Finite Group Databasebeta

← #13386017

Subgroups — #13386017

Order 1792.

1436 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C211yesmaximal, characteristic, complemented
2C2×#89818C211yesmaximal, characteristic, complemented
3#11739848C211yesmaximal, characteristic
4C22×#6383C211yesmaximal, characteristic, complemented
5#11729497C212yesmaximal, complemented
6#11729453C212yesmaximal, complemented
7#11729450C211yesmaximal, characteristic, complemented
8#11729387C212yesmaximal, complemented
9#11729364C211yesmaximal, characteristic, complemented
10#11729340C212yesmaximal, complemented
11#11729307C211yesmaximal, characteristic, complemented
12C23×D28C2211yescharacteristic
13C7×#552—24no—
14C7×#552—24no—
15C2×C7×#126—21no—
16C2×C7×#125—21no—
17C2×C7×#125C2211yescharacteristic, complemented
18C2×C7×#123—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×C22⋊Q8C2211yescharacteristic, complemented
21C2×C4×C7×Q8C2211yescharacteristic, complemented
22C2×C4×C7×D4C2211yescharacteristic
23C2×C7×#117—21no—
24C2×C7×#117C2211yescharacteristic, complemented
25C22×C7×C4⋊C4C2211yescharacteristic
26C22×#1143C2212yes—
27C2×#6500—21no—
28C2×#6493C2212yes—
29C23×C7⋊C8C2211yescharacteristic
30C2×#6409C2212yescomplemented
31#89984C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2212yescomplemented
34#89818—24no—
35#89763—21no—
36C2×#6383—24no—
37C2×#6383C2212yes—
38C2×#6383C2212yes—
39C2×#6383C2212yes—
40#89732—21no—
41#89727—21no—
42#89711C2212yes—
43#89324C2212yescomplemented
44#89324C2212yescomplemented
45#89324C2212yescomplemented
46#89319C2212yescomplemented
47#23966—71nomaximal, Sylow
48C24×D7—21no—
49C22×C7×Q8C2311yescharacteristic
50C22×C7×D4D411yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C22×C22⋊D7—22no—
55C22×D28—22no—
56C22×D28—24no—
57C22×D28—22no—
58C22×D28—21no—
59C22×D28—42no—
60C22×D28C2312yes—
61C7×#126—28no—
62C7×#125—28no—
63C7×#125—22no—
64C7×#125—22no—
65C7×#123—28no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8D414yescomplemented
70C4×C7×Q8—22no—
71C4×C7×Q8—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C7×#117—28no—
75C7×#117—22no—
76C7×#117—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4D411yescharacteristic
85C2×C7×C4⋊C4D414yescomplemented
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4D411yescharacteristic
93C2×C7×C22⋊C4D411yescharacteristic
94C2×C42×C7—21no—
95C2×C42×C7—21no—
96C2×C42×C7C2311yescharacteristic
97C2×C22⋊Dic7—21no—
98C2×#1143—24no—
99C2×#1143—24no—
100C2×#1143—42no—
101#6500—44no—
102#6493—24no—
103#6493—42no—
104C2×(C2×D7)⋊C4—22no—
105C2×(C2×D7)⋊C4—22no—
106C2×(C2×D7)⋊C4—22no—
107C2×C4⋊Dic7C2312yes—
108C22×C7⋊C8—22no—
109C22×C7⋊C8—42no—
110C22×C7⋊C8C2312yes—
111C22×C7⋊C8C2312yes—
112#6409—42no—
113C7⋊#95—21no—
114C22⋊(C7⋊C8)—22no—
115C22⋊(C7⋊C8)—42no—
116#6383—24no—
117#6383—24no—
118#6383—24no—
119C2×#552D711yescharacteristic, complemented
120C2×#450—71no—
121C22×#102—71no—
122#2408—71no—
123#2025—72no—
124#2000—72no—
125#1892—71no—
126#1880—72no—
127#1875—72no—
128#1869—71no—
129#1823—71no—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C23×D7—22no—
133C23×D7—44no—
134C23×D7—42no—
135C23×D7—42no—
136C23×D7—42no—
137C23×D7—82no—
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8C2×D412yes—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—24no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2411yescharacteristic
165C2×C22⋊D7—44no—
166C2×C22⋊D7—44no—
167C2×C22⋊D7—82no—
168C22×Dic7—22no—
169C2×D28—28no—
170C2×D28—22no—
171C2×D28—22no—
172C2×D28—22no—
173C2×D28—22no—
174C2×D28—22no—
175C2×D28—48no—
176C2×D28—44no—
177C2×D28—44no—
178C2×D28—42no—
179C2×D28#4114yes—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—24no—
184C7×C4⋊C4—24no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4C2×D412yes—
191C7×C4⋊C4C2×D412yes—
192C7×C4⋊C4C2×D412yes—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C42×C7—24no—
199C42×C7—24no—
200C42×C7—21no—
201C42×C7—21no—
202C22⋊Dic7—42no—
203#1143—48no—
204(C2×D7)⋊C4—44no—
205(C2×D7)⋊C4—44no—
206(C2×D7)⋊C4—82no—
207C4⋊Dic7—22no—
208C4⋊Dic7—22no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8—22no—
212C2×C7⋊C8—22no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—22no—
215C2×C7⋊C8—44no—
216C23×D4—71no—
217C22×SD16—72no—
218C23×C8—71no—
219#552—24no—
220#552—24no—
221C2×#126—21no—
222C2×#125—21no—
223C2×#125C2×D711yescharacteristic, complemented
224C2×#123—21no—
225C2×C22⋊Q8C2×D711yescharacteristic
226C2×C22⋊Q8C2×D711yescharacteristic, complemented
227C2×#121—72no—
228C2×#120—141no—
229C2×C4×Q8C2×D711yescharacteristic, complemented
230C2×C4×D4C2×D711yescharacteristic
231C2×#117—21no—
232C2×#117C2×D711yescharacteristic, complemented
233C22×C4⋊C4C2×D711yescharacteristic
234#450—144no—
235C2×#103—72no—
236C2×#102—72no—
237C2×#102—72no—
238C2×#102—72no—
239C2×#102—144no—
240C2×C22⋊C8—71no—
241C2×C22⋊C8—72no—
242#394—141no—
243#392—141no—
244#390—72no—
245#380—71no—
246#379—141no—
247#340—72no—
248#340—72no—
249#340—72no—
250#336—72no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—21no—
254C23×C7—22no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—22no—
258C23×C7#12012yes—
259C23×C7#13512yes—
260C23×C7AGL(1,ℤ/8)12yes—
261C23×C7C22×D411yescharacteristic
262C22×D7—24no—
263C22×D7—48no—
264C22×D7—42no—
265C22×D7—42no—
266C22×D7—42no—
267C22×D7—42no—
268C22×D7—42no—
269C22×D7—42no—
270C22×D7—88no—
271C22×D7—88no—
272C22×D7—84no—
273C22×D7—84no—
274C7×Q8—24no—
275C7×Q8—41no—
276C7×Q8—41no—
277C7×D4—44no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—24no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—21no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—22no—
293C2×C4×C7—42no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—42no—
299C2×C4×C7—48no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—42no—
303C2×C4×C7#12012yes—
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C2×#4112yes—
307C2×C4×C7H(5,𝔽2)12yes—
308C22⋊D7—88no—
309C2×Dic7—42no—
310C2×Dic7—42no—
311C2×Dic7—42no—
312D28—24no—
313D28—24no—
314D28—48no—
315D28—42no—
316D28—42no—
317C7⋊C8—42no—
318C7⋊C8—42no—
319C25—141no—
320C22×Q8C22×D711yescharacteristic
321C22×D4—72no—
322C22×D4—144no—
323C22×D4—142no—
324C22×D4—142no—
325C22×D4—142no—
326C22×D4—141no—
327C22×D4—282no—
328C22×D4C22⋊D711yescharacteristic
329C23×C4—21no—
330C23×C4C22×D711yescharacteristic
331C23×C4C22×D711yescharacteristic
332C2×SD16—144no—
333C2×SD16—144no—
334C2×SD16—282no—
335C22×C8—72no—
336C22×C8—72no—
337C22×C8—142no—
338C22×C8—282no—
339#126—28no—
340#125—28no—
341#125—22no—
342#125—22no—
343#123—28no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—22no—
347C22⋊Q8C22⋊D714yescomplemented
348#121—144no—
349#121—282no—
350#120—284no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×D4—22no—
354C4×D4—22no—
355#117—28no—
356#117—22no—
357#117—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—72no—
366C2×C4⋊C4C22⋊D711yescharacteristic
367C2×C4⋊C4C22⋊D714yescomplemented
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—141no—
375C2×C22⋊C4—142no—
376C2×C22⋊C4—142no—
377C2×C22⋊C4—142no—
378C2×C22⋊C4C22⋊D711yescharacteristic
379C2×C22⋊C4C22⋊D711yescharacteristic
380C2×C42—21no—
381C2×C42—21no—
382C2×C42C22×D711yescharacteristic
383#103—282no—
384#102—144no—
385#102—144no—
386#102—144no—
387C22⋊C8—142no—
388C22⋊C8—282no—
389#95—141no—
390C22×C7—22no—
391C22×C7—22no—
392C22×C7—24no—
393C22×C7—24no—
394C22×C7—41no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7#44914yes—
400C22×C7C2×#12011yescharacteristic
401C22×C7#54612yes—
402C22×C7C2×#13511yescharacteristic
403C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C7#57712yes—
405C2×D7—44no—
406C2×D7—44no—
407C2×D7—44no—
408C2×D7—816no—
409C2×D7—82no—
410C2×D7—82no—
411C2×D7—82no—
412C2×D7—82no—
413C4×C7—22no—
414C4×C7—41no—
415C4×C7—42no—
416C4×C7—41no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7—41no—
420C4×C7—42no—
421C4×C7#52511yescharacteristic
422C4×C7#53511yescharacteristic
423C4×C7#54612yes—
424Dic7—82no—
425C24—142no—
426C24—284no—
427C24—282no—
428C24—282no—
429C24—282no—
430C24—562no—
431C24C2×C22⋊D711yescharacteristic
432C24C2×C22⋊D711yescharacteristic
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8C2×C22⋊D712yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—74no—
441C2×D4—148no—
442C2×D4—142no—
443C2×D4—142no—
444C2×D4—142no—
445C2×D4—142no—
446C2×D4—142no—
447C2×D4—288no—
448C2×D4—284no—
449C2×D4—284no—
450C2×D4—284no—
451C2×D4—284no—
452C2×D4—282no—
453C2×D4—562no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—142no—
468C22×C4C2×C22⋊D711yescharacteristic
469C22×C4C2×C22⋊D711yescharacteristic
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C2×C22⋊D712yes—
473C22×C4C2×C22⋊D712yes—
474C22×C4C23×D711yescharacteristic
475SD16—288no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8—142no—
479C2×C8—142no—
480C2×C8—142no—
481C2×C8—142no—
482C2×C8—284no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—142no—
494C4⋊C4—142no—
495C4⋊C4C2×C22⋊D712yes—
496C4⋊C4C2×C22⋊D712yes—
497C4⋊C4C2×C22⋊D712yes—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—284no—
504C22⋊C4—284no—
505C22⋊C4—282no—
506C22⋊C4—562no—
507C42—24no—
508C42—24no—
509C42—21no—
510C42—21no—
511C2×C7—24no—
512C2×C7—81no—
513C2×C7#240411yescharacteristic
514C2×C7C2×#44911yescharacteristic
515C2×C7#301011yescharacteristic
516C2×C7#329612yes—
517C2×C7#331012yes—
518D7—84no—
519C23—22no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—144no—
527C23—288no—
528C23—282no—
529C23—282no—
530C23—282no—
531C23—282no—
532C23—282no—
533C23—282no—
534C23—568no—
535C23—568no—
536C23—564no—
537C23—564no—
538C23#651212yes—
539C23#651312yes—
540C23#651612yes—
541C23C22×C22⋊D711yescharacteristic
542Q8—24no—
543Q8—41no—
544Q8—41no—
545D4—44no—
546D4—144no—
547D4—144no—
548D4—288no—
549D4—282no—
550D4—282no—
551D4—568no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—282no—
578C2×C4—282no—
579C2×C4—282no—
580C2×C4#651612yes—
581C2×C4D7×#4112yes—
582C2×C4#655312yes—
583C2×C4C22×C22⋊D711yescharacteristic
584C2×C4C22×C22⋊D711yescharacteristic
585C8—282no—
586C8—282no—
587C7#2396611yescharacteristic, complemented, Sylow
588C22—22no—
589C22—24no—
590C22—24no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—284no—
598C22—284no—
599C22—284no—
600C22—5616no—
601C22—562no—
602C22—562no—
603C22—562no—
604C22—562no—
605C22#9001014yes—
606C22C2×#651211yescharacteristic
607C22C2×#651311yescharacteristic
608C22#9052012yes—
609C22#9052812yes—
610C22C2×#651611yescharacteristic
611C4—22no—
612C4—41no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—562no—
620C4#9051911yescharacteristic
621C4#9052012yes—
622C4#9052111yescharacteristic
623C2—24no—
624C2—81no—
625C2—564no—
626C2#1174104312yes—
627C2#1174104412yes—
628C2C2×#9001011yescharacteristic
629C2#1174154111yescharacteristic
630C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#552C2
socleC23×C7C22×D4
prime radical#11739848C2
hypercenterC2×#552D7
hypocenterC7#23966
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. C2×C7×#552C2semidirect
2. C2×#89818C2semidirect
4. C22×#6383C2semidirect
5. #11729497C2semidirect
6. #11729453C2semidirect
7. #11729450C2semidirect
8. #11729387C2semidirect
9. #11729364C2semidirect
10. #11729340C2semidirect
11. #11729307C2semidirect
17. C2×C7×#125C22semidirect
20. C2×C7×C22⋊Q8C22semidirect
21. C2×C4×C7×Q8C22semidirect
24. C2×C7×#117C22semidirect
30. C2×#6409C22semidirect
33. C2×C22⋊(C7⋊C8)C22semidirect
43. #89324C22semidirect
44. #89324C22semidirect
45. #89324C22semidirect
46. #89319C22semidirect
69. C7×C22⋊Q8D4semidirect
85. C2×C7×C4⋊C4D4semidirect
119. C2×#552D7semidirect
223. C2×#125C2×D7semidirect
226. C2×C22⋊Q8C2×D7semidirect
229. C2×C4×Q8C2×D7semidirect
232. C2×#117C2×D7semidirect
347. C22⋊Q8C22⋊D7semidirect
367. C2×C4⋊C4C22⋊D7semidirect
587. C7#23966semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C2×#898182
#117398482
C22×#63832
#117294972
#117294532
#117294502
#117293872
#117293642
#117293402
#117293072
#239667