Finite Group Databasebeta

← #13355501

Subgroups — #13355501

Order 1792.

1986 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#525C211yesmaximal, characteristic
2C23×#1136C211yesmaximal, characteristic, complemented
3C2×#90004C212yesmaximal, complemented
4C2×#89984C211yesmaximal, characteristic, complemented
5#11729451C212yesmaximal, complemented
6#11729450C212yesmaximal, complemented
7#11729436C212yesmaximal, complemented
8#11729435C212yesmaximal, complemented
9#11729335C212yesmaximal, complemented
10C22×C7×#41C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C7×#525—24no—
13C7×#525C418yescomplemented
14C2×C7×#123—21no—
15C2×C7×C22⋊Q8C2211yescharacteristic
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C7×#121C2211yescharacteristic
18C2×C7×#121C2211yescharacteristic
19C2×C7×#120—21no—
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4—21no—
22C2×C7×#117C2211yescharacteristic
23C22×C22⋊Dic7—21no—
24C22×C4⋊Dic7C2212yes—
25C22×#1136—24no—
26C22×#1136—28no—
27C22×#1136C2212yes—
28C22×#1136C2212yes—
29C23×C7⋊C8C2212yes—
30#90004—28no—
31C2×#6409C2214yescomplemented
32C2×#6406C2214yescomplemented
33#89984—21no—
34#89984—21no—
35#89984C2212yes—
36#89984C2212yes—
37C2×C7⋊#95—21no—
38#89324C2214yescomplemented
39#89324C2214yescomplemented
40#20971—71nomaximal, Sylow
41C25×C7D411yescharacteristic
42C2×C7×#41—22no—
43C2×C7×#41—24no—
44C2×C7×#41—24no—
45C2×C7×#41C2×C414yes—
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4—21no—
48C22×C7×D4—21no—
49C22×C7×D4—21no—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—24no—
52C23×C4×C7—24no—
53C23×C4×C7—21no—
54C23×C4×C7C2×C414yes—
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7D411yescharacteristic
58C23×C4×C7D414yes—
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—22no—
63C7×#123—28no—
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8C2×C414yescomplemented
67C7×C22⋊Q8C2×C414yescomplemented
68C7×#121—22no—
69C7×#121—22no—
70C7×#121C2×C414yescomplemented
71C7×#121C2×C414yescomplemented
72C7×#120—28no—
73C4×C7×D4—28no—
74C4×C7×D4—28no—
75C7×#117—22no—
76C7×#117C2×C414yes—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C42×C7—21no—
87C2×C22⋊Dic7—22no—
88C2×C22⋊Dic7—22no—
89C2×C22⋊Dic7—44no—
90C2×C4⋊Dic7—24no—
91C2×C4⋊Dic7—24no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7C2314yes—
94C2×#1136—232no—
95C2×#1136—24no—
96C2×#1136—24no—
97C2×#1136—24no—
98C2×#1136—28no—
99C2×#1136D418yescomplemented
100C22×C7⋊C8—24no—
101C22×C7⋊C8—28no—
102C22×C7⋊C8C2314yes—
103#6409—28no—
104#6406—28no—
105C7⋊#95—22no—
106C7⋊#95—22no—
107C23×M16—71no—
108C2×#525D711yescharacteristic
109C2×#417—72no—
110C2×#380—71no—
111#1895—72no—
112#1894—72no—
113#1893—72no—
114#1892—72no—
115#1736—72no—
116C24×C7—24no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7—21no—
120C24×C7—42no—
121C24×C7C22⋊C414yes—
122C24×C7C2×D411yescharacteristic
123C24×C7C2×D411yescharacteristic
124C24×C7C2×D411yescharacteristic
125C7×#41—216no—
126C7×#41—44no—
127C7×#41—44no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—22no—
131C2×C7×Q8C22×C412yes—
132C2×C7×D4—24no—
133C2×C7×D4—24no—
134C2×C7×D4—24no—
135C2×C7×D4—22no—
136C2×C7×D4—21no—
137C2×C7×D4—22no—
138C2×C7×D4—44no—
139C2×C7×D4—44no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4C22×C412yes—
143C22×C4×C7—24no—
144C22×C4×C7—216no—
145C22×C4×C7—28no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—28no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—24no—
157C22×C4×C7—44no—
158C22×C4×C7C22⋊C414yes—
159C22×C4×C7C22⋊C414yes—
160C22×C4×C7C22⋊C414yes—
161C22×C4×C7C22×C412yes—
162C22×C4×C7C22×C412yes—
163C22×C4×C7C22×C412yes—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D414yes—
171C22×C4×C7#4114yes—
172C22×C4×C7C2411yescharacteristic
173C22×Dic7—24no—
174C22×Dic7—44no—
175C22×Dic7—44no—
176C22×Dic7—44no—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4C22×C412yes—
182C7×C4⋊C4C22×C412yes—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—24no—
188C42×C7—24no—
189C22⋊Dic7—44no—
190C22⋊Dic7—44no—
191C4⋊Dic7—24no—
192C4⋊Dic7—24no—
193C4⋊Dic7—44no—
194#1136—216no—
195#1136—216no—
196C2×C7⋊C8—216no—
197C2×C7⋊C8—24no—
198C2×C7⋊C8—24no—
199C2×C7⋊C8C2×D418yes—
200C22×#41C2×D711yescharacteristic
201C24×C4C2×D711yescharacteristic
202C22×M16—72no—
203C22×M16—72no—
204C22×M16—148no—
205C22×M16—144no—
206C23×C8—72no—
207#525—24no—
208#525Dic718yescomplemented
209C2×#123—21no—
210C2×C22⋊Q8C2×D711yescharacteristic
211C2×C22⋊Q8C2×D711yescharacteristic
212C2×#121C2×D711yescharacteristic
213C2×#121C2×D711yescharacteristic
214C2×#120—21no—
215C2×C4×D4—21no—
216C2×C4×D4—21no—
217C2×#117C2×D711yescharacteristic
218C22×C4⋊C4—72no—
219C22×C22⋊C4—141no—
220#417—148no—
221C2×#103—74no—
222C2×#102—74no—
223#380—72no—
224#380—72no—
225#380—141no—
226#380—141no—
227C2×#95—141no—
228#340—74no—
229#340—74no—
230C23×C7—22no—
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—22no—
239C23×C7—28no—
240C23×C7—28no—
241C23×C7—22no—
242C23×C7—44no—
243C23×C7—48no—
244C23×C7—48no—
245C23×C7—42no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—42no—
249C23×C7C2×C22⋊C412yes—
250C23×C7C2×C22⋊C412yes—
251C23×C7C2×C22⋊C412yes—
252C23×C7#12114yes—
253C23×C7AGL(1,ℤ/8)12yes—
254C23×C7#13712yes—
255C23×C7C22×D411yescharacteristic
256C7×Q8—24no—
257C7×Q8—41no—
258C7×Q8—41no—
259C7×D4—24no—
260C7×D4—48no—
261C7×D4—48no—
262C7×D4—48no—
263C7×D4—41no—
264C7×D4—41no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—28no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—28no—
275C2×C4×C7—28no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—42no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—48no—
284C2×C4×C7C2×C22⋊C412yes—
285C2×C4×C7C2×C22⋊C412yes—
286C2×C4×C7C2×C22⋊C412yes—
287C2×C4×C7C4×D418yes—
288C2×C4×C7#12014yes—
289C2×C4×C7#12114yes—
290C2×C4×C7#12314yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×D412yes—
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C2×#4112yes—
295C2×Dic7—44no—
296C2×Dic7—44no—
297C2×Dic7—44no—
298C2×Dic7—88no—
299C7⋊C8—28no—
300C25C22⋊D711yescharacteristic
301C2×#41—22no—
302C2×#41—24no—
303C2×#41—24no—
304C2×#41C2×Dic714yes—
305C22×Q8C22×D711yescharacteristic
306C22×D4—21no—
307C22×D4—21no—
308C22×D4—21no—
309C22×D4C22×D711yescharacteristic
310C23×C4—24no—
311C23×C4—24no—
312C23×C4—21no—
313C23×C4—142no—
314C23×C4C2×Dic714yes—
315C23×C4C22⋊D711yescharacteristic
316C23×C4C22⋊D711yescharacteristic
317C23×C4C22⋊D711yescharacteristic
318C23×C4C22⋊D714yes—
319C23×C4C22×D711yescharacteristic
320C23×C4C22×D711yescharacteristic
321C23×C4C22×D711yescharacteristic
322C2×M16—78no—
323C2×M16—148no—
324C2×M16—1432no—
325C2×M16—144no—
326C2×M16—144no—
327C2×M16—144no—
328C22×C8—74no—
329C22×C8—148no—
330C22×C8—144no—
331#123—28no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8C2×Dic714yescomplemented
335C22⋊Q8C2×Dic714yescomplemented
336#121—22no—
337#121—22no—
338#121C2×Dic714yescomplemented
339#121C2×Dic714yescomplemented
340#120—28no—
341C4×D4—28no—
342C4×D4—28no—
343#117—22no—
344#117C2×Dic714yes—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—74no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—144no—
350C2×C4⋊C4—142no—
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—142no—
359C2×C22⋊C4—142no—
360C2×C22⋊C4—284no—
361C2×C42—21no—
362#103—148no—
363#102—148no—
364#95—142no—
365#95—142no—
366C22×C7—22no—
367C22×C7—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—24no—
371C22×C7—24no—
372C22×C7—24no—
373C22×C7—24no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—416no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—82no—
384C22×C7—82no—
385C22×C7#38614yes—
386C22×C7#41714yescomplemented
387C22×C7C22×C22⋊C411yescharacteristic
388C22×C7C2×#12112yes—
389C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
390C22×C7C2×#13711yescharacteristic
391C22×C7#57512yes—
392C4×C7—22no—
393C4×C7—22no—
394C4×C7—42no—
395C4×C7—42no—
396C4×C7—42no—
397C4×C7#38614yes—
398C4×C7C2×C4×D412yes—
399C4×C7C2×#12011yescharacteristic
400C4×C7C2×#12311yescharacteristic
401Dic7—84no—
402C24—24no—
403C24—22no—
404C24—22no—
405C24—21no—
406C24—42no—
407C24C22⋊Dic714yes—
408C24C2×C22⋊D711yescharacteristic
409C24C2×C22⋊D711yescharacteristic
410C24C2×C22⋊D711yescharacteristic
411#41—216no—
412#41—44no—
413#41—44no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8C22×Dic712yes—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C22×Dic712yes—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—144no—
445C22×C4—284no—
446C22×C4—284no—
447C22×C4—284no—
448C22×C4C22⋊Dic714yes—
449C22×C4C22⋊Dic714yes—
450C22×C4C22⋊Dic714yes—
451C22×C4D4×D714yes—
452C22×C4#115914yes—
453C22×C4C22×Dic712yes—
454C22×C4C22×Dic712yes—
455C22×C4C22×Dic712yes—
456C22×C4C2×C22⋊D711yescharacteristic
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C2×C22⋊D712yes—
459C22×C4C2×C22⋊D712yes—
460C22×C4C2×C22⋊D712yes—
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C23×D711yescharacteristic
463M16—1416no—
464M16—1416no—
465C2×C8—78no—
466C2×C8—1416no—
467C2×C8—144no—
468C2×C8—144no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—144no—
474C4⋊C4—144no—
475C4⋊C4—284no—
476C4⋊C4C22×Dic712yes—
477C4⋊C4C22×Dic712yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—284no—
484C22⋊C4—284no—
485C42—24no—
486C2×C7—24no—
487C2×C7—24no—
488C2×C7—42no—
489C2×C7—82no—
490C2×C7C2×#38611yescharacteristic
491C2×C7C2×#41711yescharacteristic
492C2×C7#289611yescharacteristic
493C2×C7#305512yes—
494C2×C7#305612yes—
495C23—22no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—21no—
500C23—22no—
501C23—28no—
502C23—21no—
503C23—22no—
504C23—21no—
505C23—28no—
506C23—22no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23#650214yes—
515C23#651212yes—
516C23#651412yes—
517C23C2×C22⋊Dic712yes—
518C23C2×C22⋊Dic712yes—
519C23C2×C22⋊Dic712yes—
520C23C22×C22⋊D711yescharacteristic
521Q8—24no—
522Q8—41no—
523Q8—41no—
524D4—24no—
525D4—48no—
526D4—48no—
527D4—48no—
528D4—41no—
529D4—41no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—284no—
550C2×C4—284no—
551C2×C4—284no—
552C2×C4—568no—
553C2×C4D4×Dic718yes—
554C2×C4#649814yes—
555C2×C4#650014yes—
556C2×C4#650214yes—
557C2×C4C2×C22⋊Dic712yes—
558C2×C4C2×C22⋊Dic712yes—
559C2×C4C2×C22⋊Dic712yes—
560C2×C4C2×D4×D712yes—
561C2×C4C2×#115912yes—
562C2×C4C23×Dic711yescharacteristic
563C2×C4C22×C22⋊D711yescharacteristic
564C8—148no—
565C7#2097111yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—416no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—82no—
585C22#8999214yes—
586C22#9000414yescomplemented
587C22C2×#650212yes—
588C22C2×#651211yescharacteristic
589C22#9051312yes—
590C22C2×#651411yescharacteristic
591C22C22×C22⋊Dic711yescharacteristic
592C4—22no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—564no—
598C4#8999214yes—
599C4C2×D4×Dic712yes—
600C4C2×#649811yescharacteristic
601C4C2×#650011yescharacteristic
602C2—24no—
603C2—24no—
604C2—42no—
605C2—82no—
606C2#1174036912yes—
607C2#1174037012yes—
608C2C2×#8999211yescharacteristic
609C2C2×#9000411yescharacteristic
610C2#1174151711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#525C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#525D7
hypocenterC7#20971
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
2. C23×#1136C2semidirect
3. C2×#90004C2semidirect
4. C2×#89984C2semidirect
5. #11729451C2semidirect
6. #11729450C2semidirect
7. #11729436C2semidirect
8. #11729435C2semidirect
9. #11729335C2semidirect
13. C7×#525C4semidirect
31. C2×#6409C22semidirect
32. C2×#6406C22semidirect
38. #89324C22semidirect
39. #89324C22semidirect
66. C7×C22⋊Q8C2×C4semidirect
67. C7×C22⋊Q8C2×C4semidirect
70. C7×#121C2×C4semidirect
71. C7×#121C2×C4semidirect
99. C2×#1136D4semidirect
208. #525Dic7semidirect
334. C22⋊Q8C2×Dic7semidirect
335. C22⋊Q8C2×Dic7semidirect
338. #121C2×Dic7semidirect
339. #121C2×Dic7semidirect
386. C22×C7#417semidirect
565. C7#20971semidirect
586. C22#90004semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5252
C23×#11362
C2×#900042
C2×#899842
#117294512
#117294502
#117294362
#117294352
#117293352
#209717