Finite Group Databasebeta

← #33120296

Subgroups — #33120296

Order 18688.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988874C211yesmaximal, characteristic, complemented
2C2×#17988789C211yesmaximal, characteristic, complemented
3C4×#14556565C211yesmaximal, characteristic
4#22611124C212yesmaximal, complemented
5#22610932C212yesmaximal
6#22610880C211yesmaximal, characteristic, complemented
7#22610855C211yesmaximal, characteristic, complemented
8#22610286C212yesmaximal, complemented
9#22610199C212yesmaximal, complemented
10#22609911C211yesmaximal, characteristic, complemented
11C2×#533×C73C211yesmaximal, characteristic, complemented
12C22×#11924015C2212yes—
13C2×#14556565C2211yescharacteristic
14C2×C4×D292C2211yescharacteristic
15C2×C4×C22⋊D73—21no—
16#17988874—24no—
17#17988789—24no—
18C2×#14556522C2212yes—
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556515C2212yes—
22C2×#14556515C2212yes—
23C2×#14556515C2212yes—
24C2×#14556511C2212yes—
25C2×C22⋊#10596831C2212yes—
26#17988489C2211yescharacteristic
27#17988485C2211yescharacteristic
28#17988478—21no—
29#17988466—21no—
30#17988447C2211yescharacteristic
31C4×C4⋊Dic73C2211yescharacteristic
32C4×#11924005—21no—
33#17988349C2212yes—
34#17988349C2212yes—
35#17988342C2212yes—
36#533×C73—24no—
37#533×C73—24no—
38C2×C4⋊Q8×C73C2211yescharacteristic
39C2×C4⋊Q8×C73C2211yescharacteristic
40C2×#125×C73—21no—
41C2×C22⋊Q8×C73—21no—
42C2×C22⋊Q8×C73C2212yes—
43C2×C4×Q8×C73—21no—
44C2×C4×Q8×C73C2212yes—
45C2×#117×C73—21no—
46C22×C42×C73C2211yescharacteristic
47C22×D292C2311yescharacteristic
48C22×C22⋊D73—21no—
49C22×C4×D73—21no—
50C2×#11924015—24no—
51C2×#11924015—24no—
52C2×#11924015—42no—
53C22×#10596831C2312yes—
54C22×#10596831C2312yes—
55#14556565—22no—
56#14556565—22no—
57C4×D292—22no—
58C4×D292—22no—
59C4×C22⋊D73—44no—
60C2×C4⋊Dic73C2311yescharacteristic
61C2×C4⋊Dic73C2311yescharacteristic
62C2×C4⋊Dic73C2311yescharacteristic
63C2×#11924007—21no—
64C2×C22⋊Dic73—21no—
65C2×#11924005—21no—
66C2×#11924005—21no—
67C2×#11924005—21no—
68C2×#11924005—21no—
69C2×C4×Dic73—21no—
70#14556522—24no—
71C4⋊#10596831—22no—
72C4⋊#10596831—22no—
73#14556515—24no—
74#14556515—24no—
75#14556515—42no—
76#14556511—24no—
77C22⋊#10596831—42no—
78#14556498—21no—
79#14556498—21no—
80#14556498—21no—
81C22×Q8×C73—21no—
82C22×Q8×C73C2312yes—
83C23×C4×C73D411yescharacteristic
84C23×C4×C73D411yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C4⋊Q8×C73—22no—
87C4⋊Q8×C73—22no—
88C4⋊Q8×C73D414yescomplemented
89C4⋊Q8×C73D414yescomplemented
90#125×C73—28no—
91C22⋊Q8×C73—28no—
92C22⋊Q8×C73—24no—
93C22⋊Q8×C73—24no—
94C4×Q8×C73—28no—
95C4×Q8×C73—24no—
96C4×Q8×C73—24no—
97#117×C73—28no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—22no—
102C2×C4⋊C4×C73—22no—
103C2×C4⋊C4×C73C2312yes—
104C2×C22⋊C4×C73—22no—
105C2×C42×C73—22no—
106C2×C42×C73—22no—
107C2×C42×C73—22no—
108C2×C42×C73D411yescharacteristic
109C2×C42×C73D411yescharacteristic
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112C23×D73—21no—
113C2×D292—21no—
114C2×D292—21no—
115C2×D292—21no—
116C2×D292—21no—
117C2×D292—21no—
118C2×D292#4112yes—
119C2×C22⋊D73—24no—
120C2×C22⋊D73—42no—
121C2×C22⋊D73—42no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C2×C4×D73—42no—
126C2×C4×D73—42no—
127C2×C4×D73—42no—
128#11924015—48no—
129C2×#10596831—22no—
130C2×#10596831—22no—
131C2×#10596831—22no—
132C2×#10596831—22no—
133C2×#10596831—22no—
134C2×#10596831—22no—
135C4⋊Dic73—21no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73#4112yes—
141#11924007—42no—
142C22⋊Dic73—42no—
143#11924005—24no—
144#11924005—42no—
145#11924005—42no—
146#11924005—42no—
147C4×Dic73—42no—
148C24×C73C2×D411yescharacteristic
149C2×Q8×C73—24no—
150C2×Q8×C73—22no—
151C2×Q8×C73—22no—
152C2×Q8×C73—22no—
153C2×Q8×C73—22no—
154C2×Q8×C73—22no—
155C2×Q8×C73—42no—
156C2×Q8×C73—42no—
157C2×Q8×C73C2×D414yes—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—21no—
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—41no—
170C22×C4×C73—41no—
171C22×C4×C73SD1614yescomplemented
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2411yescharacteristic
178C4⋊C4×C73—28no—
179C4⋊C4×C73—28no—
180C4⋊C4×C73—24no—
181C4⋊C4×C73—24no—
182C4⋊C4×C73—24no—
183C4⋊C4×C73—22no—
184C4⋊C4×C73C2×D414yes—
185C22⋊C4×C73—28no—
186C42×C73—28no—
187C42×C73—22no—
188C42×C73—22no—
189C42×C73—21no—
190C42×C73—21no—
191C42×C73C2×D412yes—
192C42×C73C2×D412yes—
193C23×C73—21no—
194C23×C73—21no—
195C23×C73—21no—
196C23×C73—21no—
197C23×C73—21no—
198C23×C73—21no—
199C23×C73C2×SD1612yes—
200C23×C73C22×D411yescharacteristic
201C22×D73—22no—
202C22×D73—41no—
203C22×D73—41no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73—41no—
207C22×D73—41no—
208Q8×C73—24no—
209Q8×C73—24no—
210Q8×C73—48no—
211Q8×C73—42no—
212Q8×C73—42no—
213C2×C4×C73—21no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—28no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—44no—
227C2×C4×C73—42no—
228C2×C4×C73—42no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73#12012yes—
236C2×C4×C73#12012yes—
237C2×C4×C73C2×SD1612yes—
238C2×C4×C73C2×SD1612yes—
239C2×C4×C73#13712yes—
240C2×C4×C73C22×D411yescharacteristic
241C2×C4×C73C22×D411yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73H(5,𝔽2)11yescharacteristic
245C2×C4×C732−1+411yescharacteristic
246C22⋊D73—48no—
247C2×Dic73—22no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256D292—22no—
257D292—22no—
258D292—41no—
259D292—41no—
260C4×D73—84no—
261#10596831—42no—
262#10596831—42no—
263C22×C73—22no—
264C22×C73—22no—
265C22×C73—22no—
266C22×C73—41no—
267C22×C73—41no—
268C22×C73—41no—
269C22×C73—41no—
270C22×C73C2×#12011yescharacteristic
271C22×C73#52511yescharacteristic
272C22×C73#53711yescharacteristic
273C22×C73C22×SD1611yescharacteristic
274C22×C73C2×#13711yescharacteristic
275C22×C73#57511yescharacteristic
276C22×C73#57511yescharacteristic
277C2×D73—42no—
278C2×D73—42no—
279C2×D73—42no—
280C2×D73—81no—
281C2×D73—81no—
282C2×D73—81no—
283C2×D73—81no—
284C4×C73—22no—
285C4×C73—22no—
286C4×C73—41no—
287C4×C73—44no—
288C4×C73—41no—
289C4×C73—44no—
290C4×C73#44814yes—
291C4×C73#54611yescharacteristic
292C4×C73#54711yescharacteristic
293C4×C73#54811yescharacteristic
294C4×C73#55011yescharacteristic
295Dic73—42no—
296Dic73—81no—
297Dic73—81no—
298#24151—731nomaximal, Sylow
299C2×C73—42no—
300C2×C73#240711yescharacteristic
301C2×C73C2×#44811yescharacteristic
302C2×C73#300511yescharacteristic
303C2×C73#310111yescharacteristic
304C2×C73#311611yescharacteristic
305C2×C73#324611yescharacteristic
306C2×C73#325211yescharacteristic
307D73—82no—
308C2×#533D7311yescharacteristic, complemented
309C2×#476—731no—
310C2×#461—731no—
311C4×#121—731no—
312#2035—732no—
313#2000—732no—
314#1987—731no—
315#1981—731no—
316#1880—732no—
317#1865—732no—
318#1797—731no—
319C73#2415111yescharacteristic, complemented, Sylow
320C22×SD16—732no—
321#533—24no—
322#533—24no—
323C2×C4⋊Q8C2×D7311yescharacteristic
324C2×C4⋊Q8C2×D7311yescharacteristic
325C2×#125—21no—
326C2×C22⋊Q8—21no—
327C2×C22⋊Q8C2×D7312yes—
328C2×#121—731no—
329C2×C4×Q8—21no—
330C2×C4×Q8C2×D7312yes—
331C2×C4×D4—731no—
332C2×C4×D4—1461no—
333C2×#117—21no—
334C22×C42C2×D7311yescharacteristic
335#476—1464no—
336#461—1464no—
337C2×#106—732no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×#103—732no—
341C2×#103—732no—
342C2×#103—732no—
343C2×#102—732no—
344C2×C22⋊C8—732no—
345#390—731no—
346#389—731no—
347#388—1461no—
348#386—1461no—
349#380—731no—
350C4×C4⋊C4—731no—
351C4×C22⋊C4—1461no—
352#340—732no—
353#340—732no—
354#336—732no—
355C22×Q8—21no—
356C22×Q8C22×D7312yes—
357C22×D4—731no—
358C22×D4—1461no—
359C23×C4—1461no—
360C23×C4C22⋊D7311yescharacteristic
361C23×C4C22⋊D7311yescharacteristic
362C23×C4C22×D7311yescharacteristic
363C2×SD16—2922no—
364C2×SD16—1464no—
365C2×SD16—1464no—
366C22×C8—732no—
367C22×C8—732no—
368C4⋊Q8—22no—
369C4⋊Q8—22no—
370C4⋊Q8C22⋊D7314yescomplemented
371C4⋊Q8C22⋊D7314yescomplemented
372#125—28no—
373C22⋊Q8—28no—
374C22⋊Q8—24no—
375C22⋊Q8—24no—
376#121—1462no—
377#121—1462no—
378C4×Q8—28no—
379C4×Q8—24no—
380C4×Q8—24no—
381C4×D4—2924no—
382C4×D4—1462no—
383C4×D4—1462no—
384#117—28no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—731no—
391C2×C4⋊C4—731no—
392C2×C4⋊C4—731no—
393C2×C4⋊C4—1461no—
394C2×C4⋊C4C22×D7312yes—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—1461no—
397C2×C22⋊C4—1461no—
398C2×C22⋊C4—1461no—
399C2×C22⋊C4—1461no—
400C2×C22⋊C4—1461no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—1461no—
405C2×C42C22⋊D7311yescharacteristic
406C2×C42C22⋊D7311yescharacteristic
407C2×C42C22×D7311yescharacteristic
408C2×C42C22×D7311yescharacteristic
409#106—1464no—
410C4⋊C8—1462no—
411C4⋊C8—1462no—
412#103—2922no—
413#103—1464no—
414#103—1464no—
415#102—1464no—
416C22⋊C8—2922no—
417#95—1461no—
418#95—1461no—
419#95—1461no—
420C24—1461no—
421C24C2×C22⋊D7311yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8C2×C22⋊D7314yes—
431C2×D4—2922no—
432C2×D4—2922no—
433C2×D4—732no—
434C2×D4—1464no—
435C2×D4—1461no—
436C2×D4—1461no—
437C2×D4—1461no—
438C2×D4—1461no—
439C2×D4—1461no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—2922no—
454C22×C4—2922no—
455C22×C4—2922no—
456C22×C4—1461no—
457C22×C4—1461no—
458C22×C4—1461no—
459C22×C4#1192401514yescomplemented
460C22×C4C2×C22⋊D7311yescharacteristic
461C22×C4C2×C22⋊D7311yescharacteristic
462C22×C4C2×C22⋊D7311yescharacteristic
463C22×C4C2×C22⋊D7311yescharacteristic
464C22×C4C2×C22⋊D7311yescharacteristic
465C22×C4C23×D7311yescharacteristic
466SD16—2928no—
467C2×C8—1462no—
468C2×C8—1462no—
469C2×C8—1462no—
470C2×C8—1462no—
471C2×C8—1462no—
472C2×C8—1462no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—2922no—
480C4⋊C4—732no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4C2×C22⋊D7314yes—
487C22⋊C4—28no—
488C22⋊C4—2922no—
489C22⋊C4—2922no—
490C22⋊C4—2922no—
491C22⋊C4—2922no—
492C22⋊C4—1464no—
493C42—28no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—2922no—
499C42C2×C22⋊D7312yes—
500C42C2×C22⋊D7312yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—2921no—
508C23—2921no—
509C23—2921no—
510C23—2921no—
511C23—2921no—
512C23—2921no—
513C23—1462no—
514C23C2×#1192401512yes—
515C23C22×C22⋊D7311yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—48no—
519Q8—42no—
520Q8—42no—
521D4—2928no—
522D4—2921no—
523D4—2921no—
524D4—1462no—
525D4—1462no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—5844no—
557C2×C4—1462no—
558C2×C4#1455656012yes—
559C2×C4#1455656012yes—
560C2×C4C2×#1192401512yes—
561C2×C4C2×#1192401512yes—
562C2×C4#1455662212yes—
563C2×C4C22×C22⋊D7311yescharacteristic
564C2×C4C22×C22⋊D7311yescharacteristic
565C2×C4#41×D7311yescharacteristic
566C2×C4#41×D7311yescharacteristic
567C2×C4#1455663711yescharacteristic
568C2×C4#1455663811yescharacteristic
569C8—2922no—
570C8—2922no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—2922no—
579C22—2922no—
580C22—2922no—
581C22—5841no—
582C22—5841no—
583C22—5841no—
584C22—5841no—
585C22C2×#1455656011yescharacteristic
586C22#1798909411yescharacteristic
587C22#1798916511yescharacteristic
588C22C22×#1192401511yescharacteristic
589C22C2×#1455662211yescharacteristic
590C22#1798938511yescharacteristic
591C22#1798938511yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—44no—
596C4—41no—
597C4—44no—
598C4—2922no—
599C4—5841no—
600C4—5841no—
601C4#1798872614yes—
602C4#1798922511yescharacteristic
603C4#1798923611yescharacteristic
604C4#1798924411yescharacteristic
605C4#1798925611yescharacteristic
606C2—42no—
607C2—5842no—
608C2#2261327011yescharacteristic
609C2C2×#1798872611yescharacteristic
610C2#2261917311yescharacteristic
611C2#2261993511yescharacteristic
612C2#2262008711yescharacteristic
613C2#2262124311yescharacteristic
614C2#2262129811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#533×C73C2
socleC23×C73C22×D4
prime radicalC2×#14556565C22
hypercenterC2×#533D73
hypocenterC73#24151
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#17988874C2semidirect
2. C2×#17988789C2semidirect
4. #22611124C2semidirect
6. #22610880C2semidirect
7. #22610855C2semidirect
8. #22610286C2semidirect
9. #22610199C2semidirect
10. #22609911C2semidirect
11. C2×#533×C73C2semidirect
88. C4⋊Q8×C73D4semidirect
89. C4⋊Q8×C73D4semidirect
171. C22×C4×C73SD16semidirect
308. C2×#533D73semidirect
319. C73#24151semidirect
370. C4⋊Q8C22⋊D73semidirect
371. C4⋊Q8C22⋊D73semidirect
459. C22×C4#11924015semidirect

Maximal subgroups

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

subgroupindex
C2×#179888742
C2×#179887892
C4×#145565652
#226111242
#226109322
#226108802
#226108552
#226102862
#226101992
#226099112
C2×#533×C732
#2415173