Finite Group Databasebeta

← #33089047

Subgroups — #33089047

Order 18688.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988689C211yesmaximal, characteristic
2#22612395C211yesmaximal, characteristic
3C42×Dic146C211yesmaximal, characteristic, complemented
4#22610559C211yesmaximal, characteristic
5#22610552C211yesmaximal, characteristic
6#22610524C211yesmaximal, characteristic, complemented
7#22610503C211yesmaximal, characteristic, complemented
8#22610285C211yesmaximal, characteristic
9#22610285C211yesmaximal, characteristic
10#22610202C211yesmaximal, characteristic, complemented
11#22610188C211yesmaximal, characteristic, complemented
12#22609828C211yesmaximal, characteristic, complemented
13#22609762C211yesmaximal, characteristic, complemented
14#22609757C211yesmaximal, characteristic
15C73×#1745C211yesmaximal, characteristic, complemented
16C22×#11924017C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×D292C2211yescharacteristic
21C2×#14556548—21no—
22#17988689C418yescomplemented
23C2×#14556525C2211yescharacteristic
24C2×#14556525C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28C2×#14556516C2211yescharacteristic
29C2×#14556513C2211yescharacteristic
30C2×#14556513C2211yescharacteristic
31C2×#14556513C2211yescharacteristic
32#17988489C2211yescharacteristic
33#17988489C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988485C2211yescharacteristic
36#17988478—21no—
37#17988478—21no—
38C4×#11924007—21no—
39C4×#11924007—21no—
40C4×#11924007—21no—
41C4×C4⋊Dic73C2211yescharacteristic
42C4×C4⋊Dic73C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45C4×#11924005—21no—
46C42×Dic73—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—
63C2×#11924017—22no—
64C2×#11924017—22no—
65C2×#11924017C2×C414yes—
66C4×Dic146—22no—
67C4×Dic146—22no—
68C4×Dic146—22no—
69C4×Dic146—22no—
70C4×Dic146—22no—
71C4×Dic146C2×C414yes—
72C4×D292—22no—
73C4×D292C2×C414yes—
74#14556548—44no—
75C2×C4⋊Dic73C2311yescharacteristic
76C2×C4⋊Dic73C2311yescharacteristic
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924005—21no—
87C2×#11924005—21no—
88C2×#11924005—21no—
89C2×#11924005—21no—
90C2×#11924005—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73—21no—
97C2×C4×Dic73—21no—
98#14556525—22no—
99#14556525C2×C414yes—
100#14556523—22no—
101#14556516—22no—
102#14556516—22no—
103#14556516C2×C414yes—
104#14556513—22no—
105#14556513—22no—
106#14556513C2×C414yes—
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×C414yes—
126C23×D73—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146C22×C412yes—
131C2×Dic146#4112yes—
132C2×Dic146#4112yes—
133C2×D292—21no—
134C2×D292—21no—
135C2×D292—21no—
136C2×D292—21no—
137C2×D292—21no—
138C2×D292C22×C412yes—
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—42no—
148C2×C4×D73—42no—
149#11924017—28no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73C22×C412yes—
159C4⋊Dic73C22×C412yes—
160C4⋊Dic73#4112yes—
161C4⋊Dic73#4112yes—
162#11924007—24no—
163#11924007—24no—
164#11924007—24no—
165#11924007—42no—
166#11924007—42no—
167#11924005—24no—
168#11924005—42no—
169#11924005—42no—
170#11924005—42no—
171#11924005—42no—
172C4×Dic73—24no—
173C4×Dic73—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—41no—
211C22×D73—41no—
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—22no—
246C2×Dic73—22no—
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—41no—
264D292—41no—
265C4×D73—48no—
266Dic146—22no—
267Dic146—22no—
268Dic146—22no—
269Dic146#11714yescomplemented
270C8×C73—22no—
271C8×C73—22no—
272C8×C73—41no—
273C8×C73—41no—
274C22×C73C2×C4×D411yescharacteristic
275C22×C73#52511yescharacteristic
276C22×C73#52511yescharacteristic
277C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C73C2×#13711yescharacteristic
279C22×C73#57511yescharacteristic
280C22×C73#57511yescharacteristic
281C2×D73—42no—
282C2×D73—42no—
283C2×D73—42no—
284C2×D73—81no—
285C2×D73—81no—
286C2×D73—81no—
287C2×D73—81no—
288C4×C73—22no—
289C4×C73—22no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73C2×#11711yescharacteristic
295C4×C73C4×#4111yescharacteristic
296C4×C73#52911yescharacteristic
297C4×C73#52911yescharacteristic
298Dic73—24no—
299Dic73—42no—
300Dic73—42no—
301Dic73—42no—
302Dic73—42no—
303Dic73—81no—
304Dic73—81no—
305#21124—731nomaximal, Sylow
306C2×C73#229011yescharacteristic
307C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
308C2×C73C4×#13711yescharacteristic
309C2×C73#311111yescharacteristic
310C2×C73#311211yescharacteristic
311C2×C73#311511yescharacteristic
312C2×C73#311711yescharacteristic
313D73—82no—
314C2×#440—731no—
315#2293—731no—
316C42×Q8—731no—
317#1926—731no—
318#1925—731no—
319#1921—731no—
320#1918—731no—
321#1880—731no—
322#1880—731no—
323#1865—731no—
324#1863—731no—
325#1776—731no—
326#1763—731no—
327#1762—731no—
328#1745D7311yescharacteristic, complemented
329C73#2112411yescharacteristic, complemented, Sylow
330C22×SD16—731no—
331C2×C4×Q8—731no—
332C2×C4×Q8—731no—
333C2×C4×Q8—731no—
334C2×C4×D4—731no—
335C2×#117—1461no—
336#440—738no—
337C2×#107—731no—
338C2×#107—731no—
339C2×#106—731no—
340C2×#103—731no—
341C2×#103—731no—
342C2×#103—731no—
343C2×#102—731no—
344C2×#102—731no—
345C2×#102—731no—
346C2×#97C2×D7311yescharacteristic
347C2×#97C2×D7311yescharacteristic
348#390—731no—
349#390—731no—
350#389—731no—
351#389—731no—
352#388—1461no—
353#388—1461no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—731no—
357C4×C4⋊C4—731no—
358C4×C4⋊C4—1461no—
359C4×C4⋊C4—1461no—
360C4×C4⋊C4—1461no—
361C4×C22⋊C4—1461no—
362C43—1461no—
363C43C2×D7311yescharacteristic
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#336C2×D7311yescharacteristic
371#336C2×D7311yescharacteristic
372#336C2×D7311yescharacteristic
373#336C2×D7311yescharacteristic
374C22×Q8—731no—
375C22×D4—731no—
376C23×C4—1461no—
377C2×SD16—734no—
378C2×SD16—1462no—
379C2×SD16—1462no—
380C22×C8C22×D7311yescharacteristic
381C22×C8C22×D7311yescharacteristic
382C22×C8C22×D7311yescharacteristic
383C22×C8C22×D7311yescharacteristic
384C4×Q8—734no—
385C4×Q8—1462no—
386C4×Q8—1462no—
387C4×Q8—1462no—
388C4×Q8—1462no—
389C4×Q8—1462no—
390C4×D4—734no—
391C4×D4—1462no—
392#117—2924no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C22⋊C4—1461no—
405C2×C22⋊C4—1461no—
406C2×C22⋊C4—1461no—
407C2×C22⋊C4—1461no—
408C2×C22⋊C4—1461no—
409C2×C42—1461no—
410C2×C42—1461no—
411C2×C42—1461no—
412C2×C42—1461no—
413C2×C42—1461no—
414C2×C42—1461no—
415C2×C42—1461no—
416C2×C42D29211yescharacteristic
417C2×C42D29211yescharacteristic
418C2×C42D29211yescharacteristic
419C2×C42D29211yescharacteristic
420C2×C42C22×D7311yescharacteristic
421C2×C42C22×D7311yescharacteristic
422C2×C42C22×D7311yescharacteristic
423#107—734no—
424#107—1462no—
425#106—1462no—
426#103—734no—
427#103—1462no—
428#103—1462no—
429#102—734no—
430#102—1462no—
431#102—1462no—
432#97—22no—
433#97C4×D7314yes—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95—1461no—
440C24—1461no—
441C2×Q8—732no—
442C2×Q8—732no—
443C2×Q8—732no—
444C2×Q8—1461no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×D4—732no—
448C2×D4—1461no—
449C2×D4—1461no—
450C2×D4—1461no—
451C2×D4—1461no—
452C2×D4—1461no—
453C22×C4—2922no—
454C22×C4—2922no—
455C22×C4—1461no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1464no—
462C22×C4—1461no—
463C22×C4C2×D29211yescharacteristic
464C22×C4C2×D29211yescharacteristic
465C22×C4C2×D29211yescharacteristic
466C22×C4C2×D29211yescharacteristic
467C22×C4C2×D29211yescharacteristic
468C22×C4C2×D29211yescharacteristic
469C22×C4C23×D7311yescharacteristic
470SD16—1468no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8C2×C4×D7312yes—
482C2×C8C2×C4×D7312yes—
483C4⋊C4—2922no—
484C4⋊C4—2922no—
485C4⋊C4—732no—
486C4⋊C4—732no—
487C4⋊C4—732no—
488C4⋊C4—732no—
489C4⋊C4—1464no—
490C4⋊C4—1464no—
491C4⋊C4—1461no—
492C4⋊C4—1461no—
493C4⋊C4—1461no—
494C4⋊C4—1461no—
495C4⋊C4—1464no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C22⋊C4—2922no—
501C22⋊C4—2922no—
502C22⋊C4—2922no—
503C22⋊C4—2922no—
504C22⋊C4—1464no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—41no—
513C42—2922no—
514C42—2922no—
515C42—2922no—
516C42—2922no—
517C42—1464no—
518C42—1464no—
519C42—1464no—
520C42C2×C4×D7312yes—
521C42#1192402812yes—
522C23—2921no—
523C23—2921no—
524C23—2921no—
525C23—2921no—
526C23—2921no—
527C23—2921no—
528C23—1462no—
529C23C22×D29211yescharacteristic
530Q8—734no—
531Q8—1462no—
532Q8—1462no—
533Q8—1462no—
534D4—2921no—
535D4—2921no—
536D4—1462no—
537D4—1462no—
538C2×C4—21no—
539C2×C4—21no—
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—2928no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
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—1462no—
568C2×C4—1462no—
569C2×C4—1462no—
570C2×C4—1462no—
571C2×C4—1462no—
572C2×C4—1462no—
573C2×C4—1462no—
574C2×C4C4×D29212yes—
575C2×C4C4×D29212yes—
576C2×C4#1455661912yes—
577C2×C4#1455662412yes—
578C2×C4C22×C4×D7311yescharacteristic
579C2×C4#41×D7311yescharacteristic
580C2×C4#41×D7311yescharacteristic
581C2×C4#41×D7311yescharacteristic
582C2×C4#41×D7311yescharacteristic
583C2×C4C2×#1192402811yescharacteristic
584C8—22no—
585C8—22no—
586C8—41no—
587C8—41no—
588C22—2922no—
589C22—2922no—
590C22—2922no—
591C22—5841no—
592C22—5841no—
593C22—5841no—
594C22—5841no—
595C22C2×C4×D29211yescharacteristic
596C22#1798909911yescharacteristic
597C22#1798909911yescharacteristic
598C22C2×#1455661911yescharacteristic
599C22C2×#1455662411yescharacteristic
600C22#1798938711yescharacteristic
601C22#1798938711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—2922no—
609C4—2922no—
610C4—2922no—
611C4—2922no—
612C4—5841no—
613C4—5841no—
614C4—1464no—
615C4#117×D7311yescharacteristic
616C4#1798905511yescharacteristic
617C4#1798911511yescharacteristic
618C4#1798911511yescharacteristic
619C2—5842no—
620C2#2261237511yescharacteristic
621C2C4×#1455661911yescharacteristic
622C2C4×#1455662411yescharacteristic
623C2#2262004211yescharacteristic
624C2#2262005411yescharacteristic
625C2#2262008311yescharacteristic
626C2#2262010411yescharacteristic
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#21124
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
3. C42×Dic146C2semidirect
6. #22610524C2semidirect
7. #22610503C2semidirect
10. #22610202C2semidirect
11. #22610188C2semidirect
12. #22609828C2semidirect
13. #22609762C2semidirect
15. C73×#1745C2semidirect
22. #17988689C4semidirect
269. Dic146#117semidirect
328. #1745D73semidirect
329. C73#21124semidirect

Maximal subgroups

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

subgroupindex
C2×#179886892
#226123952
C42×Dic1462
#226105592
#226105522
#226105242
#226105032
#226102852
#226102022
#226101882
#226098282
#226097622
#226097572
C73×#17452
#2112473