)×C7311yescharacteristic
582C8—41no—
583C8—41no—
584C8—41no—
585C8—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#525×C7311yescharacteristic
594C22#546×C7311yescharacteristic
595C22#546×C7311yescharacteristic
596C22C2×#135×C7311yescharacteristic
597C22#575×C7311yescharacteristic
598C22#578×C7311yescharacteristic
599C22#578×C7311yescharacteristic
600C4—22no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4#529×C7311yescharacteristic
614C4#538×C7311yescharacteristic
615C4#555×C7311yescharacteristic
616C4#559×C7311yescharacteristic
617C2—81no—
618C2C73×#245911yescharacteristic
619C2C73×#310211yescharacteristic
620C2C73×#312711yescharacteristic
621C2C73×#330011yescharacteristic
622C2C73×#330211yescharacteristic
623C2C73×#331511yescharacteristic
624C2C73×#332711yescharacteristic
subgroupG/H
center, socleC23×C73C22×D4
commutator, Frattini, cosocleC22×C4C24×C73
prime radicalC24×C73C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C73 ◁ C4×C22⋊C4×C73 ◁ G

Complements

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

NH ≅ G/Ntype
3. C73×#2056C2semidirect
5. C73×#2026C2semidirect
7. C73×#1989C2semidirect
8. C73×#1933C2semidirect
9. C73×#1928C2semidirect
11. C73×#1918C2semidirect
12. C73×#1878C2semidirect
15. C4×#103×C73C2semidirect
304. #26732C73direct (N × H)
315. #2056C2×C73semidirect
317. #2026C2×C73semidirect
319. #1989C2×C73semidirect
320. #1933C2×C73semidirect
321. #1928C2×C73semidirect
323. #1918C2×C73semidirect
324. #1878C2×C73semidirect
327. C4×#103C2×C73semidirect
328. C73#26732direct (N × H)

Maximal subgroups

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

subgroupindex
C73×#24892
C73×#23622
C73×#20562
C73×#20362
C73×#20262
C73×#20012
C73×#19892
C73×#19332
C73×#19282
C73×#19232
C73×#19182
C73×#18782
C73×#18732
C73×#18252
C4×#103×C732
#2673273