Finite Group Databasebeta

← #32991523

Subgroups — #32991523

Order 18688.

1186 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
1C2×#17989353C211yesmaximal, characteristic
2C2×#17988603C211yesmaximal, characteristic
3C2×#17988586C211yesmaximal, characteristic
4C2×#17988359C211yesmaximal, characteristic, complemented
5#22609650C211yesmaximal, characteristic, complemented
6C2×#17988345C211yesmaximal, characteristic, complemented
7C2×#343×C73C211yesmaximal, characteristic, complemented
8C22×#11924028C2211yescharacteristic
9#17989353—22no—
10#17989353—22no—
11#17989353—22no—
12#17989353C414yescomplemented
13C2×#14556599—22no—
14C2×#14556597—22no—
15C22×#11924012C2211yescharacteristic
16C22×#11924012C2211yescharacteristic
17C2×M16×D73—22no—
18C2×#14556550C2211yescharacteristic
19#17988603—22no—
20#17988603C414yescomplemented
21C2×#14556517—22no—
22#17988586—22no—
23#17988586—22no—
24C2×#14556514C412yescomplemented
25C2×#14556511C412yescomplemented
26#17988359—22no—
27#17988349—22no—
28#17988345—22no—
29C22×M16×C73C2211yescharacteristic
30C22×M16×C73C2211yescharacteristic
31C22×C4⋊C4×C73C2211yescharacteristic
32#343×C73—22no—
33C2×#11924028—21no—
34C2×#11924028—21no—
35C2×#11924028—21no—
36C2×#11924028—21no—
37C2×#11924028—21no—
38C2×#11924028—21no—
39C2×#11924028C2×C412yes—
40C2×#11924028C2×C412yes—
41C2×#11924028D412yes—
42C2×#11924028D412yes—
43C2×#11924028D412yes—
44C2×#11924028Q812yes—
45C22×Dic146C2×C411yescharacteristic
46C22×D292C2×C411yescharacteristic
47C22×C22⋊D73—21no—
48C22×C4×D73—21no—
49#14556599—28no—
50#14556599—28no—
51#14556597—28no—
52#14556597—28no—
53C2×#11924012—21no—
54C2×#11924012—21no—
55C2×#11924012—21no—
56C2×#11924012—21no—
57C2×#11924012—21no—
58C2×#11924012—21no—
59C2×#11924012—21no—
60C2×#11924012—21no—
61C2×#11924012—21no—
62C2×#11924012—21no—
63C2×#11924012C2×C412yes—
64C2×#11924012D412yes—
65C2×#11924011—24no—
66M16×D73—28no—
67M16×D73—28no—
68C22×#10596831C2×C412yes—
69C22×#10596831D412yes—
70C2×C8×D73—24no—
71#14556550—21no—
72#14556550—21no—
73#14556550—21no—
74#14556550C2×C412yes—
75C2×C4⋊Dic73—21no—
76C2×C22⋊Dic73—21no—
77C2×C4×Dic73—21no—
78#14556517—28no—
79#14556514—24no—
80#14556511—24no—
81C23×C4×C73—21no—
82C23×C4×C73C2311yescharacteristic
83C2×M16×C73—21no—
84C2×M16×C73—21no—
85C2×M16×C73—21no—
86C2×M16×C73—21no—
87C2×M16×C73—21no—
88C2×M16×C73—21no—
89C2×M16×C73—21no—
90C2×M16×C73—21no—
91C2×M16×C73C2×C412yes—
92C2×M16×C73C2×C412yes—
93C2×M16×C73D412yes—
94C2×M16×C73D412yes—
95C22×C8×C73—21no—
96C22×C8×C73—21no—
97C2×C4⋊C4×C73—21no—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73C2×C412yes—
103C23×D73—21no—
104#11924028—22no—
105#11924028—22no—
106#11924028—22no—
107#11924028—22no—
108#11924028—22no—
109#11924028—22no—
110#11924028—22no—
111#11924028—22no—
112#11924028—22no—
113#11924028—22no—
114#11924028—22no—
115#11924028—22no—
116#11924028C22⋊C414yes—
117#11924028C22⋊C414yes—
118#11924028C4⋊C414yes—
119#11924028C4⋊C414yes—
120C2×Dic146—21no—
121C2×Dic146—21no—
122C2×Dic146—21no—
123C2×Dic146C4212yes—
124C2×Dic146C22⋊C412yes—
125C2×Dic146C4⋊C412yes—
126C2×D292—21no—
127C2×D292—21no—
128C2×D292—21no—
129C2×D292C4212yes—
130C2×D292C22⋊C412yes—
131C2×D292C4⋊C412yes—
132C2×C22⋊D73—22no—
133C2×C22⋊D73—22no—
134C2×C22⋊D73—22no—
135C2×C22⋊D73—22no—
136C2×C22⋊D73—22no—
137C2×C22⋊D73—22no—
138C22×Dic73—21no—
139C22×Dic73—41no—
140C2×C4×D73—22no—
141C2×C4×D73—22no—
142C2×C4×D73—22no—
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—22no—
146#11924012—22no—
147#11924012—22no—
148#11924012—22no—
149#11924012—22no—
150#11924012—41no—
151#11924012—41no—
152#11924012—41no—
153#11924012—41no—
154#11924011—216no—
155C2×#10596831—22no—
156C2×#10596831—22no—
157C2×#10596831—22no—
158C2×#10596831—22no—
159C2×#10596831—22no—
160C2×#10596831C22⋊C414yes—
161C8×D73—216no—
162C4⋊Dic73—22no—
163C4⋊Dic73—41no—
164C22⋊Dic73—22no—
165C22⋊Dic73—41no—
166C4×Dic73—22no—
167C4×Dic73—41no—
168C24×C73#4111yescharacteristic
169C22×C4×C73—22no—
170C22×C4×C73—41no—
171C22×C4×C73—41no—
172C22×C4×C73—82no—
173C22×C4×C73C22×C411yescharacteristic
174C22×C4×C73C22×C411yescharacteristic
175C22×C4×C73C22×C411yescharacteristic
176C22×C4×C73C2×D411yescharacteristic
177C22×C4×C73C2×D411yescharacteristic
178C22×C4×C73C2×D411yescharacteristic
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×Q811yescharacteristic
184C22×C4×C73#4111yescharacteristic
185C22×C4×C73#4111yescharacteristic
186C22×C4×C73#4111yescharacteristic
187M16×C73—22no—
188M16×C73—22no—
189M16×C73—22no—
190M16×C73—22no—
191M16×C73—22no—
192M16×C73—41no—
193M16×C73—41no—
194M16×C73C22⋊C414yes—
195C2×C8×C73—22no—
196C2×C8×C73—22no—
197C2×C8×C73—22no—
198C2×C8×C73—22no—
199C2×C8×C73—41no—
200C2×C8×C73—41no—
201C4⋊C4×C73—22no—
202C4⋊C4×C73—22no—
203C4⋊C4×C73—41no—
204C4⋊C4×C73—41no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73#11711yescharacteristic
208C23×C73C4×D411yescharacteristic
209C23×C73C4×D411yescharacteristic
210C23×C73#12111yescharacteristic
211C23×C73C22⋊Q811yescharacteristic
212C23×C73#12311yescharacteristic
213C23×C73#12311yescharacteristic
214C22×D73—22no—
215C22×D73—22no—
216C22×D73—22no—
217C22×D73—81no—
218C22×D73—81no—
219C2×C4×C73—42no—
220C2×C4×C73—82no—
221C2×C4×C73—82no—
222C2×C4×C73—82no—
223C2×C4×C73C2×C4211yescharacteristic
224C2×C4×C73C2×C22⋊C411yescharacteristic
225C2×C4×C73C2×C22⋊C411yescharacteristic
226C2×C4×C73C2×C22⋊C411yescharacteristic
227C2×C4×C73C2×C22⋊C411yescharacteristic
228C2×C4×C73C2×C22⋊C411yescharacteristic
229C2×C4×C73C2×C4⋊C411yescharacteristic
230C2×C4×C73C2×C4⋊C411yescharacteristic
231C2×C4×C73C2×C4⋊C411yescharacteristic
232C2×C4×C73#11711yescharacteristic
233C2×C4×C73C4×D411yescharacteristic
234C2×C4×C73C4×D411yescharacteristic
235C2×C4×C73C4×D411yescharacteristic
236C2×C4×C73C4×D411yescharacteristic
237C2×C4×C73C4×D411yescharacteristic
238C2×C4×C73C4×D411yescharacteristic
239C2×C4×C73#12011yescharacteristic
240C2×C4×C73#12011yescharacteristic
241C2×C4×C73#12011yescharacteristic
242C2×C4×C73#12011yescharacteristic
243C2×C4×C73#12111yescharacteristic
244C2×C4×C73#12111yescharacteristic
245C2×C4×C73#12111yescharacteristic
246C2×C4×C73C22⋊Q811yescharacteristic
247C2×C4×C73C22⋊Q811yescharacteristic
248C2×C4×C73C22⋊Q811yescharacteristic
249C2×C4×C73#12311yescharacteristic
250C2×C4×C73#12311yescharacteristic
251C22⋊D73—24no—
252C22⋊D73—24no—
253C22⋊D73—24no—
254C22⋊D73—24no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—42no—
259C2×Dic73—81no—
260C2×Dic73—81no—
261D292—22no—
262D292—22no—
263D292—22no—
264D292#9514yes—
265C4×D73—24no—
266C4×D73—24no—
267C4×D73—24no—
268C4×D73—24no—
269Dic146—22no—
270Dic146—22no—
271Dic146—22no—
272Dic146#9514yes—
273C8×C73—24no—
274C8×C73—42no—
275#10596831—24no—
276#10596831—42no—
277C22×C73—21no—
278C22×C73—21no—
279C22×C73—21no—
280C22×C73—21no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73C4×C22⋊C411yescharacteristic
288C22×C73#38011yescharacteristic
289C22×C73#38111yescharacteristic
290C22×C73#38511yescharacteristic
291C22×C73#38511yescharacteristic
292C22×C73#38611yescharacteristic
293C22×C73#38611yescharacteristic
294C2×D73—24no—
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—82no—
298C2×D73—82no—
299C4×C73—82no—
300C4×C73C2×#9511yescharacteristic
301C4×C73C4×C22⋊C411yescharacteristic
302C4×C73#37911yescharacteristic
303C4×C73#38011yescharacteristic
304C4×C73#38511yescharacteristic
305C4×C73#38511yescharacteristic
306C4×C73#38611yescharacteristic
307C4×C73#38611yescharacteristic
308Dic73—24no—
309Dic73—82no—
310#10784—731nomaximal, Sylow
311C2×C73—21no—
312C2×C73—21no—
313C2×C73—21no—
314C2×C73—21no—
315C2×C73—21no—
316C2×C73—21no—
317C2×C73C22⋊#9511yescharacteristic
318C2×C73#190211yescharacteristic
319C2×C73#190311yescharacteristic
320D73—44no—
321C2×#568—731no—
322C2×#421—731no—
323C2×#417—731no—
324C2×#344—731no—
325C2×#343D7311yescharacteristic, complemented
326#1736—731no—
327C2×#338—731no—
328C73#1078411yescharacteristic, complemented, Sylow
329C22×#41—731no—
330#568—734no—
331#568—1462no—
332#568—1462no—
333#568—1462no—
334C2×#131—1462no—
335C2×#131—1462no—
336C22×M16—731no—
337C22×M16—731no—
338C22×M16—1462no—
339C22×M16C2×D7311yescharacteristic
340C22×M16C2×D7311yescharacteristic
341C2×#117—731no—
342C22×C4⋊C4C2×D7311yescharacteristic
343#421—734no—
344#421—1462no—
345C2×#104—1462no—
346#417—1462no—
347#417—1462no—
348C2×#103—732no—
349C2×#102—732no—
350#344—1462no—
351#343—22no—
352#340—1462no—
353#338—1462no—
354C2×#41—732no—
355C2×#41—732no—
356C2×#41—732no—
357C2×#41—732no—
358C2×#41—732no—
359C2×#41—732no—
360C2×#41—1461no—
361C2×#41—1461no—
362C2×#41—1461no—
363C2×#41—1461no—
364C2×#41—1461no—
365C2×#41—1461no—
366C22×Q8—731no—
367C22×D4—731no—
368C22×D4—1461no—
369C23×C4—21no—
370C23×C4—1461no—
371C23×C4C22×D7311yescharacteristic
372#131—1468no—
373#131—1468no—
374#131—1468no—
375#131—1468no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—732no—
385C2×M16—732no—
386C2×M16—1464no—
387C2×M16—1468no—
388C2×M16—1461no—
389C2×M16—1461no—
390C2×M16—1461no—
391C2×M16—1461no—
392C2×M16—1461no—
393C2×M16—1461no—
394C2×M16—1461no—
395C2×M16—1461no—
396C2×M16—1461no—
397C2×M16—1461no—
398C2×M16—1468no—
399C2×M16C4×D7312yes—
400C2×M16C4×D7312yes—
401C2×M16D29212yes—
402C2×M16C22⋊D7312yes—
403C22×C8—21no—
404C22×C8—21no—
405C22×C8—732no—
406C22×C8—732no—
407C22×C8—1464no—
408#117—732no—
409#117—1461no—
410#117—1461no—
411#117—1461no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—1461no—
418C2×C4⋊C4C4×D7312yes—
419C2×C22⋊C4—1461no—
420C2×C42—1461no—
421#104—1468no—
422#103—1464no—
423#102—1464no—
424C24—1461no—
425C24#1192402811yescharacteristic
426#41—734no—
427#41—734no—
428#41—734no—
429#41—734no—
430#41—1462no—
431#41—1462no—
432#41—1462no—
433#41—1462no—
434#41—1462no—
435#41—1462no—
436#41—1462no—
437#41—1462no—
438#41—1462no—
439#41—1462no—
440#41—1462no—
441#41—1462no—
442C2×Q8—732no—
443C2×Q8—732no—
444C2×Q8—732no—
445C2×Q8—1461no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×D4—732no—
449C2×D4—732no—
450C2×D4—732no—
451C2×D4—1462no—
452C2×D4—1462no—
453C2×D4—1462no—
454C2×D4—1462no—
455C2×D4—1462no—
456C2×D4—1462no—
457C2×D4—1461no—
458C2×D4—1461no—
459C2×D4—1461no—
460C22×C4—22no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—82no—
464C22×C4—2921no—
465C22×C4—1461no—
466C22×C4—1462no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1462no—
472C22×C4C2×C4×D7311yescharacteristic
473C22×C4C2×C4×D7311yescharacteristic
474C22×C4C2×C4×D7311yescharacteristic
475C22×C4D4×D7311yescharacteristic
476C22×C4D4×D7311yescharacteristic
477C22×C4D4×D7311yescharacteristic
478C22×C4D4×D7311yescharacteristic
479C22×C4D4×D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×D29211yescharacteristic
482C22×C4Q8×D7311yescharacteristic
483C22×C4#1192402711yescharacteristic
484C22×C4#1192402811yescharacteristic
485C22×C4#1192402911yescharacteristic
486M16—22no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—22no—
491M16—41no—
492M16—41no—
493M16—2921no—
494M16—2921no—
495M16—2921no—
496M16—2921no—
497M16—14616no—
498M16—1462no—
499M16—1462no—
500M16—1462no—
501M16—1462no—
502M16#1192400514yes—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—734no—
510C2×C8—14616no—
511C2×C8—1462no—
512C2×C8—1462no—
513C2×C8—1462no—
514C2×C8—1462no—
515C2×C8—1462no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—2921no—
521C4⋊C4—1462no—
522C22⋊C4—2921no—
523C22⋊C4—1462no—
524C42—2921no—
525C42—1462no—
526C23—41no—
527C23—41no—
528C23—5841no—
529C23—5841no—
530C23—1462no—
531C23—1462no—
532C23—1462no—
533C23#1455654811yescharacteristic
534C23C4×C22⋊D7311yescharacteristic
535C23C4×D29211yescharacteristic
536C23#1455656411yescharacteristic
537C23#1455656911yescharacteristic
538C23#1455657311yescharacteristic
539C23#1455657611yescharacteristic
540Q8—734no—
541Q8—1462no—
542Q8—1462no—
543Q8—1462no—
544D4—734no—
545D4—1464no—
546D4—1464no—
547D4—1464no—
548D4—1464no—
549D4—1462no—
550D4—1462no—
551D4—1462no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—2922no—
557C2×C4—5841no—
558C2×C4—5841no—
559C2×C4—1462no—
560C2×C4—1462no—
561C2×C4—1462no—
562C2×C4—1464no—
563C2×C4—1464no—
564C2×C4—1464no—
565C2×C4—1464no—
566C2×C4C42×D7311yescharacteristic
567C2×C4C22⋊C4×D7311yescharacteristic
568C2×C4C22⋊C4×D7311yescharacteristic
569C2×C4C22⋊C4×D7311yescharacteristic
570C2×C4C22⋊C4×D7311yescharacteristic
571C2×C4C2×#1192400511yescharacteristic
572C2×C4C4⋊C4×D7311yescharacteristic
573C2×C4C4⋊C4×D7311yescharacteristic
574C2×C4C4⋊C4×D7311yescharacteristic
575C2×C4#1455654911yescharacteristic
576C2×C4#1455655111yescharacteristic
577C2×C4#1455655111yescharacteristic
578C2×C4C4×C22⋊D7311yescharacteristic
579C2×C4#1455655311yescharacteristic
580C2×C4#1455655311yescharacteristic
581C2×C4C4×D29211yescharacteristic
582C2×C4#1455655911yescharacteristic
583C2×C4#1455655911yescharacteristic
584C2×C4#1455656111yescharacteristic
585C2×C4#1455656111yescharacteristic
586C2×C4#1455656211yescharacteristic
587C2×C4#1455656411yescharacteristic
588C2×C4#1455656611yescharacteristic
589C2×C4#1455656711yescharacteristic
590C2×C4#1455656911yescharacteristic
591C2×C4#1455657111yescharacteristic
592C2×C4#1455657311yescharacteristic
593C2×C4#1455657611yescharacteristic
594C8—24no—
595C8—42no—
596C8—2922no—
597C8—1464no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—2922no—
609C22—2922no—
610C22—5842no—
611C22—5842no—
612C22—1464no—
613C22C4×#1192400511yescharacteristic
614C22#1798844611yescharacteristic
615C22#1798844911yescharacteristic
616C22#1798845811yescharacteristic
617C22#1798846111yescharacteristic
618C22#1798846311yescharacteristic
619C22#1798846511yescharacteristic
620C4—82no—
621C4—5842no—
622C4—1464no—
623C4#95×D7311yescharacteristic
624C4#1798843411yescharacteristic
625C4#1798844211yescharacteristic
626C4#1798844411yescharacteristic
627C4#1798845811yescharacteristic
628C4#1798846111yescharacteristic
629C4#1798846311yescharacteristic
630C4#1798846511yescharacteristic
631C2—21no—
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—2924no—
638C2#2260815211yescharacteristic
639C2#2261040711yescharacteristic
640C2#2261041411yescharacteristic
subgroupG/H
centerC22#17988463
commutatorC2×C4×C73C2×C42
FrattiniC23×C4C22×D73
FittingC2×#343×C73C2
socleC22×C73#386
prime radicalC22×#11924028C22
cosocleC23×C4×C73C23
hypercenterC2×#343D73
hypocenterC73#10784
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. C2×#17988359C2semidirect
5. #22609650C2semidirect
6. C2×#17988345C2semidirect
7. C2×#343×C73C2semidirect
12. #17989353C4semidirect
20. #17988603C4semidirect
24. C2×#14556514C4semidirect
25. C2×#14556511C4semidirect
325. C2×#343D73semidirect
328. C73#10784semidirect

Maximal subgroups

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

subgroupindex
C2×#179893532
C2×#179886032
C2×#179885862
C2×#179883592
#226096502
C2×#179883452
C2×#343×C732
#1078473