Finite Group Databasebeta

← #33089216

Subgroups — #33089216

Order 18688.

772 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618732C211yesmaximal, characteristic
2#22618571C211yesmaximal, characteristic, complemented
3#131×Dic73C211yesmaximal, characteristic, complemented
4#22610582C211yesmaximal, characteristic, complemented
5#22610576C211yesmaximal, characteristic, complemented
6#22610531C211yesmaximal, characteristic, complemented
7#22610529C211yesmaximal, characteristic, complemented
8#22610260C211yesmaximal, characteristic
9#22610206C211yesmaximal, characteristic, complemented
10#22610204C211yesmaximal, characteristic, complemented
11#22609836C211yesmaximal, characteristic, complemented
12#22609743C211yesmaximal, characteristic, complemented
13#22609743C211yesmaximal, characteristic, complemented
14#22609681C211yesmaximal, characteristic, complemented
15C73×#1876C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17#41×Dic73C2211yescharacteristic
18#41×Dic73C2211yescharacteristic
19#17988699C411yescharacteristic, complemented
20#17988699C411yescharacteristic, complemented
21#17988694C411yescharacteristic, complemented
22#17988694C411yescharacteristic, complemented
23#17988688C411yescharacteristic, complemented
24#17988688C411yescharacteristic, complemented
25#17988688C411yescharacteristic, complemented
26#17988688C411yescharacteristic, complemented
27#17988645—21no—
28#17988645—21no—
29#17988637—21no—
30#17988637—21no—
31C2×#14556526C2211yescharacteristic
32C2×#14556526C2211yescharacteristic
33#17988625C2211yescharacteristic
34#17988615C2211yescharacteristic
35#17988613C2211yescharacteristic
36C2×#14556518C2211yescharacteristic
37C2×#14556518C2211yescharacteristic
38#17988583C2211yescharacteristic
39#17988583C2211yescharacteristic
40#17988554C2211yescharacteristic
41#17988545—21no—
42#17988545C2211yescharacteristic
43#17988542—21no—
44#17988542C2211yescharacteristic
45M16×Dic73—21no—
46M16×Dic73C2211yescharacteristic
47M16×Dic73C2211yescharacteristic
48C2×#14556501C2211yescharacteristic
49C2×C8×Dic73—21no—
50C2×C8×Dic73C2211yescharacteristic
51#17988433C2211yescharacteristic
52#17988352C2211yescharacteristic
53#17988348C2211yescharacteristic
54#17988348C2211yescharacteristic
55#17988347C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988342C2211yescharacteristic
58#17988341C2211yescharacteristic
59#17988341C2211yescharacteristic
60C2×#135×C73C2211yescharacteristic
61C2×#131×C73C2211yescharacteristic
62C2×#104×C73C2211yescharacteristic
63C2×#104×C73C2211yescharacteristic
64#408×C73C2211yescharacteristic
65#341×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67#14556598—22no—
68#14556598—22no—
69C2×#11924012—21no—
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C22×#10596831—21no—
73C22×#10596831C2311yescharacteristic
74C22×#10596831C2311yescharacteristic
75Q8×Dic73C2×C411yescharacteristic
76Q8×Dic73C2×C411yescharacteristic
77Q8×Dic73C2×C411yescharacteristic
78Q8×Dic73C2×C411yescharacteristic
79D4×Dic73—21no—
80D4×Dic73—21no—
81D4×Dic73—21no—
82D4×Dic73—21no—
83D4×Dic73C2×C411yescharacteristic
84D4×Dic73C2×C411yescharacteristic
85D4×Dic73C2×C411yescharacteristic
86D4×Dic73C2×C411yescharacteristic
87#14556550—21no—
88#14556550—21no—
89#14556550C2311yescharacteristic
90#14556550C2311yescharacteristic
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73C2311yescharacteristic
94C2×C4×Dic73C2311yescharacteristic
95#14556526—22no—
96#14556526—22no—
97#14556525C2×C411yescharacteristic
98#14556525C2×C411yescharacteristic
99#14556523C2×C411yescharacteristic
100#14556523C2×C411yescharacteristic
101#14556520—21no—
102#14556520—21no—
103#14556520—21no—
104#14556520—21no—
105#14556518—22no—
106#14556518—22no—
107#14556515C2×C411yescharacteristic
108#14556515C2×C411yescharacteristic
109#14556515C2×C411yescharacteristic
110#14556515C2×C411yescharacteristic
111#14556512C2×C411yescharacteristic
112#14556512C2×C411yescharacteristic
113#14556512C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115C22⋊#10596831—21no—
116C22⋊#10596831—21no—
117#14556501—21no—
118#14556501—21no—
119#14556501—21no—
120#14556501—21no—
121#14556501C2×C411yescharacteristic
122#14556501C2×C411yescharacteristic
123#14556501C2×C411yescharacteristic
124#14556501C2×C411yescharacteristic
125C8×Dic73—21no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73—21no—
129C8×Dic73D411yescharacteristic
130C8×Dic73D411yescharacteristic
131C8×Dic73D411yescharacteristic
132C8×Dic73D411yescharacteristic
133#14556498—21no—
134#14556498—21no—
135C2×#41×C73C2311yescharacteristic
136C2×#41×C73C2311yescharacteristic
137#135×C73—22no—
138#135×C73—22no—
139C2×Q16×C73C2×C411yescharacteristic
140C2×SD16×C73C2×C411yescharacteristic
141C2×SD16×C73C2×C411yescharacteristic
142C2×D8×C73C2×C411yescharacteristic
143#131×C73—22no—
144#131×C73—22no—
145C2×M16×C73—21no—
146C2×M16×C73C2311yescharacteristic
147C2×M16×C73C2311yescharacteristic
148C22×C8×C73—21no—
149C22×C8×C73C2311yescharacteristic
150C22×C8×C73C2311yescharacteristic
151C2×C42×C73C2311yescharacteristic
152#104×C73—22no—
153#104×C73—22no—
154C22⋊C8×C73—21no—
155C22⋊C8×C73—21no—
156C22×Dic73—21no—
157C22×Dic73—21no—
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73—21no—
161C22×Dic73#4111yescharacteristic
162C22×Dic73#4111yescharacteristic
163#11924012—24no—
164#11924012—21no—
165#11924012—21no—
166#11924012—21no—
167#11924012#4112yes—
168C2×#10596831—24no—
169C2×#10596831—21no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—21no—
173C2×#10596831—21no—
174C2×#10596831—21no—
175C2×#10596831C22×C411yescharacteristic
176C2×#10596831C22×C411yescharacteristic
177C2×#10596831C2×D411yescharacteristic
178C2×#10596831C2×D411yescharacteristic
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73—21no—
182C4⋊Dic73—21no—
183C4⋊Dic73C22×C411yescharacteristic
184C4⋊Dic73C22×C411yescharacteristic
185C4⋊Dic73C22×C411yescharacteristic
186C4⋊Dic73C22×C411yescharacteristic
187C22⋊Dic73—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—21no—
199C4×Dic73—21no—
200C4×Dic73—21no—
201C4×Dic73C22×C411yescharacteristic
202C4×Dic73C22×C411yescharacteristic
203C4×Dic73C2×D411yescharacteristic
204C4×Dic73C2×D411yescharacteristic
205#41×C73—21no—
206#41×C73—21no—
207#41×C73—21no—
208#41×C73—21no—
209#41×C73—21no—
210#41×C73—21no—
211#41×C73#4112yes—
212#41×C73#4112yes—
213C2×Q8×C73C22×C411yescharacteristic
214C2×Q8×C73C22×C411yescharacteristic
215C2×D4×C73—21no—
216C2×D4×C73—21no—
217C2×D4×C73C22×C411yescharacteristic
218C2×D4×C73C22×C411yescharacteristic
219C22×C4×C73—21no—
220C22×C4×C73—21no—
221C22×C4×C73—21no—
222C22×C4×C73C2411yescharacteristic
223Q16×C73—22no—
224SD16×C73—22no—
225SD16×C73—22no—
226D8×C73—22no—
227M16×C73—24no—
228M16×C73—21no—
229M16×C73—21no—
230M16×C73—21no—
231M16×C73#4112yes—
232C2×C8×C73—24no—
233C2×C8×C73—21no—
234C2×C8×C73—21no—
235C2×C8×C73—21no—
236C2×C8×C73—21no—
237C2×C8×C73—21no—
238C2×C8×C73—21no—
239C2×C8×C73C22×C411yescharacteristic
240C2×C8×C73C22×C411yescharacteristic
241C2×C8×C73C2×D411yescharacteristic
242C2×C8×C73C2×D411yescharacteristic
243C42×C73—21no—
244C42×C73—21no—
245C23×C73—21no—
246C23×C73—21no—
247C23×C73C2×#4111yescharacteristic
248Q8×C73—21no—
249Q8×C73—21no—
250Q8×C73—21no—
251Q8×C73#11712yes—
252D4×C73—22no—
253D4×C73—22no—
254D4×C73—21no—
255D4×C73—21no—
256D4×C73—21no—
257D4×C73—41no—
258D4×C73—41no—
259D4×C73#11712yes—
260C2×C4×C73—21no—
261C2×C4×C73—22no—
262C2×C4×C73—22no—
263C2×C4×C73—21no—
264C2×C4×C73—41no—
265C2×C4×C73—41no—
266C2×C4×C73—41no—
267C2×C4×C73—41no—
268C2×C4×C73—41no—
269C2×C4×C73C23×C411yescharacteristic
270C2×C4×C73C22×D411yescharacteristic
271C2×C4×C73C2×#4111yescharacteristic
272C2×C4×C73C2×#4111yescharacteristic
273C2×C4×C73C2×#4111yescharacteristic
274C2×C4×C73C2×#4111yescharacteristic
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—21no—
278C2×Dic73—21no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—42no—
282C2×Dic73—42no—
283C2×Dic73—42no—
284C2×Dic73—42no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73C4×D411yescharacteristic
289C2×Dic73C4×D411yescharacteristic
290C2×Dic73C4×D411yescharacteristic
291C2×Dic73C4×D411yescharacteristic
292C8×C73—21no—
293C8×C73—21no—
294C8×C73—21no—
295C8×C73—22no—
296C8×C73—21no—
297C8×C73—41no—
298#10596831—21no—
299#10596831—21no—
300#10596831—21no—
301#10596831—22no—
302#10596831—21no—
303#10596831—41no—
304C22×C73—21no—
305C22×C73—21no—
306C22×C73—22no—
307C22×C73—41no—
308C22×C73—41no—
309C22×C73—41no—
310C22×C73—41no—
311C22×C73—41no—
312C22×C73C2×C4×D411yescharacteristic
313C22×C73#52911yescharacteristic
314C22×C73#52911yescharacteristic
315C4×C73—22no—
316C4×C73—41no—
317C4×C73—81no—
318C4×C73C2×#11711yescharacteristic
319C4×C73C4×#4111yescharacteristic
320C4×C73#52511yescharacteristic
321C4×C73#52511yescharacteristic
322Dic73—21no—
323Dic73—21no—
324Dic73—21no—
325Dic73—21no—
326Dic73—41no—
327Dic73—41no—
328Dic73—41no—
329Dic73—41no—
330Dic73—81no—
331#21138—731nomaximal, Sylow
332C2×C73—21no—
333C2×C73—21no—
334C2×C73—42no—
335C2×C73—81no—
336C2×C73#229011yescharacteristic
337C2×C73#295511yescharacteristic
338C2×C73#295511yescharacteristic
339#2943—731no—
340#2922—731no—
341C4×#131—731no—
342#1929—731no—
343#1929—731no—
344#1922—731no—
345#1922—731no—
346#1876—731no—
347#1876D7311yescharacteristic
348#1866—731no—
349#1866—731no—
350#1778—731no—
351#1759—731no—
352#1759—731no—
353#1746—731no—
354C73#2113811yescharacteristic, complemented, Sylow
355C2×#135C2×D7311yescharacteristic
356C2×#131—731no—
357C2×#131C2×D7311yescharacteristic
358C4×#41—731no—
359C4×#41—731no—
360#442—731no—
361#442—731no—
362#441—731no—
363#441—731no—
364#440—731no—
365#440—731no—
366#440—731no—
367#440—731no—
368#433—1461no—
369#433—1461no—
370#431—1461no—
371#431—1461no—
372C2×#108—731no—
373C2×#108—731no—
374#427—731no—
375#424—731no—
376#424—731no—
377C2×#104—731no—
378C2×#104—731no—
379C2×#104C2×D7311yescharacteristic
380C2×#104C2×D7311yescharacteristic
381#416—731no—
382#416—731no—
383#408—731no—
384#408C2×D7311yescharacteristic
385#405—731no—
386#405—731no—
387#405—1461no—
388#405—1461no—
389C4×M16—731no—
390C4×M16—731no—
391C4×M16—1461no—
392C2×#97—731no—
393C2×C4×C8—731no—
394C2×C4×C8—1461no—
395#376—731no—
396#341—731no—
397#341C2×D7311yescharacteristic
398#339—731no—
399#339—731no—
400#339—731no—
401#339—731no—
402#336—731no—
403#336—731no—
404#336—731no—
405#336C2×D7311yescharacteristic
406C2×#41C22×D7311yescharacteristic
407C2×#41C22×D7311yescharacteristic
408#135—22no—
409#135—22no—
410C2×Q16C4×D7311yescharacteristic
411C2×SD16C4×D7311yescharacteristic
412C2×SD16C4×D7311yescharacteristic
413C2×D8C4×D7311yescharacteristic
414#131—22no—
415#131—22no—
416#131—1462no—
417#131—1462no—
418C2×M16—21no—
419C2×M16—731no—
420C2×M16—731no—
421C2×M16—1461no—
422C2×M16C22×D7311yescharacteristic
423C2×M16C22×D7311yescharacteristic
424C22×C8—21no—
425C22×C8—731no—
426C22×C8—731no—
427C22×C8—1461no—
428C22×C8C22×D7311yescharacteristic
429C22×C8C22×D7311yescharacteristic
430C4×Q8—731no—
431C4×Q8—731no—
432C4×Q8—731no—
433C4×Q8—731no—
434C4×D4—731no—
435C4×D4—731no—
436C4×D4—731no—
437C4×D4—731no—
438C4×D4—1461no—
439C4×D4—1461no—
440C4×D4—1461no—
441C4×D4—1461no—
442#117—731no—
443#117—731no—
444#117—1461no—
445#117—1461no—
446C2×C42—731no—
447C2×C42—731no—
448C2×C42—1461no—
449C2×C42—1461no—
450C2×C42C22×D7311yescharacteristic
451#108—1462no—
452#108—1462no—
453#107—731no—
454#107—731no—
455#106—731no—
456#106—731no—
457C4⋊C8—1461no—
458C4⋊C8—1461no—
459C4⋊C8—1461no—
460C4⋊C8—1461no—
461#104—22no—
462#104—22no—
463#104—1462no—
464#104—1462no—
465#103—731no—
466#103—731no—
467#103—731no—
468#103—731no—
469#102—731no—
470#102—731no—
471#102—731no—
472#102—731no—
473C22⋊C8—21no—
474C22⋊C8—21no—
475C22⋊C8—1461no—
476C22⋊C8—1461no—
477#97—731no—
478#97—731no—
479#97—731no—
480#97—731no—
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—1461no—
490C4×C8—1461no—
491C4×C8—1461no—
492C4×C8—1461no—
493#95—1461no—
494#95—1461no—
495#41—21no—
496#41—21no—
497#41—21no—
498#41—21no—
499#41—21no—
500#41—21no—
501#41#1192402712yes—
502#41#1192402712yes—
503C2×Q8C2×C4×D7311yescharacteristic
504C2×Q8C2×C4×D7311yescharacteristic
505C2×D4—21no—
506C2×D4—21no—
507C2×D4C2×C4×D7311yescharacteristic
508C2×D4C2×C4×D7311yescharacteristic
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—731no—
513C22×C4—731no—
514C22×C4—1461no—
515C22×C4—1461no—
516C22×C4—1461no—
517C22×C4—1461no—
518C22×C4—1461no—
519C22×C4C23×D7311yescharacteristic
520Q16—22no—
521SD16—22no—
522SD16—22no—
523D8—22no—
524M16—24no—
525M16—21no—
526M16—21no—
527M16—21no—
528M16—732no—
529M16—1461no—
530M16—1461no—
531M16—1464no—
532M16—1461no—
533M16#1192402712yes—
534C2×C8—24no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—731no—
542C2×C8—731no—
543C2×C8—731no—
544C2×C8—731no—
545C2×C8—1461no—
546C2×C8—1464no—
547C2×C8—1461no—
548C2×C8—1461no—
549C2×C8—1461no—
550C2×C8—1461no—
551C2×C8—1461no—
552C2×C8C2×C4×D7311yescharacteristic
553C2×C8C2×C4×D7311yescharacteristic
554C2×C8D4×D7311yescharacteristic
555C2×C8D4×D7311yescharacteristic
556C4⋊C4—731no—
557C4⋊C4—731no—
558C4⋊C4—731no—
559C4⋊C4—731no—
560C4⋊C4—1461no—
561C4⋊C4—1461no—
562C4⋊C4—1461no—
563C4⋊C4—1461no—
564C22⋊C4—1461no—
565C22⋊C4—1461no—
566C22⋊C4—1461no—
567C22⋊C4—1461no—
568C22⋊C4—1461no—
569C22⋊C4—1461no—
570C42—21no—
571C42—21no—
572C42—731no—
573C42—731no—
574C42—731no—
575C42—731no—
576C42—1461no—
577C42—1461no—
578C42—1461no—
579C42—1461no—
580C42—1461no—
581C42—1461no—
582C42—1461no—
583C42—1461no—
584C23—21no—
585C23—21no—
586C23#41×D7311yescharacteristic
587Q8—21no—
588Q8—21no—
589Q8—21no—
590Q8#1455654712yes—
591D4—22no—
592D4—22no—
593D4—21no—
594D4—21no—
595D4—21no—
596D4—41no—
597D4—41no—
598D4#1455654712yes—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—2922no—
609C2×C4—2922no—
610C2×C4—2922no—
611C2×C4—2922no—
612C2×C4—2921no—
613C2×C4—2921no—
614C2×C4—2921no—
615C2×C4—2921no—
616C2×C4—2921no—
617C2×C4—731no—
618C2×C4—731no—
619C2×C4—731no—
620C2×C4—731no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4C22×C4×D7311yescharacteristic
626C2×C4C2×D4×D7311yescharacteristic
627C2×C4#41×D7311yescharacteristic
628C2×C4C2×#1192402711yescharacteristic
629C2×C4C2×#1192402711yescharacteristic
630C2×C4C2×#1192402711yescharacteristic
631C8—21no—
632C8—21no—
633C8—21no—
634C8—22no—
635C8—21no—
636C8—41no—
637C8—2921no—
638C8—1461no—
639C8—1461no—
640C8—1461no—
641C8—1461no—
642C8—1462no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C4×D4×D7311yescharacteristic
652C22#1798911411yescharacteristic
653C22#1798911811yescharacteristic
654C4—22no—
655C4—41no—
656C4—81no—
657C4—2921no—
658C4—2921no—
659C4—2921no—
660C4—2921no—
661C4—5841no—
662C4—1461no—
663C4—1461no—
664C4—1461no—
665C4—1461no—
666C4C2×#1455654711yescharacteristic
667C4C4×#1192402711yescharacteristic
668C4#1798909311yescharacteristic
669C4#1798909711yescharacteristic
670C2—21no—
671C2—21no—
672C2—42no—
673C2—81no—
674C2#2261237111yescharacteristic
675C2#2261882911yescharacteristic
676C2#2261883111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1876C2
socleC22×C73C2×C4×D4
prime radicalC2×#135×C73C22
cosocleC22×C4×C73C24
hypercenter#1876D73
hypocenterC73#21138
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1876

Complements

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

NH ≅ G/Ntype
2. #22618571C2semidirect
3. #131×Dic73C2semidirect
4. #22610582C2semidirect
5. #22610576C2semidirect
6. #22610531C2semidirect
7. #22610529C2semidirect
9. #22610206C2semidirect
10. #22610204C2semidirect
11. #22609836C2semidirect
12. #22609743C2semidirect
13. #22609743C2semidirect
14. #22609681C2semidirect
19. #17988699C4semidirect
20. #17988699C4semidirect
21. #17988694C4semidirect
22. #17988694C4semidirect
23. #17988688C4semidirect
24. #17988688C4semidirect
25. #17988688C4semidirect
26. #17988688C4semidirect
354. C73#21138semidirect

Maximal subgroups

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

subgroupindex
#226187322
#226185712
#131×Dic732
#226105822
#226105762
#226105312
#226105292
#226102602
#226102062
#226102042
#226098362
#226097432
#226096812
C73×#18762
#2113873