Finite Group Databasebeta

← #33086718

Subgroups — #33086718

Order 18688.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D73C211yesmaximal, characteristic
2C2×#17988626C212yesmaximal, complemented
3C2×#17988549C211yesmaximal, characteristic
4#22612313C211yesmaximal, characteristic
5#22610123C211yesmaximal, characteristic, complemented
6#22610122C212yesmaximal, complemented
7#22610121C211yesmaximal, characteristic, complemented
8#22610120C212yesmaximal, complemented
9#22610119C211yesmaximal, characteristic, complemented
10#22609649C212yesmaximal, complemented
11C73×#1853C211yesmaximal, characteristic, complemented
12C23×C4×D73—21no—
13C23×C4×D73C2211yescharacteristic
14C22×#11924011C2212yes—
15C2×#14556549C2211yescharacteristic
16C2×#14556549C2211yescharacteristic
17C22×C4⋊Dic73C2211yescharacteristic
18C22×#11924007—21no—
19C2×C4⋊C4×D73—21no—
20C2×C4⋊C4×D73—21no—
21C2×C4⋊C4×D73—28no—
22C2×C4⋊C4×D73C2212yes—
23C2×C4⋊C4×D73C2212yes—
24#17988626—28no—
25C2×#14556525C2212yescomplemented
26C2×#14556524C2212yescomplemented
27C2×#14556522C2212yescomplemented
28#17988549C418yescomplemented
29C2×C22⋊#10596831C2211yescharacteristic
30C2×#14556503C2212yes—
31#17988449—21no—
32#17988447C2211yescharacteristic
33#17988444C2211yescharacteristic
34#17988444C2211yescharacteristic
35#17988442—21no—
36C22⋊C4×Dic73—21no—
37#17988434—21no—
38#17988350C2212yescomplemented
39#17988349C2212yescomplemented
40#17988349C2212yescomplemented
41C22×C4⋊C4×C73C2211yescharacteristic
42C2×#106×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—44no—
55C22×C4×D73—44no—
56C22×C4×D73C2×C414yes—
57C22×C4×D73D411yescharacteristic
58C22×C4×D73D411yescharacteristic
59C22×C4×D73Q814yes—
60C22×C4×D73C2311yescharacteristic
61C22×C4×D73C2311yescharacteristic
62C2×#11924011—24no—
63C2×#11924011—24no—
64C2×#11924011C2×C418yescomplemented
65C22×#10596831C2312yes—
66#14556549—24no—
67#14556549—24no—
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—
76C4⋊C4×D73—28no—
77C4⋊C4×D73—28no—
78C4⋊C4×D73—24no—
79C4⋊C4×D73—24no—
80C4⋊C4×D73—24no—
81C4⋊C4×D73D418yes—
82C2×C22⋊Dic73—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89#14556525—24no—
90#14556524—24no—
91#14556522—24no—
92C22⋊#10596831C2×C414yescomplemented
93#14556503C2×C418yescomplemented
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97C23×C4×C73—21no—
98C23×C4×C73C2311yescharacteristic
99C22×C8×C73C2312yes—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—24no—
102C2×C4⋊C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73C2311yescharacteristic
104C2×C4⋊C4×C73C2312yes—
105C2×C22⋊C4×C73—21no—
106#106×C73—24no—
107C22⋊C8×C73C2×C414yescomplemented
108#95×C73—21no—
109C23×D73—22no—
110C23×D73—22no—
111C23×D73—22no—
112C23×D73—22no—
113C23×D73—22no—
114C23×D73—22no—
115C23×D73C4⋊C414yes—
116C23×D73C2×D411yescharacteristic
117C23×D73C2×D411yescharacteristic
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—24no—
121C22×Dic73—22no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—42no—
125C22×Dic73—42no—
126C22×Dic73C4⋊C414yes—
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C2×C4×D73—22no—
130C2×C4×D73—22no—
131C2×C4×D73—22no—
132C2×C4×D73—22no—
133C2×C4×D73—22no—
134C2×C4×D73—22no—
135C2×C4×D73—216no—
136C2×C4×D73—216no—
137C2×C4×D73—28no—
138C2×C4×D73—28no—
139C2×C4×D73—432no—
140C2×C4×D73—44no—
141C2×C4×D73—44no—
142C2×C4×D73—44no—
143C2×C4×D73—44no—
144C2×C4×D73C4⋊C414yes—
145C2×C4×D73C4⋊C414yes—
146C2×C4×D73C22×C412yes—
147C2×C4×D73C22×C412yes—
148C2×C4×D73C2×D414yes—
149C2×C4×D73C2×Q812yes—
150C2×C4×D73C2×Q812yes—
151C2×C4×D73#4114yes—
152#11924011—216no—
153C2×#10596831—22no—
154C2×#10596831—22no—
155C2×#10596831C22×C414yescomplemented
156C4⋊Dic73—24no—
157C4⋊Dic73—22no—
158C4⋊Dic73—22no—
159C4⋊Dic73—22no—
160C4⋊Dic73C2×D414yes—
161#11924007—28no—
162#11924007—28no—
163C22⋊Dic73—42no—
164#11924005—42no—
165#11924005—42no—
166#11924005—42no—
167#11924005—42no—
168C4×Dic73—42no—
169C4×Dic73—42no—
170C24×C73C2×D411yescharacteristic
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—24no—
174C22×C4×C73—22no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—42no—
178C22×C4×C73—42no—
179C22×C4×C73C22×C412yes—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×Q812yes—
182C22×C4×C73C2411yescharacteristic
183C2×C8×C73—22no—
184C2×C8×C73—22no—
185C2×C8×C73C22×C414yescomplemented
186C4⋊C4×C73—24no—
187C4⋊C4×C73—22no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—22no—
190C4⋊C4×C73C2×D414yes—
191C22⋊C4×C73—42no—
192C23×C73—21no—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73C2×C4⋊C412yes—
197C23×C73AGL(1,ℤ/8)12yes—
198C23×C73#13712yes—
199C23×C73C22×D411yescharacteristic
200C22×D73—28no—
201C22×D73—28no—
202C22×D73—28no—
203C22×D73—28no—
204C22×D73—21no—
205C22×D73—21no—
206C22×D73—21no—
207C22×D73—21no—
208C22×D73—21no—
209C22×D73—21no—
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—28no—
213C22×D73—28no—
214C22×D73—28no—
215C22×D73C2×C4⋊C412yes—
216C22×D73C2×C4⋊C412yes—
217C22×D73#12012yes—
218C22×D73#12312yes—
219C22×D73AGL(1,ℤ/8)12yes—
220C22×D73#13712yes—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—24no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—48no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73C2×C4⋊C412yes—
236C2×C4×C73C23×C411yescharacteristic
237C2×C4×C73C22×D412yes—
238C2×C4×C73C22×Q811yescharacteristic
239C2×C4×C73C2×#4112yes—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—28no—
243C2×Dic73—24no—
244C2×Dic73—48no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73C2×C4⋊C412yes—
254C2×Dic73C2×C4⋊C412yes—
255C2×Dic73#12012yes—
256C2×Dic73#12312yes—
257C4×D73—28no—
258C4×D73—24no—
259C4×D73—24no—
260C4×D73—28no—
261C4×D73—416no—
262C4×D73—416no—
263C4×D73C4×D418yes—
264C4×D73C22⋊Q818yes—
265C8×C73—24no—
266#10596831—24no—
267C22×C73—22no—
268C22×C73—22no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73#42814yescomplemented
274C22×C73C22×C4⋊C411yescharacteristic
275C22×C73C2×#12011yescharacteristic
276C22×C73C2×#12311yescharacteristic
277C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C73C2×#13711yescharacteristic
280C22×C73C2×#13711yescharacteristic
281C2×D73—232no—
282C2×D73—22no—
283C2×D73—22no—
284C2×D73—22no—
285C2×D73—22no—
286C2×D73—22no—
287C2×D73—22no—
288C2×D73—22no—
289C2×D73—22no—
290C2×D73—22no—
291C2×D73—22no—
292C2×D73—22no—
293C2×D73—22no—
294C2×D73#38514yes—
295C2×D73#42814yescomplemented
296C4×C73—22no—
297C4×C73—44no—
298C4×C73—81no—
299C4×C73—81no—
300C4×C73C2×C4×D412yes—
301C4×C73C2×C22⋊Q812yes—
302Dic73—22no—
303Dic73—44no—
304Dic73—81no—
305Dic73—81no—
306Dic73#38514yes—
307#20844—731nomaximal, Sylow
308C2×C73—24no—
309C2×C73C2×#38511yescharacteristic
310C2×C73C2×#42811yescharacteristic
311C2×C73C2×#42811yescharacteristic
312C2×C73#300311yescharacteristic
313C2×C73#300411yescharacteristic
314C2×C73#309211yescharacteristic
315C2×C73#309211yescharacteristic
316D73—24no—
317D73—24no—
318C23×C4⋊C4—731no—
319C2×#428—732no—
320C2×#407—731no—
321#2280—731no—
322#1853—732no—
323#1853—731no—
324#1853D7311yescharacteristic, complemented
325#1852—731no—
326#1852—732no—
327#1852—731no—
328#1736—732no—
329C73#2084411yescharacteristic, complemented, Sylow
330C24×C4—731no—
331C24×C4—1461no—
332C22×M16—732no—
333C2×#117—731no—
334C2×#117—731no—
335C22×C4⋊C4—732no—
336C22×C4⋊C4—731no—
337C22×C4⋊C4—732no—
338C22×C4⋊C4—1468no—
339C22×C4⋊C4—1461no—
340C22×C4⋊C4—1461no—
341C22×C4⋊C4—1461no—
342C22×C4⋊C4C2×D7311yescharacteristic
343#428—1468no—
344C2×#107—732no—
345C2×#107—732no—
346C2×#106—732no—
347C2×#106C2×D7312yescomplemented
348#407—738no—
349C2×C22⋊C8—731no—
350C2×C22⋊C8—732no—
351C2×C22⋊C8C2×D7311yescharacteristic
352#381—1461no—
353#380—731no—
354#380—731no—
355#380—731no—
356#380C2×D7311yescharacteristic
357#379—1461no—
358C4×C22⋊C4—1461no—
359C4×C22⋊C4—1461no—
360#340—732no—
361#340—732no—
362#340—732no—
363#340C2×D7312yescomplemented
364C25—731no—
365C23×C4—21no—
366C23×C4—2924no—
367C23×C4—2924no—
368C23×C4—731no—
369C23×C4—731no—
370C23×C4—734no—
371C23×C4—731no—
372C23×C4—731no—
373C23×C4—734no—
374C23×C4—731no—
375C23×C4—1462no—
376C23×C4—1462no—
377C23×C4—1468no—
378C23×C4—1464no—
379C23×C4—1464no—
380C23×C4—1461no—
381C23×C4C22×D7311yescharacteristic
382C2×M16—738no—
383C2×M16—1464no—
384C2×M16—1464no—
385C22×C8—732no—
386C22×C8C22×D7312yes—
387#117—1464no—
388#117—1464no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—732no—
392C2×C4⋊C4—738no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—1464no—
396C2×C4⋊C4—1464no—
397C2×C4⋊C4—1468no—
398C2×C4⋊C4—1468no—
399C2×C4⋊C4—1464no—
400C2×C4⋊C4—1462no—
401C2×C4⋊C4—1462no—
402C2×C4⋊C4—1468no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1464no—
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7312yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1461no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4—1461no—
413C2×C22⋊C4—1461no—
414C2×C42—1461no—
415C2×C42—1461no—
416#107—1464no—
417#107—1464no—
418#106—24no—
419#106—1464no—
420C22⋊C8—734no—
421C22⋊C8—738no—
422C22⋊C8C4×D7314yescomplemented
423#95—21no—
424#95—1461no—
425#95—1461no—
426#95—1461no—
427C24—731no—
428C24—731no—
429C24—734no—
430C24—1462no—
431C24—1462no—
432C24—1462no—
433C24—1462no—
434C24—1462no—
435C24—1462no—
436C24D4×D7311yescharacteristic
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—2924no—
446C22×C4—29232no—
447C22×C4—2924no—
448C22×C4—2924no—
449C22×C4—2924no—
450C22×C4—2922no—
451C22×C4—2922no—
452C22×C4—731no—
453C22×C4—732no—
454C22×C4—734no—
455C22×C4—732no—
456C22×C4—734no—
457C22×C4—734no—
458C22×C4—731no—
459C22×C4—734no—
460C22×C4—734no—
461C22×C4—732no—
462C22×C4—732no—
463C22×C4—1468no—
464C22×C4—1468no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4—14616no—
468C22×C4—14616no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1464no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4C2×C4×D7312yes—
480C22×C4D4×D7311yescharacteristic
481C22×C4Q8×D7312yes—
482C22×C4C23×D7311yescharacteristic
483M16—14616no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—734no—
487C2×C8—1462no—
488C2×C8—1462no—
489C2×C8C2×C4×D7314yescomplemented
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—734no—
495C4⋊C4—1468no—
496C4⋊C4—1468no—
497C4⋊C4—1464no—
498C4⋊C4—1462no—
499C4⋊C4—1462no—
500C4⋊C4—1462no—
501C4⋊C4D4×D7314yes—
502C22⋊C4—42no—
503C22⋊C4—2922no—
504C22⋊C4—2922no—
505C22⋊C4—2922no—
506C22⋊C4—2922no—
507C22⋊C4—2922no—
508C42—2922no—
509C42—2922no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—732no—
515C23—732no—
516C23—732no—
517C23—732no—
518C23—732no—
519C23—732no—
520C23—1468no—
521C23—1468no—
522C23—1468no—
523C23—1468no—
524C23—1468no—
525C23—1468no—
526C23—1461no—
527C23—1461no—
528C23—1461no—
529C23—1468no—
530C23—1461no—
531C23—1461no—
532C23—1461no—
533C23—1461no—
534C23—1461no—
535C23C4⋊C4×D7312yes—
536C23#1455661812yes—
537C23#1455662012yes—
538C23C2×D4×D7311yescharacteristic
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—29216no—
554C2×C4—29216no—
555C2×C4—2928no—
556C2×C4—2922no—
557C2×C4—2922no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—732no—
565C2×C4—732no—
566C2×C4—732no—
567C2×C4—738no—
568C2×C4—738no—
569C2×C4—732no—
570C2×C4—1468no—
571C2×C4—1468no—
572C2×C4—1464no—
573C2×C4—1468no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1464no—
577C2×C4—1464no—
578C2×C4C4⋊C4×D7312yes—
579C2×C4C22×C4×D7311yescharacteristic
580C2×C4C2×D4×D7312yes—
581C2×C4C2×Q8×D7311yescharacteristic
582C2×C4#41×D7312yes—
583C8—24no—
584C8—1464no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—734no—
592C22—734no—
593C22—14632no—
594C22—1462no—
595C22—1462no—
596C22—1462no—
597C22—1462no—
598C22—1462no—
599C22—1462no—
600C22—1462no—
601C22—1462no—
602C22—1462no—
603C22—1462no—
604C22—1462no—
605C22—1462no—
606C22#1798862614yescomplemented
607C22C2×C4⋊C4×D7311yescharacteristic
608C22#120×D7311yescharacteristic
609C22#123×D7311yescharacteristic
610C22AGL(1,ℤ/8)×D7311yescharacteristic
611C22C2×#1455661811yescharacteristic
612C22#137×D7311yescharacteristic
613C22C2×#1455662011yescharacteristic
614C4—22no—
615C4—44no—
616C4—81no—
617C4—81no—
618C4—2924no—
619C4—5841no—
620C4—5841no—
621C4—734no—
622C4—1462no—
623C4C4×D4×D7312yes—
624C4C22⋊Q8×D7312yes—
625C2—24no—
626C2—1464no—
627C2—1464no—
628C2#385×D7311yescharacteristic
629C2#428×D7311yescharacteristic
630C2C2×#1798862611yescharacteristic
631C2#2261915911yescharacteristic
632C2#2261916211yescharacteristic
633C2#2261985311yescharacteristic
634C2#2261985511yescharacteristic
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#20844
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
2. C2×#17988626C2semidirect
5. #22610123C2semidirect
6. #22610122C2semidirect
7. #22610121C2semidirect
8. #22610120C2semidirect
9. #22610119C2semidirect
10. #22609649C2semidirect
11. C73×#1853C2semidirect
25. C2×#14556525C22semidirect
26. C2×#14556524C22semidirect
27. C2×#14556522C22semidirect
28. #17988549C4semidirect
38. #17988350C22semidirect
39. #17988349C22semidirect
40. #17988349C22semidirect
42. C2×#106×C73C22semidirect
45. #340×C73C22semidirect
64. C2×#11924011C2×C4semidirect
92. C22⋊#10596831C2×C4semidirect
93. #14556503C2×C4semidirect
107. C22⋊C8×C73C2×C4semidirect
155. C2×#10596831C22×C4semidirect
185. C2×C8×C73C22×C4semidirect
273. C22×C73#428semidirect
295. C2×D73#428semidirect
324. #1853D73semidirect
329. C73#20844semidirect
347. C2×#106C2×D73semidirect
363. #340C2×D73semidirect
422. C22⋊C8C4×D73semidirect
489. C2×C8C2×C4×D73semidirect
606. C22#17988626semidirect

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×#179886262
C2×#179885492
#226123132
#226101232
#226101222
#226101212
#226101202
#226101192
#226096492
C73×#18532
#2084473