Finite Group Databasebeta

← #32991378

Subgroups — #32991378

Order 18688.

1210 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989353C211yesmaximal, characteristic
2C2×#17988569C211yesmaximal, characteristic
3#22612524C211yesmaximal, characteristic
4C2×#17988359C211yesmaximal, characteristic, complemented
5C2×#17988358C211yesmaximal, characteristic, complemented
6C2×#17988356C211yesmaximal, characteristic, complemented
7C2×#343×C73C211yesmaximal, characteristic, complemented
8C22×#11924028C2211yescharacteristic
9#17989353—22no—
10#17989353—22no—
11#17989353—22no—
12#17989353C414yescomplemented
13C2×#14556599—22no—
14C2×#14556597—22no—
15C22×#11924012C2211yescharacteristic
16C22×#11924012C2211yescharacteristic
17C2×M16×D73—22no—
18C2×#14556550C2211yescharacteristic
19#17988569—22no—
20#17988569C414yescomplemented
21C2×#14556509—21no—
22C2×#14556507—21no—
23#17988492—21no—
24#17988488—21no—
25#17988466C414yes—
26#17988444—22no—
27#17988359—22no—
28#17988358—22no—
29#17988356—22no—
30C22×M16×C73C2211yescharacteristic
31C22×M16×C73C2211yescharacteristic
32C22×C4⋊C4×C73C2211yescharacteristic
33#343×C73—22no—
34C2×#11924028—21no—
35C2×#11924028—21no—
36C2×#11924028—21no—
37C2×#11924028—21no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028—21no—
41C2×#11924028—21no—
42C2×#11924028C2×C412yes—
43C2×#11924028C2×C412yes—
44C2×#11924028D412yes—
45C2×#11924028Q812yes—
46C22×Dic146D411yescharacteristic
47C22×D292D411yescharacteristic
48C22×C22⋊D73C2×C412yes—
49C22×C4×D73D412yes—
50#14556599—28no—
51#14556599—28no—
52#14556597—28no—
53#14556597—28no—
54C2×#11924012—21no—
55C2×#11924012—21no—
56C2×#11924012—21no—
57C2×#11924012—21no—
58C2×#11924012—21no—
59C2×#11924012—21no—
60C2×#11924012—21no—
61C2×#11924012—21no—
62C2×#11924012C2×C412yes—
63C2×#11924012C2×C412yes—
64C2×#11924012D412yes—
65C2×#11924012D412yes—
66C2×#11924011—24no—
67M16×D73—28no—
68M16×D73—28no—
69C22×#10596831—21no—
70C22×#10596831—21no—
71C2×C8×D73—24no—
72#14556550—21no—
73#14556550—21no—
74#14556550—21no—
75#14556550—21no—
76C2×C4⋊Dic73—21no—
77C2×C22⋊Dic73C2×C412yes—
78C2×#11924005—24no—
79C2×C4×Dic73—21no—
80#14556509—24no—
81#14556507—24no—
82#14556498—24no—
83C23×C4×C73C2×C412yes—
84C23×C4×C73C2311yescharacteristic
85C2×M16×C73—21no—
86C2×M16×C73—21no—
87C2×M16×C73—21no—
88C2×M16×C73—21no—
89C2×M16×C73—21no—
90C2×M16×C73—21no—
91C2×M16×C73—21no—
92C2×M16×C73—21no—
93C2×M16×C73C2×C412yes—
94C2×M16×C73C2×C412yes—
95C2×M16×C73D412yes—
96C2×M16×C73D412yes—
97C22×C8×C73—21no—
98C22×C8×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C23×D73C22⋊C412yes—
106#11924028—22no—
107#11924028—22no—
108#11924028—22no—
109#11924028—22no—
110#11924028—22no—
111#11924028—22no—
112#11924028—22no—
113#11924028—22no—
114#11924028—22no—
115#11924028—22no—
116#11924028—22no—
117#11924028—22no—
118#11924028—22no—
119#11924028—22no—
120#11924028—22no—
121#11924028—22no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×Dic146—21no—
125C2×Dic146—21no—
126C2×Dic146C22⋊C412yes—
127C2×Dic146C4⋊C412yes—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292—21no—
131C2×D292—21no—
132C2×D292C22⋊C412yes—
133C2×D292C4⋊C412yes—
134C2×C22⋊D73—22no—
135C2×C22⋊D73—22no—
136C2×C22⋊D73—22no—
137C2×C22⋊D73—22no—
138C2×C22⋊D73C4214yes—
139C2×C22⋊D73C4⋊C414yes—
140C22×Dic73—22no—
141C22×Dic73C22⋊C412yes—
142C2×C4×D73—22no—
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—22no—
146C2×C4×D73C22⋊C414yes—
147C2×C4×D73C4⋊C414yes—
148#11924012—22no—
149#11924012—22no—
150#11924012—22no—
151#11924012—22no—
152#11924012—22no—
153#11924012—41no—
154#11924012—41no—
155#11924012C22⋊C414yescomplemented
156#11924011—216no—
157C2×#10596831—22no—
158C2×#10596831—22no—
159C2×#10596831—22no—
160C2×#10596831—22no—
161C2×#10596831—41no—
162C2×#10596831—41no—
163C8×D73—216no—
164C4⋊Dic73—41no—
165C4⋊Dic73—41no—
166C22⋊Dic73—22no—
167C22⋊Dic73—22no—
168#11924005—84no—
169C4×Dic73—41no—
170C4×Dic73—41no—
171C24×C73C22×C411yescharacteristic
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—82no—
176C22×C4×C73C22×C411yescharacteristic
177C22×C4×C73C22×C411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×Q811yescharacteristic
186C22×C4×C73#4111yescharacteristic
187C22×C4×C73#4111yescharacteristic
188C22×C4×C73#4111yescharacteristic
189C22×C4×C73#4111yescharacteristic
190M16×C73—22no—
191M16×C73—22no—
192M16×C73—22no—
193M16×C73—22no—
194M16×C73—22no—
195M16×C73—41no—
196M16×C73—41no—
197M16×C73C22⋊C414yescomplemented
198C2×C8×C73—22no—
199C2×C8×C73—22no—
200C2×C8×C73—22no—
201C2×C8×C73—22no—
202C2×C8×C73—41no—
203C2×C8×C73—41no—
204C4⋊C4×C73—41no—
205C4⋊C4×C73—41no—
206C4⋊C4×C73—41no—
207C4⋊C4×C73—41no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73C2×C4211yescharacteristic
211C23×C73C2×C22⋊C411yescharacteristic
212C23×C73C2×C4⋊C411yescharacteristic
213C23×C73C4×D411yescharacteristic
214C23×C73C4×D411yescharacteristic
215C23×C73C4×D411yescharacteristic
216C23×C73C4×D411yescharacteristic
217C22×D73—22no—
218C22×D73—22no—
219C22×D73—42no—
220C22×D73—42no—
221C22×D73#9514yes—
222C2×C4×C73—44no—
223C2×C4×C73—42no—
224C2×C4×C73—82no—
225C2×C4×C73—82no—
226C2×C4×C73C2×C22⋊C411yescharacteristic
227C2×C4×C73C2×C22⋊C411yescharacteristic
228C2×C4×C73C2×C22⋊C411yescharacteristic
229C2×C4×C73C2×C22⋊C411yescharacteristic
230C2×C4×C73C2×C4⋊C411yescharacteristic
231C2×C4×C73C2×C4⋊C411yescharacteristic
232C2×C4×C73#11711yescharacteristic
233C2×C4×C73#11711yescharacteristic
234C2×C4×C73C4×D411yescharacteristic
235C2×C4×C73C4×D411yescharacteristic
236C2×C4×C73C4×D411yescharacteristic
237C2×C4×C73C4×D411yescharacteristic
238C2×C4×C73#12011yescharacteristic
239C2×C4×C73#12011yescharacteristic
240C2×C4×C73#12011yescharacteristic
241C2×C4×C73#12011yescharacteristic
242C2×C4×C73#12111yescharacteristic
243C2×C4×C73#12111yescharacteristic
244C2×C4×C73#12111yescharacteristic
245C2×C4×C73#12111yescharacteristic
246C2×C4×C73C22⋊Q811yescharacteristic
247C2×C4×C73C22⋊Q811yescharacteristic
248C2×C4×C73C22⋊Q811yescharacteristic
249C2×C4×C73C22⋊Q811yescharacteristic
250C2×C4×C73#12311yescharacteristic
251C2×C4×C73#12311yescharacteristic
252C2×C4×C73#12311yescharacteristic
253C2×C4×C73#12311yescharacteristic
254C22⋊D73—24no—
255C22⋊D73—24no—
256C22⋊D73—24no—
257C22⋊D73—24no—
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73#9514yes—
264D292—22no—
265D292—22no—
266D292—22no—
267D292—22no—
268C4×D73—24no—
269C4×D73—24no—
270C4×D73—24no—
271C4×D73—24no—
272Dic146—22no—
273Dic146—22no—
274Dic146—22no—
275Dic146—22no—
276C8×C73—24no—
277C8×C73—42no—
278#10596831—24no—
279#10596831—42no—
280C22×C73—21no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—21no—
284C22×C73—21no—
285C22×C73—21no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73C2×#9511yescharacteristic
291C22×C73C4×C22⋊C411yescharacteristic
292C22×C73C4×C22⋊C411yescharacteristic
293C22×C73#38511yescharacteristic
294C22×C73#38511yescharacteristic
295C22×C73#38611yescharacteristic
296C22×C73#38611yescharacteristic
297C2×D73—24no—
298C2×D73—42no—
299C2×D73—42no—
300C2×D73—44no—
301C2×D73—82no—
302C4×C73—82no—
303C4×C73#37911yescharacteristic
304C4×C73#38011yescharacteristic
305C4×C73#38011yescharacteristic
306C4×C73#38111yescharacteristic
307C4×C73#38511yescharacteristic
308C4×C73#38511yescharacteristic
309C4×C73#38611yescharacteristic
310C4×C73#38611yescharacteristic
311Dic73—24no—
312Dic73—82no—
313#10755—731nomaximal, Sylow
314C2×C73—21no—
315C2×C73—21no—
316C2×C73—21no—
317C2×C73—21no—
318C2×C73—21no—
319C2×C73—21no—
320C2×C73C22⋊#9511yescharacteristic
321C2×C73#191411yescharacteristic
322C2×C73#191511yescharacteristic
323D73—44no—
324C2×#568—731no—
325C2×#413—731no—
326#2311—731no—
327C2×#344—731no—
328C2×#344—731no—
329C2×#343—731no—
330C2×#343D7311yescharacteristic, complemented
331C73#1075511yescharacteristic, complemented, Sylow
332C22×#41—731no—
333#568—734no—
334#568—1462no—
335#568—1462no—
336#568—1462no—
337C2×#131—1462no—
338C2×#131—1462no—
339C22×M16—731no—
340C22×M16—731no—
341C22×M16—1462no—
342C22×M16C2×D7311yescharacteristic
343C22×M16C2×D7311yescharacteristic
344C2×#117—731no—
345C22×C4⋊C4C2×D7311yescharacteristic
346#413—734no—
347#413—1462no—
348C2×#101—1461no—
349C2×#100—1461no—
350#391—1461no—
351#390—1461no—
352#386—734no—
353#380—1462no—
354#344—1462no—
355#344—1462no—
356#343—22no—
357#343—1462no—
358C2×#41—732no—
359C2×#41—732no—
360C2×#41—732no—
361C2×#41—732no—
362C2×#41—1461no—
363C2×#41—1461no—
364C2×#41—1461no—
365C2×#41—1461no—
366C2×#41—1461no—
367C2×#41—1461no—
368C2×#41—1461no—
369C2×#41—1461no—
370C22×Q8—731no—
371C22×D4—732no—
372C22×D4—731no—
373C23×C4—732no—
374C23×C4C4×D7312yes—
375C23×C4C22×D7311yescharacteristic
376#131—1468no—
377#131—1468no—
378#131—1468no—
379#131—1468no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—732no—
389C2×M16—732no—
390C2×M16—732no—
391C2×M16—732no—
392C2×M16—1464no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16—1461no—
396C2×M16—1461no—
397C2×M16—1468no—
398C2×M16—1468no—
399C2×M16—1461no—
400C2×M16—1461no—
401C2×M16—1461no—
402C2×M16—1461no—
403C2×M16C4×D7312yes—
404C2×M16C4×D7312yes—
405C2×M16D29212yes—
406C2×M16C22⋊D7312yes—
407C22×C8—21no—
408C22×C8—21no—
409C22×C8—1464no—
410C22×C8—1461no—
411C22×C8—1461no—
412#117—1461no—
413#117—1461no—
414#117—1461no—
415#117—1461no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—1461no—
423C2×C22⋊C4—732no—
424C2×C22⋊C4—1464no—
425C2×C42—1461no—
426#101—1464no—
427#100—1464no—
428#95—1464no—
429C24—732no—
430C24C2×C4×D7311yescharacteristic
431#41—1462no—
432#41—1462no—
433#41—1462no—
434#41—1462no—
435#41—1462no—
436#41—1462no—
437#41—1462no—
438#41—1462no—
439#41—1462no—
440#41—1462no—
441#41—1462no—
442#41—1462no—
443#41—1462no—
444#41—1462no—
445#41—1462no—
446#41—1462no—
447C2×Q8—732no—
448C2×Q8—732no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×Q8—1461no—
452C2×Q8—1461no—
453C2×D4—734no—
454C2×D4—734no—
455C2×D4—732no—
456C2×D4—732no—
457C2×D4—1462no—
458C2×D4—1462no—
459C2×D4—1462no—
460C2×D4—1462no—
461C2×D4—1461no—
462C2×D4—1461no—
463C2×D4—1461no—
464C2×D4—1461no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—82no—
469C22×C4—732no—
470C22×C4—734no—
471C22×C4—734no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4C2×C4×D7311yescharacteristic
478C22×C4C2×C4×D7311yescharacteristic
479C22×C4D4×D7311yescharacteristic
480C22×C4D4×D7311yescharacteristic
481C22×C4D4×D7311yescharacteristic
482C22×C4D4×D7311yescharacteristic
483C22×C4D4×D7311yescharacteristic
484C22×C4C2×C22⋊D7311yescharacteristic
485C22×C4C2×D29211yescharacteristic
486C22×C4Q8×D7311yescharacteristic
487C22×C4#1192402711yescharacteristic
488C22×C4#1192402811yescharacteristic
489C22×C4#1192402811yescharacteristic
490C22×C4#1192402911yescharacteristic
491M16—22no—
492M16—22no—
493M16—22no—
494M16—22no—
495M16—22no—
496M16—41no—
497M16—41no—
498M16—2921no—
499M16—2921no—
500M16—734no—
501M16—14616no—
502M16—1462no—
503M16—1462no—
504M16—1462no—
505M16—1462no—
506M16—1462no—
507M16#1192400514yescomplemented
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—2921no—
515C2×C8—2921no—
516C2×C8—14616no—
517C2×C8—1462no—
518C2×C8—1462no—
519C2×C8—1462no—
520C2×C8—1462no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—2921no—
526C4⋊C4—2921no—
527C22⋊C4—5844no—
528C22⋊C4—1462no—
529C22⋊C4—1462no—
530C42—2921no—
531C42—2921no—
532C23—41no—
533C23—41no—
534C23—2922no—
535C23—2922no—
536C23—734no—
537C23—1462no—
538C23—1462no—
539C23C42×D7311yescharacteristic
540C23C22⋊C4×D7311yescharacteristic
541C23C4⋊C4×D7311yescharacteristic
542C23#1455655111yescharacteristic
543C23C4×C22⋊D7311yescharacteristic
544C23#1455655311yescharacteristic
545C23C4×D29211yescharacteristic
546Q8—1462no—
547Q8—1462no—
548Q8—1462no—
549Q8—1462no—
550D4—1464no—
551D4—1464no—
552D4—1464no—
553D4—1464no—
554D4—1462no—
555D4—1462no—
556D4—1462no—
557D4—1462no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—2922no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—734no—
566C2×C4—1462no—
567C2×C4—1462no—
568C2×C4—1464no—
569C2×C4—1464no—
570C2×C4—1464no—
571C2×C4—1464no—
572C2×C4C22⋊C4×D7311yescharacteristic
573C2×C4C22⋊C4×D7311yescharacteristic
574C2×C4C22⋊C4×D7311yescharacteristic
575C2×C4C2×#1192400511yescharacteristic
576C2×C4C4⋊C4×D7311yescharacteristic
577C2×C4C4⋊C4×D7311yescharacteristic
578C2×C4#1455654811yescharacteristic
579C2×C4#1455654911yescharacteristic
580C2×C4#1455655111yescharacteristic
581C2×C4C4×C22⋊D7311yescharacteristic
582C2×C4#1455655311yescharacteristic
583C2×C4C4×D29211yescharacteristic
584C2×C4#1455655911yescharacteristic
585C2×C4#1455655911yescharacteristic
586C2×C4#1455656111yescharacteristic
587C2×C4#1455656111yescharacteristic
588C2×C4#1455656211yescharacteristic
589C2×C4#1455656411yescharacteristic
590C2×C4#1455656411yescharacteristic
591C2×C4#1455656611yescharacteristic
592C2×C4#1455656711yescharacteristic
593C2×C4#1455656911yescharacteristic
594C2×C4#1455656911yescharacteristic
595C2×C4#1455657111yescharacteristic
596C2×C4#1455657311yescharacteristic
597C2×C4#1455657311yescharacteristic
598C2×C4#1455657611yescharacteristic
599C2×C4#1455657611yescharacteristic
600C8—24no—
601C8—42no—
602C8—2922no—
603C8—1464no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—2924no—
615C22—2922no—
616C22—2922no—
617C22—5842no—
618C22—1464no—
619C22#95×D7311yescharacteristic
620C22#1798843411yescharacteristic
621C22C4×#1192400511yescharacteristic
622C22#1798845811yescharacteristic
623C22#1798846111yescharacteristic
624C22#1798846311yescharacteristic
625C22#1798846511yescharacteristic
626C4—82no—
627C4—5842no—
628C4—1464no—
629C4#1798844211yescharacteristic
630C4#1798844411yescharacteristic
631C4#1798844611yescharacteristic
632C4#1798844911yescharacteristic
633C4#1798845811yescharacteristic
634C4#1798846111yescharacteristic
635C4#1798846311yescharacteristic
636C4#1798846511yescharacteristic
637C2—21no—
638C2—21no—
639C2—21no—
640C2—21no—
641C2—21no—
642C2—21no—
643C2—2924no—
644C2#2260815211yescharacteristic
645C2#2261047511yescharacteristic
646C2#2261048011yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC23×C73C2×C42
FrattiniC23×C4C22×D73
FittingC2×#343×C73C2
socleC22×C73#386
prime radicalC22×#11924028C22
cosocleC23×C4×C73C23
hypercenterC2×#343D73
hypocenterC73#10755
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. C2×#17988359C2semidirect
5. C2×#17988358C2semidirect
6. C2×#17988356C2semidirect
7. C2×#343×C73C2semidirect
12. #17989353C4semidirect
20. #17988569C4semidirect
155. #11924012C22⋊C4semidirect
197. M16×C73C22⋊C4semidirect
330. C2×#343D73semidirect
331. C73#10755semidirect
507. M16#11924005semidirect

Maximal subgroups

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

subgroupindex
C2×#179893532
C2×#179885692
#226125242
C2×#179883592
C2×#179883582
C2×#179883562
C2×#343×C732
#1075573