Finite Group Databasebeta

← C73×#16399 #32925803

Subgroups — C73×#16399 #32925803

Order 18688.

1538 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398×C73C211yesmaximal, characteristic, complemented
2C2×#397×C73C212yesmaximal
3C2×#385×C73C211yesmaximal, characteristic
4C2×#380×C73C211yesmaximal, characteristic
5C2×#380×C73C211yesmaximal, characteristic
6C73×#1445C211yesmaximal, characteristic, complemented
7C73×#1442C211yesmaximal, characteristic, complemented
8C73×#1441C212yesmaximal, complemented
9C73×#1441C211yesmaximal, characteristic, complemented
10C73×#1441C212yesmaximal, complemented
11C73×#1440C212yesmaximal
12C24×C4×C73C2211yescharacteristic
13C22×C4⋊C4×C73C2211yescharacteristic
14C22×C4⋊C4×C73C2211yescharacteristic
15C22×C4⋊C4×C73C2211yescharacteristic
16C22×C4⋊C4×C73C2212yes—
17C22×C4⋊C4×C73C2212yes—
18C22×C4⋊C4×C73C2211yescharacteristic
19C22×C22⋊C4×C73C2212yes—
20C22×C22⋊C4×C73C2212yes—
21C22×C22⋊C4×C73C2211yescharacteristic
22C22×C22⋊C4×C73C2211yescharacteristic
23#398×C73—24no—
24#397×C73—28no—
25#385×C73—24no—
26#380×C73—24no—
27#380×C73—24no—
28C2×#95×C73C2212yes—
29C2×#95×C73C2212yes—
30C2×#95×C73C2211yescharacteristic
31C2×#95×C73C2212yes—
32C2×#95×C73C2212yes—
33C2×#95×C73C2211yescharacteristic
34C2×#95×C73C2212yes—
35C2×#95×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C2×#95×C73C2212yes—
38C2×#95×C73C2212yes—
39C2×#95×C73C2211yescharacteristic
40C25×C73C2311yescharacteristic
41C23×C4×C73—22no—
42C23×C4×C73—22no—
43C23×C4×C73—24no—
44C23×C4×C73—42no—
45C23×C4×C73D414yes—
46C23×C4×C73C2311yescharacteristic
47C23×C4×C73C2312yes—
48C23×C4×C73C2312yes—
49C23×C4×C73C2312yes—
50C23×C4×C73C2311yescharacteristic
51C23×C4×C73C2311yescharacteristic
52C23×C4×C73C2312yes—
53C23×C4×C73C2311yescharacteristic
54C23×C4×C73C2311yescharacteristic
55C23×C4×C73C2311yescharacteristic
56C2×C4⋊C4×C73—24no—
57C2×C4⋊C4×C73—22no—
58C2×C4⋊C4×C73—22no—
59C2×C4⋊C4×C73—22no—
60C2×C4⋊C4×C73—22no—
61C2×C4⋊C4×C73—24no—
62C2×C4⋊C4×C73—24no—
63C2×C4⋊C4×C73—22no—
64C2×C4⋊C4×C73—22no—
65C2×C4⋊C4×C73—24no—
66C2×C4⋊C4×C73—22no—
67C2×C4⋊C4×C73—22no—
68C2×C4⋊C4×C73—24no—
69C2×C4⋊C4×C73—42no—
70C2×C4⋊C4×C73—42no—
71C2×C4⋊C4×C73—42no—
72C2×C22⋊C4×C73—22no—
73C2×C22⋊C4×C73—22no—
74C2×C22⋊C4×C73—24no—
75C2×C22⋊C4×C73—22no—
76C2×C22⋊C4×C73—24no—
77C2×C22⋊C4×C73—24no—
78C2×C22⋊C4×C73—24no—
79C2×C22⋊C4×C73—24no—
80C2×C22⋊C4×C73—42no—
81C2×C22⋊C4×C73—42no—
82C2×C22⋊C4×C73D414yes—
83#95×C73—22no—
84#95×C73—24no—
85#95×C73—22no—
86#95×C73—22no—
87#95×C73—24no—
88#95×C73—24no—
89#95×C73—24no—
90#95×C73—24no—
91#95×C73—42no—
92#95×C73—42no—
93#95×C73—42no—
94#95×C73D414yes—
95C24×C73—21no—
96C24×C73—21no—
97C24×C73—22no—
98C24×C73—21no—
99C24×C73—22no—
100C24×C73—22no—
101C24×C73—22no—
102C24×C73—21no—
103C24×C73C2×D412yes—
104C24×C73C2×D412yes—
105C24×C73#4112yes—
106C24×C73C2411yescharacteristic
107C22×C4×C73—21no—
108C22×C4×C73—21no—
109C22×C4×C73—22no—
110C22×C4×C73—22no—
111C22×C4×C73—22no—
112C22×C4×C73—21no—
113C22×C4×C73—21no—
114C22×C4×C73—22no—
115C22×C4×C73—21no—
116C22×C4×C73—21no—
117C22×C4×C73—21no—
118C22×C4×C73—22no—
119C22×C4×C73—21no—
120C22×C4×C73—22no—
121C22×C4×C73—22no—
122C22×C4×C73—22no—
123C22×C4×C73—21no—
124C22×C4×C73—22no—
125C22×C4×C73—22no—
126C22×C4×C73—22no—
127C22×C4×C73—21no—
128C22×C4×C73—28no—
129C22×C4×C73—28no—
130C22×C4×C73—22no—
131C22×C4×C73—21no—
132C22×C4×C73—22no—
133C22×C4×C73—21no—
134C22×C4×C73—22no—
135C22×C4×C73—22no—
136C22×C4×C73—22no—
137C22×C4×C73—22no—
138C22×C4×C73—22no—
139C22×C4×C73—21no—
140C22×C4×C73—22no—
141C22×C4×C73—22no—
142C22×C4×C73—22no—
143C22×C4×C73—21no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—21no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—21no—
161C22×C4×C73—44no—
162C22×C4×C73—48no—
163C22×C4×C73—48no—
164C22×C4×C73C2×D412yes—
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D414yes—
167C22×C4×C73C2×D412yes—
168C22×C4×C73C2×D412yes—
169C22×C4×C73C2×D412yes—
170C22×C4×C73#4112yes—
171C22×C4×C73#4114yes—
172C4⋊C4×C73—48no—
173C4⋊C4×C73—48no—
174C4⋊C4×C73—44no—
175C4⋊C4×C73—44no—
176C4⋊C4×C73—44no—
177C4⋊C4×C73—44no—
178C22⋊C4×C73—28no—
179C22⋊C4×C73—48no—
180C22⋊C4×C73—44no—
181C22⋊C4×C73—48no—
182C23×C73—22no—
183C23×C73—22no—
184C23×C73—22no—
185C23×C73—22no—
186C23×C73—22no—
187C23×C73—24no—
188C23×C73—22no—
189C23×C73—24no—
190C23×C73—22no—
191C23×C73—22no—
192C23×C73—24no—
193C23×C73—22no—
194C23×C73—24no—
195C23×C73—22no—
196C23×C73—42no—
197C23×C73—42no—
198C23×C73—42no—
199C23×C73—42no—
200C23×C73—42no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73#12114yes—
205C23×C73C22×D411yescharacteristic
206C23×C73C22×D411yescharacteristic
207C23×C73C22×D411yescharacteristic
208C23×C73C2×#4111yescharacteristic
209C23×C73C2×#4111yescharacteristic
210C23×C73C2×#4112yes—
211C23×C73H(5,𝔽2)12yes—
212C23×C73H(5,𝔽2)11yescharacteristic
213C23×C732−1+412yes—
214C23×C732−1+412yes—
215C23×C732−1+411yescharacteristic
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—24no—
227C2×C4×C73—24no—
228C2×C4×C73—22no—
229C2×C4×C73—24no—
230C2×C4×C73—22no—
231C2×C4×C73—24no—
232C2×C4×C73—416no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—41no—
241C2×C4×C73—42no—
242C2×C4×C73—41no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—42no—
246C2×C4×C73—41no—
247C2×C4×C73—42no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×C4×C73—42no—
254C2×C4×C73—41no—
255C2×C4×C73—42no—
256C2×C4×C73—42no—
257C2×C4×C73—42no—
258C2×C4×C73—42no—
259C2×C4×C73—41no—
260C2×C4×C73—42no—
261C2×C4×C73—42no—
262C2×C4×C73—42no—
263C2×C4×C73—42no—
264C2×C4×C73—42no—
265C2×C4×C73—42no—
266C2×C4×C73—42no—
267C2×C4×C73—42no—
268C2×C4×C73—42no—
269C2×C4×C73—42no—
270C2×C4×C73—42no—
271C2×C4×C73—42no—
272C2×C4×C73—42no—
273C2×C4×C73—42no—
274C2×C4×C73—42no—
275C2×C4×C73#12114yes—
276C2×C4×C73#12314yes—
277C22×C73—24no—
278C22×C73—24no—
279C22×C73—24no—
280C22×C73—42no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—44no—
284C22×C73—44no—
285C22×C73—44no—
286C22×C73—44no—
287C22×C73—42no—
288C22×C73C2×#12111yescharacteristic
289C22×C73C2×#12111yescharacteristic
290C22×C73C2×#12311yescharacteristic
291C22×C73#53411yescharacteristic
292C22×C73#53611yescharacteristic
293C22×C73#53612yes—
294C22×C73#53611yescharacteristic
295C22×C73#53712yes—
296C22×C73#53711yescharacteristic
297C22×C73#53912yes—
298C22×C73#54112yes—
299C22×C73#54112yes—
300C22×C73#54212yes—
301C22×C73D4211yescharacteristic
302C22×C73#54612yes—
303C22×C73#54711yescharacteristic
304C22×C73#54712yes—
305C22×C73#54811yescharacteristic
306C22×C73#55112yes—
307C22×C73#55212yes—
308C22×C73#55311yescharacteristic
309C22×C73#56212yes—
310C22×C73#56312yes—
311C4×C73—24no—
312C4×C73—24no—
313C4×C73—42no—
314C4×C73—44no—
315C4×C73—42no—
316C4×C73—82no—
317C4×C73—81no—
318C4×C73—82no—
319C4×C73—82no—
320C4×C73—81no—
321#16399C7311yesmaximal, characteristic, direct product factor, Sylow
322C2×C73—44no—
323C2×C73#241211yescharacteristic
324C2×C73#241611yescharacteristic
325C2×C73#248611yescharacteristic
326C2×C73#261811yescharacteristic
327C2×C73#262111yescharacteristic
328C2×C73#262511yescharacteristic
329C2×C73#267212yes—
330C2×C73#269012yes—
331C2×C73#272212yes—
332C2×C73#274712yes—
333C2×C73#281411yescharacteristic
334C2×#398C2×C7311yescharacteristic, complemented
335C2×#397C2×C7312yes—
336C2×#385C2×C7311yescharacteristic
337C2×#380C2×C7311yescharacteristic
338C2×#380C2×C7311yescharacteristic
339#1445C2×C7311yescharacteristic, complemented
340#1442C2×C7311yescharacteristic, complemented
341#1441C2×C7311yescharacteristic, complemented
342#1441C2×C7312yescomplemented
343#1441C2×C7312yescomplemented
344#1440C2×C7312yes—
345C73#1639911yescharacteristic, direct product factor, Sylow
346C24×C4C22×C7311yescharacteristic
347C22×C4⋊C4C22×C7311yescharacteristic
348C22×C4⋊C4C22×C7311yescharacteristic
349C22×C4⋊C4C22×C7311yescharacteristic
350C22×C4⋊C4C22×C7312yes—
351C22×C4⋊C4C22×C7312yes—
352C22×C4⋊C4C22×C7311yescharacteristic
353C22×C22⋊C4C22×C7312yes—
354C22×C22⋊C4C22×C7312yes—
355C22×C22⋊C4C22×C7311yescharacteristic
356C22×C22⋊C4C22×C7311yescharacteristic
357#398—24no—
358#397—28no—
359#385—24no—
360#380—24no—
361#380—24no—
362C2×#95C22×C7312yes—
363C2×#95C22×C7312yes—
364C2×#95C22×C7312yes—
365C2×#95C22×C7312yes—
366C2×#95C22×C7312yes—
367C2×#95C22×C7311yescharacteristic
368C2×#95C22×C7311yescharacteristic
369C2×#95C22×C7311yescharacteristic
370C2×#95C22×C7312yes—
371C2×#95C22×C7312yes—
372C2×#95C22×C7312yes—
373C2×#95C22×C7311yescharacteristic
374C25C23×C7311yescharacteristic
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—24no—
378C23×C4—42no—
379C23×C4D4×C7314yes—
380C23×C4C23×C7311yescharacteristic
381C23×C4C23×C7312yes—
382C23×C4C23×C7312yes—
383C23×C4C23×C7312yes—
384C23×C4C23×C7311yescharacteristic
385C23×C4C23×C7311yescharacteristic
386C23×C4C23×C7312yes—
387C23×C4C23×C7311yescharacteristic
388C23×C4C23×C7311yescharacteristic
389C23×C4C23×C7311yescharacteristic
390C2×C4⋊C4—24no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—24no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—22no—
398C2×C4⋊C4—22no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—24no—
403C2×C4⋊C4—42no—
404C2×C4⋊C4—42no—
405C2×C4⋊C4—42no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4—24no—
412C2×C22⋊C4—24no—
413C2×C22⋊C4—24no—
414C2×C22⋊C4—42no—
415C2×C22⋊C4—42no—
416C2×C22⋊C4D4×C7314yes—
417#95—22no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—42no—
426#95—42no—
427#95—42no—
428#95D4×C7314yes—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—21no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24C2×D4×C7312yes—
438C24C2×D4×C7312yes—
439C24#41×C7312yes—
440C24C24×C7311yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—44no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4C2×D4×C7312yes—
499C22×C4C2×D4×C7312yes—
500C22×C4C2×D4×C7314yes—
501C22×C4C2×D4×C7312yes—
502C22×C4C2×D4×C7312yes—
503C22×C4C2×D4×C7312yes—
504C22×C4#41×C7312yes—
505C22×C4#41×C7314yes—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23#121×C7314yes—
539C23C22×D4×C7311yescharacteristic
540C23C22×D4×C7311yescharacteristic
541C23C22×D4×C7311yescharacteristic
542C23C2×#41×C7311yescharacteristic
543C23C2×#41×C7311yescharacteristic
544C23C2×#41×C7312yes—
545C23H(5,𝔽2)×C7312yes—
546C23H(5,𝔽2)×C7311yescharacteristic
547C232−1+4×C7312yes—
548C232−1+4×C7312yes—
549C232−1+4×C7311yescharacteristic
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—416no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4#121×C7314yes—
610C2×C4#123×C7314yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22C2×#121×C7311yescharacteristic
623C22C2×#121×C7311yescharacteristic
624C22C2×#123×C7311yescharacteristic
625C22#534×C7311yescharacteristic
626C22#536×C7311yescharacteristic
627C22#536×C7311yescharacteristic
628C22#536×C7312yes—
629C22#537×C7312yes—
630C22#537×C7311yescharacteristic
631C22#539×C7312yes—
632C22#541×C7312yes—
633C22#541×C7312yes—
634C22#542×C7312yes—
635C22D42×C7311yescharacteristic
636C22#546×C7312yes—
637C22#547×C7311yescharacteristic
638C22#547×C7312yes—
639C22#548×C7311yescharacteristic
640C22#551×C7312yes—
641C22#552×C7312yes—
642C22#553×C7311yescharacteristic
643C22#562×C7312yes—
644C22#563×C7312yes—
645C4—24no—
646C4—24no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—81no—
651C4—82no—
652C4—81no—
653C4—82no—
654C4—82no—
655C2—44no—
656C2C73×#241211yescharacteristic
657C2C73×#241611yescharacteristic
658C2C73×#248611yescharacteristic
659C2C73×#261811yescharacteristic
660C2C73×#262111yescharacteristic
661C2C73×#262511yescharacteristic
662C2C73×#267212yes—
663C2C73×#269012yes—
664C2C73×#272212yes—
665C2C73×#274712yes—
666C2C73×#281411yescharacteristic
subgroupG/H
center, socleC24×C73C24
commutator, Frattini, cosocleC24C24×C73
prime radicalC25×C73C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C73 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#398×C73C2semidirect
6. C73×#1445C2semidirect
7. C73×#1442C2semidirect
8. C73×#1441C2semidirect
9. C73×#1441C2semidirect
10. C73×#1441C2semidirect
321. #16399C73direct (N × H)
334. C2×#398C2×C73semidirect
339. #1445C2×C73semidirect
340. #1442C2×C73semidirect
341. #1441C2×C73semidirect
342. #1441C2×C73semidirect
343. #1441C2×C73semidirect
345. C73#16399direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#398×C732
C2×#397×C732
C2×#385×C732
C2×#380×C732
C73×#14452
C73×#14422
C73×#14412
C73×#14402
#1639973