Finite Group Databasebeta

← #33056688

Subgroups — #33056688

Order 18688.

726 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620507C211yesmaximal, characteristic
2#22618640C211yesmaximal, characteristic, complemented
3#22618638C211yesmaximal, characteristic, complemented
4#22608763C211yesmaximal, characteristic, complemented
5#22608762C211yesmaximal, characteristic, complemented
6#22608761C211yesmaximal, characteristic, complemented
7#22608760C211yesmaximal, characteristic, complemented
8#22608759C211yesmaximal, characteristic, complemented
9#22608758C211yesmaximal, characteristic, complemented
10#22608757C211yesmaximal, characteristic, complemented
11#22608688C211yesmaximal, characteristic, complemented
12#22608380C211yesmaximal, characteristic, complemented
13#22608380C211yesmaximal, characteristic, complemented
14#322×D73C211yesmaximal, characteristic, complemented
15C73×#1581C211yesmaximal, characteristic, complemented
16C2×#14556599—21no—
17C2×#14556597—21no—
18#17989134C2211yescharacteristic
19#17989134C2211yescharacteristic
20#17988991C411yescharacteristic
21#17988962C411yescharacteristic
22#17988932C411yescharacteristic
23#17988932C411yescharacteristic
24#17988824C411yescharacteristic
25#17988822C411yescharacteristic
26#17988822C411yescharacteristic
27#17988820C411yescharacteristic
28C4×D584C2211yescharacteristic
29C4×#11924013C2211yescharacteristic
30C4×#11924013C2211yescharacteristic
31#17988666C2211yescharacteristic
32#17988665C2211yescharacteristic
33#17988664C2211yescharacteristic
34#17988663—21no—
35#17988662—21no—
36C8×D292C2211yescharacteristic
37D4×#10596831C2211yescharacteristic
38#17988652C2211yescharacteristic
39C8×C22⋊D73—21no—
40#17988650—21no—
41#17988609C2211yescharacteristic
42#17988609C2211yescharacteristic
43#17988552—21no—
44#17988552—21no—
45C4×C8×D73C2211yescharacteristic
46C4×C8×D73C2211yescharacteristic
47C4×C8×D73C2211yescharacteristic
48#17988340C2211yescharacteristic
49#17988339C2211yescharacteristic
50#17988339C2211yescharacteristic
51#17988316C2211yescharacteristic
52#17988316C2211yescharacteristic
53D4⋊#10596831C2211yescharacteristic
54D4⋊#10596831C2211yescharacteristic
55#17988314C2211yescharacteristic
56#17988314C2211yescharacteristic
57#17988309C2211yescharacteristic
58#17988309C2211yescharacteristic
59#17988308C2211yescharacteristic
60C4×D8×C73C2211yescharacteristic
61#436×C73C2211yescharacteristic
62C8×D4×C73C2211yescharacteristic
63#335×C73C2211yescharacteristic
64D4⋊C8×C73C2211yescharacteristic
65D4⋊C8×C73C2211yescharacteristic
66#322×C73C2211yescharacteristic
67C2×#11924028—21no—
68C2×#11924028—21no—
69C2×D584C2×C411yescharacteristic
70C2×#11924013C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72#14556599—44no—
73#14556597—44no—
74C2×#11924012—21no—
75C2×#11924011—21no—
76C2×#11924011—21no—
77C2×#11924011—21no—
78C2×#11924011—21no—
79C22×#10596831—21no—
80C2×C8×D73—21no—
81C2×C8×D73—21no—
82C2×C8×D73—21no—
83C2×C8×D73C2×C411yescharacteristic
84C2×C8×D73C2×C411yescharacteristic
85C2×C8×D73D411yescharacteristic
86C2×C8×D73D411yescharacteristic
87#14556590C2×C411yescharacteristic
88#14556590C2×C411yescharacteristic
89#14556587C2×C411yescharacteristic
90#14556587C2×C411yescharacteristic
91#14556577—21no—
92#14556577—21no—
93#14556566C2×C411yescharacteristic
94#14556566C2×C411yescharacteristic
95#14556564—21no—
96#14556564—21no—
97#14556562C2×C411yescharacteristic
98#14556562C2×C411yescharacteristic
99C4×D292C2311yescharacteristic
100C4×D292C2311yescharacteristic
101C4×C22⋊D73—21no—
102C4×C22⋊D73—21no—
103C42×D73C2311yescharacteristic
104#14556525C2×C411yescharacteristic
105#14556524C2×C411yescharacteristic
106#14556524C2×C411yescharacteristic
107C4⋊#10596831C2311yescharacteristic
108C4⋊#10596831C2311yescharacteristic
109#14556520—21no—
110#14556520—21no—
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—
123C4×#10596831C2311yescharacteristic
124C4×#10596831C2311yescharacteristic
125C4×#10596831C2311yescharacteristic
126C8×Dic73—21no—
127C8×Dic73C2×C411yescharacteristic
128C8×Dic73C2×C411yescharacteristic
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
146#11924028—42no—
147#11924028—42no—
148#11924028—42no—
149#11924028—42no—
150C2×Dic146—21no—
151C2×Dic146—21no—
152C2×D292—21no—
153C2×D292—21no—
154C2×D292C22×C411yescharacteristic
155C2×D292C22×C411yescharacteristic
156C2×C22⋊D73—21no—
157C2×C22⋊D73—21no—
158C2×C22⋊D73—21no—
159C2×C22⋊D73—21no—
160C2×C22⋊D73—21no—
161C2×C22⋊D73—21no—
162C2×C4×D73—21no—
163C2×C4×D73—21no—
164C2×C4×D73C22×C411yescharacteristic
165C2×C4×D73C2×D411yescharacteristic
166C2×C4×D73#4111yescharacteristic
167D584—41no—
168#11924013—41no—
169#11924013—41no—
170#11924012—42no—
171#11924011—42no—
172#11924011—42no—
173#11924011—42no—
174#11924011—42no—
175C2×#10596831—21no—
176C2×#10596831—21no—
177C2×#10596831—21no—
178C2×#10596831—42no—
179C2×#10596831C22×C411yescharacteristic
180C2×#10596831C22×C411yescharacteristic
181C2×#10596831C2×D411yescharacteristic
182C2×#10596831C2×D411yescharacteristic
183C8×D73—22no—
184C8×D73—41no—
185C8×D73—41no—
186C8×D73—42no—
187C8×D73—42no—
188C8×D73—42no—
189C8×D73D814yescomplemented
190C4⋊Dic73C22×C411yescharacteristic
191C4⋊Dic73C22×C411yescharacteristic
192#11924007—21no—
193#11924007—21no—
194C22⋊Dic73—21no—
195C22⋊Dic73—21no—
196#11924005—21no—
197#11924005—21no—
198#11924005—21no—
199#11924005—21no—
200C4×Dic73C22×C411yescharacteristic
201C4×Dic73C2×D411yescharacteristic
202C4×Dic73#4111yescharacteristic
203C2×D4×C73C22×C411yescharacteristic
204C2×D4×C73C22×C411yescharacteristic
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207D8×C73—41no—
208M16×C73—42no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—42no—
213C2×C8×C73C22×C411yescharacteristic
214C2×C8×C73C22×C411yescharacteristic
215C2×C8×C73C2×D411yescharacteristic
216C2×C8×C73C2×D411yescharacteristic
217C4⋊C4×C73C22×C411yescharacteristic
218C4⋊C4×C73C22×C411yescharacteristic
219C22⋊C4×C73—21no—
220C22⋊C4×C73—21no—
221C42×C73C2411yescharacteristic
222C23×C73—21no—
223C23×C73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73C4×D411yescharacteristic
227D4×C73—21no—
228D4×C73—21no—
229D4×C73—21no—
230D4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73C23×C411yescharacteristic
238C2×C4×C73C22×D411yescharacteristic
239C2×C4×C73C2×#4111yescharacteristic
240C22⋊D73—42no—
241C22⋊D73—42no—
242C22⋊D73—42no—
243C22⋊D73—42no—
244C22⋊D73—42no—
245C22⋊D73—42no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73C4×D411yescharacteristic
249C2×Dic73C4×D411yescharacteristic
250C2×Dic73C4×D411yescharacteristic
251D292—21no—
252D292—21no—
253D292—21no—
254D292—21no—
255D292—42no—
256D292—42no—
257C4×D73—21no—
258C4×D73—21no—
259C4×D73—22no—
260C4×D73—41no—
261C4×D73—41no—
262C4×D73—41no—
263C4×D73—41no—
264C4×D73C2×D812yes—
265C4×D73#13512yes—
266Dic146—42no—
267Dic146—42no—
268C8×C73—21no—
269C8×C73—21no—
270C8×C73—21no—
271C8×C73—41no—
272C8×C73—41no—
273C8×C73—41no—
274C8×C73C2×D812yes—
275#10596831—21no—
276#10596831—21no—
277#10596831—21no—
278#10596831—41no—
279#10596831—41no—
280#10596831—41no—
281#10596831C2×D812yes—
282C22×C73—41no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73C2×C4×D411yescharacteristic
289C2×D73—21no—
290C2×D73—21no—
291C2×D73—41no—
292C2×D73—41no—
293C2×D73—41no—
294C2×D73—41no—
295C2×D73—41no—
296C2×D73—41no—
297C2×D73C4×D812yes—
298C4×C73—21no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73#56811yescharacteristic
302C4×C73#56811yescharacteristic
303C4×C73C22×D811yescharacteristic
304C4×C73C2×#13511yescharacteristic
305Dic73—21no—
306Dic73—41no—
307Dic73—41no—
308Dic73C4×D812yes—
309Dic73#44412yes—
310#17975—731nomaximal, Sylow
311C2×C73—81no—
312C2×C73—81no—
313C2×C73#293211yescharacteristic
314C2×C73C2×C4×D811yescharacteristic
315C2×C73C2×#44411yescharacteristic
316D73—22no—
317D73—81no—
318D73—81no—
319#3169—731no—
320#2931—731no—
321#2931—731no—
322#1581—731no—
323#1581—731no—
324#1581—731no—
325#1581—731no—
326#1581—731no—
327#1581—731no—
328#1581—731no—
329#1581D7311yescharacteristic, complemented
330#1567—731no—
331#1492—731no—
332#1492—731no—
333C2×#322—731no—
334C73#1797511yescharacteristic, complemented, Sylow
335C2×#131—1461no—
336C2×#131—1461no—
337#532—731no—
338#532—731no—
339#500—731no—
340#493—731no—
341#486—731no—
342#486—731no—
343#466—731no—
344#466—731no—
345#466—731no—
346#466—731no—
347C4×D8—731no—
348C4×D8—731no—
349C4×D8—731no—
350C4×D8C2×D7311yescharacteristic
351#436—731no—
352#436—731no—
353#436—731no—
354#436—1461no—
355#436—1461no—
356#436C2×D7311yescharacteristic
357C8×D4—731no—
358C8×D4—731no—
359C8×D4—731no—
360C8×D4—1461no—
361C8×D4—1461no—
362C8×D4C2×D7311yescharacteristic
363#423—731no—
364#423—731no—
365#408—1461no—
366#408—1461no—
367C2×C4×C8—731no—
368C2×C4×C8—731no—
369C2×C4×C8—731no—
370#335—731no—
371#335—731no—
372#335—731no—
373#335C2×D7311yescharacteristic
374D4⋊C8—731no—
375D4⋊C8—731no—
376D4⋊C8—731no—
377D4⋊C8—731no—
378D4⋊C8—731no—
379D4⋊C8—731no—
380D4⋊C8C2×D7311yescharacteristic
381D4⋊C8C2×D7311yescharacteristic
382#322—731no—
383#322—731no—
384#322—731no—
385#322C2×D7311yescharacteristic
386C2×#41—1461no—
387C2×#41—1461no—
388C2×D8—731no—
389C2×D8—731no—
390C2×D8—731no—
391C2×D8C4×D7311yescharacteristic
392#131—2924no—
393#131—2924no—
394C2×M16—21no—
395C2×M16—1461no—
396C2×M16—1461no—
397C2×M16—1461no—
398C2×M16—1461no—
399C2×M16—1461no—
400C22×C8—21no—
401C22×C8—731no—
402C22×C8—731no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—1461no—
406C22×C8—1461no—
407C22×C8—1461no—
408C22×C8—1461no—
409C4⋊Q8—731no—
410C4⋊Q8—731no—
411#127—731no—
412#127—731no—
413#124—1461no—
414#124—1461no—
415#121—731no—
416#121—731no—
417#121—731no—
418#121—731no—
419#121—1461no—
420#121—1461no—
421C4×D4—731no—
422C4×D4—731no—
423C4×D4—1461no—
424C4×D4—1461no—
425C4×D4C22×D7311yescharacteristic
426C4×D4C22×D7311yescharacteristic
427C2×C42—731no—
428#107—731no—
429#107—731no—
430#107—731no—
431#107C4×D7311yescharacteristic
432C4⋊C8—731no—
433C4⋊C8—731no—
434C4⋊C8—1461no—
435C4⋊C8—1461no—
436C4⋊C8C22×D7311yescharacteristic
437C4⋊C8C22×D7311yescharacteristic
438#102—731no—
439#102—731no—
440#102—731no—
441#102—731no—
442#102—731no—
443#102—731no—
444#102C4×D7311yescharacteristic
445#102C4×D7311yescharacteristic
446C22⋊C8—21no—
447C22⋊C8—21no—
448C22⋊C8—1461no—
449C22⋊C8—1461no—
450C22⋊C8—1461no—
451C22⋊C8—1461no—
452C22⋊C8—1461no—
453C22⋊C8—1461no—
454C4×C8—731no—
455C4×C8—731no—
456C4×C8—731no—
457C4×C8—731no—
458C4×C8—731no—
459C4×C8—731no—
460C4×C8—731no—
461C4×C8—1461no—
462C4×C8C22×D7311yescharacteristic
463C4×C8C22×D7311yescharacteristic
464C4×C8C22×D7311yescharacteristic
465#41—2922no—
466#41—2922no—
467#41—2922no—
468#41—2922no—
469C2×Q8—1461no—
470C2×Q8—1461no—
471C2×D4—731no—
472C2×D4—731no—
473C2×D4—1461no—
474C2×D4—1461no—
475C2×D4—1461no—
476C2×D4—1461no—
477C2×D4—1461no—
478C2×D4—1461no—
479C2×D4—1461no—
480C2×D4—1461no—
481C2×D4C2×C4×D7311yescharacteristic
482C2×D4C2×C4×D7311yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—731no—
488C22×C4—1461no—
489C22×C4—1461no—
490D8—41no—
491D8—2921no—
492D8—2921no—
493D8—2921no—
494M16—42no—
495M16—2922no—
496M16—2922no—
497M16—2922no—
498M16—2922no—
499M16—2922no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—42no—
504C2×C8—2921no—
505C2×C8—2921no—
506C2×C8—2922no—
507C2×C8—2922no—
508C2×C8—2922no—
509C2×C8—2922no—
510C2×C8—731no—
511C2×C8—734no—
512C2×C8—731no—
513C2×C8—731no—
514C2×C8—731no—
515C2×C8—1462no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8C2×C4×D7311yescharacteristic
520C2×C8C2×C4×D7311yescharacteristic
521C2×C8D4×D7311yescharacteristic
522C2×C8D4×D7311yescharacteristic
523C4⋊C4—731no—
524C4⋊C4—731no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4C2×C4×D7311yescharacteristic
528C4⋊C4C2×C4×D7311yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—1461no—
532C22⋊C4—1461no—
533C22⋊C4—1461no—
534C22⋊C4—1461no—
535C22⋊C4—1461no—
536C22⋊C4—1461no—
537C42—731no—
538C42—731no—
539C42—731no—
540C42C23×D7311yescharacteristic
541C23—21no—
542C23—21no—
543C23—731no—
544C23—1461no—
545C23—1461no—
546Q8—2922no—
547Q8—2922no—
548D4—21no—
549D4—21no—
550D4—21no—
551D4—21no—
552D4—2922no—
553D4—2922no—
554D4—2922no—
555D4—2922no—
556D4—2922no—
557D4—2922no—
558D4—2922no—
559D4—2922no—
560D4—1461no—
561D4—1461no—
562D4—1461no—
563D4—1461no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—731no—
575C2×C4—731no—
576C2×C4—732no—
577C2×C4—732no—
578C2×C4—731no—
579C2×C4—1462no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4C22×C4×D7311yescharacteristic
585C2×C4C2×D4×D7311yescharacteristic
586C2×C4#41×D7311yescharacteristic
587C8—21no—
588C8—21no—
589C8—21no—
590C8—41no—
591C8—41no—
592C8—41no—
593C8—2921no—
594C8—2921no—
595C8—2921no—
596C8—732no—
597C8—1461no—
598C8—1461no—
599C8—1461no—
600C8D8×D7312yes—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—2921no—
608C22—2921no—
609C22—2921no—
610C22—2921no—
611C22—2921no—
612C22—2921no—
613C22—732no—
614C22—1461no—
615C22—1461no—
616C22C4×D4×D7311yescharacteristic
617C4—21no—
618C4—41no—
619C4—41no—
620C4—2921no—
621C4—2921no—
622C4—732no—
623C4—732no—
624C4—1461no—
625C4#1798935211yescharacteristic
626C4#1798935311yescharacteristic
627C4C2×D8×D7311yescharacteristic
628C4#135×D7311yescharacteristic
629C2—81no—
630C2—81no—
631C2—5841no—
632C2—5841no—
633C2—1462no—
634C2#2261864511yescharacteristic
635C2C4×D8×D7311yescharacteristic
636C2#444×D7311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC73×#1581C2
socleC22×C73C2×C4×D4
prime radical#22620507C2
cosocleC42×C73C24
hypercenter#1581D73
hypocenterC73#17975
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #22618640C2semidirect
3. #22618638C2semidirect
4. #22608763C2semidirect
5. #22608762C2semidirect
6. #22608761C2semidirect
7. #22608760C2semidirect
8. #22608759C2semidirect
9. #22608758C2semidirect
10. #22608757C2semidirect
11. #22608688C2semidirect
12. #22608380C2semidirect
13. #22608380C2semidirect
14. #322×D73C2semidirect
15. C73×#1581C2semidirect
189. C8×D73D8semidirect
329. #1581D73semidirect
334. C73#17975semidirect

Maximal subgroups

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

subgroupindex
#226205072
#226186402
#226186382
#226087632
#226087622
#226087612
#226087602
#226087592
#226087582
#226087572
#226086882
#226083802
#322×D732
C73×#15812
#1797573