Finite Group Databasebeta

← #13369459

Subgroups — #13369459

Order 1792.

1582 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#550C211yesmaximal, characteristic
2C2×#90004C212yesmaximal, complemented
3D4×C4⋊Dic7C211yesmaximal, characteristic, complemented
4C2×#89784C211yesmaximal, characteristic, complemented
5#11729889C212yesmaximal, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11729515C211yesmaximal, characteristic, complemented
8#11729436C212yesmaximal, complemented
9#11729431C212yesmaximal, complemented
10C4×#6409C211yesmaximal, characteristic, complemented
11#11729380C211yesmaximal, characteristic, complemented
12C22×C7×#41C2212yes—
13C7×#550—24no—
14C7×#550C418yescomplemented
15C2×C7×#127—21no—
16C2×C7×#127C2211yescharacteristic
17C2×C7×#121—22no—
18C2×C7×#121C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×D4×Dic7—21no—
23C22×C4⋊Dic7C2212yes—
24C22×#1136C2212yes—
25#90004—28no—
26C2×#6409C2212yes—
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6406C2212yes—
30C2×#6406C2212yes—
31C2×#6406C2212yes—
32C2×#6406C2212yes—
33#89984C2212yes—
34C2×C22⋊(C7⋊C8)C2212yes—
35#89784—24no—
36#89757—21no—
37#89722—22no—
38#89707C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8C2211yescharacteristic
42#89324C2211yescharacteristic
43#89324C2212yes—
44#89324C2211yescharacteristic
45#22432—71nomaximal, Sylow
46C2×C7×#41—24no—
47C2×C7×#41—24no—
48C2×C7×#41—24no—
49C2×C7×#41—42no—
50C2×C7×#41—42no—
51C2×C7×#41C2×C418yescomplemented
52C22×C7×Q8C2311yescharacteristic
53C22×C7×D4—22no—
54C22×C7×D4—22no—
55C22×C7×D4—21no—
56C22×C7×D4—21no—
57C22×C7×D4D411yescharacteristic
58C22×C7×D4C2312yes—
59C23×C4×C7—22no—
60C23×C4×C7C2312yes—
61C23×Dic7—22no—
62C7×#127—28no—
63C7×#127—22no—
64C7×#127C2×C414yescomplemented
65C7×#121—24no—
66C7×#121—216no—
67C7×#121C2×C418yescomplemented
68C4×C7×Q8—22no—
69C4×C7×Q8C2×C414yescomplemented
70C4×C7×D4—28no—
71C4×C7×D4C2×C414yes—
72C4×C7×D4D414yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4D412yes—
78C2×C42×C7—21no—
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—22no—
81D4×Dic7—44no—
82C2×C4⋊Dic7—24no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—42no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2312yes—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4×Dic7—21no—
91C2×#1136—24no—
92C2×#1136—24no—
93C2×#1136—42no—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2312yes—
96C22×C7⋊C8C2311yescharacteristic
97#6409—24no—
98#6409—22no—
99#6409D414yescomplemented
100#6406—24no—
101#6406—24no—
102#6406—42no—
103#6406—42no—
104C7⋊#95—21no—
105C7⋊#95—22no—
106C22⋊(C7⋊C8)—24no—
107C4⋊(C7⋊C8)—22no—
108C4×C7⋊C8—22no—
109C2×#550D711yescharacteristic
110C2×#436—71no—
111C2×#417—72no—
112D4×C4⋊C4—71no—
113#1982—71no—
114#1957—71no—
115#1895—72no—
116#1874—72no—
117#1803—71no—
118#1788—72no—
119C4×#103—71no—
120C24×C7—22no—
121C24×C7C2×D412yes—
122C7×#41—216no—
123C7×#41—48no—
124C7×#41—48no—
125C7×#41—48no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8—21no—
129C2×C7×Q8C22×C412yes—
130C2×C7×Q8C2×D412yes—
131C2×C7×Q8#4112yes—
132C2×C7×D4—28no—
133C2×C7×D4—28no—
134C2×C7×D4—24no—
135C2×C7×D4—24no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—44no—
144C2×C7×D4—44no—
145C2×C7×D4—42no—
146C2×C7×D4—42no—
147C2×C7×D4—81no—
148C2×C7×D4—81no—
149C2×C7×D4—82no—
150C2×C7×D4—82no—
151C2×C7×D4C22⋊C414yes—
152C2×C7×D4C22×C414yes—
153C22×C4×C7—21no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—21no—
159C22×C4×C7—28no—
160C22×C4×C7—44no—
161C22×C4×C7—44no—
162C22×C4×C7C22×C414yes—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D414yes—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—22no—
171C22×Dic7—44no—
172C22×Dic7—44no—
173C22×Dic7—82no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4C22⋊C414yes—
177C7×C4⋊C4C22×C412yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4C22⋊C418yes—
180C42×C7—24no—
181C42×C7C22×C412yes—
182C42×C7C2×D412yes—
183C22⋊Dic7—44no—
184C4⋊Dic7—21no—
185C4⋊Dic7—22no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—22no—
189C4⋊Dic7—44no—
190C4⋊Dic7—42no—
191C4⋊Dic7—81no—
192C4⋊Dic7C2×D412yes—
193C4×Dic7—42no—
194#1136—48no—
195C2×C7⋊C8—21no—
196C2×C7⋊C8—21no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8C2×D412yes—
204C22×#41C2×D712yes—
205C22×M16—72no—
206#550—24no—
207#550Dic718yescomplemented
208C2×#127—21no—
209C2×#127C2×D711yescharacteristic
210C2×#121—22no—
211C2×#121C2×D712yes—
212C2×C4×Q8C2×D711yescharacteristic
213C2×C4×D4—21no—
214C2×C4×D4—141no—
215C2×C4×D4C2×D711yescharacteristic
216C22×C4⋊C4—72no—
217#436—144no—
218C2×C4⋊C8—71no—
219#417—148no—
220C2×#103—71no—
221C2×#103—72no—
222C2×#103—71no—
223C2×#102—72no—
224C2×#102—72no—
225C2×#102—72no—
226C2×#102—72no—
227C2×C22⋊C8—72no—
228C2×C4×C8—71no—
229#389—141no—
230#385—142no—
231#384—71no—
232#380—72no—
233C4×C4⋊C4—71no—
234#340—72no—
235#340—71no—
236#340—71no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—24no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7C2×C22⋊C414yes—
251C23×C7C22×D411yescharacteristic
252C7×Q8—22no—
253C7×Q8—22no—
254C7×Q8—22no—
255C7×Q8C4×D414yescomplemented
256C7×D4—24no—
257C7×D4—24no—
258C7×D4—28no—
259C7×D4—416no—
260C7×D4—42no—
261C7×D4—42no—
262C7×D4—48no—
263C7×D4—84no—
264C7×D4—88no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—22no—
277C2×C4×C7—416no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7C2×C22⋊C412yes—
281C2×C4×C7C2×C22⋊C414yes—
282C2×C4×C7C2×C22⋊C412yes—
283C2×C4×C7AGL(1,ℤ/8)12yes—
284C2×C4×C7#13712yes—
285C2×C4×C7C23×C411yescharacteristic
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7H(5,𝔽2)11yescharacteristic
290C2×C4×C7H(5,𝔽2)11yescharacteristic
291C2×Dic7—22no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—42no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—41no—
300C2×Dic7—88no—
301C7⋊C8—22no—
302C7⋊C8—42no—
303C7⋊C8—41no—
304C2×#41—24no—
305C2×#41—24no—
306C2×#41—24no—
307C2×#41—42no—
308C2×#41—42no—
309C2×#41C2×Dic718yescomplemented
310C22×Q8C22×D711yescharacteristic
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4C22⋊D711yescharacteristic
316C22×D4C22×D712yes—
317C23×C4—22no—
318C23×C4—142no—
319C23×C4C22×D712yes—
320C2×M16—144no—
321C2×M16—144no—
322C2×M16—282no—
323C22×C8—71no—
324C22×C8—72no—
325C22×C8—71no—
326#127—28no—
327#127—22no—
328#127C2×Dic714yescomplemented
329#121—24no—
330#121—216no—
331#121C2×Dic718yescomplemented
332C4×Q8—22no—
333C4×Q8C2×Dic714yescomplemented
334C4×D4—28no—
335C4×D4—284no—
336C4×D4C2×Dic714yes—
337C4×D4C22⋊D714yes—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—144no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—282no—
347C2×C4⋊C4C22⋊D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C22⋊C4—22no—
350C2×C22⋊C4—142no—
351C2×C22⋊C4C22⋊D712yes—
352C2×C42—21no—
353C2×C42—141no—
354C2×C42C22×D711yescharacteristic
355C4⋊C8—142no—
356#103—74no—
357#103—144no—
358#103—142no—
359#102—144no—
360#102—144no—
361#102—282no—
362#102—282no—
363C22⋊C8—144no—
364C4×C8—142no—
365#95—141no—
366#95—142no—
367C22×C7—24no—
368C22×C7—24no—
369C22×C7—24no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—44no—
375C22×C7—44no—
376C22×C7—44no—
377C22×C7—82no—
378C22×C7—82no—
379C22×C7—82no—
380C22×C7—82no—
381C22×C7C22×C22⋊C411yescharacteristic
382C22×C7D4211yescharacteristic
383C22×C7#54611yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7C2×#13711yescharacteristic
386C22×C7#57611yescharacteristic
387C22×C7#57711yescharacteristic
388C4×C7—24no—
389C4×C7—24no—
390C4×C7—22no—
391C4×C7—42no—
392C4×C7#41714yescomplemented
393C4×C7C2×C4×D411yescharacteristic
394C4×C7#51811yescharacteristic
395C4×C7D4211yescharacteristic
396C4×C7#54611yescharacteristic
397Dic7—42no—
398Dic7—81no—
399Dic7—82no—
400C24—22no—
401C24C2×C22⋊D712yes—
402#41—216no—
403#41—48no—
404#41—48no—
405#41—48no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8D4×D712yes—
410C2×Q8#115912yes—
411C2×Q8C22×Dic712yes—
412C2×D4—28no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4C22⋊Dic714yes—
432C2×D4C22×Dic714yes—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—28no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—142no—
445C22×C4—284no—
446C22×C4—284no—
447C22×C4—562no—
448C22×C4C22×Dic714yes—
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D712yes—
451C22×C4C2×C22⋊D714yes—
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C23×D711yescharacteristic
454M16—288no—
455C2×C8—72no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—142no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—142no—
462C2×C8—141no—
463C2×C8—142no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—72no—
467C4⋊C4—141no—
468C4⋊C4—142no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—142no—
472C4⋊C4—284no—
473C4⋊C4—282no—
474C4⋊C4—561no—
475C4⋊C4C22⋊Dic714yes—
476C4⋊C4C22×Dic712yes—
477C22⋊C4—28no—
478C22⋊C4—284no—
479C22⋊C4C22⋊Dic718yes—
480C42—24no—
481C42—282no—
482C42C22×Dic712yes—
483C42C2×C22⋊D712yes—
484C2×C7—44no—
485C2×C7—84no—
486C2×C7D4×C22⋊C411yescharacteristic
487C2×C7C2×#41711yescharacteristic
488C2×C7#289711yescharacteristic
489C2×C7#327411yescharacteristic
490C2×C7#327811yescharacteristic
491C2×C7#333211yescharacteristic
492C2×C7#333311yescharacteristic
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C2×C22⋊Dic714yes—
507C23C22×C22⋊D711yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—22no—
511Q8D4×Dic714yescomplemented
512D4—24no—
513D4—24no—
514D4—28no—
515D4—416no—
516D4—42no—
517D4—42no—
518D4—48no—
519D4—84no—
520D4—88no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—416no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—142no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—282no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—282no—
543C2×C4—282no—
544C2×C4—281no—
545C2×C4—568no—
546C2×C4#651212yes—
547C2×C4#651412yes—
548C2×C4C2×C22⋊Dic712yes—
549C2×C4C2×C22⋊Dic714yes—
550C2×C4C2×C22⋊Dic712yes—
551C2×C4C2×D4×D711yescharacteristic
552C2×C4C2×#115911yescharacteristic
553C2×C4#654811yescharacteristic
554C2×C4#654811yescharacteristic
555C2×C4C23×Dic711yescharacteristic
556C2×C4C22×C22⋊D711yescharacteristic
557C8—142no—
558C8—282no—
559C8—281no—
560C7#2243211yescharacteristic, complemented, Sylow
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22D4×C22⋊D711yescharacteristic
576C22#9049711yescharacteristic
577C22C2×#651211yescharacteristic
578C22C2×#651411yescharacteristic
579C22#9052611yescharacteristic
580C22#9052711yescharacteristic
581C22C22×C22⋊Dic711yescharacteristic
582C4—24no—
583C4—24no—
584C4—22no—
585C4—42no—
586C4—282no—
587C4—561no—
588C4—562no—
589C4#9000414yescomplemented
590C4C2×D4×Dic711yescharacteristic
591C4#9048911yescharacteristic
592C4D4×C22⋊D711yescharacteristic
593C4#9049711yescharacteristic
594C2—44no—
595C2—84no—
596C2#1174101911yescharacteristic
597C2#1174102011yescharacteristic
598C2#1174102311yescharacteristic
599C2#1174102911yescharacteristic
600C2D4×C22⋊Dic711yescharacteristic
601C2C2×#9000411yescharacteristic
602C2#1174154411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#550C2
socleC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenterC2×#550D7
hypocenterC7#22432
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
2. C2×#90004C2semidirect
3. D4×C4⋊Dic7C2semidirect
4. C2×#89784C2semidirect
5. #11729889C2semidirect
6. #11729518C2semidirect
7. #11729515C2semidirect
8. #11729436C2semidirect
9. #11729431C2semidirect
10. C4×#6409C2semidirect
11. #11729380C2semidirect
14. C7×#550C4semidirect
51. C2×C7×#41C2×C4semidirect
64. C7×#127C2×C4semidirect
67. C7×#121C2×C4semidirect
69. C4×C7×Q8C2×C4semidirect
99. #6409D4semidirect
207. #550Dic7semidirect
255. C7×Q8C4×D4semidirect
309. C2×#41C2×Dic7semidirect
328. #127C2×Dic7semidirect
331. #121C2×Dic7semidirect
333. C4×Q8C2×Dic7semidirect
392. C4×C7#417semidirect
511. Q8D4×Dic7semidirect
560. C7#22432semidirect
589. C4#90004semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5502
C2×#900042
D4×C4⋊Dic72
C2×#897842
#117298892
#117295182
#117295152
#117294362
#117294312
C4×#64092
#117293802
#224327