Finite Group Databasebeta

← #13355035

Subgroups — #13355035

Order 1792.

1934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×C22⋊Q8C211yesmaximal, characteristic
2C2×D7×#117C211yesmaximal, characteristic
3C2×#89574C212yesmaximal, complemented
4C2×#89498C211yesmaximal, characteristic
5C7×#1873C211yesmaximal, characteristic, complemented
6#11729447C211yesmaximal, characteristic, complemented
7#11729339C211yesmaximal, characteristic, complemented
8#11728022C212yesmaximal, complemented
9#11727685C211yesmaximal, characteristic, complemented
10#11726632C212yesmaximal, complemented
11#11726621C212yesmaximal, complemented
12C22×Q8×D7C2211yescharacteristic
13C23×C4×D7C2211yescharacteristic
14C2×C7×C22⋊Q8C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C2×#6509C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C2×#6487C2211yescharacteristic
19C22×#1131C2212yes—
20D7×C22⋊Q8—24no—
21D7×C22⋊Q8C418yescomplemented
22D7×#117—28no—
23C2×#6460C2211yescharacteristic
24C2×#6459—21no—
25C2×#6455C2212yes—
26C2×D7×C4⋊C4—21no—
27C2×D7×C4⋊C4C2212yes—
28C2×D7×C4⋊C4C2211yescharacteristic
29C2×D7×C22⋊C4—21no—
30C2×D7×C22⋊C4—21no—
31C2×#6441—21no—
32C2×#6440—21no—
33C2×#6435—21no—
34C2×C42×D7—21no—
35C2×C7×#103C2212yescomplemented
36C2×C7×C22⋊C8C2211yescharacteristic
37C2×#6409C2212yescomplemented
38C2×C22⋊(C7⋊C8)C2211yescharacteristic
39C2×(C2×D7)⋊C8C2212yes—
40C2×#6390C2212yescomplemented
41C2×#6384C2212yescomplemented
42#89574—28no—
43#89498—24no—
44C7×#340C2212yescomplemented
45#89346C2212yescomplemented
46#89324C2212yescomplemented
47#89324C2212yescomplemented
48#20911—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×C414yescomplemented
57C22×C4×D7—24no—
58C22×C4×D7—24no—
59C22×C4×D7—22no—
60C22×C4×D7—21no—
61C22×C4×D7—21no—
62C22×C4×D7C2×C414yes—
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D411yescharacteristic
65C22×C4×D7D414yes—
66C22×C4×D7C2311yescharacteristic
67C22×C4×D7C2311yescharacteristic
68C22×Dic14—21no—
69C22×Dic14C2311yescharacteristic
70C22×C7×C8C2312yes—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8C2×C414yescomplemented
73C7×#117—24no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C42×C7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82#6509—22no—
83#6509C2×C414yes—
84C2×(C2×D7)⋊C4—22no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87#6489—24no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2312yes—
91#6487—22no—
92#6487C2×C414yescomplemented
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—22no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—22no—
99C22×C7⋊C8C2312yes—
100C2×#1131—24no—
101C2×#1131—24no—
102C2×#1131—24no—
103#6460—22no—
104#6460C2×C414yes—
105#6459—28no—
106#6455—24no—
107#6455D418yes—
108D7×C4⋊C4—28no—
109D7×C4⋊C4—22no—
110D7×C4⋊C4—24no—
111D7×C4⋊C4—24no—
112D7×C4⋊C4C2×C414yescomplemented
113D7×C22⋊C4—28no—
114D7×C22⋊C4—28no—
115#6441—28no—
116#6440—28no—
117#6435—28no—
118C42×D7—28no—
119C7×#103—24no—
120C7×C22⋊C8—22no—
121#6409—24no—
122C22⋊(C7⋊C8)—22no—
123(C2×D7)⋊C8—24no—
124#6390—24no—
125#6384—24no—
126C22×C22⋊Q8—71no—
127C22×#117—71no—
128C2×#419—72no—
129C2×#407—71no—
130#1873—71no—
131#1873—71no—
132#1873—71no—
133#1873D711yescharacteristic, complemented
134#1872—72no—
135#1872—72no—
136#1736—72no—
137C24×C7C2×D411yescharacteristic
138C23×D7—22no—
139C23×D7—22no—
140C23×D7—24no—
141C23×D7—42no—
142C23×D7C22⋊C414yes—
143C23×D7C2×D411yescharacteristic
144C23×D7C2×D411yescharacteristic
145C2×C7×Q8—21no—
146C2×C7×Q8—22no—
147C2×C7×Q8—22no—
148C2×C7×Q8C22×C412yes—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7C22×C412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7C22⋊C414yes—
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166Q8×D7—216no—
167Q8×D7—48no—
168C2×C4×D7—22no—
169C2×C4×D7—22no—
170C2×C4×D7—22no—
171C2×C4×D7—22no—
172C2×C4×D7—22no—
173C2×C4×D7—22no—
174C2×C4×D7—216no—
175C2×C4×D7—216no—
176C2×C4×D7—28no—
177C2×C4×D7—24no—
178C2×C4×D7—24no—
179C2×C4×D7—44no—
180C2×C4×D7—44no—
181C2×C4×D7—42no—
182C2×C4×D7—42no—
183C2×C4×D7—44no—
184C2×C4×D7C22⋊C414yes—
185C2×C4×D7C22⋊C414yes—
186C2×C4×D7C22×C412yes—
187C2×C4×D7C22×C412yes—
188C2×C4×D7C2×D412yes—
189C2×C4×D7C2×D414yes—
190C2×C4×D7C2×D412yes—
191C2×C4×D7#4114yes—
192C2×Dic14—22no—
193C2×Dic14—22no—
194C2×Dic14—21no—
195C2×Dic14—24no—
196C2×Dic14—42no—
197C2×Dic14—42no—
198C2×Dic14C22×C412yes—
199C2×C7×C8—22no—
200C2×C7×C8—22no—
201C2×C7×C8—22no—
202C7×C4⋊C4—24no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4C22×C412yes—
206C7×C4⋊C4C2×D414yes—
207C7×C22⋊C4—24no—
208C7×C22⋊C4—24no—
209C42×C7—24no—
210C22⋊Dic7—24no—
211C22⋊Dic7—24no—
212(C2×D7)⋊C4—24no—
213(C2×D7)⋊C4—24no—
214(C2×D7)⋊C4—28no—
215C4⋊Dic7—22no—
216C4⋊Dic7—21no—
217C4⋊Dic7—24no—
218C4⋊Dic7C22×C412yes—
219C4⋊Dic7C2×D414yes—
220Dic7⋊C4—28no—
221Dic7⋊C4—24no—
222Dic7⋊C4—24no—
223Dic7⋊C4—24no—
224C4×Dic7—24no—
225C4×Dic7—28no—
226C2×C7⋊C8—22no—
227C2×C7⋊C8—22no—
228C2×C7⋊C8—22no—
229#1131—48no—
230C23×Q8—71no—
231C24×C4—71no—
232C22×M16—72no—
233C2×C22⋊Q8—71no—
234C2×C22⋊Q8—71no—
235C2×C22⋊Q8—78no—
236C2×C22⋊Q8—71no—
237C2×C22⋊Q8—141no—
238C2×C22⋊Q8—141no—
239C2×C22⋊Q8—144no—
240C2×C22⋊Q8C2×D711yescharacteristic
241C2×#117—71no—
242C2×#117—72no—
243C2×#117—141no—
244C2×#117—141no—
245C2×#117—148no—
246C2×#117C2×D711yescharacteristic
247C22×C4⋊C4—71no—
248C22×C4⋊C4—72no—
249C22×C4⋊C4—141no—
250C22×C22⋊C4—141no—
251C22×C22⋊C4—141no—
252C22×C42—141no—
253#419—148no—
254C2×#103—72no—
255C2×#103—72no—
256C2×#103—72no—
257C2×#103C2×D712yescomplemented
258#407—144no—
259C2×C22⋊C8—71no—
260C2×C22⋊C8—72no—
261C2×C22⋊C8C2×D711yescharacteristic
262#340—72no—
263#340—72no—
264#340—72no—
265#340C2×D712yescomplemented
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—22no—
269C23×C7—22no—
270C23×C7C2×C22⋊C412yes—
271C23×C7C22×D411yescharacteristic
272C22×D7—21no—
273C22×D7—21no—
274C22×D7—22no—
275C22×D7—22no—
276C22×D7—22no—
277C22×D7—28no—
278C22×D7—28no—
279C22×D7—44no—
280C22×D7—48no—
281C22×D7—48no—
282C22×D7C2×C22⋊C412yes—
283C22×D7C2×C22⋊C412yes—
284C22×D7#12012yes—
285C22×D7#12312yes—
286C22×D7#13714yes—
287C7×Q8—24no—
288C7×Q8—42no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—24no—
292C2×C4×C7—24no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—24no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7C2×C22⋊C412yes—
302C2×C4×C7C23×C411yescharacteristic
303C2×C4×C7C22×D412yes—
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C2×#4112yes—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—28no—
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—24no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—42no—
315C2×Dic7—42no—
316C2×Dic7—42no—
317C2×Dic7C2×C22⋊C412yes—
318C2×Dic7C2×C22⋊C412yes—
319C2×Dic7#12012yes—
320C2×Dic7#12312yes—
321C4×D7—28no—
322C4×D7—24no—
323C4×D7—24no—
324C4×D7—28no—
325C4×D7—48no—
326C4×D7—416no—
327C4×D7—48no—
328C4×D7C4×D418yes—
329C4×D7#12118yes—
330Dic14—24no—
331Dic14—42no—
332Dic14—48no—
333C7×C8—42no—
334C7⋊C8—42no—
335C25—71no—
336C22×Q8—74no—
337C22×Q8—71no—
338C22×Q8—141no—
339C22×Q8—142no—
340C22×Q8—144no—
341C22×Q8—144no—
342C22×Q8C22×D711yescharacteristic
343C23×C4—71no—
344C23×C4—71no—
345C23×C4—74no—
346C23×C4—71no—
347C23×C4—74no—
348C23×C4—71no—
349C23×C4—71no—
350C23×C4—144no—
351C23×C4—144no—
352C23×C4—141no—
353C23×C4—142no—
354C23×C4—141no—
355C23×C4C22×D711yescharacteristic
356C2×M16—144no—
357C2×M16—144no—
358C2×M16—144no—
359C22×C8—72no—
360C22×C8C22×D712yes—
361C22⋊Q8—22no—
362C22⋊Q8—74no—
363C22⋊Q8—74no—
364C22⋊Q8—74no—
365C22⋊Q8—148no—
366C22⋊Q8—148no—
367C22⋊Q8—142no—
368C22⋊Q8—142no—
369C22⋊Q8—142no—
370C22⋊Q8C4×D714yescomplemented
371#117—24no—
372#117—78no—
373#117—148no—
374#117—148no—
375#117—144no—
376#117—144no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—72no—
379C2×C4⋊C4—71no—
380C2×C4⋊C4—74no—
381C2×C4⋊C4—144no—
382C2×C4⋊C4—142no—
383C2×C4⋊C4—148no—
384C2×C4⋊C4—142no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4—144no—
390C2×C4⋊C4C22×D711yescharacteristic
391C2×C4⋊C4C22×D712yes—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—148no—
395C2×C22⋊C4—148no—
396C2×C22⋊C4—142no—
397C2×C22⋊C4—141no—
398C2×C22⋊C4—141no—
399C2×C22⋊C4—141no—
400C2×C22⋊C4—141no—
401C2×C42—21no—
402C2×C42—148no—
403C2×C42—142no—
404C2×C42—141no—
405#103—24no—
406#103—144no—
407#103—144no—
408#103—144no—
409C22⋊C8—22no—
410C22⋊C8—144no—
411C22⋊C8—142no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—22no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7C22×C22⋊C411yescharacteristic
418C22×C7C2×#12011yescharacteristic
419C22×C7C2×#12311yescharacteristic
420C22×C7C2×#13712yes—
421C22×C7#57512yes—
422C2×D7—24no—
423C2×D7—24no—
424C2×D7—22no—
425C2×D7—22no—
426C2×D7—22no—
427C2×D7—24no—
428C2×D7—24no—
429C2×D7—22no—
430C2×D7—416no—
431C2×D7#38614yes—
432C2×D7#41914yescomplemented
433C4×C7—22no—
434C4×C7—42no—
435C4×C7—44no—
436C4×C7—42no—
437C4×C7C2×C4×D412yes—
438C4×C7C2×#12112yes—
439Dic7—22no—
440Dic7—42no—
441Dic7—44no—
442Dic7—42no—
443Dic7#38614yes—
444C24—71no—
445C24—71no—
446C24—74no—
447C24—142no—
448C24—142no—
449C24—144no—
450C24—282no—
451C24D4×D711yescharacteristic
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—72no—
456C2×Q8—144no—
457C2×Q8—1416no—
458C2×Q8—141no—
459C2×Q8—142no—
460C2×Q8—142no—
461C2×Q8—282no—
462C2×Q8—282no—
463C2×Q8—288no—
464C2×Q8C2×C4×D712yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—71no—
471C22×C4—72no—
472C22×C4—72no—
473C22×C4—74no—
474C22×C4—74no—
475C22×C4—74no—
476C22×C4—71no—
477C22×C4—74no—
478C22×C4—74no—
479C22×C4—72no—
480C22×C4—72no—
481C22×C4—142no—
482C22×C4—142no—
483C22×C4—1416no—
484C22×C4—1416no—
485C22×C4—142no—
486C22×C4—142no—
487C22×C4—144no—
488C22×C4—144no—
489C22×C4—148no—
490C22×C4—142no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4—142no—
494C22×C4—142no—
495C22×C4—141no—
496C22×C4—141no—
497C22×C4—284no—
498C22×C4—282no—
499C22×C4—282no—
500C22×C4—284no—
501C22×C4—284no—
502C22×C4C2×C4×D712yes—
503C22×C4D4×D711yescharacteristic
504C22×C4D4×D712yes—
505C22×C4C23×D711yescharacteristic
506M16—288no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—142no—
511C2×C8—142no—
512C2×C8—142no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—72no—
517C4⋊C4—74no—
518C4⋊C4—144no—
519C4⋊C4—144no—
520C4⋊C4—144no—
521C4⋊C4—148no—
522C4⋊C4—144no—
523C4⋊C4—142no—
524C4⋊C4—141no—
525C4⋊C4C2×C4×D712yes—
526C4⋊C4D4×D714yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—148no—
530C22⋊C4—144no—
531C22⋊C4—144no—
532C22⋊C4—144no—
533C22⋊C4—144no—
534C42—24no—
535C42—148no—
536C42—144no—
537C2×C7—42no—
538C2×C7C2×#38611yescharacteristic
539C2×C7C2×#41911yescharacteristic
540C2×C7#289611yescharacteristic
541C2×C7#300712yes—
542C2×C7#309512yes—
543D7—24no—
544D7—24no—
545C23—22no—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—72no—
550C23—72no—
551C23—74no—
552C23—72no—
553C23—72no—
554C23—148no—
555C23—148no—
556C23—142no—
557C23—142no—
558C23—142no—
559C23—141no—
560C23—141no—
561C23—284no—
562C23—288no—
563C23—288no—
564C23D7×C22⋊C412yes—
565C23C2×D4×D711yescharacteristic
566Q8—24no—
567Q8—42no—
568Q8—144no—
569Q8—288no—
570Q8—282no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—72no—
584C2×C4—72no—
585C2×C4—72no—
586C2×C4—78no—
587C2×C4—78no—
588C2×C4—72no—
589C2×C4—148no—
590C2×C4—148no—
591C2×C4—148no—
592C2×C4—141no—
593C2×C4—141no—
594C2×C4—144no—
595C2×C4—144no—
596C2×C4—142no—
597C2×C4—142no—
598C2×C4—144no—
599C2×C4—288no—
600C2×C4—2816no—
601C2×C4—288no—
602C2×C4—282no—
603C2×C4—282no—
604C2×C4—282no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4D7×C22⋊C412yes—
608C2×C4C22×C4×D711yescharacteristic
609C2×C4C2×D4×D712yes—
610C2×C4C2×D4×D711yescharacteristic
611C2×C4D7×#4112yes—
612C8—42no—
613C8—282no—
614C7#2091111yescharacteristic, complemented, Sylow
615C22—22no—
616C22—22no—
617C22—22no—
618C22—42no—
619C22—42no—
620C22—74no—
621C22—74no—
622C22—144no—
623C22—144no—
624C22—142no—
625C22—144no—
626C22—144no—
627C22—142no—
628C22—142no—
629C22—142no—
630C22—2816no—
631C22C2×D7×C22⋊C411yescharacteristic
632C22D7×#12011yescharacteristic
633C22D7×#12311yescharacteristic
634C22#9045912yes—
635C22D7×#13712yes—
636C4—22no—
637C4—44no—
638C4—42no—
639C4—42no—
640C4—74no—
641C4—142no—
642C4—284no—
643C4—282no—
644C4—282no—
645C4C4×D4×D712yes—
646C4D7×#12112yes—
647C2—42no—
648C2—144no—
649C2—144no—
650C2D7×#38611yescharacteristic
651C2#1173627511yescharacteristic
652C2D7×#41911yescharacteristic
653C2#1173767312yes—
654C2#1173896212yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1873C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenter#1873D7
hypocenterC7#20911
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
3. C2×#89574C2semidirect
5. C7×#1873C2semidirect
6. #11729447C2semidirect
7. #11729339C2semidirect
8. #11728022C2semidirect
9. #11727685C2semidirect
10. #11726632C2semidirect
11. #11726621C2semidirect
21. D7×C22⋊Q8C4semidirect
35. C2×C7×#103C22semidirect
37. C2×#6409C22semidirect
40. C2×#6390C22semidirect
41. C2×#6384C22semidirect
44. C7×#340C22semidirect
45. #89346C22semidirect
46. #89324C22semidirect
47. #89324C22semidirect
56. C2×Q8×D7C2×C4semidirect
72. C7×C22⋊Q8C2×C4semidirect
92. #6487C2×C4semidirect
112. D7×C4⋊C4C2×C4semidirect
133. #1873D7semidirect
257. C2×#103C2×D7semidirect
265. #340C2×D7semidirect
370. C22⋊Q8C4×D7semidirect
432. C2×D7#419semidirect
614. C7#20911semidirect

Maximal subgroups

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

subgroupindex
C2×D7×C22⋊Q82
C2×D7×#1172
C2×#895742
C2×#894982
C7×#18732
#117294472
#117293392
#117280222
#117276852
#117266322
#117266212
#209117