Finite Group Databasebeta

← #13355502

Subgroups — #13355502

Order 1792.

1894 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1131C211yesmaximal, characteristic
2C2×D7×C22⋊Q8C211yesmaximal, characteristic
3C2×#89574C212yesmaximal, complemented
4#11731307C211yesmaximal, characteristic
5C7×#1892C211yesmaximal, characteristic, complemented
6#11729726C211yesmaximal, characteristic, complemented
7#11729450C211yesmaximal, characteristic, complemented
8#11728025C212yesmaximal, complemented
9#11727268C211yesmaximal, characteristic, complemented
10#11726638C212yesmaximal, complemented
11#11726621C212yesmaximal, complemented
12C22×Q8×D7C2211yescharacteristic
13C23×C4×D7C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C2×C7×C22⋊Q8C2211yescharacteristic
16C2×#6509C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19C22×#1131—24no—
20C22×#1131—28no—
21C22×#1131C2212yes—
22C22×#1131C2212yes—
23D7×C22⋊Q8—24no—
24D7×C22⋊Q8C418yescomplemented
25C2×#6460C2211yescharacteristic
26C2×#6459—21no—
27C2×#6455C2212yes—
28C2×D7×C4⋊C4—21no—
29C2×D7×C4⋊C4C2211yescharacteristic
30C2×D7×C22⋊C4—21no—
31C2×#6441—21no—
32C2×C7×#103C2212yescomplemented
33C7×#380C2211yescharacteristic
34C2×#6409C2212yescomplemented
35#89984C2211yescharacteristic
36C2×#6390C2212yescomplemented
37#89759C2212yes—
38C2×#6384C2212yescomplemented
39#89727—21no—
40C22⋊C4×Dic7—21no—
41#89574—28no—
42#89438—21no—
43#89436—21no—
44C7×#340C2212yescomplemented
45#89346C2212yescomplemented
46#89324C2212yescomplemented
47#89324C2212yescomplemented
48#20972—71nomaximal, Sylow
49C24×D7D411yescharacteristic
50C22×C7×Q8C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C23×Dic7D411yescharacteristic
53C2×Q8×D7—22no—
54C2×Q8×D7—24no—
55C2×Q8×D7—24no—
56C2×Q8×D7C2×C414yes—
57C22×C4×D7—24no—
58C22×C4×D7—24no—
59C22×C4×D7—21no—
60C22×C4×D7—21no—
61C22×C4×D7C2×C414yes—
62C22×C4×D7D411yescharacteristic
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D414yes—
65C22×C4×D7C2311yescharacteristic
66C22×C4×D7C2311yescharacteristic
67C22×Dic14—21no—
68C22×Dic14C2311yescharacteristic
69C22×C7×C8—22no—
70C22×C7×C8—24no—
71C22×C7×C8C2312yes—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8C2×C414yes—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2312yes—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81#6509—22no—
82#6509C2×C414yes—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—22no—
86C2×(C2×D7)⋊C4—22no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7C2312yes—
89C2×C4⋊Dic7C2311yescharacteristic
90#6487—22no—
91#6487C2×C414yes—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—22no—
96C22×C7⋊C8—22no—
97C22×C7⋊C8—24no—
98C22×C7⋊C8C2312yes—
99C2×#1131—232no—
100C2×#1131—24no—
101C2×#1131—24no—
102C2×#1131—24no—
103C2×#1131—24no—
104C2×#1131—24no—
105C2×#1131D418yes—
106#6460—22no—
107#6460C2×C414yes—
108#6459—28no—
109#6455—24no—
110#6455—24no—
111D7×C4⋊C4—28no—
112D7×C4⋊C4—22no—
113D7×C4⋊C4C2×C414yes—
114D7×C22⋊C4—28no—
115#6441—28no—
116C7×#103—24no—
117C7×#95—21no—
118#6409—24no—
119C7⋊#95—21no—
120C7⋊#95—22no—
121#6390—24no—
122#6384—24no—
123C23×M16—71no—
124C22×C22⋊Q8—71no—
125C2×#419—72no—
126#2280—71no—
127#1894—72no—
128#1894—72no—
129#1892—71no—
130#1892—71no—
131#1892—71no—
132#1892D711yescharacteristic, complemented
133#1736—72no—
134C24×C7C2×D411yescharacteristic
135C23×D7—22no—
136C23×D7—22no—
137C23×D7—24no—
138C23×D7—42no—
139C23×D7C22⋊C414yes—
140C23×D7C2×D411yescharacteristic
141C23×D7C2×D411yescharacteristic
142C2×C7×Q8—21no—
143C2×C7×Q8—22no—
144C2×C7×Q8—22no—
145C2×C7×Q8C22×C412yes—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—24no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—22no—
159C22×Dic7C22⋊C414yes—
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162Q8×D7—216no—
163Q8×D7—44no—
164Q8×D7—44no—
165C2×C4×D7—24no—
166C2×C4×D7—22no—
167C2×C4×D7—22no—
168C2×C4×D7—24no—
169C2×C4×D7—216no—
170C2×C4×D7—28no—
171C2×C4×D7—28no—
172C2×C4×D7—24no—
173C2×C4×D7—24no—
174C2×C4×D7—44no—
175C2×C4×D7—44no—
176C2×C4×D7C22⋊C414yes—
177C2×C4×D7C22⋊C414yes—
178C2×C4×D7C22×C412yes—
179C2×C4×D7C22×C412yes—
180C2×C4×D7C2×D412yes—
181C2×C4×D7C2×D414yes—
182C2×C4×D7C2×D412yes—
183C2×C4×D7#4114yes—
184C2×Dic14—22no—
185C2×Dic14—22no—
186C2×Dic14—21no—
187C2×Dic14—24no—
188C2×Dic14—44no—
189C2×Dic14C22×C412yes—
190C2×C7×C8—28no—
191C2×C7×C8—22no—
192C2×C7×C8—22no—
193C2×C7×C8C2×D414yes—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4C22×C412yes—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—42no—
201C22⋊Dic7—24no—
202C22⋊Dic7—42no—
203(C2×D7)⋊C4—24no—
204(C2×D7)⋊C4—24no—
205(C2×D7)⋊C4—44no—
206(C2×D7)⋊C4—44no—
207C4⋊Dic7—21no—
208C4⋊Dic7—24no—
209C4⋊Dic7—22no—
210C4⋊Dic7—22no—
211C4⋊Dic7C22×C412yes—
212Dic7⋊C4—24no—
213Dic7⋊C4—24no—
214Dic7⋊C4—24no—
215C4×Dic7—44no—
216C2×C7⋊C8—28no—
217C2×C7⋊C8—22no—
218C2×C7⋊C8—22no—
219C2×C7⋊C8C2×D414yes—
220#1131—216no—
221#1131—216no—
222C23×Q8—71no—
223C24×C4—71no—
224C22×M16—72no—
225C22×M16—72no—
226C22×M16—148no—
227C22×M16—144no—
228C23×C8—71no—
229C23×C8C2×D711yescharacteristic
230C2×C22⋊Q8—71no—
231C2×C22⋊Q8—71no—
232C2×C22⋊Q8—78no—
233C2×C22⋊Q8—71no—
234C2×C22⋊Q8—141no—
235C2×C22⋊Q8—141no—
236C2×C22⋊Q8—144no—
237C2×C22⋊Q8C2×D711yescharacteristic
238C2×#117—72no—
239C22×C4⋊C4—71no—
240C22×C4⋊C4—141no—
241C22×C22⋊C4—141no—
242#419—148no—
243C2×#103—72no—
244C2×#103—72no—
245C2×#103—72no—
246C2×#103C2×D712yescomplemented
247#381—141no—
248#380—71no—
249#380—72no—
250#380C2×D711yescharacteristic
251#379—141no—
252C4×C22⋊C4—141no—
253C4×C22⋊C4—141no—
254#340—72no—
255#340—72no—
256#340—72no—
257#340C2×D712yescomplemented
258C23×C7—22no—
259C23×C7—22no—
260C23×C7C2×C22⋊C412yes—
261C23×C7#13712yes—
262C23×C7#13712yes—
263C23×C7C22×D411yescharacteristic
264C22×D7—21no—
265C22×D7—21no—
266C22×D7—22no—
267C22×D7—22no—
268C22×D7—21no—
269C22×D7—21no—
270C22×D7—22no—
271C22×D7—28no—
272C22×D7—28no—
273C22×D7—44no—
274C22×D7—48no—
275C22×D7—48no—
276C22×D7C2×C22⋊C412yes—
277C22×D7C2×C22⋊C412yes—
278C22×D7#12114yes—
279C7×Q8—24no—
280C7×Q8—41no—
281C7×Q8—41no—
282C2×C4×C7—22no—
283C2×C4×C7—24no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7C2×C22⋊C412yes—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D412yes—
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4112yes—
298C2×Dic7—22no—
299C2×Dic7—28no—
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7C2×C22⋊C412yes—
308C2×Dic7C2×C22⋊C412yes—
309C2×Dic7#12114yes—
310C4×D7—28no—
311C4×D7—24no—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—28no—
315C4×D7—48no—
316C4×D7—48no—
317C4×D7C4×D418yes—
318C4×D7#12014yes—
319C4×D7#12314yes—
320Dic14—24no—
321Dic14—41no—
322Dic14—41no—
323Dic14—48no—
324C7×C8—24no—
325C7⋊C8—24no—
326C25—71no—
327C22×Q8—74no—
328C22×Q8—71no—
329C22×Q8—141no—
330C22×Q8—142no—
331C22×Q8—144no—
332C22×Q8—144no—
333C22×Q8C22×D711yescharacteristic
334C23×C4—71no—
335C23×C4—71no—
336C23×C4—74no—
337C23×C4—71no—
338C23×C4—74no—
339C23×C4—71no—
340C23×C4—71no—
341C23×C4—144no—
342C23×C4—144no—
343C23×C4—141no—
344C23×C4—141no—
345C23×C4C22×D711yescharacteristic
346C2×M16—78no—
347C2×M16—144no—
348C2×M16—144no—
349C2×M16—144no—
350C2×M16—144no—
351C2×M16—1432no—
352C2×M16—144no—
353C22×C8—22no—
354C22×C8—24no—
355C22×C8—72no—
356C22×C8—144no—
357C22×C8—142no—
358C22×C8C22×D712yes—
359C22⋊Q8—22no—
360C22⋊Q8—74no—
361C22⋊Q8—74no—
362C22⋊Q8—74no—
363C22⋊Q8—148no—
364C22⋊Q8—148no—
365C22⋊Q8—142no—
366C22⋊Q8—142no—
367C22⋊Q8—142no—
368C22⋊Q8C4×D714yes—
369#117—144no—
370#117—144no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—74no—
374C2×C4⋊C4—72no—
375C2×C4⋊C4—142no—
376C2×C4⋊C4—148no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4C22×D712yes—
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—148no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4—141no—
388C2×C22⋊C4—141no—
389C2×C22⋊C4—142no—
390C2×C22⋊C4—142no—
391C2×C22⋊C4—141no—
392C2×C42—142no—
393#103—24no—
394#103—144no—
395#103—144no—
396#103—144no—
397#95—21no—
398#95—142no—
399#95—141no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—24no—
403C22×C7—24no—
404C22×C7#41914yescomplemented
405C22×C7C22×C22⋊C411yescharacteristic
406C22×C7C2×#12112yes—
407C22×C7C2×#13711yescharacteristic
408C22×C7C2×#13711yescharacteristic
409C22×C7#57512yes—
410C2×D7—22no—
411C2×D7—22no—
412C2×D7—24no—
413C2×D7—24no—
414C2×D7—22no—
415C2×D7—22no—
416C2×D7—22no—
417C2×D7—416no—
418C2×D7—42no—
419C2×D7—42no—
420C2×D7#38614yes—
421C4×C7—22no—
422C4×C7—42no—
423C4×C7—42no—
424C4×C7—82no—
425C4×C7C2×C4×D412yes—
426C4×C7C2×#12011yescharacteristic
427C4×C7C2×#12311yescharacteristic
428Dic7—22no—
429Dic7—42no—
430Dic7—42no—
431Dic7—82no—
432Dic7#38614yes—
433C24—71no—
434C24—71no—
435C24—74no—
436C24—144no—
437C24—142no—
438C24—142no—
439C24—282no—
440C24D4×D711yescharacteristic
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—72no—
445C2×Q8—144no—
446C2×Q8—1416no—
447C2×Q8—141no—
448C2×Q8—142no—
449C2×Q8—142no—
450C2×Q8—284no—
451C2×Q8—284no—
452C2×Q8—284no—
453C2×Q8C2×C4×D712yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—71no—
460C22×C4—72no—
461C22×C4—72no—
462C22×C4—74no—
463C22×C4—74no—
464C22×C4—74no—
465C22×C4—71no—
466C22×C4—74no—
467C22×C4—74no—
468C22×C4—72no—
469C22×C4—72no—
470C22×C4—148no—
471C22×C4—144no—
472C22×C4—1416no—
473C22×C4—148no—
474C22×C4—144no—
475C22×C4—144no—
476C22×C4—144no—
477C22×C4—144no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—142no—
483C22×C4—284no—
484C22×C4—284no—
485C22×C4C2×C4×D712yes—
486C22×C4D4×D711yescharacteristic
487C22×C4D4×D712yes—
488C22×C4C23×D711yescharacteristic
489M16—1416no—
490M16—1416no—
491C2×C8—28no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—74no—
495C2×C8—148no—
496C2×C8—142no—
497C2×C8—142no—
498C2×C8D4×D714yes—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—72no—
504C4⋊C4—144no—
505C4⋊C4—144no—
506C4⋊C4—144no—
507C4⋊C4—144no—
508C4⋊C4—141no—
509C4⋊C4—142no—
510C4⋊C4—142no—
511C4⋊C4C2×C4×D712yes—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—144no—
515C22⋊C4—144no—
516C22⋊C4—144no—
517C22⋊C4—284no—
518C22⋊C4—284no—
519C22⋊C4—282no—
520C42—284no—
521C2×C7—24no—
522C2×C7C2×#38611yescharacteristic
523C2×C7C2×#41911yescharacteristic
524C2×C7#289611yescharacteristic
525C2×C7#305612yes—
526C2×C7#305612yes—
527D7—24no—
528D7—42no—
529C23—22no—
530C23—22no—
531C23—72no—
532C23—74no—
533C23—72no—
534C23—148no—
535C23—141no—
536C23—141no—
537C23—142no—
538C23—141no—
539C23—141no—
540C23—142no—
541C23—142no—
542C23—148no—
543C23—284no—
544C23—288no—
545C23—288no—
546C23D7×C22⋊C412yes—
547C23#647812yes—
548C23#648112yes—
549C23C2×D4×D711yescharacteristic
550Q8—24no—
551Q8—41no—
552Q8—41no—
553Q8—144no—
554Q8—288no—
555Q8—281no—
556Q8—281no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—72no—
569C2×C4—74no—
570C2×C4—78no—
571C2×C4—74no—
572C2×C4—74no—
573C2×C4—72no—
574C2×C4—148no—
575C2×C4—148no—
576C2×C4—148no—
577C2×C4—142no—
578C2×C4—144no—
579C2×C4—142no—
580C2×C4—142no—
581C2×C4—142no—
582C2×C4—142no—
583C2×C4—288no—
584C2×C4—288no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4—282no—
589C2×C4—282no—
590C2×C4D7×C22⋊C412yes—
591C2×C4C22×C4×D711yescharacteristic
592C2×C4C2×D4×D712yes—
593C2×C4C2×D4×D711yescharacteristic
594C2×C4D7×#4112yes—
595C8—24no—
596C8—144no—
597C7#2097211yescharacteristic, complemented, Sylow
598C22—24no—
599C22—22no—
600C22—22no—
601C22—24no—
602C22—74no—
603C22—144no—
604C22—142no—
605C22—142no—
606C22—144no—
607C22—142no—
608C22—142no—
609C22—142no—
610C22—2816no—
611C22—282no—
612C22—282no—
613C22#8957414yescomplemented
614C22C2×D7×C22⋊C411yescharacteristic
615C22D7×#12112yes—
616C22C2×#647811yescharacteristic
617C22C2×#648111yescharacteristic
618C22#9045912yes—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—74no—
623C4—82no—
624C4—142no—
625C4—282no—
626C4—282no—
627C4—562no—
628C4C4×D4×D712yes—
629C4D7×#12011yescharacteristic
630C4D7×#12311yescharacteristic
631C2—24no—
632C2—144no—
633C2—282no—
634C2D7×#38611yescharacteristic
635C2C2×#8957411yescharacteristic
636C2#1173627511yescharacteristic
637C2#1173810312yes—
638C2#1173820412yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1892C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1892D7
hypocenterC7#20972
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
3. C2×#89574C2semidirect
5. C7×#1892C2semidirect
6. #11729726C2semidirect
7. #11729450C2semidirect
8. #11728025C2semidirect
9. #11727268C2semidirect
10. #11726638C2semidirect
11. #11726621C2semidirect
24. D7×C22⋊Q8C4semidirect
32. C2×C7×#103C22semidirect
34. C2×#6409C22semidirect
36. C2×#6390C22semidirect
38. C2×#6384C22semidirect
44. C7×#340C22semidirect
45. #89346C22semidirect
46. #89324C22semidirect
47. #89324C22semidirect
132. #1892D7semidirect
246. C2×#103C2×D7semidirect
257. #340C2×D7semidirect
404. C22×C7#419semidirect
597. C7#20972semidirect
613. C22#89574semidirect

Maximal subgroups

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

subgroupindex
C23×#11312
C2×D7×C22⋊Q82
C2×#895742
#117313072
C7×#18922
#117297262
#117294502
#117280252
#117272682
#117266382
#117266212
#209727