Finite Group Databasebeta

← C73×#22919 #32932323

Subgroups — C73×#22919 #32932323

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485×C73C211yesmaximal, characteristic
2C73×#2362C211yesmaximal, characteristic
3C73×#2347C211yesmaximal, characteristic
4C73×#1942C211yesmaximal, characteristic, complemented
5C73×#1933C211yesmaximal, characteristic, complemented
6C73×#1928C211yesmaximal, characteristic, complemented
7C73×#1919C211yesmaximal, characteristic, complemented
8C73×#1918C211yesmaximal, characteristic, complemented
9C73×#1892C211yesmaximal, characteristic
10C73×#1865C211yesmaximal, characteristic, complemented
11C73×#1854C211yesmaximal, characteristic
12C4×#106×C73C211yesmaximal, characteristic, complemented
13C4×#103×C73C211yesmaximal, characteristic, complemented
14C8×C22⋊C4×C73C211yesmaximal, characteristic
15C73×#1735C211yesmaximal, characteristic
16C23×C8×C73C2211yescharacteristic
17C2×C22⋊Q8×C73C2211yescharacteristic
18C2×C4×Q8×C73C2211yescharacteristic
19C2×#117×C73C2211yescharacteristic
20C2×#117×C73C2211yescharacteristic
21#485×C73C418yescomplemented
22C2×#107×C73C2211yescharacteristic
23C2×#107×C73C2211yescharacteristic
24C2×#106×C73C2211yescharacteristic
25C2×#106×C73C2211yescharacteristic
26C2×#103×C73C2211yescharacteristic
27C2×#103×C73C2211yescharacteristic
28C2×#103×C73C2211yescharacteristic
29C2×#103×C73C2211yescharacteristic
30C2×C22⋊C8×C73C2211yescharacteristic
31C2×C4×C8×C73C2211yescharacteristic
32C2×C4×C8×C73C2211yescharacteristic
33#391×C73C2211yescharacteristic
34#389×C73C2211yescharacteristic
35#389×C73C2211yescharacteristic
36#389×C73C2211yescharacteristic
37#388×C73—21no—
38#388×C73—21no—
39#388×C73—21no—
40#388×C73—21no—
41#387×C73—21no—
42#383×C73—21no—
43#380×C73C2211yescharacteristic
44C4×C4⋊C4×C73—21no—
45C4×C4⋊C4×C73—21no—
46C4×C4⋊C4×C73C2211yescharacteristic
47C4×C4⋊C4×C73C2211yescharacteristic
48C4×C4⋊C4×C73C2211yescharacteristic
49C4×C22⋊C4×C73C2211yescharacteristic
50#340×C73C2211yescharacteristic
51#340×C73C2211yescharacteristic
52#340×C73C2211yescharacteristic
53#340×C73C2211yescharacteristic
54#340×C73C2211yescharacteristic
55#340×C73C2211yescharacteristic
56#340×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Q8×C73C2311yescharacteristic
61C23×C4×C73C2311yescharacteristic
62C22×C8×C73—22no—
63C22×C8×C73—22no—
64C22×C8×C73—41no—
65C22×C8×C73C2311yescharacteristic
66C22×C8×C73C2311yescharacteristic
67C22×C8×C73C2311yescharacteristic
68C22×C8×C73C2311yescharacteristic
69C22⋊Q8×C73—22no—
70C22⋊Q8×C73C2×C414yes—
71C4×Q8×C73—22no—
72C4×Q8×C73—41no—
73#117×C73—22no—
74#117×C73—22no—
75#117×C73—41no—
76#117×C73C2×C414yes—
77C2×C4⋊C4×C73—21no—
78C2×C4⋊C4×C73—21no—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73C2311yescharacteristic
83C2×C4⋊C4×C73C2311yescharacteristic
84C2×C4⋊C4×C73C2311yescharacteristic
85C2×C4⋊C4×C73C2311yescharacteristic
86C2×C4⋊C4×C73C2311yescharacteristic
87C2×C4⋊C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73C2311yescharacteristic
89C2×C22⋊C4×C73—21no—
90C2×C22⋊C4×C73—21no—
91C2×C22⋊C4×C73—21no—
92C2×C22⋊C4×C73—21no—
93C2×C22⋊C4×C73D411yescharacteristic
94C2×C22⋊C4×C73D411yescharacteristic
95C2×C42×C73—21no—
96C2×C42×C73—21no—
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—41no—
104#107×C73C2×C414yes—
105#106×C73—22no—
106#106×C73C2×C414yes—
107#103×C73—22no—
108#103×C73—41no—
109#103×C73C2×C414yes—
110#103×C73C2×C414yes—
111C22⋊C8×C73C2×C414yes—
112C4×C8×C73—22no—
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×C73D411yescharacteristic
124#95×C73D411yescharacteristic
125C24×C73C2×D411yescharacteristic
126C2×Q8×C73—21no—
127C2×Q8×C73—21no—
128C2×Q8×C73—21no—
129C2×Q8×C73—21no—
130C2×Q8×C73—21no—
131C2×Q8×C73C22×C412yes—
132C22×C4×C73—21no—
133C22×C4×C73—21no—
134C22×C4×C73—21no—
135C22×C4×C73—21no—
136C22×C4×C73—21no—
137C22×C4×C73—21no—
138C22×C4×C73—21no—
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—21no—
144C22×C4×C73C22×C412yes—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73C2×D411yescharacteristic
148C22×C4×C73C2×D411yescharacteristic
149C22×C4×C73C2×D411yescharacteristic
150C22×C4×C73#4112yes—
151C22×C4×C73C2411yescharacteristic
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73—21no—
155C2×C8×C73—21no—
156C2×C8×C73—21no—
157C2×C8×C73—21no—
158C2×C8×C73—21no—
159C2×C8×C73—21no—
160C2×C8×C73—21no—
161C2×C8×C73—44no—
162C2×C8×C73C22×C412yes—
163C2×C8×C73C22×C412yes—
164C2×C8×C73#4112yes—
165C4⋊C4×C73—24no—
166C4⋊C4×C73—21no—
167C4⋊C4×C73—21no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—21no—
171C4⋊C4×C73—21no—
172C4⋊C4×C73—21no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73—21no—
176C4⋊C4×C73—42no—
177C4⋊C4×C73—42no—
178C4⋊C4×C73—42no—
179C4⋊C4×C73—42no—
180C4⋊C4×C73C22×C412yes—
181C4⋊C4×C73C22×C412yes—
182C4⋊C4×C73C22×C412yes—
183C4⋊C4×C73#4112yes—
184C22⋊C4×C73—24no—
185C22⋊C4×C73—24no—
186C22⋊C4×C73—22no—
187C22⋊C4×C73—22no—
188C22⋊C4×C73—42no—
189C22⋊C4×C73—81no—
190C42×C73—24no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—42no—
196C42×C73—42no—
197C42×C73—42no—
198C42×C73—81no—
199C42×C73—81no—
200C23×C73—21no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73C4×D412yes—
207C23×C73C22×D411yescharacteristic
208Q8×C73—22no—
209Q8×C73—22no—
210Q8×C73—41no—
211Q8×C73—41no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
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×C73C4×D412yes—
253C2×C4×C73#13512yes—
254C2×C4×C73#13512yes—
255C2×C4×C73C23×C411yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73H(5,𝔽2)11yescharacteristic
260C2×C4×C732−1+411yescharacteristic
261C8×C73—22no—
262C8×C73—22no—
263C8×C73—22no—
264C8×C73—41no—
265C22×C73—22no—
266C22×C73—22no—
267C22×C73—22no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73—41no—
271C22×C73—41no—
272C22×C73C2×C4×D411yescharacteristic
273C22×C73#54611yescharacteristic
274C22×C73#54711yescharacteristic
275C22×C73C2×#13511yescharacteristic
276C22×C73C2×#13511yescharacteristic
277C22×C73#57711yescharacteristic
278C22×C73#57811yescharacteristic
279C4×C73—22no—
280C4×C73—22no—
281C4×C73—22no—
282C4×C73—41no—
283C4×C73—42no—
284C4×C73—42no—
285C4×C73—42no—
286C4×C73—42no—
287C4×C73—41no—
288C4×C73—42no—
289C4×C73—81no—
290C4×C73—81no—
291C4×C73—81no—
292C4×C73C4×#4111yescharacteristic
293C4×C73#52011yescharacteristic
294C4×C73#55111yescharacteristic
295C4×C73#55211yescharacteristic
296#22919C7311yesmaximal, characteristic, direct product factor, Sylow
297C2×C73—42no—
298C2×C73#235911yescharacteristic
299C2×C73C4×#13511yescharacteristic
300C2×C73#294711yescharacteristic
301C2×C73#330111yescharacteristic
302C2×C73#330211yescharacteristic
303C2×C73#333711yescharacteristic
304C2×C73#334211yescharacteristic
305C2×#485C2×C7311yescharacteristic
306#2362C2×C7311yescharacteristic
307#2347C2×C7311yescharacteristic
308#1942C2×C7311yescharacteristic, complemented
309#1933C2×C7311yescharacteristic, complemented
310#1928C2×C7311yescharacteristic, complemented
311#1919C2×C7311yescharacteristic, complemented
312#1918C2×C7311yescharacteristic, complemented
313#1892C2×C7311yescharacteristic
314#1865C2×C7311yescharacteristic, complemented
315#1854C2×C7311yescharacteristic
316C4×#106C2×C7311yescharacteristic, complemented
317C4×#103C2×C7311yescharacteristic, complemented
318C8×C22⋊C4C2×C7311yescharacteristic
319#1735C2×C7311yescharacteristic
320C73#2291911yescharacteristic, direct product factor, Sylow
321C23×C8C22×C7311yescharacteristic
322C2×C22⋊Q8C22×C7311yescharacteristic
323C2×C4×Q8C22×C7311yescharacteristic
324C2×#117C22×C7311yescharacteristic
325C2×#117C22×C7311yescharacteristic
326#485C4×C7318yescomplemented
327C2×#107C22×C7311yescharacteristic
328C2×#107C22×C7311yescharacteristic
329C2×#106C22×C7311yescharacteristic
330C2×#106C22×C7311yescharacteristic
331C2×#103C22×C7311yescharacteristic
332C2×#103C22×C7311yescharacteristic
333C2×#103C22×C7311yescharacteristic
334C2×#103C22×C7311yescharacteristic
335C2×C22⋊C8C22×C7311yescharacteristic
336C2×C4×C8C22×C7311yescharacteristic
337C2×C4×C8C22×C7311yescharacteristic
338#391C22×C7311yescharacteristic
339#389C22×C7311yescharacteristic
340#389C22×C7311yescharacteristic
341#389C22×C7311yescharacteristic
342#388—21no—
343#388—21no—
344#388—21no—
345#388—21no—
346#387—21no—
347#383—21no—
348#380C22×C7311yescharacteristic
349C4×C4⋊C4—21no—
350C4×C4⋊C4—21no—
351C4×C4⋊C4C22×C7311yescharacteristic
352C4×C4⋊C4C22×C7311yescharacteristic
353C4×C4⋊C4C22×C7311yescharacteristic
354C4×C22⋊C4C22×C7311yescharacteristic
355#340C22×C7311yescharacteristic
356#340C22×C7311yescharacteristic
357#340C22×C7311yescharacteristic
358#340C22×C7311yescharacteristic
359#340C22×C7311yescharacteristic
360#340C22×C7311yescharacteristic
361#340C22×C7311yescharacteristic
362#340C22×C7311yescharacteristic
363#336C22×C7311yescharacteristic
364#336C22×C7311yescharacteristic
365C22×Q8C23×C7311yescharacteristic
366C23×C4C23×C7311yescharacteristic
367C22×C8—22no—
368C22×C8—22no—
369C22×C8—41no—
370C22×C8C23×C7311yescharacteristic
371C22×C8C23×C7311yescharacteristic
372C22×C8C23×C7311yescharacteristic
373C22×C8C23×C7311yescharacteristic
374C22⋊Q8—22no—
375C22⋊Q8C2×C4×C7314yes—
376C4×Q8—22no—
377C4×Q8—41no—
378#117—22no—
379#117—22no—
380#117—41no—
381#117C2×C4×C7314yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4C23×C7311yescharacteristic
388C2×C4⋊C4C23×C7311yescharacteristic
389C2×C4⋊C4C23×C7311yescharacteristic
390C2×C4⋊C4C23×C7311yescharacteristic
391C2×C4⋊C4C23×C7311yescharacteristic
392C2×C4⋊C4C23×C7311yescharacteristic
393C2×C4⋊C4C23×C7311yescharacteristic
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4D4×C7311yescharacteristic
399C2×C22⋊C4D4×C7311yescharacteristic
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C23×C7311yescharacteristic
407C2×C42C23×C7311yescharacteristic
408#107—41no—
409#107C2×C4×C7314yes—
410#106—22no—
411#106C2×C4×C7314yes—
412#103—22no—
413#103—41no—
414#103C2×C4×C7314yes—
415#103C2×C4×C7314yes—
416C22⋊C8C2×C4×C7314yes—
417C4×C8—22no—
418C4×C8—41no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95D4×C7311yescharacteristic
429#95D4×C7311yescharacteristic
430C24C2×D4×C7311yescharacteristic
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8C22×C4×C7312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4C22×C4×C7312yes—
450C22×C4C2×D4×C7311yescharacteristic
451C22×C4C2×D4×C7311yescharacteristic
452C22×C4C2×D4×C7311yescharacteristic
453C22×C4C2×D4×C7311yescharacteristic
454C22×C4C2×D4×C7311yescharacteristic
455C22×C4#41×C7312yes—
456C22×C4C24×C7311yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—44no—
467C2×C8C22×C4×C7312yes—
468C2×C8C22×C4×C7312yes—
469C2×C8#41×C7312yes—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4C22×C4×C7312yes—
486C4⋊C4C22×C4×C7312yes—
487C4⋊C4C22×C4×C7312yes—
488C4⋊C4#41×C7312yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C42—24no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42—81no—
504C42—81no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4×D4×C7312yes—
512C23C22×D4×C7311yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4C4×D4×C7312yes—
558C2×C4#135×C7312yes—
559C2×C4#135×C7312yes—
560C2×C4C23×C4×C7311yescharacteristic
561C2×C4C2×#41×C7311yescharacteristic
562C2×C4C2×#41×C7311yescharacteristic
563C2×C4C2×#41×C7311yescharacteristic
564C2×C4H(5,𝔽2)×C7311yescharacteristic
565C2×C42−1+4×C7311yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8—41no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×C4×D4×C7311yescharacteristic
578C22#546×C7311yescharacteristic
579C22#547×C7311yescharacteristic
580C22C2×#135×C7311yescharacteristic
581C22C2×#135×C7311yescharacteristic
582C22#577×C7311yescharacteristic
583C22#578×C7311yescharacteristic
584C4—22no—
585C4—22no—
586C4—22no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4C4×#41×C7311yescharacteristic
598C4#520×C7311yescharacteristic
599C4#551×C7311yescharacteristic
600C4#552×C7311yescharacteristic
601C2—42no—
602C2C73×#235911yescharacteristic
603C2C4×#135×C7311yescharacteristic
604C2C73×#294711yescharacteristic
605C2C73×#330111yescharacteristic
606C2C73×#330211yescharacteristic
607C2C73×#333711yescharacteristic
608C2C73×#334211yescharacteristic
subgroupG/H
center, socleC23×C73C22×D4
commutatorC2×C4C23×C4×C73
Frattini, cosocleC22×C4C24×C73
prime radicalC24×C73C2×D4
LCSG ▷ C2×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
4. C73×#1942C2semidirect
5. C73×#1933C2semidirect
6. C73×#1928C2semidirect
7. C73×#1919C2semidirect
8. C73×#1918C2semidirect
10. C73×#1865C2semidirect
12. C4×#106×C73C2semidirect
13. C4×#103×C73C2semidirect
21. #485×C73C4semidirect
296. #22919C73direct (N × H)
308. #1942C2×C73semidirect
309. #1933C2×C73semidirect
310. #1928C2×C73semidirect
311. #1919C2×C73semidirect
312. #1918C2×C73semidirect
314. #1865C2×C73semidirect
316. C4×#106C2×C73semidirect
317. C4×#103C2×C73semidirect
320. C73#22919direct (N × H)
326. #485C4×C73semidirect

Maximal subgroups

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

subgroupindex
C2×#485×C732
C73×#23622
C73×#23472
C73×#19422
C73×#19332
C73×#19282
C73×#19192
C73×#19182
C73×#18922
C73×#18652
C73×#18542
C4×#106×C732
C4×#103×C732
C8×C22⋊C4×C732
C73×#17352
#2291973