Finite Group Databasebeta

← #33089201

Subgroups — #33089201

Order 18688.

800 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618699C211yesmaximal, characteristic
2#22618574C211yesmaximal, characteristic, complemented
3#131×Dic73C211yesmaximal, characteristic, complemented
4#22610580C211yesmaximal, characteristic, complemented
5#22610578C211yesmaximal, characteristic, complemented
6#22610533C211yesmaximal, characteristic, complemented
7#22610527C211yesmaximal, characteristic, complemented
8#22610258C211yesmaximal, characteristic
9#22610256C211yesmaximal, characteristic
10#22610208C211yesmaximal, characteristic, complemented
11#108×Dic73C211yesmaximal, characteristic, complemented
12#104×Dic73C211yesmaximal, characteristic, complemented
13#22609735C211yesmaximal, characteristic, complemented
14#22609675C211yesmaximal, characteristic, complemented
15C73×#1866C211yesmaximal, characteristic, complemented
16C2×#14556611C2211yescharacteristic
17C2×#14556598C2211yescharacteristic
18#41×Dic73C2211yescharacteristic
19#17989055C2211yescharacteristic
20#17988679C411yescharacteristic, complemented
21#17988679C411yescharacteristic, complemented
22#17988678C411yescharacteristic, complemented
23#17988678C411yescharacteristic, complemented
24#17988672C411yescharacteristic, complemented
25#17988672C411yescharacteristic, complemented
26#17988667C411yescharacteristic, complemented
27#17988667C411yescharacteristic, complemented
28#17988647—21no—
29#17988647—21no—
30#17988639—21no—
31#17988639—21no—
32C2×#14556527C2211yescharacteristic
33#17988623C2211yescharacteristic
34#17988614C2211yescharacteristic
35C2×#14556519C2211yescharacteristic
36C2×#14556518C2211yescharacteristic
37C2×#14556517C2211yescharacteristic
38#17988584C2211yescharacteristic
39#17988555C2211yescharacteristic
40#17988553C2211yescharacteristic
41#17988545—21no—
42#17988545C2211yescharacteristic
43#17988542—21no—
44#17988542C2211yescharacteristic
45M16×Dic73—21no—
46M16×Dic73C2211yescharacteristic
47M16×Dic73C2211yescharacteristic
48C2×C8×Dic73—21no—
49C2×C8×Dic73C2211yescharacteristic
50C2×C8×Dic73C2211yescharacteristic
51C42×Dic73C2211yescharacteristic
52#17988351C2211yescharacteristic
53#17988351C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988347C2211yescharacteristic
56#17988346C2211yescharacteristic
57#17988342C2211yescharacteristic
58#17988341C2211yescharacteristic
59#17988341C2211yescharacteristic
60C2×#131×C73C2211yescharacteristic
61C2×#108×C73C2211yescharacteristic
62#424×C73C2211yescharacteristic
63C2×#104×C73C2211yescharacteristic
64#416×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67C2×#11924028C2311yescharacteristic
68#14556611—22no—
69#14556611—22no—
70C2×#11924018C2×C411yescharacteristic
71C2×#11924016C2×C411yescharacteristic
72C2×#11924015C2×C411yescharacteristic
73C2×#11924013C2×C411yescharacteristic
74#14556598—22no—
75#14556598—22no—
76C2×#11924012—21no—
77C2×#11924012C2311yescharacteristic
78C2×#11924012C2311yescharacteristic
79C22×#10596831—21no—
80C22×#10596831C2311yescharacteristic
81C22×#10596831C2311yescharacteristic
82Q8×Dic73C2×C411yescharacteristic
83Q8×Dic73C2×C411yescharacteristic
84#14556556C2×C411yescharacteristic
85#14556556C2×C411yescharacteristic
86D4×Dic73—21no—
87D4×Dic73—21no—
88D4×Dic73C2×C411yescharacteristic
89D4×Dic73C2×C411yescharacteristic
90#14556553C2×C411yescharacteristic
91#14556553C2×C411yescharacteristic
92#14556551—21no—
93#14556551—21no—
94#14556550—21no—
95#14556550C2311yescharacteristic
96#14556548—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73—21no—
100C2×C4×Dic73C2311yescharacteristic
101C2×C4×Dic73C2311yescharacteristic
102C42×D73—21no—
103#14556527—22no—
104#14556524C2×C411yescharacteristic
105#14556524C2×C411yescharacteristic
106#14556522C2×C411yescharacteristic
107#14556522C2×C411yescharacteristic
108C4⋊#10596831—21no—
109C4⋊#10596831—21no—
110#14556519—22no—
111#14556518—22no—
112#14556517—22no—
113#14556516C2×C411yescharacteristic
114#14556516C2×C411yescharacteristic
115#14556513C2×C411yescharacteristic
116#14556513C2×C411yescharacteristic
117#14556503—21no—
118#14556503—21no—
119#14556503—21no—
120#14556503—21no—
121#14556501—21no—
122#14556501—21no—
123#14556501—21no—
124#14556501—21no—
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73—21no—
129C8×Dic73C2×C411yescharacteristic
130C8×Dic73C2×C411yescharacteristic
131C8×Dic73C2×C411yescharacteristic
132C8×Dic73C2×C411yescharacteristic
133C8×Dic73D411yescharacteristic
134C8×Dic73D411yescharacteristic
135C8×Dic73D411yescharacteristic
136C8×Dic73D411yescharacteristic
137C2×#41×C73C2311yescharacteristic
138#131×C73—22no—
139#131×C73—22no—
140C2×M16×C73—21no—
141C2×M16×C73C2311yescharacteristic
142C2×M16×C73C2311yescharacteristic
143C22×C8×C73—21no—
144C22×C8×C73C2311yescharacteristic
145C22×C8×C73C2311yescharacteristic
146#117×C73C2311yescharacteristic
147C2×C42×C73C2311yescharacteristic
148#108×C73—22no—
149C4⋊C8×C73—21no—
150C4⋊C8×C73—21no—
151#104×C73—22no—
152#103×C73C2×C411yescharacteristic
153#103×C73C2×C411yescharacteristic
154#102×C73C2×C411yescharacteristic
155#102×C73C2×C411yescharacteristic
156#11924028—21no—
157#11924028—21no—
158#11924028—21no—
159#11924028—21no—
160C2×Dic146C22×C411yescharacteristic
161C2×D292C22×C411yescharacteristic
162C2×C22⋊D73—21no—
163C22×Dic73—21no—
164C22×Dic73—21no—
165C22×Dic73—21no—
166C22×Dic73#4111yescharacteristic
167C22×Dic73#4111yescharacteristic
168C2×C4×D73—21no—
169C2×C4×D73—21no—
170C2×C4×D73—21no—
171#11924018—22no—
172#11924016—22no—
173#11924015—22no—
174#11924013—22no—
175#11924012—24no—
176#11924012—21no—
177#11924012—21no—
178#11924012—21no—
179#11924012#4112yes—
180C2×#10596831—24no—
181C2×#10596831—21no—
182C2×#10596831—21no—
183C2×#10596831—21no—
184C2×#10596831—21no—
185C2×#10596831—21no—
186C2×#10596831—21no—
187C2×#10596831C22×C411yescharacteristic
188C2×#10596831C22×C411yescharacteristic
189C2×#10596831C2×D411yescharacteristic
190C2×#10596831C2×D411yescharacteristic
191C4⋊Dic73—21no—
192C4⋊Dic73—21no—
193C4⋊Dic73C22×C411yescharacteristic
194C4⋊Dic73C22×C411yescharacteristic
195#11924007—21no—
196#11924007—21no—
197C22⋊Dic73—21no—
198C22⋊Dic73—21no—
199C22⋊Dic73—21no—
200#11924005—21no—
201#11924005—21no—
202C4×Dic73—24no—
203C4×Dic73—21no—
204C4×Dic73—21no—
205C4×Dic73—21no—
206C4×Dic73—21no—
207C4×Dic73—21no—
208C4×Dic73—21no—
209C4×Dic73—21no—
210C4×Dic73—42no—
211C4×Dic73C22×C411yescharacteristic
212C4×Dic73C22×C411yescharacteristic
213C4×Dic73C2×D411yescharacteristic
214C4×Dic73C2×D411yescharacteristic
215#41×C73—21no—
216#41×C73—21no—
217#41×C73#4112yes—
218#41×C73#4112yes—
219C2×Q8×C73C22×C411yescharacteristic
220C2×D4×C73—21no—
221C2×D4×C73C22×C411yescharacteristic
222C22×C4×C73—21no—
223C22×C4×C73—21no—
224C22×C4×C73C2411yescharacteristic
225M16×C73—24no—
226M16×C73—21no—
227M16×C73—21no—
228M16×C73—21no—
229M16×C73#4112yes—
230C2×C8×C73—24no—
231C2×C8×C73—21no—
232C2×C8×C73—21no—
233C2×C8×C73—21no—
234C2×C8×C73—21no—
235C2×C8×C73—21no—
236C2×C8×C73—21no—
237C2×C8×C73C22×C411yescharacteristic
238C2×C8×C73C22×C411yescharacteristic
239C2×C8×C73C2×D411yescharacteristic
240C2×C8×C73C2×D411yescharacteristic
241C4⋊C4×C73C22×C411yescharacteristic
242C4⋊C4×C73C22×C411yescharacteristic
243C22⋊C4×C73—21no—
244C42×C73—21no—
245C42×C73—21no—
246C42×C73—21no—
247C23×C73—21no—
248C23×C73C2×#4111yescharacteristic
249C22×D73—21no—
250Q8×C73—21no—
251Q8×C73#11712yes—
252D4×C73—22no—
253D4×C73—22no—
254D4×C73—21no—
255D4×C73#11712yes—
256C2×C4×C73—21no—
257C2×C4×C73—22no—
258C2×C4×C73—22no—
259C2×C4×C73—22no—
260C2×C4×C73—21no—
261C2×C4×C73—21no—
262C2×C4×C73—41no—
263C2×C4×C73—41no—
264C2×C4×C73C23×C411yescharacteristic
265C2×C4×C73C22×D411yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×C4×C73C2×#4111yescharacteristic
268C2×C4×C73C2×#4111yescharacteristic
269C2×C4×C73C2×#4111yescharacteristic
270C22⋊D73—41no—
271C22⋊D73—41no—
272C2×Dic73—22no—
273C2×Dic73—22no—
274C2×Dic73—22no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—42no—
281C2×Dic73—42no—
282C2×Dic73C4×D411yescharacteristic
283C2×Dic73C4×D411yescharacteristic
284C2×Dic73C4×D411yescharacteristic
285C2×Dic73C4×D411yescharacteristic
286D292—21no—
287D292—21no—
288C4×D73—41no—
289C4×D73—41no—
290C4×D73—42no—
291C4×D73—42no—
292Dic146—21no—
293Dic146—21no—
294C8×C73—21no—
295C8×C73—21no—
296C8×C73—21no—
297C8×C73—22no—
298C8×C73—21no—
299C8×C73—41no—
300#10596831—21no—
301#10596831—21no—
302#10596831—21no—
303#10596831—22no—
304#10596831—21no—
305#10596831—41no—
306C22×C73—21no—
307C22×C73—21no—
308C22×C73—22no—
309C22×C73—41no—
310C22×C73—41no—
311C22×C73C2×C4×D411yescharacteristic
312C22×C73#52911yescharacteristic
313C22×C73#52911yescharacteristic
314C2×D73—41no—
315C2×D73—41no—
316C2×D73—41no—
317C4×C73—22no—
318C4×C73—42no—
319C4×C73—41no—
320C4×C73—41no—
321C4×C73C2×#11711yescharacteristic
322C4×C73C4×#4111yescharacteristic
323C4×C73#52511yescharacteristic
324C4×C73#52511yescharacteristic
325Dic73—21no—
326Dic73—21no—
327Dic73—42no—
328Dic73—41no—
329Dic73—41no—
330Dic73—41no—
331Dic73#44312yes—
332Dic73#44412yes—
333#21137—731nomaximal, Sylow
334C2×C73—21no—
335C2×C73—21no—
336C2×C73—42no—
337C2×C73#229011yescharacteristic
338C2×C73C2×#44311yescharacteristic
339C2×C73C2×#44411yescharacteristic
340D73—81no—
341C4×#135—731no—
342#2922—731no—
343C4×#131—731no—
344#1929—731no—
345#1929—731no—
346#1922—731no—
347#1922—731no—
348#1876—731no—
349#1876—731no—
350#1866—731no—
351#1866D7311yescharacteristic, complemented
352C4×#108—731no—
353C4×#104—731no—
354C4×#104—731no—
355#1744—731no—
356C73#2113711yescharacteristic, complemented, Sylow
357C2×#135—731no—
358C2×#131—731no—
359C2×#131C2×D7311yescharacteristic
360C4×#41—731no—
361C4×#41—731no—
362C4×Q16—731no—
363C4×Q16—731no—
364C4×SD16—731no—
365C4×SD16—731no—
366C4×SD16—731no—
367C4×SD16—731no—
368C4×D8—731no—
369C4×D8—731no—
370#433—1461no—
371#433—1461no—
372#431—1461no—
373#431—1461no—
374C2×#108—731no—
375C2×#108C2×D7311yescharacteristic
376#427—731no—
377#424—731no—
378#424C2×D7311yescharacteristic
379C2×#104—731no—
380C2×#104—731no—
381C2×#104—731no—
382C2×#104C2×D7311yescharacteristic
383#416—731no—
384#416C2×D7311yescharacteristic
385#408—731no—
386#408—731no—
387#405—731no—
388#405—731no—
389#405—1461no—
390#405—1461no—
391C4×M16—731no—
392C4×M16—731no—
393C4×M16—1461no—
394C2×C4×C8—731no—
395C2×C4×C8—731no—
396C2×C4×C8—1461no—
397C43—731no—
398#341—731no—
399#341—731no—
400#339—731no—
401#339—731no—
402#339—731no—
403#339C2×D7311yescharacteristic
404#336—731no—
405#336—731no—
406#336—731no—
407#336C2×D7311yescharacteristic
408C2×#41—731no—
409C2×#41C22×D7311yescharacteristic
410#135—1462no—
411#135—1462no—
412C2×Q16—731no—
413C2×SD16—731no—
414C2×SD16—731no—
415C2×D8—731no—
416#131—22no—
417#131—22no—
418#131—1462no—
419#131—1462no—
420C2×M16—21no—
421C2×M16—731no—
422C2×M16—731no—
423C2×M16—1461no—
424C2×M16C22×D7311yescharacteristic
425C2×M16C22×D7311yescharacteristic
426C22×C8—21no—
427C22×C8—731no—
428C22×C8—731no—
429C22×C8—1461no—
430C22×C8C22×D7311yescharacteristic
431C22×C8C22×D7311yescharacteristic
432C4×Q8—731no—
433C4×Q8—731no—
434C4×Q8—731no—
435C4×Q8—731no—
436C4×D4—731no—
437C4×D4—731no—
438C4×D4—731no—
439C4×D4—731no—
440C4×D4—1461no—
441C4×D4—1461no—
442C4×D4—1461no—
443C4×D4—1461no—
444#117—731no—
445#117—1461no—
446#117—1461no—
447#117C22×D7311yescharacteristic
448C2×C42—731no—
449C2×C42—731no—
450C2×C42—1461no—
451C2×C42—1461no—
452C2×C42—1461no—
453C2×C42—1461no—
454C2×C42C22×D7311yescharacteristic
455#108—22no—
456#108—1462no—
457#107—731no—
458#107—731no—
459#106—731no—
460#106—731no—
461C4⋊C8—21no—
462C4⋊C8—21no—
463C4⋊C8—1461no—
464C4⋊C8—1461no—
465#104—22no—
466#104—1462no—
467#104—1462no—
468#104—1462no—
469#103—731no—
470#103—731no—
471#103C4×D7311yescharacteristic
472#103C4×D7311yescharacteristic
473#102—731no—
474#102—731no—
475#102C4×D7311yescharacteristic
476#102C4×D7311yescharacteristic
477C22⋊C8—1461no—
478C22⋊C8—1461no—
479C22⋊C8—1461no—
480C22⋊C8—1461no—
481#97—1461no—
482#97—1461no—
483#97—1461no—
484#97—1461no—
485C4×C8—731no—
486C4×C8—731no—
487C4×C8—731no—
488C4×C8—731no—
489C4×C8—731no—
490C4×C8—731no—
491C4×C8—731no—
492C4×C8—731no—
493C4×C8—1461no—
494C4×C8—1461no—
495C4×C8—1461no—
496C4×C8—1461no—
497#41—21no—
498#41—21no—
499#41—1461no—
500#41—1461no—
501#41—1461no—
502#41—1461no—
503#41#1192402712yes—
504#41#1192402912yes—
505C2×Q8—731no—
506C2×Q8C2×C4×D7311yescharacteristic
507C2×D4—21no—
508C2×D4—731no—
509C2×D4—1461no—
510C2×D4C2×C4×D7311yescharacteristic
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—731no—
514C22×C4—731no—
515C22×C4—1461no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4—1461no—
519C22×C4—1461no—
520C22×C4—1461no—
521C22×C4C23×D7311yescharacteristic
522Q16—1462no—
523SD16—1462no—
524SD16—1462no—
525D8—1462no—
526M16—24no—
527M16—21no—
528M16—21no—
529M16—21no—
530M16—732no—
531M16—1461no—
532M16—1461no—
533M16—1464no—
534M16—1461no—
535M16#1192402712yes—
536C2×C8—24no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—731no—
544C2×C8—731no—
545C2×C8—731no—
546C2×C8—731no—
547C2×C8—1461no—
548C2×C8—1464no—
549C2×C8—1461no—
550C2×C8—1461no—
551C2×C8—1461no—
552C2×C8—1461no—
553C2×C8—1461no—
554C2×C8C2×C4×D7311yescharacteristic
555C2×C8C2×C4×D7311yescharacteristic
556C2×C8D4×D7311yescharacteristic
557C2×C8D4×D7311yescharacteristic
558C4⋊C4—731no—
559C4⋊C4—731no—
560C4⋊C4—1461no—
561C4⋊C4—1461no—
562C4⋊C4—1461no—
563C4⋊C4—1461no—
564C4⋊C4C2×C4×D7311yescharacteristic
565C4⋊C4C2×C4×D7311yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—1461no—
568C22⋊C4—1461no—
569C22⋊C4—1461no—
570C22⋊C4—1461no—
571C22⋊C4—1461no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—2922no—
576C42—731no—
577C42—731no—
578C42—731no—
579C42—731no—
580C42—1461no—
581C42—1461no—
582C42—1461no—
583C42—1461no—
584C42—1464no—
585C42—1461no—
586C42—1461no—
587C42—1461no—
588C23—21no—
589C23—1461no—
590C23#41×D7311yescharacteristic
591Q8—21no—
592Q8—1461no—
593Q8—1461no—
594Q8#1455654912yes—
595D4—22no—
596D4—22no—
597D4—21no—
598D4—2921no—
599D4—2921no—
600D4—1461no—
601D4—1461no—
602D4#1455654912yes—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—2922no—
612C2×C4—2922no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2922no—
616C2×C4—2922no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4—731no—
620C2×C4—731no—
621C2×C4—731no—
622C2×C4—731no—
623C2×C4—1462no—
624C2×C4—1461no—
625C2×C4—1461no—
626C2×C4—1461no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4C22×C4×D7311yescharacteristic
630C2×C4C2×D4×D7311yescharacteristic
631C2×C4#41×D7311yescharacteristic
632C2×C4C2×#1192402711yescharacteristic
633C2×C4C2×#1192402711yescharacteristic
634C2×C4C2×#1192402911yescharacteristic
635C8—21no—
636C8—21no—
637C8—21no—
638C8—22no—
639C8—21no—
640C8—41no—
641C8—2921no—
642C8—1461no—
643C8—1461no—
644C8—1461no—
645C8—1461no—
646C8—1462no—
647C22—21no—
648C22—21no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—2921no—
653C22—2921no—
654C22—2921no—
655C22C4×D4×D7311yescharacteristic
656C22#1798911411yescharacteristic
657C22#1798912311yescharacteristic
658C4—22no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—2922no—
663C4—2921no—
664C4—2921no—
665C4—2921no—
666C4—732no—
667C4—732no—
668C4—1461no—
669C4—1461no—
670C4C2×#1455654911yescharacteristic
671C4C4×#1192402711yescharacteristic
672C4#1798909311yescharacteristic
673C4#1798910011yescharacteristic
674C2—21no—
675C2—21no—
676C2—42no—
677C2—5841no—
678C2#2261236711yescharacteristic
679C2#443×D7311yescharacteristic
680C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1866C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556611C22
cosocleC22×C4×C73C24
hypercenter#1866D73
hypocenterC73#21137
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #22618574C2semidirect
3. #131×Dic73C2semidirect
4. #22610580C2semidirect
5. #22610578C2semidirect
6. #22610533C2semidirect
7. #22610527C2semidirect
10. #22610208C2semidirect
11. #108×Dic73C2semidirect
12. #104×Dic73C2semidirect
13. #22609735C2semidirect
14. #22609675C2semidirect
15. C73×#1866C2semidirect
20. #17988679C4semidirect
21. #17988679C4semidirect
22. #17988678C4semidirect
23. #17988678C4semidirect
24. #17988672C4semidirect
25. #17988672C4semidirect
26. #17988667C4semidirect
27. #17988667C4semidirect
351. #1866D73semidirect
356. C73#21137semidirect

Maximal subgroups

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

subgroupindex
#226186992
#226185742
#131×Dic732
#226105802
#226105782
#226105332
#226105272
#226102582
#226102562
#226102082
#108×Dic732
#104×Dic732
#226097352
#226096752
C73×#18662
#2113773