Finite Group Databasebeta

← #34031578

Subgroups — #34031578

Order 18688.

1878 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#22626753C214yesmaximal, complemented
2#22626750C214yesmaximal, complemented
3#22626746C214yesmaximal, complemented
4#22626744C214yesmaximal, complemented
5#22626723C211yesmaximal, characteristic, complemented
6#22626721C211yesmaximal, characteristic, complemented
7#22626660C211yesmaximal, characteristic, complemented
8#22626653C211yesmaximal, characteristic, complemented
9#22626649C212yesmaximal, complemented
10#22626645C212yesmaximal, complemented
11#22626618C212yesmaximal, complemented
12#22626612C211yesmaximal, characteristic, complemented
13C4×#14556678C212yesmaximal, complemented
14#22626491C211yesmaximal, characteristic, complemented
15#22623315C211yesmaximal, characteristic
16C2×#14556678C2212yes—
17C2×#14556678C2212yes—
18C2×D73⋊M16C2211yescharacteristic
19C22×#11924036C2211yescharacteristic
20#17989639C2214yescomplemented
21#17989639C2214yescomplemented
22#17989637C2214yescomplemented
23#17989636C2214yescomplemented
24#17989622C2214yes—
25#17989622C2214yescomplemented
26#17989620C2214yes—
27#17989619C2214yes—
28#17989618C2214yes—
29#17989618C2214yes—
30#17989618C2214yescomplemented
31#17989618C2214yescomplemented
32#17989616C2214yes—
33#17989614C2214yes—
34D4×#10596833C2214yes—
35D4×#10596833C2214yescomplemented
36#17989612C2214yes—
37#17989611C2214yes—
38#17989611C2214yescomplemented
39#17989611C2214yescomplemented
40#17989608C2214yescomplemented
41#17989607C2214yescomplemented
42#17989594C2212yes—
43#17989594C2212yes—
44#17989592C2211yescharacteristic
45#17989591C2211yescharacteristic
46#17989561C2212yes—
47#17989561C2212yes—
48#17989558C2212yes—
49#17989552C2212yes—
50#17989552C2212yes—
51#17989552C2212yes—
52#17989552C2212yes—
53#17989551C2212yes—
54#17989551C2212yes—
55#17989551C2214yescomplemented
56#17989551C2214yescomplemented
57C4×#11924038C2212yes—
58C4×#11924037C2214yescomplemented
59#17989546C2211yescharacteristic
60#17989546C2211yescharacteristic
61C4×#11924036C2211yescharacteristic
62C4×#11924036C2211yescharacteristic
63C2×C4×#10596833C2214yescomplemented
64#17989204C412yes—
65#17989202C412yes—
66#17989200C411yescharacteristic
67#17989199C411yescharacteristic
68#17989191C411yescharacteristic
69#17989180C414yes—
70#17989169C414yes—
71#17989143C2211yescharacteristic
72#17989135C2211yescharacteristic
73#17989118C2214yes—
74#17989114C2214yes—
75#17989097C2212yes—
76C4×#11924027C2212yes—
77#117×D73C2211yescharacteristic
78#540×C73C411yescharacteristic
79#14556678—28no—
80#14556678—28no—
81C2×#11924038C2312yes—
82C2×#11924038C2312yes—
83C2×#11924038C2312yes—
84D73⋊M16—24no—
85C2×#11924037C2312yes—
86C2×#11924037C2314yes—
87C2×#11924037C2314yes—
88C2×#11924036—24no—
89C2×#11924036C2311yescharacteristic
90C2×#11924036C2311yescharacteristic
91C2×#11924036C2311yescharacteristic
92C2×#11924036C2311yescharacteristic
93C2×#11924036C2311yescharacteristic
94C2×#11924036C2311yescharacteristic
95C22×#10596833C2311yescharacteristic
96C22×#10596833C2311yescharacteristic
97C22×#10596833C2314yes—
98C22×#10596833C2314yes—
99#14556658C2314yes—
100#14556658C2314yes—
101#14556658C2314yes—
102C4⋊#10596833C2314yes—
103C4⋊#10596833C2314yes—
104C4⋊#10596833C2314yes—
105#14556648C2314yes—
106C22⋊#10596833C2314yes—
107C22⋊#10596833C2314yes—
108C22⋊#10596833C2314yes—
109C22⋊#10596833C2314yes—
110C22⋊#10596833C2314yes—
111#14556646C2312yes—
112#14556646C2314yes—
113#14556646C2312yes—
114#14556646C2314yes—
115#14556646C2312yes—
116#14556646C2312yes—
117#14556646C2314yes—
118C4×#10596833C2314yes—
119C4×#10596833C2312yes—
120C4×#10596833C2314yes—
121C4×#10596833C2312yes—
122C4×#10596833C2312yes—
123C4×#10596833C2314yes—
124C4×#10596833C2312yes—
125C2×#11924027C2312yes—
126C22×C4×D73C2311yescharacteristic
127#14556591C2×C411yescharacteristic
128#14556590C2×C412yes—
129#14556586C2×C414yes—
130#14556584C2×C414yes—
131#14556584C2×C414yes—
132#14556584C2×C414yes—
133#14556583C2×C411yescharacteristic
134#14556582C2×C414yes—
135#14556581C2×C414yes—
136#14556578C2×C412yes—
137#14556578C2×C414yes—
138#14556577C2×C414yes—
139#14556574C2×C414yes—
140#14556573C2×C414yes—
141#14556572C2×C414yes—
142#14556572C2×C414yes—
143#14556571C2×C412yes—
144#14556570C2×C414yes—
145#14556570C2×C412yes—
146#14556569C2×C414yes—
147#14556568C2×C414yes—
148#14556563C2×C414yes—
149#14556562C2×C414yes—
150#14556562C2×C412yes—
151#14556559C2×C414yes—
152C4×Dic146C2×C412yes—
153#14556556C2×C412yes—
154#14556556C2314yes—
155D4×Dic73C2×C412yes—
156D4×Dic73C2314yes—
157D4×Dic73C2314yes—
158C4×C22⋊D73C2×C414yes—
159#14556551C2×C414yes—
160#14556551C2314yes—
161#14556550C2×C411yescharacteristic
162#14556549C2312yes—
163#14556549C2312yes—
164#14556548C2×C411yescharacteristic
165#14556548C2×C411yescharacteristic
166#14556547C2312yes—
167#14556547C2314yes—
168C4⋊C4×D73C2×C412yes—
169C22⋊C4×D73C2×C412yes—
170C2×C4×Dic73C2314yes—
171C42×D73C2311yescharacteristic
172C42×D73C2311yescharacteristic
173#127×C73C2×C411yescharacteristic
174#125×C73C2×C411yescharacteristic
175#123×C73C2×C414yes—
176#121×C73C2×C414yes—
177#121×C73C2×C412yes—
178C4×D4×C73C2×C412yes—
179#117×C73C2×C411yescharacteristic
180#11924038—24no—
181#11924038—24no—
182#11924038—24no—
183#11924037—24no—
184#11924037—28no—
185#11924037—28no—
186#11924036—22no—
187#11924036—22no—
188#11924036—22no—
189#11924036—22no—
190#11924036#4114yes—
191#11924036#4114yes—
192C2×#10596833—22no—
193C2×#10596833—22no—
194C2×#10596833—28no—
195C2×#10596833—28no—
196C2×#10596833C2411yescharacteristic
197C2×#10596833C2411yescharacteristic
198C2×#10596833C2414yes—
199C2×#10596833C2414yes—
200C2×#10596833C2412yes—
201C2×#10596833C2411yescharacteristic
202C2×#10596833C2412yes—
203C2×#10596833C2411yescharacteristic
204C23×D73C22×C411yescharacteristic
205#11924027—24no—
206#11924027—24no—
207C2×Dic146C22×C412yes—
208C2×Dic146C22×C414yes—
209C2×C22⋊D73C22×C414yes—
210C2×C22⋊D73C22×C414yes—
211C22×Dic73C22×C414yes—
212C22×Dic73C2414yes—
213C22×Dic73C2411yescharacteristic
214C2×C4×D73—22no—
215C2×C4×D73—22no—
216C2×C4×D73C22×C411yescharacteristic
217C2×C4×D73C22×C411yescharacteristic
218C2×C4×D73C22×C412yes—
219C2×C4×D73C22×C412yes—
220C2×C4×D73C2411yescharacteristic
221C2×C4×D73C2411yescharacteristic
222C4⋊Dic73C22×C412yes—
223C4⋊Dic73C22×C412yes—
224C4⋊Dic73C22×C414yes—
225C4⋊Dic73C22×C414yes—
226#11924007C22×C412yes—
227#11924007C22×C412yes—
228#11924007C22×C414yes—
229#11924007C22×C414yes—
230#11924007C22×C414yes—
231#11924007C22×C414yes—
232#11924007C22×C414yes—
233C22⋊Dic73C22×C412yes—
234C22⋊Dic73C22×C414yes—
235C22⋊Dic73C22×C414yes—
236C22⋊Dic73C22×C414yes—
237C22⋊Dic73C22×C414yes—
238C22⋊Dic73C22×C414yes—
239#11924005C22×C412yes—
240#11924005C22×C412yes—
241#11924005C22×C414yes—
242#11924005C22×C414yes—
243C4×Dic73C22×C411yescharacteristic
244C4×Dic73C22×C411yescharacteristic
245C4×Dic73C22×C414yes—
246C4×Dic73C22×C414yes—
247C4×Dic73C2412yes—
248C4×Dic73C2412yes—
249C4×Dic73C2414yes—
250C2×D4×C73C22×C412yes—
251C2×D4×C73C22×C414yes—
252C2×D4×C73C22×C414yes—
253C22×C4×C73C22×C411yescharacteristic
254C22×C4×C73C22×C414yes—
255C4⋊C4×C73C22×C412yes—
256C4⋊C4×C73C22×C412yes—
257C4⋊C4×C73C22×C414yes—
258C22⋊C4×C73C22×C412yes—
259C22⋊C4×C73C22×C414yes—
260C22⋊C4×C73C22×C414yes—
261C42×C73C22×C411yescharacteristic
262C42×C73C22×C411yescharacteristic
263C23×C73C23×C411yescharacteristic
264C23×C73C23×C414yes—
265C22×D73—22no—
266C22×D73—42no—
267C22×D73C23×C411yescharacteristic
268C22×D73C23×C411yescharacteristic
269D4×C73—24no—
270D4×C73—44no—
271D4×C73—44no—
272C2×C4×C73—22no—
273C2×C4×C73—44no—
274C2×C4×C73C23×C412yes—
275C2×C4×C73C23×C412yes—
276C2×C4×C73C23×C411yescharacteristic
277C2×C4×C73C23×C414yes—
278C2×C4×C73C23×C411yescharacteristic
279C22⋊D73—28no—
280C22⋊D73—44no—
281C2×Dic73—21no—
282C2×Dic73—21no—
283C2×Dic73—24no—
284C2×Dic73—24no—
285C2×Dic73—44no—
286C2×Dic73C23×C412yes—
287C2×Dic73C23×C414yes—
288C2×Dic73C23×C412yes—
289C2×Dic73C23×C414yes—
290C2×Dic73C23×C411yescharacteristic
291C2×Dic73C2511yescharacteristic
292C4×D73—22no—
293C4×D73—22no—
294C4×D73—21no—
295C4×D73—21no—
296C4×D73—42no—
297C4×D73—42no—
298C4×D73C2×#4112yes—
299C4×D73C2×#4112yes—
300Dic146—24no—
301Dic146—44no—
302#10596833—21no—
303#10596833—21no—
304#10596833—24no—
305#10596833—24no—
306#10596833—22no—
307#10596833C2×#4112yes—
308#10596833C2×#4114yes—
309#10596833C2×#4112yes—
310C22×C73—24no—
311C22×C73—21no—
312C22×C73—24no—
313C22×C73—24no—
314C22×C73—22no—
315C22×C73C24×C411yescharacteristic
316C2×D73—22no—
317C2×D73—21no—
318C2×D73—22no—
319C2×D73—44no—
320C2×D73C4×#4112yes—
321C4×C73—21no—
322C4×C73—24no—
323C4×C73—22no—
324C4×C73—22no—
325C4×C73C4×#4112yes—
326Dic73—24no—
327Dic73—21no—
328Dic73—22no—
329Dic73—24no—
330Dic73—22no—
331Dic73C22×#4111yescharacteristic
332Dic73C22×#4111yescharacteristic
333#60206—731nomaximal, Sylow
334C2×C73—41no—
335C2×C73—44no—
336C2×C73C2×C4×#4111yescharacteristic
337C2×C73#357312yes—
338D73—22no—
339D73—41no—
340#3455—731no—
341#2967—734no—
342#2967—734no—
343#2966—734no—
344#2966—734no—
345#2961—731no—
346#2960—731no—
347#2932—731no—
348#2931—731no—
349#2930—732no—
350#2929—732no—
351#2923—732no—
352#2921—731no—
353C4×#131—732no—
354C2×#405—731no—
355C73#6020611yescharacteristic, complemented, Sylow
356C2×#131—732no—
357C2×#131—732no—
358C22×M16—731no—
359C23×C8—731no—
360#542—732no—
361#542—732no—
362#541—731no—
363#541—731no—
364#540—731no—
365#540#6440811yescharacteristic
366#539—734no—
367#538—734no—
368#533—731no—
369#532—731no—
370#529—734no—
371#529—734no—
372#525—732no—
373C4×#41—732no—
374C2×#117—731no—
375#446—734no—
376#446—734no—
377#446—734no—
378C8×Q8—734no—
379#436—734no—
380#436—734no—
381#436—734no—
382#435—734no—
383#435—734no—
384#435—734no—
385#435—734no—
386#435—734no—
387#435—734no—
388C8×D4—734no—
389C8×D4—734no—
390C8×D4—734no—
391#433—734no—
392#433—734no—
393#433—734no—
394#433—734no—
395#432—734no—
396#432—734no—
397#424—732no—
398#424—732no—
399#423—731no—
400#423—731no—
401#408—732no—
402#408—732no—
403#407—732no—
404#405—732no—
405#405—732no—
406#405—732no—
407#405—732no—
408#405—732no—
409#405—732no—
410#405—734no—
411#405—734no—
412C4×M16—732no—
413C4×M16—734no—
414C2×#97—731no—
415C2×#97—731no—
416C2×C4×C8—731no—
417C2×C4×C8—731no—
418C2×C4×C8—734no—
419C2×#41—732no—
420C23×C4—731no—
421#131—1468no—
422#131—1468no—
423C2×M16—732no—
424C2×M16—734no—
425C2×M16—734no—
426C2×M16—732no—
427C2×M16—732no—
428C2×M16—732no—
429C2×M16—1464no—
430C22×C8—731no—
431C22×C8—731no—
432C22×C8—731no—
433C22×C8—731no—
434C22×C8—734no—
435C22×C8—734no—
436C22×C8—731no—
437C22×C8—731no—
438C22×C8—731no—
439C22×C8—731no—
440C22×C8—1464no—
441C4⋊Q8—732no—
442C4⋊Q8—731no—
443#127C2×#6440811yescharacteristic
444#126—734no—
445#126—734no—
446#126—734no—
447#126—734no—
448#125—734no—
449#125—734no—
450#125—731no—
451#125C2×#6440811yescharacteristic
452#124—734no—
453#124—732no—
454#124—734no—
455#123—734no—
456#123—734no—
457#123—734no—
458#123—734no—
459#123C2×#6440814yes—
460C22⋊Q8—732no—
461C22⋊Q8—734no—
462C22⋊Q8—732no—
463C22⋊Q8—734no—
464C22⋊Q8—734no—
465#121—732no—
466#121—734no—
467#121—734no—
468#121C2×#6440812yes—
469#121C2×#6440814yes—
470#120—734no—
471C4×Q8—732no—
472C4×Q8—734no—
473C4×Q8—732no—
474C4×D4—732no—
475C4×D4—734no—
476C4×D4—734no—
477C4×D4—734no—
478C4×D4—734no—
479C4×D4—734no—
480C4×D4C2×#6440812yes—
481#117—731no—
482#117—732no—
483#117—732no—
484#117—732no—
485#117—734no—
486#117—731no—
487#117—731no—
488#117C2×#6440811yescharacteristic
489C2×C4⋊C4—732no—
490C2×C22⋊C4—732no—
491C2×C42—734no—
492C2×C42—731no—
493C2×C42—731no—
494C4⋊C8—734no—
495C4⋊C8—734no—
496C4⋊C8—734no—
497C4⋊C8—734no—
498C4⋊C8—734no—
499C4⋊C8—734no—
500C22⋊C8—734no—
501C22⋊C8—734no—
502C22⋊C8—734no—
503C22⋊C8—734no—
504C22⋊C8—734no—
505C22⋊C8—734no—
506#97—732no—
507#97—732no—
508#97—732no—
509#97—732no—
510#97—734no—
511#97—734no—
512#97—734no—
513C4×C8—732no—
514C4×C8—732no—
515C4×C8—734no—
516C4×C8—732no—
517C4×C8—734no—
518C4×C8—732no—
519C4×C8—734no—
520C24—731no—
521#41—1464no—
522#41—1464no—
523C2×Q8—734no—
524C2×Q8—732no—
525C2×D4—734no—
526C2×D4—734no—
527C2×D4C22×#6440812yes—
528C2×D4C22×#6440814yes—
529C2×D4C22×#6440814yes—
530C22×C4—734no—
531C22×C4—732no—
532C22×C4—732no—
533C22×C4—731no—
534C22×C4—734no—
535C22×C4—731no—
536C22×C4—731no—
537C22×C4—731no—
538C22×C4—731no—
539C22×C4—1462no—
540C22×C4—1462no—
541C22×C4C22×#6440811yescharacteristic
542C22×C4C22×#6440814yes—
543M16—1464no—
544M16—1468no—
545M16—1468no—
546M16—1464no—
547M16—1464no—
548M16—1464no—
549C2×C8—731no—
550C2×C8—732no—
551C2×C8—731no—
552C2×C8—734no—
553C2×C8—734no—
554C2×C8—734no—
555C2×C8—734no—
556C2×C8—732no—
557C2×C8—731no—
558C2×C8—731no—
559C2×C8—1462no—
560C2×C8—1462no—
561C2×C8—1468no—
562C2×C8—1468no—
563C2×C8—1462no—
564C2×C8—1462no—
565C2×C8—1462no—
566C2×C8—1462no—
567C4⋊C4—732no—
568C4⋊C4—732no—
569C4⋊C4—734no—
570C4⋊C4—734no—
571C4⋊C4—734no—
572C4⋊C4—734no—
573C4⋊C4—734no—
574C4⋊C4—732no—
575C4⋊C4—732no—
576C4⋊C4—734no—
577C4⋊C4—734no—
578C4⋊C4C22×#6440812yes—
579C4⋊C4C22×#6440812yes—
580C4⋊C4C22×#6440814yes—
581C22⋊C4—732no—
582C22⋊C4—734no—
583C22⋊C4—734no—
584C22⋊C4—734no—
585C22⋊C4—734no—
586C22⋊C4—734no—
587C22⋊C4—732no—
588C22⋊C4—732no—
589C22⋊C4—734no—
590C22⋊C4—734no—
591C22⋊C4C22×#6440812yes—
592C22⋊C4C22×#6440814yes—
593C22⋊C4C22×#6440814yes—
594C42—731no—
595C42—731no—
596C42—734no—
597C42—734no—
598C42—732no—
599C42—732no—
600C42—734no—
601C42C22×#6440811yescharacteristic
602C42C22×#6440811yescharacteristic
603C23—2922no—
604C23—731no—
605C23—731no—
606C23—1462no—
607C23C23×#6440811yescharacteristic
608C23C23×#6440814yes—
609Q8—2924no—
610Q8—1464no—
611D4—24no—
612D4—44no—
613D4—44no—
614D4—2924no—
615D4—1468no—
616C2×C4—22no—
617C2×C4—44no—
618C2×C4—2922no—
619C2×C4—2922no—
620C2×C4—2924no—
621C2×C4—731no—
622C2×C4—732no—
623C2×C4—732no—
624C2×C4—732no—
625C2×C4—732no—
626C2×C4—734no—
627C2×C4—734no—
628C2×C4—731no—
629C2×C4—1461no—
630C2×C4—1461no—
631C2×C4—1462no—
632C2×C4—1461no—
633C2×C4—1464no—
634C2×C4—1464no—
635C2×C4—1461no—
636C2×C4—1462no—
637C2×C4C23×#6440812yes—
638C2×C4C23×#6440814yes—
639C2×C4C23×#6440812yes—
640C2×C4C23×#6440811yescharacteristic
641C2×C4C23×#6440811yescharacteristic
642C8—732no—
643C8—732no—
644C8—734no—
645C8—1461no—
646C8—1461no—
647C8—1464no—
648C8—1464no—
649C8—1462no—
650C22—21no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—22no—
655C22—2924no—
656C22—732no—
657C22—1461no—
658C22—1462no—
659C22—1462no—
660C22C24×#6440811yescharacteristic
661C4—21no—
662C4—22no—
663C4—24no—
664C4—22no—
665C4—731no—
666C4—731no—
667C4—1461no—
668C4—1464no—
669C4—1462no—
670C4—1464no—
671C4—1462no—
672C4#41×#6440812yes—
673C2—41no—
674C2—44no—
675C2—2921no—
676C2—1462no—
677C2C2×#41×#6440811yescharacteristic
678C2#2262692612yes—
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#540×C73C4
prime radical#22623315C2
cosocleC2×Dic73C25
hypercenter#540#64408
hypocenterC73#60206
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #22626753C2semidirect
2. #22626750C2semidirect
3. #22626746C2semidirect
4. #22626744C2semidirect
5. #22626723C2semidirect
6. #22626721C2semidirect
7. #22626660C2semidirect
8. #22626653C2semidirect
9. #22626649C2semidirect
10. #22626645C2semidirect
11. #22626618C2semidirect
12. #22626612C2semidirect
13. C4×#14556678C2semidirect
14. #22626491C2semidirect
20. #17989639C22semidirect
21. #17989639C22semidirect
22. #17989637C22semidirect
23. #17989636C22semidirect
25. #17989622C22semidirect
30. #17989618C22semidirect
31. #17989618C22semidirect
35. D4×#10596833C22semidirect
38. #17989611C22semidirect
39. #17989611C22semidirect
40. #17989608C22semidirect
41. #17989607C22semidirect
55. #17989551C22semidirect
56. #17989551C22semidirect
58. C4×#11924037C22semidirect
63. C2×C4×#10596833C22semidirect
355. C73#60206semidirect

Maximal subgroups

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

subgroupindex
#226267532
#226267502
#226267462
#226267442
#226267232
#226267212
#226266602
#226266532
#226266492
#226266452
#226266182
#226266122
C4×#145566782
#226264912
#226233152
#6020673