Finite Group Databasebeta

← #33158650

Subgroups — #33158650

Order 18688.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614342C211yesmaximal, characteristic, complemented
2#22611344C211yesmaximal, characteristic, complemented
3#22611305C211yesmaximal, characteristic, complemented
4#22611117C211yesmaximal, characteristic
5#22611110C211yesmaximal, characteristic
6#22610699C211yesmaximal, characteristic, complemented
7#22610656C211yesmaximal, characteristic, complemented
8#22610539C211yesmaximal, characteristic, complemented
9#22610509C211yesmaximal, characteristic, complemented
10#22610499C211yesmaximal, characteristic, complemented
11#22610192C211yesmaximal, characteristic, complemented
12#22610124C211yesmaximal, characteristic, complemented
13#22610021C211yesmaximal, characteristic, complemented
14C4×#14556512C211yesmaximal, characteristic, complemented
15C73×#2361C211yesmaximal, characteristic
16C2×#14556524C2211yescharacteristic
17C2×#14556522C2211yescharacteristic
18C2×#14556522C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556512C2211yescharacteristic
22C2×#14556512C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C22⋊#10596831C2211yescharacteristic
27C2×C4×#10596831C2211yescharacteristic
28#17988528—21no—
29#17988515—21no—
30#17988485C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988475—21no—
33#17988454—21no—
34#17988453C2211yescharacteristic
35#17988447C2211yescharacteristic
36#17988447C2211yescharacteristic
37C4×C4⋊Dic73C2211yescharacteristic
38#17988349C2211yescharacteristic
39#17988349C2211yescharacteristic
40#17988349C2211yescharacteristic
41#17988349C2211yescharacteristic
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48C2×#121×C73C2211yescharacteristic
49C2×C4×D4×C73—21no—
50C2×C4×D4×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2211yescharacteristic
52#390×C73C2211yescharacteristic
53#389×C73C2211yescharacteristic
54#388×C73—21no—
55#386×C73—21no—
56#385×C73—21no—
57C4×C4⋊C4×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59C22×#10596831C2311yescharacteristic
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62C22×#10596831C2311yescharacteristic
63C2×C4⋊Dic73C2311yescharacteristic
64C2×C4⋊Dic73C2311yescharacteristic
65C2×C4⋊Dic73C2311yescharacteristic
66C2×C4⋊Dic73C2311yescharacteristic
67C2×#11924007—21no—
68C2×#11924007—21no—
69C2×C22⋊Dic73—21no—
70C2×C22⋊Dic73—21no—
71C2×C4×Dic73—21no—
72#14556524—22no—
73#14556522—22no—
74#14556522—22no—
75#14556522—41no—
76C4⋊#10596831—22no—
77#14556512—22no—
78#14556512—22no—
79#14556512—22no—
80#14556512—41no—
81C22⋊#10596831—22no—
82C22⋊#10596831—41no—
83C4×#10596831—41no—
84#14556498—21no—
85#14556498—21no—
86#14556498—21no—
87#14556498—21no—
88#14556498—21no—
89#14556498—21no—
90#14556498—21no—
91C22×D4×C73—21no—
92C22×D4×C73C2311yescharacteristic
93C23×C4×C73—21no—
94C23×C4×C73—21no—
95C23×C4×C73C2311yescharacteristic
96#121×C73—22no—
97#121×C73—22no—
98C4×D4×C73—22no—
99C4×D4×C73—41no—
100C4×D4×C73—44no—
101C2×C4⋊C4×C73—22no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—41no—
105C2×C4⋊C4×C73C2311yescharacteristic
106C2×C4⋊C4×C73C2311yescharacteristic
107C2×C4⋊C4×C73C2311yescharacteristic
108C2×C22⋊C4×C73—21no—
109C2×C22⋊C4×C73—21no—
110C2×C22⋊C4×C73—21no—
111C2×C22⋊C4×C73—21no—
112C2×C22⋊C4×C73D411yescharacteristic
113C2×C22⋊C4×C73D411yescharacteristic
114C2×C42×C73—21no—
115C2×C42×C73C2311yescharacteristic
116C2×C42×C73C2311yescharacteristic
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#4112yes—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—21no—
145#11924007—81no—
146#11924007—81no—
147C22⋊Dic73—42no—
148C22⋊Dic73—81no—
149C4×Dic73—42no—
150C24×C73—21no—
151C24×C73C2×D411yescharacteristic
152C2×D4×C73—24no—
153C2×D4×C73—21no—
154C2×D4×C73—21no—
155C2×D4×C73—21no—
156C2×D4×C73—21no—
157C2×D4×C73—21no—
158C2×D4×C73—42no—
159C2×D4×C73—42no—
160C2×D4×C73#4112yes—
161C22×C4×C73—21no—
162C22×C4×C73—21no—
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—21no—
170C22×C4×C73—42no—
171C22×C4×C73—42no—
172C22×C4×C73—42no—
173C22×C4×C73—42no—
174C22×C4×C73—81no—
175C22×C4×C73—81no—
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2411yescharacteristic
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—42no—
187C4⋊C4×C73—42no—
188C4⋊C4×C73#4112yes—
189C4⋊C4×C73#4112yes—
190C22⋊C4×C73—24no—
191C22⋊C4×C73—22no—
192C22⋊C4×C73—22no—
193C22⋊C4×C73—42no—
194C22⋊C4×C73—42no—
195C22⋊C4×C73—81no—
196C42×C73—21no—
197C42×C73—21no—
198C42×C73—21no—
199C42×C73—21no—
200C42×C73—42no—
201C23×C73—21no—
202C23×C73—21no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—21no—
206C23×C73—21no—
207C23×C73—22no—
208C23×C73—41no—
209C23×C73—41no—
210C23×C73—41no—
211C23×C73—41no—
212C23×C73—41no—
213C23×C73—41no—
214C23×C73#13512yes—
215C23×C73C22×D411yescharacteristic
216D4×C73—22no—
217D4×C73—22no—
218D4×C73—48no—
219D4×C73—41no—
220D4×C73—41no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—21no—
226C2×C4×C73—22no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—84no—
249C2×C4×C73—84no—
250C2×C4×C73AGL(1,ℤ/8)12yes—
251C2×C4×C73C2×#4111yescharacteristic
252C2×C4×C73C2×#4111yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73H(5,𝔽2)11yescharacteristic
256C2×C4×C732−1+411yescharacteristic
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269#10596831—22no—
270#10596831—41no—
271#10596831—41no—
272#10596831—41no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—22no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—42no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—81no—
284C22×C73—81no—
285C22×C73—81no—
286C22×C73—81no—
287C22×C73#52511yescharacteristic
288C22×C73#54611yescharacteristic
289C22×C73#54711yescharacteristic
290C22×C73C2×#13511yescharacteristic
291C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
292C22×C73#57711yescharacteristic
293C22×C73#57711yescharacteristic
294C4×C73—22no—
295C4×C73—41no—
296C4×C73—41no—
297C4×C73—42no—
298C4×C73—41no—
299C4×C73—41no—
300C4×C73—42no—
301C4×C73—81no—
302C4×C73#52911yescharacteristic
303C4×C73#53911yescharacteristic
304C4×C73#55311yescharacteristic
305C4×C73#55611yescharacteristic
306Dic73—81no—
307Dic73—81no—
308Dic73—81no—
309Dic73—81no—
310#27031—731nomaximal, Sylow
311C2×C73—42no—
312C2×C73—82no—
313C2×C73#246611yescharacteristic
314C2×C73#310511yescharacteristic
315C2×C73#312511yescharacteristic
316C2×C73#330111yescharacteristic
317C2×C73#331011yescharacteristic
318C2×C73#333711yescharacteristic
319C2×C73#335411yescharacteristic
320#2495—731no—
321#2361D7311yescharacteristic
322#2073—731no—
323#2065—731no—
324#2034—731no—
325#2033—731no—
326#1949—731no—
327#1941—731no—
328#1923—731no—
329#1919—731no—
330#1918—731no—
331#1864—731no—
332#1853—731no—
333#1824—731no—
334C4×#102—731no—
335C73#2703111yescharacteristic, complemented, Sylow
336C2×#121C2×D7311yescharacteristic
337C2×C4×D4—21no—
338C2×C4×D4C2×D7311yescharacteristic
339C22×C4⋊C4C2×D7311yescharacteristic
340C2×#107—731no—
341C2×#106—731no—
342C2×#106—731no—
343C2×#106—731no—
344C2×C4⋊C8—731no—
345C2×#102—731no—
346C2×#102—731no—
347C2×#102—731no—
348C2×#102—731no—
349C2×C22⋊C8—731no—
350C2×C22⋊C8—731no—
351C2×C4×C8—731no—
352#400—1461no—
353#397—1461no—
354#390C2×D7311yescharacteristic
355#389—731no—
356#389—731no—
357#389C2×D7311yescharacteristic
358#388—21no—
359#388—1461no—
360#386—21no—
361#385—21no—
362#383—1461no—
363#382—731no—
364#380—731no—
365#380—731no—
366C4×C4⋊C4—731no—
367C4×C4⋊C4C2×D7311yescharacteristic
368C4×C22⋊C4C2×D7311yescharacteristic
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#336—731no—
378#336—731no—
379C22×D4—21no—
380C22×D4C22×D7311yescharacteristic
381C23×C4—21no—
382C23×C4—21no—
383C23×C4C22×D7311yescharacteristic
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—731no—
388#121—22no—
389#121—22no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—44no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4C22×D7311yescharacteristic
404C2×C4⋊C4C22×D7311yescharacteristic
405C2×C4⋊C4C22×D7311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1461no—
411C2×C22⋊C4—1461no—
412C2×C22⋊C4C22⋊D7311yescharacteristic
413C2×C22⋊C4C22⋊D7311yescharacteristic
414C2×C42—21no—
415C2×C42—1461no—
416C2×C42C22×D7311yescharacteristic
417C2×C42C22×D7311yescharacteristic
418#107—1462no—
419#106—2921no—
420#106—1462no—
421#106—1462no—
422C4⋊C8—1462no—
423#102—2921no—
424#102—1462no—
425#102—1462no—
426#102—1462no—
427C22⋊C8—2921no—
428C22⋊C8—1462no—
429C4×C8—2921no—
430#95—21no—
431#95—21no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95C22⋊D7311yescharacteristic
440#95C22⋊D7311yescharacteristic
441C24—21no—
442C24C2×C22⋊D7311yescharacteristic
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4#1192402812yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—81no—
466C22×C4—81no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4—1461no—
471C22×C4C2×C22⋊D7311yescharacteristic
472C22×C4C2×C22⋊D7311yescharacteristic
473C22×C4C2×C22⋊D7311yescharacteristic
474C22×C4C2×C22⋊D7311yescharacteristic
475C22×C4C2×C22⋊D7311yescharacteristic
476C22×C4C23×D7311yescharacteristic
477C2×C8—732no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—5841no—
496C4⋊C4—5841no—
497C4⋊C4—1461no—
498C4⋊C4—1461no—
499C4⋊C4—1461no—
500C4⋊C4—1461no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4#1192402712yes—
506C4⋊C4#1192402812yes—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—2922no—
514C22⋊C4—5841no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—2922no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23#1455661112yes—
535C23C22×C22⋊D7311yescharacteristic
536D4—22no—
537D4—22no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4#1455661712yes—
583C2×C4#41×D7311yescharacteristic
584C2×C4C2×#1192402711yescharacteristic
585C2×C4C2×#1192402811yescharacteristic
586C2×C4C2×#1192402811yescharacteristic
587C2×C4#1455663611yescharacteristic
588C2×C4#1455663811yescharacteristic
589C8—2921no—
590C8—2921no—
591C8—2921no—
592C8—1462no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#1798909811yescharacteristic
608C22#1798923111yescharacteristic
609C22#1798923611yescharacteristic
610C22C2×#1455661111yescharacteristic
611C22C2×#1455661711yescharacteristic
612C22#1798939511yescharacteristic
613C22#1798939811yescharacteristic
614C4—22no—
615C4—42no—
616C4—41no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—5841no—
623C4—5841no—
624C4—5841no—
625C4—5841no—
626C4#1798911711yescharacteristic
627C4#1798917911yescharacteristic
628C4#1798928711yescharacteristic
629C4#1798930711yescharacteristic
630C2—42no—
631C2—82no—
632C2#2261396111yescharacteristic
633C2#2261997811yescharacteristic
634C2#2262020811yescharacteristic
635C2#2262183311yescharacteristic
636C2#2262195211yescharacteristic
637C2#2262230411yescharacteristic
638C2#2262252611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2361C2
socleC23×C73C22×D4
prime radicalC2×#121×C73C22
hypercenter#2361D73
hypocenterC73#27031
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. #22614342C2semidirect
2. #22611344C2semidirect
3. #22611305C2semidirect
6. #22610699C2semidirect
7. #22610656C2semidirect
8. #22610539C2semidirect
9. #22610509C2semidirect
10. #22610499C2semidirect
11. #22610192C2semidirect
12. #22610124C2semidirect
13. #22610021C2semidirect
14. C4×#14556512C2semidirect
335. C73#27031semidirect

Maximal subgroups

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

subgroupindex
#226143422
#226113442
#226113052
#226111172
#226111102
#226106992
#226106562
#226105392
#226105092
#226104992
#226101922
#226101242
#226100212
C4×#145565122
C73×#23612
#2703173