Finite Group Databasebeta

← #33119301

Subgroups — #33119301

Order 18688.

1176 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988915C211yesmaximal, characteristic, complemented
2C2×#17988898C211yesmaximal, characteristic
3C2×#17988485C211yesmaximal, characteristic, complemented
4#22611266C212yesmaximal, complemented
5#22611167C212yesmaximal, complemented
6#22610249C211yesmaximal, characteristic
7#22610239C211yesmaximal, characteristic, complemented
8#22610192C212yesmaximal, complemented
9#22610185C212yesmaximal, complemented
10#22609999C211yesmaximal, characteristic, complemented
11C2×#529×C73C211yesmaximal, characteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007—21no—
15#17988915—24no—
16#17988898—24no—
17C2×#14556524C2212yes—
18C2×#14556522C2212yes—
19C2×#14556515C2212yes—
20C2×#14556515C2212yes—
21C2×#14556512C2212yes—
22C2×#14556512C2212yes—
23C2×C22⋊#10596831C2211yescharacteristic
24C2×C22⋊#10596831C2211yescharacteristic
25#17988485—21no—
26#17988485—21no—
27#17988485C2212yes—
28#17988485C2212yes—
29C2×#14556498—21no—
30#17988349C2212yes—
31#17988349C2212yes—
32#17988342C2212yes—
33#17988342C2212yes—
34#529×C73—24no—
35#529×C73—24no—
36C2×#126×C73—21no—
37C2×#125×C73C2211yescharacteristic
38C2×#124×C73C2211yescharacteristic
39C2×#123×C73—21no—
40C2×C22⋊Q8×C73C2211yescharacteristic
41C2×#121×C73C2211yescharacteristic
42C2×C4×Q8×C73C2211yescharacteristic
43C2×C4×D4×C73—21no—
44C2×C4×D4×C73C2211yescharacteristic
45C2×#117×C73—21no—
46C22×C42×C73C2211yescharacteristic
47C23×Dic73—21no—
48C23×Dic73—21no—
49C22×#10596831C2312yes—
50C22×#10596831C2312yes—
51C2×C4⋊Dic73—24no—
52C2×C4⋊Dic73—21no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—22no—
55C2×C4⋊Dic73—21no—
56C2×C4⋊Dic73C2312yes—
57C2×C4⋊Dic73C2312yes—
58C2×#11924007—22no—
59C2×#11924007—22no—
60C2×#11924007—44no—
61#14556524—24no—
62#14556522—24no—
63#14556515—24no—
64#14556515—24no—
65#14556512—24no—
66#14556512—24no—
67C22⋊#10596831—22no—
68C22⋊#10596831—22no—
69#14556498—22no—
70#14556498—22no—
71C22×Q8×C73C2311yescharacteristic
72C22×D4×C73—21no—
73C22×D4×C73C2311yescharacteristic
74C23×C4×C73—21no—
75C23×C4×C73D411yescharacteristic
76C23×C4×C73D411yescharacteristic
77C23×C4×C73C2311yescharacteristic
78#126×C73—28no—
79#125×C73—22no—
80#125×C73—22no—
81#124×C73—22no—
82#124×C73—22no—
83#123×C73—28no—
84C22⋊Q8×C73—22no—
85C22⋊Q8×C73D414yes—
86#121×C73—22no—
87#121×C73D414yes—
88C4×Q8×C73—22no—
89C4×Q8×C73—22no—
90C4×D4×C73—28no—
91C4×D4×C73—22no—
92C4×D4×C73—22no—
93#117×C73—28no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73—21no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73C2311yescharacteristic
99C2×C4⋊C4×C73C2311yescharacteristic
100C2×C22⋊C4×C73—21no—
101C2×C22⋊C4×C73—21no—
102C2×C22⋊C4×C73—21no—
103C2×C22⋊C4×C73—21no—
104C2×C22⋊C4×C73—21no—
105C2×C42×C73—22no—
106C2×C42×C73—22no—
107C2×C42×C73—21no—
108C2×C42×C73D411yescharacteristic
109C2×C42×C73D411yescharacteristic
110C2×C42×C73C2311yescharacteristic
111C2×C42×C73C2311yescharacteristic
112C22×Dic73—24no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—42no—
116C22×Dic73—42no—
117C22×Dic73—42no—
118C22×Dic73—42no—
119C22×Dic73—42no—
120C2×#10596831—22no—
121C2×#10596831—22no—
122C2×#10596831—22no—
123C2×#10596831—22no—
124C2×#10596831—22no—
125C2×#10596831—22no—
126C4⋊Dic73—24no—
127C4⋊Dic73—22no—
128C4⋊Dic73—22no—
129C4⋊Dic73—22no—
130C4⋊Dic73—42no—
131C4⋊Dic73#4114yes—
132#11924007—44no—
133#11924007—44no—
134C24×C73—21no—
135C24×C73C2×D411yescharacteristic
136C2×Q8×C73—21no—
137C2×Q8×C73—22no—
138C2×Q8×C73—22no—
139C2×Q8×C73C2×D412yes—
140C2×D4×C73—24no—
141C2×D4×C73—21no—
142C2×D4×C73—22no—
143C2×D4×C73—22no—
144C2×D4×C73—42no—
145C2×D4×C73—42no—
146C2×D4×C73C2×D412yes—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—24no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—21no—
156C22×C4×C73—21no—
157C22×C4×C73—22no—
158C22×C4×C73—41no—
159C22×C4×C73—42no—
160C22×C4×C73—42no—
161C22×C4×C73C2×D411yescharacteristic
162C22×C4×C73C2×D411yescharacteristic
163C22×C4×C73C2×D411yescharacteristic
164C22×C4×C73C2×D411yescharacteristic
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D412yes—
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2411yescharacteristic
169C4⋊C4×C73—24no—
170C4⋊C4×C73—24no—
171C4⋊C4×C73—24no—
172C4⋊C4×C73—24no—
173C4⋊C4×C73—21no—
174C4⋊C4×C73—21no—
175C4⋊C4×C73C2×D412yes—
176C4⋊C4×C73C2×D412yes—
177C22⋊C4×C73—24no—
178C22⋊C4×C73—24no—
179C22⋊C4×C73—24no—
180C22⋊C4×C73—24no—
181C22⋊C4×C73—24no—
182C42×C73—24no—
183C42×C73—22no—
184C42×C73—22no—
185C42×C73—21no—
186C42×C73—21no—
187C42×C73—21no—
188C42×C73—21no—
189C23×C73—21no—
190C23×C73—21no—
191C23×C73—21no—
192C23×C73—21no—
193C23×C73—22no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73—41no—
200C23×C73#12012yes—
201C23×C73#13512yes—
202C23×C73AGL(1,ℤ/8)12yes—
203C23×C73C22×D411yescharacteristic
204Q8×C73—24no—
205Q8×C73—42no—
206D4×C73—24no—
207D4×C73—48no—
208D4×C73—42no—
209C2×C4×C73—21no—
210C2×C4×C73—21no—
211C2×C4×C73—21no—
212C2×C4×C73—21no—
213C2×C4×C73—24no—
214C2×C4×C73—24no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—22no—
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—44no—
228C2×C4×C73—48no—
229C2×C4×C73—41no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—44no—
236C2×C4×C73#12012yes—
237C2×C4×C73#13712yes—
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C22×D411yescharacteristic
240C2×C4×C73C2×#4112yes—
241C2×C4×C732−1+412yes—
242C2×Dic73—24no—
243C2×Dic73—48no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—84no—
250#10596831—42no—
251#10596831—42no—
252C22×C73—22no—
253C22×C73—22no—
254C22×C73—22no—
255C22×C73—22no—
256C22×C73—22no—
257C22×C73—22no—
258C22×C73—42no—
259C22×C73—42no—
260C22×C73—42no—
261C22×C73—81no—
262C22×C73—81no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73#44914yes—
266C22×C73C2×#12011yescharacteristic
267C22×C73#52511yescharacteristic
268C22×C73#53611yescharacteristic
269C22×C73C2×#13511yescharacteristic
270C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C73C2×#13711yescharacteristic
272C22×C73#57511yescharacteristic
273C4×C73—22no—
274C4×C73—22no—
275C4×C73—42no—
276C4×C73—41no—
277C4×C73—41no—
278C4×C73—42no—
279C4×C73—41no—
280C4×C73—42no—
281C4×C73#54712yes—
282C4×C73#54712yes—
283Dic73—44no—
284Dic73—82no—
285#24066—731nomaximal, Sylow
286C2×C73—24no—
287C2×C73—82no—
288C2×C73#240611yescharacteristic
289C2×C73C2×#44911yescharacteristic
290C2×C73#300711yescharacteristic
291C2×C73#310811yescharacteristic
292C2×C73#311411yescharacteristic
293C2×C73#323211yescharacteristic
294C2×C73#323811yescharacteristic
295C2×#529D7311yescharacteristic
296C2×#483—731no—
297C2×#480—731no—
298C2×#389—731no—
299#2059—732no—
300#2042—732no—
301#1874—731no—
302#1872—731no—
303#1864—732no—
304#1863—732no—
305#1817—731no—
306C73#2406611yescharacteristic, complemented, Sylow
307#529—24no—
308#529—24no—
309C2×#126—21no—
310C2×#125C2×D7311yescharacteristic
311C2×#124C2×D7311yescharacteristic
312C2×#123—21no—
313C2×C22⋊Q8C2×D7311yescharacteristic
314C2×#121C2×D7311yescharacteristic
315C2×C4×Q8C2×D7311yescharacteristic
316C2×C4×D4—21no—
317C2×C4×D4C2×D7311yescharacteristic
318C2×#117—21no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—1461no—
322C22×C42C2×D7311yescharacteristic
323#483—1464no—
324#480—1464no—
325C2×#107—732no—
326C2×#106—732no—
327C2×#103—732no—
328C2×#103—732no—
329C2×#102—732no—
330C2×#102—732no—
331C2×C22⋊C8—731no—
332C2×C22⋊C8—731no—
333#389—732no—
334#389—732no—
335#389—1461no—
336#389—1461no—
337C2×#95—1461no—
338#340—732no—
339#340—732no—
340#336—732no—
341#336—732no—
342C22×Q8C22×D7311yescharacteristic
343C22×D4—21no—
344C22×D4C22×D7311yescharacteristic
345C23×C4—21no—
346C23×C4—1461no—
347C23×C4—1461no—
348C23×C4C22⋊D7311yescharacteristic
349C23×C4C22⋊D7311yescharacteristic
350C23×C4C22×D7311yescharacteristic
351C22×C8—732no—
352C22×C8—732no—
353#126—28no—
354#125—22no—
355#125—22no—
356#124—22no—
357#124—22no—
358#123—28no—
359C22⋊Q8—22no—
360C22⋊Q8C22⋊D7314yes—
361#121—22no—
362#121C22⋊D7314yes—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×D4—28no—
366C4×D4—22no—
367C4×D4—22no—
368#117—28no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—2924no—
374C2×C4⋊C4—732no—
375C2×C4⋊C4—732no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4—1462no—
379C2×C4⋊C4—1462no—
380C2×C4⋊C4—1462no—
381C2×C4⋊C4—1464no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4C22×D7311yescharacteristic
384C2×C4⋊C4C22×D7311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C42—21no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42C22⋊D7311yescharacteristic
394C2×C42C22⋊D7311yescharacteristic
395C2×C42C22×D7311yescharacteristic
396C2×C42C22×D7311yescharacteristic
397#107—1464no—
398#106—1464no—
399#103—1464no—
400#103—1464no—
401#102—1464no—
402#102—1464no—
403C22⋊C8—1462no—
404C22⋊C8—1462no—
405#95—1462no—
406#95—1462no—
407C24—21no—
408C24C2×C22⋊D7311yescharacteristic
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8C2×C22⋊D7312yes—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4C2×C22⋊D7312yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—2922no—
435C22×C4—2922no—
436C22×C4—2922no—
437C22×C4—2922no—
438C22×C4—2922no—
439C22×C4—1462no—
440C22×C4—1464no—
441C22×C4—1462no—
442C22×C4C2×C22⋊D7311yescharacteristic
443C22×C4C2×C22⋊D7311yescharacteristic
444C22×C4C2×C22⋊D7311yescharacteristic
445C22×C4C2×C22⋊D7311yescharacteristic
446C22×C4C2×C22⋊D7312yes—
447C22×C4C2×C22⋊D7312yes—
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C23×D7311yescharacteristic
450C2×C8—1462no—
451C2×C8—1462no—
452C2×C8—1462no—
453C2×C8—1462no—
454C2×C8—1462no—
455C2×C8—1462no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—2924no—
463C4⋊C4—2924no—
464C4⋊C4—2922no—
465C4⋊C4—734no—
466C4⋊C4—1464no—
467C4⋊C4—1462no—
468C4⋊C4—1462no—
469C4⋊C4—1462no—
470C4⋊C4C2×C22⋊D7312yes—
471C4⋊C4C2×C22⋊D7312yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#1455656012yes—
496C23#1455661112yes—
497C23#1455661612yes—
498C23C22×C22⋊D7311yescharacteristic
499Q8—24no—
500Q8—42no—
501D4—24no—
502D4—48no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—2928no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2922no—
537C2×C4—5844no—
538C2×C4—1464no—
539C2×C4#1455656012yes—
540C2×C4#1455662112yes—
541C2×C4C22×C22⋊D7311yescharacteristic
542C2×C4C22×C22⋊D7311yescharacteristic
543C2×C4#41×D7312yes—
544C2×C4#1455663812yes—
545C8—2922no—
546C8—2922no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#1798873214yes—
561C22C2×#1455656011yescharacteristic
562C22#1798909411yescharacteristic
563C22#1798915811yescharacteristic
564C22C2×#1455661111yescharacteristic
565C22C2×#1455661611yescharacteristic
566C22C2×#1455662111yescharacteristic
567C22#1798938511yescharacteristic
568C4—22no—
569C4—22no—
570C4—42no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—2924no—
577C4—5842no—
578C4#1798923612yes—
579C4#1798923612yes—
580C2—24no—
581C2—82no—
582C2#2261326211yescharacteristic
583C2C2×#1798873211yescharacteristic
584C2#2261919111yescharacteristic
585C2#2262000711yescharacteristic
586C2#2262006511yescharacteristic
587C2#2262109511yescharacteristic
588C2#2262114911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#529×C73C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenterC2×#529D73
hypocenterC73#24066
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
1. C2×#17988915C2semidirect
3. C2×#17988485C2semidirect
4. #22611266C2semidirect
5. #22611167C2semidirect
7. #22610239C2semidirect
8. #22610192C2semidirect
9. #22610185C2semidirect
10. #22609999C2semidirect
306. C73#24066semidirect

Maximal subgroups

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

subgroupindex
C2×#179889152
C2×#179888982
C2×#179884852
#226112662
#226111672
#226102492
#226102392
#226101922
#226101852
#226099992
C2×#529×C732
#2406673