Finite Group Databasebeta

← #33088248

Subgroups — #33088248

Order 18688.

790 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#427×D73C211yesmaximal, characteristic
2#22618492C211yesmaximal, characteristic
3#22618347C211yesmaximal, characteristic
4#22610533C211yesmaximal, characteristic, complemented
5#22610532C211yesmaximal, characteristic, complemented
6#22610531C211yesmaximal, characteristic, complemented
7#22610530C211yesmaximal, characteristic, complemented
8#22610529C211yesmaximal, characteristic, complemented
9#22610528C211yesmaximal, characteristic, complemented
10#22610527C211yesmaximal, characteristic, complemented
11#341×D73C211yesmaximal, characteristic
12#22609639C211yesmaximal, characteristic
13#22609639C211yesmaximal, characteristic
14#22609626C211yesmaximal, characteristic
15C73×#1922C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×M16×D73C2211yescharacteristic
18C2×M16×D73C2211yescharacteristic
19C22×C8×D73C2211yescharacteristic
20C2×#14556548C2211yescharacteristic
21#117×D73C2211yescharacteristic
22#117×D73C2211yescharacteristic
23#17988625C2211yescharacteristic
24#17988624C411yescharacteristic, complemented
25#17988624C411yescharacteristic, complemented
26#17988623C2211yescharacteristic
27#17988623C2211yescharacteristic
28#17988622C411yescharacteristic, complemented
29#17988622C411yescharacteristic, complemented
30#107×D73C411yescharacteristic, complemented
31#107×D73C411yescharacteristic, complemented
32#106×D73C411yescharacteristic, complemented
33#106×D73C411yescharacteristic, complemented
34#17988615C2211yescharacteristic
35#17988614C2211yescharacteristic
36#17988613C2211yescharacteristic
37#17988612—21no—
38#17988612—21no—
39#17988609—21no—
40#17988609—21no—
41#17988545C2211yescharacteristic
42#17988544—21no—
43#17988543C2211yescharacteristic
44#17988542C2211yescharacteristic
45#17988541—21no—
46#17988539—21no—
47C4×#11924011—21no—
48#17988352C2211yescharacteristic
49#17988351C2211yescharacteristic
50#17988351C2211yescharacteristic
51#17988348C2211yescharacteristic
52#17988348C2211yescharacteristic
53#17988347C2211yescharacteristic
54#17988347C2211yescharacteristic
55#17988346C2211yescharacteristic
56#17988346C2211yescharacteristic
57#17988342C2211yescharacteristic
58#17988341C2211yescharacteristic
59#17988341C2211yescharacteristic
60#427×C73C2211yescharacteristic
61#424×C73C2211yescharacteristic
62#405×C73C2211yescharacteristic
63#341×C73C2211yescharacteristic
64#339×C73C2211yescharacteristic
65#339×C73C2211yescharacteristic
66#336×C73C2211yescharacteristic
67C22×C4×D73C2311yescharacteristic
68C2×#11924012C2311yescharacteristic
69C2×#11924012C2311yescharacteristic
70C2×#11924011—21no—
71C2×#11924011—22no—
72C2×#11924011—22no—
73C2×#11924011—21no—
74C2×#11924011D411yescharacteristic
75C2×#11924011D411yescharacteristic
76C2×#11924011D411yescharacteristic
77C2×#11924011D411yescharacteristic
78M16×D73—22no—
79M16×D73—22no—
80M16×D73—22no—
81M16×D73Q814yes—
82C22×#10596831C2311yescharacteristic
83C22×#10596831C2311yescharacteristic
84C2×C8×D73—21no—
85C2×C8×D73—22no—
86C2×C8×D73—22no—
87C2×C8×D73—21no—
88C2×C8×D73C2×C411yescharacteristic
89C2×C8×D73C2×C411yescharacteristic
90C2×C8×D73C2×C411yescharacteristic
91C2×C8×D73C2×C411yescharacteristic
92#14556550C2311yescharacteristic
93#14556550C2311yescharacteristic
94#14556549C2×C411yescharacteristic
95#14556549C2×C411yescharacteristic
96#14556549C2×C411yescharacteristic
97#14556549C2×C411yescharacteristic
98#14556548—21no—
99#14556548—21no—
100#14556548—22no—
101#14556548—22no—
102#14556547—21no—
103#14556547—21no—
104C2×#11924007—21no—
105C4⋊C4×D73C2×C411yescharacteristic
106C4⋊C4×D73C2×C411yescharacteristic
107C4⋊C4×D73C2×C411yescharacteristic
108C4⋊C4×D73C2×C411yescharacteristic
109C2×#11924005—21no—
110C22⋊C4×D73—21no—
111C22⋊C4×D73—21no—
112C2×C4×Dic73C2311yescharacteristic
113C42×D73—21no—
114C42×D73—21no—
115#14556525C2×C411yescharacteristic
116#14556525C2×C411yescharacteristic
117#14556524C2×C411yescharacteristic
118#14556524C2×C411yescharacteristic
119#14556524C2×C411yescharacteristic
120#14556524C2×C411yescharacteristic
121#14556523C2×C411yescharacteristic
122#14556523C2×C411yescharacteristic
123#14556522C2×C411yescharacteristic
124#14556522C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556522C2×C411yescharacteristic
127C4⋊#10596831—21no—
128C4⋊#10596831—21no—
129#14556520—21no—
130#14556520—21no—
131#14556520—21no—
132#14556520—21no—
133#14556502—21no—
134#14556501—21no—
135#14556501—21no—
136C4×#10596831—21no—
137C8×Dic73—21no—
138C8×Dic73—21no—
139C2×M16×C73C2311yescharacteristic
140C2×M16×C73C2311yescharacteristic
141C22×C8×C73C2311yescharacteristic
142C22×C8×C73C2311yescharacteristic
143#117×C73C2311yescharacteristic
144#117×C73C2311yescharacteristic
145C2×C42×C73C2311yescharacteristic
146#107×C73C2×C411yescharacteristic
147#107×C73C2×C411yescharacteristic
148#106×C73C2×C411yescharacteristic
149#106×C73C2×C411yescharacteristic
150C4⋊C8×C73—21no—
151C4⋊C8×C73—21no—
152#97×C73—21no—
153C4×C8×C73—21no—
154C23×D73#4111yescharacteristic
155C22×Dic73—21no—
156C22×Dic73#4111yescharacteristic
157C2×C4×D73—21no—
158C2×C4×D73—21no—
159C2×C4×D73—21no—
160C2×C4×D73—21no—
161C2×C4×D73C22×C411yescharacteristic
162C2×C4×D73C22×C411yescharacteristic
163C2×C4×D73C2×D411yescharacteristic
164C2×C4×D73C2×D411yescharacteristic
165C2×C4×D73C2×Q812yes—
166C2×C4×D73#4112yes—
167C2×C4×D73#4112yes—
168C2×C4×D73#4112yes—
169#11924012—21no—
170#11924012—21no—
171#11924012—21no—
172#11924012C2×Q812yes—
173#11924011—24no—
174#11924011—22no—
175#11924011—22no—
176#11924011—22no—
177#11924011—22no—
178#11924011—42no—
179C2×#10596831—21no—
180C2×#10596831—21no—
181C2×#10596831—21no—
182C2×#10596831—21no—
183C2×#10596831—21no—
184C2×#10596831—21no—
185C2×#10596831C22×C411yescharacteristic
186C2×#10596831C22×C411yescharacteristic
187C2×#10596831C2×D411yescharacteristic
188C2×#10596831C2×D411yescharacteristic
189C8×D73—22no—
190C8×D73—22no—
191C8×D73—22no—
192C8×D73—22no—
193C8×D73—24no—
194C8×D73—42no—
195C4⋊Dic73C22×C411yescharacteristic
196C4⋊Dic73C22×C411yescharacteristic
197C4⋊Dic73C22×C411yescharacteristic
198C4⋊Dic73C22×C411yescharacteristic
199#11924007—21no—
200#11924007—21no—
201#11924007—21no—
202#11924007—21no—
203#11924007—42no—
204C22⋊Dic73—21no—
205C22⋊Dic73—21no—
206#11924005—21no—
207#11924005—21no—
208#11924005—21no—
209#11924005—21no—
210#11924005—42no—
211C4×Dic73—21no—
212C4×Dic73—21no—
213C4×Dic73—21no—
214C4×Dic73—21no—
215C4×Dic73—21no—
216C4×Dic73—21no—
217C4×Dic73—21no—
218C4×Dic73—21no—
219C22×C4×C73—21no—
220C22×C4×C73C2411yescharacteristic
221M16×C73—21no—
222M16×C73—21no—
223M16×C73—21no—
224M16×C73C2×Q812yes—
225C2×C8×C73—21no—
226C2×C8×C73—21no—
227C2×C8×C73—21no—
228C2×C8×C73—21no—
229C2×C8×C73—21no—
230C2×C8×C73—21no—
231C2×C8×C73C22×C411yescharacteristic
232C2×C8×C73C22×C411yescharacteristic
233C2×C8×C73C2×D411yescharacteristic
234C2×C8×C73C2×D411yescharacteristic
235C4⋊C4×C73C22×C411yescharacteristic
236C4⋊C4×C73C22×C411yescharacteristic
237C4⋊C4×C73C22×C411yescharacteristic
238C4⋊C4×C73C22×C411yescharacteristic
239C22⋊C4×C73—21no—
240C22⋊C4×C73—21no—
241C42×C73—21no—
242C42×C73—21no—
243C42×C73—21no—
244C42×C73—21no—
245C23×C73C2×#4111yescharacteristic
246C22×D73—41no—
247C22×D73—41no—
248C22×D73C4×D411yescharacteristic
249C22×D73C4×D411yescharacteristic
250C22×D73#12512yes—
251C22×D73#12612yes—
252C2×C4×C73—21no—
253C2×C4×C73—21no—
254C2×C4×C73—21no—
255C2×C4×C73—21no—
256C2×C4×C73—41no—
257C2×C4×C73—41no—
258C2×C4×C73—41no—
259C2×C4×C73C23×C411yescharacteristic
260C2×C4×C73C22×D411yescharacteristic
261C2×C4×C73C22×Q811yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
263C2×C4×C73C2×#4111yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73C4×D411yescharacteristic
273C2×Dic73C4×D411yescharacteristic
274C2×Dic73#12512yes—
275C2×Dic73#12612yes—
276C4×D73—81no—
277C4×D73—81no—
278C4×D73—81no—
279C4×D73—81no—
280C4×D73#11712yes—
281C4×D73#11712yes—
282C4×D73C4×Q812yes—
283C4×D73C4×Q812yes—
284C4×D73C22⋊Q812yes—
285C4×D73C22⋊Q812yes—
286C4×D73#12312yes—
287C4×D73#12312yes—
288C8×C73—21no—
289C8×C73—21no—
290C8×C73—22no—
291C8×C73—21no—
292C8×C73—21no—
293C8×C73—41no—
294#10596831—21no—
295#10596831—21no—
296#10596831—22no—
297#10596831—21no—
298#10596831—21no—
299#10596831—41no—
300C22×C73—21no—
301C22×C73—21no—
302C22×C73C2×C4×D411yescharacteristic
303C22×C73C2×#12511yescharacteristic
304C22×C73C2×#12611yescharacteristic
305C2×D73—21no—
306C2×D73—21no—
307C2×D73—21no—
308C2×D73—21no—
309C2×D73—42no—
310C2×D73—42no—
311C2×D73#38712yes—
312C2×D73#38712yes—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—41no—
317C4×C73—81no—
318C4×C73C2×#11711yescharacteristic
319C4×C73C2×C4×Q811yescharacteristic
320C4×C73C2×C22⋊Q811yescharacteristic
321C4×C73C2×#12311yescharacteristic
322Dic73—41no—
323Dic73—41no—
324Dic73—41no—
325Dic73—41no—
326Dic73—81no—
327Dic73#38712yes—
328Dic73#38712yes—
329#21043—731nomaximal, Sylow
330C2×C73—21no—
331C2×C73—21no—
332C2×C73C2×#38711yescharacteristic
333C2×C73#295711yescharacteristic
334C2×C73#295711yescharacteristic
335D73—22no—
336D73—22no—
337C2×#427—731no—
338#2905—731no—
339#2873—731no—
340#1922—731no—
341#1922—731no—
342#1922—731no—
343#1922—731no—
344#1922—731no—
345#1922—731no—
346#1922—731no—
347#1922D7311yescharacteristic, complemented
348C2×#341—731no—
349#1733—731no—
350#1733—731no—
351#1729—731no—
352C73#2104311yescharacteristic, complemented, Sylow
353C22×M16—731no—
354C22×M16—731no—
355C22×M16—731no—
356C23×C8—731no—
357C2×#117—731no—
358C2×#117—731no—
359C2×#117—731no—
360#427—731no—
361#427—731no—
362#427—731no—
363#427—731no—
364#427—731no—
365#427—731no—
366#427—731no—
367#427C2×D7311yescharacteristic
368C2×#107—731no—
369C2×#107—731no—
370C2×#106—731no—
371C2×#106—731no—
372#424—731no—
373#424—731no—
374#424—731no—
375#424—1461no—
376#424—1461no—
377#424C2×D7311yescharacteristic
378#423—1461no—
379#423—1461no—
380#405—731no—
381#405—731no—
382#405—731no—
383#405—1461no—
384#405—1461no—
385#405C2×D7311yescharacteristic
386C4×M16—1461no—
387C4×M16—1461no—
388#341—731no—
389#341—731no—
390#341—731no—
391#341C2×D7311yescharacteristic
392#339—731no—
393#339—731no—
394#339—731no—
395#339—731no—
396#339—731no—
397#339—731no—
398#339C2×D7311yescharacteristic
399#339C2×D7311yescharacteristic
400#336—731no—
401#336—731no—
402#336—731no—
403#336C2×D7311yescharacteristic
404C23×C4—731no—
405C2×M16—731no—
406C2×M16—731no—
407C2×M16—731no—
408C2×M16—731no—
409C2×M16—731no—
410C2×M16—734no—
411C2×M16—731no—
412C2×M16—1461no—
413C2×M16—1461no—
414C2×M16—1462no—
415C2×M16—1462no—
416C2×M16—1462no—
417C2×M16—1462no—
418C2×M16—1462no—
419C2×M16C22×D7311yescharacteristic
420C2×M16C22×D7311yescharacteristic
421C22×C8—731no—
422C22×C8—731no—
423C22×C8—731no—
424C22×C8—731no—
425C22×C8—731no—
426C22×C8—731no—
427C22×C8—1461no—
428C22×C8—1462no—
429C22×C8—1462no—
430C22×C8—1461no—
431C22×C8C22×D7311yescharacteristic
432C22×C8C22×D7311yescharacteristic
433#117—731no—
434#117—731no—
435#117—731no—
436#117—731no—
437#117—731no—
438#117—731no—
439#117—1462no—
440#117—1462no—
441#117—1461no—
442#117—1461no—
443#117—1461no—
444#117—1461no—
445#117C22×D7311yescharacteristic
446#117C22×D7311yescharacteristic
447C2×C4⋊C4—731no—
448C2×C4⋊C4—731no—
449C2×C4⋊C4—731no—
450C2×C4⋊C4—731no—
451C2×C4⋊C4—1461no—
452C2×C22⋊C4—1461no—
453C2×C22⋊C4—1461no—
454C2×C22⋊C4—1461no—
455C2×C42—731no—
456C2×C42—1461no—
457C2×C42—1461no—
458C2×C42C22×D7311yescharacteristic
459#107—731no—
460#107—731no—
461#107—731no—
462#107—731no—
463#107—731no—
464#107—731no—
465#107C4×D7311yescharacteristic
466#107C4×D7311yescharacteristic
467#106—731no—
468#106—731no—
469#106—731no—
470#106—731no—
471#106—731no—
472#106—731no—
473#106C4×D7311yescharacteristic
474#106C4×D7311yescharacteristic
475C4⋊C8—21no—
476C4⋊C8—21no—
477C4⋊C8—1461no—
478C4⋊C8—1461no—
479C4⋊C8—1461no—
480C4⋊C8—1461no—
481C4⋊C8—1461no—
482C4⋊C8—1461no—
483#97—21no—
484#97—1461no—
485#97—1461no—
486#97—1461no—
487C4×C8—21no—
488C4×C8—1461no—
489C4×C8—1461no—
490C4×C8—1461no—
491C24—731no—
492C22×C4—21no—
493C22×C4—731no—
494C22×C4—731no—
495C22×C4—731no—
496C22×C4—731no—
497C22×C4—731no—
498C22×C4—732no—
499C22×C4—732no—
500C22×C4—732no—
501C22×C4—732no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4C23×D7311yescharacteristic
508M16—21no—
509M16—21no—
510M16—21no—
511M16—2922no—
512M16—732no—
513M16—1464no—
514M16—1462no—
515M16—1462no—
516M16—1462no—
517M16—1462no—
518M16—1461no—
519M16—1461no—
520M16—1461no—
521M16Q8×D7312yes—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—2922no—
529C2×C8—731no—
530C2×C8—731no—
531C2×C8—731no—
532C2×C8—731no—
533C2×C8—1462no—
534C2×C8—1462no—
535C2×C8—1462no—
536C2×C8—1462no—
537C2×C8—1464no—
538C2×C8—1461no—
539C2×C8—1461no—
540C2×C8—1461no—
541C2×C8—1461no—
542C2×C8—1461no—
543C2×C8—1461no—
544C2×C8C2×C4×D7311yescharacteristic
545C2×C8C2×C4×D7311yescharacteristic
546C2×C8D4×D7311yescharacteristic
547C2×C8D4×D7311yescharacteristic
548C4⋊C4—2922no—
549C4⋊C4—731no—
550C4⋊C4—731no—
551C4⋊C4—731no—
552C4⋊C4—731no—
553C4⋊C4—1461no—
554C4⋊C4—1461no—
555C4⋊C4—1461no—
556C4⋊C4—1461no—
557C4⋊C4C2×C4×D7311yescharacteristic
558C4⋊C4C2×C4×D7311yescharacteristic
559C4⋊C4C2×C4×D7311yescharacteristic
560C4⋊C4C2×C4×D7311yescharacteristic
561C22⋊C4—21no—
562C22⋊C4—21no—
563C22⋊C4—2922no—
564C22⋊C4—1461no—
565C22⋊C4—1461no—
566C22⋊C4—1461no—
567C22⋊C4—1461no—
568C22⋊C4—1461no—
569C22⋊C4—1461no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—1461no—
575C42—1461no—
576C42—1461no—
577C42—1461no—
578C42—1461no—
579C42—1461no—
580C42—1461no—
581C42—1461no—
582C23—2921no—
583C23—2921no—
584C23—732no—
585C23—732no—
586C23—731no—
587C23—731no—
588C23#41×D7311yescharacteristic
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—5841no—
600C2×C4—5841no—
601C2×C4—5841no—
602C2×C4—5841no—
603C2×C4—731no—
604C2×C4—731no—
605C2×C4—732no—
606C2×C4—732no—
607C2×C4—732no—
608C2×C4—732no—
609C2×C4—732no—
610C2×C4—732no—
611C2×C4—732no—
612C2×C4—732no—
613C2×C4—732no—
614C2×C4—732no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1461no—
618C2×C4—1461no—
619C2×C4C22×C4×D7311yescharacteristic
620C2×C4C2×D4×D7311yescharacteristic
621C2×C4C2×Q8×D7311yescharacteristic
622C2×C4#41×D7311yescharacteristic
623C2×C4#41×D7311yescharacteristic
624C2×C4#41×D7311yescharacteristic
625C8—21no—
626C8—21no—
627C8—22no—
628C8—21no—
629C8—21no—
630C8—41no—
631C8—2921no—
632C8—1461no—
633C8—1461no—
634C8—1462no—
635C8—1461no—
636C8—1461no—
637C22—21no—
638C22—21no—
639C22—2922no—
640C22—2922no—
641C22—732no—
642C22—732no—
643C22—1461no—
644C22—1461no—
645C22—1461no—
646C22—1461no—
647C22C4×D4×D7311yescharacteristic
648C22#125×D7311yescharacteristic
649C22#126×D7311yescharacteristic
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—81no—
655C4—2921no—
656C4—2921no—
657C4—2921no—
658C4—2921no—
659C4—5841no—
660C4—732no—
661C4—732no—
662C4#117×D7311yescharacteristic
663C4C4×Q8×D7311yescharacteristic
664C4C22⋊Q8×D7311yescharacteristic
665C4#123×D7311yescharacteristic
666C2—21no—
667C2—21no—
668C2—1462no—
669C2—1462no—
670C2#387×D7311yescharacteristic
671C2#2261884311yescharacteristic
672C2#2261884511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1922C2
socleC22×C73C2×C4×D4
prime radicalC23×D73#41
cosocleC22×C4×C73C24
hypercenter#1922D73
hypocenterC73#21043
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

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

NH ≅ G/Ntype
4. #22610533C2semidirect
5. #22610532C2semidirect
6. #22610531C2semidirect
7. #22610530C2semidirect
8. #22610529C2semidirect
9. #22610528C2semidirect
10. #22610527C2semidirect
15. C73×#1922C2semidirect
24. #17988624C4semidirect
25. #17988624C4semidirect
28. #17988622C4semidirect
29. #17988622C4semidirect
30. #107×D73C4semidirect
31. #107×D73C4semidirect
32. #106×D73C4semidirect
33. #106×D73C4semidirect
347. #1922D73semidirect
352. C73#21043semidirect

Maximal subgroups

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

subgroupindex
#427×D732
#226184922
#226183472
#226105332
#226105322
#226105312
#226105302
#226105292
#226105282
#226105272
#341×D732
#226096392
#226096262
C73×#19222
#2104373