Finite Group Databasebeta

← #13321573

Subgroups — #13321573

Order 1792.

1074 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743039C211yesmaximal, characteristic
2#11736047C211yesmaximal, characteristic, complemented
3#11730814C211yesmaximal, characteristic, complemented
4C2×C7×#328C211yesmaximal, characteristic, complemented
5#11729444C211yesmaximal, characteristic, complemented
6#11727698C211yesmaximal, characteristic, complemented
7C2×#89246C211yesmaximal, characteristic, complemented
8#11726091C212yesmaximal, complemented
9#11726088C212yesmaximal, complemented
10#11726066C212yesmaximal, complemented
11#11726065C212yesmaximal, complemented
12C2×C7×C22⋊Q8C2211yescharacteristic
13#90508C412yescomplemented
14C2×#6493C2211yescharacteristic
15C2×C4×C22⋊D7—21no—
16C2×C4×C22⋊D7C2211yescharacteristic
17#90336—21no—
18#90335—21no—
19#90333C412yescomplemented
20#90328—22no—
21#90321C412yescomplemented
22#90318C2212yes—
23#90315C2212yes—
24#90205—22no—
25#90205C412yescomplemented
26#90184—21no—
27#90177—21no—
28C2×C7×C22⋊C8C2211yescharacteristic
29C2×C7×#95C2211yescharacteristic
30#89984C2211yescharacteristic
31#89982—21no—
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33#89883—22no—
34#89502—22no—
35#89498C2212yescomplemented
36#89491C2212yescomplemented
37#89440—21no—
38#89439C2211yescharacteristic, complemented
39D7×#95C2211yescharacteristic, complemented
40#89422C2211yescharacteristic, complemented
41#89421C2211yescharacteristic, complemented
42#89419—21no—
43C7×#328C2212yescomplemented
44C7×#328C2212yescomplemented
45#89333C2212yescomplemented
46#89333C2212yescomplemented
47#89265C2212yes—
48#89265C2212yes—
49#89246C2212yes—
50#89246C2212yes—
51#17622—71nomaximal, Sylow
52C22×C7×Q8C2×C411yescharacteristic, complemented
53C23×C4×C7—21no—
54C23×C4×C7C2311yescharacteristic
55C22×C22⋊D7—21no—
56C22×C22⋊D7D411yescharacteristic
57C2×#1161C2×C412yescomplemented
58C22×D28C2×C411yescharacteristic, complemented
59C22×C4×D7—21no—
60C22×C4×D7C2311yescharacteristic
61C22×C7×C8—21no—
62C7×C22⋊Q8—21no—
63C7×C22⋊Q8—21no—
64C7×C22⋊Q8C2×C412yescomplemented
65C7×C22⋊Q8C2312yes—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
68C2×C7×C22⋊C4—21no—
69C2×C22⋊Dic7—21no—
70C2×C22⋊Dic7D411yescharacteristic
71#6510—22no—
72#6510C2×C412yescomplemented
73#6509—22no—
74#6509C2×C412yescomplemented
75Q8×Dic7C2×C412yescomplemented
76#6493—21no—
77#6493—21no—
78#6493—22no—
79#6493—22no—
80#6493C2×C412yescomplemented
81#6493C2312yes—
82C4×C22⋊D7—22no—
83C4×C22⋊D7—22no—
84C4×C22⋊D7—44no—
85C4×C22⋊D7C2×C412yes—
86C4×C22⋊D7C2×C412yes—
87C4×C22⋊D7D412yes—
88C4×C22⋊D7D412yes—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4D411yescharacteristic
92C2×C4⋊Dic7—21no—
93C2×C4⋊Dic7C2×C411yescharacteristic, complemented
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4D411yescharacteristic
98C2×C4×Dic7—21no—
99C2×C4×Dic7C2311yescharacteristic
100C22×C7⋊C8—21no—
101C2×#1131—22no—
102#6461—22no—
103#6461—22no—
104#6461—22no—
105#6458—22no—
106#6458C2×C412yescomplemented
107#6457—22no—
108#6457—22no—
109#6457—22no—
110#6456—22no—
111#6456C2×C412yescomplemented
112#6455—22no—
113#6455C2×C412yescomplemented
114#6453—22no—
115#6453C2×C412yescomplemented
116#6449—22no—
117#6449—22no—
118#6445—22no—
119#6445—22no—
120(C22⋊D7)⋊C4—22no—
121(C22⋊D7)⋊C4—22no—
122D7×C22⋊C4—22no—
123#6440—22no—
124C7×C22⋊C8—21no—
125C7×C22⋊C8C2312yes—
126C7×#95—21no—
127C7×#95C2311yescharacteristic
128C7×#95C2311yescharacteristic
129C7⋊#95—21no—
130C7⋊#95—21no—
131C7⋊#95C2311yescharacteristic
132C7⋊#95C2311yescharacteristic
133C22⋊(C7⋊C8)—21no—
134C22⋊(C7⋊C8)C2312yes—
135(C2×D7)⋊C8—22no—
136#6389—22no—
137Dic7⋊C8—22no—
138#3505—71no—
139#2973—71no—
140#2349—71no—
141#1520—71no—
142#1519—71no—
143#1513—72no—
144#1512—72no—
145#1509—72no—
146#1504—72no—
147C2×#328D711yescharacteristic, complemented
148C2×#327—71no—
149C24×C7C2×D411yescharacteristic
150C23×D7—21no—
151C23×D7C2×D411yescharacteristic
152C2×C7×Q8—21no—
153C2×C7×Q8—81no—
154C2×C7×Q8C22×C412yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—81no—
162C22×C4×C7—81no—
163C22×C4×C7C22×C411yescharacteristic
164C22×C4×C7C22×C411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2411yescharacteristic
169C2×C22⋊D7—22no—
170C2×C22⋊D7—22no—
171C2×C22⋊D7—22no—
172C2×C22⋊D7—22no—
173C2×C22⋊D7—42no—
174C2×C22⋊D7—42no—
175C2×C22⋊D7C22⋊C412yes—
176C2×C22⋊D7C22⋊C412yes—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7C2×D411yescharacteristic
181C22×Dic7C2×D411yescharacteristic
182C22×Dic7C2×D411yescharacteristic
183#1161—82no—
184C2×D28—22no—
185C2×D28—22no—
186C2×D28—21no—
187C2×D28—81no—
188C2×D28C22×C412yes—
189C2×C4×D7—22no—
190C2×C4×D7—22no—
191C2×C4×D7—42no—
192C2×C4×D7—42no—
193C2×C4×D7—81no—
194C2×C4×D7—81no—
195C2×C4×D7C22×C412yes—
196C2×C4×D7C2×D412yes—
197C2×C7×C8—22no—
198C2×C7×C8—81no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4C22×C412yes—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C22⋊Dic7—42no—
206C22⋊Dic7C22⋊C412yes—
207C22⋊Dic7C22⋊C412yes—
208(C2×D7)⋊C4—22no—
209(C2×D7)⋊C4—22no—
210(C2×D7)⋊C4—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213(C2×D7)⋊C4—22no—
214(C2×D7)⋊C4—22no—
215(C2×D7)⋊C4—22no—
216(C2×D7)⋊C4—42no—
217(C2×D7)⋊C4C22⋊C412yes—
218(C2×D7)⋊C4C22⋊C412yes—
219C4⋊Dic7—21no—
220C4⋊Dic7—22no—
221C4⋊Dic7—22no—
222C4⋊Dic7—81no—
223C4⋊Dic7C22×C412yes—
224Dic7⋊C4—22no—
225Dic7⋊C4—22no—
226Dic7⋊C4—42no—
227Dic7⋊C4—42no—
228Dic7⋊C4—81no—
229Dic7⋊C4C22⋊C412yes—
230Dic7⋊C4C22⋊C412yes—
231C4×Dic7—22no—
232C4×Dic7—22no—
233C4×Dic7—42no—
234C4×Dic7C22×C412yes—
235C4×Dic7C2×D412yes—
236C2×C7⋊C8—22no—
237C2×C7⋊C8—81no—
238#1131—82no—
239#553—72no—
240#553—142no—
241#552—72no—
242#551—142no—
243#548—72no—
244#546—72no—
245#539—141no—
246#538—141no—
247#534—141no—
248#529—72no—
249#525—72no—
250C2×C22⋊Q8C2×D711yescharacteristic
251C2×#121—71no—
252#518—141no—
253C2×C4×D4—71no—
254C2×C4×D4—141no—
255#435—142no—
256#435—142no—
257#432—72no—
258#407—72no—
259C2×C22⋊C8—71no—
260C2×C22⋊C8C2×D711yescharacteristic
261#389—141no—
262#385—141no—
263#385—141no—
264#382—71no—
265#380—71no—
266#380—71no—
267C4×C4⋊C4—71no—
268C2×#95—71no—
269C2×#95C2×D711yescharacteristic
270#328—72no—
271#328—72no—
272#328C2×D712yescomplemented
273#328C2×D712yescomplemented
274#327—72no—
275#327—72no—
276#327—72no—
277#327—72no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7C2×C22⋊C411yescharacteristic
281C23×C7C2×C22⋊C411yescharacteristic
282C23×C7AGL(1,ℤ/8)12yes—
283C23×C7#13712yes—
284C23×C7C22×D411yescharacteristic
285C22×D7—21no—
286C22×D7—21no—
287C22×D7—22no—
288C22×D7—41no—
289C22×D7—41no—
290C22×D7—42no—
291C22×D7—81no—
292C22×D7—81no—
293C22×D7C2×C22⋊C412yes—
294C7×Q8—42no—
295C7×Q8—81no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—81no—
311C2×C4×C7—81no—
312C2×C4×C7—84no—
313C2×C4×C7C2×C22⋊C411yescharacteristic
314C2×C4×C7C2×C22⋊C411yescharacteristic
315C2×C4×C7C23×C411yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C22⋊D7—24no—
318C22⋊D7—24no—
319C22⋊D7—42no—
320C22⋊D7—42no—
321C22⋊D7—44no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7—22no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—42no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7C2×C22⋊C412yes—
335C2×Dic7C2×C22⋊C412yes—
336C2×Dic7C2×C22⋊C412yes—
337D28—42no—
338D28—81no—
339D28—82no—
340D28—82no—
341C4×D7—42no—
342C4×D7—42no—
343C4×D7—82no—
344C4×D7—82no—
345C4×D7—84no—
346C7×C8—42no—
347C7⋊C8—42no—
348C2×#41—72no—
349C22×Q8C4×D711yescharacteristic, complemented
350C22×D4—71no—
351C22×D4—71no—
352C22×D4—141no—
353C23×C4—21no—
354C23×C4—71no—
355C23×C4—141no—
356C23×C4C22×D711yescharacteristic
357C2×M16—142no—
358C22×C8—21no—
359C22×C8—141no—
360#126—142no—
361#126—142no—
362#126—142no—
363#125—72no—
364#125—142no—
365#124—72no—
366#124—142no—
367#123—142no—
368#123—142no—
369#123—142no—
370#123—142no—
371#123—142no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—72no—
375C22⋊Q8—142no—
376C22⋊Q8C4×D712yescomplemented
377C22⋊Q8C22×D712yes—
378#121—72no—
379#121—72no—
380#121—72no—
381#121—142no—
382#121—142no—
383#121—142no—
384#121—141no—
385#121—141no—
386#120—142no—
387#120—142no—
388C4×Q8—72no—
389C4×D4—72no—
390C4×D4—72no—
391C4×D4—72no—
392C4×D4—72no—
393C4×D4—72no—
394C4×D4—142no—
395C4×D4—142no—
396C4×D4—142no—
397C4×D4—142no—
398C4×D4—142no—
399C4×D4—284no—
400#117—72no—
401#117—142no—
402#117—142no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—71no—
405C2×C4⋊C4—71no—
406C2×C4⋊C4—141no—
407C2×C4⋊C4—141no—
408C2×C4⋊C4—141no—
409C2×C4⋊C4—141no—
410C2×C4⋊C4C4×D711yescharacteristic, complemented
411C2×C22⋊C4—21no—
412C2×C22⋊C4—71no—
413C2×C22⋊C4—71no—
414C2×C22⋊C4—142no—
415C2×C22⋊C4—141no—
416C2×C22⋊C4—141no—
417C2×C22⋊C4—141no—
418C2×C42—71no—
419C2×C42—141no—
420C4⋊C8—142no—
421C22⋊C8—21no—
422C22⋊C8—72no—
423C22⋊C8—142no—
424C22⋊C8—141no—
425C22⋊C8C22×D712yes—
426#97—142no—
427#95—21no—
428#95—71no—
429#95—71no—
430#95—141no—
431#95—141no—
432#95C22×D711yescharacteristic
433#95C22×D711yescharacteristic
434C22×C7—21no—
435C22×C7—21no—
436C22×C7—21no—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7—21no—
440C22×C7—42no—
441C22×C7—42no—
442C22×C7#41712yes—
443C22×C7#41712yes—
444C22×C7C22×C22⋊C411yescharacteristic
445C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
446C22×C7C2×#13711yescharacteristic
447C2×D7—22no—
448C2×D7—22no—
449C2×D7—22no—
450C2×D7—42no—
451C2×D7—42no—
452C2×D7—42no—
453C2×D7—42no—
454C2×D7—42no—
455C2×D7—162no—
456C4×C7—21no—
457C4×C7—21no—
458C4×C7—21no—
459C4×C7—21no—
460C4×C7—42no—
461C4×C7—42no—
462C4×C7—42no—
463C4×C7—81no—
464Dic7—22no—
465Dic7—22no—
466Dic7—22no—
467Dic7—42no—
468Dic7—81no—
469Dic7—81no—
470C24—71no—
471C24—141no—
472C24D4×D711yescharacteristic
473#41—562no—
474C2×Q8—21no—
475C2×Q8—81no—
476C2×Q8C2×C4×D712yes—
477C2×D4—72no—
478C2×D4—72no—
479C2×D4—72no—
480C2×D4—142no—
481C2×D4—142no—
482C2×D4—142no—
483C2×D4—142no—
484C2×D4—142no—
485C2×D4—142no—
486C2×D4—141no—
487C2×D4—282no—
488C2×D4—282no—
489C2×D4—561no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—71no—
497C22×C4—71no—
498C22×C4—71no—
499C22×C4—72no—
500C22×C4—72no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—142no—
504C22×C4—142no—
505C22×C4—141no—
506C22×C4—141no—
507C22×C4—141no—
508C22×C4—282no—
509C22×C4—282no—
510C22×C4—561no—
511C22×C4—561no—
512C22×C4C2×C4×D711yescharacteristic
513C22×C4C2×C4×D711yescharacteristic
514C22×C4D4×D711yescharacteristic
515C22×C4D4×D711yescharacteristic
516C22×C4D4×D711yescharacteristic
517C22×C4C23×D711yescharacteristic
518M16—562no—
519C2×C8—22no—
520C2×C8—81no—
521C2×C8—142no—
522C2×C8—561no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—72no—
527C4⋊C4—72no—
528C4⋊C4—72no—
529C4⋊C4—142no—
530C4⋊C4—142no—
531C4⋊C4—142no—
532C4⋊C4—142no—
533C4⋊C4—141no—
534C4⋊C4—282no—
535C4⋊C4—282no—
536C4⋊C4—561no—
537C4⋊C4—561no—
538C4⋊C4C2×C4×D712yes—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—72no—
542C22⋊C4—72no—
543C22⋊C4—72no—
544C22⋊C4—72no—
545C22⋊C4—142no—
546C22⋊C4—142no—
547C22⋊C4—142no—
548C22⋊C4—142no—
549C22⋊C4—142no—
550C22⋊C4—142no—
551C22⋊C4—142no—
552C22⋊C4—142no—
553C22⋊C4—282no—
554C22⋊C4—282no—
555C42—72no—
556C42—72no—
557C42—142no—
558C42—142no—
559C42—282no—
560C2×C7—21no—
561C2×C7—21no—
562C2×C7—22no—
563C2×C7—22no—
564C2×C7#288311yescharacteristic
565C2×C7C2×#41711yescharacteristic
566C2×C7#290111yescharacteristic
567D7—42no—
568D7—82no—
569C23—41no—
570C23—41no—
571C23—72no—
572C23—141no—
573C23—141no—
574C23—142no—
575C23—281no—
576C23—281no—
577C23—282no—
578C23—561no—
579C23—561no—
580C23D7×C22⋊C411yescharacteristic
581C23D7×C22⋊C411yescharacteristic
582C23#647712yes—
583C23#648112yes—
584C23C2×D4×D711yescharacteristic
585Q8—42no—
586Q8—81no—
587D4—144no—
588D4—144no—
589D4—282no—
590D4—282no—
591D4—282no—
592D4—284no—
593D4—561no—
594D4—562no—
595D4—562no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—72no—
611C2×C4—72no—
612C2×C4—72no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—84no—
616C2×C4—141no—
617C2×C4—141no—
618C2×C4—142no—
619C2×C4—282no—
620C2×C4—282no—
621C2×C4—281no—
622C2×C4—281no—
623C2×C4—282no—
624C2×C4—281no—
625C2×C4—281no—
626C2×C4—281no—
627C2×C4—281no—
628C2×C4—281no—
629C2×C4—281no—
630C2×C4—562no—
631C2×C4—562no—
632C2×C4—564no—
633C2×C4D7×C22⋊C411yescharacteristic
634C2×C4D7×C22⋊C411yescharacteristic
635C2×C4C22×C4×D711yescharacteristic
636C2×C4C2×D4×D711yescharacteristic
637C8—42no—
638C8—282no—
639C7#1762211yescharacteristic, complemented, Sylow
640C22—21no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—42no—
647C22—42no—
648C22—142no—
649C22—142no—
650C22—142no—
651C22—282no—
652C22—282no—
653C22—282no—
654C22—282no—
655C22—282no—
656C22—1122no—
657C22#8957512yes—
658C22#8957512yes—
659C22C2×D7×C22⋊C411yescharacteristic
660C22C2×#647711yescharacteristic
661C22C2×#648111yescharacteristic
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—81no—
670C4—142no—
671C4—142no—
672C4—142no—
673C4—282no—
674C4—561no—
675C4—561no—
676C2—21no—
677C2—21no—
678C2—22no—
679C2—22no—
680C2—282no—
681C2—562no—
682C2#1173613311yescharacteristic
683C2C2×#8957511yescharacteristic
684C2#1173642811yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#328C2
socleC22×C7C22×C22⋊C4
prime radical#11743039C2
cosocleC22×C4×C7C24
hypercenterC2×#328D7
hypocenterC7#17622
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. #11736047C2semidirect
3. #11730814C2semidirect
4. C2×C7×#328C2semidirect
5. #11729444C2semidirect
6. #11727698C2semidirect
7. C2×#89246C2semidirect
8. #11726091C2semidirect
9. #11726088C2semidirect
10. #11726066C2semidirect
11. #11726065C2semidirect
13. #90508C4semidirect
19. #90333C4semidirect
21. #90321C4semidirect
25. #90205C4semidirect
35. #89498C22semidirect
36. #89491C22semidirect
38. #89439C22semidirect
39. D7×#95C22semidirect
40. #89422C22semidirect
41. #89421C22semidirect
43. C7×#328C22semidirect
44. C7×#328C22semidirect
45. #89333C22semidirect
46. #89333C22semidirect
52. C22×C7×Q8C2×C4semidirect
57. C2×#1161C2×C4semidirect
58. C22×D28C2×C4semidirect
64. C7×C22⋊Q8C2×C4semidirect
67. C2×C7×C4⋊C4C2×C4semidirect
72. #6510C2×C4semidirect
74. #6509C2×C4semidirect
75. Q8×Dic7C2×C4semidirect
80. #6493C2×C4semidirect
93. C2×C4⋊Dic7C2×C4semidirect
106. #6458C2×C4semidirect
111. #6456C2×C4semidirect
113. #6455C2×C4semidirect
115. #6453C2×C4semidirect
147. C2×#328D7semidirect
272. #328C2×D7semidirect
273. #328C2×D7semidirect
349. C22×Q8C4×D7semidirect
376. C22⋊Q8C4×D7semidirect
410. C2×C4⋊C4C4×D7semidirect
639. C7#17622semidirect

Maximal subgroups

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

subgroupindex
#117430392
#117360472
#117308142
C2×C7×#3282
#117294442
#117276982
C2×#892462
#117260912
#117260882
#117260662
#117260652
#176227