Finite Group Databasebeta

← #33051895

Subgroups — #33051895

Order 18688.

1210 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989066C211yesmaximal, characteristic
2#22618960C211yesmaximal, characteristic, complemented
3#22608274C212yesmaximal, complemented
4#22608267C212yesmaximal, complemented
5#22608263C212yesmaximal, complemented
6#22608260C212yesmaximal, complemented
7#22608247C212yesmaximal, complemented
8C2×#17988312C212yesmaximal, complemented
9C73×#2973C211yesmaximal, characteristic, complemented
10C22×#11924027C411yescharacteristic
11#17989066—24no—
12#17989066—24no—
13C2×C4×Dic146C2211yescharacteristic
14C2×#14556556C411yescharacteristic
15C2×D4×Dic73C411yescharacteristic
16C2×C4×C22⋊D73C2211yescharacteristic
17C2×C4×C22⋊D73C2211yescharacteristic
18C2×#14556551C411yescharacteristic
19C2×#14556551C411yescharacteristic
20C2×#14556548C2211yescharacteristic
21C2×#14556547C411yescharacteristic
22C2×#14556547C411yescharacteristic
23C22×#11924007C2211yescharacteristic
24C22×#11924007C2211yescharacteristic
25C2×C4⋊C4×D73C411yescharacteristic
26#17988659—24no—
27#17988644C2212yes—
28#17988551C2212yes—
29C2×C22⋊#10596831C2212yes—
30#17988312C2214yes—
31#17988312C2214yes—
32#17988312C2214yes—
33#17988310C2214yes—
34C2×C4×D4×C73C2211yescharacteristic
35#435×C73—24no—
36#432×C73C2212yes—
37#407×C73C2212yes—
38C2×C22⋊C8×C73C2212yes—
39C2×#11924027—24no—
40C2×#11924027—24no—
41C2×#11924027—24no—
42C22×Dic146C2×C411yescharacteristic
43C22×C22⋊D73C2×C411yescharacteristic
44C22×C22⋊D73C2×C411yescharacteristic
45C23×Dic73C2×C411yescharacteristic
46C23×Dic73C2×C411yescharacteristic
47C23×Dic73C2×C411yescharacteristic
48C23×Dic73C2×C411yescharacteristic
49C22×C4×D73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C22×C4×D73C2×C411yescharacteristic
52C2×#11924012—22no—
53C22×#10596831—22no—
54C4×Dic146—22no—
55C4×Dic146—22no—
56#14556556—24no—
57D4×Dic73—24no—
58C4×C22⋊D73—22no—
59C4×C22⋊D73—22no—
60C4×C22⋊D73—22no—
61C4×C22⋊D73—22no—
62#14556551—24no—
63#14556551—24no—
64#14556548—22no—
65#14556548—22no—
66#14556547—24no—
67#14556547—24no—
68C2×C4⋊Dic73C2×C411yescharacteristic
69C2×#11924007—22no—
70C2×#11924007—22no—
71C2×#11924007—22no—
72C2×#11924007—22no—
73C2×#11924007C2×C411yescharacteristic
74C2×#11924007C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2311yescharacteristic
79C2×#11924007C2311yescharacteristic
80C2×#11924007C2311yescharacteristic
81C4⋊C4×D73—24no—
82C2×C22⋊Dic73C2×C411yescharacteristic
83C2×C22⋊Dic73C2×C411yescharacteristic
84C2×#11924005C2×C411yescharacteristic
85C2×#11924005C2311yescharacteristic
86C2×C4×Dic73C2×C411yescharacteristic
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C2×C4×Dic73C2×C411yescharacteristic
90C2×C4×Dic73C2×C411yescharacteristic
91C4⋊#10596831—22no—
92C22⋊#10596831—22no—
93C22⋊#10596831—22no—
94C22⋊#10596831C2314yes—
95#14556502—22no—
96C22×D4×C73C2×C411yescharacteristic
97C23×C4×C73C2311yescharacteristic
98C23×C4×C73C2311yescharacteristic
99C2×M16×C73—22no—
100C22×C8×C73—22no—
101C4×D4×C73D414yes—
102C4×D4×C73D414yes—
103C2×C4⋊C4×C73C2×C411yescharacteristic
104C2×C22⋊C4×C73C2×C411yescharacteristic
105C2×C22⋊C4×C73C2×C411yescharacteristic
106C2×C42×C73C2311yescharacteristic
107C4⋊C8×C73—22no—
108C22⋊C8×C73—22no—
109C22⋊C8×C73—22no—
110C22⋊C8×C73C2314yes—
111#97×C73—22no—
112C23×D73C22×C411yescharacteristic
113#11924027—232no—
114C2×Dic146—22no—
115C2×Dic146—22no—
116C2×Dic146—22no—
117C2×C22⋊D73—22no—
118C2×C22⋊D73—22no—
119C2×C22⋊D73—22no—
120C2×C22⋊D73—22no—
121C2×C22⋊D73—22no—
122C2×C22⋊D73M1614yescomplemented
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—81no—
131C22×Dic73—81no—
132C22×Dic73M1614yescomplemented
133C22×Dic73C22×C411yescharacteristic
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C2×C4×D73—22no—
141C2×C4×D73—22no—
142C2×C4×D73—22no—
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—81no—
146C2×C4×D73—81no—
147#11924012—82no—
148C2×#10596831—24no—
149C2×#10596831—82no—
150C4⋊Dic73—22no—
151#11924007—22no—
152#11924007—22no—
153#11924007—22no—
154#11924007—22no—
155#11924007—22no—
156#11924007—21no—
157#11924007—21no—
158#11924007—21no—
159#11924007—21no—
160#11924007—21no—
161#11924007—21no—
162C22⋊Dic73—22no—
163C22⋊Dic73—22no—
164#11924005—22no—
165#11924005—21no—
166#11924005—21no—
167C4×Dic73—22no—
168C4×Dic73—22no—
169C4×Dic73—22no—
170C4×Dic73—22no—
171C4×Dic73—22no—
172C24×C73C22×C411yescharacteristic
173C24×C73C22×C411yescharacteristic
174C2×D4×C73—22no—
175C2×D4×C73—22no—
176C2×D4×C73C22⋊C414yes—
177C22×C4×C73—81no—
178C22×C4×C73—81no—
179C22×C4×C73C22×C411yescharacteristic
180C22×C4×C73C22×C411yescharacteristic
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C22×C4×C73C2×D412yes—
184C22×C4×C73C2×D412yes—
185C22×C4×C73C2×D412yes—
186C22×C4×C73C2×D412yes—
187C22×C4×C73C2411yescharacteristic
188M16×C73—82no—
189C2×C8×C73—24no—
190C2×C8×C73—82no—
191C4⋊C4×C73C22⋊C414yes—
192C22⋊C4×C73C22⋊C414yes—
193C22⋊C4×C73C22⋊C414yes—
194C42×C73C2×D412yes—
195C42×C73C2×D412yes—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73C2×C22⋊C412yes—
203C23×C73C2×C22⋊C412yes—
204C23×C73C2×M1612yes—
205C23×C73C2×M1612yes—
206C23×C73C23×C411yescharacteristic
207C22×D73—21no—
208C22×D73—21no—
209C22×D73—41no—
210C22×D73—41no—
211C22×D73C2×M1612yes—
212D4×C73—28no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—41no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—41no—
223C2×C4×C73C2×C22⋊C412yes—
224C2×C4×C73C2×C22⋊C412yes—
225C2×C4×C73C2×C22⋊C412yes—
226C2×C4×C73C2×C22⋊C412yes—
227C2×C4×C73C22×D411yescharacteristic
228C2×C4×C73C22×D411yescharacteristic
229C22⋊D73—28no—
230C22⋊D73—28no—
231C2×Dic73—28no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—28no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—82no—
248C2×Dic73—82no—
249C2×Dic73C2×M1612yes—
250C2×Dic73C2×M1612yes—
251C2×Dic73C2×M1612yes—
252C4×D73—28no—
253C4×D73—82no—
254C4×D73—82no—
255Dic146—28no—
256C8×C73—44no—
257#10596831—44no—
258C22×C73—21no—
259C22×C73—22no—
260C22×C73—21no—
261C22×C73—22no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73#40714yes—
270C22×C73C22×C22⋊C411yescharacteristic
271C22×C73C22×M1611yescharacteristic
272C22×C73C22×M1611yescharacteristic
273C2×D73—22no—
274C2×D73—42no—
275C2×D73—41no—
276C2×D73—41no—
277C2×D73—42no—
278C4×C73—22no—
279C4×C73—21no—
280C4×C73—21no—
281C4×C73—22no—
282C4×C73—22no—
283C4×C73—22no—
284Dic73—22no—
285Dic73—22no—
286Dic73—22no—
287Dic73—41no—
288Dic73—41no—
289Dic73—41no—
290Dic73—41no—
291#17368—731nomaximal, Sylow
292C2×C73—21no—
293C2×C73—21no—
294C2×C73—24no—
295C2×C73—42no—
296C2×C73C2×#40711yescharacteristic
297C2×C73#288311yescharacteristic
298C2×C73#288911yescharacteristic
299D73—42no—
300C2×#520—731no—
301#2973—731no—
302#2973D7311yescharacteristic, complemented
303#1467—732no—
304#1465—732no—
305#1464—732no—
306#1463—732no—
307#1459—732no—
308C2×#324—732no—
309C73#1736811yescharacteristic, complemented, Sylow
310C22×#41—731no—
311#520—1464no—
312#520—1464no—
313C2×C4×Q8—731no—
314C2×C4×Q8—731no—
315C2×C4×D4—731no—
316C2×C4×D4—731no—
317C2×C4×D4—731no—
318C2×C4×D4—731no—
319C2×C4×D4—731no—
320C2×C4×D4C2×D7311yescharacteristic
321C2×#117—731no—
322C2×#117—731no—
323C2×#117—731no—
324C22×C4⋊C4—731no—
325C22×C4⋊C4—731no—
326C22×C4⋊C4—731no—
327#435—24no—
328#435—1464no—
329#432—732no—
330#432C2×D7312yes—
331#407—732no—
332#407C2×D7312yes—
333C2×C22⋊C8—732no—
334C2×C22⋊C8C2×D7312yes—
335#324—734no—
336#324—734no—
337#324—734no—
338#323—734no—
339C2×#41—1464no—
340C2×#41—1464no—
341C2×#41—1464no—
342C22×Q8—731no—
343C22×D4—731no—
344C22×D4—731no—
345C22×D4C4×D7311yescharacteristic
346C23×C4—731no—
347C23×C4—731no—
348C23×C4—731no—
349C23×C4—731no—
350C23×C4—731no—
351C23×C4—731no—
352C23×C4—731no—
353C23×C4C22×D7311yescharacteristic
354C23×C4C22×D7311yescharacteristic
355C2×M16—22no—
356C2×M16—1462no—
357C22×C8—22no—
358C22×C8—1462no—
359C4×Q8—1464no—
360C4×Q8—1462no—
361C4×Q8—1462no—
362C4×D4—1464no—
363C4×D4—1462no—
364C4×D4—1462no—
365C4×D4—1462no—
366C4×D4—1462no—
367C4×D4—1464no—
368C4×D4—1464no—
369C4×D4D29214yes—
370C4×D4C22⋊D7314yes—
371#117—1464no—
372#117—1464no—
373#117—1462no—
374#117—1462no—
375C2×C4⋊C4—731no—
376C2×C4⋊C4—731no—
377C2×C4⋊C4—731no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—731no—
384C2×C4⋊C4—1462no—
385C2×C4⋊C4—1462no—
386C2×C4⋊C4—1462no—
387C2×C4⋊C4—1462no—
388C2×C4⋊C4—1464no—
389C2×C4⋊C4C4×D7311yescharacteristic
390C2×C22⋊C4—731no—
391C2×C22⋊C4—731no—
392C2×C22⋊C4—731no—
393C2×C22⋊C4—731no—
394C2×C22⋊C4C4×D7311yescharacteristic
395C2×C22⋊C4C4×D7311yescharacteristic
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—731no—
399C2×C42—731no—
400C2×C42—731no—
401C2×C42C22×D7311yescharacteristic
402C4⋊C8—22no—
403C4⋊C8—1462no—
404C22⋊C8—22no—
405C22⋊C8—22no—
406C22⋊C8—734no—
407C22⋊C8—1462no—
408C22⋊C8—1462no—
409C22⋊C8C22×D7314yes—
410#97—22no—
411#97—1462no—
412C24—731no—
413C24C2×C4×D7311yescharacteristic
414C24C2×C4×D7311yescharacteristic
415#41—14632no—
416C2×Q8—1462no—
417C2×Q8—1462no—
418C2×Q8—1462no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—734no—
422C2×D4—1462no—
423C2×D4—1462no—
424C2×D4—1462no—
425C2×D4—1462no—
426C2×D4—1462no—
427C2×D4#1192400514yes—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—5841no—
431C22×C4—5841no—
432C22×C4—5841no—
433C22×C4—5841no—
434C22×C4—731no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—731no—
438C22×C4—734no—
439C22×C4—731no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4C2×C4×D7311yescharacteristic
455C22×C4C2×C4×D7311yescharacteristic
456C22×C4C2×C4×D7311yescharacteristic
457C22×C4C2×C4×D7311yescharacteristic
458C22×C4C2×C22⋊D7312yes—
459C22×C4C2×C22⋊D7312yes—
460C22×C4C2×D29212yes—
461C22×C4C2×D29212yes—
462C22×C4C23×D7311yescharacteristic
463M16—82no—
464M16—5842no—
465C2×C8—24no—
466C2×C8—82no—
467C2×C8—5842no—
468C2×C8—1464no—
469C4⋊C4—1462no—
470C4⋊C4—1462no—
471C4⋊C4—1462no—
472C4⋊C4—1462no—
473C4⋊C4—1462no—
474C4⋊C4—1462no—
475C4⋊C4—1461no—
476C4⋊C4—1461no—
477C4⋊C4—1461no—
478C4⋊C4—1461no—
479C4⋊C4—1461no—
480C4⋊C4—1461no—
481C4⋊C4#1192400514yes—
482C22⋊C4—1462no—
483C22⋊C4—1462no—
484C22⋊C4—1462no—
485C22⋊C4—1461no—
486C22⋊C4—1461no—
487C22⋊C4#1192400514yes—
488C22⋊C4#1192400514yes—
489C42—1462no—
490C42—1462no—
491C42—1462no—
492C42—1462no—
493C42—1462no—
494C42C2×C22⋊D7312yes—
495C42C2×D29212yes—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—2921no—
503C23—2921no—
504C23—732no—
505C23—1461no—
506C23—1461no—
507C23C2×#1192400512yes—
508C23C2×#1192400512yes—
509C23M16×D7312yes—
510C23M16×D7312yes—
511C23C22×C4×D7311yescharacteristic
512Q8—1468no—
513D4—28no—
514D4—1468no—
515D4—1468no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—5842no—
531C2×C4—5842no—
532C2×C4—5842no—
533C2×C4—5842no—
534C2×C4—732no—
535C2×C4—732no—
536C2×C4—732no—
537C2×C4—1468no—
538C2×C4—1461no—
539C2×C4—1461no—
540C2×C4—1461no—
541C2×C4—1461no—
542C2×C4—1468no—
543C2×C4—1461no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1468no—
547C2×C4—1461no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4C2×#1192400512yes—
551C2×C4C2×#1192400512yes—
552C2×C4C2×#1192400512yes—
553C2×C4C2×#1192400512yes—
554C2×C4C22×C22⋊D7311yescharacteristic
555C2×C4C22×D29211yescharacteristic
556C8—44no—
557C8—2924no—
558C22—21no—
559C22—22no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—2922no—
570C22—2922no—
571C22—2921no—
572C22—2921no—
573C22—1462no—
574C22#1798855014yes—
575C22C22×#1192400511yescharacteristic
576C22C2×M16×D7311yescharacteristic
577C22C2×M16×D7311yescharacteristic
578C4—21no—
579C4—21no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—2921no—
585C4—2921no—
586C4—2921no—
587C4—2921no—
588C4—1462no—
589C4—1462no—
590C4—1462no—
591C2—21no—
592C2—21no—
593C2—24no—
594C2—42no—
595C2—2922no—
596C2C2×#1798855011yescharacteristic
597C2#2261839111yescharacteristic
598C2#2261841311yescharacteristic
subgroupG/H
centerC22C22×#11924005
commutatorC23×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2973C2
socleC22×C73C22×C22⋊C4
prime radicalC22×#11924027C4
cosocleC22×C4×C73C24
hypercenter#2973D73
hypocenterC73#17368
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #22618960C2semidirect
3. #22608274C2semidirect
4. #22608267C2semidirect
5. #22608263C2semidirect
6. #22608260C2semidirect
7. #22608247C2semidirect
8. C2×#17988312C2semidirect
9. C73×#2973C2semidirect
122. C2×C22⋊D73M16semidirect
132. C22×Dic73M16semidirect
302. #2973D73semidirect
309. C73#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#179890662
#226189602
#226082742
#226082672
#226082632
#226082602
#226082472
C2×#179883122
C73×#29732
#1736873