Finite Group Databasebeta

← #13407144

Subgroups — #13407144

Order 1792.

1262 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#552C211yesmaximal, characteristic, complemented
2C2×#89839C211yesmaximal, characteristic, complemented
3C2×#89799C211yesmaximal, characteristic, complemented
4#11739823C211yesmaximal, characteristic
5#11729492C211yesmaximal, characteristic, complemented
6#11729453C212yesmaximal, complemented
7#11729453C212yesmaximal, complemented
8#11729385C211yesmaximal, characteristic, complemented
9#11729369C211yesmaximal, characteristic, complemented
10#11729340C212yesmaximal, complemented
11#11729314C212yesmaximal, complemented
12C7×#552—24no—
13C7×#552—24no—
14C2×C7×#126—21no—
15C2×C7×#125—21no—
16C2×C7×#125C2211yescharacteristic
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8C2211yescharacteristic
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C7×#117—21no—
23C2×C7×#117C2211yescharacteristic
24C22×C7×C4⋊C4C2211yescharacteristic
25C22×#1143C2212yes—
26C2×#6503—21no—
27C2×#6493C2212yes—
28C2×#6493C2212yes—
29C2×#6437C2211yescharacteristic
30C2×#6409C2212yes—
31C2×C22⋊(C7⋊C8)C2212yes—
32C2×C22⋊(C7⋊C8)C2212yes—
33#89839—24no—
34#89799—24no—
35#89760—21no—
36C2×#6383C2212yes—
37C2×#6383C2212yes—
38C2×#6383C2212yes—
39#89732—21no—
40#89732—21no—
41C4×C4⋊Dic7C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44#89324C2212yes—
45#89324C2212yes—
46#89324C2212yes—
47#25583—71nomaximal, Sylow
48C22×C7×Q8C2311yescharacteristic
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7C2311yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×C22⋊D7—22no—
54C22×C22⋊D7—22no—
55C22×D28—21no—
56C22×D28—21no—
57C22×D28C2312yes—
58C7×#126—28no—
59C7×#125—28no—
60C7×#125—22no—
61C7×#125D414yescomplemented
62C7×#123—28no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8D414yescomplemented
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C7×#117—28no—
72C7×#117—22no—
73C7×#117—22no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4D411yescharacteristic
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4D411yescharacteristic
90C2×C7×C22⋊C4D411yescharacteristic
91C2×C42×C7—21no—
92C2×C42×C7—21no—
93C2×C42×C7C2311yescharacteristic
94C2×#1143—24no—
95C2×#1143—42no—
96C2×#1143—42no—
97#6503—44no—
98#6493—24no—
99#6493—24no—
100#6493—42no—
101#6493—42no—
102C2×(C2×D7)⋊C4—22no—
103C2×(C2×D7)⋊C4—22no—
104C2×C4⋊Dic7C2312yes—
105C2×C4×Dic7—21no—
106C22×C7⋊C8C2312yes—
107C22×C7⋊C8C2312yes—
108#6437—22no—
109#6437—22no—
110#6409—24no—
111C7⋊#95—21no—
112C22⋊(C7⋊C8)—42no—
113C22⋊(C7⋊C8)—42no—
114#6383—24no—
115#6383—24no—
116#6383—24no—
117C4⋊(C7⋊C8)—22no—
118#6377—22no—
119C2×#552D711yescharacteristic, complemented
120C2×#490—71no—
121C2×#460—71no—
122#2541—71no—
123#2000—72no—
124#2000—72no—
125#1989—71no—
126#1955—71no—
127#1875—72no—
128#1874—72no—
129#1763—71no—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C23×D7—22no—
133C2×C7×Q8—21no—
134C2×C7×Q8—22no—
135C2×C7×Q8—22no—
136C2×C7×Q8C2×D412yes—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—41no—
140C22×C4×C7—24no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C2×C22⋊D7—44no—
161C2×C22⋊D7—44no—
162C2×C22⋊D7—44no—
163C2×C22⋊D7—44no—
164C2×C22⋊D7—82no—
165C2×C22⋊D7—82no—
166C22×Dic7—22no—
167C2×D28—22no—
168C2×D28—22no—
169C2×D28—22no—
170C2×D28—22no—
171C2×D28—22no—
172C2×D28—22no—
173C2×D28—42no—
174C2×D28—42no—
175C2×D28—42no—
176C2×D28—42no—
177C2×D28—82no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—24no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4C2×D412yes—
189C7×C4⋊C4C2×D412yes—
190C7×C4⋊C4C2×D412yes—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C42×C7—24no—
197C42×C7—24no—
198C42×C7C2×D412yes—
199C42×C7C2×D412yes—
200#1143—48no—
201(C2×D7)⋊C4—44no—
202(C2×D7)⋊C4—44no—
203C4⋊Dic7—22no—
204C4⋊Dic7#4114yes—
205C4×Dic7—42no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—22no—
210C2×C7⋊C8—22no—
211C2×C7⋊C8—22no—
212C22×SD16—72no—
213#552—24no—
214#552—24no—
215C2×#127—71no—
216C2×#127—141no—
217C2×#126—21no—
218C2×#125—21no—
219C2×#125C2×D711yescharacteristic
220C2×#123—21no—
221C2×C22⋊Q8C2×D711yescharacteristic
222C2×C22⋊Q8C2×D711yescharacteristic
223C2×#121—72no—
224C2×#121—72no—
225C2×C4×Q8C2×D711yescharacteristic
226C2×C4×D4C2×D711yescharacteristic
227C2×#117—21no—
228C2×#117C2×D711yescharacteristic
229C22×C4⋊C4C2×D711yescharacteristic
230#490—144no—
231#460—144no—
232C2×C4⋊C8—71no—
233C2×#103—72no—
234C2×#102—72no—
235C2×#102—72no—
236C2×#102—72no—
237C2×C22⋊C8—72no—
238C2×C22⋊C8—72no—
239C2×#97—71no—
240#392—141no—
241#392—141no—
242#390—141no—
243C4×C4⋊C4—71no—
244#340—72no—
245#340—72no—
246#340—72no—
247C23×C7—21no—
248C23×C7—22no—
249C23×C7—21no—
250C23×C7—22no—
251C23×C7—21no—
252C23×C7—22no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—22no—
256C23×C7AGL(1,ℤ/8)12yes—
257C23×C7C22×D411yescharacteristic
258C22×D7—42no—
259C22×D7—42no—
260C22×D7—42no—
261C22×D7—42no—
262C22×D7—42no—
263C22×D7—42no—
264C22×D7—42no—
265C7×Q8—22no—
266C7×Q8—22no—
267C7×Q8—42no—
268C7×D4—44no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—48no—
281C2×C4×C7—42no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—42no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7#12012yes—
294C2×C4×C7#12012yes—
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4112yes—
298C2×C4×C7H(5,𝔽2)12yes—
299C22⋊D7—88no—
300C22⋊D7—88no—
301C2×Dic7—24no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304D28—42no—
305D28—42no—
306D28—42no—
307D28—42no—
308D28—84no—
309D28—84no—
310C7⋊C8—42no—
311C7⋊C8—42no—
312C22×Q8C22×D711yescharacteristic
313C22×D4—72no—
314C22×D4—142no—
315C22×D4—142no—
316C22×D4—141no—
317C22×D4—141no—
318C22×D4C22⋊D711yescharacteristic
319C23×C4—21no—
320C23×C4C22×D711yescharacteristic
321C23×C4C22×D711yescharacteristic
322C2×SD16—144no—
323C2×SD16—282no—
324C2×SD16—282no—
325C22×C8—72no—
326C22×C8—72no—
327#127—142no—
328#127—142no—
329#127—284no—
330#126—28no—
331#125—28no—
332#125—22no—
333#125C22⋊D714yescomplemented
334#123—28no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339#121—144no—
340#121—144no—
341#121—282no—
342#121—282no—
343C4×Q8—22no—
344C4×Q8C22⋊D714yescomplemented
345C4×D4—22no—
346C4×D4—22no—
347#117—28no—
348#117—22no—
349#117—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—72no—
359C2×C4⋊C4C22⋊D711yescharacteristic
360C2×C4⋊C4C22×D711yescharacteristic
361C2×C4⋊C4C22×D711yescharacteristic
362C2×C4⋊C4C22×D711yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—142no—
367C2×C22⋊C4—142no—
368C2×C22⋊C4C22⋊D711yescharacteristic
369C2×C22⋊C4C22⋊D711yescharacteristic
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—141no—
373C2×C42C22×D711yescharacteristic
374C4⋊C8—142no—
375#103—144no—
376#102—144no—
377#102—144no—
378#102—144no—
379C22⋊C8—282no—
380C22⋊C8—282no—
381#97—142no—
382#95—141no—
383C22×C7—22no—
384C22×C7—24no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7—42no—
388C22×C7—41no—
389C22×C7—41no—
390C22×C7—41no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7C2×#12011yescharacteristic
394C22×C7#54612yes—
395C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
396C22×C7#57511yescharacteristic
397C22×C7#57712yes—
398C2×D7—82no—
399C2×D7—82no—
400C2×D7—82no—
401C2×D7—82no—
402C2×D7—82no—
403C2×D7—82no—
404C2×D7—82no—
405C4×C7—22no—
406C4×C7—22no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7—42no—
410C4×C7—42no—
411C4×C7—42no—
412C4×C7—42no—
413C4×C7#53211yescharacteristic
414C4×C7#53511yescharacteristic
415C4×C7#55012yes—
416Dic7—44no—
417C24—142no—
418C24C2×C22⋊D711yescharacteristic
419C24C2×C22⋊D711yescharacteristic
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8C2×C22⋊D712yes—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—41no—
427C2×D4—142no—
428C2×D4—142no—
429C2×D4—142no—
430C2×D4—142no—
431C2×D4—142no—
432C2×D4—142no—
433C2×D4—282no—
434C2×D4—282no—
435C2×D4—284no—
436C2×D4—284no—
437C2×D4—284no—
438C2×D4—284no—
439C2×D4—282no—
440C2×D4—282no—
441C2×D4—562no—
442C2×D4—562no—
443C2×D4—562no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—142no—
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C23×D711yescharacteristic
465SD16—288no—
466C2×C8—142no—
467C2×C8—142no—
468C2×C8—142no—
469C2×C8—142no—
470C2×C8—142no—
471C2×C8—142no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—74no—
483C4⋊C4—142no—
484C4⋊C4C2×C22⋊D712yes—
485C4⋊C4C2×C22⋊D712yes—
486C4⋊C4C2×C22⋊D712yes—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—284no—
493C22⋊C4—284no—
494C42—24no—
495C42—24no—
496C42—282no—
497C42C2×C22⋊D712yes—
498C42C2×C22⋊D712yes—
499C2×C7—42no—
500C2×C7—81no—
501C2×C7#243111yescharacteristic
502C2×C7#300611yescharacteristic
503C2×C7#301011yescharacteristic
504C2×C7#331012yes—
505C2×C7#331712yes—
506D7—162no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—282no—
517C23—282no—
518C23—282no—
519C23—282no—
520C23—282no—
521C23—282no—
522C23—282no—
523C23#651212yes—
524C23C22×C22⋊D711yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—42no—
528D4—44no—
529D4—282no—
530D4—282no—
531D4—282no—
532D4—282no—
533D4—564no—
534D4—564no—
535D4—568no—
536D4—568no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—48no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—144no—
562C2×C4—282no—
563C2×C4—282no—
564C2×C4#651612yes—
565C2×C4#651612yes—
566C2×C4D7×#4112yes—
567C2×C4#655312yes—
568C2×C4C22×C22⋊D711yescharacteristic
569C2×C4C22×C22⋊D711yescharacteristic
570C8—282no—
571C8—282no—
572C7#2558311yescharacteristic, complemented, Sylow
573C22—22no—
574C22—24no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—562no—
584C22—562no—
585C22—562no—
586C22—562no—
587C22—562no—
588C22—562no—
589C22—562no—
590C22C2×#651211yescharacteristic
591C22#9051311yescharacteristic
592C22#9052012yes—
593C22#9052812yes—
594C22C2×#651611yescharacteristic
595C4—22no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—42no—
603C4—284no—
604C4#9052111yescharacteristic
605C4#9052311yescharacteristic
606C4#9052512yes—
607C2—42no—
608C2—81no—
609C2—1122no—
610C2#1174104412yes—
611C2#1174104912yes—
612C2#1174152711yescharacteristic
613C2#1174153711yescharacteristic
614C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#552C2
socleC23×C7C22×D4
prime radical#11739823C2
hypercenterC2×#552D7
hypocenterC7#25583
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×#89839C2semidirect
3. C2×#89799C2semidirect
5. #11729492C2semidirect
6. #11729453C2semidirect
7. #11729453C2semidirect
8. #11729385C2semidirect
9. #11729369C2semidirect
10. #11729340C2semidirect
11. #11729314C2semidirect
61. C7×#125D4semidirect
68. C4×C7×Q8D4semidirect
119. C2×#552D7semidirect
333. #125C22⋊D7semidirect
344. C4×Q8C22⋊D7semidirect
572. C7#25583semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5522
C2×#898392
C2×#897992
#117398232
#117294922
#117294532
#117293852
#117293692
#117293402
#117293142
#255837