Finite Group Databasebeta

← #33096272

Subgroups — #33096272

Order 18688.

1162 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988549C211yesmaximal, characteristic
2#22612313C211yesmaximal, characteristic
3C2×#17988444C211yesmaximal, characteristic
4#22610133C212yesmaximal, complemented
5#22610129C212yesmaximal, complemented
6#22610123C211yesmaximal, characteristic, complemented
7#22610121C211yesmaximal, characteristic, complemented
8#22610119C211yesmaximal, characteristic, complemented
9#22609963C212yesmaximal, complemented
10#22609961C212yesmaximal, complemented
11C73×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×#11924011C2212yes—
14C2×#14556549C2211yescharacteristic
15C2×#14556549C2211yescharacteristic
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924005—21no—
18C2×#14556525C2212yescomplemented
19C2×#14556524C2212yescomplemented
20C2×#14556522C2212yescomplemented
21#17988549C418yescomplemented
22C2×C22⋊#10596831C2211yescharacteristic
23C2×#14556503C2212yes—
24#17988449—21no—
25#17988447C2211yescharacteristic
26#17988444—21no—
27#17988444—21no—
28#17988444C2211yescharacteristic
29#17988444C2212yes—
30#17988444C2211yescharacteristic
31#17988444C2212yes—
32#17988442—21no—
33C22⋊C4×Dic73—21no—
34#17988434—21no—
35C2×#14556498—21no—
36#17988350C2212yescomplemented
37#17988349C2212yescomplemented
38#17988349C2212yescomplemented
39C22×C4⋊C4×C73C2211yescharacteristic
40C2×#106×C73C2212yescomplemented
41C2×C22⋊C8×C73C2211yescharacteristic
42#380×C73C2211yescharacteristic
43#340×C73C2212yescomplemented
44C24×D73D411yescharacteristic
45C23×Dic73—21no—
46C23×Dic73D411yescharacteristic
47C22×C4×D73—42no—
48C22×C4×D73—42no—
49C22×C4×D73C2×C414yes—
50C22×C4×D73D411yescharacteristic
51C22×C4×D73D411yescharacteristic
52C22×C4×D73Q814yes—
53C22×C4×D73C2311yescharacteristic
54C22×C4×D73C2311yescharacteristic
55C2×#11924011—42no—
56C2×#11924011—42no—
57C2×#11924011C2×C418yescomplemented
58C22×#10596831C2312yes—
59#14556549—42no—
60#14556549—42no—
61C2×C4⋊Dic73—21no—
62C2×C4⋊Dic73—42no—
63C2×C4⋊Dic73C2312yes—
64C2×C4⋊Dic73C2311yescharacteristic
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C22⋊Dic73—21no—
67C2×#11924005—21no—
68C2×#11924005—21no—
69C2×#11924005—21no—
70C2×#11924005—21no—
71C2×#11924005—22no—
72C2×#11924005—22no—
73C2×#11924005—44no—
74C2×C4×Dic73—21no—
75C2×C4×Dic73—21no—
76#14556525—42no—
77#14556524—42no—
78#14556522—42no—
79C22⋊#10596831C2×C414yes—
80#14556503C2×C418yescomplemented
81#14556498—22no—
82#14556498—22no—
83#14556498—21no—
84#14556498—21no—
85#14556498—21no—
86C23×C4×C73—21no—
87C23×C4×C73C2311yescharacteristic
88C22×C8×C73C2312yes—
89C2×C4⋊C4×C73—21no—
90C2×C4⋊C4×C73—42no—
91C2×C4⋊C4×C73C2311yescharacteristic
92C2×C4⋊C4×C73C2311yescharacteristic
93C2×C4⋊C4×C73C2312yes—
94C2×C22⋊C4×C73—21no—
95#106×C73—42no—
96C22⋊C8×C73C2×C414yes—
97#95×C73—21no—
98C23×D73—22no—
99C23×D73—22no—
100C23×D73—41no—
101C23×D73—41no—
102C23×D73—41no—
103C23×D73—41no—
104C23×D73C4⋊C414yes—
105C23×D73C2×D411yescharacteristic
106C23×D73C2×D411yescharacteristic
107C22×Dic73—21no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—41no—
111C22×Dic73—41no—
112C22×Dic73—42no—
113C22×Dic73—42no—
114C22×Dic73—42no—
115C22×Dic73C4⋊C414yes—
116C22×Dic73C2×D411yescharacteristic
117C22×Dic73C2×D411yescharacteristic
118C2×C4×D73—22no—
119C2×C4×D73—22no—
120C2×C4×D73—22no—
121C2×C4×D73—22no—
122C2×C4×D73—48no—
123C2×C4×D73—48no—
124C2×C4×D73—41no—
125C2×C4×D73—41no—
126C2×C4×D73—41no—
127C2×C4×D73—41no—
128C2×C4×D73C4⋊C414yes—
129C2×C4×D73C4⋊C414yes—
130C2×C4×D73C22×C412yes—
131C2×C4×D73C22×C412yes—
132C2×C4×D73C2×Q812yes—
133C2×C4×D73C2×Q812yes—
134#11924011—48no—
135C2×#10596831—22no—
136C2×#10596831—22no—
137C2×#10596831C22×C414yes—
138C4⋊Dic73—22no—
139C4⋊Dic73—22no—
140C4⋊Dic73—22no—
141C4⋊Dic73—22no—
142C4⋊Dic73—42no—
143C22⋊Dic73—42no—
144#11924005—42no—
145#11924005—42no—
146#11924005—44no—
147#11924005—44no—
148#11924005—42no—
149#11924005—42no—
150C4×Dic73—42no—
151C4×Dic73—42no—
152C24×C73C2×D411yescharacteristic
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73—42no—
159C22×C4×C73—42no—
160C22×C4×C73—42no—
161C22×C4×C73C22×C412yes—
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73C2×Q812yes—
164C22×C4×C73C2411yescharacteristic
165C2×C8×C73—22no—
166C2×C8×C73—22no—
167C2×C8×C73C22×C414yes—
168C4⋊C4×C73—22no—
169C4⋊C4×C73—22no—
170C4⋊C4×C73—22no—
171C4⋊C4×C73—22no—
172C4⋊C4×C73—42no—
173C22⋊C4×C73—42no—
174C23×C73—21no—
175C23×C73—21no—
176C23×C73—21no—
177C23×C73—21no—
178C23×C73C2×C4⋊C412yes—
179C23×C73AGL(1,ℤ/8)12yes—
180C23×C73#13712yes—
181C23×C73C22×D411yescharacteristic
182C22×D73—21no—
183C22×D73—21no—
184C22×D73—21no—
185C22×D73—21no—
186C22×D73—21no—
187C22×D73—21no—
188C22×D73—21no—
189C22×D73—21no—
190C22×D73—21no—
191C22×D73—21no—
192C22×D73—21no—
193C22×D73—21no—
194C22×D73—28no—
195C22×D73—44no—
196C22×D73—44no—
197C22×D73—44no—
198C22×D73—44no—
199C22×D73—44no—
200C22×D73—44no—
201C22×D73C2×C4⋊C412yes—
202C22×D73C2×C4⋊C412yes—
203C2×C4×C73—21no—
204C2×C4×C73—21no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73—42no—
208C2×C4×C73—41no—
209C2×C4×C73—41no—
210C2×C4×C73—42no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—42no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—84no—
217C2×C4×C73C2×C4⋊C412yes—
218C2×C4×C73C23×C411yescharacteristic
219C2×C4×C73C22×Q811yescharacteristic
220C2×C4×C73H(5,𝔽2)12yes—
221C2×C4×C73H(5,𝔽2)12yes—
222C2×Dic73—21no—
223C2×Dic73—21no—
224C2×Dic73—21no—
225C2×Dic73—21no—
226C2×Dic73—44no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73—42no—
230C2×Dic73—42no—
231C2×Dic73—41no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—41no—
235C2×Dic73—42no—
236C2×Dic73—84no—
237C2×Dic73C2×C4⋊C412yes—
238C2×Dic73C2×C4⋊C412yes—
239C4×D73—24no—
240C4×D73—24no—
241C4×D73—24no—
242C4×D73—24no—
243C4×D73—44no—
244C4×D73—44no—
245C8×C73—24no—
246#10596831—24no—
247C22×C73—22no—
248C22×C73—22no—
249C22×C73—22no—
250C22×C73—22no—
251C22×C73—22no—
252C22×C73—22no—
253C22×C73#42814yes—
254C22×C73C22×C4⋊C411yescharacteristic
255C22×C73#53411yescharacteristic
256C22×C73#53411yescharacteristic
257C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
258C22×C73C2×#13711yescharacteristic
259C22×C73#57511yescharacteristic
260C22×C73#57511yescharacteristic
261C2×D73—22no—
262C2×D73—22no—
263C2×D73—22no—
264C2×D73—22no—
265C2×D73—22no—
266C2×D73—22no—
267C2×D73—416no—
268C2×D73—41no—
269C2×D73—41no—
270C2×D73—41no—
271C2×D73—41no—
272C2×D73—41no—
273C2×D73—41no—
274C2×D73—41no—
275C2×D73—41no—
276C4×C73—22no—
277C4×C73—82no—
278C4×C73—81no—
279C4×C73—81no—
280C4×C73#51812yes—
281C4×C73#54312yes—
282Dic73—22no—
283Dic73—22no—
284Dic73—81no—
285Dic73—81no—
286Dic73—82no—
287#21944—731nomaximal, Sylow
288C2×C73—24no—
289C2×C73#231511yescharacteristic
290C2×C73C2×#42811yescharacteristic
291C2×C73#291111yescharacteristic
292C2×C73#319411yescharacteristic
293C2×C73#319411yescharacteristic
294C2×C73#319611yescharacteristic
295C2×C73#319611yescharacteristic
296D73—42no—
297D73—42no—
298C2×#407—731no—
299#2280—731no—
300C2×#380—731no—
301#1854—732no—
302#1854—732no—
303#1853—731no—
304#1853D7311yescharacteristic, complemented
305#1852—731no—
306#1852—731no—
307#1809—732no—
308#1809—732no—
309C73#2194411yescharacteristic, complemented, Sylow
310C24×C4—731no—
311C22×M16—732no—
312C2×#117—731no—
313C2×#117—731no—
314C22×C4⋊C4—731no—
315C22×C4⋊C4C2×D7311yescharacteristic
316C22×C22⋊C4—1461no—
317C2×#107—732no—
318C2×#107—732no—
319C2×#106—732no—
320C2×#106C2×D7312yescomplemented
321#407—738no—
322C2×C22⋊C8—731no—
323C2×C22⋊C8—732no—
324C2×C22⋊C8C2×D7311yescharacteristic
325#381—1461no—
326#380—731no—
327#380—732no—
328#380—731no—
329#380—731no—
330#380—732no—
331#380—1461no—
332#380—1461no—
333#380C2×D7311yescharacteristic
334#379—1461no—
335C4×C22⋊C4—1461no—
336C4×C22⋊C4—1461no—
337C2×#95—1461no—
338#340—732no—
339#340—732no—
340#340—732no—
341#340C2×D7312yescomplemented
342C25—731no—
343C23×C4—21no—
344C23×C4—2922no—
345C23×C4—2922no—
346C23×C4—731no—
347C23×C4—734no—
348C23×C4—731no—
349C23×C4—731no—
350C23×C4—734no—
351C23×C4—731no—
352C23×C4—731no—
353C23×C4—1461no—
354C23×C4C22×D7311yescharacteristic
355C2×M16—2922no—
356C2×M16—2922no—
357C2×M16—738no—
358C22×C8—732no—
359C22×C8C22×D7312yes—
360#117—2922no—
361#117—2922no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—42no—
364C2×C4⋊C4—2922no—
365C2×C4⋊C4—732no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—1461no—
369C2×C4⋊C4C22×D7312yes—
370C2×C4⋊C4C22×D7311yescharacteristic
371C2×C4⋊C4C22×D7311yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—2924no—
374C2×C22⋊C4—1462no—
375C2×C22⋊C4—1462no—
376C2×C22⋊C4—1461no—
377C2×C22⋊C4—1461no—
378C2×C22⋊C4—1461no—
379C2×C22⋊C4—1461no—
380C2×C22⋊C4—1461no—
381C2×C42—1461no—
382C2×C42—1461no—
383#107—2922no—
384#107—2922no—
385#106—42no—
386#106—2922no—
387C22⋊C8—734no—
388C22⋊C8—738no—
389C22⋊C8C4×D7314yes—
390#95—21no—
391#95—1462no—
392#95—1462no—
393#95—1461no—
394#95—1461no—
395#95—1461no—
396C24—2921no—
397C24—2921no—
398C24—2921no—
399C24—2921no—
400C24—734no—
401C24—731no—
402C24—731no—
403C24—1462no—
404C24—1462no—
405C24D4×D7311yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—2921no—
415C22×C4—2921no—
416C22×C4—2921no—
417C22×C4—2921no—
418C22×C4—2928no—
419C22×C4—2928no—
420C22×C4—2921no—
421C22×C4—2921no—
422C22×C4—2922no—
423C22×C4—2922no—
424C22×C4—2922no—
425C22×C4—734no—
426C22×C4—734no—
427C22×C4—734no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—732no—
431C22×C4—732no—
432C22×C4—732no—
433C22×C4—732no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4C2×C4×D7312yes—
442C22×C4D4×D7311yescharacteristic
443C22×C4Q8×D7312yes—
444C22×C4C23×D7311yescharacteristic
445M16—2928no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—734no—
449C2×C8—1462no—
450C2×C8—1462no—
451C2×C8C2×C4×D7314yes—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—2922no—
458C4⋊C4—1462no—
459C4⋊C4—1462no—
460C4⋊C4—1462no—
461C4⋊C4—1462no—
462C22⋊C4—42no—
463C22⋊C4—2924no—
464C22⋊C4—2924no—
465C22⋊C4—2922no—
466C22⋊C4—2922no—
467C22⋊C4—2922no—
468C22⋊C4—2922no—
469C22⋊C4—2922no—
470C42—2922no—
471C42—2922no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—2924no—
477C23—2924no—
478C23—2924no—
479C23—2924no—
480C23—2924no—
481C23—2924no—
482C23—732no—
483C23—732no—
484C23—1461no—
485C23—1461no—
486C23—1461no—
487C23—1461no—
488C23—1461no—
489C23—1461no—
490C23—1468no—
491C23—1461no—
492C23—1461no—
493C23—1461no—
494C23—1461no—
495C23—1461no—
496C23—1461no—
497C23C4⋊C4×D7312yes—
498C23#1455661812yes—
499C23#1455662012yes—
500C23C2×D4×D7311yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—2924no—
516C2×C4—2924no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2922no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—2924no—
527C2×C4—5844no—
528C2×C4—732no—
529C2×C4—732no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1464no—
535C2×C4—1464no—
536C2×C4—1464no—
537C2×C4—1464no—
538C2×C4C4⋊C4×D7312yes—
539C2×C4C22×C4×D7311yescharacteristic
540C2×C4C2×Q8×D7311yescharacteristic
541C2×C4#1455663612yes—
542C2×C4#1455663712yes—
543C8—24no—
544C8—1464no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—29216no—
552C22—2921no—
553C22—2921no—
554C22—2921no—
555C22—2921no—
556C22—2921no—
557C22—2921no—
558C22—2921no—
559C22—2921no—
560C22—1462no—
561C22—1462no—
562C22—1462no—
563C22—1462no—
564C22—1462no—
565C22—1462no—
566C22#1798862614yes—
567C22C2×C4⋊C4×D7311yescharacteristic
568C22#1798914711yescharacteristic
569C22#1798914811yescharacteristic
570C22C2×#1455661811yescharacteristic
571C22C2×#1455662011yescharacteristic
572C22#1798938311yescharacteristic
573C22#1798938611yescharacteristic
574C4—22no—
575C4—82no—
576C4—81no—
577C4—81no—
578C4—5841no—
579C4—5842no—
580C4—5841no—
581C4—1462no—
582C4—1462no—
583C4#1798905912yes—
584C4#1798921212yes—
585C2—24no—
586C2—2922no—
587C2—2922no—
588C2#2261254211yescharacteristic
589C2C2×#1798862611yescharacteristic
590C2#2261851811yescharacteristic
591C2#2262074811yescharacteristic
592C2#2262075011yescharacteristic
593C2#2262075811yescharacteristic
594C2#2262076011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1853C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenter#1853D73
hypocenterC73#21944
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
4. #22610133C2semidirect
5. #22610129C2semidirect
6. #22610123C2semidirect
7. #22610121C2semidirect
8. #22610119C2semidirect
9. #22609963C2semidirect
10. #22609961C2semidirect
11. C73×#1853C2semidirect
18. C2×#14556525C22semidirect
19. C2×#14556524C22semidirect
20. C2×#14556522C22semidirect
21. #17988549C4semidirect
36. #17988350C22semidirect
37. #17988349C22semidirect
38. #17988349C22semidirect
40. C2×#106×C73C22semidirect
43. #340×C73C22semidirect
57. C2×#11924011C2×C4semidirect
80. #14556503C2×C4semidirect
304. #1853D73semidirect
309. C73#21944semidirect
320. C2×#106C2×D73semidirect
341. #340C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179885492
#226123132
C2×#179884442
#226101332
#226101292
#226101232
#226101212
#226101192
#226099632
#226099612
C73×#18532
#2194473