Finite Group Databasebeta

← #33085569

Subgroups — #33085569

Order 18688.

1590 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×M16×D73C211yesmaximal, characteristic
2C2×#17988630C211yesmaximal, characteristic, complemented
3C2×#17988627C211yesmaximal, characteristic, complemented
4#22612310C211yesmaximal, characteristic
5C2×#17988444C211yesmaximal, characteristic
6#22609963C212yesmaximal, complemented
7#22609963C212yesmaximal, complemented
8#22609959C212yesmaximal, complemented
9#22609955C212yesmaximal, complemented
10#22609651C211yesmaximal, characteristic, complemented
11C73×#1736C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×#11924012C2211yescharacteristic
14C22×#11924011C2212yes—
15C2×M16×D73—24no—
16C2×M16×D73—24no—
17C2×M16×D73C418yescomplemented
18C22×C8×D73C2212yes—
19C2×#14556550C2211yescharacteristic
20C22×C4⋊Dic73C2211yescharacteristic
21C22×#11924005—21no—
22#17988630—24no—
23#17988627—24no—
24C2×#14556524C2212yescomplemented
25C2×#14556524C2212yescomplemented
26C2×#14556522C2212yescomplemented
27C2×#14556522C2212yescomplemented
28#17988448—21no—
29#17988444—21no—
30#17988444—21no—
31#17988444C2211yescharacteristic
32#17988444C2212yes—
33#17988444C2211yescharacteristic
34#17988444C2212yes—
35#17988444C2211yescharacteristic
36#17988444C2211yescharacteristic
37#17988442—21no—
38C4×#11924005—21no—
39C4×#11924005—21no—
40C2×#14556498—21no—
41#17988350C2212yescomplemented
42#17988350C2212yescomplemented
43C22×M16×C73C2211yescharacteristic
44C2×#117×C73C2211yescharacteristic
45C22×C4⋊C4×C73C2211yescharacteristic
46#340×C73C2212yescomplemented
47#340×C73C2212yescomplemented
48C24×D73D411yescharacteristic
49C23×Dic73—21no—
50C23×Dic73D411yescharacteristic
51C22×C4×D73—22no—
52C22×C4×D73—22no—
53C22×C4×D73—22no—
54C22×C4×D73—22no—
55C22×C4×D73C2×C414yes—
56C22×C4×D73D411yescharacteristic
57C22×C4×D73D411yescharacteristic
58C22×C4×D73Q814yes—
59C22×C4×D73C2311yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C2×#11924012C2×C414yes—
62C2×#11924012D414yes—
63C2×#11924012D414yes—
64C2×#11924011—24no—
65C2×#11924011—24no—
66C2×#11924011C2×C418yescomplemented
67M16×D73—232no—
68C22×#10596831C2312yes—
69C2×C8×D73—24no—
70C2×C8×D73—24no—
71C2×C8×D73C2×C418yescomplemented
72#14556550—22no—
73#14556550—22no—
74C2×C4⋊Dic73—22no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—22no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2312yes—
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C22⋊Dic73—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—22no—
86C2×#11924005—22no—
87C2×#11924005—44no—
88C2×C4×Dic73—21no—
89#14556524—24no—
90#14556524—24no—
91#14556522—24no—
92#14556522—24no—
93#14556498—22no—
94#14556498—22no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99C23×C4×C73—21no—
100C23×C4×C73C2311yescharacteristic
101C2×M16×C73—22no—
102C2×M16×C73—22no—
103C2×M16×C73C2×C414yes—
104C22×C8×C73C2312yes—
105#117×C73—22no—
106#117×C73—22no—
107C2×C4⋊C4×C73—22no—
108C2×C4⋊C4×C73—22no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C4⋊C4×C73C2312yes—
112C2×C4⋊C4×C73C2311yescharacteristic
113C2×C22⋊C4×C73—21no—
114C2×C42×C73—21no—
115C23×D73—22no—
116C23×D73—22no—
117C23×D73—22no—
118C23×D73—22no—
119C23×D73—41no—
120C23×D73—41no—
121C23×D73C4⋊C414yes—
122C23×D73C2×D411yescharacteristic
123C23×D73C2×D411yescharacteristic
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—21no—
127C22×Dic73—21no—
128C22×Dic73—22no—
129C22×Dic73—42no—
130C22×Dic73—42no—
131C22×Dic73—42no—
132C22×Dic73C4⋊C414yes—
133C22×Dic73C2×D411yescharacteristic
134C22×Dic73C2×D411yescharacteristic
135C2×C4×D73—28no—
136C2×C4×D73—28no—
137C2×C4×D73—22no—
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—22no—
141C2×C4×D73—28no—
142C2×C4×D73—28no—
143C2×C4×D73—21no—
144C2×C4×D73—21no—
145C2×C4×D73—21no—
146C2×C4×D73—21no—
147C2×C4×D73—21no—
148C2×C4×D73—21no—
149C2×C4×D73—21no—
150C2×C4×D73—21no—
151C2×C4×D73C4⋊C414yes—
152C2×C4×D73C4⋊C414yes—
153C2×C4×D73C22×C412yes—
154C2×C4×D73C22×C412yes—
155C2×C4×D73C2×Q812yes—
156C2×C4×D73C2×Q812yes—
157#11924012C22⋊C4116yescomplemented
158#11924011—216no—
159C2×#10596831C22×C414yes—
160C2×#10596831C2×D414yes—
161C2×#10596831C2×D414yes—
162C8×D73—216no—
163C4⋊Dic73—21no—
164C4⋊Dic73—22no—
165C4⋊Dic73—21no—
166C4⋊Dic73—22no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—42no—
170C22⋊Dic73—42no—
171#11924005—42no—
172#11924005—42no—
173#11924005—44no—
174#11924005—42no—
175#11924005—44no—
176#11924005—42no—
177C4×Dic73—42no—
178C24×C73C2×D411yescharacteristic
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—22no—
182C22×C4×C73—42no—
183C22×C4×C73—42no—
184C22×C4×C73—42no—
185C22×C4×C73C22×C412yes—
186C22×C4×C73C2×D412yes—
187C22×C4×C73C2×D412yes—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×Q812yes—
190C22×C4×C73#4112yes—
191C22×C4×C73#4112yes—
192C22×C4×C73C2411yescharacteristic
193M16×C73—28no—
194C2×C8×C73—22no—
195C2×C8×C73—22no—
196C2×C8×C73C22×C414yes—
197C4⋊C4×C73—22no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—22no—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—42no—
204C22⋊C4×C73—42no—
205C42×C73—42no—
206C23×C73—21no—
207C23×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73C2×C4⋊C412yes—
211C23×C73#12112yes—
212C23×C73#12112yes—
213C23×C73C22×D411yescharacteristic
214C22×D73—21no—
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—21no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73—21no—
221C22×D73—21no—
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—28no—
225C22×D73—28no—
226C22×D73—44no—
227C22×D73—44no—
228C22×D73—44no—
229C22×D73—44no—
230C22×D73—44no—
231C22×D73C2×C4⋊C412yes—
232C22×D73C2×C4⋊C412yes—
233C22×D73AGL(1,ℤ/8)12yes—
234C22×D73#13712yes—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—42no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—84no—
245C2×C4×C73C2×C22⋊C414yes—
246C2×C4×C73C2×C4⋊C412yes—
247C2×C4×C73#11714yes—
248C2×C4×C73#12112yes—
249C2×C4×C73#12112yes—
250C2×C4×C73C22⋊Q814yes—
251C2×C4×C73C22⋊Q814yes—
252C2×C4×C73C23×C411yescharacteristic
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C22×Q811yescharacteristic
256C2×C4×C73C2×#4111yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—28no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—42no—
272C2×Dic73—84no—
273C2×Dic73C2×C4⋊C412yes—
274C2×Dic73C2×C4⋊C412yes—
275C4×D73—22no—
276C4×D73—22no—
277C4×D73—22no—
278C4×D73—22no—
279C4×D73—22no—
280C4×D73—22no—
281C4×D73—22no—
282C4×D73—22no—
283C4×D73—28no—
284C4×D73—28no—
285C8×C73—24no—
286#10596831C2×C22⋊C418yescomplemented
287C22×C73—22no—
288C22×C73—22no—
289C22×C73—22no—
290C22×C73—22no—
291C22×C73—22no—
292C22×C73—22no—
293C22×C73#38014yes—
294C22×C73C22×C4⋊C411yescharacteristic
295C22×C73C2×#12111yescharacteristic
296C22×C73C2×#12111yescharacteristic
297C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C73C2×#13711yescharacteristic
299C22×C73#57511yescharacteristic
300C22×C73#57511yescharacteristic
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—22no—
304C2×D73—22no—
305C2×D73—22no—
306C2×D73—22no—
307C2×D73—22no—
308C2×D73—22no—
309C2×D73—22no—
310C2×D73—416no—
311C2×D73—41no—
312C2×D73—41no—
313C2×D73—41no—
314C2×D73—41no—
315C2×D73#42814yescomplemented
316C4×C73—81no—
317C4×C73—82no—
318C4×C73—81no—
319C4×C73#38014yes—
320C4×C73C22×C22⋊C411yescharacteristic
321C4×C73C2×#11711yescharacteristic
322C4×C73C2×C22⋊Q811yescharacteristic
323C4×C73C2×C22⋊Q811yescharacteristic
324Dic73—22no—
325Dic73—22no—
326Dic73—81no—
327Dic73—82no—
328Dic73—81no—
329#20661—731nomaximal, Sylow
330C2×C73—24no—
331C2×C73C2×#38011yescharacteristic
332C2×C73C2×#42811yescharacteristic
333C2×C73#291111yescharacteristic
334C2×C73#305511yescharacteristic
335C2×C73#305511yescharacteristic
336C2×C73#305611yescharacteristic
337C2×C73#305611yescharacteristic
338D73—24no—
339D73—42no—
340C23×M16—731no—
341C2×#428—731no—
342C2×#428—731no—
343#2280—731no—
344C2×#380—731no—
345#1809—732no—
346#1809—732no—
347#1808—732no—
348#1807—732no—
349#1736—731no—
350#1736D7311yescharacteristic, complemented
351C73#2066111yescharacteristic, complemented, Sylow
352C24×C4—731no—
353C22×M16—731no—
354C22×M16—738no—
355C22×M16—732no—
356C22×M16—1464no—
357C22×M16—1464no—
358C22×M16C2×D7311yescharacteristic
359C23×C8—732no—
360C2×#117—731no—
361C2×#117C2×D7311yescharacteristic
362C22×C4⋊C4—731no—
363C22×C4⋊C4C2×D7311yescharacteristic
364C22×C22⋊C4—1461no—
365#428—1464no—
366#428—1464no—
367C2×#107—732no—
368C2×#107—732no—
369C2×#106—732no—
370C2×#106—732no—
371#381—1461no—
372#380—731no—
373#380—732no—
374#380—731no—
375#380—732no—
376#380—731no—
377#380—731no—
378#380—1461no—
379#380—1461no—
380#379—1461no—
381C4×C22⋊C4—1461no—
382C4×C22⋊C4—1461no—
383C2×#95—1461no—
384#340—732no—
385#340—732no—
386#340C2×D7312yescomplemented
387#340C2×D7312yescomplemented
388C25—731no—
389C23×C4—21no—
390C23×C4—731no—
391C23×C4—734no—
392C23×C4—731no—
393C23×C4—731no—
394C23×C4—734no—
395C23×C4—731no—
396C23×C4—731no—
397C23×C4—1461no—
398C23×C4—1462no—
399C23×C4—1462no—
400C23×C4—1462no—
401C23×C4—1462no—
402C23×C4C22×D7311yescharacteristic
403C2×M16—22no—
404C2×M16—22no—
405C2×M16—734no—
406C2×M16—734no—
407C2×M16—734no—
408C2×M16—738no—
409C2×M16—1464no—
410C2×M16—1464no—
411C2×M16—14632no—
412C2×M16C4×D7314yes—
413C22×C8—732no—
414C22×C8—738no—
415C22×C8—1464no—
416C22×C8—1464no—
417C22×C8C22×D7312yes—
418#117—22no—
419#117—22no—
420#117—1462no—
421#117—1462no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—732no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1462no—
430C2×C4⋊C4—1462no—
431C2×C4⋊C4C22×D7312yes—
432C2×C4⋊C4C22×D7311yescharacteristic
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—2924no—
436C2×C22⋊C4—1462no—
437C2×C22⋊C4—1462no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C22⋊C4—1461no—
442C2×C22⋊C4—1461no—
443C2×C42—21no—
444C2×C42—1461no—
445#107—1464no—
446#107—1464no—
447#106—1464no—
448#106—1464no—
449#95—1462no—
450#95—1462no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455C24—2921no—
456C24—2921no—
457C24—734no—
458C24—731no—
459C24—731no—
460C24—1462no—
461C24—1462no—
462C24—1462no—
463C24—1462no—
464C24D4×D7311yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—2922no—
472C22×C4—2922no—
473C22×C4—2922no—
474C22×C4—734no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—734no—
478C22×C4—734no—
479C22×C4—732no—
480C22×C4—732no—
481C22×C4—732no—
482C22×C4—732no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1468no—
489C22×C4—1468no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1468no—
495C22×C4—1468no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4C2×C4×D7312yes—
505C22×C4D4×D7312yes—
506C22×C4D4×D7312yes—
507C22×C4D4×D7311yescharacteristic
508C22×C4Q8×D7312yes—
509C22×C4#1192402812yes—
510C22×C4#1192402812yes—
511C22×C4C23×D7311yescharacteristic
512M16—28no—
513M16—7316no—
514M16—14616no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—734no—
518C2×C8—734no—
519C2×C8—734no—
520C2×C8—14616no—
521C2×C8C2×C4×D7314yes—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—2922no—
530C4⋊C4—1461no—
531C4⋊C4—1462no—
532C4⋊C4—1461no—
533C4⋊C4—1462no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C22⋊C4—42no—
537C22⋊C4—2924no—
538C22⋊C4—2922no—
539C22⋊C4—2922no—
540C22⋊C4—2924no—
541C22⋊C4—2922no—
542C22⋊C4—2922no—
543C22⋊C4—2922no—
544C42—42no—
545C42—2922no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—2924no—
551C23—2924no—
552C23—2924no—
553C23—2924no—
554C23—2924no—
555C23—732no—
556C23—732no—
557C23—732no—
558C23—732no—
559C23—1468no—
560C23—1461no—
561C23—1461no—
562C23—1461no—
563C23—1468no—
564C23—1461no—
565C23—1461no—
566C23—1461no—
567C23—1461no—
568C23—1461no—
569C23—1461no—
570C23—1461no—
571C23C4⋊C4×D7312yes—
572C23#1455656412yes—
573C23#1455656412yes—
574C23C2×D4×D7311yescharacteristic
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2922no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—2921no—
594C2×C4—5844no—
595C2×C4—732no—
596C2×C4—732no—
597C2×C4—1468no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1468no—
603C2×C4—1468no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1462no—
612C2×C4C22⋊C4×D7314yes—
613C2×C4C4⋊C4×D7312yes—
614C2×C4#1455654814yes—
615C2×C4#1455656412yes—
616C2×C4#1455656412yes—
617C2×C4#1455656914yes—
618C2×C4#1455656914yes—
619C2×C4C22×C4×D7311yescharacteristic
620C2×C4C2×D4×D7311yescharacteristic
621C2×C4C2×D4×D7311yescharacteristic
622C2×C4C2×Q8×D7311yescharacteristic
623C2×C4C2×#1192402811yescharacteristic
624C2×C4C2×#1192402811yescharacteristic
625C8—24no—
626C8—738no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—29216no—
634C22—2921no—
635C22—2921no—
636C22—2921no—
637C22—2921no—
638C22—734no—
639C22—1462no—
640C22—1462no—
641C22—1462no—
642C22—1462no—
643C22—1462no—
644C22—1462no—
645C22—1462no—
646C22—1462no—
647C22—1462no—
648C22#1798844614yes—
649C22C2×C4⋊C4×D7311yescharacteristic
650C22C2×#1455656411yescharacteristic
651C22C2×#1455656411yescharacteristic
652C22AGL(1,ℤ/8)×D7311yescharacteristic
653C22#137×D7311yescharacteristic
654C22#1798938311yescharacteristic
655C22#1798938611yescharacteristic
656C4—82no—
657C4—81no—
658C4—81no—
659C4—5841no—
660C4—5842no—
661C4—5841no—
662C4—1462no—
663C4—1462no—
664C4#1798844614yes—
665C4C2×C22⋊C4×D7311yescharacteristic
666C4C2×#1455654811yescharacteristic
667C4C2×#1455656911yescharacteristic
668C4C2×#1455656911yescharacteristic
669C2—24no—
670C2—2922no—
671C2—1464no—
672C2C2×#1798844611yescharacteristic
673C2#428×D7311yescharacteristic
674C2#2261851811yescharacteristic
675C2#2261954811yescharacteristic
676C2#2261955311yescharacteristic
677C2#2261955911yescharacteristic
678C2#2261956411yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1736C2
socleC23×C73C22×D4
prime radicalC24×D73D4
cosocleC22×C4×C73C24
hypercenter#1736D73
hypocenterC73#20661
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C2×#17988630C2semidirect
3. C2×#17988627C2semidirect
6. #22609963C2semidirect
7. #22609963C2semidirect
8. #22609959C2semidirect
9. #22609955C2semidirect
10. #22609651C2semidirect
11. C73×#1736C2semidirect
17. C2×M16×D73C4semidirect
24. C2×#14556524C22semidirect
25. C2×#14556524C22semidirect
26. C2×#14556522C22semidirect
27. C2×#14556522C22semidirect
41. #17988350C22semidirect
42. #17988350C22semidirect
46. #340×C73C22semidirect
47. #340×C73C22semidirect
66. C2×#11924011C2×C4semidirect
71. C2×C8×D73C2×C4semidirect
157. #11924012C22⋊C4semidirect
286. #10596831C2×C22⋊C4semidirect
315. C2×D73#428semidirect
350. #1736D73semidirect
351. C73#20661semidirect
386. #340C2×D73semidirect
387. #340C2×D73semidirect

Maximal subgroups

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

subgroupindex
C22×M16×D732
C2×#179886302
C2×#179886272
#226123102
C2×#179884442
#226099632
#226099592
#226099552
#226096512
C73×#17362
#2066173