Finite Group Databasebeta

← #33089006

Subgroups — #33089006

Order 18688.

966 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988685C211yesmaximal, characteristic
2#22612409C211yesmaximal, characteristic, complemented
3C4×#14556553C211yesmaximal, characteristic
4#22610575C211yesmaximal, characteristic, complemented
5#22610568C211yesmaximal, characteristic, complemented
6#22610526C211yesmaximal, characteristic, complemented
7#22610505C211yesmaximal, characteristic, complemented
8#22610262C211yesmaximal, characteristic
9#22610262C211yesmaximal, characteristic
10#22610193C211yesmaximal, characteristic
11#22610179C211yesmaximal, characteristic
12#22609829C211yesmaximal, characteristic, complemented
13#22609757C211yesmaximal, characteristic
14#22609679C211yesmaximal, characteristic, complemented
15C73×#1763C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×C4×D292C2211yescharacteristic
18C2×#14556553C2211yescharacteristic
19C2×#14556553C2211yescharacteristic
20C2×C42×D73—21no—
21#17988685C418yescomplemented
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556522C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556513C2211yescharacteristic
27C2×#14556513C2211yescharacteristic
28C2×#14556511C2211yescharacteristic
29C2×#14556502C2211yescharacteristic
30#17988494C2211yescharacteristic
31#17988494C2211yescharacteristic
32#17988486C2211yescharacteristic
33#17988486C2211yescharacteristic
34#17988473—21no—
35#17988473—21no—
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4×C4⋊Dic73—21no—
39C4×C4⋊Dic73C2211yescharacteristic
40C4×#11924005—21no—
41#17988434—21no—
42#17988434—21no—
43#17988433—21no—
44C42×Dic73C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C2×C4×Q8×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#340×C73C2211yescharacteristic
60C22×D292C2311yescharacteristic
61C22×C4×D73—21no—
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C2×#11924015—22no—
65C2×#11924015—22no—
66C2×#11924015C2×C414yes—
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C4×D292—22no—
70C4×D292C2×C414yes—
71#14556553—22no—
72#14556553—22no—
73#14556553—22no—
74#14556553—22no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924005—21no—
81C2×#11924005—21no—
82C2×#11924005—21no—
83C2×C4×Dic73—21no—
84C2×C4×Dic73—21no—
85C2×C4×Dic73—21no—
86C2×C4×Dic73D411yescharacteristic
87C2×C4×Dic73D411yescharacteristic
88C2×C4×Dic73D411yescharacteristic
89C2×C4×Dic73D411yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92C42×D73—44no—
93#14556524—22no—
94#14556524C2×C414yescomplemented
95#14556522—22no—
96#14556515C2×C414yescomplemented
97#14556513—22no—
98#14556513—22no—
99#14556511C2×C414yes—
100#14556502C2×C414yescomplemented
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110C22×Q8×C73C2311yescharacteristic
111C22×C8×C73C2311yescharacteristic
112C22×C8×C73C2311yescharacteristic
113C4×Q8×C73—22no—
114C4×Q8×C73C2×C414yescomplemented
115C2×C4⋊C4×C73—21no—
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73—21no—
121C2×C42×C73C2311yescharacteristic
122#103×C73—22no—
123#103×C73—22no—
124#97×C73—22no—
125#95×C73—21no—
126C23×D73—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292—21no—
130C2×D292C22×C412yes—
131C2×D292#4112yes—
132C2×D292#4112yes—
133C22×Dic73—21no—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73C2×D411yescharacteristic
137C22×Dic73C2×D411yescharacteristic
138C22×Dic73C2×D411yescharacteristic
139C22×Dic73C2×D411yescharacteristic
140C2×C4×D73—24no—
141C2×C4×D73—24no—
142C2×C4×D73—24no—
143C2×C4×D73—42no—
144C2×C4×D73—42no—
145C2×C4×D73—42no—
146C2×C4×D73—42no—
147C2×C4×D73—42no—
148C2×C4×D73—42no—
149#11924015—28no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831C22×C412yes—
155C2×#10596831C22×C412yes—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—42no—
162C4⋊Dic73—42no—
163C4⋊Dic73C22×C412yes—
164#11924005—24no—
165#11924005—24no—
166#11924005—24no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—22no—
172C4×Dic73—22no—
173C4×Dic73—41no—
174C4×Dic73—41no—
175C4×Dic73—42no—
176C4×Dic73—42no—
177C4×Dic73—42no—
178C2×Q8×C73—21no—
179C2×Q8×C73—21no—
180C2×Q8×C73—21no—
181C2×Q8×C73—21no—
182C2×Q8×C73—21no—
183C2×Q8×C73C22×C412yes—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73—21no—
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2411yescharacteristic
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C4⋊C4×C73—24no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—21no—
200C4⋊C4×C73—21no—
201C4⋊C4×C73C22×C412yes—
202C4⋊C4×C73#4112yes—
203C4⋊C4×C73#4112yes—
204C42×C73—24no—
205C42×C73—42no—
206C42×C73C22×C412yes—
207C42×C73#4112yes—
208C23×C73C22×D411yescharacteristic
209C22×D73—22no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—41no—
213C22×D73—41no—
214C22×D73—41no—
215C22×D73—41no—
216Q8×C73—22no—
217Q8×C73—22no—
218Q8×C73—41no—
219Q8×C73—41no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73C4×D412yes—
237C2×C4×C73C4×D412yes—
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×C4×C73C2×#4111yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—22no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73AGL(1,ℤ/8)12yes—
264C2×Dic73#13712yes—
265D292—22no—
266D292—22no—
267D292—22no—
268D292#11714yes—
269C4×D73—48no—
270C4×D73—48no—
271C4×D73—48no—
272C8×C73—41no—
273C8×C73—41no—
274#10596831—22no—
275#10596831—22no—
276C22×C73C2×C4×D411yescharacteristic
277C22×C73#52511yescharacteristic
278C22×C73#52511yescharacteristic
279C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C73C2×#13711yescharacteristic
281C22×C73#57511yescharacteristic
282C22×C73#57511yescharacteristic
283C2×D73—24no—
284C2×D73—42no—
285C2×D73—42no—
286C2×D73—42no—
287C2×D73—42no—
288C2×D73—42no—
289C2×D73—42no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—42no—
293C4×C73—42no—
294C4×C73—42no—
295C4×C73—42no—
296C4×C73C2×#11711yescharacteristic
297C4×C73C4×#4111yescharacteristic
298C4×C73#52911yescharacteristic
299C4×C73#52911yescharacteristic
300Dic73—22no—
301Dic73—22no—
302Dic73—41no—
303Dic73—41no—
304Dic73—42no—
305Dic73—81no—
306Dic73—81no—
307#21121—731nomaximal, Sylow
308C2×C73#229011yescharacteristic
309C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
310C2×C73C4×#13711yescharacteristic
311C2×C73#311011yescharacteristic
312C2×C73#311111yescharacteristic
313C2×C73#311411yescharacteristic
314C2×C73#311611yescharacteristic
315D73—44no—
316C2×#440—731no—
317#2295—731no—
318C42×D4—731no—
319#1928—731no—
320#1927—731no—
321#1921—731no—
322#1918—731no—
323#1877—731no—
324#1877—731no—
325#1864—731no—
326#1862—731no—
327#1776—731no—
328#1763D7311yescharacteristic, complemented
329#1762—731no—
330#1745—731no—
331C73#2112111yescharacteristic, complemented, Sylow
332C22×SD16—731no—
333C2×C4×Q8C2×D7311yescharacteristic
334C2×C4×D4—731no—
335C2×C4×D4—731no—
336C2×C4×D4—731no—
337C22×C42—1461no—
338#440—738no—
339C2×#107—731no—
340C2×#107—731no—
341C2×#106—731no—
342C2×#103—731no—
343C2×#103C2×D7311yescharacteristic
344C2×#103C2×D7311yescharacteristic
345C2×#102—731no—
346C2×#102—731no—
347C2×#102—731no—
348C2×#97—731no—
349C2×#97C2×D7311yescharacteristic
350#391—731no—
351#391—731no—
352#389—731no—
353#389—731no—
354#387—1461no—
355#387—1461no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—731no—
359C4×C4⋊C4—1461no—
360C4×C4⋊C4C2×D7311yescharacteristic
361C4×C22⋊C4—1461no—
362C4×C22⋊C4—1461no—
363C4×C22⋊C4—1461no—
364#376—1461no—
365C43—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340C2×D7311yescharacteristic
371#340C2×D7311yescharacteristic
372#336—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376C22×Q8C22×D7311yescharacteristic
377C22×D4—731no—
378C23×C4—1461no—
379C23×C4—1461no—
380C23×C4—1461no—
381C2×SD16—734no—
382C2×SD16—1462no—
383C2×SD16—1462no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8C22×D7311yescharacteristic
387C22×C8C22×D7311yescharacteristic
388C4×Q8—22no—
389C4×Q8C4×D7314yescomplemented
390C4×D4—734no—
391C4×D4—1462no—
392C4×D4—1462no—
393C4×D4—1462no—
394C4×D4—1462no—
395C4×D4—1462no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4C22×D7311yescharacteristic
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—2924no—
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—731no—
414C2×C42—731no—
415C2×C42—731no—
416C2×C42—731no—
417C2×C42—1461no—
418C2×C42—1461no—
419C2×C42—1461no—
420C2×C42C22×D7311yescharacteristic
421#107—734no—
422#107—1462no—
423#106—1462no—
424#103—22no—
425#103—22no—
426#103—734no—
427#102—734no—
428#102—1462no—
429#102—1462no—
430#97—22no—
431#97—734no—
432#95—21no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442C24—1461no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8C2×C4×D7312yes—
449C2×D4—732no—
450C2×D4—732no—
451C2×D4—732no—
452C2×D4—1461no—
453C2×D4—1461no—
454C2×D4—1461no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—2922no—
462C22×C4—2922no—
463C22×C4—2922no—
464C22×C4—2922no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—1464no—
470C22×C4—1464no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1464no—
475C22×C4D4×D7311yescharacteristic
476C22×C4D4×D7311yescharacteristic
477C22×C4C23×D7311yescharacteristic
478SD16—1468no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—732no—
486C2×C8—732no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—2922no—
496C4⋊C4—2922no—
497C4⋊C4—732no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4C2×C4×D7312yes—
504C4⋊C4#1192402812yes—
505C4⋊C4#1192402812yes—
506C22⋊C4—1464no—
507C22⋊C4—1464no—
508C22⋊C4—1464no—
509C42—24no—
510C42—42no—
511C42—2922no—
512C42—2921no—
513C42—2921no—
514C42—2922no—
515C42—2922no—
516C42—1462no—
517C42—1462no—
518C42—1461no—
519C42—1461no—
520C42—1461no—
521C42—1461no—
522C42C2×C4×D7312yes—
523C42#1192402712yes—
524C23—2921no—
525C23—2921no—
526C23—2921no—
527C23—2921no—
528C23—1462no—
529C23—1462no—
530C23—1462no—
531C23C2×D4×D7311yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536D4—734no—
537D4—1462no—
538D4—1462no—
539D4—1462no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—2928no—
557C2×C4—2928no—
558C2×C4—2928no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—732no—
568C2×C4—732no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1462no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4#1455655112yes—
581C2×C4#1455655112yes—
582C2×C4C22×C4×D7311yescharacteristic
583C2×C4#41×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4C2×#1192402711yescharacteristic
586C2×C4C2×#1192402811yescharacteristic
587C2×C4C2×#1192402811yescharacteristic
588C8—41no—
589C8—41no—
590C8—1462no—
591C8—1462no—
592C22—2922no—
593C22—2922no—
594C22—2922no—
595C22—2922no—
596C22—2922no—
597C22—2922no—
598C22—1464no—
599C22C2×#1455655111yescharacteristic
600C22#1798909611yescharacteristic
601C22#1798909611yescharacteristic
602C22AGL(1,ℤ/8)×D7311yescharacteristic
603C22#137×D7311yescharacteristic
604C22#1798938611yescharacteristic
605C22#1798938611yescharacteristic
606C4—22no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—2922no—
613C4—2921no—
614C4—2921no—
615C4—5841no—
616C4—5841no—
617C4—1462no—
618C4—1462no—
619C4#117×D7311yescharacteristic
620C4C4×#1192402811yescharacteristic
621C4#1798911911yescharacteristic
622C4#1798911911yescharacteristic
623C2—2924no—
624C2#2261237711yescharacteristic
625C2#2261874411yescharacteristic
626C2#2261874611yescharacteristic
627C2#2262002811yescharacteristic
628C2#2262004611yescharacteristic
629C2#2262007311yescharacteristic
630C2#2262009111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1763C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1763D73
hypocenterC73#21121
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. #22612409C2semidirect
4. #22610575C2semidirect
5. #22610568C2semidirect
6. #22610526C2semidirect
7. #22610505C2semidirect
12. #22609829C2semidirect
14. #22609679C2semidirect
15. C73×#1763C2semidirect
21. #17988685C4semidirect
94. #14556524C2×C4semidirect
96. #14556515C2×C4semidirect
100. #14556502C2×C4semidirect
114. C4×Q8×C73C2×C4semidirect
328. #1763D73semidirect
331. C73#21121semidirect
389. C4×Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886852
#226124092
C4×#145565532
#226105752
#226105682
#226105262
#226105052
#226102622
#226101932
#226101792
#226098292
#226097572
#226096792
C73×#17632
#2112173