Finite Group Databasebeta

← #33055918

Subgroups — #33055918

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619929C211yesmaximal, characteristic
2#22618640C211yesmaximal, characteristic, complemented
3C8×D4×D73C211yesmaximal, characteristic, complemented
4#22608797C211yesmaximal, characteristic, complemented
5#22608795C211yesmaximal, characteristic, complemented
6#22608793C211yesmaximal, characteristic, complemented
7#22608755C211yesmaximal, characteristic, complemented
8#22608754C211yesmaximal, characteristic, complemented
9#22608752C211yesmaximal, characteristic, complemented
10#22608752C211yesmaximal, characteristic, complemented
11#22608700C211yesmaximal, characteristic, complemented
12#22608354C211yesmaximal, characteristic, complemented
13#22608328C211yesmaximal, characteristic, complemented
14#22608222C211yesmaximal, characteristic, complemented
15C73×#1586C211yesmaximal, characteristic, complemented
16C2×#14556599—21no—
17C22×C8×D73—21no—
18#17989134C2211yescharacteristic
19C4×D4×D73C2211yescharacteristic
20#17988902C411yescharacteristic
21#17988899C411yescharacteristic
22#17988866C411yescharacteristic
23#17988865C411yescharacteristic
24#17988781C411yescharacteristic
25#17988780C411yescharacteristic
26#17988722C411yescharacteristic
27#17988719C411yescharacteristic
28C4×D584C2211yescharacteristic
29C4×#11924013C2211yescharacteristic
30C4×#11924013C2211yescharacteristic
31#17988666C2211yescharacteristic, complemented
32#17988663—21no—
33C8×D292C2211yescharacteristic
34D4×#10596831C2211yescharacteristic
35D4×#10596831C2211yescharacteristic, complemented
36#17988652C2211yescharacteristic
37#17988652C2211yescharacteristic, complemented
38C8×C22⋊D73—21no—
39#17988650—21no—
40#17988650—21no—
41#17988638C2211yescharacteristic
42#17988638C2211yescharacteristic
43#17988609C2211yescharacteristic
44C4⋊C8×D73C2211yescharacteristic
45#17988552—21no—
46C22⋊C8×D73—21no—
47C4×C8×D73C2211yescharacteristic
48#17988338C2211yescharacteristic, complemented
49#17988337C2211yescharacteristic, complemented
50#17988337C2211yescharacteristic, complemented
51#17988316C2211yescharacteristic
52#17988316C2211yescharacteristic, complemented
53D4⋊#10596831C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic, complemented
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic, complemented
57#17988309C2211yescharacteristic, complemented
58#17988309C2211yescharacteristic, complemented
59#17988308C2211yescharacteristic, complemented
60C4×D8×C73C2211yescharacteristic
61#436×C73C2211yescharacteristic, complemented
62C8×D4×C73C2211yescharacteristic
63#334×C73C2211yescharacteristic, complemented
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic, complemented
66#322×C73C2211yescharacteristic, complemented
67C2×#11924028—21no—
68C2×D4×D73C2×C411yescharacteristic
69C22×C4×D73—21no—
70C2×D584C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72C2×#11924013C2×C411yescharacteristic
73#14556599—44no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C22×#10596831—21no—
77C22×#10596831—21no—
78C2×C8×D73—21no—
79C2×C8×D73—21no—
80C2×C8×D73—21no—
81C2×C8×D73—21no—
82C2×C8×D73—44no—
83C2×C8×D73D411yescharacteristic
84C2×C8×D73D411yescharacteristic
85#14556590C2×C411yescharacteristic
86#14556587C2×C411yescharacteristic
87#14556577—21no—
88#14556566C2×C411yescharacteristic
89#14556564—21no—
90#14556562C2×C411yescharacteristic
91C4×D292C2311yescharacteristic
92C4×D292C2311yescharacteristic
93D4×Dic73C2×C411yescharacteristic
94#14556553C2×C411yescharacteristic
95C4×C22⋊D73—21no—
96C4×C22⋊D73—21no—
97#14556551—21no—
98C4⋊C4×D73C2×C411yescharacteristic
99C22⋊C4×D73—21no—
100C42×D73C2311yescharacteristic
101#14556525C2×C411yescharacteristic
102#14556524C2×C411yescharacteristic
103#14556524C2×C411yescharacteristic
104C4⋊#10596831C2311yescharacteristic
105C4⋊#10596831C2311yescharacteristic
106#14556520—21no—
107#14556520—21no—
108#14556520—21no—
109#14556520C2×C411yescharacteristic
110#14556520C2×C411yescharacteristic
111#14556513C2×C411yescharacteristic
112#14556513C2×C411yescharacteristic
113#14556512C2×C411yescharacteristic
114#14556512C2×C411yescharacteristic
115#14556511C2×C411yescharacteristic
116#14556511C2×C411yescharacteristic
117C22⋊#10596831—21no—
118C22⋊#10596831—21no—
119#14556503—21no—
120#14556503—21no—
121#14556503—21no—
122#14556503—21no—
123#14556503—21no—
124#14556503C2×C411yescharacteristic
125#14556503C2×C411yescharacteristic
126C4×#10596831C2311yescharacteristic
127C4×#10596831C2311yescharacteristic
128C4×#10596831C2311yescharacteristic
129C8×Dic73D411yescharacteristic
130C8×Dic73D411yescharacteristic
131C2×D8×C73C2×C411yescharacteristic
132C2×M16×C73—21no—
133C22×C8×C73—21no—
134C4×D4×C73C2311yescharacteristic
135C4×D4×C73C2311yescharacteristic
136#107×C73C2×C411yescharacteristic
137C4⋊C8×C73C2311yescharacteristic
138C4⋊C8×C73C2311yescharacteristic
139#102×C73C2×C411yescharacteristic
140#102×C73C2×C411yescharacteristic
141C22⋊C8×C73—21no—
142C22⋊C8×C73—21no—
143C4×C8×C73C2311yescharacteristic
144C4×C8×C73C2311yescharacteristic
145C4×C8×C73C2311yescharacteristic
146C23×D73—21no—
147#11924028—42no—
148#11924028—42no—
149C2×Dic146—21no—
150C2×D292—21no—
151C2×D292C22×C411yescharacteristic
152C2×D292C22×C411yescharacteristic
153C2×C22⋊D73—21no—
154C2×C22⋊D73—21no—
155C2×C22⋊D73—21no—
156C2×C22⋊D73—21no—
157D4×D73—22no—
158D4×D73—22no—
159C22×Dic73—21no—
160C2×C4×D73—21no—
161C2×C4×D73—21no—
162C2×C4×D73—21no—
163C2×C4×D73—21no—
164C2×C4×D73—42no—
165C2×C4×D73—42no—
166C2×C4×D73C22×C411yescharacteristic
167C2×C4×D73C2×D411yescharacteristic
168C2×C4×D73#4111yescharacteristic
169D584—22no—
170#11924013—22no—
171#11924013—22no—
172#11924011—42no—
173#11924011—42no—
174C2×#10596831—21no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—42no—
178C2×#10596831—42no—
179C2×#10596831C22×C411yescharacteristic
180C2×#10596831C22×C411yescharacteristic
181C2×#10596831C2×D411yescharacteristic
182C2×#10596831C2×D411yescharacteristic
183C8×D73—22no—
184C8×D73—22no—
185C8×D73—42no—
186C8×D73—42no—
187C8×D73—42no—
188C8×D73—42no—
189C4⋊Dic73C22×C411yescharacteristic
190C4⋊Dic73C22×C411yescharacteristic
191#11924007—21no—
192#11924007—21no—
193C22⋊Dic73—21no—
194C22⋊Dic73—21no—
195#11924005—21no—
196#11924005—21no—
197#11924005—21no—
198#11924005—21no—
199C4×Dic73C22×C411yescharacteristic
200C4×Dic73C2×D411yescharacteristic
201C4×Dic73#4111yescharacteristic
202C2×D4×C73C22×C411yescharacteristic
203C2×D4×C73C22×C411yescharacteristic
204C22×C4×C73—21no—
205C22×C4×C73—21no—
206D8×C73—22no—
207M16×C73—42no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—42no—
212C2×C8×C73C22×C411yescharacteristic
213C2×C8×C73C22×C411yescharacteristic
214C2×C8×C73C2×D411yescharacteristic
215C2×C8×C73C2×D411yescharacteristic
216C4⋊C4×C73C22×C411yescharacteristic
217C4⋊C4×C73C22×C411yescharacteristic
218C22⋊C4×C73—21no—
219C22⋊C4×C73—21no—
220C42×C73C2411yescharacteristic
221C23×C73—21no—
222C23×C73—21no—
223C22×D73—24no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—42no—
227C22×D73—42no—
228C22×D73C4×D411yescharacteristic
229D4×C73—21no—
230D4×C73—21no—
231D4×C73—21no—
232D4×C73#13112yescomplemented
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73C23×C411yescharacteristic
240C2×C4×C73C22×D411yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C22⋊D73—24no—
243C22⋊D73—42no—
244C22⋊D73—42no—
245C22⋊D73—42no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—42no—
249C2×Dic73C4×D411yescharacteristic
250C2×Dic73C4×D411yescharacteristic
251C2×Dic73C4×D411yescharacteristic
252D292—21no—
253D292—21no—
254D292—21no—
255D292—42no—
256D292#13112yescomplemented
257C4×D73—21no—
258C4×D73—21no—
259C4×D73—41no—
260C4×D73—42no—
261C4×D73—41no—
262C4×D73—41no—
263C4×D73—41no—
264C4×D73—41no—
265C4×D73—42no—
266C4×D73AGL(1,ℤ/8)12yes—
267C4×D73AGL(1,ℤ/8)12yes—
268Dic146—42no—
269C8×C73—21no—
270C8×C73—21no—
271C8×C73—21no—
272C8×C73—21no—
273C8×C73—41no—
274C8×C73—41no—
275C8×C73—41no—
276#10596831—21no—
277#10596831—21no—
278#10596831—21no—
279#10596831—21no—
280#10596831—41no—
281#10596831—41no—
282#10596831—41no—
283C22×C73—22no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73C2×C4×D411yescharacteristic
290C2×D73—21no—
291C2×D73—21no—
292C2×D73—22no—
293C2×D73—48no—
294C2×D73—41no—
295C2×D73—41no—
296C2×D73—41no—
297C2×D73—41no—
298C2×D73—41no—
299C2×D73#44212yes—
300C4×C73—21no—
301C4×C73—41no—
302C4×C73—41no—
303C4×C73C2×#13111yescharacteristic
304C4×C73C2×#13111yescharacteristic
305C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
306C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
307Dic73—21no—
308Dic73—21no—
309Dic73—41no—
310Dic73—41no—
311Dic73#44212yes—
312#17907—731nomaximal, Sylow
313C2×C73—42no—
314C2×C73—81no—
315C2×C73#292911yescharacteristic
316C2×C73C2×#44211yescharacteristic
317C2×C73#295511yescharacteristic
318D73—22no—
319D73—42no—
320D73—81no—
321#3100—731no—
322#2931—731no—
323C2×C8×D4—731no—
324#1586—731no—
325#1586—731no—
326#1586—731no—
327#1586D7311yescharacteristic, complemented
328#1580—731no—
329#1580—731no—
330#1580—731no—
331#1580—731no—
332#1570—731no—
333#1486—731no—
334#1480—731no—
335#1452—731no—
336C73#1790711yescharacteristic, complemented, Sylow
337C2×#131—1461no—
338C23×C8—1461no—
339#532—731no—
340C2×C4×D4—731no—
341#480—731no—
342#480—731no—
343#474—731no—
344#474—731no—
345#459—731no—
346#459—731no—
347#447—731no—
348#447—731no—
349C4×D8—731no—
350C4×D8—731no—
351C4×D8—731no—
352C4×D8C2×D7311yescharacteristic
353#436—731no—
354#436—1461no—
355#436C2×D7311yescharacteristic, complemented
356C8×D4—731no—
357C8×D4—731no—
358C8×D4—731no—
359C8×D4—731no—
360C8×D4—731no—
361C8×D4—1461no—
362C8×D4—1461no—
363C8×D4—1461no—
364C8×D4C2×D7311yescharacteristic
365#431—731no—
366#431—731no—
367#423—731no—
368C2×C4⋊C8—731no—
369#408—1461no—
370C2×C22⋊C8—1461no—
371C2×C4×C8—731no—
372#334—731no—
373#334—731no—
374#334—731no—
375#334C2×D7311yescharacteristic, complemented
376D4⋊C8—731no—
377D4⋊C8—731no—
378D4⋊C8—731no—
379D4⋊C8—731no—
380D4⋊C8—731no—
381D4⋊C8—731no—
382D4⋊C8C2×D7311yescharacteristic
383D4⋊C8C2×D7311yescharacteristic, complemented
384#322—731no—
385#322—731no—
386#322—731no—
387#322C2×D7311yescharacteristic, complemented
388C2×#41—1461no—
389C22×D4—731no—
390C23×C4—1461no—
391C2×D8—731no—
392C2×D8—731no—
393C2×D8—731no—
394C2×D8C4×D7311yescharacteristic
395#131—2924no—
396C2×M16—21no—
397C2×M16—1461no—
398C2×M16—1461no—
399C22×C8—21no—
400C22×C8—2924no—
401C22×C8—731no—
402C22×C8—731no—
403C22×C8—1461no—
404C22×C8—1461no—
405C22×C8—1461no—
406C22×C8—1461no—
407C22×C8—1461no—
408C22×C8—1461no—
409C4⋊Q8—731no—
410#127—731no—
411#124—1461no—
412#121—731no—
413#121—731no—
414#121—1461no—
415C4×D4—731no—
416C4×D4—731no—
417C4×D4—731no—
418C4×D4—731no—
419C4×D4—1461no—
420C4×D4—1461no—
421C4×D4—1461no—
422C4×D4C22×D7311yescharacteristic
423C4×D4C22×D7311yescharacteristic
424C2×C4⋊C4—731no—
425C2×C22⋊C4—1461no—
426C2×C42—731no—
427#107—731no—
428#107—731no—
429#107—731no—
430#107C4×D7311yescharacteristic
431C4⋊C8—731no—
432C4⋊C8—731no—
433C4⋊C8—731no—
434C4⋊C8—731no—
435C4⋊C8—1461no—
436C4⋊C8—1461no—
437C4⋊C8—1461no—
438C4⋊C8C22×D7311yescharacteristic
439C4⋊C8C22×D7311yescharacteristic
440#102—731no—
441#102—731no—
442#102—731no—
443#102—731no—
444#102—731no—
445#102—731no—
446#102C4×D7311yescharacteristic
447#102C4×D7311yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—731no—
451C22⋊C8—731no—
452C22⋊C8—1461no—
453C22⋊C8—1461no—
454C22⋊C8—1461no—
455C22⋊C8—1461no—
456C22⋊C8—1461no—
457C22⋊C8—1461no—
458C22⋊C8—1461no—
459C4×C8—731no—
460C4×C8—731no—
461C4×C8—731no—
462C4×C8—731no—
463C4×C8—731no—
464C4×C8C22×D7311yescharacteristic
465C4×C8C22×D7311yescharacteristic
466C4×C8C22×D7311yescharacteristic
467C24—1461no—
468#41—2922no—
469#41—2922no—
470C2×Q8—1461no—
471C2×D4—731no—
472C2×D4—731no—
473C2×D4—1461no—
474C2×D4—1461no—
475C2×D4—1461no—
476C2×D4—1461no—
477C2×D4—1462no—
478C2×D4—1461no—
479C2×D4—1462no—
480C2×D4C2×C4×D7311yescharacteristic
481C2×D4C2×C4×D7311yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—731no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494D8—22no—
495D8—1462no—
496D8—1462no—
497D8—1462no—
498M16—42no—
499M16—2922no—
500M16—2922no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—42no—
505C2×C8—2922no—
506C2×C8—2922no—
507C2×C8—2922no—
508C2×C8—2922no—
509C2×C8—2922no—
510C2×C8—2922no—
511C2×C8—731no—
512C2×C8—731no—
513C2×C8—731no—
514C2×C8—731no—
515C2×C8—1462no—
516C2×C8—1462no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8C2×C4×D7311yescharacteristic
521C2×C8C2×C4×D7311yescharacteristic
522C2×C8D4×D7311yescharacteristic
523C2×C8D4×D7311yescharacteristic
524C4⋊C4—731no—
525C4⋊C4—731no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4C2×C4×D7311yescharacteristic
529C4⋊C4C2×C4×D7311yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—1461no—
533C22⋊C4—1461no—
534C22⋊C4—1461no—
535C22⋊C4—1461no—
536C22⋊C4—1461no—
537C22⋊C4—1461no—
538C42—731no—
539C42—731no—
540C42—731no—
541C42C23×D7311yescharacteristic
542C23—21no—
543C23—21no—
544C23—2922no—
545C23—2922no—
546C23—731no—
547C23—1464no—
548C23—1461no—
549C23—1461no—
550Q8—2922no—
551D4—21no—
552D4—21no—
553D4—21no—
554D4—2922no—
555D4—2922no—
556D4—2922no—
557D4—2922no—
558D4—732no—
559D4—1464no—
560D4—1461no—
561D4—1461no—
562D4—1461no—
563D4#1455659912yescomplemented
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—2921no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—2922no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—731no—
579C2×C4—732no—
580C2×C4—732no—
581C2×C4—731no—
582C2×C4—731no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4C22×C4×D7311yescharacteristic
588C2×C4C2×D4×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C8—21no—
591C8—21no—
592C8—21no—
593C8—21no—
594C8—41no—
595C8—41no—
596C8—41no—
597C8—2921no—
598C8—2921no—
599C8—2921no—
600C8—1461no—
601C8—1461no—
602C8—1461no—
603C8—1461no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—2928no—
611C22—2921no—
612C22—2921no—
613C22—2921no—
614C22—2921no—
615C22—2921no—
616C22—732no—
617C22—1462no—
618C22—1461no—
619C22—1461no—
620C22C4×D4×D7311yescharacteristic
621C4—21no—
622C4—41no—
623C4—41no—
624C4—2921no—
625C4—2921no—
626C4—732no—
627C4—1461no—
628C4—1461no—
629C4#131×D7311yescharacteristic
630C4C2×#1455659911yescharacteristic
631C4AGL(1,ℤ/8)×D7311yescharacteristic
632C4AGL(1,ℤ/8)×D7311yescharacteristic
633C2—42no—
634C2—81no—
635C2—2922no—
636C2—5841no—
637C2—1462no—
638C2#2261862211yescharacteristic
639C2#442×D7311yescharacteristic
640C2#2261882911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1586C2
socleC22×C73C2×C4×D4
prime radical#22619929C2
cosocleC42×C73C24
hypercenter#1586D73
hypocenterC73#17907
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1586

Complements

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

NH ≅ G/Ntype
2. #22618640C2semidirect
3. C8×D4×D73C2semidirect
4. #22608797C2semidirect
5. #22608795C2semidirect
6. #22608793C2semidirect
7. #22608755C2semidirect
8. #22608754C2semidirect
9. #22608752C2semidirect
10. #22608752C2semidirect
11. #22608700C2semidirect
12. #22608354C2semidirect
13. #22608328C2semidirect
14. #22608222C2semidirect
15. C73×#1586C2semidirect
31. #17988666C22semidirect
35. D4×#10596831C22semidirect
37. #17988652C22semidirect
48. #17988338C22semidirect
49. #17988337C22semidirect
50. #17988337C22semidirect
52. #17988316C22semidirect
54. D4⋊#10596831C22semidirect
56. #17988314C22semidirect
57. #17988309C22semidirect
58. #17988309C22semidirect
59. #17988308C22semidirect
61. #436×C73C22semidirect
63. #334×C73C22semidirect
65. D4⋊C8×C73C22semidirect
66. #322×C73C22semidirect
232. D4×C73#131semidirect
256. D292#131semidirect
327. #1586D73semidirect
336. C73#17907semidirect
355. #436C2×D73semidirect
375. #334C2×D73semidirect
383. D4⋊C8C2×D73semidirect
387. #322C2×D73semidirect
563. D4#14556599semidirect

Maximal subgroups

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

subgroupindex
#226199292
#226186402
C8×D4×D732
#226087972
#226087952
#226087932
#226087552
#226087542
#226087522
#226087002
#226083542
#226083282
#226082222
C73×#15862
#1790773