Finite Group Databasebeta

← #1852×D73 #33086686

Subgroups — #1852×D73 #33086686

Order 18688.

1966 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D73C211yesmaximal, characteristic
2C2×#107×D73C212yesmaximal, complemented
3C2×C22⋊C8×D73C211yesmaximal, characteristic
4#380×D73C211yesmaximal, characteristic
5#22610121C211yesmaximal, characteristic, complemented
6#22610120C212yesmaximal, complemented
7#22610119C211yesmaximal, characteristic, complemented
8#22610118C211yesmaximal, characteristic, complemented
9#22610117C212yesmaximal, complemented
10#340×D73C212yesmaximal, complemented
11C73×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D73—21no—
13C23×C4×D73C2211yescharacteristic
14C22×C8×D73C2212yes—
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924007—21no—
17C2×C4⋊C4×D73—21no—
18C2×C4⋊C4×D73—21no—
19C2×C4⋊C4×D73—28no—
20C2×C4⋊C4×D73C2211yescharacteristic
21C2×C4⋊C4×D73C2211yescharacteristic
22C2×C4⋊C4×D73C2212yes—
23C2×C4⋊C4×D73C2212yes—
24C2×C22⋊C4×D73—21no—
25C2×#14556525C2212yescomplemented
26C2×#14556524C2212yescomplemented
27C2×#14556524C2212yescomplemented
28#107×D73—28no—
29C2×C22⋊#10596831C2211yescharacteristic
30C2×#14556503C2212yes—
31C22⋊C8×D73C418yescomplemented
32#17988447C2211yescharacteristic
33#17988446—21no—
34#17988445—21no—
35#17988444C2211yescharacteristic
36#17988444C2211yescharacteristic
37#95×D73—21no—
38#17988350C2212yescomplemented
39#17988349C2212yescomplemented
40#17988349C2212yescomplemented
41C22×C4⋊C4×C73C2211yescharacteristic
42C2×#107×C73C2212yescomplemented
43C2×C22⋊C8×C73C2211yescharacteristic
44#380×C73C2211yescharacteristic
45#340×C73C2212yescomplemented
46C24×D73D411yescharacteristic
47C23×Dic73—21no—
48C23×Dic73D411yescharacteristic
49C22×C4×D73—24no—
50C22×C4×D73—24no—
51C22×C4×D73—28no—
52C22×C4×D73—22no—
53C22×C4×D73—22no—
54C22×C4×D73—21no—
55C22×C4×D73—21no—
56C22×C4×D73—44no—
57C22×C4×D73—44no—
58C22×C4×D73C2×C414yes—
59C22×C4×D73D411yescharacteristic
60C22×C4×D73D411yescharacteristic
61C22×C4×D73Q814yes—
62C22×C4×D73C2311yescharacteristic
63C22×C4×D73C2311yescharacteristic
64C22×#10596831C2312yes—
65C2×C8×D73—24no—
66C2×C8×D73—24no—
67C2×C8×D73C2×C418yescomplemented
68C2×C4⋊Dic73—24no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73C2311yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2312yes—
73C2×#11924007—28no—
74C2×#11924007—22no—
75C2×#11924007—22no—
76C2×#11924007—21no—
77C2×#11924007—21no—
78C4⋊C4×D73—28no—
79C4⋊C4×D73—28no—
80C4⋊C4×D73—24no—
81C4⋊C4×D73—24no—
82C4⋊C4×D73—24no—
83C4⋊C4×D73—24no—
84C4⋊C4×D73—24no—
85C4⋊C4×D73D418yes—
86C2×C22⋊Dic73—21no—
87C2×#11924005—21no—
88C2×#11924005—21no—
89C22⋊C4×D73—44no—
90#14556525—24no—
91#14556524—24no—
92#14556524—24no—
93C22⋊#10596831C2×C414yescomplemented
94#14556503C2×C418yescomplemented
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98C23×C4×C73—21no—
99C23×C4×C73C2311yescharacteristic
100C22×C8×C73C2312yes—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—24no—
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2312yes—
106C2×C22⋊C4×C73—21no—
107#107×C73—24no—
108C22⋊C8×C73C2×C414yescomplemented
109#95×C73—21no—
110C23×D73—22no—
111C23×D73—22no—
112C23×D73—22no—
113C23×D73—22no—
114C23×D73C4⋊C414yes—
115C23×D73D814yes—
116C23×D73Q1614yes—
117C23×D73C2×D411yescharacteristic
118C23×D73C2×D411yescharacteristic
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—24no—
122C22×Dic73—22no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—42no—
126C22×Dic73—42no—
127C22×Dic73C4⋊C414yes—
128C22×Dic73C2×D411yescharacteristic
129C22×Dic73C2×D411yescharacteristic
130C2×C4×D73—22no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—22no—
135C2×C4×D73—22no—
136C2×C4×D73—216no—
137C2×C4×D73—216no—
138C2×C4×D73—28no—
139C2×C4×D73—28no—
140C2×C4×D73—44no—
141C2×C4×D73—432no—
142C2×C4×D73—44no—
143C2×C4×D73—44no—
144C2×C4×D73—44no—
145C2×C4×D73—42no—
146C2×C4×D73—42no—
147C2×C4×D73—42no—
148C2×C4×D73—42no—
149C2×C4×D73—42no—
150C2×C4×D73—42no—
151C2×C4×D73C4⋊C414yes—
152C2×C4×D73C4⋊C414yes—
153C2×C4×D73C22×C412yes—
154C2×C4×D73C22×C412yes—
155C2×C4×D73C2×D414yes—
156C2×C4×D73C2×Q812yes—
157C2×C4×D73C2×Q812yes—
158C2×C4×D73#4114yes—
159C2×#10596831—22no—
160C2×#10596831—22no—
161C2×#10596831C22×C414yescomplemented
162C8×D73—216no—
163C4⋊Dic73—24no—
164C4⋊Dic73—22no—
165C4⋊Dic73—22no—
166C4⋊Dic73—22no—
167C4⋊Dic73C2×D414yes—
168#11924007—28no—
169#11924007—28no—
170#11924007—42no—
171#11924007—42no—
172C22⋊Dic73—42no—
173#11924005—42no—
174#11924005—42no—
175C24×C73C2×D411yescharacteristic
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—24no—
179C22×C4×C73—22no—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—42no—
183C22×C4×C73—42no—
184C22×C4×C73C22×C412yes—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×Q812yes—
187C22×C4×C73C2411yescharacteristic
188C2×C8×C73—22no—
189C2×C8×C73—22no—
190C2×C8×C73C22×C414yescomplemented
191C4⋊C4×C73—24no—
192C4⋊C4×C73—22no—
193C4⋊C4×C73—22no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73C2×D414yes—
196C22⋊C4×C73—42no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73—21no—
200C23×C73—21no—
201C23×C73C2×C4⋊C412yes—
202C23×C73C2×D812yes—
203C23×C73C2×Q1612yes—
204C23×C73C22×D411yescharacteristic
205C22×D73—21no—
206C22×D73—21no—
207C22×D73—21no—
208C22×D73—21no—
209C22×D73—28no—
210C22×D73—28no—
211C22×D73—28no—
212C22×D73—28no—
213C22×D73—28no—
214C22×D73—28no—
215C22×D73#107116yescomplemented
216C22×D73C2×C4⋊C412yes—
217C22×D73C2×C4⋊C412yes—
218C22×D73#12012yes—
219C22×D73#12312yes—
220C22×D73C2×D812yes—
221C22×D73C2×D812yes—
222C22×D73C2×Q1612yes—
223C22×D73C2×Q1612yes—
224C22×D73AGL(1,ℤ/8)12yes—
225C22×D73#13712yes—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—24no—
231C2×C4×C73—48no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
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×C73C2×C4⋊C412yes—
241C2×C4×C73C23×C411yescharacteristic
242C2×C4×C73C22×D412yes—
243C2×C4×C73C22×Q811yescharacteristic
244C2×C4×C73C2×#4112yes—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—28no—
248C2×Dic73—24no—
249C2×Dic73—48no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73C2×C4⋊C412yes—
259C2×Dic73C2×C4⋊C412yes—
260C2×Dic73#12012yes—
261C2×Dic73#12312yes—
262C4×D73—28no—
263C4×D73—24no—
264C4×D73—24no—
265C4×D73—28no—
266C4×D73—416no—
267C4×D73—416no—
268C4×D73—84no—
269C4×D73—84no—
270C4×D73C4×D418yes—
271C4×D73C22⋊Q818yes—
272C8×C73—24no—
273#10596831—24no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—22no—
280C22×C73C2×#10714yescomplemented
281C22×C73C22×C4⋊C411yescharacteristic
282C22×C73C2×#12011yescharacteristic
283C22×C73C2×#12311yescharacteristic
284C22×C73C22×D811yescharacteristic
285C22×C73C22×Q1611yescharacteristic
286C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C73C2×#13711yescharacteristic
288C2×D73—22no—
289C2×D73—22no—
290C2×D73—22no—
291C2×D73—22no—
292C2×D73—22no—
293C2×D73—22no—
294C2×D73—232no—
295C2×D73#38514yes—
296C2×D73C2×#10714yes—
297C2×D73C2×#10714yescomplemented
298C2×D73#42814yes—
299C2×D73#44714yes—
300C2×D73#45114yes—
301C2×D73#48014yes—
302C2×D73#48414yes—
303C4×C73—22no—
304C4×C73—44no—
305C4×C73—81no—
306C4×C73—81no—
307C4×C73C2×C4×D412yes—
308C4×C73C2×C22⋊Q812yes—
309Dic73—22no—
310Dic73—44no—
311Dic73—81no—
312Dic73—81no—
313Dic73#38514yes—
314C2×#1852—731nomaximal, Sylow
315C2×C73—24no—
316C2×C73C2×#38511yescharacteristic
317C2×C73C22×#10711yescharacteristic
318C2×C73C2×#42811yescharacteristic
319C2×C73C2×#44711yescharacteristic
320C2×C73C2×#45111yescharacteristic
321C2×C73C2×#48011yescharacteristic
322C2×C73C2×#48411yescharacteristic
323D73—24no—
324D73#185218yesdirect product factor
325C23×C4⋊C4—731no—
326C22×#107—732no—
327C22×C22⋊C8—731no—
328C2×#380—731no—
329#1852—732no—
330#1852—731no—
331#1852—731no—
332#1852—732no—
333#1852—731no—
334#1852D7311yescharacteristic, direct product factor
335C2×#340—732no—
336C73C2×#185211yescharacteristic, complemented, Sylow
337C24×C4—731no—
338C24×C4—1461no—
339C23×C8—732no—
340C22×C4⋊C4—732no—
341C22×C4⋊C4—731no—
342C22×C4⋊C4—732no—
343C22×C4⋊C4—731no—
344C22×C4⋊C4—731no—
345C22×C4⋊C4—1468no—
346C22×C4⋊C4—1461no—
347C22×C4⋊C4—1461no—
348C22×C4⋊C4—1461no—
349C22×C4⋊C4C2×D7311yescharacteristic
350C22×C22⋊C4—1461no—
351C2×#107—732no—
352C2×#107—732no—
353C2×#107—732no—
354C2×#107—1468no—
355C2×#107C2×D7312yescomplemented
356C2×C22⋊C8—731no—
357C2×C22⋊C8—732no—
358C2×C22⋊C8—738no—
359C2×C22⋊C8C2×D7311yescharacteristic
360#380—731no—
361#380—731no—
362#380—731no—
363#380—1461no—
364#380—1461no—
365#380C2×D7311yescharacteristic
366C2×#95—1461no—
367#340—732no—
368#340—732no—
369#340—732no—
370#340C2×D7312yescomplemented
371C25—731no—
372C23×C4—21no—
373C23×C4—2924no—
374C23×C4—2924no—
375C23×C4—731no—
376C23×C4—731no—
377C23×C4—734no—
378C23×C4—731no—
379C23×C4—731no—
380C23×C4—734no—
381C23×C4—731no—
382C23×C4—1461no—
383C23×C4—1461no—
384C23×C4—1462no—
385C23×C4—1462no—
386C23×C4—1468no—
387C23×C4—1464no—
388C23×C4—1464no—
389C23×C4—1461no—
390C23×C4C22×D7311yescharacteristic
391C22×C8—732no—
392C22×C8—738no—
393C22×C8—1464no—
394C22×C8—1464no—
395C22×C8C22×D7312yes—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—732no—
399C2×C4⋊C4—738no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—1464no—
403C2×C4⋊C4—1464no—
404C2×C4⋊C4—1464no—
405C2×C4⋊C4—1464no—
406C2×C4⋊C4—1468no—
407C2×C4⋊C4—1468no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4—1464no—
411C2×C4⋊C4—1462no—
412C2×C4⋊C4—1462no—
413C2×C4⋊C4—1468no—
414C2×C4⋊C4—1461no—
415C2×C4⋊C4—1464no—
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C4⋊C4C22×D7312yes—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—2924no—
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424#107—24no—
425#107—1464no—
426#107—1464no—
427#107—1464no—
428C22⋊C8—734no—
429C22⋊C8—738no—
430C22⋊C8C4×D7314yescomplemented
431#95—21no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435C24—731no—
436C24—734no—
437C24—734no—
438C24—734no—
439C24—731no—
440C24—1462no—
441C24—1462no—
442C24—1462no—
443C24—1462no—
444C24D4×D7311yescharacteristic
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—2924no—
454C22×C4—2924no—
455C22×C4—2924no—
456C22×C4—29232no—
457C22×C4—2924no—
458C22×C4—2922no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—2922no—
462C22×C4—2922no—
463C22×C4—2922no—
464C22×C4—2922no—
465C22×C4—2922no—
466C22×C4—731no—
467C22×C4—732no—
468C22×C4—734no—
469C22×C4—732no—
470C22×C4—734no—
471C22×C4—734no—
472C22×C4—731no—
473C22×C4—734no—
474C22×C4—734no—
475C22×C4—732no—
476C22×C4—732no—
477C22×C4—1468no—
478C22×C4—1468no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—14616no—
482C22×C4—14616no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1464no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4C2×C4×D7312yes—
494C22×C4D4×D7311yescharacteristic
495C22×C4Q8×D7312yes—
496C22×C4C23×D7311yescharacteristic
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—734no—
500C2×C8—14616no—
501C2×C8—1462no—
502C2×C8—1462no—
503C2×C8C2×C4×D7314yescomplemented
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—2922no—
509C4⋊C4—2922no—
510C4⋊C4—734no—
511C4⋊C4—1468no—
512C4⋊C4—1468no—
513C4⋊C4—1464no—
514C4⋊C4—1462no—
515C4⋊C4—1462no—
516C4⋊C4—1462no—
517C4⋊C4D4×D7314yes—
518C22⋊C4—42no—
519C22⋊C4—2922no—
520C22⋊C4—2922no—
521C22⋊C4—2922no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—732no—
527C23—732no—
528C23—732no—
529C23—732no—
530C23—732no—
531C23—732no—
532C23—732no—
533C23—7316no—
534C23—732no—
535C23—732no—
536C23—732no—
537C23—1468no—
538C23—1468no—
539C23—1468no—
540C23—1468no—
541C23—1468no—
542C23—1468no—
543C23—1461no—
544C23—1461no—
545C23—1461no—
546C23—1461no—
547C23C4⋊C4×D7312yes—
548C23D8×D7312yes—
549C23Q16×D7312yes—
550C23C2×D4×D7311yescharacteristic
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—29216no—
566C2×C4—29216no—
567C2×C4—2928no—
568C2×C4—2922no—
569C2×C4—2922no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—5844no—
577C2×C4—5844no—
578C2×C4—732no—
579C2×C4—732no—
580C2×C4—732no—
581C2×C4—738no—
582C2×C4—738no—
583C2×C4—732no—
584C2×C4—1468no—
585C2×C4—1468no—
586C2×C4—1464no—
587C2×C4—1468no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1464no—
591C2×C4—1464no—
592C2×C4C4⋊C4×D7312yes—
593C2×C4C22×C4×D7311yescharacteristic
594C2×C4C2×D4×D7312yes—
595C2×C4C2×Q8×D7311yescharacteristic
596C2×C4#41×D7312yes—
597C8—24no—
598C8—1464no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—734no—
606C22—734no—
607C22—734no—
608C22—734no—
609C22—734no—
610C22—734no—
611C22—734no—
612C22—734no—
613C22—14632no—
614C22—1462no—
615C22—1462no—
616C22—1462no—
617C22—1462no—
618C22—1462no—
619C22—1462no—
620C22#107×D7314yescomplemented
621C22C2×C4⋊C4×D7311yescharacteristic
622C22#120×D7311yescharacteristic
623C22#123×D7311yescharacteristic
624C22C2×D8×D7311yescharacteristic
625C22C2×Q16×D7311yescharacteristic
626C22AGL(1,ℤ/8)×D7311yescharacteristic
627C22#137×D7311yescharacteristic
628C4—22no—
629C4—44no—
630C4—81no—
631C4—81no—
632C4—2924no—
633C4—5841no—
634C4—5841no—
635C4—734no—
636C4—1462no—
637C4C4×D4×D7312yes—
638C4C22⋊Q8×D7312yes—
639C2—24no—
640C2—738no—
641C2—1464no—
642C2#385×D7311yescharacteristic
643C2C2×#107×D7311yescharacteristic
644C2#428×D7311yescharacteristic
645C2#447×D7311yescharacteristic
646C2#451×D7311yescharacteristic
647C2#480×D7311yescharacteristic
648C2#484×D7311yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1852C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenter#1852D73
hypocenterC73C2×#1852
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#107×D73C2semidirect
5. #22610121C2semidirect
6. #22610120C2semidirect
7. #22610119C2semidirect
8. #22610118C2semidirect
9. #22610117C2semidirect
10. #340×D73C2semidirect
11. C73×#1852C2semidirect
25. C2×#14556525C22semidirect
26. C2×#14556524C22semidirect
27. C2×#14556524C22semidirect
31. C22⋊C8×D73C4semidirect
38. #17988350C22semidirect
39. #17988349C22semidirect
40. #17988349C22semidirect
42. C2×#107×C73C22semidirect
45. #340×C73C22semidirect
67. C2×C8×D73C2×C4semidirect
93. C22⋊#10596831C2×C4semidirect
94. #14556503C2×C4semidirect
108. C22⋊C8×C73C2×C4semidirect
161. C2×#10596831C22×C4semidirect
190. C2×C8×C73C22×C4semidirect
215. C22×D73#107semidirect
280. C22×C73C2×#107semidirect
297. C2×D73C2×#107semidirect
324. D73#1852direct (N × H)
334. #1852D73direct (N × H)
336. C73C2×#1852semidirect
355. C2×#107C2×D73semidirect
370. #340C2×D73semidirect
430. C22⋊C8C4×D73semidirect
503. C2×C8C2×C4×D73semidirect
620. C22#107×D73semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D732
C2×#107×D732
C2×C22⋊C8×D732
#380×D732
#226101212
#226101202
#226101192
#226101182
#226101172
#340×D732
C73×#18522
C2×#185273