Finite Group Databasebeta

← #33119204

Subgroups — #33119204

Order 18688.

1338 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×#17988915C211yesmaximal, characteristic, complemented
2C2×#17988588C211yesmaximal, characteristic, complemented
3#22614248C211yesmaximal, characteristic, complemented
4#22611174C212yesmaximal, complemented
5#22611124C212yesmaximal, complemented
6#22610365C211yesmaximal, characteristic, complemented
7#22610229C211yesmaximal, characteristic, complemented
8#22610192C212yesmaximal, complemented
9#22609871C212yesmaximal, complemented
10#22609871C211yesmaximal, characteristic, complemented
11C2×#547×C73C211yesmaximal, characteristic
12C22×#11924012C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14#17988915—24no—
15C2×#14556522C2212yes—
16#17988588—24no—
17C2×#14556515C2212yes—
18C2×#14556515C2212yes—
19C2×#14556515C2212yes—
20C2×#14556515C2212yes—
21C2×#14556512C2212yes—
22C2×#14556512C2212yes—
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2212yes—
25#17988526—21no—
26#17988523—21no—
27#17988485C2212yes—
28#17988467—21no—
29#17988459—21no—
30#17988447C2212yes—
31#17988447C2211yescharacteristic
32#17988349C2212yes—
33#17988342C2212yes—
34C22×#41×C73C2211yescharacteristic
35C22×#41×C73C2211yescharacteristic
36#547×C73—24no—
37#547×C73—24no—
38C2×C4⋊Q8×C73C2211yescharacteristic
39C2×#123×C73—21no—
40C2×#123×C73—21no—
41C2×C22⋊Q8×C73C2211yescharacteristic
42C2×C22⋊Q8×C73C2211yescharacteristic
43C2×#121×C73—21no—
44C2×C4×D4×C73C2211yescharacteristic
45C2×C4×D4×C73C2211yescharacteristic
46C22×C4⋊C4×C73—21no—
47C23×Dic73—21no—
48C2×#11924012—22no—
49C2×#11924012—42no—
50C22×#10596831C2312yes—
51C22×#10596831C2312yes—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—21no—
55C2×C4⋊Dic73C2312yes—
56C2×C4⋊Dic73C2312yes—
57C2×#11924007—22no—
58C2×C22⋊Dic73—21no—
59C2×C22⋊Dic73—22no—
60#14556522—24no—
61#14556515—24no—
62#14556515—24no—
63#14556515—42no—
64#14556515—42no—
65#14556512—24no—
66#14556512—24no—
67C22⋊#10596831—22no—
68C22⋊#10596831—42no—
69#14556498—22no—
70#14556498—21no—
71#14556498—22no—
72C2×#41×C73—22no—
73C2×#41×C73—22no—
74C2×#41×C73—22no—
75C2×#41×C73—24no—
76C2×#41×C73—24no—
77C2×#41×C73—42no—
78C2×#41×C73—42no—
79C2×#41×C73D414yes—
80C22×Q8×C73C2311yescharacteristic
81C22×Q8×C73C2311yescharacteristic
82C22×D4×C73—21no—
83C22×D4×C73—21no—
84C22×D4×C73D411yescharacteristic
85C22×D4×C73C2311yescharacteristic
86C23×C4×C73—21no—
87C23×C4×C73—21no—
88C23×C4×C73—21no—
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2311yescharacteristic
91C4⋊Q8×C73—22no—
92C4⋊Q8×C73—22no—
93#123×C73—28no—
94#123×C73—28no—
95C22⋊Q8×C73—22no—
96C22⋊Q8×C73—22no—
97C22⋊Q8×C73—22no—
98C22⋊Q8×C73D414yes—
99#121×C73—28no—
100C4×D4×C73—22no—
101C4×D4×C73—22no—
102C4×D4×C73—22no—
103C4×D4×C73—22no—
104C2×C4⋊C4×C73—28no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73—21no—
108C2×C4⋊C4×C73—21no—
109C2×C4⋊C4×C73D411yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73—21no—
113C2×C22⋊C4×C73—21no—
114C2×C22⋊C4×C73D411yescharacteristic
115C2×C22⋊C4×C73D411yescharacteristic
116C2×C42×C73C2311yescharacteristic
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—42no—
120C22×Dic73—82no—
121#11924012—44no—
122C2×#10596831—22no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C2×#10596831—22no—
127C2×#10596831—22no—
128C4⋊Dic73—22no—
129C4⋊Dic73—22no—
130C4⋊Dic73—22no—
131C4⋊Dic73—22no—
132C4⋊Dic73—42no—
133#11924007—82no—
134C22⋊Dic73—42no—
135C22⋊Dic73—82no—
136C24×C73—21no—
137C24×C73C2×D411yescharacteristic
138C24×C73C2×D411yescharacteristic
139#41×C73—28no—
140#41×C73—44no—
141#41×C73—48no—
142#41×C73—48no—
143#41×C73—44no—
144#41×C73—44no—
145C2×Q8×C73—21no—
146C2×Q8×C73—21no—
147C2×Q8×C73—22no—
148C2×Q8×C73—22no—
149C2×Q8×C73—22no—
150C2×Q8×C73—22no—
151C2×Q8×C73C2×D412yes—
152C2×Q8×C73C2×D412yes—
153C2×D4×C73—24no—
154C2×D4×C73—22no—
155C2×D4×C73—24no—
156C2×D4×C73—22no—
157C2×D4×C73—21no—
158C2×D4×C73—22no—
159C2×D4×C73—41no—
160C2×D4×C73—44no—
161C2×D4×C73—44no—
162C2×D4×C73C2×D412yes—
163C2×D4×C73#4114yes—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—24no—
167C22×C4×C73—24no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—21no—
171C22×C4×C73—24no—
172C22×C4×C73—21no—
173C22×C4×C73—21no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—44no—
177C22×C4×C73—44no—
178C22×C4×C73—42no—
179C22×C4×C73—42no—
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D412yes—
184C22×C4×C73C2×D412yes—
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2411yescharacteristic
187C4⋊C4×C73—24no—
188C4⋊C4×C73—24no—
189C4⋊C4×C73—24no—
190C4⋊C4×C73—22no—
191C4⋊C4×C73—24no—
192C4⋊C4×C73—21no—
193C4⋊C4×C73C2×D412yes—
194C22⋊C4×C73—24no—
195C22⋊C4×C73—24no—
196C22⋊C4×C73—24no—
197C22⋊C4×C73—22no—
198C22⋊C4×C73—22no—
199C42×C73—21no—
200C42×C73—21no—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73—24no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—22no—
207C23×C73—21no—
208C23×C73—22no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—22no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73—42no—
215C23×C73#12012yes—
216C23×C73C22×D411yescharacteristic
217Q8×C73—22no—
218Q8×C73—22no—
219Q8×C73—24no—
220Q8×C73—41no—
221Q8×C73—41no—
222Q8×C73—42no—
223D4×C73—22no—
224D4×C73—24no—
225D4×C73—44no—
226D4×C73—48no—
227D4×C73—48no—
228D4×C73#12314yes—
229C2×C4×C73—21no—
230C2×C4×C73—22no—
231C2×C4×C73—21no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—24no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—21no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—48no—
243C2×C4×C73—48no—
244C2×C4×C73—48no—
245C2×C4×C73—42no—
246C2×C4×C73—42no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—42no—
252C2×C4×C73#12012yes—
253C2×C4×C73#13712yes—
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C22×D411yescharacteristic
256C2×C4×C73C2×#4112yes—
257C2×C4×C73H(5,𝔽2)12yes—
258C2×Dic73—42no—
259C2×Dic73—42no—
260C2×Dic73—42no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—84no—
265#10596831—42no—
266#10596831—42no—
267C22×C73—22no—
268C22×C73—24no—
269C22×C73—22no—
270C22×C73—22no—
271C22×C73—42no—
272C22×C73—44no—
273C22×C73—42no—
274C22×C73—42no—
275C22×C73—42no—
276C22×C73—44no—
277C22×C73—42no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—42no—
282C22×C73C2×#12011yescharacteristic
283C22×C73#54612yes—
284C22×C73C2×#13711yescharacteristic
285C22×C73#57511yescharacteristic
286C22×C73#57712yes—
287C4×C73—22no—
288C4×C73—22no—
289C4×C73—41no—
290C4×C73—42no—
291C4×C73—42no—
292C4×C73—42no—
293C4×C73—41no—
294C4×C73C2×#12311yescharacteristic
295C4×C73#53411yescharacteristic
296C4×C73#54612yes—
297Dic73—82no—
298Dic73—82no—
299#24055—731nomaximal, Sylow
300C2×C73—42no—
301C2×C73—44no—
302C2×C73—81no—
303C2×C73#240511yescharacteristic
304C2×C73#300711yescharacteristic
305C2×C73#301011yescharacteristic
306C2×C73#330712yes—
307C2×C73#333112yes—
308C2×#547D7311yescharacteristic
309C2×#483—731no—
310C2×#417—731no—
311#2487—731no—
312#2043—732no—
313#2035—732no—
314#1894—731no—
315#1870—731no—
316#1864—732no—
317#1788—732no—
318#1788—731no—
319C73#2405511yescharacteristic, complemented, Sylow
320C22×#41C2×D7311yescharacteristic
321C22×#41C2×D7311yescharacteristic
322C22×M16—731no—
323#547—24no—
324#547—24no—
325C2×C4⋊Q8C2×D7311yescharacteristic
326C2×#123—21no—
327C2×#123—21no—
328C2×C22⋊Q8C2×D7311yescharacteristic
329C2×C22⋊Q8C2×D7311yescharacteristic
330C2×#121—21no—
331C2×C4×D4C2×D7311yescharacteristic
332C2×C4×D4C2×D7311yescharacteristic
333C22×C4⋊C4—21no—
334C22×C4⋊C4—731no—
335#483—1464no—
336C2×#106—732no—
337#417—1464no—
338C2×#103—732no—
339C2×#103—732no—
340C2×#103—732no—
341C2×#103—732no—
342C2×#102—732no—
343C2×#102—732no—
344C2×C22⋊C8—732no—
345C2×C22⋊C8—731no—
346#400—1461no—
347#399—1461no—
348#389—732no—
349#386—1461no—
350#385—1461no—
351#380—731no—
352#380—732no—
353#340—732no—
354#336—732no—
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—24no—
359C2×#41—24no—
360C2×#41—42no—
361C2×#41—42no—
362C2×#41C22⋊D7314yes—
363C22×Q8C22×D7311yescharacteristic
364C22×Q8C22×D7311yescharacteristic
365C22×D4—21no—
366C22×D4—21no—
367C22×D4C22⋊D7311yescharacteristic
368C22×D4C22×D7311yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4—21no—
372C23×C4—1461no—
373C23×C4C22×D7311yescharacteristic
374C23×C4C22×D7311yescharacteristic
375C2×M16—2922no—
376C2×M16—1462no—
377C22×C8—732no—
378C22×C8—732no—
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381#123—28no—
382#123—28no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8C22⋊D7314yes—
387#121—28no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—732no—
398C2×C4⋊C4—732no—
399C2×C4⋊C4—1461no—
400C2×C4⋊C4—1462no—
401C2×C4⋊C4—1462no—
402C2×C4⋊C4—1462no—
403C2×C4⋊C4C22⋊D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1461no—
409C2×C22⋊C4—1462no—
410C2×C22⋊C4C22⋊D7311yescharacteristic
411C2×C22⋊C4C22⋊D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413#106—1464no—
414#103—2922no—
415#103—2922no—
416#103—1464no—
417#103—1464no—
418#102—1464no—
419#102—1464no—
420C22⋊C8—2922no—
421C22⋊C8—1462no—
422#95—1462no—
423#95—1461no—
424#95—1462no—
425C24—21no—
426C24C2×C22⋊D7311yescharacteristic
427C24C2×C22⋊D7311yescharacteristic
428#41—28no—
429#41—44no—
430#41—48no—
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8C2×C22⋊D7312yes—
441C2×Q8C2×C22⋊D7312yes—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4C2×C22⋊D7312yes—
452C2×D4#1192402714yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—2922no—
470C22×C4—5842no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C2×C22⋊D7312yes—
477C22×C4C2×C22⋊D7312yes—
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C23×D7311yescharacteristic
480M16—2924no—
481C2×C8—1462no—
482C2×C8—1462no—
483C2×C8—1462no—
484C2×C8—1462no—
485C2×C8—1462no—
486C2×C8—1462no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—2922no—
494C4⋊C4—5842no—
495C4⋊C4—1462no—
496C4⋊C4—1462no—
497C4⋊C4—1462no—
498C4⋊C4—1462no—
499C4⋊C4C2×C22⋊D7312yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—2922no—
506C22⋊C4—5842no—
507C42—21no—
508C42—21no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—21no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23#1455656012yes—
524C23C22×C22⋊D7311yescharacteristic
525Q8—22no—
526Q8—24no—
527Q8—22no—
528Q8—41no—
529Q8—42no—
530Q8—41no—
531D4—22no—
532D4—24no—
533D4—44no—
534D4—48no—
535D4—48no—
536D4#1455657214yes—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—2922no—
561C2×C4—2922no—
562C2×C4—2922no—
563C2×C4—2922no—
564C2×C4—2922no—
565C2×C4—2922no—
566C2×C4—5844no—
567C2×C4#1455656012yes—
568C2×C4#1455662112yes—
569C2×C4C22×C22⋊D7311yescharacteristic
570C2×C4C22×C22⋊D7311yescharacteristic
571C2×C4C2×#1192402712yes—
572C2×C4#1455663612yes—
573C8—2922no—
574C8—2922no—
575C22—22no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—42no—
590C22C2×#1455656011yescharacteristic
591C22#1798923112yes—
592C22C2×#1455662111yescharacteristic
593C22#1798938511yescharacteristic
594C22#1798939512yes—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—5842no—
603C4—5842no—
604C4C2×#1455657211yescharacteristic
605C4#1798914611yescharacteristic
606C4#1798923112yes—
607C2—42no—
608C2—44no—
609C2—81no—
610C2#2261325411yescharacteristic
611C2#2261919111yescharacteristic
612C2#2261921711yescharacteristic
613C2#2262190712yes—
614C2#2262224712yes—
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
Fitting, prime radicalC2×#547×C73C2
socleC23×C73C22×D4
hypercenterC2×#547D73
hypocenterC73#24055
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#17988915C2semidirect
2. C2×#17988588C2semidirect
3. #22614248C2semidirect
4. #22611174C2semidirect
5. #22611124C2semidirect
6. #22610365C2semidirect
7. #22610229C2semidirect
8. #22610192C2semidirect
9. #22609871C2semidirect
10. #22609871C2semidirect
319. C73#24055semidirect

Maximal subgroups

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

subgroupindex
C2×#179889152
C2×#179885882
#226142482
#226111742
#226111242
#226103652
#226102292
#226101922
#226098712
C2×#547×C732
#2405573