Finite Group Databasebeta

← #33090468

Subgroups — #33090468

Order 18688.

1154 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D73C211yesmaximal, characteristic
2C4×M16×D73C211yesmaximal, characteristic
3#22612335C211yesmaximal, characteristic
4#22610876C211yesmaximal, characteristic, complemented
5#22610875C211yesmaximal, characteristic, complemented
6#22610874C211yesmaximal, characteristic, complemented
7#22610873C214yesmaximal, complemented
8#22609639C214yesmaximal
9C73×#1986C211yesmaximal, characteristic, complemented
10C2×M16×D73C2212yes—
11#17989144C2211yescharacteristic
12#17989143C412yescomplemented
13#17989142C411yescharacteristic, complemented
14#17989142C411yescharacteristic, complemented
15#17989141—21no—
16#17989141—21no—
17#17989140C2212yes—
18#17989139—21no—
19#17989139—21no—
20C4⋊Q8×D73C411yescharacteristic, complemented
21C4⋊Q8×D73C411yescharacteristic, complemented
22#125×D73C412yescomplemented
23C22⋊Q8×D73—21no—
24C22⋊Q8×D73—21no—
25C4×Q8×D73—21no—
26C4×Q8×D73—21no—
27C2×#14556548C2212yes—
28#117×D73C2212yes—
29C2×C42×D73C2211yescharacteristic
30C4×#11924012C2211yescharacteristic
31#17988539—21no—
32M16×Dic73C2212yes—
33C4×#11924011—21no—
34#97×D73—21no—
35C4×C8×D73—21no—
36#17988456—21no—
37#17988451—21no—
38#17988446—21no—
39#17988444—21no—
40C4×#11924007C2212yes—
41C4×C4⋊Dic73C2211yescharacteristic
42#17988348C2214yes—
43#17988347C2214yes—
44#17988346C2214yes—
45#533×C73C2211yescharacteristic
46C4×M16×C73C2211yescharacteristic
47C4×C4⋊C4×C73C2211yescharacteristic
48#339×C73C2214yes—
49C2×Q8×D73—21no—
50C2×Q8×D73—21no—
51C22×C4×D73D411yescharacteristic
52C22×C4×D73D411yescharacteristic
53C22×C4×D73C2311yescharacteristic
54C2×#11924012C2312yes—
55C2×#11924011—22no—
56M16×D73—24no—
57M16×D73Q818yescomplemented
58C2×C8×D73—22no—
59#14556591C2×C411yescharacteristic
60#14556591C2×C411yescharacteristic
61#14556590—21no—
62#14556590—21no—
63#14556590C2×C412yes—
64#14556590C2×C412yes—
65#14556589—21no—
66#14556589—21no—
67#14556583C2×C412yes—
68#14556582C2×C412yes—
69#14556582C2×C412yes—
70#14556581—22no—
71#14556581—22no—
72#14556571—21no—
73#14556571—21no—
74#14556570—21no—
75#14556570—21no—
76#14556569—22no—
77#14556569—22no—
78#14556568—22no—
79#14556568—22no—
80#14556567—21no—
81#14556567—21no—
82Q8×Dic73—21no—
83Q8×Dic73—21no—
84C4×Dic146—22no—
85C4×Dic146—22no—
86C4×Dic146—21no—
87C4×Dic146—21no—
88#14556556—21no—
89#14556556—21no—
90#14556556—22no—
91#14556556—22no—
92#14556550C2312yes—
93#14556549C2×C412yes—
94#14556549C2×C412yes—
95#14556548—22no—
96#14556548—28no—
97#14556547—22no—
98C2×C4⋊Dic73—21no—
99C2×#11924007—22no—
100C2×#11924007—22no—
101C4⋊C4×D73—21no—
102C4⋊C4×D73—21no—
103C4⋊C4×D73—22no—
104C4⋊C4×D73—22no—
105C4⋊C4×D73C2×C412yes—
106C4⋊C4×D73C2×C412yes—
107C2×#11924005—22no—
108C22⋊C4×D73—22no—
109C2×C4×Dic73D411yescharacteristic
110C2×C4×Dic73D411yescharacteristic
111C2×C4×Dic73C2312yes—
112C2×C4×Dic73C2311yescharacteristic
113C42×D73—22no—
114C42×D73C2×C411yescharacteristic
115C42×D73C2×C411yescharacteristic
116C42×D73C2×C411yescharacteristic
117C42×D73C2×C411yescharacteristic
118C42×D73D411yescharacteristic
119C42×D73D411yescharacteristic
120C42×D73D411yescharacteristic
121C42×D73D411yescharacteristic
122#14556502—21no—
123#14556501—22no—
124C4×#10596831—21no—
125C8×Dic73—22no—
126#14556498—22no—
127#14556498—21no—
128C2×M16×C73C2312yes—
129C4⋊Q8×C73C2×C411yescharacteristic
130C4⋊Q8×C73C2×C411yescharacteristic
131#125×C73C2×C412yes—
132C22⋊Q8×C73—21no—
133C22⋊Q8×C73—21no—
134C4×Q8×C73—21no—
135C4×Q8×C73—21no—
136#117×C73C2312yes—
137C2×C4⋊C4×C73—21no—
138C2×C42×C73C2312yes—
139C2×C42×C73C2311yescharacteristic
140#97×C73—21no—
141C4×C8×C73—21no—
142#95×C73—21no—
143C23×D73C2×D411yescharacteristic
144C2×Dic146—21no—
145C2×Dic146—21no—
146C2×Dic146—22no—
147C2×Dic146—22no—
148Q8×D73—44no—
149Q8×D73—44no—
150C22×Dic73—22no—
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154C2×C4×D73—22no—
155C2×C4×D73—22no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73C22⋊C411yescharacteristic
159C2×C4×D73C22⋊C411yescharacteristic
160C2×C4×D73C22⋊C411yescharacteristic
161C2×C4×D73C22⋊C411yescharacteristic
162C2×C4×D73C22⋊C411yescharacteristic
163C2×C4×D73C22⋊C411yescharacteristic
164C2×C4×D73C22⋊C411yescharacteristic
165C2×C4×D73C22⋊C411yescharacteristic
166C2×C4×D73C22×C411yescharacteristic
167C2×C4×D73C22×C411yescharacteristic
168C2×C4×D73C2×D411yescharacteristic
169C2×C4×D73C2×D411yescharacteristic
170C2×C4×D73C2×Q814yes—
171C2×C4×D73#4114yes—
172#11924012—22no—
173#11924012C2×Q814yescomplemented
174#11924011—28no—
175C2×#10596831—22no—
176C8×D73—28no—
177C4⋊Dic73—21no—
178C4⋊Dic73—21no—
179C4⋊Dic73—22no—
180C4⋊Dic73—22no—
181C4⋊Dic73—42no—
182C4⋊Dic73C22×C412yes—
183C4⋊Dic73C22×C412yes—
184#11924007—22no—
185#11924007—22no—
186#11924007—22no—
187#11924007—22no—
188#11924007—22no—
189#11924007—22no—
190#11924007—22no—
191#11924007—22no—
192#11924007—44no—
193#11924007—44no—
194C22⋊Dic73—22no—
195#11924005—22no—
196#11924005—22no—
197#11924005—44no—
198C4×Dic73—22no—
199C4×Dic73—22no—
200C4×Dic73—22no—
201C4×Dic73—24no—
202C4×Dic73C22⋊C411yescharacteristic
203C4×Dic73C22⋊C411yescharacteristic
204C4×Dic73C22⋊C411yescharacteristic
205C4×Dic73C22⋊C411yescharacteristic
206C4×Dic73C22⋊C411yescharacteristic
207C4×Dic73C22⋊C411yescharacteristic
208C4×Dic73C22⋊C411yescharacteristic
209C4×Dic73C22⋊C411yescharacteristic
210C4×Dic73C22×C411yescharacteristic
211C4×Dic73C22×C411yescharacteristic
212C4×Dic73C2×D411yescharacteristic
213C4×Dic73C2×D411yescharacteristic
214C2×Q8×C73—21no—
215C2×Q8×C73—21no—
216C22×C4×C73—22no—
217C22×C4×C73C2×D411yescharacteristic
218C22×C4×C73C2×D411yescharacteristic
219C22×C4×C73C2411yescharacteristic
220M16×C73—22no—
221M16×C73C2×Q814yescomplemented
222C2×C8×C73—22no—
223C4⋊C4×C73—22no—
224C4⋊C4×C73—22no—
225C4⋊C4×C73—21no—
226C4⋊C4×C73—21no—
227C4⋊C4×C73—42no—
228C4⋊C4×C73C22×C412yes—
229C4⋊C4×C73C22×C412yes—
230C22⋊C4×C73—22no—
231C42×C73—22no—
232C42×C73—24no—
233C42×C73C22×C411yescharacteristic
234C42×C73C22×C411yescharacteristic
235C42×C73C2×D411yescharacteristic
236C42×C73C2×D411yescharacteristic
237C23×C73C22×D411yescharacteristic
238C22×D73—41no—
239C22×D73—41no—
240C22×D73C2×C22⋊C411yescharacteristic
241C22×D73C2×C22⋊C411yescharacteristic
242C22×D73#12412yes—
243C22×D73C4⋊Q812yes—
244Q8×C73—42no—
245Q8×C73—42no—
246C2×C4×C73—22no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—41no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×C4×C73C2×C22⋊C411yescharacteristic
254C2×C4×C73C2×C22⋊C411yescharacteristic
255C2×C4×C73C2×C22⋊C411yescharacteristic
256C2×C4×C73C2×C22⋊C411yescharacteristic
257C2×C4×C73C23×C411yescharacteristic
258C2×C4×C73C22×D411yescharacteristic
259C2×C4×C73C22×Q812yes—
260C2×C4×C73C2×#4112yes—
261C2×Dic73—22no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—41no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73C2×C22⋊C411yescharacteristic
269C2×Dic73C2×C22⋊C411yescharacteristic
270C2×Dic73C2×C22⋊C411yescharacteristic
271C2×Dic73C2×C22⋊C411yescharacteristic
272C2×Dic73C2×C22⋊C411yescharacteristic
273C2×Dic73C2×C22⋊C411yescharacteristic
274C2×Dic73#12412yes—
275C2×Dic73C4⋊Q812yes—
276C4×D73—22no—
277C4×D73—22no—
278C4×D73—22no—
279C4×D73—22no—
280C4×D73—41no—
281C4×D73—41no—
282C4×D73—41no—
283C4×D73—41no—
284C4×D73—44no—
285C4×D73—44no—
286C4×D73C4×Q814yes—
287C4×D73C4×Q814yes—
288C4×D73C22⋊Q814yes—
289C4×D73C22⋊Q814yes—
290Dic146—44no—
291Dic146—42no—
292Dic146—42no—
293Dic146—44no—
294C8×C73—24no—
295#10596831—24no—
296C22×C73—21no—
297C22×C73—21no—
298C22×C73C22×C22⋊C411yescharacteristic
299C22×C73C2×#12411yescharacteristic
300C22×C73C2×C4⋊Q811yescharacteristic
301C2×D73—21no—
302C2×D73—21no—
303C2×D73—21no—
304C2×D73—21no—
305C2×D73—42no—
306C2×D73—42no—
307C2×D73#39112yes—
308C2×D73#39112yes—
309C4×C73—21no—
310C4×C73—21no—
311C4×C73—21no—
312C4×C73—42no—
313C4×C73—42no—
314C4×C73—82no—
315C4×C73#42112yes—
316C4×C73C2×C4×Q812yes—
317C4×C73C2×C22⋊Q812yes—
318Dic73—21no—
319Dic73—21no—
320Dic73—21no—
321Dic73—42no—
322Dic73—42no—
323Dic73—82no—
324Dic73#39112yes—
325Dic73#39112yes—
326Dic73#42112yes—
327#21306—731nomaximal, Sylow
328C2×C73—21no—
329C2×C73—21no—
330C2×C73C2×#39111yescharacteristic
331C2×C73C2×#42111yescharacteristic
332C2×C73C2×#42111yescharacteristic
333D73—22no—
334D73—22no—
335C2×#533—731no—
336C2×C4×M16—731no—
337#2285—731no—
338#1986—731no—
339#1986—731no—
340#1986—734no—
341#1986—731no—
342#1986D7311yescharacteristic, complemented
343#1733—734no—
344C73#2130611yescharacteristic, complemented, Sylow
345C22×M16—732no—
346#533—731no—
347#533—731no—
348#533—732no—
349#533—732no—
350#533—731no—
351#533—1461no—
352#533—1461no—
353#533—1461no—
354#533—1461no—
355#533C2×D7311yescharacteristic
356C2×C4⋊Q8—731no—
357C2×C4⋊Q8—731no—
358C2×#125—732no—
359C2×C22⋊Q8—1461no—
360C2×C22⋊Q8—1461no—
361C2×C4×Q8—1461no—
362C2×C4×Q8—1461no—
363C2×#117—732no—
364C2×#117—732no—
365C22×C42—731no—
366C4×M16—732no—
367C4×M16—731no—
368C4×M16—1461no—
369C4×M16—1461no—
370C4×M16C2×D7311yescharacteristic
371C2×#97—1461no—
372C2×C4×C8—1461no—
373#384—1461no—
374#382—1461no—
375#380—1461no—
376#380—1461no—
377C4×C4⋊C4—732no—
378C4×C4⋊C4—731no—
379C4×C4⋊C4C2×D7311yescharacteristic
380#339—734no—
381#339—734no—
382#339—734no—
383#339C2×D7314yes—
384C22×Q8—1461no—
385C22×Q8—1461no—
386C23×C4—731no—
387C23×C4—731no—
388C23×C4—731no—
389C2×M16—732no—
390C2×M16—738no—
391C2×M16—1462no—
392C2×M16—1464no—
393C2×M16C22×D7312yes—
394C22×C8—1462no—
395C4⋊Q8—732no—
396C4⋊Q8—732no—
397C4⋊Q8—731no—
398C4⋊Q8—731no—
399C4⋊Q8—1461no—
400C4⋊Q8—1461no—
401C4⋊Q8—1461no—
402C4⋊Q8—1461no—
403C4⋊Q8C4×D7311yescharacteristic
404C4⋊Q8C4×D7311yescharacteristic
405#125—732no—
406#125—732no—
407#125—732no—
408#125—1462no—
409#125—1462no—
410#125C4×D7312yes—
411C22⋊Q8—21no—
412C22⋊Q8—21no—
413C22⋊Q8—1462no—
414C22⋊Q8—1462no—
415C22⋊Q8—1462no—
416C22⋊Q8—1462no—
417C22⋊Q8—1461no—
418C22⋊Q8—1461no—
419C22⋊Q8—1461no—
420C22⋊Q8—1461no—
421C22⋊Q8—1461no—
422C22⋊Q8—1461no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—1462no—
426C4×Q8—1462no—
427C4×Q8—1462no—
428C4×Q8—1462no—
429C4×Q8—1461no—
430C4×Q8—1461no—
431C4×Q8—1461no—
432C4×Q8—1461no—
433C4×Q8—1461no—
434C4×Q8—1461no—
435#117—732no—
436#117—732no—
437#117—732no—
438#117—1468no—
439#117—1462no—
440#117—1462no—
441#117C22×D7312yes—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—732no—
444C2×C4⋊C4—732no—
445C2×C4⋊C4—1462no—
446C2×C4⋊C4—1462no—
447C2×C4⋊C4—1461no—
448C2×C4⋊C4—1461no—
449C2×C4⋊C4—1462no—
450C2×C4⋊C4—1462no—
451C2×C4⋊C4—1461no—
452C2×C22⋊C4—1462no—
453C2×C22⋊C4—1462no—
454C2×C42—731no—
455C2×C42—731no—
456C2×C42—731no—
457C2×C42—731no—
458C2×C42—731no—
459C2×C42—731no—
460C2×C42—731no—
461C2×C42—731no—
462C2×C42—731no—
463C2×C42—731no—
464C2×C42—731no—
465C2×C42—732no—
466C2×C42—1462no—
467C2×C42C22×D7312yes—
468C2×C42C22×D7311yescharacteristic
469#97—21no—
470#97—1462no—
471#97—1461no—
472C4×C8—21no—
473C4×C8—1462no—
474C4×C8—1461no—
475#95—21no—
476#95—1462no—
477#95—1461no—
478C24—731no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—2924no—
482C2×Q8—2924no—
483C2×Q8—1462no—
484C2×Q8—1462no—
485C2×Q8—1461no—
486C2×Q8—1461no—
487C22×C4—22no—
488C22×C4—2922no—
489C22×C4—2922no—
490C22×C4—731no—
491C22×C4—731no—
492C22×C4—731no—
493C22×C4—731no—
494C22×C4—731no—
495C22×C4—731no—
496C22×C4—731no—
497C22×C4—731no—
498C22×C4—731no—
499C22×C4—731no—
500C22×C4—734no—
501C22×C4—734no—
502C22×C4—731no—
503C22×C4—731no—
504C22×C4—731no—
505C22×C4—731no—
506C22×C4—731no—
507C22×C4—1462no—
508C22×C4—1462no—
509C22×C4—1462no—
510C22×C4D4×D7311yescharacteristic
511C22×C4D4×D7311yescharacteristic
512C22×C4C23×D7311yescharacteristic
513M16—22no—
514M16—734no—
515M16—1468no—
516M16—1462no—
517M16Q8×D7314yescomplemented
518C2×C8—22no—
519C2×C8—1468no—
520C2×C8—1462no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—2922no—
527C4⋊C4—2924no—
528C4⋊C4—2924no—
529C4⋊C4—732no—
530C4⋊C4—732no—
531C4⋊C4—1462no—
532C4⋊C4—1462no—
533C4⋊C4—1462no—
534C4⋊C4—1462no—
535C4⋊C4—1462no—
536C4⋊C4—1462no—
537C4⋊C4—1462no—
538C4⋊C4—1462no—
539C4⋊C4—1462no—
540C4⋊C4—1462no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C4⋊C4C2×C4×D7312yes—
544C4⋊C4C2×C4×D7312yes—
545C22⋊C4—22no—
546C22⋊C4—2924no—
547C22⋊C4—1462no—
548C22⋊C4—1462no—
549C22⋊C4—1462no—
550C42—22no—
551C42—24no—
552C42—731no—
553C42—731no—
554C42—731no—
555C42—731no—
556C42—731no—
557C42—731no—
558C42—731no—
559C42—731no—
560C42—731no—
561C42—731no—
562C42—731no—
563C42—731no—
564C42—1462no—
565C42—1462no—
566C42—1464no—
567C42—1462no—
568C42C2×C4×D7311yescharacteristic
569C42C2×C4×D7311yescharacteristic
570C42D4×D7311yescharacteristic
571C42D4×D7311yescharacteristic
572C23—2921no—
573C23—2921no—
574C23—732no—
575C23—732no—
576C23—731no—
577C23—731no—
578C23C2×D4×D7311yescharacteristic
579Q8—42no—
580Q8—42no—
581Q8—2922no—
582Q8—2924no—
583Q8—2924no—
584Q8—2922no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2924no—
597C2×C4—2924no—
598C2×C4—2922no—
599C2×C4—2922no—
600C2×C4—2922no—
601C2×C4—2921no—
602C2×C4—732no—
603C2×C4—731no—
604C2×C4—731no—
605C2×C4—731no—
606C2×C4—732no—
607C2×C4—734no—
608C2×C4—734no—
609C2×C4—734no—
610C2×C4—734no—
611C2×C4—731no—
612C2×C4—731no—
613C2×C4—731no—
614C2×C4—1462no—
615C2×C4—1462no—
616C2×C4—1462no—
617C2×C4—1462no—
618C2×C4—1462no—
619C2×C4—1462no—
620C2×C4—1462no—
621C2×C4C22⋊C4×D7311yescharacteristic
622C2×C4C22⋊C4×D7311yescharacteristic
623C2×C4C22⋊C4×D7311yescharacteristic
624C2×C4C22⋊C4×D7311yescharacteristic
625C2×C4C22×C4×D7311yescharacteristic
626C2×C4C2×D4×D7311yescharacteristic
627C2×C4C2×Q8×D7312yes—
628C2×C4#41×D7312yes—
629C8—24no—
630C8—1464no—
631C22—21no—
632C22—21no—
633C22—2922no—
634C22—2922no—
635C22—732no—
636C22—732no—
637C22—1461no—
638C22—1461no—
639C22—1461no—
640C22—1461no—
641C22C2×C22⋊C4×D7311yescharacteristic
642C22#124×D7311yescharacteristic
643C22C4⋊Q8×D7311yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—2922no—
651C4—2922no—
652C4—5842no—
653C4—732no—
654C4—732no—
655C4—732no—
656C4—1461no—
657C4—1461no—
658C4—1461no—
659C4#1798860212yes—
660C4C4×Q8×D7312yes—
661C4C22⋊Q8×D7312yes—
662C2—21no—
663C2—21no—
664C2—1462no—
665C2—1462no—
666C2#391×D7311yescharacteristic
667C2#421×D7311yescharacteristic
668C2C2×#1798860211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1986C2
socleC22×C73C22×C22⋊C4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1986D73
hypocenterC73#21306
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #22610876C2semidirect
5. #22610875C2semidirect
6. #22610874C2semidirect
7. #22610873C2semidirect
9. C73×#1986C2semidirect
12. #17989143C4semidirect
13. #17989142C4semidirect
14. #17989142C4semidirect
20. C4⋊Q8×D73C4semidirect
21. C4⋊Q8×D73C4semidirect
22. #125×D73C4semidirect
57. M16×D73Q8semidirect
173. #11924012C2×Q8semidirect
221. M16×C73C2×Q8semidirect
342. #1986D73semidirect
344. C73#21306semidirect
517. M16Q8×D73semidirect

Maximal subgroups

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

subgroupindex
#533×D732
C4×M16×D732
#226123352
#226108762
#226108752
#226108742
#226108732
#226096392
C73×#19862
#2130673