Finite Group Databasebeta

← #33088240

Subgroups — #33088240

Order 18688.

822 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618519C211yesmaximal, 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#22609656C211yesmaximal, characteristic
12#339×D73C211yesmaximal, characteristic
13#339×D73C211yesmaximal, characteristic
14#22609623C211yesmaximal, characteristic
15C73×#1922C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C22×#11924011C2211yescharacteristic
18C2×M16×D73C2211yescharacteristic
19C2×M16×D73C2211yescharacteristic
20#117×D73C2211yescharacteristic
21#117×D73C2211yescharacteristic
22C2×C42×D73C2211yescharacteristic
23#17988628C411yescharacteristic, complemented
24#17988628C411yescharacteristic, complemented
25#17988628C411yescharacteristic, complemented
26#17988628C411yescharacteristic, complemented
27#17988626C411yescharacteristic, complemented
28#17988626C411yescharacteristic, complemented
29#17988626C411yescharacteristic, complemented
30#17988626C411yescharacteristic, complemented
31#17988625C2211yescharacteristic
32#17988623C2211yescharacteristic
33#17988623C2211yescharacteristic
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×D73—21no—
68C22×C4×D73C2311yescharacteristic
69C2×#11924012C2311yescharacteristic
70C2×#11924012C2311yescharacteristic
71C2×#11924011—21no—
72C2×#11924011—22no—
73C2×#11924011—22no—
74C2×#11924011—22no—
75C2×#11924011—22no—
76C2×#11924011—21no—
77C2×#11924011C2×C411yescharacteristic
78C2×#11924011C2×C411yescharacteristic
79C2×#11924011C2×C411yescharacteristic
80C2×#11924011C2×C411yescharacteristic
81C2×#11924011D411yescharacteristic
82C2×#11924011D411yescharacteristic
83C2×#11924011D411yescharacteristic
84C2×#11924011D411yescharacteristic
85M16×D73—22no—
86M16×D73—22no—
87M16×D73—22no—
88M16×D73Q814yes—
89C22×#10596831C2311yescharacteristic
90C22×#10596831C2311yescharacteristic
91C2×C8×D73—21no—
92C2×C8×D73—21no—
93#14556550C2311yescharacteristic
94#14556550C2311yescharacteristic
95#14556549C2×C411yescharacteristic
96#14556549C2×C411yescharacteristic
97#14556549C2×C411yescharacteristic
98#14556549C2×C411yescharacteristic
99#14556548—21no—
100#14556548—21no—
101#14556547—21no—
102#14556547—21no—
103C4⋊C4×D73C2×C411yescharacteristic
104C4⋊C4×D73C2×C411yescharacteristic
105C4⋊C4×D73C2×C411yescharacteristic
106C4⋊C4×D73C2×C411yescharacteristic
107C22⋊C4×D73—21no—
108C22⋊C4×D73—21no—
109C2×C4×Dic73—21no—
110C2×C4×Dic73C2311yescharacteristic
111C42×D73—21no—
112C42×D73—21no—
113C42×D73—22no—
114C42×D73—22no—
115#14556525C2×C411yescharacteristic, complemented
116#14556525C2×C411yescharacteristic, complemented
117#14556524C2×C411yescharacteristic, complemented
118#14556524C2×C411yescharacteristic, complemented
119#14556524C2×C411yescharacteristic, complemented
120#14556524C2×C411yescharacteristic, complemented
121#14556523C2×C411yescharacteristic, complemented
122#14556523C2×C411yescharacteristic, complemented
123#14556522C2×C411yescharacteristic, complemented
124#14556522C2×C411yescharacteristic, complemented
125#14556522C2×C411yescharacteristic, complemented
126#14556522C2×C411yescharacteristic, complemented
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, complemented
147#107×C73C2×C411yescharacteristic, complemented
148#106×C73C2×C411yescharacteristic, complemented
149#106×C73C2×C411yescharacteristic, complemented
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×D73—42no—
162C2×C4×D73—42no—
163C2×C4×D73—42no—
164C2×C4×D73C22×C411yescharacteristic
165C2×C4×D73C22×C411yescharacteristic
166C2×C4×D73C2×D411yescharacteristic
167C2×C4×D73C2×D411yescharacteristic
168C2×C4×D73C2×Q812yes—
169C2×C4×D73#4112yes—
170C2×C4×D73#4112yes—
171C2×C4×D73#4112yes—
172#11924012—21no—
173#11924012—21no—
174#11924012—21no—
175#11924012C2×Q812yes—
176#11924011—22no—
177#11924011—22no—
178#11924011—22no—
179#11924011—22no—
180#11924011—24no—
181#11924011—22no—
182#11924011—22no—
183#11924011—22no—
184#11924011—22no—
185#11924011—42no—
186C2×#10596831—21no—
187C2×#10596831—21no—
188C2×#10596831—21no—
189C2×#10596831—21no—
190C2×#10596831—21no—
191C2×#10596831—21no—
192C2×#10596831C22×C411yescharacteristic
193C2×#10596831C22×C411yescharacteristic
194C2×#10596831C2×D411yescharacteristic
195C2×#10596831C2×D411yescharacteristic
196C8×D73—24no—
197C8×D73—42no—
198C4⋊Dic73C22×C411yescharacteristic
199C4⋊Dic73C22×C411yescharacteristic
200C4⋊Dic73C22×C411yescharacteristic
201C4⋊Dic73C22×C411yescharacteristic
202#11924007—21no—
203#11924007—21no—
204#11924007—21no—
205#11924007—21no—
206C22⋊Dic73—21no—
207C22⋊Dic73—21no—
208#11924005—21no—
209#11924005—21no—
210#11924005—21no—
211#11924005—21no—
212C4×Dic73—21no—
213C4×Dic73—21no—
214C4×Dic73—21no—
215C4×Dic73—21no—
216C4×Dic73—21no—
217C4×Dic73—21no—
218C4×Dic73—21no—
219C4×Dic73—21no—
220C4×Dic73—42no—
221C22×C4×C73—21no—
222C22×C4×C73C2411yescharacteristic
223M16×C73—21no—
224M16×C73—21no—
225M16×C73—21no—
226M16×C73C2×Q812yes—
227C2×C8×C73—21no—
228C2×C8×C73—21no—
229C2×C8×C73—21no—
230C2×C8×C73—21no—
231C2×C8×C73—21no—
232C2×C8×C73—21no—
233C2×C8×C73C22×C411yescharacteristic
234C2×C8×C73C22×C411yescharacteristic
235C2×C8×C73C2×D411yescharacteristic
236C2×C8×C73C2×D411yescharacteristic
237C4⋊C4×C73C22×C411yescharacteristic
238C4⋊C4×C73C22×C411yescharacteristic
239C4⋊C4×C73C22×C411yescharacteristic
240C4⋊C4×C73C22×C411yescharacteristic
241C22⋊C4×C73—21no—
242C22⋊C4×C73—21no—
243C42×C73—21no—
244C42×C73—21no—
245C42×C73—21no—
246C42×C73—21no—
247C23×C73C2×#4111yescharacteristic
248C22×D73—41no—
249C22×D73—41no—
250C22×D73C4×D411yescharacteristic
251C22×D73C4×D411yescharacteristic
252C22×D73#12512yes—
253C22×D73#12612yes—
254C2×C4×C73—22no—
255C2×C4×C73—21no—
256C2×C4×C73—21no—
257C2×C4×C73—21no—
258C2×C4×C73—21no—
259C2×C4×C73—41no—
260C2×C4×C73—41no—
261C2×C4×C73C23×C411yescharacteristic
262C2×C4×C73C22×D411yescharacteristic
263C2×C4×C73C22×Q811yescharacteristic
264C2×C4×C73C2×#4111yescharacteristic
265C2×C4×C73C2×#4111yescharacteristic
266C2×C4×C73C2×#4111yescharacteristic
267C2×Dic73—22no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73C4×D411yescharacteristic
275C2×Dic73C4×D411yescharacteristic
276C2×Dic73#12512yes—
277C2×Dic73#12612yes—
278C4×D73—48no—
279C4×D73—81no—
280C4×D73—81no—
281C4×D73—81no—
282C4×D73—81no—
283C4×D73#11712yes—
284C4×D73#11712yes—
285C4×D73C4×Q812yes—
286C4×D73C4×Q812yes—
287C4×D73C22⋊Q812yes—
288C4×D73C22⋊Q812yes—
289C4×D73#12312yes—
290C4×D73#12312yes—
291C8×C73—21no—
292C8×C73—21no—
293C8×C73—22no—
294C8×C73—21no—
295C8×C73—21no—
296C8×C73—41no—
297#10596831—21no—
298#10596831—21no—
299#10596831—22no—
300#10596831—21no—
301#10596831—21no—
302#10596831—41no—
303C22×C73—21no—
304C22×C73—21no—
305C22×C73C2×C4×D411yescharacteristic
306C22×C73C2×#12511yescharacteristic
307C22×C73C2×#12611yescharacteristic
308C2×D73—21no—
309C2×D73—21no—
310C2×D73—21no—
311C2×D73—21no—
312C2×D73—42no—
313C2×D73—42no—
314C2×D73#38712yes—
315C2×D73#38712yes—
316C4×C73—42no—
317C4×C73—41no—
318C4×C73—41no—
319C4×C73—41no—
320C4×C73—41no—
321C4×C73C2×#11711yescharacteristic
322C4×C73C2×C4×Q811yescharacteristic
323C4×C73C2×C22⋊Q811yescharacteristic
324C4×C73C2×#12311yescharacteristic
325Dic73—42no—
326Dic73—41no—
327Dic73—41no—
328Dic73—41no—
329Dic73—41no—
330Dic73#38712yes—
331Dic73#38712yes—
332#21042—731nomaximal, Sylow
333C2×C73—21no—
334C2×C73—21no—
335C2×C73C2×#38711yescharacteristic
336C2×C73#295611yescharacteristic
337C2×C73#295611yescharacteristic
338D73—22no—
339D73—22no—
340#2911—731no—
341#2905—731no—
342#2873—731no—
343#1922—731no—
344#1922—731no—
345#1922—731no—
346#1922—731no—
347#1922—731no—
348#1922—731no—
349#1922—731no—
350#1922D7311yescharacteristic, complemented
351#1738—731no—
352C2×#339—731no—
353C2×#339—731no—
354#1728—731no—
355C73#2104211yescharacteristic, complemented, Sylow
356C22×M16—731no—
357C22×M16—731no—
358C22×M16—731no—
359C22×M16—731no—
360C2×#117—731no—
361C2×#117—731no—
362C22×C42—731no—
363#428—731no—
364#428—731no—
365#428—731no—
366#428—731no—
367#428—731no—
368#428—731no—
369#428—731no—
370#428—731no—
371#427—731no—
372#427—731no—
373#427—731no—
374#427C2×D7311yescharacteristic
375#424—731no—
376#424—731no—
377#424—731no—
378#424—1461no—
379#424—1461no—
380#424C2×D7311yescharacteristic
381#423—1461no—
382#423—1461no—
383#405—731no—
384#405—731no—
385#405—731no—
386#405—1461no—
387#405—1461no—
388#405C2×D7311yescharacteristic
389C4×M16—1461no—
390C4×M16—1461no—
391#341—731no—
392#341—731no—
393#341—731no—
394#341C2×D7311yescharacteristic
395#339—731no—
396#339—731no—
397#339—731no—
398#339—731no—
399#339—731no—
400#339—731no—
401#339C2×D7311yescharacteristic
402#339C2×D7311yescharacteristic
403#336—731no—
404#336—731no—
405#336—731no—
406#336C2×D7311yescharacteristic
407C23×C4—731no—
408C23×C4—1461no—
409C2×M16—731no—
410C2×M16—731no—
411C2×M16—731no—
412C2×M16—731no—
413C2×M16—731no—
414C2×M16—731no—
415C2×M16—731no—
416C2×M16—731no—
417C2×M16—731no—
418C2×M16—734no—
419C2×M16—731no—
420C2×M16—1461no—
421C2×M16—1461no—
422C2×M16—1462no—
423C2×M16—1462no—
424C2×M16—1462no—
425C2×M16—1462no—
426C2×M16—1462no—
427C2×M16—1462no—
428C2×M16—1462no—
429C2×M16C22×D7311yescharacteristic
430C2×M16C22×D7311yescharacteristic
431C22×C8—731no—
432C22×C8—731no—
433C22×C8—1461no—
434C22×C8—1461no—
435C22×C8C22×D7311yescharacteristic
436C22×C8C22×D7311yescharacteristic
437#117—731no—
438#117—731no—
439#117—731no—
440#117—731no—
441#117—731no—
442#117—731no—
443#117—1461no—
444#117—1461no—
445#117—1461no—
446#117—1461no—
447#117C22×D7311yescharacteristic
448#117C22×D7311yescharacteristic
449C2×C4⋊C4—731no—
450C2×C4⋊C4—731no—
451C2×C4⋊C4—731no—
452C2×C4⋊C4—731no—
453C2×C22⋊C4—1461no—
454C2×C22⋊C4—1461no—
455C2×C42—731no—
456C2×C42—1462no—
457C2×C42—1462no—
458C2×C42—1461no—
459C2×C42—1461no—
460C2×C42—1461no—
461C2×C42C22×D7311yescharacteristic
462#107—731no—
463#107—731no—
464#107—731no—
465#107—731no—
466#107—731no—
467#107—731no—
468#107C4×D7311yescharacteristic, complemented
469#107C4×D7311yescharacteristic, complemented
470#106—731no—
471#106—731no—
472#106—731no—
473#106—731no—
474#106—731no—
475#106—731no—
476#106C4×D7311yescharacteristic, complemented
477#106C4×D7311yescharacteristic, complemented
478C4⋊C8—21no—
479C4⋊C8—21no—
480C4⋊C8—1461no—
481C4⋊C8—1461no—
482C4⋊C8—1461no—
483C4⋊C8—1461no—
484C4⋊C8—1461no—
485C4⋊C8—1461no—
486#97—21no—
487#97—1461no—
488#97—1461no—
489#97—1461no—
490C4×C8—21no—
491C4×C8—1461no—
492C4×C8—1461no—
493C4×C8—1461no—
494C24—731no—
495C22×C4—21no—
496C22×C4—2922no—
497C22×C4—2922no—
498C22×C4—2922no—
499C22×C4—731no—
500C22×C4—731no—
501C22×C4—731no—
502C22×C4—731no—
503C22×C4—731no—
504C22×C4—732no—
505C22×C4—732no—
506C22×C4—732no—
507C22×C4—732no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4—1461no—
511C22×C4—1461no—
512C22×C4—1461no—
513C22×C4C23×D7311yescharacteristic
514M16—21no—
515M16—21no—
516M16—21no—
517M16—2922no—
518M16—732no—
519M16—1462no—
520M16—1462no—
521M16—1462no—
522M16—1462no—
523M16—1464no—
524M16—1462no—
525M16—1462no—
526M16—1462no—
527M16—1462no—
528M16—1461no—
529M16—1461no—
530M16—1461no—
531M16Q8×D7312yes—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—2922no—
539C2×C8—731no—
540C2×C8—731no—
541C2×C8—731no—
542C2×C8—731no—
543C2×C8—1464no—
544C2×C8—1461no—
545C2×C8—1461no—
546C2×C8—1461no—
547C2×C8—1461no—
548C2×C8—1461no—
549C2×C8—1461no—
550C2×C8C2×C4×D7311yescharacteristic
551C2×C8C2×C4×D7311yescharacteristic
552C2×C8D4×D7311yescharacteristic
553C2×C8D4×D7311yescharacteristic
554C4⋊C4—731no—
555C4⋊C4—731no—
556C4⋊C4—731no—
557C4⋊C4—731no—
558C4⋊C4—1461no—
559C4⋊C4—1461no—
560C4⋊C4—1461no—
561C4⋊C4—1461no—
562C4⋊C4C2×C4×D7311yescharacteristic
563C4⋊C4C2×C4×D7311yescharacteristic
564C4⋊C4C2×C4×D7311yescharacteristic
565C4⋊C4C2×C4×D7311yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—1461no—
569C22⋊C4—1461no—
570C22⋊C4—1461no—
571C22⋊C4—1461no—
572C22⋊C4—1461no—
573C22⋊C4—1461no—
574C42—21no—
575C42—21no—
576C42—21no—
577C42—21no—
578C42—2922no—
579C42—1461no—
580C42—1461no—
581C42—1461no—
582C42—1461no—
583C42—1461no—
584C42—1461no—
585C42—1461no—
586C42—1461no—
587C23—2921no—
588C23—2921no—
589C23—732no—
590C23—732no—
591C23—731no—
592C23—731no—
593C23#41×D7311yescharacteristic
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—2928no—
602C2×C4—2921no—
603C2×C4—2921no—
604C2×C4—5841no—
605C2×C4—5841no—
606C2×C4—5841no—
607C2×C4—5841no—
608C2×C4—731no—
609C2×C4—731no—
610C2×C4—732no—
611C2×C4—732no—
612C2×C4—732no—
613C2×C4—732no—
614C2×C4—732no—
615C2×C4—732no—
616C2×C4—732no—
617C2×C4—732no—
618C2×C4—732no—
619C2×C4—732no—
620C2×C4—1462no—
621C2×C4—1461no—
622C2×C4—1461no—
623C2×C4—1461no—
624C2×C4—1461no—
625C2×C4C22×C4×D7311yescharacteristic
626C2×C4C2×D4×D7311yescharacteristic
627C2×C4C2×Q8×D7311yescharacteristic
628C2×C4#41×D7311yescharacteristic
629C2×C4#41×D7311yescharacteristic
630C2×C4#41×D7311yescharacteristic
631C8—21no—
632C8—21no—
633C8—22no—
634C8—21no—
635C8—21no—
636C8—41no—
637C8—2921no—
638C8—1461no—
639C8—1461no—
640C8—1462no—
641C8—1461no—
642C8—1461no—
643C22—21no—
644C22—21no—
645C22—2922no—
646C22—2922no—
647C22—732no—
648C22—732no—
649C22—1461no—
650C22—1461no—
651C22—1461no—
652C22—1461no—
653C22C4×D4×D7311yescharacteristic
654C22#125×D7311yescharacteristic
655C22#126×D7311yescharacteristic
656C4—42no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—2922no—
662C4—2921no—
663C4—2921no—
664C4—2921no—
665C4—2921no—
666C4—732no—
667C4—732no—
668C4#117×D7311yescharacteristic
669C4C4×Q8×D7311yescharacteristic
670C4C22⋊Q8×D7311yescharacteristic
671C4#123×D7311yescharacteristic
672C2—21no—
673C2—21no—
674C2—1462no—
675C2—1462no—
676C2#387×D7311yescharacteristic
677C2#2261883611yescharacteristic
678C2#2261883911yescharacteristic
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#21042
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
23. #17988628C4semidirect
24. #17988628C4semidirect
25. #17988628C4semidirect
26. #17988628C4semidirect
27. #17988626C4semidirect
28. #17988626C4semidirect
29. #17988626C4semidirect
30. #17988626C4semidirect
115. #14556525C2×C4semidirect
116. #14556525C2×C4semidirect
117. #14556524C2×C4semidirect
118. #14556524C2×C4semidirect
119. #14556524C2×C4semidirect
120. #14556524C2×C4semidirect
121. #14556523C2×C4semidirect
122. #14556523C2×C4semidirect
123. #14556522C2×C4semidirect
124. #14556522C2×C4semidirect
125. #14556522C2×C4semidirect
126. #14556522C2×C4semidirect
146. #107×C73C2×C4semidirect
147. #107×C73C2×C4semidirect
148. #106×C73C2×C4semidirect
149. #106×C73C2×C4semidirect
350. #1922D73semidirect
355. C73#21042semidirect
468. #107C4×D73semidirect
469. #107C4×D73semidirect
476. #106C4×D73semidirect
477. #106C4×D73semidirect

Maximal subgroups

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

subgroupindex
#226185192
#226184922
#226183472
#226105332
#226105322
#226105312
#226105302
#226105292
#226105282
#226105272
#226096562
#339×D732
#226096232
C73×#19222
#2104273