Finite Group Databasebeta

← #33089002

Subgroups — #33089002

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×#17988689C211yesmaximal, characteristic
2#22612408C211yesmaximal, characteristic, complemented
3C42×D292C211yesmaximal, characteristic
4#22610573C211yesmaximal, characteristic, complemented
5#22610566C211yesmaximal, characteristic, complemented
6#22610524C211yesmaximal, characteristic, complemented
7#22610503C211yesmaximal, characteristic, complemented
8#22610264C211yesmaximal, characteristic
9#22610264C211yesmaximal, characteristic
10#22610195C211yesmaximal, characteristic
11#22610181C211yesmaximal, characteristic
12#22609828C211yesmaximal, characteristic, complemented
13#22609762C211yesmaximal, characteristic, complemented
14#22609757C211yesmaximal, characteristic
15C73×#1745C211yesmaximal, characteristic, complemented
16C22×#11924017C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×D292C2211yescharacteristic
19C2×C4×D292C2211yescharacteristic
20C2×C4×D292C2211yescharacteristic
21C2×C42×D73—21no—
22#17988689C418yescomplemented
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29C2×#14556513C2211yescharacteristic
30C2×#14556513C2211yescharacteristic
31C2×#14556513C2211yescharacteristic
32#17988493C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988471—21no—
37#17988471—21no—
38C4×#11924007—21no—
39C4×C4⋊Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C4×#11924005—21no—
44C4×#11924005—21no—
45C4×#11924005—21no—
46#17988433—21no—
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53C2×#97×C73C2211yescharacteristic
54C2×#97×C73C2211yescharacteristic
55C43×C73C2211yescharacteristic
56#336×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×Dic146C2311yescharacteristic
61C22×D292C2311yescharacteristic
62C22×C4×D73—21no—
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C2×#11924017—22no—
66C2×#11924017—22no—
67C2×#11924017C2×C414yes—
68C4×Dic146—22no—
69C4×Dic146C2×C414yescomplemented
70C4×D292—22no—
71C4×D292—22no—
72C4×D292—22no—
73C4×D292—22no—
74C4×D292—22no—
75C4×D292C2×C414yes—
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C42×D73—44no—
94#14556525—22no—
95#14556525C2×C414yescomplemented
96#14556523—22no—
97#14556516—22no—
98#14556516—22no—
99#14556516C2×C414yescomplemented
100#14556513—22no—
101#14556513—22no—
102#14556513C2×C414yes—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C22×C8×C73C2311yescharacteristic
117C2×C42×C73D411yescharacteristic
118C2×C42×C73D411yescharacteristic
119C2×C42×C73D411yescharacteristic
120C2×C42×C73D411yescharacteristic
121C2×C42×C73C2311yescharacteristic
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124#97×C73—22no—
125#97×C73C2×C414yescomplemented
126C23×D73—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—21no—
131C2×Dic146—21no—
132C2×Dic146C22×C412yes—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292—21no—
136C2×D292C22×C412yes—
137C2×D292#4112yes—
138C2×D292#4112yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C2×C4×D73—24no—
147C2×C4×D73—24no—
148C2×C4×D73—24no—
149C2×C4×D73—42no—
150C2×C4×D73—42no—
151C2×C4×D73—42no—
152C2×C4×D73—42no—
153C2×C4×D73—42no—
154C2×C4×D73—42no—
155#11924017—28no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73C22×C412yes—
165C4⋊Dic73C22×C412yes—
166C4⋊Dic73#4112yes—
167C4⋊Dic73#4112yes—
168#11924007—24no—
169#11924007—42no—
170#11924007—42no—
171#11924005—24no—
172#11924005—24no—
173#11924005—24no—
174C4×Dic73—24no—
175C4×Dic73—42no—
176C4×Dic73—42no—
177C4×Dic73—42no—
178C4×Dic73—42no—
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×C73C2411yescharacteristic
186C2×C8×C73—21no—
187C2×C8×C73—21no—
188C2×C8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73C22×C412yes—
197C2×C8×C73C22×C412yes—
198C42×C73—22no—
199C42×C73—22no—
200C42×C73—21no—
201C42×C73—21no—
202C42×C73—21no—
203C42×C73—21no—
204C42×C73—41no—
205C42×C73—41no—
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—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73C4×D412yes—
231C2×C4×C73C4×D412yes—
232C2×C4×C73AGL(1,ℤ/8)12yes—
233C2×C4×C73#13712yes—
234C2×C4×C73C23×C411yescharacteristic
235C2×C4×C73C2×#4111yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C2×Dic73—22no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261D292—22no—
262D292—22no—
263D292—22no—
264D292#11714yes—
265C4×D73—48no—
266C4×D73—48no—
267C4×D73—48no—
268Dic146—22no—
269Dic146—22no—
270Dic146—41no—
271Dic146—41no—
272C8×C73—22no—
273C8×C73—22no—
274C8×C73—41no—
275C8×C73—41no—
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—22no—
293C4×C73—22no—
294C4×C73—41no—
295C4×C73—41no—
296C4×C73C2×#11711yescharacteristic
297C4×C73C4×#4111yescharacteristic
298C4×C73#52911yescharacteristic
299C4×C73#52911yescharacteristic
300Dic73—42no—
301Dic73—42no—
302Dic73—42no—
303Dic73—42no—
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#1763—731no—
329#1762—731no—
330#1745D7311yescharacteristic, complemented
331C73#2112111yescharacteristic, complemented, Sylow
332C22×SD16—731no—
333C2×C4×Q8—731no—
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×#103—731no—
344C2×#103—731no—
345C2×#102—731no—
346C2×#102—731no—
347C2×#102—731no—
348C2×#97C2×D7311yescharacteristic
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—731no—
360C4×C4⋊C4—1461no—
361C4×C22⋊C4—1461no—
362C4×C22⋊C4—1461no—
363C4×C22⋊C4—1461no—
364#376—1461no—
365C43C2×D7311yescharacteristic
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#336C2×D7311yescharacteristic
373#336C2×D7311yescharacteristic
374#336C2×D7311yescharacteristic
375#336C2×D7311yescharacteristic
376C22×Q8—731no—
377C22×D4—731no—
378C23×C4—1461no—
379C23×C4—1461no—
380C23×C4—1461no—
381C2×SD16—734no—
382C2×SD16—1462no—
383C2×SD16—1462no—
384C22×C8C22×D7311yescharacteristic
385C22×C8C22×D7311yescharacteristic
386C22×C8C22×D7311yescharacteristic
387C22×C8C22×D7311yescharacteristic
388C4×Q8—734no—
389C4×Q8—1462no—
390C4×D4—734no—
391C4×D4—1462no—
392C4×D4—1462no—
393C4×D4—1462no—
394C4×D4—1462no—
395C4×D4—1462no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C42—2924no—
409C2×C42—1461no—
410C2×C42—1461no—
411C2×C42—1461no—
412C2×C42—1461no—
413C2×C42—1461no—
414C2×C42D29211yescharacteristic
415C2×C42D29211yescharacteristic
416C2×C42D29211yescharacteristic
417C2×C42D29211yescharacteristic
418C2×C42C22×D7311yescharacteristic
419C2×C42C22×D7311yescharacteristic
420C2×C42C22×D7311yescharacteristic
421#107—734no—
422#107—1462no—
423#106—1462no—
424#103—734no—
425#103—1462no—
426#103—1462no—
427#102—734no—
428#102—1462no—
429#102—1462no—
430#97—22no—
431#97C4×D7314yescomplemented
432#95—1461no—
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—732no—
444C2×Q8—1461no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×D4—732no—
450C2×D4—732no—
451C2×D4—732no—
452C2×D4—1461no—
453C2×D4—1461no—
454C2×D4—1461no—
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—1464no—
462C22×C4—1464no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1464no—
471C22×C4C2×D29211yescharacteristic
472C22×C4C2×D29211yescharacteristic
473C22×C4C2×D29211yescharacteristic
474C22×C4C2×D29211yescharacteristic
475C22×C4C2×D29211yescharacteristic
476C22×C4C2×D29211yescharacteristic
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—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8C2×C4×D7312yes—
490C2×C8C2×C4×D7312yes—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—732no—
494C4⋊C4—732no—
495C4⋊C4—732no—
496C4⋊C4—732no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1464no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C22⋊C4—1464no—
507C22⋊C4—1464no—
508C22⋊C4—1464no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—41no—
517C42—2922no—
518C42—2922no—
519C42—2922no—
520C42—2922no—
521C42—1464no—
522C42C2×C4×D7312yes—
523C42#1192402812yes—
524C23—2921no—
525C23—2921no—
526C23—2921no—
527C23—2921no—
528C23—1462no—
529C23—1462no—
530C23—1462no—
531C23C22×D29211yescharacteristic
532Q8—2921no—
533Q8—2921no—
534Q8—1462no—
535Q8—1462no—
536D4—734no—
537D4—1462no—
538D4—1462no—
539D4—1462no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—2928no—
555C2×C4—2928no—
556C2×C4—2928no—
557C2×C4—2921no—
558C2×C4—2921no—
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—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—1462no—
574C2×C4—1462no—
575C2×C4—1462no—
576C2×C4—1462no—
577C2×C4—1462no—
578C2×C4C4×D29212yes—
579C2×C4C4×D29212yes—
580C2×C4#1455661912yes—
581C2×C4#1455662412yes—
582C2×C4C22×C4×D7311yescharacteristic
583C2×C4#41×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4#41×D7311yescharacteristic
586C2×C4#41×D7311yescharacteristic
587C2×C4C2×#1192402811yescharacteristic
588C8—22no—
589C8—22no—
590C8—41no—
591C8—41no—
592C22—2922no—
593C22—2922no—
594C22—2922no—
595C22—2922no—
596C22—2922no—
597C22—2922no—
598C22—1464no—
599C22C2×C4×D29211yescharacteristic
600C22#1798909911yescharacteristic
601C22#1798909911yescharacteristic
602C22C2×#1455661911yescharacteristic
603C22C2×#1455662411yescharacteristic
604C22#1798938711yescharacteristic
605C22#1798938711yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—2922no—
613C4—2922no—
614C4—2922no—
615C4—2922no—
616C4—2922no—
617C4—5841no—
618C4—5841no—
619C4#117×D7311yescharacteristic
620C4#1798905511yescharacteristic
621C4#1798911511yescharacteristic
622C4#1798911511yescharacteristic
623C2—2924no—
624C2#2261237511yescharacteristic
625C2C4×#1455661911yescharacteristic
626C2C4×#1455662411yescharacteristic
627C2#2262003111yescharacteristic
628C2#2262004211yescharacteristic
629C2#2262007111yescharacteristic
630C2#2262009411yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1745C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1745D73
hypocenterC73#21121
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C43 ◁ #1745

Complements

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

NH ≅ G/Ntype
2. #22612408C2semidirect
4. #22610573C2semidirect
5. #22610566C2semidirect
6. #22610524C2semidirect
7. #22610503C2semidirect
12. #22609828C2semidirect
13. #22609762C2semidirect
15. C73×#1745C2semidirect
22. #17988689C4semidirect
69. C4×Dic146C2×C4semidirect
95. #14556525C2×C4semidirect
99. #14556516C2×C4semidirect
125. #97×C73C2×C4semidirect
330. #1745D73semidirect
331. C73#21121semidirect
431. #97C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886892
#226124082
C42×D2922
#226105732
#226105662
#226105242
#226105032
#226102642
#226101952
#226101812
#226098282
#226097622
#226097572
C73×#17452
#2112173