Finite Group Databasebeta

← #13355497

Subgroups — #13355497

Order 1792.

1986 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1131C211yesmaximal, characteristic, complemented
2C2×#90297C211yesmaximal, characteristic
3C2×#89543C212yesmaximal, complemented
4D7×#380C211yesmaximal, characteristic, complemented
5C7×#1893C211yesmaximal, characteristic, complemented
6#11729726C211yesmaximal, characteristic, complemented
7#11729435C211yesmaximal, characteristic, complemented
8#11727957C212yesmaximal, complemented
9#11727268C211yesmaximal, characteristic, complemented
10#11726638C212yesmaximal, complemented
11#11726621C212yesmaximal, complemented
12C22×#1159C2211yescharacteristic
13C23×C4×D7C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C2×C7×#121C2211yescharacteristic
16C2×#6501C2211yescharacteristic
17C2×#6500—21no—
18C2×D4×Dic7—21no—
19C2×#6487C2211yescharacteristic
20C23×C7⋊C8C2211yescharacteristic
21C22×#1131—24no—
22C22×#1131—28no—
23C22×#1131C2212yes—
24C22×#1131C2212yes—
25#90297—24no—
26#90297C418yescomplemented
27C2×#6460C2211yescharacteristic
28C2×#6455C2211yescharacteristic
29C2×D7×C4⋊C4C2212yes—
30C2×#6446—21no—
31C2×(C22⋊D7)⋊C4—21no—
32C2×D7×C22⋊C4—21no—
33C2×C7×#102C2212yescomplemented
34C7×#380C2211yescharacteristic
35C2×#6406C2212yescomplemented
36#89984C2211yescharacteristic
37C2×#6390C2212yescomplemented
38#89759C2212yes—
39C2×#6384C2212yescomplemented
40#89714—21no—
41#89543—28no—
42#89439—21no—
43D7×#95—21no—
44C7×#340C2212yescomplemented
45#89346C2212yescomplemented
46#89324C2212yescomplemented
47#89324C2212yescomplemented
48#20971—71nomaximal, Sylow
49C24×D7D411yescharacteristic
50C22×C7×D4—21no—
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—21no—
54C22×C22⋊D7—21no—
55C23×Dic7—21no—
56C23×Dic7D411yescharacteristic
57C2×#1159—24no—
58C2×#1159—22no—
59C2×#1159—24no—
60C2×#1159C2×C414yes—
61C22×C4×D7—24no—
62C22×C4×D7—24no—
63C22×C4×D7—22no—
64C22×C4×D7C2×C414yes—
65C22×C4×D7D411yescharacteristic
66C22×C4×D7D411yescharacteristic
67C22×C4×D7D414yes—
68C22×C4×D7C2311yescharacteristic
69C22×C4×D7C2311yescharacteristic
70C22×Dic14C2311yescharacteristic
71C22×C7×C8—22no—
72C22×C7×C8—24no—
73C22×C7×C8C2312yes—
74C7×#121—22no—
75C7×#121C2×C414yescomplemented
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83#6501—22no—
84#6501C2×C414yescomplemented
85#6500—28no—
86D4×Dic7—28no—
87C2×(C2×D7)⋊C4—21no—
88C2×(C2×D7)⋊C4—21no—
89C2×(C2×D7)⋊C4—22no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7C2312yes—
92C2×C4⋊Dic7C2311yescharacteristic
93#6487—22no—
94#6487C2×C414yescomplemented
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—22no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8—22no—
99C22×C7⋊C8—24no—
100C22×C7⋊C8C2312yes—
101C2×#1131—232no—
102C2×#1131—24no—
103C2×#1131—24no—
104C2×#1131—24no—
105C2×#1131—24no—
106C2×#1131—24no—
107C2×#1131D418yescomplemented
108#6460—22no—
109#6460C2×C414yescomplemented
110#6455—22no—
111#6455C2×C414yes—
112D7×C4⋊C4—24no—
113D7×C4⋊C4—24no—
114#6446—28no—
115(C22⋊D7)⋊C4—28no—
116D7×C22⋊C4—44no—
117C7×#102—24no—
118C7×#95—21no—
119#6406—24no—
120C7⋊#95—21no—
121C7⋊#95—22no—
122#6390—24no—
123#6384—24no—
124C23×M16—71no—
125C2×#525—71no—
126C2×#417—72no—
127C2×#380—71no—
128#1895—72no—
129#1894—72no—
130#1893—71no—
131#1893D711yescharacteristic, complemented
132#1892—71no—
133#1892—71no—
134#1736—72no—
135C24×C7—21no—
136C24×C7C2×D411yescharacteristic
137C23×D7—22no—
138C23×D7—22no—
139C23×D7—24no—
140C23×D7—42no—
141C23×D7C22⋊C414yes—
142C23×D7C2×D411yescharacteristic
143C23×D7C2×D411yescharacteristic
144C2×C7×D4—24no—
145C2×C7×D4—22no—
146C2×C7×D4—21no—
147C2×C7×D4—22no—
148C2×C7×D4—42no—
149C2×C7×D4—42no—
150C2×C7×D4C22×C412yes—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7C22×C412yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D412yes—
158C22×C4×C7C2411yescharacteristic
159C2×C22⋊D7—24no—
160C2×C22⋊D7—24no—
161C2×C22⋊D7—44no—
162C2×C22⋊D7—44no—
163C22×Dic7—24no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—24no—
167C22×Dic7—22no—
168C22×Dic7—44no—
169C22×Dic7C22⋊C414yes—
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172#1159—216no—
173#1159—44no—
174#1159—44no—
175C2×C4×D7—24no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×C4×D7—24no—
179C2×C4×D7—216no—
180C2×C4×D7—28no—
181C2×C4×D7—28no—
182C2×C4×D7—44no—
183C2×C4×D7—44no—
184C2×C4×D7—44no—
185C2×C4×D7C22⋊C414yes—
186C2×C4×D7C22⋊C414yes—
187C2×C4×D7C22×C412yes—
188C2×C4×D7C22×C412yes—
189C2×C4×D7C2×D412yes—
190C2×C4×D7C2×D414yes—
191C2×C4×D7C2×D412yes—
192C2×C4×D7#4114yes—
193C2×Dic14—22no—
194C2×Dic14—22no—
195C2×Dic14—21no—
196C2×Dic14C22×C412yes—
197C2×C7×C8—28no—
198C2×C7×C8—22no—
199C2×C7×C8—22no—
200C2×C7×C8C2×D414yes—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4C22×C412yes—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—42no—
207C22⋊Dic7—24no—
208C22⋊Dic7—24no—
209C22⋊Dic7—42no—
210(C2×D7)⋊C4—24no—
211(C2×D7)⋊C4—24no—
212(C2×D7)⋊C4—44no—
213C4⋊Dic7—21no—
214C4⋊Dic7—24no—
215C4⋊Dic7—22no—
216C4⋊Dic7—22no—
217C4⋊Dic7C22×C412yes—
218Dic7⋊C4—24no—
219Dic7⋊C4—44no—
220C4×Dic7—24no—
221C2×C7⋊C8—28no—
222C2×C7⋊C8—22no—
223C2×C7⋊C8—22no—
224C2×C7⋊C8C2×D414yes—
225#1131—216no—
226#1131—216no—
227C22×#41—71no—
228C24×C4—71no—
229C22×M16—72no—
230C22×M16—72no—
231C22×M16—148no—
232C22×M16—144no—
233C23×C8—71no—
234C23×C8C2×D711yescharacteristic
235#525—78no—
236#525—144no—
237C2×#123—141no—
238C2×C22⋊Q8—71no—
239C2×C22⋊Q8—71no—
240C2×#121—71no—
241C2×#121C2×D711yescharacteristic
242C2×#120—141no—
243C2×C4×D4—141no—
244C2×C4×D4—141no—
245C2×#117—71no—
246C22×C4⋊C4—72no—
247C22×C22⋊C4—141no—
248#417—148no—
249C2×#103—72no—
250C2×#103—72no—
251C2×#102—72no—
252C2×#102C2×D712yescomplemented
253#380—71no—
254#380—72no—
255#380—141no—
256#380—141no—
257#380C2×D711yescharacteristic
258C2×#95—141no—
259#340—72no—
260#340—72no—
261#340—72no—
262#340C2×D712yescomplemented
263C23×C7—22no—
264C23×C7—22no—
265C23×C7—22no—
266C23×C7—42no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—42no—
270C23×C7C2×C22⋊C412yes—
271C23×C7AGL(1,ℤ/8)12yes—
272C23×C7#13712yes—
273C23×C7C22×D411yescharacteristic
274C22×D7—21no—
275C22×D7—21no—
276C22×D7—22no—
277C22×D7—22no—
278C22×D7—21no—
279C22×D7—21no—
280C22×D7—22no—
281C22×D7—28no—
282C22×D7—28no—
283C22×D7—44no—
284C22×D7—48no—
285C22×D7—48no—
286C22×D7C2×C22⋊C412yes—
287C22×D7C2×C22⋊C412yes—
288C22×D7#12114yes—
289C7×D4—24no—
290C7×D4—48no—
291C7×D4—41no—
292C7×D4—41no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7C2×C22⋊C412yes—
304C2×C4×C7C23×C411yescharacteristic
305C2×C4×C7C22×D412yes—
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C2×#4112yes—
308C22⋊D7—48no—
309C22⋊D7—48no—
310C2×Dic7—22no—
311C2×Dic7—28no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—48no—
317C2×Dic7—42no—
318C2×Dic7—42no—
319C2×Dic7—42no—
320C2×Dic7C2×C22⋊C412yes—
321C2×Dic7C2×C22⋊C412yes—
322C2×Dic7#12114yes—
323C4×D7—28no—
324C4×D7—24no—
325C4×D7—22no—
326C4×D7—22no—
327C4×D7—28no—
328C4×D7—88no—
329C4×D7C4×D418yes—
330C4×D7#12014yes—
331C4×D7#12314yes—
332Dic14—24no—
333Dic14—41no—
334Dic14—41no—
335C7×C8—24no—
336C7⋊C8—24no—
337C25—71no—
338C2×#41—74no—
339C2×#41—142no—
340C2×#41—144no—
341C2×#41—144no—
342C22×Q8—71no—
343C22×D4—21no—
344C22×D4—141no—
345C22×D4—141no—
346C22×D4C22×D711yescharacteristic
347C23×C4—71no—
348C23×C4—71no—
349C23×C4—74no—
350C23×C4—71no—
351C23×C4—71no—
352C23×C4—74no—
353C23×C4—71no—
354C23×C4—144no—
355C23×C4—144no—
356C23×C4—142no—
357C23×C4—141no—
358C23×C4C22×D711yescharacteristic
359C2×M16—78no—
360C2×M16—144no—
361C2×M16—144no—
362C2×M16—144no—
363C2×M16—144no—
364C2×M16—1432no—
365C2×M16—144no—
366C22×C8—22no—
367C22×C8—24no—
368C22×C8—72no—
369C22×C8—144no—
370C22×C8—142no—
371C22×C8C22×D712yes—
372#123—148no—
373C22⋊Q8—74no—
374C22⋊Q8—74no—
375C22⋊Q8—142no—
376C22⋊Q8—142no—
377#121—22no—
378#121—74no—
379#121—142no—
380#121C4×D714yescomplemented
381#120—148no—
382C4×D4—148no—
383C4×D4—148no—
384#117—74no—
385#117—142no—
386C2×C4⋊C4—71no—
387C2×C4⋊C4—72no—
388C2×C4⋊C4—144no—
389C2×C4⋊C4—144no—
390C2×C4⋊C4—141no—
391C2×C4⋊C4—141no—
392C2×C4⋊C4—142no—
393C2×C4⋊C4C22×D712yes—
394C2×C4⋊C4C22×D711yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—141no—
399C2×C22⋊C4—141no—
400C2×C22⋊C4—141no—
401C2×C22⋊C4—142no—
402C2×C22⋊C4—141no—
403C2×C22⋊C4—284no—
404C2×C42—141no—
405#103—144no—
406#103—144no—
407#102—24no—
408#102—144no—
409#95—21no—
410#95—142no—
411#95—141no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—24no—
415C22×C7—24no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—82no—
420C22×C7—82no—
421C22×C7#41714yescomplemented
422C22×C7C22×C22⋊C411yescharacteristic
423C22×C7C2×#12112yes—
424C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
425C22×C7C2×#13711yescharacteristic
426C22×C7#57512yes—
427C2×D7—22no—
428C2×D7—22no—
429C2×D7—24no—
430C2×D7—24no—
431C2×D7—22no—
432C2×D7—22no—
433C2×D7—22no—
434C2×D7—416no—
435C2×D7—42no—
436C2×D7—42no—
437C2×D7#38614yes—
438C4×C7—22no—
439C4×C7—42no—
440C4×C7—82no—
441C4×C7C2×C4×D412yes—
442C4×C7C2×#12011yescharacteristic
443C4×C7C2×#12311yescharacteristic
444Dic7—22no—
445Dic7—42no—
446Dic7—42no—
447Dic7—82no—
448Dic7#38614yes—
449C24—21no—
450C24—71no—
451C24—71no—
452C24—74no—
453C24—144no—
454C24—142no—
455C24—142no—
456C24—282no—
457C24D4×D711yescharacteristic
458#41—1416no—
459#41—284no—
460#41—284no—
461C2×Q8—72no—
462C2×Q8—141no—
463C2×Q8—142no—
464C2×Q8—142no—
465C2×D4—24no—
466C2×D4—22no—
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—144no—
472C2×D4—144no—
473C2×D4—284no—
474C2×D4—284no—
475C2×D4C2×C4×D712yes—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—71no—
481C22×C4—72no—
482C22×C4—74no—
483C22×C4—72no—
484C22×C4—74no—
485C22×C4—74no—
486C22×C4—71no—
487C22×C4—74no—
488C22×C4—74no—
489C22×C4—72no—
490C22×C4—72no—
491C22×C4—148no—
492C22×C4—144no—
493C22×C4—1416no—
494C22×C4—148no—
495C22×C4—144no—
496C22×C4—144no—
497C22×C4—144no—
498C22×C4—142no—
499C22×C4—142no—
500C22×C4—141no—
501C22×C4—141no—
502C22×C4—142no—
503C22×C4—284no—
504C22×C4—284no—
505C22×C4—284no—
506C22×C4—284no—
507C22×C4C2×C4×D712yes—
508C22×C4D4×D711yescharacteristic
509C22×C4D4×D712yes—
510C22×C4C23×D711yescharacteristic
511M16—1416no—
512M16—1416no—
513C2×C8—28no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—74no—
517C2×C8—148no—
518C2×C8—142no—
519C2×C8—142no—
520C2×C8D4×D714yes—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—72no—
525C4⋊C4—144no—
526C4⋊C4—144no—
527C4⋊C4—141no—
528C4⋊C4—142no—
529C4⋊C4—142no—
530C4⋊C4—284no—
531C4⋊C4C2×C4×D712yes—
532C22⋊C4—24no—
533C22⋊C4—42no—
534C22⋊C4—144no—
535C22⋊C4—144no—
536C22⋊C4—144no—
537C22⋊C4—144no—
538C22⋊C4—284no—
539C22⋊C4—282no—
540C42—144no—
541C2×C7—24no—
542C2×C7—82no—
543C2×C7C2×#38611yescharacteristic
544C2×C7C2×#41711yescharacteristic
545C2×C7#289611yescharacteristic
546C2×C7#305512yes—
547C2×C7#305612yes—
548D7—24no—
549D7—42no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—42no—
556C23—42no—
557C23—72no—
558C23—74no—
559C23—72no—
560C23—148no—
561C23—141no—
562C23—141no—
563C23—142no—
564C23—141no—
565C23—141no—
566C23—142no—
567C23—142no—
568C23—148no—
569C23—284no—
570C23—288no—
571C23—288no—
572C23D7×C22⋊C412yes—
573C23#647412yes—
574C23#647812yes—
575C23C2×D4×D711yescharacteristic
576Q8—144no—
577Q8—281no—
578Q8—281no—
579D4—24no—
580D4—48no—
581D4—41no—
582D4—41no—
583D4—288no—
584D4—288no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—72no—
596C2×C4—74no—
597C2×C4—78no—
598C2×C4—74no—
599C2×C4—74no—
600C2×C4—72no—
601C2×C4—148no—
602C2×C4—148no—
603C2×C4—148no—
604C2×C4—142no—
605C2×C4—144no—
606C2×C4—142no—
607C2×C4—142no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—288no—
611C2×C4—282no—
612C2×C4—282no—
613C2×C4—282no—
614C2×C4—282no—
615C2×C4—282no—
616C2×C4—568no—
617C2×C4D7×C22⋊C412yes—
618C2×C4C22×C4×D711yescharacteristic
619C2×C4C2×D4×D712yes—
620C2×C4C2×D4×D711yescharacteristic
621C2×C4D7×#4112yes—
622C8—24no—
623C8—144no—
624C7#2097111yescharacteristic, complemented, Sylow
625C22—24no—
626C22—22no—
627C22—22no—
628C22—24no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—74no—
633C22—82no—
634C22—82no—
635C22—144no—
636C22—142no—
637C22—142no—
638C22—144no—
639C22—142no—
640C22—142no—
641C22—142no—
642C22—2816no—
643C22—282no—
644C22—282no—
645C22#8954314yescomplemented
646C22C2×D7×C22⋊C411yescharacteristic
647C22D7×#12112yes—
648C22C2×#647411yescharacteristic
649C22C2×#647811yescharacteristic
650C22#9045912yes—
651C4—22no—
652C4—42no—
653C4—74no—
654C4—82no—
655C4—142no—
656C4—282no—
657C4—282no—
658C4—562no—
659C4C4×D4×D712yes—
660C4D7×#12011yescharacteristic
661C4D7×#12311yescharacteristic
662C2—24no—
663C2—82no—
664C2—144no—
665C2—282no—
666C2D7×#38611yescharacteristic
667C2C2×#8954311yescharacteristic
668C2#1173627511yescharacteristic
669C2#1173810312yes—
670C2#1173816012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1893C2
socleC23×C7C22×D4
prime radicalC2×#90297C2
cosocleC22×C4×C7C24
hypercenter#1893D7
hypocenterC7#20971
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
1. C23×#1131C2semidirect
3. C2×#89543C2semidirect
4. D7×#380C2semidirect
5. C7×#1893C2semidirect
6. #11729726C2semidirect
7. #11729435C2semidirect
8. #11727957C2semidirect
9. #11727268C2semidirect
10. #11726638C2semidirect
11. #11726621C2semidirect
26. #90297C4semidirect
33. C2×C7×#102C22semidirect
35. C2×#6406C22semidirect
37. C2×#6390C22semidirect
39. C2×#6384C22semidirect
44. C7×#340C22semidirect
45. #89346C22semidirect
46. #89324C22semidirect
47. #89324C22semidirect
75. C7×#121C2×C4semidirect
84. #6501C2×C4semidirect
94. #6487C2×C4semidirect
107. C2×#1131D4semidirect
109. #6460C2×C4semidirect
131. #1893D7semidirect
252. C2×#102C2×D7semidirect
262. #340C2×D7semidirect
380. #121C4×D7semidirect
421. C22×C7#417semidirect
624. C7#20971semidirect
645. C22#89543semidirect

Maximal subgroups

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

subgroupindex
C23×#11312
C2×#902972
C2×#895432
D7×#3802
C7×#18932
#117297262
#117294352
#117279572
#117272682
#117266382
#117266212
#209717