Finite Group Databasebeta

← #33966698

Subgroups — #33966698

Order 18688.

1706 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623753C212yesmaximal, complemented
2#22622881C214yesmaximal, complemented
3#22622877C214yesmaximal, complemented
4#22622822C212yesmaximal, complemented
5#22622820C212yesmaximal, complemented
6#22622798C212yesmaximal, complemented
7#22622796C212yesmaximal, complemented
8#22620470C211yesmaximal, characteristic, complemented
9#22620467C211yesmaximal, characteristic, complemented
10C2×#17988988C211yesmaximal, characteristic, complemented
11#22619797C212yesmaximal, complemented
12#22619793C212yesmaximal, complemented
13#22619751C212yesmaximal, complemented
14#22618863C211yesmaximal, characteristic, complemented
15#22618807C211yesmaximal, characteristic
16#135×Dic73C211yesmaximal, characteristic
17C73×#2951C211yesmaximal, characteristic
18#17989316C2214yes—
19#17989313C2214yes—
20#17989296—24no—
21#17989296C2214yes—
22#17989294—24no—
23#17989294C2214yes—
24#17989218—22no—
25#17989218C2212yes—
26#17989215—22no—
27#17989215C2212yes—
28#17989140—22no—
29#17989140C2211yescharacteristic
30C2×#14556582—22no—
31C2×#14556582C2212yes—
32#17989070C2212yes—
33#41×Dic73C2212yes—
34#17988996C2214yes—
35#17988995C2214yes—
36#17988992C2212yes—
37#17988991C2212yes—
38#17988989C2212yes—
39#17988988C2212yes—
40#17988988C2212yes—
41#17988988C2212yes—
42#17988988C2212yes—
43#17988988C2212yes—
44#17988896C2214yes—
45#17988896C2214yes—
46#17988894C2214yes—
47#17988892C2214yes—
48#17988889C2214yes—
49#17988887C2214yes—
50#17988885C2214yes—
51#17988885C2214yes—
52#17988877C2214yes—
53#17988875C2214yes—
54#17988872C2214yes—
55#17988870C2214yes—
56#17988715C2212yes—
57#17988715C2212yes—
58C8×Dic146C2212yes—
59C8×Dic146C2212yes—
60#17988688C2214yes—
61Q16×Dic73C2211yescharacteristic
62Q16×Dic73C2211yescharacteristic
63Q16×Dic73C2212yes—
64SD16×Dic73C2212yes—
65SD16×Dic73C2212yes—
66D8×Dic73C2211yescharacteristic
67D8×Dic73C2212yes—
68D8×Dic73C2211yescharacteristic
69#17988629C2212yes—
70#17988627C2212yes—
71#17988625C2211yescharacteristic
72C2×#14556525C2211yescharacteristic
73C2×#14556522C2212yes—
74#17988615C2211yescharacteristic
75#17988614C2211yescharacteristic
76#17988610C2211yescharacteristic
77#17988588C2212yes—
78#17988583C2212yes—
79#17988542C2211yescharacteristic
80C2×C8×Dic73C2211yescharacteristic
81C2×#135×C73C2211yescharacteristic
82#520×C73C2212yes—
83#440×C73C2214yes—
84C4×Q16×C73C2212yes—
85C4×D8×C73C2212yes—
86C2×#107×C73C2211yescharacteristic
87#417×C73C2212yes—
88#405×C73C2211yescharacteristic
89C2×#11924012C2311yescharacteristic
90C22×#10596831C2311yescharacteristic
91#14556591—24no—
92#14556591C2314yes—
93#14556590—24no—
94#14556590—22no—
95#14556590—22no—
96#14556590D411yescharacteristic
97#14556590D411yescharacteristic
98#14556590C2314yes—
99#14556583—24no—
100#14556583C2314yes—
101#14556582—22no—
102#14556582—24no—
103#14556582—24no—
104#14556582—24no—
105#14556582—22no—
106#14556582—22no—
107#14556582—22no—
108#14556582D411yescharacteristic
109#14556582D411yescharacteristic
110#14556582C2314yes—
111#14556582C2314yes—
112#14556582C2314yes—
113#14556570—24no—
114#14556570—24no—
115#14556570—24no—
116#14556568—24no—
117#14556568—24no—
118#14556568—24no—
119#14556568—24no—
120#14556568—24no—
121#14556568D412yes—
122#14556568D412yes—
123Q8×Dic73C2314yes—
124Q8×Dic73C2312yes—
125Q8×Dic73C2312yes—
126C4×Dic146—24no—
127C4×Dic146—24no—
128C4×Dic146C2312yes—
129C4×Dic146C2312yes—
130D4×Dic73—22no—
131D4×Dic73—22no—
132D4×Dic73—24no—
133D4×Dic73C2314yes—
134D4×Dic73C2312yes—
135D4×Dic73C2312yes—
136#14556550—22no—
137#14556550—22no—
138#14556550C2312yes—
139#14556550C2311yescharacteristic
140C2×C4⋊Dic73—22no—
141C2×C4⋊Dic73—22no—
142C2×C4⋊Dic73—24no—
143C2×C4⋊Dic73C2312yes—
144C2×#11924007—22no—
145C2×#11924007—24no—
146C2×C4×Dic73—22no—
147C2×C4×Dic73C2311yescharacteristic
148#14556525C2312yes—
149#14556525C2312yes—
150#14556525C2311yescharacteristic
151#14556525C2311yescharacteristic
152#14556525C2314yes—
153#14556524C2314yes—
154#14556523C2314yes—
155#14556523C2312yes—
156#14556522C2314yes—
157#14556522C2314yes—
158#14556522C2314yes—
159C4⋊#10596831C2312yes—
160C4⋊#10596831C2312yes—
161#14556520C2312yes—
162#14556520C2312yes—
163#14556520C2312yes—
164#14556520C2312yes—
165#14556515C2312yes—
166#14556515C2312yes—
167#14556515C2314yes—
168#14556512C2312yes—
169#14556512C2312yes—
170#14556512C2314yes—
171#14556501C2312yes—
172C8×Dic73C2311yescharacteristic
173C8×Dic73C2311yescharacteristic
174C8×Dic73C2312yes—
175C8×Dic73C2311yescharacteristic
176C8×Dic73C2311yescharacteristic
177C2×#41×C73C2312yes—
178#135×C73Q818yescomplemented
179C2×Q16×C73C2311yescharacteristic
180C2×SD16×C73C2312yes—
181C2×D8×C73C2311yescharacteristic
182C2×M16×C73C2311yescharacteristic
183C22×C8×C73C2311yescharacteristic
184C4×Q8×C73C2314yes—
185C4×D4×C73—24no—
186C4×D4×C73C2314yes—
187#117×C73—22no—
188#117×C73C2311yescharacteristic
189C2×C4⋊C4×C73—22no—
190C2×C4⋊C4×C73C2312yes—
191#107×C73C2312yes—
192#107×C73C2312yes—
193#103×C73C2314yes—
194#102×C73C2314yes—
195#97×C73C2312yes—
196C4×C8×C73C2312yes—
197C2×Dic146—24no—
198C2×Dic146—24no—
199C2×Dic146C2×D412yes—
200C2×Dic146C2×D412yes—
201C22×Dic73—22no—
202C22×Dic73—22no—
203C22×Dic73—22no—
204C22×Dic73—24no—
205C22×Dic73C2×D411yescharacteristic
206C22×Dic73C2×D411yescharacteristic
207#11924012—22no—
208C2×#10596831—22no—
209C2×#10596831C2411yescharacteristic
210C2×#10596831C2411yescharacteristic
211C2×#10596831C2412yes—
212C4⋊Dic73—22no—
213C4⋊Dic73—24no—
214C4⋊Dic73—24no—
215C4⋊Dic73—24no—
216C4⋊Dic73—24no—
217C4⋊Dic73—24no—
218C4⋊Dic73—22no—
219C4⋊Dic73—22no—
220C4⋊Dic73C2×D411yescharacteristic
221C4⋊Dic73C2×D412yes—
222C4⋊Dic73C2×D411yescharacteristic
223C4⋊Dic73C2×D412yes—
224C4⋊Dic73C2414yes—
225C4⋊Dic73C2412yes—
226C4⋊Dic73C2412yes—
227#11924007—24no—
228#11924007—24no—
229#11924007—24no—
230#11924007—24no—
231#11924007—24no—
232#11924007—24no—
233#11924007C2×D412yes—
234#11924007C2×D412yes—
235#11924007C2×D412yes—
236#11924007C2×D412yes—
237C22⋊Dic73—22no—
238C22⋊Dic73—22no—
239C22⋊Dic73—22no—
240C22⋊Dic73—24no—
241C22⋊Dic73C2×D412yes—
242C4×Dic73—22no—
243C4×Dic73—22no—
244C4×Dic73—22no—
245C4×Dic73—24no—
246C4×Dic73C2×D411yescharacteristic
247C4×Dic73C2×D411yescharacteristic
248C4×Dic73C2411yescharacteristic
249C4×Dic73C2411yescharacteristic
250C4×Dic73C2412yes—
251#41×C73—24no—
252#41×C73C2×Q818yes—
253C2×Q8×C73C2412yes—
254C2×D4×C73—22no—
255C2×D4×C73C2412yes—
256C22×C4×C73—22no—
257C22×C4×C73—22no—
258C22×C4×C73—24no—
259C22×C4×C73C2411yescharacteristic
260Q16×C73C2×Q814yes—
261SD16×C73C2×Q818yes—
262D8×C73C2×Q814yes—
263M16×C73—22no—
264C2×C8×C73C2×Q814yes—
265C2×C8×C73C2412yes—
266C2×C8×C73C2411yescharacteristic
267C2×C8×C73C2411yescharacteristic
268C4⋊C4×C73—22no—
269C4⋊C4×C73—24no—
270C4⋊C4×C73C2×D411yescharacteristic
271C4⋊C4×C73C2×D411yescharacteristic
272C4⋊C4×C73C2414yes—
273C22⋊C4×C73—24no—
274C22⋊C4×C73C2×D412yes—
275C42×C73—24no—
276C42×C73C2412yes—
277C23×C73—22no—
278C23×C73C22×D411yescharacteristic
279Q8×C73—22no—
280Q8×C73C22×Q814yes—
281D4×C73—28no—
282D4×C73—22no—
283D4×C73C22×Q814yes—
284C2×C4×C73—22no—
285C2×C4×C73—28no—
286C2×C4×C73—24no—
287C2×C4×C73—84no—
288C2×C4×C73—82no—
289C2×C4×C73C22×D412yes—
290C2×C4×C73C22×D412yes—
291C2×C4×C73C22×D411yescharacteristic
292C2×C4×C73C22×Q812yes—
293C2×C4×C732−1+412yes—
294C2×C4×C73C2511yescharacteristic
295C2×Dic73—22no—
296C2×Dic73—22no—
297C2×Dic73—24no—
298C2×Dic73—41no—
299C2×Dic73—41no—
300C2×Dic73—84no—
301C2×Dic73—82no—
302C2×Dic73—82no—
303C2×Dic73—82no—
304C2×Dic73C22×D411yescharacteristic
305C2×Dic73C22×D411yescharacteristic
306C2×Dic73C22×D412yes—
307C2×Dic73C22×D411yescharacteristic
308C2×Dic73C22×D411yescharacteristic
309C2×Dic73C22×D412yes—
310Dic146—42no—
311Dic146—42no—
312Dic146—84no—
313Dic146—84no—
314C8×C73—22no—
315C8×C73C22×Q812yes—
316C8×C73C22×Q812yes—
317#10596831—21no—
318#10596831—21no—
319#10596831—22no—
320C22×C73—21no—
321C22×C73—21no—
322C22×C73—24no—
323C22×C73—42no—
324C22×C73—42no—
325C22×C73D4×Q812yes—
326C22×C73C23×D411yescharacteristic
327C4×C73—22no—
328C4×C73—22no—
329C4×C73—24no—
330C4×C73—44no—
331C4×C73D4×Q812yes—
332C4×C73C23×Q811yescharacteristic
333C4×C73C2×2−1+411yescharacteristic
334Dic73—21no—
335Dic73—21no—
336Dic73—22no—
337Dic73—22no—
338Dic73—21no—
339Dic73—21no—
340Dic73—42no—
341Dic73—42no—
342Dic73—44no—
343#61376—731nomaximal, Sylow
344C2×C73—22no—
345C2×C73—44no—
346C2×C73C2×D4×Q811yescharacteristic
347C2×C73#358411yescharacteristic
348C2×C73#358411yescharacteristic
349#3504—732no—
350#3388—734no—
351#3388—734no—
352#3383—732no—
353#3383—732no—
354#3380—732no—
355#3380—732no—
356#3164—731no—
357#3164—731no—
358C2×#499—731no—
359#3084—732no—
360#3084—732no—
361#3080—732no—
362#2960—731no—
363#2951—731no—
364#2951D7311yescharacteristic
365C4×#135—731no—
366C73#6137611yescharacteristic, complemented, Sylow
367C2×#135C2×D7311yescharacteristic
368#557—734no—
369#557—734no—
370#554—734no—
371#554—734no—
372#554—1464no—
373#554—1464no—
374#544—732no—
375#544—732no—
376#544—1462no—
377#544—1462no—
378#533—731no—
379#533—1462no—
380C2×#125—732no—
381C2×#125—1462no—
382#520—732no—
383#520C2×D7312yes—
384C4×#41—732no—
385#501—734no—
386#501—734no—
387#500—732no—
388#500—732no—
389#499—732no—
390#499—732no—
391#499—732no—
392#499—732no—
393#499—732no—
394#499—732no—
395#479—734no—
396#479—734no—
397#479—734no—
398#479—734no—
399#478—734no—
400#478—734no—
401#478—734no—
402#478—734no—
403#476—734no—
404#476—734no—
405#475—734no—
406#475—734no—
407#446—732no—
408#446—732no—
409C8×Q8—732no—
410C8×Q8—732no—
411#440—734no—
412#440C2×D7314yes—
413C4×Q16—731no—
414C4×Q16—731no—
415C4×Q16—732no—
416C4×Q16C2×D7312yes—
417C4×SD16—732no—
418C4×SD16—732no—
419C4×D8—731no—
420C4×D8—731no—
421C4×D8—732no—
422C4×D8C2×D7312yes—
423#428—732no—
424#428—732no—
425#427—731no—
426C2×#107—731no—
427C2×#107C2×D7311yescharacteristic
428C2×#106—732no—
429#424—731no—
430#424—731no—
431#423—731no—
432#417—732no—
433#417C2×D7312yes—
434#416—732no—
435#405—731no—
436#405C2×D7311yescharacteristic
437C2×C4×C8—731no—
438C2×#41C22×D7312yes—
439#135Dic14618yescomplemented
440C2×Q16C22×D7311yescharacteristic
441C2×SD16C22×D7312yes—
442C2×D8C22×D7311yescharacteristic
443C2×M16—731no—
444C2×M16C22×D7311yescharacteristic
445C22×C8—731no—
446C22×C8C22×D7311yescharacteristic
447C4⋊Q8—731no—
448C4⋊Q8—731no—
449C4⋊Q8—734no—
450C4⋊Q8—734no—
451C4⋊Q8—1464no—
452C4⋊Q8—1462no—
453C4⋊Q8—1464no—
454C4⋊Q8—1462no—
455#125—731no—
456#125—731no—
457#125—734no—
458#125—734no—
459#125—734no—
460#125—734no—
461#125—1464no—
462#125—1462no—
463#125—1462no—
464#125—1462no—
465#125—1464no—
466#125—1464no—
467#125—1464no—
468#125—1462no—
469C22⋊Q8—732no—
470C22⋊Q8—732no—
471C22⋊Q8—1464no—
472C22⋊Q8—1464no—
473C22⋊Q8—1464no—
474C22⋊Q8—1464no—
475C22⋊Q8—1464no—
476C22⋊Q8—1464no—
477C22⋊Q8—1464no—
478C22⋊Q8—1464no—
479C4×Q8—732no—
480C4×Q8—732no—
481C4×Q8—732no—
482C4×Q8—732no—
483C4×Q8—734no—
484C4×Q8—1464no—
485C4×Q8—1464no—
486C4×Q8C22×D7314yes—
487C4×D4—24no—
488C4×D4—732no—
489C4×D4—732no—
490C4×D4—734no—
491C4×D4—1464no—
492C4×D4—1462no—
493C4×D4—1462no—
494C4×D4C22×D7314yes—
495#117—22no—
496#117—731no—
497#117—732no—
498#117—1462no—
499#117—1462no—
500#117C22×D7311yescharacteristic
501C2×C4⋊C4—22no—
502C2×C4⋊C4—732no—
503C2×C4⋊C4—1462no—
504C2×C4⋊C4—1464no—
505C2×C4⋊C4—1464no—
506C2×C4⋊C4—1462no—
507C2×C4⋊C4—1462no—
508C2×C4⋊C4C22×D7312yes—
509C2×C42—731no—
510C2×C42—1462no—
511#107—732no—
512#107—732no—
513#107—731no—
514#107—731no—
515#107—734no—
516#107—734no—
517#107C22×D7312yes—
518#107C22×D7312yes—
519#106—732no—
520#106—734no—
521#106—734no—
522#106—734no—
523#106—734no—
524C4⋊C8—732no—
525C4⋊C8—732no—
526C4⋊C8—732no—
527C4⋊C8—732no—
528C4⋊C8—732no—
529C4⋊C8—732no—
530#103—732no—
531#103—732no—
532#103—734no—
533#103C22×D7314yes—
534#102—732no—
535#102—732no—
536#102—734no—
537#102C22×D7314yes—
538#97—732no—
539#97C22×D7312yes—
540C4×C8—731no—
541C4×C8—731no—
542C4×C8—731no—
543C4×C8—731no—
544C4×C8—732no—
545C4×C8C22×D7312yes—
546#41—24no—
547#41C2×Dic14618yes—
548C2×Q8—732no—
549C2×Q8—732no—
550C2×Q8—1464no—
551C2×Q8—1464no—
552C2×Q8C23×D7312yes—
553C2×D4—22no—
554C2×D4C23×D7312yes—
555C22×C4—22no—
556C22×C4—22no—
557C22×C4—24no—
558C22×C4—731no—
559C22×C4—731no—
560C22×C4—1462no—
561C22×C4—1462no—
562C22×C4—1464no—
563C22×C4—1462no—
564C22×C4C23×D7311yescharacteristic
565Q16C2×Dic14614yes—
566SD16C2×Dic14618yes—
567D8C2×Dic14614yes—
568M16—22no—
569M16—1462no—
570C2×C8—731no—
571C2×C8—731no—
572C2×C8—732no—
573C2×C8—1462no—
574C2×C8C2×Dic14614yes—
575C2×C8C23×D7312yes—
576C2×C8C23×D7311yescharacteristic
577C2×C8C23×D7311yescharacteristic
578C4⋊C4—22no—
579C4⋊C4—24no—
580C4⋊C4—732no—
581C4⋊C4—732no—
582C4⋊C4—732no—
583C4⋊C4—732no—
584C4⋊C4—731no—
585C4⋊C4—732no—
586C4⋊C4—731no—
587C4⋊C4—732no—
588C4⋊C4—734no—
589C4⋊C4—732no—
590C4⋊C4—732no—
591C4⋊C4—1464no—
592C4⋊C4—1464no—
593C4⋊C4—1464no—
594C4⋊C4—1464no—
595C4⋊C4—1464no—
596C4⋊C4—1464no—
597C4⋊C4—1464no—
598C4⋊C4—1464no—
599C4⋊C4—1462no—
600C4⋊C4—1462no—
601C4⋊C4—1464no—
602C4⋊C4—1464no—
603C4⋊C4—1462no—
604C4⋊C4—1464no—
605C4⋊C4D4×D7311yescharacteristic
606C4⋊C4D4×D7311yescharacteristic
607C4⋊C4C23×D7314yes—
608C22⋊C4—24no—
609C22⋊C4—732no—
610C22⋊C4—1462no—
611C22⋊C4—1462no—
612C22⋊C4—1464no—
613C22⋊C4—1462no—
614C22⋊C4D4×D7312yes—
615C42—24no—
616C42—731no—
617C42—731no—
618C42—731no—
619C42—731no—
620C42—732no—
621C42—1462no—
622C42—1462no—
623C42—1464no—
624C42—1462no—
625C42C23×D7312yes—
626C23—22no—
627C23C2×D4×D7311yescharacteristic
628Q8—22no—
629Q8—2922no—
630Q8—2922no—
631Q8—5844no—
632Q8—5844no—
633Q8C22×Dic14614yes—
634D4—28no—
635D4—22no—
636D4C22×Dic14614yes—
637C2×C4—22no—
638C2×C4—28no—
639C2×C4—24no—
640C2×C4—84no—
641C2×C4—82no—
642C2×C4—2921no—
643C2×C4—2921no—
644C2×C4—5842no—
645C2×C4—5842no—
646C2×C4—5844no—
647C2×C4—5842no—
648C2×C4—731no—
649C2×C4—731no—
650C2×C4—731no—
651C2×C4—731no—
652C2×C4—732no—
653C2×C4—732no—
654C2×C4—1462no—
655C2×C4—1462no—
656C2×C4—1464no—
657C2×C4C2×D4×D7312yes—
658C2×C4C2×D4×D7312yes—
659C2×C4C2×D4×D7311yescharacteristic
660C2×C4C22×Dic14612yes—
661C2×C4#1455663812yes—
662C2×C4C24×D7311yescharacteristic
663C8—22no—
664C8—1461no—
665C8—1461no—
666C8—1462no—
667C8C22×Dic14612yes—
668C8C22×Dic14612yes—
669C22—21no—
670C22—21no—
671C22—24no—
672C22—42no—
673C22—42no—
674C22D4×Dic14612yes—
675C22C22×D4×D7311yescharacteristic
676C4—24no—
677C4—22no—
678C4—22no—
679C4—44no—
680C4—2922no—
681C4—2924no—
682C4—2922no—
683C4—1461no—
684C4—1461no—
685C4—1462no—
686C4—1461no—
687C4—1461no—
688C4—1462no—
689C4D4×Dic14612yes—
690C4C23×Dic14611yescharacteristic
691C4C2×#1455663811yescharacteristic
692C2—22no—
693C2—44no—
694C2C2×D4×Dic14611yescharacteristic
695C2#2262469911yescharacteristic
696C2#2262470111yescharacteristic
subgroupG/H
centerC22C22×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC73×#2951C2
socleC22×C73C23×D4
prime radicalC2×#135×C73C22
hypercenter#2951D73
hypocenterC73#61376
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #22623753C2semidirect
2. #22622881C2semidirect
3. #22622877C2semidirect
4. #22622822C2semidirect
5. #22622820C2semidirect
6. #22622798C2semidirect
7. #22622796C2semidirect
8. #22620470C2semidirect
9. #22620467C2semidirect
10. C2×#17988988C2semidirect
11. #22619797C2semidirect
12. #22619793C2semidirect
13. #22619751C2semidirect
14. #22618863C2semidirect
178. #135×C73Q8semidirect
366. C73#61376semidirect
439. #135Dic146semidirect

Maximal subgroups

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

subgroupindex
#226237532
#226228812
#226228772
#226228222
#226228202
#226227982
#226227962
#226204702
#226204672
C2×#179889882
#226197972
#226197932
#226197512
#226188632
#226188072
#135×Dic732
C73×#29512
#6137673