Finite Group Databasebeta

← #13355504

Subgroups — #13355504

Order 1792.

1894 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D7×M16C211yesmaximal, characteristic
2C2×D7×C22⋊Q8C211yesmaximal, characteristic
3C2×#89574C212yesmaximal, complemented
4#11731307C211yesmaximal, characteristic
5C7×#1894C211yesmaximal, characteristic, complemented
6#11729743C211yesmaximal, characteristic, complemented
7#11729451C211yesmaximal, characteristic, complemented
8#11728024C212yesmaximal, complemented
9#11727270C211yesmaximal, characteristic, complemented
10#11726636C212yesmaximal, complemented
11#11726621C212yesmaximal, complemented
12C22×Q8×D7C2211yescharacteristic
13C23×C4×D7C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C2×C7×C22⋊Q8C2211yescharacteristic
16C2×#6509C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×D7×M16—24no—
20C2×D7×M16—28no—
21C22×#1131C2212yes—
22C22×C8×D7C2212yes—
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
69C2×C7×M16—22no—
70C2×C7×M16—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—
96C2×#1136—22no—
97C2×#1136—24no—
98C22×C7⋊C8C2312yes—
99D7×M16—232no—
100C2×#1131—24no—
101C2×#1131—24no—
102C2×#1131D418yes—
103C2×C8×D7—24no—
104C2×C8×D7—24no—
105C2×C8×D7—24no—
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—71no—
128#1894—71no—
129#1894—71no—
130#1894D711yescharacteristic, complemented
131#1892—72no—
132#1892—72no—
133#1736—72no—
134C24×C7C2×D411yescharacteristic
135C23×D7—24no—
136C23×D7—24no—
137C23×D7—41no—
138C23×D7—41no—
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—42no—
175C2×C4×D7—42no—
176C2×C4×D7—44no—
177C2×C4×D7C22⋊C414yes—
178C2×C4×D7C22⋊C414yes—
179C2×C4×D7C22×C412yes—
180C2×C4×D7C22×C412yes—
181C2×C4×D7C2×D412yes—
182C2×C4×D7C2×D414yes—
183C2×C4×D7C2×D412yes—
184C2×C4×D7#4114yes—
185C2×Dic14—22no—
186C2×Dic14—22no—
187C2×Dic14—21no—
188C2×Dic14—24no—
189C2×Dic14—44no—
190C2×Dic14C22×C412yes—
191C7×M16—28no—
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C2×C7×C8C2×D414yes—
195C7×C4⋊C4—24no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4C22×C412yes—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—42no—
202C22⋊Dic7—24no—
203C22⋊Dic7—42no—
204(C2×D7)⋊C4—24no—
205(C2×D7)⋊C4—24no—
206(C2×D7)⋊C4—44no—
207(C2×D7)⋊C4—44no—
208C4⋊Dic7—21no—
209C4⋊Dic7—24no—
210C4⋊Dic7—22no—
211C4⋊Dic7—22no—
212C4⋊Dic7C22×C412yes—
213Dic7⋊C4—24no—
214Dic7⋊C4—24no—
215Dic7⋊C4—24no—
216C4×Dic7—44no—
217#1136—28no—
218C2×C7⋊C8—22no—
219C2×C7⋊C8—22no—
220C2×C7⋊C8C2×D414yes—
221#1131—216no—
222C8×D7—216no—
223C23×Q8—71no—
224C24×C4—71no—
225C22×M16—71no—
226C22×M16—72no—
227C22×M16—148no—
228C22×M16—144no—
229C22×M16C2×D711yescharacteristic
230C23×C8—72no—
231C2×C22⋊Q8—71no—
232C2×C22⋊Q8—71no—
233C2×C22⋊Q8—78no—
234C2×C22⋊Q8—71no—
235C2×C22⋊Q8—141no—
236C2×C22⋊Q8—141no—
237C2×C22⋊Q8—144no—
238C2×C22⋊Q8C2×D711yescharacteristic
239C2×#117—72no—
240C22×C4⋊C4—71no—
241C22×C4⋊C4—141no—
242C22×C22⋊C4—141no—
243#419—148no—
244C2×#103—72no—
245C2×#103—72no—
246C2×#103—72no—
247C2×#103C2×D712yescomplemented
248#381—141no—
249#380—71no—
250#380—72no—
251#380C2×D711yescharacteristic
252#379—141no—
253C4×C22⋊C4—141no—
254C4×C22⋊C4—141no—
255#340—72no—
256#340—72no—
257#340—72no—
258#340C2×D712yescomplemented
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—21no—
262C23×C7—21no—
263C23×C7C2×C22⋊C412yes—
264C23×C7C22×D411yescharacteristic
265C22×D7—22no—
266C22×D7—22no—
267C22×D7—21no—
268C22×D7—21no—
269C22×D7—21no—
270C22×D7—21no—
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—
279C22×D7#13714yes—
280C7×Q8—24no—
281C7×Q8—41no—
282C7×Q8—41no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—42no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7C2×C22⋊C412yes—
296C2×C4×C7C23×C411yescharacteristic
297C2×C4×C7C22×D412yes—
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C2×#4112yes—
300C2×Dic7—22no—
301C2×Dic7—28no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—42no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—42no—
308C2×Dic7—42no—
309C2×Dic7—42no—
310C2×Dic7C2×C22⋊C412yes—
311C2×Dic7C2×C22⋊C412yes—
312C2×Dic7#12114yes—
313C4×D7—28no—
314C4×D7—24no—
315C4×D7—22no—
316C4×D7—22no—
317C4×D7—28no—
318C4×D7—48no—
319C4×D7—48no—
320C4×D7C4×D418yes—
321C4×D7#12014yes—
322C4×D7#12314yes—
323Dic14—24no—
324Dic14—41no—
325Dic14—41no—
326Dic14—48no—
327C7×C8—24no—
328C7⋊C8—24no—
329C25—71no—
330C22×Q8—74no—
331C22×Q8—71no—
332C22×Q8—141no—
333C22×Q8—142no—
334C22×Q8—144no—
335C22×Q8—144no—
336C22×Q8C22×D711yescharacteristic
337C23×C4—71no—
338C23×C4—71no—
339C23×C4—74no—
340C23×C4—71no—
341C23×C4—74no—
342C23×C4—71no—
343C23×C4—71no—
344C23×C4—144no—
345C23×C4—144no—
346C23×C4—141no—
347C23×C4—141no—
348C23×C4C22×D711yescharacteristic
349C2×M16—22no—
350C2×M16—24no—
351C2×M16—78no—
352C2×M16—144no—
353C2×M16—144no—
354C2×M16—144no—
355C2×M16—1432no—
356C2×M16—142no—
357C22×C8—72no—
358C22×C8—144no—
359C22×C8—144no—
360C22×C8—144no—
361C22×C8C22×D712yes—
362C22⋊Q8—22no—
363C22⋊Q8—74no—
364C22⋊Q8—74no—
365C22⋊Q8—74no—
366C22⋊Q8—148no—
367C22⋊Q8—148no—
368C22⋊Q8—142no—
369C22⋊Q8—142no—
370C22⋊Q8—142no—
371C22⋊Q8C4×D714yes—
372#117—144no—
373#117—144no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—74no—
377C2×C4⋊C4—72no—
378C2×C4⋊C4—142no—
379C2×C4⋊C4—148no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4C22×D712yes—
385C2×C4⋊C4C22×D711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—148no—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C22⋊C4—142no—
393C2×C22⋊C4—142no—
394C2×C22⋊C4—141no—
395C2×C42—142no—
396#103—24no—
397#103—144no—
398#103—144no—
399#103—144no—
400#95—21no—
401#95—142no—
402#95—141no—
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—22no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7C22×C22⋊C411yescharacteristic
409C22×C7C2×#12112yes—
410C22×C7C2×#13712yes—
411C22×C7#57511yescharacteristic
412C22×C7#57511yescharacteristic
413C2×D7—22no—
414C2×D7—22no—
415C2×D7—22no—
416C2×D7—22no—
417C2×D7—24no—
418C2×D7—24no—
419C2×D7—24no—
420C2×D7—24no—
421C2×D7—416no—
422C2×D7#38614yes—
423C2×D7#41914yescomplemented
424C4×C7—22no—
425C4×C7—42no—
426C4×C7—42no—
427C4×C7—82no—
428C4×C7C2×C4×D412yes—
429C4×C7C2×#12011yescharacteristic
430C4×C7C2×#12311yescharacteristic
431Dic7—22no—
432Dic7—42no—
433Dic7—42no—
434Dic7—82no—
435Dic7#38614yes—
436C24—71no—
437C24—71no—
438C24—74no—
439C24—144no—
440C24—144no—
441C24—281no—
442C24—281no—
443C24D4×D711yescharacteristic
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—72no—
448C2×Q8—144no—
449C2×Q8—1416no—
450C2×Q8—141no—
451C2×Q8—142no—
452C2×Q8—142no—
453C2×Q8—284no—
454C2×Q8—284no—
455C2×Q8—284no—
456C2×Q8C2×C4×D712yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—71no—
463C22×C4—72no—
464C22×C4—72no—
465C22×C4—74no—
466C22×C4—74no—
467C22×C4—74no—
468C22×C4—71no—
469C22×C4—74no—
470C22×C4—74no—
471C22×C4—72no—
472C22×C4—72no—
473C22×C4—148no—
474C22×C4—144no—
475C22×C4—1416no—
476C22×C4—148no—
477C22×C4—144no—
478C22×C4—144no—
479C22×C4—144no—
480C22×C4—144no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—141no—
484C22×C4—141no—
485C22×C4—142no—
486C22×C4—284no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4C2×C4×D712yes—
490C22×C4D4×D711yescharacteristic
491C22×C4D4×D712yes—
492C22×C4C23×D711yescharacteristic
493M16—28no—
494M16—1416no—
495M16—148no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—74no—
499C2×C8—1416no—
500C2×C8—142no—
501C2×C8—142no—
502C2×C8D4×D714yes—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—72no—
508C4⋊C4—144no—
509C4⋊C4—144no—
510C4⋊C4—144no—
511C4⋊C4—144no—
512C4⋊C4—141no—
513C4⋊C4—142no—
514C4⋊C4—142no—
515C4⋊C4C2×C4×D712yes—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—144no—
519C22⋊C4—144no—
520C22⋊C4—144no—
521C22⋊C4—284no—
522C22⋊C4—284no—
523C22⋊C4—282no—
524C42—284no—
525C2×C7—42no—
526C2×C7C2×#38611yescharacteristic
527C2×C7C2×#41911yescharacteristic
528C2×C7#289611yescharacteristic
529C2×C7#305612yes—
530C2×C7#305612yes—
531D7—24no—
532D7—24no—
533C23—21no—
534C23—22no—
535C23—21no—
536C23—22no—
537C23—72no—
538C23—72no—
539C23—74no—
540C23—74no—
541C23—148no—
542C23—148no—
543C23—141no—
544C23—141no—
545C23—142no—
546C23—141no—
547C23—141no—
548C23—142no—
549C23—284no—
550C23—288no—
551C23—288no—
552C23D7×C22⋊C412yes—
553C23C2×D4×D711yescharacteristic
554Q8—24no—
555Q8—41no—
556Q8—41no—
557Q8—144no—
558Q8—288no—
559Q8—281no—
560Q8—281no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—72no—
574C2×C4—74no—
575C2×C4—78no—
576C2×C4—74no—
577C2×C4—74no—
578C2×C4—72no—
579C2×C4—148no—
580C2×C4—148no—
581C2×C4—148no—
582C2×C4—142no—
583C2×C4—144no—
584C2×C4—142no—
585C2×C4—142no—
586C2×C4—142no—
587C2×C4—142no—
588C2×C4—288no—
589C2×C4—288no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—282no—
593C2×C4—282no—
594C2×C4—282no—
595C2×C4—282no—
596C2×C4D7×C22⋊C412yes—
597C2×C4C22×C4×D711yescharacteristic
598C2×C4C2×D4×D712yes—
599C2×C4C2×D4×D711yescharacteristic
600C2×C4D7×#4112yes—
601C8—24no—
602C8—144no—
603C7#2097211yescharacteristic, complemented, Sylow
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—74no—
610C22—74no—
611C22—142no—
612C22—142no—
613C22—142no—
614C22—142no—
615C22—144no—
616C22—144no—
617C22—144no—
618C22—144no—
619C22—2816no—
620C22C2×D7×C22⋊C411yescharacteristic
621C22D7×#12112yes—
622C22#9046411yescharacteristic
623C22D7×#13712yes—
624C22#9046811yescharacteristic
625C4—22no—
626C4—42no—
627C4—42no—
628C4—74no—
629C4—82no—
630C4—142no—
631C4—282no—
632C4—282no—
633C4—562no—
634C4C4×D4×D712yes—
635C4D7×#12011yescharacteristic
636C4D7×#12311yescharacteristic
637C2—42no—
638C2—144no—
639C2—144no—
640C2D7×#38611yescharacteristic
641C2#1173632611yescharacteristic
642C2D7×#41911yescharacteristic
643C2#1173825112yes—
644C2#1173834912yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1894C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1894D7
hypocenterC7#20972
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
3. C2×#89574C2semidirect
5. C7×#1894C2semidirect
6. #11729743C2semidirect
7. #11729451C2semidirect
8. #11728024C2semidirect
9. #11727270C2semidirect
10. #11726636C2semidirect
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
130. #1894D7semidirect
247. C2×#103C2×D7semidirect
258. #340C2×D7semidirect
423. C2×D7#419semidirect
603. C7#20972semidirect

Maximal subgroups

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

subgroupindex
C22×D7×M162
C2×D7×C22⋊Q82
C2×#895742
#117313072
C7×#18942
#117297432
#117294512
#117280242
#117272702
#117266362
#117266212
#209727