Finite Group Databasebeta

← #33154624

Subgroups — #33154624

Order 18688.

732 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614447C211yesmaximal, characteristic, complemented
2#22611334C211yesmaximal, characteristic
3#22611305C211yesmaximal, characteristic
4#22611167C211yesmaximal, characteristic
5#22611167C211yesmaximal, characteristic
6#22610610C211yesmaximal, characteristic, complemented
7#22610562C211yesmaximal, characteristic, complemented
8#22610523C211yesmaximal, characteristic, complemented
9#22610516C211yesmaximal, characteristic, complemented
10#22610490C211yesmaximal, characteristic, complemented
11#22610185C211yesmaximal, characteristic, complemented
12#22610178C211yesmaximal, characteristic
13#22609936C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic
15C73×#2367C211yesmaximal, characteristic
16C2×#14556581—21no—
17C2×#14556524C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556515C2211yescharacteristic
21C2×#14556515C2211yescharacteristic
22C2×#14556515C2211yescharacteristic
23C2×#14556515C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×#14556512C2211yescharacteristic
26C2×#14556512C2211yescharacteristic
27C2×#14556512C2211yescharacteristic
28C2×C4×#10596831C2211yescharacteristic
29#17988527—21no—
30#17988517—21no—
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988485C2211yescharacteristic
35#17988472—21no—
36#17988457C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988342C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#124×C73C2211yescharacteristic
49C2×C4×Q8×C73C2211yescharacteristic
50C2×C4×D4×C73C2211yescharacteristic
51#391×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#388×C73—21no—
54#388×C73—21no—
55#385×C73—21no—
56C4×C4⋊C4×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73—21no—
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63#14556581—44no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73C2311yescharacteristic
67C2×C4⋊Dic73C2311yescharacteristic
68C2×C4⋊Dic73C2311yescharacteristic
69C2×C4⋊Dic73C2311yescharacteristic
70C2×#11924007—21no—
71C2×#11924007—21no—
72C2×#11924007—21no—
73C2×#11924007—21no—
74C2×C4×Dic73—21no—
75#14556524—22no—
76#14556522—22no—
77C4⋊#10596831—41no—
78#14556515—22no—
79#14556515—22no—
80#14556515—22no—
81#14556515—41no—
82#14556512—22no—
83#14556512—22no—
84#14556512—22no—
85#14556512—41no—
86C4×#10596831—41no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91#14556498—21no—
92C22×Q8×C73C2311yescharacteristic
93C22×D4×C73C2311yescharacteristic
94C23×C4×C73—21no—
95#124×C73—22no—
96#124×C73—22no—
97C4×Q8×C73—22no—
98C4×Q8×C73—41no—
99C4×D4×C73—22no—
100C4×D4×C73—41no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73D411yescharacteristic
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C4⋊C4×C73C2311yescharacteristic
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C22⋊C4×C73—21no—
107C2×C22⋊C4×C73—21no—
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C42×C73—21no—
112C2×C42×C73—21no—
113C2×C42×C73C2311yescharacteristic
114C2×C42×C73C2311yescharacteristic
115C2×C42×C73C2311yescharacteristic
116#95×C73—21no—
117#95×C73—21no—
118#95×C73—21no—
119#95×C73D411yescharacteristic
120#95×C73D411yescharacteristic
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C2×#10596831—21no—
126C2×#10596831—21no—
127C2×#10596831—21no—
128C2×#10596831—21no—
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C2×#10596831—21no—
136C2×#10596831—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—42no—
145C4⋊Dic73—42no—
146C4⋊Dic73#4112yes—
147#11924007—42no—
148#11924007—42no—
149#11924007—42no—
150#11924007—42no—
151C4×Dic73—42no—
152C24×C73—21no—
153C2×Q8×C73—21no—
154C2×Q8×C73—21no—
155C2×Q8×C73—21no—
156C2×Q8×C73—21no—
157C2×Q8×C73—21no—
158C2×Q8×C73#4112yes—
159C2×D4×C73—21no—
160C2×D4×C73—21no—
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73#4112yes—
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—42no—
169C22×C4×C73—42no—
170C22×C4×C73—81no—
171C22×C4×C73C2×D411yescharacteristic
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—22no—
179C4⋊C4×C73—22no—
180C4⋊C4×C73—21no—
181C4⋊C4×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—42no—
184C4⋊C4×C73#4112yes—
185C22⋊C4×C73—24no—
186C22⋊C4×C73—24no—
187C22⋊C4×C73—42no—
188C22⋊C4×C73—42no—
189C22⋊C4×C73—81no—
190C42×C73—21no—
191C42×C73—21no—
192C42×C73—21no—
193C42×C73—21no—
194C42×C73—21no—
195C42×C73—21no—
196C42×C73—42no—
197C42×C73—81no—
198C23×C73—22no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—41no—
202C23×C73—41no—
203C23×C73—41no—
204C23×C73—41no—
205C23×C73C22×D411yescharacteristic
206Q8×C73—22no—
207Q8×C73—22no—
208Q8×C73—41no—
209Q8×C73—41no—
210D4×C73—22no—
211D4×C73—22no—
212D4×C73—41no—
213D4×C73—41no—
214C2×C4×C73—21no—
215C2×C4×C73—21no—
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—22no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—84no—
240C2×C4×C73#13512yes—
241C2×C4×C73AGL(1,ℤ/8)12yes—
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73H(5,𝔽2)11yescharacteristic
247C2×C4×C732−1+411yescharacteristic
248C2×Dic73—22no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260#10596831—41no—
261#10596831—41no—
262#10596831—41no—
263#10596831—41no—
264C22×C73—42no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—81no—
268C22×C73—81no—
269C22×C73—81no—
270C22×C73—81no—
271C22×C73#52511yescharacteristic
272C22×C73#54611yescharacteristic
273C22×C73#54811yescharacteristic
274C22×C73C2×#13511yescharacteristic
275C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C73#57711yescharacteristic
277C22×C73#57811yescharacteristic
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—42no—
283C4×C73—41no—
284C4×C73—41no—
285C4×C73—42no—
286C4×C73—81no—
287C4×C73#52911yescharacteristic
288C4×C73#53911yescharacteristic
289C4×C73#55211yescharacteristic
290C4×C73#55311yescharacteristic
291Dic73—42no—
292Dic73—81no—
293Dic73—81no—
294Dic73—81no—
295#26761—731nomaximal, Sylow
296C2×C73—82no—
297C2×C73#246011yescharacteristic
298C2×C73#310611yescharacteristic
299C2×C73#312011yescharacteristic
300C2×C73#330011yescharacteristic
301C2×C73#330611yescharacteristic
302C2×C73#336511yescharacteristic
303C2×C73#336711yescharacteristic
304#2506—731no—
305#2367D7311yescharacteristic
306#2071—731no—
307#2065—731no—
308#2042—731no—
309#2042—731no—
310#1933—731no—
311#1927—731no—
312#1921—731no—
313#1920—731no—
314#1916—731no—
315#1863—731no—
316#1862—731no—
317#1803—731no—
318C4×#102—731no—
319C73#2676111yescharacteristic, complemented, Sylow
320C2×#125—1461no—
321C2×#124C2×D7311yescharacteristic
322C2×C4×Q8C2×D7311yescharacteristic
323C2×C4×D4C2×D7311yescharacteristic
324C2×#107—731no—
325C2×#106—731no—
326C2×C4⋊C8—731no—
327C2×#103—731no—
328C2×#103—731no—
329C2×#103—731no—
330C2×#103—731no—
331C2×#102—731no—
332C2×#102—731no—
333C2×#102—731no—
334C2×#102—731no—
335C2×C4×C8—731no—
336#400—1461no—
337#398—1461no—
338#391C2×D7311yescharacteristic
339#390C2×D7311yescharacteristic
340#389—731no—
341#389—731no—
342#389—731no—
343#389—731no—
344#388—21no—
345#388—21no—
346#387—1461no—
347#385—21no—
348#384—731no—
349C4×C4⋊C4—731no—
350C4×C4⋊C4C2×D7311yescharacteristic
351C4×C4⋊C4C2×D7311yescharacteristic
352C4×C22⋊C4—21no—
353#340—731no—
354#340—731no—
355#340—731no—
356#340—731no—
357#340—731no—
358#340—731no—
359#336—731no—
360#336—731no—
361#336—731no—
362#336—731no—
363C22×Q8C22×D7311yescharacteristic
364C22×D4C22×D7311yescharacteristic
365C23×C4—21no—
366C22×C8—731no—
367C22×C8—731no—
368C22×C8—731no—
369C22×C8—731no—
370#125—2924no—
371#124—22no—
372#124—22no—
373C4×Q8—22no—
374C4×Q8—41no—
375C4×D4—22no—
376C4×D4—41no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—731no—
379C2×C4⋊C4—731no—
380C2×C4⋊C4—731no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—1461no—
383C2×C4⋊C4—1461no—
384C2×C4⋊C4—1461no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4C22⋊D7311yescharacteristic
389C2×C4⋊C4C22⋊D7311yescharacteristic
390C2×C4⋊C4C22×D7311yescharacteristic
391C2×C4⋊C4C22×D7311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—1461no—
400C2×C42C22×D7311yescharacteristic
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7311yescharacteristic
403#107—1462no—
404#106—1462no—
405C4⋊C8—2921no—
406#103—2921no—
407#103—1462no—
408#103—1462no—
409#103—1462no—
410#102—2921no—
411#102—1462no—
412#102—1462no—
413#102—1462no—
414C4×C8—2921no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1461no—
419#95—1461no—
420#95—1461no—
421#95—1461no—
422#95—1461no—
423#95C22⋊D7311yescharacteristic
424#95C22⋊D7311yescharacteristic
425C24—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8#1192402812yes—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4#1192402812yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—1461no—
445C22×C4—1461no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4C2×C22⋊D7311yescharacteristic
449C22×C4C2×C22⋊D7311yescharacteristic
450C22×C4C2×C22⋊D7311yescharacteristic
451C22×C4C2×C22⋊D7311yescharacteristic
452C22×C4C2×C22⋊D7311yescharacteristic
453C22×C4C2×C22⋊D7311yescharacteristic
454C22×C4C23×D7311yescharacteristic
455C2×C8—1461no—
456C2×C8—1461no—
457C2×C8—1461no—
458C2×C8—1461no—
459C2×C8—1461no—
460C2×C8—1461no—
461C2×C8—1461no—
462C2×C8—1461no—
463C2×C8—1461no—
464C2×C8—1461no—
465C2×C8—1461no—
466C2×C8—1461no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—2922no—
474C4⋊C4—2922no—
475C4⋊C4—2922no—
476C4⋊C4—2922no—
477C4⋊C4—2922no—
478C4⋊C4—2922no—
479C4⋊C4—732no—
480C4⋊C4—1461no—
481C4⋊C4—1461no—
482C4⋊C4—1461no—
483C4⋊C4—1461no—
484C4⋊C4—1461no—
485C4⋊C4—1461no—
486C4⋊C4—1461no—
487C4⋊C4#1192402712yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—81no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—81no—
501C42—2922no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C22⋊D7311yescharacteristic
510Q8—22no—
511Q8—22no—
512Q8—41no—
513Q8—41no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2921no—
549C2×C4—2921no—
550C2×C4—2921no—
551C2×C4—2921no—
552C2×C4—2921no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—1462no—
556C2×C4#1455661112yes—
557C2×C4#1455661612yes—
558C2×C4#41×D7311yescharacteristic
559C2×C4C2×#1192402711yescharacteristic
560C2×C4C2×#1192402811yescharacteristic
561C2×C4C2×#1192402811yescharacteristic
562C2×C4#1455663611yescharacteristic
563C2×C4#1455663811yescharacteristic
564C8—2921no—
565C8—2921no—
566C8—2921no—
567C8—2921no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#1798909811yescharacteristic
576C22#1798923111yescharacteristic
577C22#1798924411yescharacteristic
578C22C2×#1455661111yescharacteristic
579C22C2×#1455661611yescharacteristic
580C22#1798939811yescharacteristic
581C22#1798940211yescharacteristic
582C4—22no—
583C4—22no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—42no—
589C4—41no—
590C4—81no—
591C4—2922no—
592C4—5841no—
593C4—5841no—
594C4—5841no—
595C4#1798911711yescharacteristic
596C4#1798917911yescharacteristic
597C4#1798927711yescharacteristic
598C4#1798928611yescharacteristic
599C2—82no—
600C2#2261387711yescharacteristic
601C2#2261999111yescharacteristic
602C2#2262014511yescharacteristic
603C2#2262181811yescharacteristic
604C2#2262189211yescharacteristic
605C2#2262262311yescharacteristic
606C2#2262264511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2367C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenter#2367D73
hypocenterC73#26761
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #22614447C2semidirect
6. #22610610C2semidirect
7. #22610562C2semidirect
8. #22610523C2semidirect
9. #22610516C2semidirect
10. #22610490C2semidirect
11. #22610185C2semidirect
13. #22609936C2semidirect
319. C73#26761semidirect

Maximal subgroups

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

subgroupindex
#226144472
#226113342
#226113052
#226111672
#226106102
#226105622
#226105232
#226105162
#226104902
#226101852
#226101782
#226099362
C4×#145565122
C73×#23672
#2676173