Finite Group Databasebeta

← #33012969

Subgroups — #33012969

Order 18688.

1622 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988500C211yesmaximal, characteristic
2C2×#17988493C211yesmaximal, characteristic, complemented
3C2×C4×#11924005C211yesmaximal, characteristic
4#22608201C211yesmaximal, characteristic, complemented
5#22608196C211yesmaximal, characteristic, complemented
6#22608191C212yesmaximal, complemented
7#22608177C212yesmaximal
8#22608177C211yesmaximal, characteristic
9#22608165C212yesmaximal, complemented
10#22608152C212yesmaximal
11C73×#1434C211yesmaximal, characteristic, complemented
12C23×Dic146C2211yescharacteristic
13C23×C4×D73C2211yescharacteristic
14C22×C4⋊Dic73C2212yes—
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924005C2211yescharacteristic
17C22×#11924005C2211yescharacteristic
18C22×#11924005C2212yes—
19C22×#11924005C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21#17988500C418yescomplemented
22#17988493—24no—
23C4×#11924005—24no—
24C2×#14556498C2212yes—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2211yescharacteristic
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C22×C42×C73C2211yescharacteristic
34C2×#95×C73C2212yes—
35C2×#95×C73C2211yescharacteristic
36C2×#95×C73C2212yes—
37C2×#95×C73C2211yescharacteristic
38C24×D73C2311yescharacteristic
39C22×Dic146—22no—
40C22×Dic146—22no—
41C22×Dic146—42no—
42C22×Dic146—42no—
43C22×Dic146C2×C414yescomplemented
44C23×Dic73C2312yes—
45C23×Dic73C2312yes—
46C23×Dic73C2311yescharacteristic
47C23×Dic73C2311yescharacteristic
48C23×Dic73C2311yescharacteristic
49C22×C4×D73—22no—
50C22×C4×D73—22no—
51C22×C4×D73—42no—
52C22×C4×D73—42no—
53C22×C4×D73Q814yescomplemented
54C2×C4⋊Dic73—24no—
55C2×C4⋊Dic73—22no—
56C2×C4⋊Dic73—42no—
57C2×C4⋊Dic73—42no—
58C2×C4⋊Dic73—42no—
59C2×#11924005—22no—
60C2×#11924005—22no—
61C2×#11924005—24no—
62C2×#11924005—22no—
63C2×#11924005—22no—
64C2×#11924005—22no—
65C2×#11924005—42no—
66C2×#11924005—42no—
67C2×#11924005—42no—
68C2×#11924005—42no—
69C2×#11924005C2×C414yes—
70C2×#11924005C2×C418yes—
71C2×C4×Dic73—22no—
72C2×C4×Dic73—42no—
73#14556498—22no—
74#14556498—22no—
75#14556498—24no—
76#14556498—24no—
77#14556498—42no—
78#14556498—42no—
79#14556498—42no—
80#14556498—42no—
81#14556498C2×C418yescomplemented
82C23×C4×C73C2312yes—
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C23×C4×C73C2311yescharacteristic
87C2×C42×C73—42no—
88C2×C42×C73D414yes—
89#95×C73—24no—
90#95×C73—22no—
91#95×C73—42no—
92#95×C73C2×C414yescomplemented
93C23×D73—21no—
94C23×D73—21no—
95C23×D73—21no—
96C23×D73—21no—
97C23×D73—22no—
98C23×D73—22no—
99C23×D73—22no—
100C23×D73—22no—
101C23×D73C22×C412yes—
102C23×D73C2×Q812yes—
103C23×D73#4112yes—
104C2×Dic146—28no—
105C2×Dic146—48no—
106C2×Dic146—44no—
107C2×Dic146—44no—
108C2×Dic146—48no—
109C22×Dic73—21no—
110C22×Dic73—22no—
111C22×Dic73—21no—
112C22×Dic73—21no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—21no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—22no—
132C22×Dic73—22no—
133C22×Dic73—22no—
134C22×Dic73C22×C412yes—
135C22×Dic73C22×C414yes—
136C22×Dic73C2×Q812yes—
137C22×Dic73#4112yes—
138C2×C4×D73—28no—
139C2×C4×D73—48no—
140C2×C4×D73—44no—
141C2×C4×D73—44no—
142C2×C4×D73—48no—
143C4⋊Dic73—48no—
144C4⋊Dic73—44no—
145#11924005—28no—
146#11924005—28no—
147#11924005—48no—
148#11924005—48no—
149C4×Dic73—44no—
150C24×C73C2411yescharacteristic
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73C22×C414yes—
173C22×C4×C73C22×C412yes—
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2×D412yes—
177C22×C4×C73C2×Q812yes—
178C22×C4×C73#4112yes—
179C22×C4×C73#4112yes—
180C42×C73—44no—
181C23×C73C23×C411yescharacteristic
182C23×C73C22×D411yescharacteristic
183C23×C73C22×Q811yescharacteristic
184C23×C73C2×#4111yescharacteristic
185C23×C73C2×#4111yescharacteristic
186C23×C73C2×#4111yescharacteristic
187C23×C73C2×#4111yescharacteristic
188C23×C73H(5,𝔽2)12yes—
189C23×C73H(5,𝔽2)12yes—
190C23×C73H(5,𝔽2)12yes—
191C23×C732−1+412yes—
192C22×D73—22no—
193C22×D73—22no—
194C22×D73—24no—
195C22×D73—24no—
196C22×D73—22no—
197C22×D73—22no—
198C22×D73—22no—
199C22×D73—22no—
200C22×D73—24no—
201C22×D73—24no—
202C22×D73—24no—
203C22×D73—24no—
204C22×D73—41no—
205C22×D73—41no—
206C22×D73—41no—
207C22×D73—41no—
208C22×D73—42no—
209C22×D73—42no—
210C22×D73—42no—
211C22×D73—42no—
212C22×D73—42no—
213C22×D73—42no—
214C22×D73C4×Q814yes—
215C2×C4×C73—24no—
216C2×C4×C73—22no—
217C2×C4×C73—22no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—24no—
223C2×C4×C73—24no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—42no—
230C2×C4×C73—42no—
231C2×C4×C73—42no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73C4×D414yes—
243C2×C4×C73C22⋊Q814yes—
244C2×Dic73—22no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—24no—
249C2×Dic73—22no—
250C2×Dic73—22no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—41no—
254C2×Dic73—42no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—42no—
260C2×Dic73—41no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C4×D73—416no—
274Dic146—416no—
275C22×C73C2×C4×D411yescharacteristic
276C22×C73C2×C4×Q811yescharacteristic
277C22×C73C4×#4111yescharacteristic
278C22×C73#51812yes—
279C22×C73#52012yes—
280C22×C73C2×C22⋊Q811yescharacteristic
281C22×C73#52511yescharacteristic
282C22×C73#52911yescharacteristic
283C22×C73#53311yescharacteristic
284C22×C73#53411yescharacteristic
285C22×C73#53511yescharacteristic
286C22×C73#53511yescharacteristic
287C22×C73#53611yescharacteristic
288C22×C73#53812yes—
289C22×C73#53812yes—
290C22×C73#53812yes—
291C22×C73#53812yes—
292C22×C73#53912yes—
293C22×C73#54212yes—
294C22×C73#54212yes—
295C22×C73#54212yes—
296C22×C73#54312yes—
297C22×C73#54412yes—
298C2×D73—28no—
299C2×D73—24no—
300C2×D73—42no—
301C2×D73—42no—
302C2×D73—44no—
303C2×D73—42no—
304C2×D73—42no—
305C2×D73—42no—
306C2×D73—42no—
307C2×D73—44no—
308C2×D73—44no—
309C4×C73—24no—
310C4×C73—24no—
311C4×C73—44no—
312C4×C73—42no—
313C4×C73—82no—
314Dic73—44no—
315Dic73—42no—
316Dic73—42no—
317Dic73—42no—
318Dic73—82no—
319#14319—731nomaximal, Sylow
320C2×C73C4×C22⋊Q811yescharacteristic
321C2×C73#232111yescharacteristic
322C2×C73#232611yescharacteristic
323C2×C73#232912yes—
324C2×C73#233712yes—
325C2×C73#262611yescharacteristic
326C2×C73#262811yescharacteristic
327C2×C73#263111yescharacteristic
328C2×C73#263811yescharacteristic
329C2×C73#265312yes—
330C2×C73#266512yes—
331D73—44no—
332C2×#393—731no—
333C2×#391—731no—
334C2×C4×C22⋊C4—731no—
335#1446—731no—
336#1444—731no—
337#1443—732no—
338#1439—732no—
339#1439—731no—
340#1436—732no—
341#1434D7311yescharacteristic, complemented
342C22⋊#95—732no—
343C73#1431911yescharacteristic, complemented, Sylow
344C23×Q8—731no—
345C24×C4—731no—
346C22×C4⋊C4—732no—
347C22×C4⋊C4—731no—
348C22×C22⋊C4—732no—
349C22×C22⋊C4—732no—
350C22×C22⋊C4—731no—
351C22×C22⋊C4—731no—
352C22×C42—731no—
353C22×C42C2×D7311yescharacteristic
354#393—738no—
355#391—1464no—
356C4×C22⋊C4—1464no—
357C2×#95—732no—
358C2×#95—732no—
359C2×#95—732no—
360C2×#95—732no—
361C2×#95—732no—
362C2×#95—732no—
363C2×#95—732no—
364C2×#95—731no—
365C2×#95—731no—
366C2×#95C2×D7312yes—
367C2×#95C2×D7311yescharacteristic
368C2×#95C2×D7312yes—
369C2×#95C2×D7311yescharacteristic
370C25—731no—
371C22×Q8—2922no—
372C22×Q8—2922no—
373C22×Q8—734no—
374C22×Q8—1462no—
375C22×Q8—1462no—
376C23×C4—2922no—
377C23×C4—2922no—
378C23×C4—734no—
379C23×C4—731no—
380C23×C4—732no—
381C23×C4—731no—
382C23×C4—731no—
383C23×C4—732no—
384C23×C4—1462no—
385C23×C4—1462no—
386C23×C4C22×D7312yes—
387C23×C4C22×D7312yes—
388C23×C4C22×D7311yescharacteristic
389C23×C4C22×D7311yescharacteristic
390C23×C4C22×D7311yescharacteristic
391C2×C4⋊C4—2922no—
392C2×C4⋊C4—2922no—
393C2×C4⋊C4—2922no—
394C2×C4⋊C4—1464no—
395C2×C4⋊C4—1462no—
396C2×C22⋊C4—2922no—
397C2×C22⋊C4—2922no—
398C2×C22⋊C4—2922no—
399C2×C22⋊C4—2922no—
400C2×C22⋊C4—738no—
401C2×C22⋊C4—734no—
402C2×C22⋊C4—1464no—
403C2×C22⋊C4—1462no—
404C2×C22⋊C4—1462no—
405C2×C22⋊C4—1462no—
406C2×C22⋊C4—1462no—
407C2×C22⋊C4—1462no—
408C2×C42—42no—
409C2×C42—2922no—
410C2×C42—1462no—
411C2×C42D29214yes—
412#95—24no—
413#95—22no—
414#95—42no—
415#95—2922no—
416#95—2922no—
417#95—2922no—
418#95—2922no—
419#95—738no—
420#95—1462no—
421#95—1462no—
422#95—1464no—
423#95—1464no—
424#95C4×D7314yescomplemented
425C24—732no—
426C24—732no—
427C24—732no—
428C24—1461no—
429C24—1461no—
430C24—1461no—
431C24—1461no—
432C24—1462no—
433C24—1462no—
434C24—1462no—
435C24—1462no—
436C24C23×D7311yescharacteristic
437C2×Q8—2924no—
438C2×Q8—2924no—
439C2×Q8—2928no—
440C2×Q8—2928no—
441C2×Q8—1468no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—2924no—
464C22×C4—2924no—
465C22×C4—2928no—
466C22×C4—2928no—
467C22×C4—732no—
468C22×C4—734no—
469C22×C4—732no—
470C22×C4—732no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1461no—
479C22×C4—1462no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1468no—
497C22×C4C2×C4×D7314yes—
498C22×C4C2×C4×D7312yes—
499C22×C4C2×D29212yes—
500C22×C4C2×D29212yes—
501C22×C4C2×D29212yes—
502C22×C4Q8×D7312yes—
503C22×C4#1192402712yes—
504C22×C4#1192402812yes—
505C4⋊C4—2924no—
506C4⋊C4—2928no—
507C22⋊C4—2928no—
508C22⋊C4—2928no—
509C22⋊C4—1468no—
510C22⋊C4—1468no—
511C42—44no—
512C42—2924no—
513C23—2921no—
514C23—2921no—
515C23—2921no—
516C23—2921no—
517C23—2922no—
518C23—2922no—
519C23—2922no—
520C23—2922no—
521C23—2922no—
522C23—2922no—
523C23—734no—
524C23—1462no—
525C23—1464no—
526C23—1464no—
527C23—1464no—
528C23—1464no—
529C23—1462no—
530C23—1462no—
531C23—1464no—
532C23—1464no—
533C23—1462no—
534C23—1462no—
535C23—1462no—
536C23C22×C4×D7311yescharacteristic
537C23C22×D29211yescharacteristic
538C23C2×Q8×D7311yescharacteristic
539C23#41×D7311yescharacteristic
540C23#41×D7311yescharacteristic
541C23C2×#1192402711yescharacteristic
542C23C2×#1192402811yescharacteristic
543C23#1455663612yes—
544C23#1455663712yes—
545C23#1455663712yes—
546C23#1455663812yes—
547Q8—29216no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—29216no—
576C2×C4—2921no—
577C2×C4—2922no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2922no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—2922no—
594C2×C4—2922no—
595C2×C4—2922no—
596C2×C4—1464no—
597C2×C4—1464no—
598C2×C4—1462no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1464no—
602C2×C4—1464no—
603C2×C4—1462no—
604C2×C4—1462no—
605C2×C4C4×D29214yes—
606C2×C4#1455657114yes—
607C22—2924no—
608C22—2922no—
609C22—2924no—
610C22—2922no—
611C22—2922no—
612C22—2922no—
613C22—2924no—
614C22—2922no—
615C22—2922no—
616C22—1468no—
617C22—1464no—
618C22C2×C4×D29211yescharacteristic
619C22C4×Q8×D7311yescharacteristic
620C22C4×#1192402711yescharacteristic
621C22#1798905912yes—
622C22#1798906712yes—
623C22C2×#1455657111yescharacteristic
624C22#1798909911yescharacteristic
625C22#1798911911yescharacteristic
626C22#1798914111yescharacteristic
627C22#1798914911yescharacteristic
628C22#1798915611yescharacteristic
629C22#1798915611yescharacteristic
630C22#1798916311yescharacteristic
631C22#1798917012yes—
632C22#1798917412yes—
633C22#1798917612yes—
634C22#1798917612yes—
635C22#1798918012yes—
636C22#1798920512yes—
637C22#1798920612yes—
638C22#1798920812yes—
639C22#1798921212yes—
640C22#1798921712yes—
641C4—24no—
642C4—24no—
643C4—44no—
644C4—42no—
645C4—82no—
646C4—2924no—
647C4—2922no—
648C4—2922no—
649C4—2922no—
650C4—5842no—
651C2—2924no—
652C2C4×#1455657111yescharacteristic
653C2#2261259111yescharacteristic
654C2#2261264211yescharacteristic
655C2#2261265812yes—
656C2#2261273812yes—
657C2#2261573911yescharacteristic
658C2#2261575511yescharacteristic
659C2#2261578611yescharacteristic
660C2#2261585111yescharacteristic
661C2#2261599812yes—
662C2#2261610712yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1434C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenter#1434D73
hypocenterC73#14319
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C2×#17988493C2semidirect
4. #22608201C2semidirect
5. #22608196C2semidirect
6. #22608191C2semidirect
9. #22608165C2semidirect
11. C73×#1434C2semidirect
21. #17988500C4semidirect
43. C22×Dic146C2×C4semidirect
53. C22×C4×D73Q8semidirect
81. #14556498C2×C4semidirect
92. #95×C73C2×C4semidirect
341. #1434D73semidirect
343. C73#14319semidirect
424. #95C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179885002
C2×#179884932
C2×C4×#119240052
#226082012
#226081962
#226081912
#226081772
#226081652
#226081522
C73×#14342
#1431973